Level I Math Black Line Masters Toolkit


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1 Level I Math Black Line Masters Toolkit NSSAL (Draft) C. David Pilmer 2012 (Last Updated: February, 2013)
2 This resource is the intellectual property of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. The following are permitted to use and reproduce this resource for classroom purposes. Nova Scotia instructors delivering the Nova Scotia Adult Learning Program Canadian public school teachers delivering public school curriculum Canadian nonprofit tuitionfree adult basic education programs The following are not permitted to use or reproduce this resource without the written authorization of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. Upgrading programs at postsecondary institutions Core programs at postsecondary institutions Public or private schools outside of Canada Basic adult education programs outside of Canada Individuals, not including teachers or instructors, are permitted to use this resource for their own learning. They are not permitted to make multiple copies of the resource for distribution. Nor are they permitted to use this resource under the direction of a teacher or instructor at a learning institution. Acknowledgments The Adult Education Division would like to thank the following ALP instructors for piloting this resource and offering suggestions during its development. Andre Davey (Metroworks) Shannon Davis (YCLA) Andrea Fitzgerald (CLANS) Elizabeth Grzesik (EHALA) Cheryl Mycroft (GALA) Joyce Power (Metroworks) David Sweeny (YCLA) Kirsteen Thomson (CanU)
3 Table of Contents Introduction iv Difficulty Game or Puzzle Number Magnitude (Whole Numbers) Expanded Form (A and B) 2 Write the Number (A to C) 4 Write the Number (D and E) 8 Place Value 12 Before, After, or Between (A and B) 13 Closer To, and Odd or Even (A and B) 15 Whole Numbers and Number Lines (A and B) 17 Give an Example.. 19 Closer To (A and B).. 20 Operations with Whole Numbers. 22 Connect Four Addition Game (A to C) Connect Four Subtraction Game (A to D) Subtraction Search Multiplication Models 31 Multiplication Array Game 33 Connect Four Multiplication Game (A to E). 34 Connect Four Multiples Game (A to C) 39 Capture the Flag Multiplication Game. 42 Random Multiplication Facts Quizzes.. 44 Multiplying on Your Hands Put the Number in the Right Box Investigation: Multiplying by Multiples of 10, 100, and Multiplying by Multiples of 10, 100, and Multiplying Two Digit Numbers, Part 1 (Expanded Form).. 54 Multiplying Two Digit Numbers, Part 2 (Lattice Method) 59 Multiplying MultiDigit Numbers.. 65 Connect Four Division Game Division Search.. 70 Divisibility Chart 71 More Divisibility (A and B).. 72 Divisibility or Prime Connect Four Game 75 Division with Remainders. 76 Long Division (Partial Quotient Method). 80 Prime Factorization 88 Multiple Operations (Whole Numbers) Express the Number in Multiple Ways.. 91 Name the Preceding or Next.. 92 One of these Things is Not Like the Others.. 93 NSSAL i Draft
4 Fact Family Puzzle Pathways Two of These Boxes Just Don't Belong (A and B).. 98 Equivalent Venn Diagrams and Whole Numbers 101 Whole Number Crossword Puzzle (A to D). 103 KenKen Puzzles (A and B) 107 KenKen Puzzles (C and D) 110 KenKen Puzzles (E) Find the Two Numbers Which Combination of Numbers Works?. 114 Magic Squares Addition Pyramids 116 Row Factors and Column Factors. 118 Letter and Number Sentences 119 Math Logic Puzzles 120 Number Sentences (A) Number Sentences (B). 122 Order of Operations (A) 123 Order of Operations (B) 126 Order of Operations (C) 129 Patterns. 132 What's the Pattern? (A) What's the Pattern? (B) Toothpick Patterns. 135 Create the Pattern (A and B) Number Patterns (A and B) 139 Row, Column, and Diagonal Pattern. 141 What's the Relationship? 142 Input Output (A to D) 144 Filling or Draining. 148 Travelling Towards or Away From Home 153 Weight of the Water Word Problems 165 Describing the Relationships with Words. 166 List the Numbers Based on the Written Description 167 Addition and Subtraction Crossword 168 Multiplication and Division Crossword 170 Operations Crossword 172 Word Sentence to Number Sentence to Answer (A to C) 174 What are the Possibilities? (A). 177 What are the Possibilities? (B). 180 Does It Make Sense?. 182 Insert Your Own Numbers and Words. 184 NSSAL ii Draft
5 Put the Numbers in Where It Makes Sense (A to B) 185 Put the Numbers in Where It Makes Sense (C) 187 Put the Numbers in Where It Makes Sense (D) 188 Not Enough Information is Provided 190 Word Problems with Too Much Information 191 Create Your Own Math Statement 193 Word Problems (A and B) 195 Same Numbers, Similar Context, Different Math (A). 200 Same Numbers, Similar Context, Different Math (B). 202 More than One Question 204 Food Chart (A and B) 206 Keeping Track of the New Stock (A and B) 209 Consumer Math 211 How Much Do They Have? (A to E). 212 Emptying the Junk Drawer. 217 Connect Four Money Game Find the Price. 220 Name a Product near that Price. 221 Least to Most Expensive 222 What are the Three Items Worth? What Is It Worth? (A and B). 224 Purchasing Groceries. 226 Measurement 228 Which Measurement Is Reasonable? 229 Telling Time (A to F) 230 Connect Four Time Ahead Game (A and B) 236 How Much Time Has Passed? (A to C) 238 Geometry (From 3D to 2D) 244 Isoball 245 Orthographic Projections (A) 246 Orthographic Projections (B). 250 Transfer Image to Dot Paper. 252 Isometric Projections 253 Orthographic Isometric Challenge 257 Edges, Faces, and Vertices (A and B) Construct the Geometric Figure 261 Nets 266 Describe the Faces. 268 Geometry Terminology Crossword 277 Reflections (A to C) Base Ten Blocks What's the Number? NSSAL iii Draft
6 Express the Number Different Ways 285 Express with the Fewest Number of Manipulatives. 287 Adding 295 Adding with Regrouping 299 Subtracting. 305 Subtracting with Regrouping. 310 Answers 317 NSSAL iv Draft
7 Introduction The concepts covered in Level I Math fit into one of the following five categories. Number and Operations Patterns and Relations Statistics and Probability Shape, Space, and Measurement Consumer Math The specific outcomes aligned with each of these categories can be found in the ALP Level I Math Curriculum Guide. Over the years, CommunityLearning Organizations have collected, and the Adult Education Division have supplied, a variety of print resources used in the delivery of Level I Math. Many of those resources can still be used with this new curriculum but we emphasize that there is a much greater emphasis on mathematical understanding and multiple representations of concepts in this new program. Although we want our learners to develop a level of automaticity as it pertains to operations with whole numbers, we do not want this math course, or any other ALP math course, to focus primarily on the mastery of skills. Unfortunately many of the "traditional" textbooks used in adult basic education programs do have this as their primary focus. For this reason, the Adult Education division would like all instructors to use the following ALP resources in the delivery of Level I Math, and to supplement that material with the more traditional resources they have collected over the years. Level I Math Black Line Masters Mental Math Customized Practice Number Sense These resources include activities, exercises, investigations, and games that encourage understanding and thinking, rather than solely focussing on the mastery of algorithms. Learners are ultimately better served when mathematical concepts are examined and taught in this matter. We do not expect all Level I learners to complete all the worksheets or activities in the resources above, rather instructors will use their professional judgement to choose the items that are most appropriate for their individual learners. By supplying these materials the LAE is providing a greater variety of education tools for ALP instructors; the instructors have to decide what tools are best suited for their learners, at what times, and in what sequence. For example, let's consider multiplication of two multidigit numbers. Most instructors are familiar with the traditional algorithm for such multiplication, but some instructors are unfamiliar with multiplication of multidigit numbers using the expanded forms of the numbers and/or lattice multiplication. These latter two techniques are found in this resource. Does that mean that all learners need to know all three methods? Definitely not; chose the technique that works best for your learner. Please do not view this specific resource as a textbook. Although within sections, the activity sheets are generally arranged from easiest to hardest, a seamless flow from one activity to the NSSAL v Draft
8 next was not created. This booklet is merely a collection of black line masters to be used as the instructor sees fit. All of these materials are available at the NSSAL site ( NSSAL Practitioners Website ( NSSAL vi Draft
9 Number Magnitude (Whole Numbers) NSSAL 1 Draft
10 Expanded Form (A) 1. Write the multidigit number in its expanded form. Two examples have been done for you. Number Expanded Form Number Expanded Form e.g e.g (a) 42 (b) 694 (c) 3985 (d) 569 (e) 78 (f) 4281 (g) 867 (h) 31 (i) 6497 (j) 528 (k) 826 (l) 5923 (m) 59 (n) 3045 (o) 4808 (p) 703 (q) 6420 (r) 5099 (s) 810 (t) Given the expanded form, write the multidigit number. The last eight have had their expanded forms scrambled. Expanded Form Number Expanded Form Number (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) (w) (x) NSSAL 2 Draft
11 Expanded Form (B) 1. Write the multidigit number in its expanded form. An example has been done for you. Number Expanded Form e.g (a) (b) 5685 (c) (d) (e) (f) (g) (h) (i) (j) Given the expanded form, write the multidigit number. The last four have had their expanded forms scrambled. Expanded Form (a) Number (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) NSSAL 3 Draft
12 Write the Number (A) 1. Write the number that has been described using words. Word Description Number Word Description Number e.g. fiftyseven 57 e.g. nineteen 19 (a) eightytwo (b) fortynine (c) sixteen (d) ten (e) fortysix (f) twentyseven (g) seventyfour (h) thirteen (i) fiftysix (j) eleven (k) thirtyeight (l) twentythree (m) eight (n) ninetynine (o) seventeen (p) eightythree (q) twelve (r) fiftytwo (s) ninety (t) thirtyone (u) seven (v) sixtythree (w) fifteen (x) sixtyeight 2. Write out the number using words. Number Word Description e.g. 72 seventytwo (a) 59 (b) 42 (c) 18 (d) 37 (e) 61 (f) 95 (g) 21 NSSAL 4 Draft
13 Write the Number (B) 1. Write the number that has been described using words. Word Description Number Word Description Number e.g. three hundred fortynine 349 e.g. six hundred eight 608 (a) nine hundred thirtytwo (b) two hundred fortysix (c) seven hundred twelve (d) three hundred sixty (e) seventynine (f) six hundred twentyone (g) one hundred seven (h) five hundred eightynine (i) four hundred ninety (j) fortytwo (k) eight hundred eleven (l) two hundred seventysix (m) seven hundred two (n) three hundred nineteen (o) twelve (p) five hundred thirtyone (q) six hundred seventy (r) nine (s) one hundred eightysix (t) nine hundred (u) two hundred sixteen (v) sixtyfive (w) seven hundred twenty (x) four hundred sixtytwo 2. Write out the number using words. Number Word Description e.g. 452 four hundred fiftytwo (a) 578 (b) 352 (c) 79 (d) 217 (e) 906 (f) 740 (g) 541 NSSAL 5 Draft
14 Write the Number (C) 1. Write the number that has been described using words. Word Description Number e.g. three thousand, two hundred ninetyone e.g. six thousand, fourteen (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) eight thousand, three hundred twentytwo four thousand, six hundred eightythree seven thousand, five hundred thirteen four hundred eleven nine thousand, five hundred twelve three thousand, four hundred twentynine two thousand, nine hundred fifty one thousand, seventyeight two hundred seven five thousand, nine hundred eightythree six thousand, eight hundred seven nine thousand, fortysix (m) five hundred ninetyseven (n) (o) (p) (q) (r) (s) (t) (u) (v) (w) eight thousand, two hundred seventyfour fifteen four thousand, nine hundred twentyeight one thousand, three hundred eighteen thirtyeight three thousand, seventysix six thousand, nine hundred seven thousand, three hundred eight seven hundred sixteen nine thousand, five hundred seventy NSSAL 6 Draft
15 2. Write out the number using words. Number Word Description e.g three thousand, six hundred twelve (a) (b) (c) 547 (d) (e) (f) 63 (g) (h) (i) 708 (j) NSSAL 7 Draft
16 Write the Number (D) 1. Write the number that has been described using words. Word Description Number e.g. twentythree thousand, eight hundred two e.g. one hundred fifty thousand, six hundred twentythree (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) fiftysix thousand, seven hundred fortysix two hundred thirtynine thousand, one hundred fifteen forty thousand, three hundred seventyone three thousand six hundred five five hundred twentythree thousand, ninety sixty thousand, two hundred eight ninetythree three hundred five thousand, sixtyeight nine hundred one thirteen thousand, seven hundred fifteen five hundred thirtysix thousand four hundred seven thousand, fiftytwo (m) nine thousand, four hundred sixty (n) (o) (p) (q) (r) (s) (t) (u) (v) (w) fifty thousand, six hundred nine seven hundred thirteen thousand, three hundred ninetyone twelve thousand, ninetysix six hundred thirty two hundred thousand, five hundred sixteen eighty thousand, five hundred seventy ten thousand, four hundred three hundred six thousand, one hundred eleven nine hundred fifteen eight hundred seven thousand, two NSSAL 8 Draft
17 2. Write out the number using words. Number Word Description e.g two hundred fiftyfour thousand, seven hundred three (a) (b) (c) 780 (d) (e) (f) (g) (h) (i) (j) NSSAL 9 Draft
18 Write the Number (E) 1. Write the number that has been described using words. Word Description Number e.g. six million, fiftythree thousand, eight hundred seven e.g. twentyfour million, one hundred thousand, fifty (a) ten million, ninetysix thousand, eight hundred two (b) one million, two hundred five thousand, sixteen (c) seven hundred thirtyfour million (d) eighty million, five hundred twentynine thousand, seventy (e) four hundred twelve million, six hundred seventy thousand (f) eightyfive million, fifteen thousand, nine hundred (g) ninetyseven thousand, eight hundred twelve (h) six hundred twentyseven million, seven hundred fifty (i) forty million, sixtyfive thousand, ninety (j) five hundred six million, seventy thousand, nine hundred (k) three hundred two thousand, twentyeight (l) eleven million, three thousand, fortyseven (m) nine million, three hundred thirteen thousand, four (n) five hundred twenty million, six hundred seventytwo NSSAL 10 Draft
19 2. Write out the number using words. Number Word Description e.g Eight million, two hundred ninety thousand, fortythree (a) (b) (c) (d) (e) (f) (g) (h) NSSAL 11 Draft
20 Place Value Complete each of the following. 1. For 2 345, what number is in the: (a) tens' place? (b) thousands' place? (c) ones' place? 3. For 7 890, what number is in the: (a) ones' place? (b) thousands' place? (c) hundreds' place? 5. For , what number is in the: (a) ten thousands' place? (b) hundred thousands' place? (c) ones' place? 7. For , what number is in the: (a) thousands' place? (b) hundreds' place? (c) tens' place? 9. For , what number is in the: (a) hundreds' place? (b) hundred thousands' place? (c) millions' place? 11. For , what number is in the: (a) tens' place? (b) ten thousands' place? (c) hundred thousands' place? 2. For , what number is in the: (a) ten thousands' place? (b) hundreds' place? (c) thousands' place? 4. For , what number is in the: (a) hundreds' place? (b) ten thousands' place? (c) tens' place? 6. For , what number is in the: (a) tens' place? (b) thousands' place? (c) hundred thousands' place? 8. For , what number is in the: (a) ten thousands' place? (b) ones' place? (c) hundred thousands' place? 10. For , what number is in the: (a) tens' place? (b) millions' place? (c) ten thousands' place? 12. For , what number is in the: (a) millions' place? (b) hundreds' place? (c) thousands' place? NSSAL 12 Draft
21 Before, After, or Between (A) Your Answers: Number Word Description e.g. What number is before 8? 7 seven e.g. What number is between 12 and 14? 13 thirteen e.g. What number is after 29? 30 thirty 1. What number is after 6? 2. What number is before 11? 3. What number is between 18 and 20? 4. What number is after 15? 5. What number is between 23 and 25? 6. What number is before 27? 7. What number is between 28 and 30? 8. What number is after 34? 9. What number is before 37? 10. What number is after 11? 11. What number is between 46 and 48? 12. What number is after 59? 13. What number is before 70? 14. What number is between 81 and 83? 15. What number is after 42? 16. What number is between 90 and 92? 17. What number is before 77? 18. What number is after 99? 19. What number is between 73 and 74? 20. What number is before 80? NSSAL 13 Draft
22 Before, After, or Between (B) Your Answers: Number Word Description e.g. What number is before 120? 119 one hundred nineteen e.g. What number is between 456 and 458? 457 four hundred fiftyseven e.g. What number is after 599? 600 six hundred 1. What number is after 325? 2. What number is before 421? 3. What number is between 188 and 190? 4. What number is after 239? 5. What number is between 356 and 358? 6. What number is before 650? 7. What number is between 286 and 288? 8. What number is before 700? 9. What number is before 998? 10. What number is after 437? 11. What number is between 638 and 640? 12. What number is after 399? 13. What number is before 900? 14. What number is between 513 and 515? 15. What number is after 661? 16. What number is before 712? 17. What number is between 600 and 602? 18. What number is after 807? 19. What number is after 999? 20. What number is between 499 and 501? NSSAL 14 Draft
23 Closer to, and Odd or Even (A) For each of the following, write the number based on the description, indicate whether it is closer to 0 or 100, and state whether it is an odd or even number. Word Description Number Closer to 0 or 100 Odd or Even e.g. fiftynine odd e.g. thirtysix 36 0 even (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) eighteen seventysix eightythree forty twentyfour thirtynine ninetythree sixtyfive fortysix seventyone eleven thirtyeight fortynine seventytwo ninetysix twentyseven sixteen twelve fortyfour eightyfive fiftyeight thirtyseven NSSAL 15 Draft
24 Closer to, and Odd or Even (B) For each of the following, write the number based on the description, indicate whether it is closer to 0 or 100, and state whether it is an odd or even number. Word Description Number Closer to 0 or 1000 Odd or Even e.g. three hundred seventytwo even e.g. eight hundred eleven odd (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) two hundred fortynine six hundred twentythree seven hundred ninetyfour one hundred eightysix five hundred twelve three hundred seven four hundred sixty ninetynine two hundred seventysix five hundred fortyseven one hundred eightyfour nine hundred eight fiftythree three hundred fifty nine hundred ninetysix six hundred fortyfive one thousand, thirtyone two hundred seventyseven three thousand, ten seven hundred sixtynine two thousand, three hundred eight hundred fifteen NSSAL 16 Draft
25 Whole Numbers and Number Lines (A) 1. Place each number at its approximate location on the number line Place each number at its approximate location on the number line Place each number at its approximate location on the number line Place each number at its approximate location on the number line Place each number at its approximate location on the number line NSSAL 17 Draft
26 Whole Numbers and Number Lines (B) 1. Place each number at its approximate location on the number line Place each number at its approximate location on the number line Place each number at its approximate location on the number line Place each number at its approximate location on the number line Place each number at its approximate location on the number line NSSAL 18 Draft
27 Give an Example e.g. Give an example where the number 365 is used to represent something. Answer: There are 365 days in one year. 1. Give an example where the number 10 is used to represent something. 2. Give an example where the number 12 is used to represent something. 3. Give an example where the number 18 is used to represent something. 4. Give an example where the number 24 is used to represent something. 5. Give an example where the number 25 is used to represent something. 6. Give an example where the number 30 is used to represent something. 7. Give an example where the number 50 is used to represent something. 8. Give an example where the number 60 is used to represent something. 9. Give an example where the number 100 is used to represent something. 10. Give an example where the number 1000 is used to represent something. NSSAL 19 Draft
28 Closer To (A) Three whole numbers are provided. You are asked to determine whether the first number is closer to the second number or closer to the third number. Circle the correct answer. Two examples have been provided. e.g. Is 8 closer to 5 or 10? Answer: 10 e.g. Is 43 closer to 40 or 50? Answer: Is 6 closer to 5 or 8? 2. Is 3 closer to 1 or 7? 3. Is 4 closer to 0 or 6? 4. Is 7 closer to 4 or 9? 5. Is 5 closer to 1 or 8? 6. Is 8 closer to 4 or 10? 7. Is 6 closer to 3 or 10? 8. Is 2 closer to 0 or 3? 9. Is 9 closer to 7 or 12? 10. Is 8 closer to 3 or 12? 11. Is 6 closer to 0 or 11? 12. Is 7 closer to 5 or 11? 13. Is 9 closer to 6 or 13? 14. Is 10 closer to 7 or 15? 15. Is 11 closer to 10 or 14? 16. Is 10 closer to 6 or 12? 17. Is 15 closer to 13 or 19? 18. Is 13 closer to 10 or 15? 19. Is 16 closer to 14 or 20? 20. Is 12 closer to 8 or 15? 21. Is 19 closer to 15 or 21? 22. Is 18 closer to 16 or 22? 23. Is 24 closer to 20 or 30? 24. Is 27 closer to 20 or 30? 25. Is 39 closer to 30 or 40? 26. Is 46 closer to 40 or 50? 27. Is 73 closer to 70 or 80? 28. Is 94 closer to 90 or 100? 29. Is 28 closer to 20 or 30? 30. Is 60 closer to 0 or 100? 31. Is 40 closer to 0 or 100? 32. Is 70 closer to 0 or 100? 33. Is 30 closer to 20 or 50? 34. Is 70 closer to 50 or 100? NSSAL 20 Draft
29 Closer To (B) Three whole numbers are provided. You are asked to determine whether the first number is closer to the second number or closer to the third number. Circle the correct answer. Two examples have been provided. e.g. Is 34 closer to 30 or 40? Answer: 30 e.g. Is 459 closer to 400 or 500? Answer: Is 36 closer to 30 or 40? 2. Is 44 closer to 40 or 50? 3. Is 67 closer to 60 or 70? 4. Is 50 closer to 20 or 60? 5. Is 60 closer to 10 or 90? 6. Is 80 closer to 70 or 100? 7. Is 100 closer to 80 or 150? 8. Is 150 closer to 130 or 160? 9. Is 270 closer to 250 or 300? 10. Is 420 closer to 400 or 500? 11. Is 200 closer to 0 or 300? 12. Is 500 closer to 400 or 800? 13. Is 700 closer to 600 or 750? 14. Is 600 closer to 550 or 700? 15. Is 640 closer to 600 or 700? 16. Is 870 closer to 800 or 900? 17. Is 81 closer to 20 or 100? 18. Is 67 closer to 30 or 80? 19. Is 58 closer to 0 or 90? 20. Is 37 closer to 0 or 100? 21. Is 99 closer to 0 or 200? 22. Is 230 closer to 100 or 300? 23. Is 341 closer to 300 or 350? 24. Is 789 closer to 750 or 800? 25. Is 699 closer to 600 or 750? 26. Is 219 closer to 100 or 250? 27. Is 224 closer to 200 or 300? 28. Is 547 closer to 500 or 550? 29. Is 839 closer to 800 or 900? 30. Is 658 closer to 650 or 700? 31. Is 2399 closer to 2000 or 3000? 32. Is 1837 closer to 1000 or 2000? 33. Is 5643 closer to 5000 or 6000? 34. Is 2845 closer to 2000 or 4000? NSSAL 21 Draft
30 Operations with Whole Numbers NSSAL 22 Draft
31 Connect Four Addition Game (A) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Addend Strip whose sum is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Addend Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the addend strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Addend Strip: NSSAL 23 Draft
32 Connect Four Addition Game (B) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Addend Strip whose sum is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Addend Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the addend strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Addend Strip: NSSAL 24 Draft
33 Connect Four Addition Game (C) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Addend Strip whose sum is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Addend Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the addend strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Addend Strip: NSSAL 25 Draft
34 Connect Four Subtraction Game (A) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they have chosen the two numbers, they can capture one square with that appropriate difference (i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored counter on the square. There may be other squares with that same difference but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that difference using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Value 1: Value 2: NSSAL 26 Draft
35 Connect Four Subtraction Game (B) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they have chosen the two numbers, they can capture one square with that appropriate difference (i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored counter on the square. There may be other squares with that same difference but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that difference using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Value 1: Value 2: NSSAL 27 Draft
36 Connect Four Subtraction Game (C) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they have chosen the two numbers, they can capture one square with that appropriate difference (i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored counter on the square. There may be other squares with that same difference but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that difference using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Value 1: Value 2: NSSAL 28 Draft
37 Connect Four Subtraction Game (D) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they have chosen the two numbers, they can capture one square with that appropriate difference (i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored counter on the square. There may be other squares with that same difference but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that difference using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Value 1: Value 2: NSSAL 29 Draft
38 Subtraction Search There are twenty subtraction facts (e.g = 4) hidden in this grid. Check for three adjoining numbers that produce this fact. These numbers could be oriented horizontally, vertically, or diagonally. Circle the three adjoin numbers and record the fact below. Some of the facts cross over each other (e.g. vertical facts intersect with horizontal facts) Subtraction Facts: NSSAL 30 Draft
39 Multiplication Models There are three common models used to represent the operation of multiplication. These are the area model, set model and number line model. An example of each has been provided below for the product of 3 and 4. Example: Mathematic al Sentence Area Model Set Model Number Line Model A rectangle measuring 3 units by 4 units. 12 Three sets of four For each of the following, complete the mathematical statement and draw the three models. (a) 2 7 Area Model: Set Model: Number Line Model: (b) 6 3 Area Model: Set Model: Number Line Model: NSSAL 31 Draft
40 (c) 5 5 Area Model: Set Model: Number Line Model: (d) 7 4 Area Model: Set Model: Number Line Model: (e) 3 6 Area Model: Set Model: Number Line Model: NSSAL 32 Draft
41 Multiplication Array Game Roll two dice. Multiply the two numbers that are rolled and write down the mathematical sentence (e.g. 3 4 = 12) in the space provided. In the circular array recording sheet that has been provided, create an array using the rolled numbers. (e.g. 3 rows of 4, or 3 columns of 4). Once this is done, roll the dice again and repeat the procedure. The only thing is that the new array cannot overlap other array. Continue to roll the dice, write the sentences and draw the arrays until you are unable to draw new arrays (i.e. cannot be drawn without overlapping existing arrays). See how many arrays you can draw on the recording. It is a combination of luck and skill. Mathematical Sentence: Circular Array Recording Sheet: Number of Arrays Drawn: NSSAL 33 Draft
42 Connect Four Multiplication Game A Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Factor Strip whose product is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Factor Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the fraction strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Factor Strip: NSSAL 34 Draft
43 Connect Four Multiplication Game B Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Factor Strip whose product is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Factor Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the fraction strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Factor Strip: NSSAL 35 Draft
44 Connect Four Multiplication Game C Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Factor Strip whose product is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Factor Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the fraction strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Factor Strip: NSSAL 36 Draft
45 Connect Four Multiplication Game D Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Factor Strip whose product is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Factor Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the fraction strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Factor Strip: NSSAL 37 Draft
46 Connect Four Multiplication Game E Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers on the Factor Strip whose product is that desired square. Once they have chosen the two numbers, they can capture one square with that appropriate product. They either mark the square with an X or place a colored counter on the square. There may be other squares with that same product but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on the Factor Strip. They then mark the square with that product using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved on the fraction strip in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Factor Strip: NSSAL 38 Draft
47 Connect Four Multiples Game (A) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place one paper clip on the appropriate "Multiple of" Strip and one paper clip on the appropriate "Range for Multiple" Strip. For example to capture a 32, the first paper clip could be on the "2" or "4" (because 32 is a multiple of 2 or 4), while the second paper clip must be on the "30 to 39" range (because 32 is within this range). They either mark the square with an X or place a colored counter on the square. Only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that product using an O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: "Multiple of" Strip: "Range for Multiple" Strip: to to to 39 NSSAL 39 Draft
48 Connect Four Multiples Game (B) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place one paper clip on the appropriate "Multiple of" Strip and one paper clip on the appropriate "Range for Multiple" Strip. For example to capture a 45, the first paper clip could be on the "5" or "9" (because 45 is a multiple of 5 or 9), while the second paper clip must be on the "40 to 49" range (because 45 is within this range). They either mark the square with an X or place a colored counter on the square. Only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that product using an O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: "Multiple of" Strip: "Range for Multiple" Strip: to to to 49 NSSAL 40 Draft
49 Connect Four Multiples Game (C) Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place one paper clip on the appropriate "Multiple of" Strip and one paper clip on the appropriate "Range for Multiple" Strip. For example to capture a 56, the first paper clip could be on the "7" or "8" (because 56 is a multiple of 7 or 8), while the second paper clip must be on the "50 to 59" range (because 56 is within this range). They either mark the square with an X or place a colored counter on the square. Only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that product using an O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: "Multiple of" Strip: "Range for Multiple" Strip: to to to to 69 NSSAL 41 Draft
50 Capture the Flag Multiplication Game Mission: The winner is the first player to capture the opposing player's flag on the opposite side of the board. Rules: With this game each player starts with six markers on opposing sides of the board at the designated spots. Coins can be used as markers. For example, six pennies for one player, and six nickels for the other player. Each round a player can only move one marker one square either forwards, backwards, diagonally, or sideways. However, the player must know the multiplication fact to make that move. For example if the marker is first on a "2" square and wishes to move to a "5" square, then they must tell the other player that "2 times 5 is 10." If they do not know the fact, then they must choose another marker to move. Markers can move two spaces if they are jumping an opponent's marker and thus eliminating that marker. Again the player must know the multiplication fact associated with their starting square and their landing square. Each player is required to move a marker each round. Two markers cannot share the same square. There are a few squares that have "blockers" that limit movement and the opportunities to jump and eliminate markers. You are not permitted to guard your flag by placing one of your markers on your own flag. NSSAL 42 Draft
51 Player 2 Start Player 2 Start Player 2 Start Player 2 Start Player 2 Start Player 2 Start Player 1 Start Player 1 Start Player 1 Start Player 1 Start Player 1 Start Player 1 Start NSSAL 43 Draft
52 Random Multiplication Facts Quizzes Please note that there are three different types of quizzes (A, B & C). These three types correspond to the suggested order that the multiplication facts are taught. Quiz A1 Name: Quiz A2 Name: Quiz A3 Name: Quiz A4 Name: Quiz B1 Name: Quiz B2 Name: NSSAL 44 Draft
53 Quiz B3 Name: Quiz B4 Name: Quiz C1 Name: Quiz C2 Name: Quiz C3 Name: Quiz C4 Name: NSSAL 45 Draft
54 Multiplying on Your Hands Most people are OK with the multiplication facts with the numbers 0, 1, 2, 3, 4, and 5. Examples: However, many people struggle remembering the facts for larger numbers (6, 7, 8, and 9). Examples: There is a neat way to get these multiplication facts using your hands. First we need to identify the numbers associated with the fingers on your hands Example 1 Complete the following operation. 8 8? Step 1 Touch the appropriate fingers to find the multiplication fact. For example, if you wanted to work out 8 8, take finger eight from the left hand and touch it to finger eight from the right hand. NSSAL 46 Draft
55 Step 2 Count the number of fingers above the touching fingers on the left hand. In this case there are 2. Count the number of fingers above the touching fingers on the right hand. In this case there are 2. Now multiply these two numbers together ( ) Step 3 Count the two touching fingers and all the fingers dangling below the touching fingers. In this case we have 6 fingers. Now multiply this number by 10 ( ) Step 4 Add the two numbers that were generated in steps 2 and Therefore: NSSAL 47 Draft
56 Example 2 Complete the following operation. 7 9? Step 1 Touch the correct fingers. Step 2 Count the fingers above the touching fingers left hand and count the fingers above the touching fingers on the right hand. Multiply these two numbers Step 3 Count the two touching fingers and all the fingers dangling below the touching fingers. Now multiply this number by Step 4 Add the numbers from steps 2 and 3 ( ). Therefore: NSSAL 48 Draft
57 Example 3 Complete the following operation. 6 7? Step 1 Step Step Step 4 Since , then "Hand" clipart by scarlett was downloaded on April 5, 2011 from NSSAL 49 Draft
58 Put the Number in the Right Box With each question you have been provided with a list of numbers. Place those numbers in the appropriate box found on the right. A sample question has been provided. e.g. List of Numbers Multiple of 4 Not a Multiple of 4 16, 12, 9, 5, 20, 6, Multiple of 3 12, 24, 36 9, 6, 27 24, 17, 8, 27, 22, 36 Not a Multiple of 3 16, 20, 8 5, 17, List of Numbers Multiple of 2 Not a Multiple of 2 15, 4, 11, 6, 19, 18, 10, 14, 7, 12, 9, 21 Multiple of 3 Not a Multiple of 3 2. List of Numbers Multiple of 2 Not a Multiple of 2 19, 15, 10, 9, 16, 30, 22, 25, 18, 40, 45, 21 Multiple of 5 Not a Multiple of 5 3. List of Numbers Multiple of 3 Not a Multiple of 3 24, 20, 8, 15, 30, 9, 28, 25, 27, 40, 45, 26 Multiple of 5 Not a Multiple of 5 4 List of Numbers Multiple of 4 Not a Multiple of 4 16, 24, 21, 30, 9, 12, 8, 18, 6, 28, 15, 36 Multiple of 6 Not a Multiple of 6 5. List of Numbers Multiple of 5 Not a Multiple of 5 27, 25, 20, 12, 10, 40, 12, 19, 32, 80, 35, 28 Multiple of 4 Not a Multiple of 4 NSSAL 50 Draft
59 Investigation: Multiplying by Multiples of 10, 100, and 1000 Numbers are multiples of 10 if they can be expressed as a whole number multiplied by 10 e.g. 30 = 3 10 e.g. 70 = 7 10 e.g. 340 = Numbers are multiples of 100 if they can be expressed as a whole number multiplied by 100. e.g. 300 = e.g. 700 = e.g = Numbers are multiples of 1000 if they can be expressed as a whole number multiplied by e.g = e.g = e.g = In this section we will discover how we can multiply numbers that are multiples of 10, 100 and (e.g ). Use a calculator to work out the answers to the following two sets of questions. Look for a pattern. (Hint: Look at the number of zeros in each question.) First Set of Questions Second Set of Questions Questions 1. Based on the work that you have done above, what do you think the answers to each of these is. (Only use a calculator to check your answers.) (a) (b) (c) (e) (g) (d) (f) (h) Explain the rule that you have discovered. It may be easiest to do by looking at a specific question or questions (e.g , , ) NSSAL 51 Draft
60 Multiplying by Multiples of 10, 100, and 1000 In the previous investigation, you discovered how to multiply numbers that are multiples of 10, 100, and 1000 (e.g ). Here are two examples that cover the material that you learned in that investigation. Example 1 Evaluate Answer: To work out , it is a three step process (i) Omitting the zeros, multiply the two numbers ( ) (ii) Next count the number of zeros in the original question (There are 3; two from the number 400 and one from the 70) (iii) Take the product from step (a) and tack on the number of zeros from step (b). Therefore: = Example 2 Evaluate Answer: To work out , it is a three step process. (i) Omitting the zeros, multiply the two numbers ( ) (ii) Next count the number of zeros in the original question (There are 5; two from the number 800 and three from the 6000) (iii) Take the product from step (a) and tack on the number of zeros from step (b). Therefore: = Questions: 1. Complete the operation and express the answer in written form. Two sample questions has been completed for you. Answer in Written Form e.g. e.g. (a) (b) (c) (d) (e) (f) five hundred forty thousand two thousand one hundred NSSAL 52 Draft
61 2. Complete the following operations. Do not use a calculator. (a) (b) 9 60 (c) (d) (e) (f) (g) (h) (i) 40 8 (j) (k) 9 80 (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) (w) (x) 90 8 (y) (z) NSSAL 53 Draft
62 Multiplying Two Digit Numbers, Part 1 (Expanded Form) In order to complete this activity sheet, you should already know: 1. How to express a number in its expanded form e.g e.g How to multiply numbers that are multiples of 10, 100, or e.g. To work out : (a) Omitting the zeros, multiply the two numbers ( ) (b) Next count the number of zeros in the original question (There are 3; two from the number 400 and one from the 70) (c) Take the product from step (a) and tack on the number of zeros from step (b). Therefore: = e.g. To work out : (a) Omitting the zeros, multiply the two numbers ( ) (b) Next count the number of zeros in the original question (There are 5; two from the number 800 and three from the 6000) (c) Take the product from step (a) and tack on the number of zeros from step (b). Therefore: = Let's look at multiplying two digit numbers. The easiest way to do this is work through a sample problem. You may wish to view the following video that corresponds to the examples below. (or Google Search: YouTube Multiplying Two Digit Numbers Part 1 (Expanded Form)) Example 1 Complete the following operation Answer: We first need to express the two numbers in their expanded forms. 73 = = Next we set the numbers up so that we can do the multiplication. Note that the 20 and the 4 must both be multiplied by the 70 and the 3. That means we have to do four sets of multiplication ; first set of multiplication 4 70 ; second set of multiplication 20 3; third set of multiplication ; fourth set of multiplication Therefore: NSSAL 54 Draft
63 Example 2 Complete the following operation Answer: We first need to express the two numbers in their expanded forms. 67 = = Next we set the numbers up so that we can do the multiplication Therefore: Example 3 Complete the following operation Answer: Example 4 Complete the following operation Answer: The 9 must be multiplied by both the 60 and the 7. The 40 must be multiplied by both the 60 and the 7. The 7 must be multiplied by both the 80 and the 4. The 50 must be multiplied by both the 80 and the 4. The 5 must be multiplied by both the 90 and the 1. The 30 must be multiplied by both the 90 and the 1. NSSAL 55 Draft
64 Questions 1. The answer to is partially completed. Fill in the four missing components The 5 must be multiplied by both the 60 and the 3. The 90 must be multiplied by both the 60 and the The answer to is partially completed. Fill in the five missing components The 8 must be multiplied by both the 70 and the 4. The 30 must be multiplied by both the 70 and the 4. 3 Complete each of the operations. Only use a calculator to check your answers. (a) (b) NSSAL 56 Draft
65 (c) (d) (e) (f) (g) (h) NSSAL 57 Draft
66 4. In this section we have focused on multiplying two digit numbers using a specific technique. This technique, however, can be used to multiply even larger numbers. An example has been provided below. Look at this example and then multiply the numbers 436 and 72 on your own. Example: =? The two numbers are written in their expanded forms The 7 must be multiplied by the 600, 50, and The 30 must be multiplied by the 600, 50, and Now work out in the space provided. NSSAL 58 Draft
67 Multiplying Two Digit Numbers, Part 2 (Lattice Method) Up to this point if we wanted to multiply two digit numbers, we had to write both numbers in their expanded form and then do the four sets of multiplication, and add the four numbers. For example, if we wanted to multiply 82 and 53, we would have to do the following = Expanded Forms 53 = Remember that the 3 must be multiplied by both the 80 and 2. The same is true with the 50; it must also be multiplied by the and Therefore: There is an alternate method that is directly tied to the technique but it is much easier and faster. It involves using a two column two row chart where each cell is divided in two by a diagonal. You can view explanations that correspond to examples 1 and 2 by going to the following website. (or Google Search: YouTube Multiplying Two Digit Numbers (Lattice Method)) Example 1 Complete the following operation We can use the following chart to multiply two digit numbers. We put the 82 along the top of the chart, and the 53 along the right side of the chart NSSAL 59 Draft
68 Multiply the 5 by the 8, and place the two digits of the product (40) in the two spaces in the upper left hand corner of the chart Now multiply the 5 by the 2, and place the two digits of the product (10) in the two spaces in the upper right hand corner of the chart Repeat this procedure by multiplying the 3 by the 8 (product: 24), the 3 by 2 (product: 6), and fill in the appropriate spaces in the chart NSSAL 60 Draft
69 Now ignore the 82 and 53 along the outside of our chart. Starting at the bottom, add the numbers along each diagonal placing the answer along the outer edge of the chart. If a sum exceeds 9, carry the tens digit up to the next diagonal (This does not occur in this example.) Start here and work your way up to the next diagonal. 4 6 These numbers along the outside, starting at the upper left, represent the digits of your product. Therefore: Example 2 Complete the following operation = carry 1 Example 3 Complete the following operation = carry 1 NSSAL 61 Draft
70 Example 4 Complete the following operation carry = Why Does this Work? Let's take the last example and do it another way where we express 68 as , and 95 as When we multiply these expanded forms of the numbers, make sure the 5 is multiplied by both the 60 and 8, and similarly the 90 must be multiplied by both the 60 and Notice how the columns in the first technique match up to the diagonals in the second technique. The nice thing about the second technique is that it takes care of the place values (units, tens, hundreds, and thousands) for us = 6460 NSSAL 62 Draft
71 Questions 1. Multiply the numbers using the chart provided. (a) (b) (c) (d) (e) (f) NSSAL 63 Draft
72 2. Work out using both techniques (The chart has not been provided; you must draw your own.). Which technique do you prefer? 3. Complete the following operations. (a) (b) (c) (d) NSSAL 64 Draft
73 Multiplying MultiDigit Numbers On previous activity sheets, you learned how to multiple two digit numbers using two techniques. Those same techniques can be used with numbers having three or more digits, and as before, the second technique (i.e. chart method) is far easier and faster. You may wish to view the following video. (or Google Search: YouTube Multiplying MultiDigit Numbers (Lattice Method)) Example 1 Complete the operation Answer: First Technique (Expanded Form with Repeated Multiplication) The two numbers are written in their expanded forms. The 7 must be multiplied by the 300, 80, and 2. The 50 must be multiplied by the 300, 80, and Therefore: Answer: Second Technique (Chart Method) Our chart will have three columns because one number is a three digit number. The chart will have two rows because the other number is a two digit number. Remember that each cell is divided in two by a diagonal We could have set the chart up so that it had two columns and three rows. If that was the case, then 57 would be along the top and 382 would be down along the right side. NSSAL 65 Draft
74 We will now multiply the numbers Now ignore the 382 and 57 along the outside of our chart. Starting at the bottom, add the numbers along each diagonal placing the answer along the outer edge of the chart. If a sum exceeds 9, carry the tens digit up to the next diagonal. Carry Therefore: Example 2 Complete the following operation using the chart method Answer: Set up the chart with the three digit numbers along the sides and do the multiplication NSSAL 66 Draft
75 Ignore the numbers along the sides, and add the numbers along the diagonals. Carry Carry Therefore: = Example 3 Complete the following operation using the chart method Answer: Carry 1 Carry 1 Carry Therefore: = NSSAL 67 Draft
76 Questions: 1. Complete the following operations using the chart provided. (a) (b) (c) Complete the operations. (a) (b) NSSAL 68 Draft
77 Connect Four Division Game Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place two paper clips on two numbers; one from Value 1 and one from Value 2. Once they have chosen the two numbers, they can capture one square with that appropriate quotient (i.e. Value 1 divided by Value 2). They either mark the square with an X or place a colored counter on the square. There may be other squares with that same quotient but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that quotient using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: Value 1: Value 2: NSSAL 69 Draft
78 Division Search There are twenty division facts (e.g = 2) hidden in this grid. Check for three adjoining numbers that produce this fact. These numbers could be oriented horizontally, vertically, or diagonally. Circle the three adjoin numbers and record the fact below. Some of the facts cross over each other (e.g. vertical facts intersect with horizontal facts) Division Facts: NSSAL 70 Draft
79 Divisibility Chart You need three colored pencils (red, yellow, and blue) to complete this activity. Each number between 1 and 60 has three blocks below it. Using a calculator, determine whether the number is divisible by 2, 3, and/or 5. If the number is divisible by 2, shade the first block below the number red. If the number is divisible by 3, shade the second block below the number yellow. If the number is divisible by 5, shade the third block below the number blue. For example, the number 15 is both divisible by 3 and 5, therefore its second block is shaded yellow, and its third block is shaded blue Now that the chart is complete, look for patterns. Look at all the numbers that are divisible by 2. How can you determine if a number is divisible by 2 by just looking at the number (versus using a calculator)? Look at all the numbers that are divisible by 3. How can you determine if a number is divisible by 3 by just looking at the number (versus using a calculator)? (Hint: Consider adding the digits.) Look at all the numbers that are divisible by 5. How can you determine if a number is divisible by 5 by just looking at the number (versus using a calculator)? NSSAL 71 Draft
80 More Divisibility (A) You have already learned how to identify numbers that are divisible by 2, 3, and 5. Numbers that are divisible by 2 are even numbers. e.g. 6, 34, 798, 2050, and 3942 are all divisible by 2 Questions: If the sum of the digits of a multidigit number produces a number that is divisible by three, then the original multidigit number is divisible by 3. e.g. 561 is divisible by 3 because = 12 and 12 is divisible by 3. If the ones digit is a 0 or a 5, then the number is divisible by 5. e.g. 75, 120, 375, and 2960 are all divisible by Beside each number you will find a box corresponding to a number that might divide evenly into the original number. Check off the appropriate boxes to identify whether the original number is divisible by 2, 3, and/or 5. Do not use a calculator e.g. 78 e.g. 345 (a) 64 (b) 35 (c) 81 (d) 90 (e) 40 (f) 42 (g) 105 (h) 307 (i) 208 (j) 635 (k) 915 (l) 410 (m) 720 (n) 816 (o) 1245 (p) 2036 (q) 4109 (r) 7281 (s) 9130 (t) 3075 (u) 8374 (v) (a) Create 4 threedigit numbers that are divisible by 2 and 5 (but not 3). Do not use numbers encountered in question 1. (b) Create 4 threedigit numbers that are divisible by 2 and 3 (but not 5). Do not use numbers encountered in question 1. NSSAL 72 Draft
81 More Divisibility (B) You have already learned how to identify numbers that are divisible by 2, 3, and 5. Numbers that are divisible by 2 are even numbers. If the sum of the digits of a multidigit number produces a number that is divisible by three, then the original multidigit number is divisible by 3. If the ones digit is a 0 or a 5, then the number is divisible by 5. Did you know that if a number is divisible by both 2 and 3, then that number is also divisible by 6? Did you know that if a number is divisible by both 2 and 5, then that number is also divisible by 10? Did you know that if a number is divisible by both 3 and 5, then that number is also divisible by 15? Questions: 1. Beside each number you will find a box corresponding to a number that might divide evenly into the original number. Check off the appropriate boxes to identify whether the original number is divisible by 2, 3, 5, 6, 10, and/or 15. Do not use a calculator e.g. 220 e.g. 315 (a) 12 (b) 45 (c) 30 (d) 14 (e) 36 (f) 19 (g) 430 (h) 114 (i) 207 (j) 96 (k) 225 (l) 600 (m) 704 (n) 425 NSSAL 73 Draft
82 (o) 408 (p) 570 (q) 615 (r) 1078 (s) 2310 (t) 7131 (u) 8706 (v) 5603 (w) 4700 (x) 6125 (y) (a) Create 4 threedigit numbers that are divisible by 2, 3, and 6 (but not 5, 10, or 15). Do not use numbers encountered in question 1. (b) Create 4 threedigit numbers that are divisible by 3, 5, and 15 (but not 2, 6, or 10). Do not use numbers encountered in question 1. (c) Create 4 fourdigit numbers that are divisible by 3 (but not 2, 5, 6, 10, or 15). Do not use numbers encountered in question Fill in the blanks. (a) If a number is divisible by both 2 and 7, then that number is also divisible by. (b) If a number is divisible by both 3 and 11, then that number is also divisible by. (c) If a number is divisible by both 5 and 6, then that number is also divisible by. NSSAL 74 Draft
83 Divisibility or Prime Connect Four Game Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place one paper clip on the Tens strip and one paperclip on the Ones strip. They have now generated a two digit number. That two digit number is either divisible by a single digit whole number greater than 1 (i.e. 2, 3, 4, 5, 6, 7, 8, 9), or the number is a prime. The player captures a single square that describes the number. For example if the two digit number is 14, it is divisible by 2 or 7 (of the choices we are given), then the player can capture either a square with a 2 on it, or a square with a 7 on it. If the number is prime, then a square marked P can be captured. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips on either the Tens or Ones strip. They then mark the square that describes that number using a O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: P P P P 6 P Tens Strip: Ones Strip NSSAL 75 Draft
84 Division with Remainders Up to this point, we have only worked with division questions that worked out evenly (i.e. no remainders). Examples of this are shown below But what happens when division questions do not work out evenly. 19 7? 22 3? 44 5? 67 8? 78 10? With these types of questions, we have to talk about remainders. Consider the examples that follow. Example 1 Complete the operation: 9 4 Answer: We are going to show you three ways to solve this question. You are only required to learn the third method (i.e. most efficient method). We have shown the first two methods so that you understand why the third method works. Method 1: Sharing Items Suppose you had 9 apples that you had to share evenly between 4 people. How many apples would each person get? Are there any apples left over? Each person gets 2 apples, and 1 apple is left over. This 1 apple is what remains. Therefore we can conclude that: with a remainder of 1. This can also be written as 9 4 2, R:1 Method 2: Using Cuisenaire Rods Take the Cuisenaire rod for 9 (color: blue) and figure out how many Cuisenaire rods for 4 (color: purple) fit into it. Since your exercise sheet is photocopied in black and white, we have used two different textures so that you can distinguish the two types of rods Rod Representing 9: Rod Representing 4: We can place two purple rods (i.e. rods representing 4) on top of the blue rod (i.e. rod representing 9), but they don't cover the whole thing. A small portion, which is 1 unit long, is sticking out. 4 4 NSSAL 76 Draft
85 So 2 sets of 4 fit into 9, with 1 left over. That means that 9 4 is equal to 2 with a remainder of 1. This is written as 9 4 2, R:1 (Same answer that we got when we used the sharing items method.) Method 3: Using the Rule Four does not divide evenly into 9, so start by finding the closest number to 9 that is smaller than 9 which 4 does divide evenly into. That number is Now find the difference between the 8 and 9. The difference is This difference is our remainder. Therefore: 9 4 2, R:1 Example 2 Complete the operation: 10 3 Answer: Method 1: Sharing Items Suppose you had 10 apples that you had to share evenly between 3 people. How many apples would each person get? Are there any apples left over? Method 2: Using Cuisenaire Rods Take the Cuisenaire rod for 10 (color: orange) and figure out how many Cuisenaire rods for 3 (color: lime green) fit into it. Since your exercise sheet is photocopied in black and white, we have used two different textures so that you can distinguish the two types of rods Rod Representing 10: Rod Representing 3: Each person gets 3 apples, and 1 apple is left over. This 1 apple is what remains. Therefore we can conclude that: with a remainder of 1. This can also be written as , R:1 We can place three lime green rods (i.e. rods representing 3) on top of the orange rod (i.e. rod representing 10), but they don't cover the whole thing. A small portion, which is 1 unit long, is sticking out So 3 sets of 3 fit into 10, with 1 left over. That means that 10 3 is equal to 3 with a remainder of 1. This is written as , R:1 (Same answer that we got when we used the sharing items method.) NSSAL 77 Draft
86 Method 3: Using the Rule Three does not divide evenly into 10, so start by finding the closest number to 10 that is smaller than 10 which 3 does divide evenly into. That number is Now find the difference between the 9 and 10. The difference is This difference is our remainder. Therefore: , R:1 For the remaining examples, we will use the "Rule" to obtain our answer. This is the method we would like you to use when you complete the exercise questions. Example 3 Complete the following division questions. (a) 19 7 (b) 22 3 (c) 44 5 (d) 67 8 Answers: (a) Start by finding the closest number to 19 that is smaller than 19 which 7 does divide evenly into. That number is Now find the difference between the 19 and 14. The difference is This difference is our remainder. Therefore: , R:5 (b) If , and , then , R:1 (c) If , and , then , R:4 (d) If , and , then , R:3 Questions: 1. Complete the following division. Most, but not all, of these questions involve remainders. (a) 13 5 (b) 7 2 (c) 14 3 (d) 18 3 (e) 33 7 (f) 43 8 (g) 25 6 (h) 19 2 (i) 29 3 NSSAL 78 Draft
87 (j) 40 6 (k) 42 7 (l) 22 5 (m) 60 9 (n) 31 8 (o) 13 4 (p) 38 5 (q) 48 9 (r) 32 4 (s) 18 4 (t) 19 6 (u) 54 5 (v) 35 8 (w) 50 7 (x) There are twenty apples to share equally between six people. How many apples does each person get? How many apples are left over? NSSAL 79 Draft
88 Long Division (Partial Quotient Method) Division can be thought as "splitting a number into equal parts." Another way of thinking about division is to say "how many times can one number be subtracted from another number" (i.e. repeated subtraction). e.g =? "Splitting a Number into Equal Parts" "Repeated Subtraction" = 0 We were able to split the number 12 into four equal parts of three; therefore 12 4 = 3 Taking these approaches with larger numbers is far too difficult and timeconsuming. e.g =? We were able to subtract 4 from 12, three times. Therefore 12 4 = 3 "Splitting a Number into Equal Parts" We were able to split the number 52 into four equal parts of thirteen; therefore 52 4 = 13 NSSAL 80 Draft
89 "Repeated Subtraction" = 0 We were able to subtract 4 from 52, thirteen times. Therefore 52 4 = 13 Now there is the traditional algorithm for long division that some of you may have learned in the past, but we are going to use a different approach called the Partial Quotient Method. The biggest problem with the traditional method is that on it is very easy to make a mistake, there is only one way to complete the question, and that learners often do not understand why the procedure works. The Partial Quotient Method tends to make more sense because it relies on repeated subtraction, and allows learners to use multiple ways to obtain the correct answer. To use this technique, you must have a good grasp of your multiplication facts, be able to multiply by multiples of 10, 100, and 1000, and be able to subtract multidigit numbers. These are all things that we have done in previous lessons. Example 1 Solve Answer: How many times does 8 go into 1600? This learner goes with 200 because he/she knows that (The learner could have gone with a larger number like 300, but it does not matter with this method.) Now he/she subtracted 1600 from How many times does 8 go into This learner goes with 100 because he/she knows that Now he/she subtracted 800 from NSSAL 81 Draft
90 How many times does 8 go into 456. This learner goes with 50 because he/she knows that Now he/she subtracted 400 from How many times does 8 go into 56. This learner goes with 7 because he/she knows that After we filled in the 56, we did the subtraction and found that we had a remainder of 0. That means 8 divides evenly into In our last step the learner simply adds 200, 100, 50, and 7. That means that the quotient is = 357 NSSAL 82 Draft
91 View the following YouTube video that walks you through Examples 2 and 3. (or Google Search: YouTube Long Division (Partial Quotient Method) nsccalpmath) Example 2 Solve Answer: We have shown you three solutions to this question. The first student was the most efficient because he/she did the question in the fewest number of steps, however, all of the students have correct answers. That is the great thing about the partial quotient method; there is more than one way to do it right. In this case, 7 does not go evenly into 4785; we have a remainder of 4 when we complete the division. First Learner: Second Learner: Third Learner: 683 R: R: R: NSSAL 83 Draft
92 Example 3 Solve Answer: We have again supplied multiple solutions; all of them are correct. First Learner Second Learner: Third Learner: 3517 R: R: R: Questions Complete each of the division questions. Show all of your work NSSAL 84 Draft
93 NSSAL 85 Draft
94 NSSAL 86 Draft
95 NSSAL 87 Draft
96 Prime Factorization Prime numbers are numbers that are only divisible by one or itself. The first eight prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. Factors are numbers that multiply together to get another number. For example, the numbers 3 and 4 are factors of 12 because The numbers 2 and 6 are also factors of 12 because Prime factorization is the process of finding the prime numbers that multiply together to make another number. Example Determine the prime factors of each of the following. (a) 12 (b) 40 (c) 150 Answers: We these questions, there are often multiple ways of arriving at the final answer. (a) Method 1: In this case the learner starts by expressing 12 as the product of 2 and The learner then realizes that 6 is not a prime number, so proceeds to factor the Final Answer Method 2: In this case the learner starts by expressing 12 as the product of 3 and The learner then realizes that 4 is not a prime number, so proceeds to factor the Final Answer The two learners got the same answer even though they started out on slightly different paths. (b) is not prime so it's expressed as is not prime so it's expressed as Final Answer (c) Neither 15 nor 10 is a prime number Final Answer NSSAL 88 Draft
97 Questions 1. Determine the prime factors for each of the following. (a) 6 (b) 21 (c) 10 (d) 35 (e) 49 (f) 26 (g) 33 (h) 20 (i) 44 (j) 42 (k) 45 (l) 66 (m) 30 (n) 70 (o) 27 (p) 63 (q) 110 (r) 18 (s) 16 (t) 100 (u) 36 (v) 250 (w) 81 (x) 24 NSSAL 89 Draft
98 Multiple Operations (Whole Numbers) NSSAL 90 Draft
99 Express the Number in Multiple Ways For each number, express it as: At least two number sentences involving addition, At least two number sentences involving subtraction, At least two number sentences involving multiplication, At least two number sentences involving division, and At least three written sentences. (Please note that answers will vary from learner to learner.) Example: Number 10 Two number sentences involving addition: , Two number sentences involving subtraction: , Two number sentences involving multiplication: , Two number sentences involving division: , Three written sentences: The number is three more than seven The number is half of twenty. The number is five times bigger than two. Questions (a) Number 4 (b) Number 6 (c) Number 8 (d) Number 9 NSSAL 91 Draft
100 Name the Preceding or Next Fill in the blank with the missing number. Two examples have been completed to assist you. e.g. Name the next even number to each of the following. (a) 36 : (b) 74 : (c) 88 : Answers: (a) 36 : 38 (b) 74 : 76 (c) 88 : 90 e.g. Name the preceding multiple of five. (a) 45 : (b) 30 : (c) 100 : Answers: (a) 45 : 40 (b) 30 : 25 (c) 100 : Name the next odd number to each of the following (a) 47 : (b) 63 : (c) 29 : 2. Name the preceding even number to each of the following. (a) 36 : (b) 58 : (c) 50 : 3. Name the preceding multiple of five to each of the following. (a) 35 : (b) 20 : (c) 55 : 4. Name the next multiple of ten to each of the following. (a) 60 : (b) 30 : (c) 90 : 5. Name the next multiple of three to each of the following. (a) 21 : (b) 15 : (c) 27 : 6. Name the preceding multiple of four to each of the following. (a) 36 : (b) 28 : (c) 16 : 7. Name the preceding multiple of three to each of the following. (a) 15 : (b) 9 : (c) 27 : 8. Name the next multiple of four to each of the following. (a) 28 : (b) 20 : (c) 4 : 9. Name the next multiple of six to each of the following. (a) 24 : (b) 42 : (c) 18 : NSSAL 92 Draft
101 One of these Things is Not Like the Others With each of these questions, you must identify which one of the four does not belong. You must also explain why that one does not belong and why the remaining three belong together. (Hint: Think about odd and even, prime and composite, divisibility, patterns, perfect squares ) e.g The 2 does not belong because it is an even number and the remaining numbers, 13, 15, and 9, are all odd numbers , 5, 7, 9, 11 1, 2, 4, 8, 16, 2, 7, 12, 17, 22, 1, 4, 7, 10, 13, NSSAL 93 Draft
102 , 34, 37, 40, 18, 21, 24, 27 66, 63, 60, 57, 52, 55, 58, 61, 9. dozen fourteen 10. one hundred five twentysix eightyseven fiftytwo ninetynine one less than 100 NSSAL 94 Draft
103 Fact Family Puzzle (Multiplications and Division) Print the following onto rigid paper, cut each fact family into four puzzle pieces, shuffle all the family facts together, and ask the learners to sort them into their appropriate families NSSAL 95 Draft
104 NSSAL 96 Draft
105 Pathways Find the pathway from the upper left hand corner to the lower right hand corner of each grid by moving to equivalent and adjacent squares (i.e. squares to the left, right, top, or bottom). Start Start Finish Finish Start Start Finish Finish NSSAL 97 Draft
106 Two of These Boxes Just Don't Belong (A) Three boxes in each row have the same answers; the remaining two boxes just don't belong. Circle the three boxes in each row that have the same answers NSSAL 98 Draft
107 Two of These Boxes Just Don't Belong (B) Three boxes in each row have the same answers; the remaining two boxes just don't belong. Circle the three boxes in each row that have the same answers NSSAL 99 Draft
108 Equivalent Determine the missing number. Example 1: The answer is 10 because 5 6 and 3 10 both equal 30. Example 2: The answer is 7 because 6 8 and 7 2 both equal 14. Part 1 (a) (b) (c) (d) (e) (f) Part 2 (a) (b) (c) (d) (e) (f) Part 3 (a) (b) (c) (d) (e) (f) Part 4 (a) (b) (c) (d) (e) (f) Part 5 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) (w) (x) NSSAL 100 Draft
109 Venn Diagrams and Whole Numbers A Venn diagram, which is normally comprised of overlapping circles, is used to show relationships between different things. For this activity we are going to use them to illustrate relationships between different whole numbers. Suppose a learner is given the numbers 3, 5, 6, 9, 12, 11, 14, 16, 20, 21, and 23. They are asked to take those numbers and identify those that are divisible by 2, and those that are divisible by 3. Now some of these numbers (e.g. 5, 11, 23) are not divisible by either 2 or 3. Some numbers (e.g. 14, 16, 20) are only divisible by 2, others (e.g. 3, 9, 21) are divisible by only 3, and still others (e.g. 6, 12) are divisible by both 2 and 3. A Venn diagram, like the shown on the right, can be used to illustrate this. With the questions below, you have been given an incomplete Venn diagram and a list of numbers. Your mission is to place the numbers correctly in the Venn diagram. 1. List: 4, 5, 8, 9, 10, 14, 15, 20, 21, 22, 25, Divisible by 2 Divisible by Divisible by 2 Divisible by 5 2. List: 6, 9, 10, 11, 12, 15, 16, 18, 20, 25, 27, 30 Divisible by 3 Divisible by 5 NSSAL 101 Draft
110 3. List: 2, 3, 4, 7, 9, 10, 13, 15, 18, 20, 21, 23, 25 Note: A prime number is a number that can be divided evenly only by 1 or itself. (e.g. The number 11 is a prime number because it is only divisible by 1 and 11.) Odd Prime 4. List: 3, 4, 7, 9, 10, 15, 16, 18, 25, 27, 28, 30, 36 Note: For our purposes, a perfect square is a number that can be expressed as a whole number squared. (e.g. 81 is a perfect square because 81 = 9 2 ) Perfect Square Even 5. List: 7, 9, 10, 13, 15, 18, 30, 36, 45, 48, 55, 90 Multiple of 9 Multiple of 5 NSSAL 102 Draft
111 Whole Number Crossword Puzzle (A) A B C D E F G H I J K L M N O P Q R S T U V W Across: A. Next even number after 384 C G. one thousand, four hundred twenty I. 5 more than 228 J. Double 25 L. The product of 4 and 8 O Q R. 5 times 7 T. The number of minutes in 1 hour and 34 minutes V. 4 2 W. A number between 10 and 20 that is divisible by both 5 and 3 Down: B D E F H. 7 less than 470 K. Next number in the following sequence. 70, 74, 78, 82, M. 3 sets of 9 N. increase 734 by 20 P S. Next number in the following sequence. 63, 60, 57, 54, U. The number of cents in 2 quarters, 1 dime, and 1 nickel NSSAL 103 Draft
112 Whole Number Crossword Puzzle (B) A B C D E F G H I J K L M N O P Q R S T U V W Across: A C. 5 more than 81 G I. increase 153 by 30 J L. 7 = 4 O. The number of minutes in 6 hours and 4 minutes Q. A number between 10 and 20 that is divisible by 2, 3, 6, and 9 R. decrease 70 by 7 T. The product of 2 and 7 V W. The even number before 88 Down: B. The next odd number after 51 D. six thousand, four hundred thirtynine E. 213 rounded to the nearest tens F. Next number in the following sequence 394, 399, 404, 409, H K. 7 2 M. Double 12 N P. 6 sets of 11 S. 6 less than double 20 U. Next number in the following sequence 60, 54, 48, 42, NSSAL 104 Draft
113 Whole Number Crossword Puzzle (C) A B C D E F G H I J K L M N O P Q R S T U V W Across: A C. Triple 6 plus 1 G rounded to the nearest hundreds I. increase 361 by 40 J. Number of cents in 3 quarters and 2 dimes L. 9 times 6 O Q R T. Double 13 V. Next number in the following sequence 50, 47, 44, 41, W. 6 = 8 Down: B D. nine thousand, seven hundred twelve E. Next number in the following sequence 886, 890, 894, 898, 902, F H. The next multiple of 5 that follows 130 K. 8 sets of 4 M. 8 2 N. Number of minutes in 3 hours and 16 minutes P S. 39 decreased by 6 U. A number between 20 and 30 that is divisible by 2, 4, 7, and 14 NSSAL 105 Draft
114 Whole Number Crossword Puzzle (D) A B C D E F G H I J K L M N O P Q R S T U V W Across: A. The next odd number after 769 C. 6 sets of 3 G. six thousand, three hundred seven I. Next number in the following sequence 338, 344, 350, 356, J L O Q. 6 times 7 R. 9 = 4 T. Number of cents in 2 quarters and 3 dimes V. A number between 40 and 50 that is a multiple of 3, 5, 9, and 15 W. The product of 8 and 9 Down: B. 87 decreased by 9 D E. 746 increased by 60 F. 50 less than 784 H. Number of minutes in 2 hours and 37 minutes K. triple 8 plus 4 M. Next number in the following sequence 107, 104, 101, 98, N P. 9 2 S. Double 32 U NSSAL 106 Draft
115 KenKen Puzzles (A) KenKen puzzles were invented in 2004 by Japanese math teacher Tetsuya Miyamoto. The goal is to fill a grid with numbers such that no number is repeated in the same row or column. For a 3 by 3 KenKen grid, one can only use the numbers 1, 2, and 3. For a 4 by 4 KenKen grid, one can only use the digits 1, 2, 3, and 4. In addition to this, grids are divided into heavily outlined groups of cells, called cages. The numbers in these cages have to produce the target number using the indicated operation. For example if two cells are within a cage and 5+ is shown in an upper corner, then one find two numbers that add to 5. If working with a 3 by 3 grid where we are restricted to the numbers 1, 2, and 3, the only possible numbers are 2 and 3. All of this will make more sense after viewing the following online video. (or Google Search: YouTube KenKen Will Shortz Introduces New Puzzle) Questions: Complete the following KenKen Puzzles. Since these are 3 by 3 puzzles, you can only use the numbers 1, 2, and 3. Work in pencil and make sure you have a good eraser. (a) (b) (c) (d) NSSAL 107 Draft
116 (e) (f) (g) 1 5+ (h) (i) (j) NSSAL 108 Draft
117 KenKen Puzzles (B) Insert the numbers 1, 2, and 3 into the grid such that: no number is repeated in the same row or column, and the numbers in the cages produce the target number using the indicated operation. In these questions, we have limited ourselves to the operations of addition and multiplication. (a) 2 5+ (b) (c) 4+ 6 (d) (e) 4+ 6 (f) NSSAL 109 Draft
118 KenKen Puzzles (C) Normally with 3 by 3 KenKen Puzzles, we only use the numbers 1, 2, and 3 in the grid. We are going to change this. For the puzzles below, we are going to use the numbers 3, 4, and 5. New Instructions: Insert the numbers 3, 4, and 5 into the grid such that: no number is repeated in the same row or column, and the numbers in the cages produce the target number using the indicated operation. In these questions, we have limited ourselves to the operations of addition and multiplication. (a) (b) (c) 15 4 (d) (e) (f) NSSAL 110 Draft
119 KenKen Puzzles (D) Normally with 3 by 3 KenKen Puzzles, we only use the numbers 1, 2, and 3 in the grid. We are going to change this. For the puzzles below, we are going to use the numbers 5, 6, and 7. New Instructions: Insert the numbers 5, 6, and 7 into the grid such that: no number is repeated in the same row or column, and the numbers in the cages produce the target number using the indicated operation. (a) (b) (c) (d) (e) 42 5 (f) NSSAL 111 Draft
120 KenKen Puzzles (E) Insert the numbers 1, 2, 3, and 4 into the grid such that: no number is repeated in the same row or column, and the numbers in the cages produce the target number using the indicated operation. In these questions, we have limited ourselves to the operations of addition and multiplication. (a) (b) (c) 6 12 (d) NSSAL 112 Draft
121 Find the Two Numbers Example: Find the two numbers that multiply to give 12, and add to give 7. Answer: The numbers are 3 and 4 because and Questions: Find two numbers that: (a) multiply to give 15, and add to give 8. Answer: & (d) multiply to give 12, and add to give 13. Answer: & (g) multiply to give 16, and add to give 8. Answer: & (j) multiply to give 12, and add to give 8. Answer: & (m) multiply to give 22, and add to give 13. Answer: & (p) multiply to give 40, and add to give 14. Answer: & (s) multiply to give 35, and add to give 12. (v) Answer: & multiply to give 8, and add to give 9. Answer: & (b) multiply to give 20, and add to give 12. Answer: & (e) multiply to give 18, and add to give 9. Answer: & (h) multiply to give 30, and add to give 17. Answer: & (k) multiply to give 10, and add to give 11. Answer: & (n) multiply to give 42, and add to give 13. Answer: & (q) multiply to give 63, and add to give 16. Answer: & (t) multiply to give 50, and add to give 27. Answer: & (w) multiply to give 60, and add to give 23. Answer: & (c) multiply to give 28, and add to give 11. Answer: & (f) multiply to give 40, and add to give 13. Answer: & (i) multiply to give 36, and add to give 13. Answer: & (l) multiply to give 25, and add to give 10. Answer: & (o) multiply to give 24, and add to give 14. Answer: & (r) multiply to give 32, and add to give 12. Answer: & (u) multiply to give 24, and add to give 10. Answer: & (x) multiply to give 100, and add to give 20. Answer: & NSSAL 113 Draft
122 Which Combination of Numbers Works? In each case you have been given three numbers and an incomplete calculation. Insert the numbers in the appropriate positions to make the calculation complete. (a) = 19 (b) = 17 (c) = 5 (d) = 34 (e) = 15 (f) = 10 (g) = 13 (h) = 27 (i) = 14 (j) = 2 (k) = 9 (l) = 45 (m) = 6 (n) = 26 (o) = 49 NSSAL 114 Draft
123 Magic Squares In a magic square, the numbers in each column, row, and diagonal all add up to the same number. For example, with the magic square on the right, the numbers in each column, row, and diagonal all add up to Complete each of the magic squares below. (a) 6 (b) 3 (c) (d) 4 (e) 4 (f) (g) 5 (h) 4 (i) NSSAL 115 Draft
124 Addition Pyramids With addition pyramids, the two numbers in adjoining boxes add to give the number in the box immediately above Insert the missing numbers in each of the following addition pyramids NSSAL 116 Draft
125 NSSAL 117 Draft
126 Row Factors and Column Factors In each question you have been provided with a chart that is missing four numbers. These numbers are the factors of the numbers found to the right of each row, and factors of the numbers found at the bottom of each column. Find the missing numbers. Example: Answer: Questions: (a) 10 (b) 18 (c) (d) 27 (e) 60 (f) (g) 18 (h) 18 (i) (j) 24 (k) 20 (l) (m) 54 (n) 56 (o) NSSAL 118 Draft
127 Letter and Number Sentences 1. A B 9 2. C D 5 (a) If A is 5, how much is B? (a) If C is 12, how much is D? (b) If B is 7, how much is A? (b) If D is 4, how much is C? (c) If A is 3, how much is B? (c) If C is 20, how much is D? (d) If B is 1, how much is A? (d) If D is 8, how much is C? 3. E F G H 6 (a) If E is 6, how much is F? (a) If G is 30, how much is H? (b) If F is 3, how much is E? (b) If H is 3, how much is G? (c) If E is 12, how much is F? (c) If G is 42, how much is H? (d) If F is 1, how much is E? (d) If H is 10, how much is G? 5. I J 7 6. K L 30 (a) If I is 16, how much is J? (a) If K is 10, how much is L? (b) If J is 5, how much is I? (b) If L is 5, how much is K? (c) If I is 11, how much is J? (c) If K is 2, how much is L? (d) If J is 10, how much is I? (d) If L is 30, how much is K? 7. M N 4 8. P Q 13 (a) If M is 36, how much is N? (a) If P is 9, how much is Q? (b) If N is 3, how much is M? (b) If Q is 2, how much is P? (c) If M is 28, how much is N? (c) If P is 6, how much is Q? (d) If N is 5, how much is M? (d) If Q is 8, how much is P? NSSAL 119 Draft
128 Math Logic Puzzles For each, find the numbers represented by the symbols,, and. Hint: For each of the puzzles, one of the equations, not necessarily the first equation, allows you to solve for a symbol very quickly. Puzzle 1:  = = 6 + = 8 Puzzle 2: + = 8 + = 6 + = 7 Puzzle 3: + = 8 3 = 6 + = 10 Puzzle 4:  = = 10 4 = 8 Puzzle 5:  2 = = 17 = 18 Puzzle 6:  = = 9 + = Puzzle 7:  3 = + = 2 14 = 2 Puzzle 8: = 3 = 4  = 6 Puzzle 9: 24 = = 9 + = 3 Answers (in no particular order) = 4, = 6, = 2 = 6, = 4, = 12 = 6, = 2, = 9 = 14, = 6, = 4 = 11, = 3, = 8 = 7, = 5, = 1 = 2, = 5, = 3 = 4, = 2, = 5 = 1, = 7, = 4 NSSAL 120 Draft
129 Number Sentences (A) In each case create four number sentences using the three numbers provided. Example 1: Example 2: Answer: Answer: Questions: NSSAL 121 Draft
130 Number Sentences (B) Create as many number sentences as possible using three numbers from the chart at a time and limiting yourself to the operations of addition, subtraction, multiplication, and division. Example: You can create 16 number sentences using these numbers. Answer: You can create 12 number sentences using these numbers You can create 20 number sentences using these numbers You can create 24 number sentences using these numbers NSSAL 122 Draft
131 Order of Operations (A) Below, the same question was done by four different learners. The problem was that everyone ended up with different answers. Dave's Answer: Nashi's Answer Rana's Answer Montez's Answer Dave worked from left to right. He did the multiplication first, followed by the subtraction, and then did the division Nashi started with the subtraction, followed by multiplication, and then did the division Rana started with the multiplication, followed by the division, and then did the subtraction Montez worked from right to left. He started with the division, followed by the subtraction, and then did the multiplication. All of these learners started with the same question, but ended up with very different answers based on the order they chose to do the operations. Only one of the learners is correct. Do you know which one? The correct answer is 22. Rana did the question correctly because she knew the order of operations, the rules used to clarify which mathematical operations are done first in a mathematical expression. Most people remember the proper order by using the acronym BEDMAS. B  brackets first E  then exponents (e.g. squaring, cubing) DM  followed by division and multiplication in the order they appear (i.e. from left to right) AS  followed by addition and subtraction in the order they appear (i.e. from left to right) For this activity sheet we are only going to look at questions involving division, multiplication, addition, and subtraction (i.e. only the "DMAS" portion of "BEDMAS") Example 1 Evaluate each of the following. (a) (b) (c) (d) Answers: (a) The mathematical expression only involves the operations of addition and multiplication. According to BEDMAS, we do multiplication before addition NSSAL 123 Draft
132 (b) The mathematical expression only involves the operations of division and subtraction. According to BEDMAS, we do division before subtraction (c) The mathematical expression involves the operations of subtraction, addition and multiplication. According to BEDMAS, multiplication is done before subtraction or addition. Once this is done we have to decide between the subtraction and addition. When these operations occur in the same question, we always work from left to right. That means we will do the subtraction before the addition (d) The mathematical expression involves the operations of multiplication, addition, and division. According to BEDMAS we do division and multiplication before addition. However, do we do the division before the multiplication, or vice versa? When these two operations occur in the same question, we always work from left to right Questions Evaluate each of the mathematical expressions. Show your work and do not use a calculator. (a) (b) (c) (d) (e) (f) NSSAL 124 Draft
133 (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) (w) (x) NSSAL 125 Draft
134 Order of Operations (B) Below, the same question was done by four different learners. The problem was that everyone ended up with different answers. Kendrick's Answer: Helen's Answer Jun's Answer Nasrin's Answer Kendrick worked from left to right. He did the addition first, followed by the squaring, and then did the division. Helen started with the division, followed by the squaring, and then did the addition. Jun started with the squaring, followed by the addition, and then did the division. Nasrin started with the squaring, followed by the division, and then did the addition. All of these learners started with the same question, but ended up with very different answers based on the order they chose to do the operations. Only one of the learners is correct. Do you know which one? The correct answer is 14. Nasrin did the question correctly because he knew the order of operations, the rules used to clarify which mathematical operations are done first in a mathematical expression. The proper order can be remembered using the acronym BEDMAS. B  brackets first E  then exponents (e.g. squaring, cubing) DM  followed by division and multiplication in the order they appear (i.e. from left to right) AS  followed by addition and subtraction in the order they appear (i.e. from left to right) For this activity sheet we are only going to look at questions involving exponents, division, multiplication, addition, and subtraction (i.e. only the "EDMAS" portion of "BEDMAS") Example 1 Evaluate each of the following. 2 3 (a) (b) (c) NSSAL 126 Draft (d) Answers: 2 (a) The mathematical expression involves the operations of addition, squaring, and multiplication. According to BEDMAS, we would do the squaring (i.e. exponents) first, followed by multiplication, and finally the addition Remember: 3 means
135 3 (b) The mathematical expression involves the operations of multiplication, subtraction, and cubing. According to BEDMAS, we would do the cubing (i.e. exponents) first, followed by multiplication, and finally the subtraction Remember: 2 means (c) The mathematical expression involves the operations of squaring, division, addition, and multiplication. According to BEDMAS, we would do the squaring (i.e. exponents) first. Next we have to decide between the division and multiplication. When these operations occur in the same question, we always work from left to right. That means we will do the division before the multiplication. The last operation we will complete is the addition (d) The mathematical expression involves the operations of subtraction, cubing, addition, and division. According to BEDMAS, we would do the cubing (i.e. exponents) first. Next we would do the division. Once this is done we have to decide between the subtraction and addition. When these operations occur in the same question, we always work from left to right. That means we will do the subtraction before the addition Questions Evaluate each of the mathematical expressions. Show your work and do not use a calculator. (a) (b) (c) NSSAL 127 Draft
136 (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) NSSAL 128 Draft
137 Order of Operations (C) Below, the same question was done by four different learners. The problem was that everyone ended up with different answers. Kimi's Answer: Ryan's Answer Paulette's Answer Ajay's Answer Kimi started with the multiplication, followed by the squaring, and then did the subtraction. Ryan started with the operation in the brackets, followed by the squaring, and then did the multiplication. Paulette started with the squaring, followed by the subtraction, and then did the multiplication. Ajay started with the operation in the brackets, followed by the multiplication, and then did the squaring. All of these learners started with the same question, but ended up with very different answers based on the order they chose to do the operations. Only one of the learners is correct. Do you know which one? The correct answer is 36. Ryan did the question correctly because he knew the order of operations, the rules used to clarify which mathematical operations are done first in a mathematical expression. The proper order can be remembered using the acronym BEDMAS. B  brackets first E  then exponents (e.g. squaring, cubing) DM  followed by division and multiplication in the order they appear (i.e. from left to right) AS  followed by addition and subtraction in the order they appear (i.e. from left to right) Example 1 Evaluate each of the following. (a) (b) (c) Answers: (a) With the mathematical expression we have multiplication, subtraction embedded within a set of brackets, and squaring. According to BEDMAS, we start with the operations in the brackets, followed by the squaring (i.e. exponents), and then finish off with the multiplication Remember: 5 means NSSAL 129 Draft
138 (b) With the mathematical expression we have subtraction embedded within a set of brackets, cubing, addition, and multiplication. According to BEDMAS, we start with the operations in the brackets. This will be followed by the cubing (i.e. exponents). We will then do the multiplication, and then finish up with the addition Remember: 2 means (c) With the mathematical expression we will start with the addition that is embedded within a set of brackets. Next we will work with the exponents (i.e. the squaring and the cubing). Following this we do the multiplication. That leaves us with the addition and multiplication. When these operations occur in the same question, we always work from left to right. That means we will do the addition before the subtraction Questions Evaluate each of the mathematical expressions. Show your work and do not use a calculator. (a) (b) (c) (d) (e) (f) NSSAL 130 Draft
139 (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) NSSAL 131 Draft
140 Patterns NSSAL 132 Draft
141 What's the Pattern? (A) Look for a pattern and then fill in the three missing symbols. (a),,,,,,,,,,,,,,,, (b),,,,,,,,,,,,,,,, (c),,,,,,,,,,,,,,,,, (d),,,,,,,,,,,,, (e),,,,,,,,,,,, (f) A,, B,, C,, D,, E,, F,,,, (g),,,,,,,,,,,,, (h),,,,,,,,,,,,,,, (i),,,,,,,,,,,,, (j), R,, P,, R,, P,, R,, P,, R,,,, (k),,,,,,,,,,,,,,,,,, (l),,,,,,,,,,,,,,,,, (m),,,,,,,,,,,,,,,, (n) Z, z, Y, y, X, x, W, w, V, v, U, u,,, (o),,,,,,,,,,,,,,,,,, (p),,,,,,,,,,,,,,,,,, NSSAL 133 Draft
142 What's the Pattern? (B) Look for a pattern and then fill in the three missing symbols. Hint: Sometimes more than one trait is changing. e.g. Every figure is rotated slightly counter clockwise (45 o ) and every fifth figure is not bolded.,,,,,,,,,,,,,, (a),,,,,,,,,,,,,,,, (b),,,,,,,,,,,,,,,, (c),,,,,,,,,,,,, (d),,,,,,,,,,,,,,,, (e),,,,,,,,,,,,,,,, (f),,,,,,,,,,,, (g),,,,,,,,,,, (h),,,,,,,,,,,,,,,, (i),,,,,,,,,,,,,,, (j),,,,,,,,,,,,,,,, (k),,,,,,,,,,,,,,,,, (l),,,,,,,,,,,,,,,, (m),, A,,, B,,, C,,, D,,,, (n),,,,,,,,,,,,,,,,,, (o),,,,,,,,,,,,,,, NSSAL 134 Draft
143 Toothpick Patterns You have been supplied with a sequence of shapes where each shape is created using toothpicks. For example a triangle is made up of 3 toothpicks. In these questions, you are going to be looking for a pattern in terms of the number of toothpicks as you move from one figure to the next. There are three specific parts to each question. (a) Draw the next figure in the sequence. (b) Describe the sequence in terms of numbers (i.e. numbers of toothpicks in each figure). (c) In words, describe what is happening to the numbers as you move from figure to figure in the sequence. Example: Answer: (a) Next Figure: (b) Sequence Using Numbers: 4, 7, 10, 13, 16 (c) Describe Sequence Using Works: Start at 4 and keep adding NSSAL 135 Draft
144 NSSAL 136 Draft
145 Create the Pattern (A) Using the instructions, create the first five numbers in the sequence. e.g. Start at 7 and go up by 3 each time. Answer: 7, 10, 13, 16, 19 (a) Start at 9 and go up by 2 each time. (b) Start at 24 and go down by 1 each time. (c) Start at 8 and go up by 3 each time. (d) Start at 4 and go up by 5 each time. (e) Start at 33 and go down by 3 each time. (f) Start at 29 and go down by 2 each time. (g) Start at 11 and go up by 4 each time. (h) Start at 30 and go down by 2 each time. (i) Start at 3 and go up by 6 each time. (j) Start at 2 and go up by 4 each time. (k) Start at 11 and go up by 5 each time. (l) Start at 38 and go down by 3 each time. (m) Start at 36 and go down by 4 each time. (n) Start at 23 and go up by 6 each time. (o) Start at 40 and go down by 5 each time. (p) Start at 13 and go up by 10 each time. (q) Start at 72 and go down by 2 each time. (r) Start at 99 and go down by 10 each time. (s) Start at 41 and go up by 4 each time.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, NSSAL 137 Draft
146 Create the Pattern (B) Using the instructions, create the first four numbers in the sequence. e.g. Start at 125 and go up by 6 each time. Answer: 125, 131, 137, 143 (a) Start at 63 and go up by 7 each time. (b) Start at 234 and go down by 2 each time. (c) Start at 81 and go up by 3 each time. (d) Start at 79 and go down by 3 each time. (e) Start at 126 and go up by 5 each time. (f) Start at 540 and go down by 10 each time. (g) Start at 352 and go up by 20 each time. (h) Start at 47 and go up by 4 each time. (i) Start at 68 and go down by 4 each time. (j) Start at 275 and go up by 25 each time. (k) Start at 134 and go up by 6 each time. (l) Start at 456 and go down by 100 each time. (m) Start at 99 and go down by 11 each time. (n) Start at 347 and go up by 3 each time. (o) Start at 605 and go down by 5 each time. (p) Start at 710 and go up by 30 each time. (q) Start at 670 and go down by 20 each time. (r) Start at 412 and go up by 6 each time. (s) Start at 364 and go down by 3 each time.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, NSSAL 138 Draft
147 Number Patterns (A) Look at the pattern and fill in the missing numbers. (a) 6, 8, 10, 12, 14, 16,,, (b) 23, 22, 21, 20, 19, 18, 17, 16, 15,,, (c) 5, 7, 9, 11, 13, 15, 17, 19, 21,,,? (d) 30, 28, 26, 24, 22, 20, 18,,, (e) 0, 3, 6, 9, 12, 15,,, (f) 10, 15, 20, 25, 30, 35,,, (g) 40, 36, 32, 28, 24, 20,,, (h) 31, 28, 25, 22, 19, 16, 13,,, (i) 7, 12, 17, 22, 27, 32, 37, 42,,, (j) 45, 47, 49, 51, 53, 55, 57,,, (k) 19, 22, 25, 28, 31, 34, 37,,, (l) 51, 47, 43, 39, 35, 31, 27, 23,,, (m) 67, 62, 57, 52, 47, 42, 37,,, (n) 44, 43,, 41,, 39, 38,, 36 (o) 23,, 27, 29, 31,,, 37, 39 (p) 30,,, 21, 18, 15, 12,, 6 (q) 17, 21,,, 33,, 41, 45, 49, 53 (r), 14, 19, 24,,, 39, 44, 49 (s), 33,, 39, 42,, 48, 51, 54, 57 (t),, 41, 36, 31,, 21, 16, 11, 6 (u) 91,, 93,, 95,, 97 NSSAL 139 Draft
148 Number Patterns (B) Look at the pattern and fill in the missing numbers. (a) 2, 4, 6, 8, 10, 12,,, (b) 4, 7, 10, 13, 16,,, (c) 29, 27, 25, 23, 21,,,? (d) 55, 50, 45, 40, 35,,, (e) 3, 10, 17, 24, 31, 38,,, (f) 64, 56, 48, 40, 32, 24,,, (g) 0, 6, 12, 18, 24, 30, 36,,, (h) 1, 12, 23, 34, 45, 56,,, (i) 44, 40, 36, 32, 28, 24, 20,,, (j) 100, 104, 108, 112, 116, 120,,, (k) 675, 680, 685, 690, 695, 700, 705, 710,,, (l) 190, 210, 230, 250, 270, 290, 310, 330,,, (m) 326, 324, 322, 320, 318, 316,,, (n) 6, 10, 14,, 22, 26,,, 38 (o) 40,, 34, 31,, 25, 22,, 16 (p) 56, 61,, 71, 76, 81,,, 96, 101, 106 (q), 25, 29,, 37, 41, 45,, 53, 57 (r), 245,, 233, 227, 221, 215,, 203 (s) 16,,, 40, 48,, 64, 72, 80 (t),, 292,, 284, 280, 276, 272, 268 (u) 420,, 460,, 500,, 540 NSSAL 140 Draft
149 Row, Column, and Diagonal Patterns Describe the pattern between numbers found in columns, rows and diagonals. (Note: Rows go from left to right. Columns go from top to bottom. Diagonals go from the upper left to the lower right.) e.g Row Pattern: Column Pattern: Diagonal Pattern: Add 2 Add 6 Add Row Pattern: Row Pattern: Row Pattern: Column Pattern: Column Pattern: Column Pattern: Diagonal Pattern: Diagonal Pattern: Diagonal Pattern: Row Pattern: Row Pattern: Row Pattern: Column Pattern: Column Pattern: Column Pattern: Diagonal Pattern: Diagonal Pattern: Diagonal Pattern: NSSAL 141 Draft
150 What's the Relationship? A chart containing numbers between and equal to 1 and 50 is provided. Some of the numbers are enlarged and bolded. What is the relationship amongst those enlarged and bolded numbers? Write the answer in as many ways as you can think of. Example 1: Example Answers: Multiples of 6 Divisible by 6 Divisible by both 2 and 3 Start at 6 and keep adding 6 Start at 48 and keep subtracting 6 Answers: All are prime numbers Only divisible by 1 and themself (a) (b) (c) (d) NSSAL 142 Draft
151 (e) (f) (g) (h) (i) (j) (k) (l) NSSAL 143 Draft
152 Input Output (A) A number is put in (the input number) and a different number is spit out (the output number). In each case, determine the mathematical rule that changes the input number to the output number. Three examples have been provided to help you understand what needs to be done. Example I: Input Output Example II: Input Output Example III: Input Output (a) Rule: Input 3 = Output Input Output 7 11 Rule: (b) Rule: Input  6 = Output Input Output 3 6 Rule: (c) Rule: Input 2 = Output Input Output 27 9 Rule: (d) Input Output Rule: (e) Input Output Rule: (f) Input Output Rule: (g) Input Output Rule: (h) Input Output Rule: (i) Input Output Rule: NSSAL 144 Draft
153 Input Output (B) A number is put in (the input number) and a different number is spit out (the output number). In each case, determine the mathematical rule that changes the input number to the output number. Three examples have been provided to help you understand what needs to be done. Example I: Input Output Example II: Input Output Example III: Input Output (a) Rule: Input 7 = Output Input Output 4 36 Rule: (b) Rule: Input  9 = Output Input Output 13 6 Rule: (c) Rule: Input 6 = Output Input Output 4 13 Rule: (d) Input Output Rule: (e) Input Output Rule: (f) Input Output Rule: (g) Input Output Rule: (h) Input Output Rule: (i) Input Output Rule: NSSAL 145 Draft
154 Input Output (C) A number is put in (the input number) and a different number is spit out (the output number). In each case, determine the mathematical rule that changes the input number to the output number. Three examples have been provided to help you understand what needs to be done. Example I: Input Output Example II: Input Output Example III: Input Output (a) Rule: Input 60 = Output Input Output Rule: (b) Rule: Input  20 = Output Input Output Rule: (c) Rule: Input 3 = Output Input Output Rule: (d) Input Output Rule: (e) Input Output Rule: (f) Input Output Rule: (g) Input Output Rule: (h) Input Output Rule: (i) Input Output Rule: NSSAL 146 Draft
155 Input Output (D) In each case, determine the mathematical rule that changes the input number to the output number. For these questions, the input number is multiplied by 2, 3, 4, or 5 and then has a number added to or subtracted from it. Example I: Input Output Example II: Input Output Example III: Input Output (a) Rule: (Input 2 )  1 = Output Input Output Rule: (b) Rule: (Input 4 ) + 3 = Output Input Output Rule: (c) Rule: (Input 5 )  4 = Output Input Output Rule: (d) Input Output Rule: (e) Input Output Rule: (f) Input Output Rule: (g) Input Output Rule: (h) Input Output Rule: (i) Input Output Rule: NSSAL 147 Draft
156 Filling or Draining A large container, which can hold 20 litres of water when filled to the brim, is either being filled or drained at a constant rate. You will be able to tell based on the sequence of diagrams of the container that have been supplied. Your mission is to complete the next diagram in the sequence, then describe each situation using words, using a table, and using an equation, and finally predict when the container will be full or empty. e.g. 10 L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Answer: In the last diagram, we should show the water level at the 8 L mark. Written Description: Table of Values: Time Litres The container initially had 14 L of water in it and it is being drained at a rate of 2 litres per minute. 10 L Equation: Litres = 142 Time or L = 142T If the container initially held 14 L of water and it's losing 2 L per minute, then in 7 minutes the container will be empty (i.e. hold 0 L of water) NSSAL 148 Draft
157 1. 10 L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Empty: L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Full: NSSAL 149 Draft
158 3. 10 L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Full: L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Empty: NSSAL 150 Draft
159 5. 10 L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Empty: L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Full: NSSAL 151 Draft
160 7. 10 L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Empty: L 10 L 10 L 10 L time = 0 minutes time = 1 minute time = 2 minutes time = 3 minutes Written Description: Table of Values: Time Litres Equation: Full: NSSAL 152 Draft
161 Travelling Towards or Away From Home Montez is either travelling towards or away from his home at a constant speed. You will be able to tell by looking at the sequence of diagrams that have been provided. Your mission is to describe each situation using words, using a table of values (where the times go from 0 seconds to 4 seconds) and using an equation, and finally predict how far Montez is from home at t = 7 seconds. e.g. t = 0 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Answer: Written Description: Montez is initially 7 metres from home and runs away from the home at a rate of 4 metres per second. Table of Values: Time Distance from Home Equation: distance = time or d = 7 + 4t At t = 7 seconds, Montez will be 35 metres from home. The easiest way to figure this out is to substitute 7 in for t in the equation d = 7 + 4t, and then solve for d. You could also take the table of values and expand it until you reach a time of 7 seconds or jump along the number line on the diagram four spaces for every second until you reach the desired time. NSSAL 153 Draft
162 1. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Written Description: 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Table of Values: Time Distance from Home Equation: At t = 7 seconds 2. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m NSSAL 154 Draft
163 Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds 3. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds NSSAL 155 Draft
164 4. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds 5. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m NSSAL 156 Draft
165 Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds 6. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds NSSAL 157 Draft
166 7. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds 8. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m NSSAL 158 Draft
167 Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds 9. t = 0 seconds t = 1 second 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m t = 2 seconds 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m 0 m 5 m 10 m 15 m 20 m 25 m 30 m 35 m Written Description: Table of Values: Time Distance from Home Equation: At t = 7 seconds NSSAL 159 Draft
168 Weight of the Water A spring balance is used to measure the weight of an object that is suspended below it. That weight will be measured in newtons (N). In this situation we have a container suspended below our spring balance that is either being filled or drained of water at a constant rate. Your mission is to describe each situation using words, using a table of values (where the times go from 0 seconds to 4 seconds) and using an equation, and finally predict the weight of the water at t = 6 seconds. The scale was adjusted so that the weight of the empty container is not included. e.g. time = 0 seconds time = 1 second time = 2 seconds Newtons 0 Newtons 0 Newtons Answer: Written Description: The container initially contained water weighting 5 newtons and then water was added such that the weight increased by 2 newtons per second. Table of Values: Time Weight Equation: weight = time or w = 5 + 2t At t = 6 seconds, the weight of the container and its contents should be 17 newtons. NSSAL 160 Draft
169 1. time = 0 seconds time = 1 second time = 2 seconds Newtons 0 Newtons 0 Newtons Written Description: Table of Values: Time Weight Equation: At t = 6 seconds 2. time = 0 seconds time = 1 second time = 2 seconds Newtons 0 Newtons 0 Newtons NSSAL 161 Draft
170 Written Description: Table of Values: Time Weight Equation: At t = 6 seconds 3. time = 0 seconds time = 1 second time = 2 seconds Newtons 0 Newtons 0 Newtons Written Description: Table of Values: Time Weight Equation: At t = 6 seconds NSSAL 162 Draft
171 4. time = 0 seconds time = 1 second time = 2 seconds Newtons 0 Newtons 0 Newtons Written Description: Table of Values: Time Weight Equation: At t = 6 seconds 5. time = 0 seconds time = 2 seconds time = 4 seconds Newtons 0 Newtons 0 Newtons NSSAL 163 Draft
172 Written Description: Table of Values: Time Weight Equation: At t = 6 seconds 6. time = 0 seconds time = 2 seconds time = 4 seconds Newtons 0 Newtons 0 Newtons Written Description: Table of Values: Time Weight Equation: At t = 6 seconds NSSAL 164 Draft
173 Word Problems NSSAL 165 Draft
174 Describing the Relationships with Words Example: Given the diagram on the right, describe as many mathematical relationships using the different shapes. Answer: There are 3 arrows. There are 5 lightning bolts. There are 6 hearts. There are a total of 14 shapes. There is 1 more heart than lightning bolts. There is 1 less lightning bolt than hearts. There are 2 more lightning bolts than arrows. There are 2 less arrows than lightning bolts. There are 3 more hearts than arrows. There are 3 less arrows than hearts. There are twice as many hearts as arrows. There are half as many arrows as hearts. Now do the same with this diagram that has moons, suns and hearts. NSSAL 166 Draft
175 List the Numbers Based on the Written Description 1. List all the whole numbers between 8 and List all the even numbers between and equal to 10 and List all the multiples of 10 between and equal to 1 and List all the numbers that are divisible by 5 between 13 and List all the two digit numbers whose tens digit and one digit are the same. 6. List all the numbers that are divisible by 3 between 16 and List all prime numbers between and equal to 5 and List all the odd numbers that are divisible by 3 between 2 and List all the even numbers that are divisible by 5 between 3 and List all the two digit numbers whose digits add to List all the two digit numbers greater than 30 that are multiples of List all the numbers less than 35 that are divisible by List all composite numbers (i.e. not prime) between and equal to 9 and List all the numbers that are divisible by 2 and 3 between 5 and List all the odd numbers that are divisible by 5 between 7 and 56. NSSAL 167 Draft
176 Addition and Subtraction Crossword Complete the following crossword using words to express your answer (e.g. seven). Do not use a calculator Across: 1. nine subtract seven 2. four plus eight 3. forty plus ten 5. eight subtract four 6. three add four 8. six increased by seven 10. nineteen decreased by three 13. five plus six 15. eleven minus two Down: 1. six plus four 2. five minus two 3. sixty subtract twenty 4. six add six 5. decrease sixteen by one 7. increase nine by two 9. seventeen subtract eight 10. thirty plus thirty 11. six add two NSSAL 168 Draft
177 Across: 17. twenty subtract eighteen 19. sixteen decreased by fifteen 21. subtract seventy from eighty 23. two increased by ten 25. fourteen subtract eleven 27. one plus seven 29. add eight to twelve 31. sixty decreased by fifty 33. eighteen minus twelve 34. decrease thirteen by eight 35. seventeen subtract ten 36. increase ten by thirty 37. thirteen minus four Down: 12. seventy decreased by seventy 14. add five to five 16. seventeen minus nine 18. decrease twenty by nineteen 20. increase four by seven 22. fifteen subtract seven 24. zero plus five 26. subtract ten from twentyone 28. fifty subtract twenty 30. increase eight by three 31. fifteen decreased by twelve 32. add zero to nine NSSAL 169 Draft
178 Multiplication and Division Crossword Complete the following crossword using words to express your answer (e.g. twentyseven). Include hyphens () where appropriate. Do not use a calculator Across 2. four multiplied by twenty 5. five times six Down 1. eight times four 3. four multiplied by four NSSAL 170 Draft
179 Across 7. eighteen divided by nine 8. nine times four 9. eight multiplied by three 12. seven times five 15. twentyseven divided by three 17. three multiplied by ten 19. three times five 20. six multiplied by seven 23. twentyfive divided by five 24. twentyfour divided by two 25. seven divided by seven 26. sixty divided by six 28. thirtyfive divided by five 32. three times three 33. two multiplied by eight 34. eighty divided by eight 36. fortyeight divided by six 37. fifteen divided by three 38. seven times ten 40. six divided by one 41. sixty divided by thirty 42. seven multiplied by two 44. one hundred divided by ten 45. six times nine Down 4. four times seven 5. two times six 6. forty divided by ten 10. forty divided by eight 11. eighteen divided by three 13. nine times nine 14. ten multiplied by five 16. four times five 18. twentyone divided by seven 21. fiftysix divided by eight 22. eleven times one 27. ten times nine 29. sixtythree divided by seven 30. eight times seven 31. eight multiplied by eight 34. two times six 35. thirty divided by six 37. twentyfour divided by six 38. fiftyfour divided by nine 39. twelve divided by three 40. fortynine divided by seven 41. sixty divided by twenty 43. one hundred divided by fifty NSSAL 171 Draft
180 Operations Crossword Complete the following crossword using words to express your answers (e.g. thirtyfive). Include hyphens () where appropriate. Do not use a calculator Across 2. nine times six 4. seven multiplied by three 7. thirtytwo divided by eight Down 1. sixteen minus six 2. thirtyfive divided by seven 3. fortythree plus three NSSAL 172 Draft
181 Across 9. one hundred subtract ten 11. twelve minus nine 12. fortyfive divided by nine 14. seven increased by three 15. nine times nine 16. fiftysix divided by eight 18. twentyone decreased by two 21. subtract thirty from thirty 22. seven multiplied by six 23. ten times seven 26. seventeen decreased by six 29. fortynine divided by seven 31. eight divided by eight 36. six multiplied by six 38. three times four 41. one hundred subtract ninetynine 42. eighty divided by forty 43. fourteen decreased by five 44. twenty multiplied by three 45. fortyfour divided by four Down 4. eighteen subtract five 5. sixteen divided by eight 6. four add five 8. five multiplied by two 10. three times eight 13. eight multiplied by nine 17. nine increased by four 19. twentyseven divided by nine 20. subtract fifteen from nineteen 23. eighteen divided by three 24. sixtythree divided by seven 25. twelve times zero 27. nine multiplied by nine 28. eighty minus seventy 30. two times forty 32. thirty divided by two 33. subtract six from eleven 34. thirty divided by five 35. four multiplied by two 36. one hundred divided by five 37. zero multiplied by six 39. seventeen minus eight 40. sixty divided by ten NSSAL 173 Draft
182 Word Sentence to Number Sentence to Answer (A) In each case take the word sentence, make a number sentence out of it, and then provide the answer. A few examples have been provided. Word Sentence Number Sentence Answer e.g. What is the product of seven and three? e.g. What is five increased by six? e.g. What is a third of twelve? (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) What is the sum of three and eight? What is six multiplied by five? What do you get when you double three? What is nine decreased by four? What is half of ten? Given seven and five, what is their total? What is eight times three? What is twelve divided by two? What is seven increased by six? What do you get when you triple six? What do you get when eight is taken away from ten? What is seven less four? (m) What is three plus eleven? (n) (o) (p) (q) (r) (s) (t) (u) (v) What do we get when ten is broken into five equal parts? How much more is nine compared to two? What is a quarter of eight? What is the product of eight and two? What is six combined with eight? What is six taken from thirteen? What is ten increased by six? How many threes fit into fifteen? What do you get when nine is removed from ten? NSSAL 174 Draft
183 Word Sentence to Number Sentence to Answer (B) In each case take the word sentence, make a number sentence out of it, and then provide the answer. A few examples have been provided. Word Sentence Number Sentence Answer e.g. The sum of seven and nine e.g. What is twentytwo decreased by nine e.g. What is a quarter of twenty (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) What do you get when you double eleven? What is thirtyfive divided by seven? What is seven times six? What is seventeen increased by eight? What do you get when nine is taken away from fifteen? What is the sum of twelve and seven? What do we get when ten is broken into two equal parts? What is sixteen plus ten? What is the product of nine and seven? What is half of twentyfour? What is thirtyseven less five? What is six multiplied by eight? (m) What is three taken from fortynine? (n) (o) (p) (q) (r) (s) (t) (u) (v) What is twentysix increased by eleven? What do you get when six is removed from twentythree? How many nines fit into eightyone? What is fortyfive decreased by six? What is eighteen combined with nine? What do you get when you triple twelve? Given seven and seventeen, what is their total? How much more is thirtysix compared to three? What is a third of twentyseven? NSSAL 175 Draft
184 Word Sentence to Number Sentence to Answer (C) In each case take the word sentence, make a number sentence out of it, and then provide the answer. A few examples have been provided. Word Sentence Number Sentence Answer e.g. What is the product of nine hundred and eight e.g. How many eights are in three hundred twenty? e.g. What do we get when sixty is increased by seventy (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Given five hundred and two hundred, what is their total? What is fifty multiplied by three? What is three hundred increased by four hundred twenty? What is forty removed from one hundred? What is a third of nine thousand? How many sevens fit into three hundred fifty? What is seventy times eighty? What is the sum of eleven and eighty? How much more is ninety compared to thirty? What is five thousand plus eight thousand? What is two hundred sixty decreased by twenty? What do you get when you double four thousand? (m) What is six hundred increased by two hundred thirty? (n) (o) (p) (q) (r) (s) (t) (u) (v) What do you get when you triple forty? What is one thousand nine hundred less eight hundred? What is half of sixteen thousand? What is two hundred eighty divided by seven? What do you get when ten is taken away from ninety? What is sixty combined with eighty? What do we get when six is broken into six equal parts? What is seventy taken from ninetysix? What is the product of six and seven thousand? NSSAL 176 Draft
185 What are the Possibilities? (A) The following questions have more than one answer. Find all the possible answers in each case. Example 1 The product of two whole numbers is greater than 8 and less than 13. What are the possibilities for the two numbers? Answer: The word "product" means that we are dealing with the operation of multiplication. We need to find all the pairs of whole numbers (e.g. 0, 1, 2, 3, ) that multiply to give us 9, 10, 11, or 12 (i.e. greater than 8 and less than 13) Therefore the possibilities are: 1 and 9 3 and 3 1 and 10 2 and 5 1 and 11 1 and 12 2 and 6 3 and 4 Example 2 The difference of two single digit numbers is 3. What are the possibilities for the two numbers? Answer: The word "difference" means that we are dealing with the operation of subtraction. Notice that we are told to consider only single digit numbers (i.e. number 1 through 9), rather than multidigit numbers (e.g. 13, 25, 159) Therefore the possibilities are: 3 and 0 4 and 1 5 and 2 6 and 3 7 and 4 8 and 5 9 and 6 Example 3 The sum of two even numbers is 8. What are the possibilities for the two numbers? Answer: The word "sum" means that we are dealing with the operation of addition. We are told to work only with even numbers (e.g. 2, 4, 6, 8, 10, ) whose sum is Therefore the possibilities are: 8 and 0 2 and 6 4 and 4 NSSAL 177 Draft
186 Questions 1. The sum of two single digit numbers is 5. What are the possibilities for the two numbers? 2. The product of two whole numbers is 18 or 20. What are the possibilities for the two numbers? 3. When dividing two single digit numbers, the quotient is 2. What are the possibilities for the two numbers? 4. The difference of two single digit numbers is 5. What are the possibilities for the two numbers? 5. The product of two whole numbers is a whole number that is between, or equal to, 4 and 6. What are the possibilities for the two numbers? 6. The sum of two single digit numbers is greater than 14 and less than 19. What are the possibilities for the two numbers? 7. When dividing a two digit number that is 24 or less by a single digit number, the quotient is 4. What are the possibilities for the two numbers? NSSAL 178 Draft
187 8. The difference of two whole numbers, which are both less than or equal to 6, is 2 or 3. What are the possibilities for the two numbers? 9. The product of an even and odd number is equal a whole number that is 10 or less. What are the possibilities for the two numbers? 10. The sum of two odd numbers is 8 or 10. What are the possibilities for the two numbers? 11. When dividing a two digit number by a single digit odd number, the quotient is 2. What are the possibilities for the two numbers? 12. The difference of two even numbers, which are both less than or equal to 10, is 4. What are the possibilities for the two numbers? NSSAL 179 Draft
188 What are the Possibilities? (B) The following questions have more than one answer. Find all the possible answers in each case. e.g. The product of two single digit odd numbers is greater than 8 and less than 40. What are the possibilities for the two numbers? Answer: Start by listing all single digit odd numbers. 1, 3, 5, 7, 9 The word product tells us that we are multiplying. We need to look at all the possible ways of multiplying two of those odd numbers. These are listed below Now we only want those numbers whose products are greater than 8 and less than 40. That means we are limited to the following So there are seven combinations of numbers that work. 1 and 9 3 and 3 3 and 5 3 and 7 3 and 9 5 and 5 5 and 7 1. The product of two single digit even numbers is 16 or greater. What are the possibilities for the two numbers? 2. The product of two different single digit odd numbers is less than 30. What are the possibilities for the two numbers? NSSAL 180 Draft
189 3. The sum of two odd numbers is 14, and their product is greater than 20. What are the possibilities for the two numbers? 4. Two single digit numbers differ by 3, and their product is less than 45 and greater than 5. What are the possibilities for the two numbers? 5. The product of two whole numbers is 24, and their sum is 14 or less. What are the possibilities for the two numbers? 6. The product of two even numbers is 24 or less, and their sum is greater than 8. What are the possibilities for the two numbers? NSSAL 181 Draft
190 Does It Make Sense? A statement has been supplied. You decide whether the statement makes sense. If not, explain why. 1. Jim purchased a song from itunes for $9. Before the purchase he had $20 in his itunes account; after the purchase, he had $11 in the account. 2. Janice purchased 2 litres of homogenized milk, 4 litres of skim milk, and 1 litre of chocolate milk. In total she purchased 8 litres of milk. 3. Tanya and her two sisters decided to equally share the $210 bill for their parents' anniversary gift. That meant that each girl had to pay $ Kiana ran 6 kilometres per day over 7 days. In that period of time she ran 36 kilometres. 5. Yisha had 3 quarters, 1 dime, and 1 nickel. That means that he was 10 cents short of one dollar. 6. The roommates in an apartment decided to purchase a $600 flat screen television and share in the expense equally. If there were 100 roommates, then that meant that each had to pay $6. 7. Sapphire bought 6 sweaters, each costing $15 before taxes. Her total bill for the sweaters before taxes would be $ There were 26 students in the class at the beginning of the year. Over the course of the year, 4 transferred out and 1 transferred in. That means that by the end of the year, there were 21 students in the class. NSSAL 182 Draft
191 9. Faris only works Monday through Wednesday. Each day he works 8 hours. Over 2 weeks, he works a total of 48 hours. 10. There were initial 20 people signed up for the workshop. Over the next week that number doubled. At the last minute, 6 people said that they would be unable to attend. In the end, 34 people attended the workshop. 11. Jacob bought 3 bags of potatoes and 2 bags of carrots. The each bag of potatoes had a mass of 5 kg, and a bag of carrots had a mass of 1 kg. The total mass of carrots and potatoes that were purchased by Jacob was 30 kg. 12. Each day, Anne drove at 100 km/h for 6 hours. If she maintains this, then she will travel km over 3 days. 13. The brand new 50 inch flat screen television cost $100 but the taxes came to $15, meaning that the total bill was $ The time it takes Arthur to get ready for work is 1 hour and 15 minutes. His shower takes 20 minutes. Breakfast and watching the morning news takes 25 minutes. The remaining 30 minutes is spent doing things like brushing his teeth, making his bed, getting dressed, and using the washroom. 15. There were 12 apples and 18 oranges. If they were shared equally among 3 people, then each person would get 4 oranges and 6 apples. 16. Helen was selling beverages at the fair for $2. Half of the money on each beverage was profit. If she sold 230 beverages, then her profit was $460. NSSAL 183 Draft
192 Insert Your Own Numbers and Words Below you have been given a written statement that is missing numbers (smaller blanks) and words (larger blanks). Your job is to add your own numbers and words so that the written statement makes sense. Reasonable numbers and words must be used. Naturally there are an infinite numbers of acceptable answers. e.g. Kimi purchased, each costing. These items came to a total of. Learner #1's Acceptable Answer: Kimi purchased 7 pairs of socks, each costing $3. These items came to a total of $21. Learner #2's Acceptable Answer: Kimi purchased 2 jars of peanut butter, each costing $4.50. These items came to a total of $9. 1. Tammy and Peter have and. That means that they have a total of. 2. The bag of had a mass of kg. If kg is removed, then that means kg remains. 3. Nita split the evenly amongst her friends. That meant that each friend got. 4. The children each had. That means that altogether they had. 5. The temperature of the was initially degrees Celsius. Over time, the temperature dropped by o C, so that it ended up being o C. 6. At the beginning of the day Jorell had litres of. He used litres and a friend later returned litres she had taken a few weeks ago. Jorell now had litres. NSSAL 184 Draft
193 Put the Numbers in Where It Makes Sense (A) With each question, you have been given a brief statement and you must fit the numbers (supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A number can only be used once. 1. Lei and Jun are siblings. Lei is years younger than Jun. If Jun is years old, then Lei is years old Marcus has dime and nickels in his pocket. If he has nickels and dimes, then he has a total of cents Very few men attended the show. If there were men and women, then there times as many women as men Normally people attend the neighborhood watch meeting. That number increased slightly by such that people attended Anne had dollars but spent most of her money on a dollar top (after taxes). She now has dollars left in her purse Bashir had candies to split evenly between his children. Each child got candies; enough to ruin their supper If there are minutes in an hour, then we know that there are minutes in hours Ryan, who prefers running, ran for minutes and biked for minutes. That means he trained for a total of minutes The mechanic ordered containers of engine oil at a cost of dollars per container. The total cost was dollars room mates got together to purchase an dollar couch. If they all paid the same amount, then each pays dollars NSSAL 185 Draft
194 Put the Numbers in Where It Makes Sense (B) With each question, you have been given a brief statement and you must fit the numbers (supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A number can only be used once. 1. Tanya has dimes and quarters in her purse. That means she has cents of change in her purse Bill and Ajay are friends. Bill is years older than Ajay. If Bill is years old, then Ajay is years old Kim bought apples and 4 oranges. Therefore she bought times as many apples as oranges, or more apples than oranges Three friends equally share the cost of a dollar pizza that was divided into 6 pieces. Each pays dollars and gets pieces Ryan had dollars but spent most of his money on a dollar DVD (after taxes). He now has dollars left If there are hours in one day, then there are hours in days The room temperature was degrees Celsius. If it is turned up slightly by degrees, then the new temperature is degrees There were millilitres of water in a container. If only millilitres is poured out, then the container still has millilitres There are 8 SUVs and cars in the lot. Therefore there are times as many cars as SUVs, or more cars than SUVs The store owner ordered packages of printer paper at a cost of dollars per package. His bill (before taxes) was dollars NSSAL 186 Draft
195 Put the Numbers in Where It Makes Sense (C) With each question, you have been given a brief statement and you must fit the numbers (supplied in the boxes to the right) into the appropriate blanks so that it all makes sense. A number can only be used once and there is one extra number provided in the boxes that should not be used. 1. The cereal, the more expensive item, cost dollars, and the dish soap cost dollars. The total cost was dollars The jar contained candies. If you ate, which is most of the candy, then that would leave in the jar kg of flour must be divided evenly amongst families. Each family was pleased to get kg The DVD cost dollars. The socks cost dollars. The DVD was dollars more expensive than the socks The corner store owner sold bottles of pop. If each sold for dollars, then his total pop sales were dollars Tom drove for more hours than Ed. Tom drove hours, and Ed drove for hours Kim worked hours on Monday and less on Tuesday. If she got hours on Tuesday, then her total was hours Hinto had nickels and quarters. He had a total of cents in nickels and quarters The friends on his Facebook account increased by, going from to There times as many children at the movie compared to adults. There were adults and children NSSAL 187 Draft
196 Put the Numbers in Where It Makes Sense (D) With each question, you have been given a brief story and you must fit the numbers (supplied in the box below) into the appropriate blanks in the story so that it all makes sense. A number can only be used once. 1. A cinema in a movie theatre can hold people. Unfortunately that day, only half of the cinema was full meaning only people are viewing the movie. The theatre charges dollars for child tickets and dollars for adult tickets. The total earnings for that showing in that cinema were dollars Attendance for the annual blues concert is normally people. This year, the number attending grew by, meaning that a total of people attended. If individual tickets sold for, the promoters expected to bring in dollars more than last year just in ticket sales Taylor works at a hardware store where he makes dollars per hour. Typically he works hours per week, just shy of full time hours, and brings in dollars (before deductions). If he works an additional hours a week, he will make dollars more (before deductions) Tanya has teenage children. Montez, the oldest, is years old. Tylena, the youngest, is years old. Kiana, the middle child, is years older than Tylena, making her years old. Tanya, the mother, is years old NSSAL 188 Draft
197 5. A group of seniors wants to charter buses to go on a trip. They check with the local charter company and learn that each bus can take people and that the company charges dollars a day for the bus and driver. Since seniors wish to take the trip, then that means that they will need to charter buses. Unfortunately that means that seats on the buses will be unused. If the seniors are planning on taking a two day trip, the total cost for chartering the buses is dollars (before taxes) A tank initially held litres of water. A pump that removes water from the tank at a rate of litres per minute is switched on ten minutes. That means that litres have been removed, leaving litres in the tank. If someone comes after the pump was switched off and pours litres of water into the tank, the tank will now hold litres of water Two brothers, Brian and Dave, work for the same company. Brian makes dollars more per hour than Dave. Since Brian makes dollars per hour, that means that Dave makes dollars per hour. That means that in a hour work week, Brian will make dollars before deductions, and David will make dollars before deductions There were times as many people at the Rolling Bones concert than at the Tragically Flipped concert. If people were at the Flip concert, then that means that people were at the Bones concert. The Bones charged dollars per ticket, while the Flip only charged dollars per ticket. That means that the Bone brought in dollars more in ticket sales for their concert NSSAL 189 Draft
198 Not Enough Information is Provided In each situation below, you are asked to solve a problem but not quite enough information is provided. Explain what is needed to complete the question. e.g. For a Christmas bonus, the owner of a company is going to rent a bus to take his 110 employees to a concert. How much is it going to cost to rent the buses? Answer: We need to know if the buses are the same size, how many passengers each bus holds, and how much the bus company charges for each bus for this particular round trip. 1. The jar of marbles has a mass of 40 grams. What is the mass of each marble? 2. Manish bought six cans of apple juice. How much did he pay? 3. Marcus has a $600 to pay his friends for helping him shingle his cottage roof. How much should each get? 4. If a large container of water has water being removed from it at a constant rate of 2 litres per minute, then how much water will be left in the container? 5. If at the fire hall fund raiser, volunteers are selling hotdogs for $1, chips for $1.25, and pop for $0.75, then how much money would they make by the end of the day? 6. Jun's children would like to spend a few days at an overnight outdoor adventure camp this summer. The camp charges $40 per day per child. This includes meals. How much will Jun have to pay? NSSAL 190 Draft
199 Word Problems with Too Much Information With each word problem, identify the extra information (i.e. number) that is not needed to solve the problem, and then identify the correct solution from the multiple choice selections. e.g. There are 3 children in Tammy's family and she needs to purchase 2 litres of orange juice, 5 litres of milk, and 4 litres of apple juice each week for them. How many litres in total of fluids does she purchase each week for her children? (a) (b) (c) (d) Answer: Extra Information: There are 3 children in Tammy's family Correct Solution: (d) e.g. There are 18 kg of potatoes and 12 kg of turnip. If these vegetables are to be shared equally by 3 families, how many kilograms of turnip does each family get? (a) (b) (c) (d) Answer: Extra Information: 18 kg of potatoes Correct Solution: (b) A stapler costs $5, a notebook costs $3, and a calendar costs $10. How much more is the calendar compared to the stapler? (a) (b) (c) (d) Extra Information: 2. A bottle of pop costs $2 and a bag of potato chips costs $3. How much does it cost to purchase 8 bottles of pop? (a) (b) (c) (d) Extra Information: 3. Tyrus is taking a flight. His carryon bag weighs 10 kg, and his two checkin bags weigh 16 kg and 24 kg. How much do his checkin bags weigh in total? (a) (b) (c) (d) Extra Information: NSSAL 191 Draft
200 4. Kiana has 4 children, 6 nieces, and 8 nephews. What is the difference between the number of nieces and the number of nephews Kiana has? (a) (b) (c) (d) Extra Information: 5. There are 20 pens and 10 pencils to be shared equally among 5 employees. How many pens does each employee get? (a) (b) (c) (d) Extra Information: 6. There are 7 men and 8 women in a running club. If each woman ran 6 kilometres that day, how far did the women travel in total? (a) (b) (c) (d) Extra Information: 7. After 3 hours, Ryan completed 2 pages of math homework and 6 pages of English homework. How many pages of homework did he complete within that period of time? (a) (b) (c) (d) Extra Information: 8. Dave has $30 to spend at the flea market. No sales tax is charged at the flea market. He is trying to decide if he should buy a $12 DVD or a $23 sweatshirt. If he purchases the DVD, how much change will he have? (a) (b) (c) (d) Extra Information: 9. In 3 hours, Anne could read 90 pages of a novel or run 15 km. How many pages of a novel can Anne read in an hour? (a) (b) (c) (d) Extra Information: 10. When Lei jogs, she travels at 6 kilometres per hour. When she cycles, she travels at 20 kilometres per hour. How far can she jog in 4 hours? (a) (b) (c) (d) Extra Information: NSSAL 192 Draft
201 Create Your Own Math Statement In each case you are provided with a mathematical operation and its corresponding solution. Your mission is to create your own real world statement that corresponds to that operation. Use complete sentences. Try to be creative. Do not use the same type of application more than once (e.g. don't create two math statements involving how much money is left after making a purchase). Naturally there will be a wide range of acceptable answers. e.g = 11 Possible Answer: There were 16 people attending Jeff's party. Five people had to leave at 10:00 p.m., leaving 11 people still at the party. (a) = 16 (b) 249 = 15 (c) 3 6 = 18 (d) 35 5 = 7 NSSAL 193 Draft
202 (e) = 18 (f) 6 7 = 42 (g) 18 6 = 3 (h) = 9 (i) = 10 (j) = 27 NSSAL 194 Draft
203 Word Problems (A) Answer the following questions. Show your work (i.e. Show the operation you used). e.g. Marcy went to the flea market and bought each of her 3 nephews the same toy truck. Each toy cost $8. How much did she spend on toys? Answer: 3 8 = 24 She spent $24 on toys for her nephews. 1. Last week Micheline trained 5 hours for the iron man competition, and this week she trained for 6 hours. How many hours in total did she train over this two week period? 2. Nita purchased a small bottle of overthecounter medication. The bottle initially contained 20 tablets. A few weeks later, there were only 6 tablets left in the bottle. How many tablets had she used? 3. Rhonda ran 5 kilometres each day for 7 days. How far did she run during that week? 4. Four friends have $28 to split evenly between them. How much does each person get? 5. Ajay had $35 cash but then spent $13 on a paperback novel and coffee. How much cash did he have left? 6. A building is 27 metres high. If each story is 3 metres high, how many stories does the building have? 7. Thomas grew tomato plants in his backyard. Two months later, he picked 8 tomatoes from one of the plants and 7 tomatoes from another. How many tomatoes did he pick altogether on that day? 8. A vendor is charging $3 for a hot dog and pop. If 40 customers purchased this combination, how much money did he bring in from the sales of the hot dog and pop combo. 9. Two departments in a company were combined to create one new department. If there were 10 people in the first department, and 9 people in the second department, how many are now in the new department. Assume that no one was fired or laid off. NSSAL 195 Draft
204 10. Three room mates pooled their money to buy a $900 flat screen television. How much did each pay, assuming they all paid the same amount? 11. Two neighbors live along the same lake. Kiana has 15 metres of beachfront, while Lei has 28 metres of beachfront. How much more beachfront does Lei have compared to Kiana? 12. Two people are cycling. One cyclist is travelling at a speed of 12 km/h, and the other at 19 km/h. How much faster is the second cyclist? 13. There are 5 construction sites, and each site requires 9 workers. How many workers are needed in total? 14. A century is 100 years. A decade is 10 years. How many decades are in a century? 15. Anne has $60. Dave has $39 more than Anne. How much money does Dave have? 16. The owner of a bookstore ordered 60 copies of a new hardcover book. If each book costs her $9, how much will she have to pay for all the books? 17. On Thursday, the low temperature of the day was 4 o C, while the high temperature was 17 o C. What is the difference between the high and low temperature for that day? 18. Manish went for a hike with his friend. Both carried a pack; Manish's pack weighed 12 kg, and his friend's weighed 14 kg. Part way through the hike, the friend injured his ankle, so Manish had to carry out both packs. How much did Manish have to carry out on the return trip? 19. A local rock band was organizing its own concert. They were hoping to raise $480. If the tickets to the show cost $6, how many tickets need to be sold? 20. The doctor instructed the patient to take 4 pills per day for 10 days. How many pills did the doctor prescribe to this patient? NSSAL 196 Draft
205 Word Problems (B) Answer the following questions. Show all your work in the space provided. 1. A skydiver jumps from the aircraft at an altitude of 3640 metres. At an altitude of 1780 metres, she deploys the parachute. How far did she fall before deploying the chute? 2. Sapphire had 237 books in her collection. She went to a used bookstore and bought 19 more. How many books does she now have? 3. Six campaign workers have to hand out information booklets to the 498 households in a neighborhood. If each worker must distribute the same number of booklets, how many booklets should each worker distribute? 4. Each container of drywall compound weights 27 kilograms. If a contractor needs to purchase 15 containers, what will the total weight be? 5. Jeff's band knows 24 rock songs, 19 blues songs, and 5 country songs. How many songs do they know in total? NSSAL 197 Draft
206 6. Ryan's takehome pay this month was $1920. If his rent was $650, how much is left over for other expenses? 7. A vehicle travelled 207 kilometres on 9 litres of gasoline. How far could the vehicle travel on one litre of gasoline? 8. The local professional hockey team drew 8454 people to their first game, and 7461 people to their second game. How many people in total attended the first two games? 9. Sixteen people decided to go to an outdoor concert. If each ticket cost $85, how much was spent in total by the sixteen people on tickets? 10. Three friends are driving to a vacation spot in the same car. If they have to travel 1365 kilometres and decide to share the driving equally, how far does each have to drive? NSSAL 198 Draft
207 11. Jacob has 89 DVD movies. If Sasha has 47 more DVD movies, how many does she have? 12. There are 257 employees at the company. If 118 of them are women, how many are men? 13. Originally there was 196 litres of liquid in the barrel. If 68 litres is removed, how much remains in the barrel? 14. A local charity wants to raise $1260 by selling $9 tickets to the dance at the fire hall. How many tickets need to be sold so that the charity reaches its desired goal? 15. A doggie daycare company has 31 sites across the country. If each site can serve 25 dogs on any one day, what is the maximum number of dogs that can be served by the company on one day? NSSAL 199 Draft
208 Same Numbers, Similar Context, Different Math (A) Word problems that use similar numbers in similar contexts have been grouped together. The mathematics to solve these grouped word problems, however, is quite different. Your mission is to solve each of these problems, showing the number sentence (e.g = 21) that was used solve the question. If there are any word problems involving the purchasing of product, do not worry about the taxes associated with those purchases. 1. (a) Five women went out for lunch and each spent $20. How much was spent in total? (b) Five women wished to purchase a $20 cheese plate for their afternoon party. If they shared in the cost of the cheese plate equally, how much did each woman have to pay? (c) Twenty women were supposed to attend the lunch but five cancelled out at the last moment. How many women attended the lunch? (d) Restaurant reservations were made for twenty women but five women turned up unexpectedly. How many women attended the lunch? 2. (a) Marcus has two dogs; one weighs 40 kg and the other weighs 8 kg. How many times heavier is the large dog compared to the small dog? (b) Jacob's overweight dog originally weighed 40 kg, but after being put on a new diet and exercise program, the dog's weight dropped by 8 kg. How much does the dog weigh? (c) Hanna runs a large kennel and presently owns 40 dogs. If each dog eats 8 kg of dry dog food per month, how many kilograms of dog food does she need each month? (d) At one year, Manish's dog weighed 40 kg. Over the next year the dog gained another 8 kg. How much does the dog weigh? NSSAL 200 Draft
209 3. (a) Before starting his exercise program, Jim estimated that he only did physical activity for 5 hours per month. After starting his exercise program, he estimates that he is doing 30 hours per month. How many additional hours of exercise are being received under this new program compared to when Jim had no program? (b) Rana was exercising for approximately 30 hours per month, and then decided to increase that by an additional 5 hours per month. How many hours per month of exercise is she now receiving? (c) Each month Grace exercises for 30 hours. Each month Tanya only exercises for 5 hours. How many times larger is Grace's exercise program compared to Tanya's? (d) Each month Bashir exercises for 30 hours. How many hours of exercise would he get over 5 months? 4. (a) Caledonia Elementary School was taking 50 children on a field trip. School board policy requires 10 chaperones. How many people in total should be attending the trip? (b) A college was organizing a ski day for its learners. They decided that they would need 10 buses to transport learners. If each bus can take 50 learners, how many learners in total can the college take to the ski hill? (c) Parker Middle School was going to take 50 students on a field trip, but 10 were unable to attend due to illness. How many students were able to make the trip? (d) Sampson High School was initially planning on using a bus to transport 50 students to the science fair. A bus was unavailable so they decided to rent vans. If each van can transport 10 students, how many vans would be needed? NSSAL 201 Draft
210 Similar Numbers, Similar Context, Different Math (B) Word problems that use similar numbers in similar contexts have been grouped together. The mathematics to solve these grouped word problems, however, is quite different. Your mission is to solve each of these problems, showing the number sentence (e.g = 21) that was used solve the question. If there are any word problems involving the purchasing of product, do not worry about the taxes associated with those purchases. 1. (a) Janice bought a $40 item at the store. If she received $10 change, how much money did she initially pass to the cashier? (b) Janice was hosting a celebration for her family and needed to purchase several $40 items. If she purchased 10 of these items, how much did she pay? (c) Janice wanted to purchase a $40 item but only had $10 on her. Her sister lent her the rest of the money to buy the item. How much money did her sister lend her? (d) Janice received a $40 gift from her 10 friends. If the friends equally shared in the cost of the gift, how much did each person pay? (e) If Janice bought two $40 items and three $10 items, what was the total bill? (f) In January Janice still owed $40 on the purchase of an item. Since that time she has made two monthly payments, each of $10. How much does she still owe? (g) Janice wanted to buy three $40 items but only had nine $10 bills in her purse. How much more money does she need to make the purchase? 2. (a) If Samir has to take 2 pills per day for 20 days, then how many pills did he ultimately have to take? (b) Samir's pill bottle contained 20 pills. If he was instructed to take 2 pills per day, how long would his supply of pills last? (c) Samir's pill bottle contained 20 pills. If he took 2 pills on the first day, how many pills are left for the remaining days? NSSAL 202 Draft
211 (d) Samir's new pill bottle contains 20 pills and his old pill bottle only contains 2 pills. How many pills does he have in total? (e) Samir has initially had 20 pills. If he takes 2 pills per day for three days, how many pills will be left? (f) Samir had three bottles, each containing 20 pills. He also had four sample packages, each containing 2 pills. How many pills does he have in total? (g) Samir has 20 pills and must take 2 pills twice a day. How long will his supply of pills last? 3. (a) If a business has 5 employees in one department and 30 employees in another department, then how many people do they have in the two departments? (b) If a business has 5 departments, each with 30 employees, then how many employees do they have in total? (c) A business had 30 employees but unfortunately had to lay off 5 people. How many employees do they now have? (d) A business has 30 employees that are shared equally amongst the 5 departments. How many employees does each department have? (e) A business runs three shifts each day. Each shift is made up of 5 managers and 30 assembly line workers. How many employees does this business need each day? (f) The 5 managers each make twenty dollars per hour. The 30 assembly line workers each make ten dollars per hour. Assuming that all employees are at work during the day, how much does the business pay out per hour for employee wages? (g) A business originally had 30 employees but proceeded to hire four teams, each made up of 5 employees. How many employees does the company now have? NSSAL 203 Draft
212 More than One Question Example Frank collected famous autographs. He had 5 autographs from baseball players, 8 from football players, and 10 from hockey players. (a) How many more autographs does he have football players compared to baseball players? (b) How many autographs does he have in total? (c) If Frank had not collected football player autographs, how many autographs would he have in total? (d) Suppose Frank also wanted to collect autographs of basketball players and set a goal of having 3 times as many of autographs of basketball players as compared to hockey players. If he reached this goal, how many basketball player autographs would he have? Answers: (a) autographs (b) autographs (c) autographs (d) autographs Questions 1. The large container initial held 12 litres of water. Meera first removed 5 litres of water, then removed 3 litres of water. (a) How much water did Meera remove in total? (b) How much more water did Meera first remove compared to the amount she removed the second time? (c) In the end, how much water remained in the large container? (d) If Meera had not removed water for the second time, how much would have remained in the large container. NSSAL 204 Draft
213 2. Harris retired and purchased a hobby farm. On this farm he several types of animals which included 2 cows and 14 chickens. He also had sheep and pigs. The number of sheep was 3 times the number of cows. He had half as many pigs as chickens. (a) How many sheep did Harris have? (b) How many more chickens did he have compared to cows? (c) How many pigs did Harris have? (d) How many animals did he have in total? (e) If he sold all his pigs, how many animals would he have? 3. Candice restores antique automobiles and motorcycles. She presently has 3 automobiles, and 4 times as many motorcycles. (a) How many motorcycles does she have? (b) How many antique vehicles does she have in total? (c) Assuming that each car has a spare tire, how many tires does she have for her antique automobiles? (d) How many more antique motorcycles does she have compared to antique automobiles? (e) If she purchased another automobile, but the number of motorcycles remained the same, how many times more motorcycles would she have compared to automobiles? NSSAL 205 Draft
214 Food Chart (A) The following chart shows the amounts of protein, fat and carbohydrates in different servings of foods. Food and Serving Protein (grams) Fat (grams) Carbohydrates (grams) Watermelon (1 slice) Apple Pie (1 slice) White Bread (1 slice) Tomato Juice (1 cup) Peanut Butter (1 tablespoon) Canned Tuna in Oil (3 ounces) Beef and Vegetable Stew (1 cup) Poached Egg (1 egg) Blueberries (1 cup) Corn Chips (1 ounce) Answer each of the following using the information supplied in the above chart. Include the number sentence (e.g = 4) that you used to find your answer. 1. If you were to have one slice of watermelon and one slice of white bread, then how many grams of protein would you have ingested (i.e. eaten)? 2. What's the difference in the number of grams of fat between one tablespoon of peanut butter and one poached egg? 3. If you ate two cups of blueberries, then how many grams of carbohydrates would you have ingested? 4. How many times larger is the number of grams of protein in one cup of beef and vegetable stew compared to one cup of tomato juice? 5. If you ate a poached egg, a slice of white bread, and one cup of tomato juice, then how many grams of carbohydrates would you have ingested? 6. How many times larger is the number of grams of fat in two slices of apple pie compared to one ounce of corn chips? 7. What is the difference in the number of grams of protein between one 3 ounce can of tuna (in oil) and five slices of watermelon? NSSAL 206 Draft
215 Food Chart (B) The following chart shows the amounts of protein, fat and carbohydrates in different servings of foods. Food and Serving Food Energy (kilocalories) Sodium (milligrams) Carbohydrates (grams) Watermelon (1 slice) Apple Pie (1 slice) White Bread (1 slice) Tomato Juice (1 cup) Peanut Butter (1 tablespoon) Canned Tuna in Oil (3 ounces) Beef and Vegetable Stew (1 cup) Poached Egg (1 egg) Blueberries (1 cup) Corn Chips (1 ounce) Answer each of the following using the information supplied in the above chart. Show your work. 1. If you drank a cup of tomato juice and ate one poached egg, how many kilocalories of food energy would you have ingested? 2. How many times larger is the number of grams of carbohydrates in two cups of beef and vegetable stew compared to one cup of tomato juice? 3. What is the difference in the number of milligrams of sodium in one slice of apple pie and one poached egg? 4. How many milligrams of sodium are there in four slices of white bread? NSSAL 207 Draft
216 5. If you ate one slice of watermelon, one tablespoon of peanut butter, and one ounce of corn chips, how many kilocalories would be ingested? 6. What is the difference in the number of milligrams of sodium in three ounces of corn chips and two tablespoons of peanut butter? 7. How many grams of carbohydrates would be ingested if you ate two slices of pie, three slices of watermelon, and one cup of blueberries? 8. How many times larger is the amount of sodium in two poached eggs compared to four slices of watermelon? 9. What is the difference in the number of kilocalories of a meal comprised of one cup of beef and vegetable stew and one cup of tomato juice, and a meal comprised of a three ounce can of tuna and a slice of white bread. 10. How many times larger is the amount of carbohydrates in a meal comprised of one slice of apple pie and two cups of blueberries, compared to a meal comprised of one cup of beef and vegetable stew and one slice of watermelon? NSSAL 208 Draft
217 Keeping Track of New Stock (A) The store just started selling five new products. A large order of each item was made at the beginning of week one and all of these items were placed on their shelves. The spreadsheet below shows how many units of each product are present on the shelves at the beginning of each week. Assume that the shelves were not restocked with the new items. Delicate Deluxe Dog Extra Soft Greek Olives Ceramic Pots Chocolates Food Toilet Tissue Week Week Week Use the information in the spreadsheet to answer the following questions. Show the number sentence (e.g = 11) that you used to solve the question. 1. How many times larger was the number of units of deluxe dog food compared to the number of units of ceramic pots at the beginning of week one? 2. How many more units of delicate chocolates were there on the shelves at the beginning of week one compared to units of extra soft toilet tissue? 3. In total, how many units of Greek olives and ceramic pots were on the shelves at the beginning of week two? 4. If they had tripled their order of extra soft toilet paper at the beginning of week one, how many units would they have ordered? 5. How many units of delicate chocolates were sold between the beginning of week one and the beginning of week three? 6. How many times larger is the number of units of extra soft toilet paper on week one compared to week three? 7. In total, how many units of delicate chocolates, extra soft toilet paper and Greek olives were on the shelves at the beginning of week three? 8. If five customers wanted to buy all the units of Greek olives that were present at the beginning of week three and each would purchase the same amount, how many units would each customer get? NSSAL 209 Draft
218 Keeping Track of New Stock (B) The store just started selling five new products. A large order of each item was made at the beginning of week one and all of these items were placed on their shelves. The spreadsheet below shows how many units of each product are present on the shelves at the beginning of each week. Assume that the shelves were not restocked with the new items. Delicate Deluxe Dog Extra Soft Greek Olives Ceramic Pots Chocolates Food Toilet Tissue Week Week Week Use the information in the spreadsheet to answer the following questions. Show the number sentence (e.g. (4533) + (3025) = 17) that you used to solve the question. 1. If the store doubled the number of units of deluxe dog food and ceramic pots they ordered on week one, how many units of these two items would they have in total at that time? 2. How many units of deluxe dog food and Greek olives were sold in total between the beginning of week two and the beginning of week three? 3. How many more units of deluxe dog food were sold between weeks one and two compared to units of Greek olives sold in the same time period? 4. How many more units of delicate chocolates were sold between weeks two and three compared to units of extra soft toilet tissue sold in the same time period? 5. How many more combined units of deluxe dog food and extra soft toilet tissues are on the shelves at the beginning of week two compared to the number of units of delicate chocolates at that same time? 6. If the units sold of delicate chocolates tripled between the beginning of week one and the beginning of week two, then how many units would have been sold at that time? 7. How many times larger was the number of units sold of ceramic pots between the beginning of week one and the beginning of week three compared to units sold of Greek olives over the same period? 8. How many times larger was the number of units sold of deluxe dog food between the beginning of week two and the beginning of week three compared to units sold of the same product between the beginning of week one and the beginning of week two? NSSAL 210 Draft
219 Consumer Math NSSAL 211 Draft
220 How Much Do They Have? (A) Determine the amount of money each person has. Kate s Money: Paula s Money: Meera s Money: Ryan s Money: Yoshi s Money: Your Answers: Kate s Money: Paula s Money: Meera s Money: Ryan s Money: Yoshi s Money: NSSAL 212 Draft
221 How Much Do They Have? (B) Determine the amount of money each person has. Dave s Money: Shelly s Money: Maurita s Money: Jun s Money: Lei s Money: Your Answers: Dave s Money: Shelly s Money: Maurita s Money: Jun s Money: Lei s Money: NSSAL 213 Draft
222 How Much Do They Have? (C) Determine the amount of money each person has. Andrew s Cash: Mary s Cash: Kara s Cash: Montez s Cash: Shima s Cash: Your Answers: Andrew s Cash: Mary s Cash: Kara s Cash: Montez s Cash: Shima s Cash: NSSAL 214 Draft
223 How Much Do They Have? (D) Determine the amount of money each person has. Hamid s Cash: Tanya s Cash: Samir s Cash: Mark s Cash: Hatsu s Cash: Your Answers: Hamid s Cash: Tanya s Cash: Samir s Cash: Mark s Cash: Hatsu s Cash: NSSAL 215 Draft
224 How Much Do They Have? (E) Determine the amount of money each person has. We apologize that the bills are not the appropriate size compared to the coins. Sasha s Money: Hinto s Money: Lisa s Money: Tiva s Money: Meera s Money: Your Answers: Sasha s Money: Hinto s Money: Lisa s Money: Tiva s Money: Meera s Money: NSSAL 216 Draft
225 Emptying the Junk Drawer You were emptying your junk drawer and scattered the money across the top of your table (See below.). You have few bills and lots of change. How much money do you have from that drawer? Use the chart on the next to organize the information and complete your calculations. NSSAL 217 Draft
226 Bills or Change Number of Each Value in Dollars $20 bills $10 Bills $5 Bills Toonies ($2 coins) Loonies ($1 coins) Quarters (25 coins) Dimes (10 coins) Nickels (5 coins) Total: If you found three more nickels, five more quarters, and four more loonies, how much would you have in total? NSSAL 218 Draft
227 Connect Four Money Game Number of Players: Two Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically or diagonally. Instructions: 1. Roll a die to see which player will go first. 2. The first player looks at the board and decides which square he/she wishes to capture. They place a paper clip on each of the strips below (i.e. Nickel and Dime Strip, and Quarter and Loonie Strip). Once they have chosen the coinage, they can capture one square with the appropriate total (e.g. 3 nickels plus 1 loonie is equal to $1.15). They either mark the square with an X or place a colored counter on the square. There may be other squares with that same total but only one square can be captured at a time. 3. Now the second player is ready to capture a square but he/she can only move one of the paperclips. They then mark the square with that total using an O or a different colored marker. If a player cannot move a single paperclip to capture a square, a paperclip must still be moved in order to ensure that the game can continue. 4. Play alternates until one player connects four squares. Remember that only one player clip is moved at a time. If none of the players is able to connect four, then the winner is the individual who has captured the most squares. Game Board: $0.35 $1.05 $1.10 $0.45 $0.35 $0.95 $0.45 $0.85 $0.60 $1.15 $0.65 $1.10 $1.10 $0.95 $1.20 $0.30 $0.90 $0.85 $0.65 $0.55 $0.70 $1.10 $1.20 $0.35 $0.30 $0.90 $0.35 $0.60 $0.80 $0.40 $0.60 $1.05 $0.70 $0.85 $1.15 $0.55 Nickel and Dime Strip: Quarter and Loonie Strip: 1 nickel 2 nickels 3 nickels 1 dime 2 dimes 1 quarter 2 quarters 3 quarters 1 loonie NSSAL 219 Draft
228 Find the Price Using the internet, flyers, and/or catalogues, find the price of each of the items. Assume that we are purchasing new items. If the price is a sale price, check off the appropriate box. Also identify the name of the business in which you can purchase the item at that price. Answers are going to vary because learners are likely to be using different websites, flyers, and catalogues. Milk (2 litres) Tube of Toothpaste Hamburger Meat (1 kg) Price: Sale Price: Sale Price: Sale Business: Business: Business: Cooked Chicken Head of Lettuce 20 inch Delivery Pizza Price: Sale Price: Sale Price: Sale Business: Business: Business: Woman's Winter Boots Pair of Men's Jeans Leather Gloves Price: Sale Price: Sale Price: Sale Business: Business: Business: Mountain Bike (Adult) Table Lamp Toaster Oven Price: Sale Price: Sale Price: Sale Business: Business: Business: Queen Size Mattress Gas Barbeque Portable Music Player Price: Sale Price: Sale Price: Sale Business: Business: Business: 40 inch Flatscreen TV Leather Sofa Compact Car (e.g. Corolla) Price: Sale Price: Sale Price: Sale Business: Business: Business: NSSAL 220 Draft
229 Name a Product near that Price In this activity you are to name a product near the indicated price. Naturally every learner will have different answers. You can use flyers or catalogues to assist with this activity. e.g. Name a product that worth approximately $2. Some Possible Answers: Can of Soup, Pen, Comb, Small Package of Screws Price Your Product Price Your Product 1. $1 2. $5 Price Your Product Price Your Product 3. $10 4. $20 Price Your Product Price Your Product 5. $40 6. $70 Price Your Product Price Your Product 7. $ $150 Price Your Product Price Your Product 9. $ $400 Price Your Product Price Your Product 11. $ $1000 Price Your Product Price Your Product 13. $ $2000 Price Your Product Price Your Product 15. $ $8000 NSSAL 221 Draft
230 Least to Most Expensive In each case you are supplied with four common items that one might purchase. Your mission is to number the items 1 through 4 in the space below, where 1 is the least expensive, and 4 is the most expensive. 1. Two Slice Toaster Litre of Milk Refrigerator Electric Fry Pan 2. Silver Earrings Leather Jacket Pair of Socks Pencil 3. Power Drill Winter Gloves Fingernail Clippers Mattress (Queen) 4. Hardcover Book Birthday Card Prescription Glasses Pair of Winter Boots 5. Coffee Table Kitchen Stove Can Opener Electric Kettle 6. Four Winter Tires 20 inch Television DVD Player Four Litres of Oil 7. Small Flashlight Delivery Pizza Reclining Chair Table Lamp 8. Hair Dryer Coffee Mug Chocolate Bar Toaster Oven 9. Paperback Novel Big Concert Ticket Spring Jacket Bus Toll 10. Love Seat Dish Soap 10 kg Turkey Four Litres of Milk NSSAL 222 Draft
231 What Are the Three Items Worth? In each case, you are given three items and four prices. Match the best price to the item. Item: Price Item: Price 1. Hammer 2. Box of Cereal Florescent Light Bulb Can of Pop A Pair of Jeans Tshirt Price Choices: $1, $3, $10, $50 Price Choices: $2, $6, $12, $50 Item: Price Item: Price 3. Sofa 4. Fastfood Hamburger A Pair of Sunglasses A Pair of Work Boots Microwave Oven Knapsack for School Price Choices: $2, $20, $150, $600 Price Choices: $3, $40, $90, $400 Item: Price Item: Price 5. A Man's Haircut 6. Tube of Toothpaste Loaf of Bread An Adult Bicycle Small Waste Paper Basket Winter Coat Price Choices: $2, $25, $80, $250 Price Choices: $3, $15, $35, $120 NSSAL 223 Draft
232 What Is It Worth? (A) You have been given a list of items and been asked to select a price that best matches that item. The prices have been supplied below. Only use each price once. You will also be asked to write out the value using words (e.g. $1249 is one thousand two hundred fortynine dollars). (a) Item Rideon Lawn Mower Approximate Price (Number) Approximate Price (Words) (b) DVD/Bluray Player (c) Toaster (d) HD Video Camera (e) Compact Car (f) 42 inch Flat Screen TV (g) Tank of Gas (Compact Car) Prices to Choose From: $17 $ $50 $2149 $549 $329 $ $99 NSSAL 224 Draft
233 What Is It Worth? (B) You have been given a list of items and been asked to select a price that best matches that item. The prices have been supplied below. Only use each price once. You will also be asked to write out the value using words (e.g. $1249 is one thousand two hundred fortynine dollars). (a) Item Washing Machine Approximate Price (Number) Approximate Price (Words) (b) Hair Dryer (c) Minivan (d) Fast Food Meal for One (e) Gasoline Push Mower (f) Child's Bicycle (g) Four Person Hot Tub Prices to Choose From: $239 $8 $3599 $21 $ $469 $ $69 NSSAL 225 Draft
234 Purchasing Groceries On this page and the next, you have been provided with a list of common items that can be found at a grocery store. Your job is to indicate which products you would purchase for each of the four scenarios. Scenario #1: You are single and have a limited budget of $150. Scenario #2: You are single and have a limited budget of $230. Scenario #3: You are a parent with two children (ages 7 and 10), and have a limited budget of $250. Scenario #4: You are a parent with two children (ages 7 and 10), and have a limited budget of $350. Be prepared to discuss your selections with your instructor and/or classmates. Please note that items whose weight is marked with an asterisk (*), do not have to purchase at that weight. For example, one pound (lb.) of tomatoes can be purchased for $2, but half a pound of tomatoes can be purchased for $1. You decide how much you need. In these cases just indicate the amount to be purchased using the appropriate dollar value. Item Price Scenario #1 Single, $150 budget Potatoes (10 lb. bag) $4 Potato Salad (454 g) $3 Green Peppers (four pack) $6 Tomatoes (1 lb.*) $2 Baby Carrots (340 g) $2 Cucumber $2 Cauliflower (1 lb.*) $4 Apples (5 lbs.) $5 Oranges (4 lbs.) $4 Pears (1 lb.*) $2 Peaches (1 lb.*) $3 Seedless Red Grapes (1 lb.*) $4 Bananas (1 lb.*) $2 Strawberries (1 lb.) $3 Kiwi (4) $2 Whole Pineapple $5 Loaf of Bread (675 g) $2 Crackers (450 g) $2 Cereal (445 g) $4 Milk (2 l) $4 Cheese (500 g block) $6 Cheese Slices (500 g) $6 Parmesan Cheese (250 g) $6 Margarine (454 g) $3 Scenario #2 Single, $220 budget Scenario #3 Family of 3, $250 budget Scenario #4 Family of 3, $350 budget NSSAL 226 Draft
235 Item Price Scenario #1 Single, $150 budget Dozen Small Eggs $2 Stew Beef (1 lb.) $6 Ground Beef (1 lb.*) $6 Port Tenderloin (1 lb.*) $5 Bacon (500 g) $6 Roasting Chicken (5 lb.) $14 Trout Fillets (1 lb.*) $8 Haddock Fillets (1 lb.*) $6 Salmon Portion (140 g) $5 Canned Tuna (170 g) $2 Canned Cooked Ham (454 g) $7 Chicken Noodle Soup (540 ml) $2 Canned Chili (425 g) $3 Frozen Pizza (400 g) $6 Frozen Lasagna (1 kg) $9 Frozen Meatballs (680 g) $10 Frozen French Fries (454 g) $4 Frozen Corn (750 g) $3 Frozen Mixed Vegetables (1 kg) $3 Frozen Lemonade (295 ml) $1 Peanut Butter (500 g) $4 Roasted Cashews (100 g) $2 Macaroni and Cheese (200 g) $1 Ketchup (1 L) $3 Sunflower Oil (1 L) $6 Tea (240 bags) $11 Coffee (300 g) $8 Cranberry Cocktail (1.89 L) $3 Pop (2 L) $2 Potato Chips (235 g) $4 Chocolate Bars (four pack) $4 Ice Cream (1.5 L) $4 Deodorant (150 g) $4 Toothpaste (130 ml) $3 Bathroom Tissue (8 rolls) $5 Paper Towels (2 rolls) $3 Dishwashing Liquid (950 ml) $2 Bathroom Cleaner (500 ml) $2 Total: Scenario #2 Single, $220 budget Scenario #3 Family of 3, $250 budget Scenario #4 Family of 3, $350 budget NSSAL 227 Draft
236 Measurement NSSAL 228 Draft
237 Which Measurement Is Reasonable? In each case, you have been given a situation or object in which you must consider which supplied measurement is most appropriate. 1. The length of an adult's arm (a) 65 centimetres (c) 45 kilometres 2. The length of three city blocks (a) 9 metres (c) 3 centimetres (b) 30 metres (d) 110 centimetres (b) 20 kilometres (d) 1 kilometre 3. The amount of lemonade that could be held in a pitcher (a) 30 millilitres (b) 8 litres (c) 2 litres (d) 50 millilitres 4. The weight of a ten year old boy (a) 75 grams (c) 100 kilograms 5. The amount of water in a drinking glass (a) 3 litres (c) 350 millilitres (b) 34 kilograms (d) 450 grams (b) 300 litres (d) 40 millilitres 6. The height of a threestory apartment building (a) 12 metres (b) 100 cm (c) 1 kilometre (d) 30 metres 7. The weight of a compact car (a) 900 grams (c) 300 kilograms (b) 1800 kilograms (d) 7000 grams 8. The length of a man's fingernail on his pinkie finger (a) 4 centimetres (b) 7 metres (c) 2 metres (d) 1 centimetre 9. The distance from Halifax, Nova Scotia to Truro, Nova Scotia (a) 15 kilometres (b) 4000 metres (c) 125 centimetres (d) 100 kilometres 10. The amount of gasoline that can be held in a compact car's gas tank (a) 50 millilitres (b) 8 litres (c) 40 litres (d) 130 millilitres NSSAL 229 Draft
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