Teacher s Resource Chapter 3: Number Relationships SAMPLE CHAPTER

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1 Teacher s Resource Chapter 3: Number Relationships SAMPLE CHAPTER

2 Nelson The Teachers Choice Easy to Read. Easy to Use. Student Book Student ebooks also available Teacher s Resource

3 for Student Success! Student Success Adapted Program Lesson-by-lesson support for struggling students! Unique to Nelson Math Focus! Student Workbook Solutions Manual Combined Grades Resource For more information, please contact your Nelson sales representative. (See back cover)

4 Chapter 3 Number Relationships Contents OVERVIEW Introduction Curriculum across Grades 5 to 7: Number Math Background Planning for Instruction Problem Solving Reading Strategies Connections to Literature Connections to Other Math Strands Connections to Other Curricula Connections to Home and Community Chapter 3 Planning Chart Chapter 3 Assessment Summary TEACHING NOTES Chapter Opener Getting Started: Banner Design Lesson 1: Identifying Factors Lesson 2: Identifying Multiples Curious Math: String Art Lesson 3: Prime and Composite Numbers Math Game: Colouring Factors Lesson 4: Identifying Factors by Dividing Lesson 5: Creating Composite Numbers Mid-Chapter Review Lesson 6: Solving Problems Using an Organized List Lesson 7: Representing Integers Curious Math: Countdown Clock Lesson 8: Comparing and Ordering Integers Lesson 9: Order of Operations Math Game: Four in a Row Chapter Review Chapter Task: A Block Dropping Game Chapters 1 3 Cumulative Review CHAPTER 3 BLACKLINE MASTERS Family Letter Scaffolding for Getting Started Scaffolding for Lesson 2, Question String Art Mid-Chapter Review Frequently Asked Questions Four in a Row Game Board Calculation Cards Chapter Review Frequently Asked Questions Chapter 3 Test Chapter 3 Task: A Block Dropping Game Answers for Chapter 3 Masters From Masters Booklet Review of Essential Skills: Chapter cm Grid Paper cm Grid Paper Chart Number Lines Initial Assessment Summary Assessment Rubrics for Mathematical Processes Chapter Checklist: Chapter Self-Assessment: Chapter 3 Lesson Goals Self-Assessment: Mathematical Processes Self-Assessment: What I Like Self-Assessment: How I Learn Introduction This chapter provides students with opportunities to use their understanding of number relationships to identify factors and multiples, to determine whether a number is prime or composite, to compare and order integers, and to use the rules for order of operations to calculate the value of an expression. They will build upon the mental mathematics strategies developed in Grade 5 to determine factors and multiples. Throughout the chapter, students use concrete and pictorial models to help develop an understanding of new concepts before attempting to use mental mathematics strategies. Answers and Solutions Answers to all numbered questions are provided in the Student Book. Full solutions are provided in the Solutions Manual. Selected answers are provided in the Teacher s Resource lesson notes. Contents 1

5 Curriculum across Grades 5 to 7: Number The Grade 6 outcomes and achievement indicators listed below are addressed in this chapter. When the outcome or indicator is the focus of a lesson or feature, the lesson number or feature is indicated in brackets. Strand: Number General Outcome: Develop number sense. Grade 5 Grade 6 Grade 7 Specific Outcome N3. Apply mental mathematics strategies and number properties, such as skip counting from a known fact using doubling or halving using patterns in the 9s facts using repeated doubling or halving to determine answers for basic multiplication facts to 81 and related division facts. [C, CN, ME, R, V] Specific Outcomes N3. Demonstrate an understanding of factors and multiples by determining multiples and factors of numbers less than 100 identifying prime and composite numbers solving problems involving multiples. (1, 2, CM1, 3, MG1, 4, 5, 6) [PS, R, V] Achievement Indicators Identify multiples for a given number and explain the strategy used to identify them. (2, CM1, 6) Determine all the whole-number factors of a given number using arrays. (1, MG1) Identify the factors for a given number and explain the strategy used, e.g., concrete or visual representations, repeated division by prime numbers, or factor trees. (1, MG1, 4, 5, 6) Provide an example of a prime number and explain why it is a prime number. (3, 4, 5) Provide an example of a composite number and explain why it is a composite number. (3, 4, 5) Sort a given set of numbers as prime and composite. (3) Solve a given problem involving factors or multiples. (1, 2, CM1, 3, 6) Explain why 0 and 1 are neither prime nor composite. (3) N7. Demonstrate an understanding of integers, concretely, pictorially, and symbolically. (7, CM2, 8) [C, CN, R, V] Achievement Indicators Extend a given number line by adding numbers less than zero and explain the pattern on each side of zero. (7, CM2) Place given integers on a number line and explain how integers are ordered. (8) Describe contexts in which integers are used, e.g., on a thermometer. (7, CM2) Compare two integers; represent their relationship using the symbols <, >, and =, and verify using a number line. (8) Order given integers in ascending or descending order. (8) N9. Explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers). (9 MG2) [CN, ME, PS, T] Achievement Indicators Demonstrate and explain with examples why there is a need to have a standardized order of operations. (9) Apply the order of operations to solve multi-step problems with or without technology, e.g., computer, calculator. (9, MG2) Specific Outcomes N1. Determine and explain why a number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, and why a number cannot be divided by 0. [C, R] N6. Demonstrate an understanding of addition and subtraction of integers, concretely, pictorially, and symbolically. [C, CN, PS, R, V] Mathematical Processes: C Communication, CN Connections, ME Mental Mathematics and Estimation, PS Problem Solving, R Reasoning, T Technology, V Visualization Features: CM1 (Curious Math: String Art), MG1 (Math Game: Colouring Factors), CM2 (Curious Math: Countdown Clock), MG2 (Math Game: Four in a Row) 2 Chapter 3: Number Relationships

6 Math Background An understanding of number relationships is essential to functioning in daily life. Students gain this understanding by exploring factors, multiples, and integers directly. Students also gain an intuitive understanding about numbers by relating numbers to a variety of real-world contexts. For example, students use reasoning to solve number problems in the real world. In addition, visualizing number patterns and relationships allows students to make connections and identify number relationships, further developing number sense. Throughout the chapter, students are encouraged to use mental math to determine factors and multiples and to solve complex expressions using the order of operations. It is important for students to demonstrate computational math skills as well as flexibility with numbers. Students are encouraged to use reasoning to check their answers, to analyze and evaluate their thinking, and to listen and learn from the strategies of others. See PRIME (Professional Resources and Instruction for Mathematics Educators): Number and Operations by Marian Small (Thomson Nelson, 2005) for additional math background and teaching strategies. Planning for Instruction Problem Solving In Lesson 6, students solve problems by using an organized list. Students will also solve a variety of problems throughout the chapter as they apply their understanding of factors, multiples, and integers. Assign a Problem of the Week from the selection below or from your own collection. 1. A number has nine different factors. Two of its multiples are 72 and 108. What is the number? (36: factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36; ; ) 2. The temperature on Monday was 11 C. The temperature on Tuesday was 15 C. The temperature on Wednesday was 13 C. On Thursday, it was colder than Monday but warmer than Wednesday. What was the temperature on Thursday? ( 12 C: ) 3. Maddy copied down a number sentence in math class, but she forgot to write the brackets. Where should Maddy place the brackets to make the number sentence true? (Maddy should place the brackets around the addition and subtraction. 3 (2 6) (12 8) ) Reading Strategies The reading strategies highlighted in this chapter are Monitoring Comprehension (Mid-Chapter Review) and Finding Important Information (Lesson 6). To reinforce the use of these strategies, you may apply them to other questions throughout the lessons as opportunities present themselves. Connections to Literature Expand your classroom library or math centre with books related to the math in this chapter. For example: Frasier, Debra. On the Day You Were Born. Harcourt Children s Books, Merrill, Jean. The Toothpaste Millionaire. Houghton Mifflin, Murphy, Stuart. Less Than Zero. HarperTrophy, Connections to Other Math Strands Patterns and Algebra: In the Getting Started activity, students will identify the pattern in a banner design. In Lesson 2, students will use number patterns as a way to identify multiples. Shape and Space: In the Chapter Task, students will describe how squares and rectangles with different dimensions can be used to fill a large square. Measurement: In Lesson 8, students will use their knowledge of integers to compare and order temperatures. Connections to Other Curricula Art: In Curious Math: String Art, students will use a modified version of string art to represent multiples of numbers. Science: In Lesson 2, students will use multiples to determine the years in which the comet Kojima will likely be visible from Earth. In Lesson 8, students will compare and order positive and negative temperatures. In Lesson 9, students will use formulas to calculate heart rate and lung capacity. Connections to Home and Community Have students use everyday situations to order and compare numbers, identify factors and multiples, and use the order of operations. Send home Family Letter p. 74, which contains suggestions for a variety of activities related to the math in this chapter that students can do at home. Have students complete the Nelson Math Focus 6 Workbook pages for this chapter at home. Use the suggestions for at-home activities in Follow-Up and Preparation for Next Class in various lessons. Overview 3

7 Chapter 3 Planning Chart Key Concepts* Number and Operations Numbers tell how many or how much. Classifying numbers provides information about the characteristics of the numbers. There are different, but equivalent, representations for a number. Benchmark numbers are useful for relating and estimating numbers. *PRIME (Professional Resources and Instruction for Math Educators): Number and Operations by Marian Small (Thomson Nelson, 2005) Key Principles A number can be described as the product of its factors. Describing a number as a multiple suggests thinking of it in terms of a unit other than 1; for example, since 6 is a multiple of 3, it is two 3s. Knowing that a number is prime or composite gives you information about how many factors it has, as well as about how it can be represented as an array. Integers include the natural numbers and their opposites, as well as zero. They describe amounts above, below, and including the zero benchmark. Integers can be compared by using their positions relative to the zero benchmark. Order of operations rules are used to ensure that everyone reading an expression interprets it the same way. Student Book Section Getting Started Banner Designs pp (TR pp. 9 12) Lesson Goal Activate knowledge about number relationships. Grade 6 Outcomes Pacing 13 Days Prerequisite Skills/Concepts 1 day Recall multiplication facts and related division facts to 81. Identify and extend number patterns. Lesson 1 Identifying Factors pp (TR pp ) Lesson 2 Identifying Multiples pp (TR pp ) Lesson 3 Prime and Composite Numbers pp (TR pp ) Lesson 4 Identifying Factors by Dividing pp (TR pp ) Lesson 5 Creating Composite Numbers p. 85 (TR pp ) Lesson 6 Solving Problems Using an Organized List, pp (TR pp ) Lesson 7 Representing Integers pp (TR pp ) Lesson 8 Comparing and Ordering Integers pp (TR pp ) Lesson 9 Order of Operations pp (TR pp ) Identify factors to solve problems. Identify multiples to solve problems. Identify prime and composite numbers. Identify factors by dividing composite numbers by primes. Multiply combinations of factors to create composite numbers. Use an organized list to solve problems that involve number relationships. Use integers to describe situations. Use a number line to compare and order integers. Apply the rules for order of operations with whole numbers. N3 1 day Calculate products and quotients using mental math. Divide a two-digit number by a one-digit number. Understand the meaning of the term factor. Use arrays to multiply and divide numbers. N3 1 day Identify factors of whole numbers. Extend a number pattern by multiplying or adding whole numbers. N3 1 day Identify factors and multiples of whole numbers. N3 1 day Identify prime and composite numbers. Identify factors of whole numbers. N3 1 day Multiply and divide combinations of one-digit and two-digit numbers. Identify prime and composite numbers. N3 1 day Identify factors and multiples of whole numbers. Identify prime and composite numbers. N7 1 day Locate numbers on a number line. N7 1 day Locate integers on a number line. Use the symbols <, >, and to compare numbers. N9 1 day Use mental math to add, subtract, multiply, and divide whole numbers. Curious Math 1 p. 77 (TR pp ) Math Game 1 p. 81 (TR pp ) Mid-Chapter Review pp (TR pp ) Curious Math 2 p. 93 (TR pp.51 52) Math Game 2 p. 101 (TR pp ) Chapter Review, pp (TR pp ) Chapter Task, p. 105 (TR pp ) Chapters 1 3 Cumulative Review pp (TR pp ) 3 days 4 Chapter 3: Number Relationships

8 Chapter Goals Identify prime numbers, composite numbers, factors, and multiples. Determine the factors of a composite number. Use an organized list to solve problems. Represent, order, and compare integers. Explain and apply the order of operations with whole numbers. Materials Masters Extra Practice in the Student Book and Workbook pencil crayons 2 cm Grid Paper, Masters Booklet p. 23 Optional: Scaffolding for Getting Started pp Optional: Review of Essential Skills: Chapter 3, Masters Booklet p. 5 Optional: Initial Assessment Summary, Masters Booklet p. 57 Optional: counters Optional: linking cubes Optional: 1 cm Grid Paper, Masters Booklet p. 22 Optional: Chapter Checklist: Chapter 3, Masters Booklet p. 64 Mid-Chapter Review Questions 1 & 2 Chapter Review Questions 1, 2, & 3 Workbook, p. 17 rulers Optional: counters Number Lines, Masters Booklet p. 33 Optional: Scaffolding for Lesson 2, Question 3 p. 77 counters 100 Chart, Masters Booklet p cm Grid Paper, Masters Booklet p. 23 Mid-Chapter Review Questions 3 & 4 Chapter Review Questions 4 & 5 Workbook, p. 18 Mid-Chapter Review Question 5 Chapter Review Questions 6 & 7 Workbook p. 19 number cards 40 to 50 Mid-Chapter Review Questions 6 & 7 Chapter Review Question 8 Workbook, p. 20 Optional: chart paper and markers Optional: 100 Chart, Masters Booklet p. 30 Optional: 1 cm Grid Paper, Masters Booklet p. 23 Workbook p. 21 Optional: 100 Chart, Masters Booklet p. 30 Chapter Review Question 9 Workbook, p. 22 Number Lines, Masters Booklet p. 33 Chapter Review Question 10 Workbook p. 23 Number Lines, Masters Booklet p. 33 Chapter Review Questions 11 & 12 Workbook, p. 24 calculators Chapter Review Questions 13 & 14 Workbook p. 25 For materials and masters for features, reviews, and the Chapter Task, see the TR section. Workbook p. 26 Overview 5

9 Chapter 3 Assessment Summary These charts list references to the many assessment opportunities in the chapter. Formative assessment (Assessment for Learning) provides information about students understanding of concepts and helps you adapt instruction to students needs. A key question in each lesson links to the lesson goal. Initial or diagnostic assessment ideas (also part of Assessment for Learning) are provided in Getting Started. Summative assessment (Assessment of Learning) opportunities are provided in the Mid-Chapter Review, Chapter Review, and Chapter Task. Have students self-assess their learning (Assessment as Learning) using one of the self-assessment tools provided in the Masters Booklet. Opportunities for Feedback: Assessment for Learning Student Book Section Chart Key Question Grade 6 Outcomes Mathematical Process Focus for Key Question Lesson 1 Identifying Factors pp TR p. 17 5, model, written answer N3. Demonstrate an understanding of factors and multiples by determining multiples and factors of numbers less than 100 identifying prime and composite numbers solving problems involving multiples. [PS, R, V] Reasoning, Visualization Lesson 2 Identifying Multiples pp TR p. 21 5, short answer, written answer N3 Problem Solving, Visualization Curious Math String Art p. 77 Lesson 3 Prime and Composite Numbers pp Math Game Colouring Factors p. 81 Lesson 4 Identifying Factors by Dividing pp TR p. 23 N3 Problem Solving, Reasoning, Visualization TR p. 28 4, written answer N3 Reasoning, Visualization TR p. 30 N3 Reasoning TR p. 34 4, written answer N3 Reasoning Lesson 5 Creating Composite Numbers p. 85 TR p. 38 entire exploration, investigation N3 Problem Solving, Reasoning Mid-Chapter Review pp TR p. 41 1, model, written answer N3 Visualization Lesson 6 Solving Problems Using an Organized List pp Lesson 7 Representing Integers pp Curious Math Countdown Clock p. 93 Lesson 8 Comparing and Ordering Integers pp , short answer N3 Reasoning 3, short answer N3 Problem Solving 4, short answer, written answer N3 Reasoning 5, short answer N3 Problem Solving 6, short answer N3 Reasoning, Visualization 7, written answer N3 Reasoning TR p. 46 6, written answer N3 Problem Solving, Reasoning TR p. 50 4, short answer, model N7. Demonstrate an understanding of integers, concretely, pictorially, and symbolically. [C, CN, R, V] Reasoning, Visualization TR p. 52 N7 Connections, Reasoning TR p. 57 6, model, written answer N7 Communication, Connection, Visualization Lesson 9 Order of Operations pp TR p. 61 4, short answer N9. Explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers). [CN, ME, PS, T] Connections, Mental Mathematics and Estimation, Problem Solving, Technology Math Game Countdown Clock p. 101 TR p. 63 N9 Mental Mathematics and Estimation Mathematical Processes: C Communication, CN Connections, ME Mental Mathematics and Estimation, PS Problem Solving, R Reasoning, T Technology, V Visualization 6 Chapter 3: Number Relationships

10 Assessment of Learning Student Book Section Chart Question Grade 6 Outcome Mathematical Process Focus for Question Mid-Chapter Review TR pp , model, written answer N3 Visualization pp , short answer N3 Reasoning Chapter Review pp and Chapter Test (TR pp ) Chapter Task A Block Dropping Game, p , short answer N3 Problem Solving 4, short answer, written answer N3 Reasoning 5, short answer N3 Problem Solving 6, short answer N3 Visualization 7, written answer N3 Reasoning TR pp , written answer N3 Visualization 2, written answer N3 Reasoning 3, 4, short answer N3 Reasoning 5, written answer N3 Problem Solving 6, short answer, written answer N3 Reasoning 7, written answer N3 Reasoning 8, short answer N3 Reasoning 9, short answer N3 Problem Solving, Reasoning 10, written answer N7 Communication 11, written answer, model N7 Visualization 12, short answer, model N7 Communication, Visualization 13, short answer N9 Mental Mathematics and Estimation 14, short answer, written answer N9 Problem Solving TR p. 71 entire task, investigation N3 Problem Solving, Reasoning, Visualization Assessment as Learning Student Book Section Mid-Chapter Review pp Chapter Review pp Student Self-Assessment Masters Chapter 3 Lesson Goals, Masters Booklet p. 75 Self-Assessment: Mathematical Processes, Masters Booklet p. 84 Self-Assessment: What I Like, Masters Booklet p. 85 Self-Assessment: How I Learn, Masters Booklet p. 85 Chapter 3 Lesson Goals, Masters Booklet p. 75 Self-Assessment: Mathematical Processes, Masters Booklet p. 84 Self-Assessment: What I Like, Masters Booklet p. 85 Self-Assessment: How I Learn, Masters Booklet p. 85 Overview 7

11 Chapter 3 Chapter Opener STUDENT BOOK PAGES Using the Chapter Opener Draw students attention to the photograph on Student Book pages 66 and 67. Tell students that the Craik Eco-Centre is an energy-efficient building that uses renewable energy. Together, read the opening task. Record and discuss students responses. If students have trouble getting started, have them use 12 linking cubes and make as many different rectangular prisms as they can. Encourage students to arrange prisms in multiple layers, such as 2 2 3, as well as single layers, such as Review how the length, width, and thickness of a prism can be used to identify factors of 12. As students assemble model walls with 36 linking cubes, encourage them to build walls with layers as well. Students might identify different numbers of walls depending on whether or not they distinguish between the order of the dimensions. For example, they might consider a 2 18 and an 18 2 wall to be equivalent walls. Sample Discourse Suppose a wall has a thickness of 1 cube. How many different walls can you make with 36 cubes? Five: 36 cubes long and 1 cube high, 18 cubes long and 2 cubes high, 12 cubes long and 3 cubes high, 9 cubes long and 4 cubes high, and 6 cubes long and 6 cubes high If you know two factors that multiply together to make 36, these factors represent the length and height of a wall. Suppose a wall has a thickness of 2 cubes. How many different walls can you make with 36 cubes? Three: 18 cubes long and 1 cube high, 9 cubes long and 2 cubes high, and 6 cubes long and 3 cubes high What other wall can you make with 36 linking cubes? I can make a wall 4 cubes thick, 3 cubes long, and 3 cubes high. I can make a wall 2 cubes thick, 9 cubes long, and 2 cubes high. I can make a wall 6 cubes thick, 3 cubes long, and 2 cubes high. Read and discuss the five goals of the chapter. Ask students to suggest different ways they can determine the factors of a number. Have students record in their journals their thoughts about one of the goals, using a prompt such as Examples of situations where I would need to identify the factors of a number are. At the end of the chapter, you can ask students to complete the same prompt. Then they can compare their ideas with the ones recorded at the beginning of the chapter and reflect on what they have learned. At this point, it would be appropriate to send home Family Letter p. 74 ask students to look through the chapter and add math word cards to your classroom word wall. Here are some terms related to this chapter: factor product multiple prime number composite number integer opposite integers rules for order of operations Family Letter p Chapter 3: Number Relationships

12 Chapter 3 Getting Started Banner Designs STUDENT BOOK PAGES GOAL Activate knowledge about number relationships. PREREQUISITE SKILLS/CONCEPTS Recall multiplication facts and related division facts to 81. Identify and extend number patterns. 2 cm Grid Paper, Masters Booklet p. 23 Optional: Scaffolding for Getting Started p Preparation and Planning Pacing min Activity min What Do You Think? Materials pencil crayons Masters 2 cm Grid Paper, Masters Booklet p. 23 Optional: Scaffolding for Getting Started pp Optional: Review of Essential Skills: Chapter 3, Masters Booklet p. 5 Optional: Initial Assessment Summary, Masters Booklet p. 57 Nelson Website Visit and follow the links to Nelson Math Focus 6, Chapter 3. Optional: Review of Essential Skills: Chapter 3, Masters Booklet p. 5 Optional: Initial Assessment Summary, Masters Booklet p. 57 Math Background The Getting Started activity helps students activate knowledge of number relationships and principles learned in earlier grades. Specifically, students will use number patterns, skip counting, and multiplication to determine multiples of two whole numbers. Students need a firm understanding of multiplication and division facts to help them identify both multiples and factors of whole numbers. Getting Started: Banner Designs 9

13 Using the Activity (Whole Class/Pairs/Small Groups) min Use this activity to activate knowledge of factors and multiples and number patterns and as an opportunity for initial assessment. Together, read about Daniel s Heritage Day banner and then read the central question on Student Book page 68. Distribute grid paper to students. Have students work in pairs or small groups to answer Prompts A to C. Students having difficulty sketching may prefer writing the letter E for eagle instead of drawing the symbol. Discuss the answers to these prompts as a class. Have students work in groups to answer Prompt D. Have volunteers share their banners with the class. If extra support is required, guide these students and provide copies of Scaffolding for Getting Started pp Answers to the Activity A. For example, B. For example, I saw the pattern 6, 12, 18. The pattern shows skip counting by 6s. So the next square with an eagle should be the 24th square because 18 6 = 24. C. For example, I can multiply 1, 2, and 3 by 6 to get 6 1 = 6, 6 2 = 12, and 6 3 = 18, which are the numbers of the first three red squares that have an eagle. So I can solve the equation 6 = 30 to figure out the number of red squares with an eagle. I can divide by 6 to solve the problem. There are 30 squares and 30 6 = 5, so 5 red squares will have an eagle. D. For example, I ll create a banner with 100 squares. I ll colour every second square yellow. Every fifth square will have the symbol for a horse. I ll figure out how many yellow squares will have a horse. horse horse 10 Every 10th square has a horse in a yellow square. So I predict that the number of yellow squares with a horse in 100 squares is = Chapter 3: Number Relationships

14 Using What Do You Think? (Small Groups/Whole Class) min Use this anticipation guide to activate knowledge and understanding of factors and multiples. Explain to students that the statements involve math concepts or skills they will learn about in the chapter they are not expected to know the answers at this point. Ask students to read the statements, think about each one for a few seconds, and decide whether they agree or disagree. Have volunteers explain the reasons for their choices. Students can exchange their thoughts in small groups, in groups where all agree or disagree, or in a general class discussion. Tell students they can revisit their ideas at the end of the chapter. Possible Responses for What Do You Think? Correct responses are indicated with an asterisk (*). Students should be able to give correct responses by the end of the chapter. 1. For example, agree. If you multiply 5 by 6, you get 30. You can also multiply 1 and 30 to get 30, and there are other factors of 30 too. So when you multiply two whole numbers, the product has more than two factors. *For example, disagree. When you multiply 1 by 1, you get 1, and 1 is the only factor. 2. For example, agree. The last digit is 0 so when you multiply numbers like 10 and 20, you get a 0 in the ones digit of the product. *For example, disagree = 200 and neither factor has 0 as the ones digit. 3. *For example, agree. If you extend the first pattern by adding 5 and the second pattern by adding 7, you get 35 on both lists. Then if you keep adding 5 and 7, you will get 70 as the next number on both lists. So if you continue adding both 5 and 7, you will get lots of the same numbers on both lists. For example, disagree. The three numbers in each list are different. One list of numbers goes up by 5s and the other list goes up by 7s. So you will not get many of the same numbers. 4. For example, agree. 3 has two factors, 1 and = 6. 6 has four factors: 1, 2, 3, and 6. So multiplying 3 by 2 doubled the number of factors. *For example, disagree. 4 has three factors: 1, 2, and 4; 2 4 = 8. 8 has four factors: 1, 2, 4 and 8. So when you multiply 4 by 2, you do not get double the number of factors. Getting Started: Banner Designs 11

15 Initial Assessment: Assessment for Learning What you will see students doing When students understand Prompt B Students explain how to use a number pattern to predict the next red banner square that will have an eagle symbol. Prompt C Students explain how to use a multiplication equation to figure out how many red squares will have an eagle. If students misunderstand Students may not recognize that every sixth square has both characteristics (eagle, red) and cannot extend the pattern 6, 12, 18, beyond 18. (See 3 below.) Students may not be able to connect multiplication facts with determining the number of red squares that will have an eagle. (See 4 below.) Prompts C & D Students determine the number of coloured squares that will have a symbol and explain their method. Students may not connect determining the number of squares with number patterns or multiplication facts. (See 3 and 5 below.) Differentiating Instruction: How you can respond SUPPORTING STUDENTS WHO ARE ALMOST THERE 1. Use Scaffolding for Getting Started pp Use Review of Essential Skills: Chapter 3, Masters Booklet p. 5 to activate students skills. 3. Have students number the squares from left to right and note the numbers of the coloured squares that have a symbol. Discuss the pattern in the numbers (6, 12, 18, ) and discuss the strategies that students might use to determine the next number in the pattern, for example, skip counting by Remind students that in a multiplication fact, two factors are multiplied to give a product. Help students understand that one of the factors is the number of squares from one coloured square with a symbol to the next, and the product is the total number of squares in the banner. The unknown factor is the number of coloured squares with a symbol that will be in the banner. For example, if there is a coloured square with a symbol every 5 squares and a total of 50 squares, students can use the multiplication fact 5 = 50 to calculate the number of coloured squares with a symbol that will be in the banner. 4. Remind students that a multiplication fact is another way to represent skip counting. For example, to complete the multiplication sentence 6 = 18, students can skip count by 6s until they reach 18, and count the number of skips. There are three skips, so 3 6 = 18. Suggest students use the same thinking for patterns that reach greater numbers. SUPPORTING STUDENTS WHO ARE NOT READY This chapter assumes that students are already comfortable identifying and extending number patterns and calculating the missing factor in a multiplication equation. For this activity: You may want to focus on working with number patterns and eliminate consideration of multiplication equations. In some lessons, suggestions for adapting the lesson to deal with students who are in a lower developmental phase can be found at the end of the Opportunities for Feedback: Assessment for Learning chart. 12 Chapter 3: Number Relationships

16 Chapter 3 1 Identifying Factors STUDENT BOOK PAGES GOAL Identify factors to solve problems. PREREQUISITE SKILLS/CONCEPTS Calculate products and quotients using mental math. Divide a two-digit number by a one-digit number. Understand the meaning of the term factor. Use arrays to multiply and divide numbers. SPECIFIC OUTCOME N3. Demonstrate an understanding of factors and multiples by determining multiples and factors of numbers less than 100 identifying prime and composite numbers solving problems involving multiples. [PS, R, V] Achievement Indicators Determine all the whole-number factors of a given number using arrays. Identify the factors and multiples for a given number and explain the strategy used, e.g., concrete or visual representations, repeated division by prime numbers, or factor trees. Solve a given problem involving factors or multiples. Preparation and Planning Pacing Materials Masters 5 10 min Introduction min Teaching and Learning min Consolidation Optional: counters Optional: linking cubes Optional: 1 cm Grid Paper, Masters Booklet p. 22 Optional: Chapter Checklist: Chapter 3, Masters Booklet p. 64 Recommended Questions 3, 4, 5, 6, 7, 8, & 13 Practising Questions Key Question Question 5 Extra Practice Mid-Chapter Review Questions 1 & 2 Chapter Review Questions 1, 2, & 3 Workbook p. 17 Mathematical Process Focus Nelson Website R (Reasoning) and V (Visualization) Visit and follow the links to Nelson Math Focus 6, Chapter 3. Math Background Students should be familiar with the relationship between factors of a number and division of that number. For example, 2 is a factor of 10 because the quotient (5) is a whole number and the remainder is 0. To identify all of the factors of a number and to help them visualize those factors, students can use arrays. An array is a pictorial or concrete model of a number in which the rows and columns of the array represent factors of the number. For example, a 4-by-5 array shows that 4 and 5 are factors of 20 because the array has 4 rows, 5 columns, and a total of 20 elements. As students use reasoning and mental math to identify the factors of a number, they can show the factors in a factor rainbow. A factor rainbow lists all of a number s factors in a row and pictorially links the factors that can be multiplied together to result in that number. It is important to list the factors systematically so none are forgotten. Optional: 1 cm Grid Paper, Masters Booklet, p. 22 Optional: Chapter Checklist: Chapter 3, Masters Booklet p. 64 Lesson 1: Identifying Factors 13

17 1 Introduction (Small Groups/Whole Class) 5 10 min Distribute 12 counters to each group. Have students form the counters into an array. Alternatively, have them colour arrays of 12 on grid paper. Ask volunteers to share their arrays with the class. Try to elicit all of the possible arrays for the number 12: 1-by-12, 2-by-6, and 3-by-4. Some students may also suggest reversing the order of rows and columns, for example, 12-by-1. Accept these answers but make sure students realize that the factors are still the same. Sample Discourse How did you decide how many counters would go in each row of your array? I tried to make rows that were all the same size without having any counters left over. I then counted the number of counters in each row to determine one factor. I chose a number of rows that is a factor of 12, and then put the counters into that number of rows. Can you make an array with five rows? No, because there will be two counters left over. No, because 5 is not a factor of 12. No, because 5 does not divide evenly into Teaching and Learning (Whole Class/Pairs) min Together, read about the Earth Day project and then read the central question on Student Book page 70. Work through Mai s Arrays together. Students may represent the arrays with symbols as Mai did, or they may use counters or grid paper. Some students may use pairs of factors to identify two arrays rather than one array. Work through Jason s Factor Rainbow with students to show how to systematically record all the factors of 18. For example, students may reverse the rows and columns to get 6 arrays for 18 seedlings: 1-by-18 and 18-by-1, 2-by-9 and 9-by-2, and 3-by-6 and 6-by-3. Tell students they can solve the problem either way as long as they list the number of arrays the same way for each number in the chart. They should also note that the factors 1, 2, 3, 6, 9, and 18 remain the same. Have students work in pairs to complete Prompts A to C. When students have completed the activity, discuss the answers as a class. 14 Chapter 3: Number Relationships

18 Answers to Prompts A. For example, I used a factor rainbow to record the number of factors and the number of arrays for each number of seedlings. The factors of 25 are 1, 5, and 25. So 25 seedlings can be planted in 2 arrays: 1-by-25, 5-by-5 Grade 2 1 The factors of 29 are 1 and 29. So 29 seedlings can be planted in 1 array: 1-by-29 Grade The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. So 36 seedlings can be planted in 5 arrays: 1-by-36, 2-by-18, Grade 4 3-by-12, 4-by-9, 6-by The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. So 48 seedlings can be planted in 5 arrays: 1-by-48, 2-by-24, Grade 5 3-by-16, 4-by-12, 6-by The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. So 56 seedlings can be planted in 4 arrays: 1-by-56, 2-by-28, Grade 6 4-by-14, 7-by B. For example, I chose Jason s method because I can use mental math to figure out the factors of a number. The factor rainbow helps me keep track of the factors I have figured out. I didn t use Mai s method because it takes too long to draw all the arrays for each number. C. Both 36 and 48 seedlings can be planted in 5 arrays. Reflecting (Whole Class) Here students compare and contrast arrays with factor rainbows as methods for identifying the factors of a number. Students also explain how they know when they have identified all of the factors, using each method. Ensure students understand that arrays can be used to identify factors, while factor rainbows are primarily a method for recording the factors. Students should also connect the dimensions of the arrays with the factors listed in the factor rainbow. 56 Lesson 1: Identifying Factors 15

19 Answers to Reflecting Questions D. For example, they are the same in that each of the dimensions of Mai s arrays matches a factor pair in Jason s factor rainbow. They are different because Mai has to draw rectangles to list arrays while Jason uses mental math to list factors. E. For example, Mai drew arrays with 1, 2, and 3 rows. She knew that 4 and 5 aren t factors of 18, so she couldn t plant 18 seedlings in 4 or 5 rows. She knew that a 3-by-6 array can be arranged in either 3 rows of 6 or 6 rows of 3. She knew that 7 and 8 aren t factors of 18, so she couldn t plant 18 seedlings in 7 or 8 rows. She knew that a 2-by-9 array can be arranged in either 2 rows of 9 or 9 rows of 2. So she knew that there are no other possible arrays for 18 seedlings. Jason s factor rainbow shows he identified the matching factors of 1 and 18, 2 and 9, and 3 and 6. He only had to see if 4 or 5 is a factor because he had already figured out factors of 18 that are 6 or greater. Because 4 and 5 are not factors, he knew he had identified all factors of Consolidation min Checking (Pairs) Students can use either arrays or factor rainbows to identify the factors. Refer students to Mai s and Jason s methods for guidance. Have counters and grid paper available for students to use to model arrays. Practising (Individual) These questions provide students with practice in using arrays and factor rainbows to identify and record factors. Provide counters or grid paper to students to help them model the arrays. 6. Students should recognize that the number of coins can only be divided by 1 and itself. In Lesson 3, students will formalize this understanding as they learn about prime and composite numbers. Answers to Key Question 5. a) 3-by-8 4-by-6 b) For example, each number of rows and columns in an array represents a factor of 24. So the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Closing (Whole Class) Question 13 allows students to reflect on and consolidate their learning for this lesson as they connect the numbers of rows and columns in an array to the factors of the number. Answer to Closing Question 13. For example, if you want to identify the factors of 26, you can draw arrays. The numbers of rows and columns of the arrays are the factors of 26. So 1, 2, 13, and 26 are factors of 26. You can also use mental math to identify the factors and use a rainbow to help you keep track Follow-Up and Preparation for Next Class Have students follow up on the lesson at home using a group of small items such as toothpicks. Suggest that students arrange the group into an array. Using the array, students should identify factors of the number used in the array. 1-by-24 2-by Chapter 3: Number Relationships

20 Opportunities for Feedback: Assessment for Learning What you will see students doing When students understand Students use arrays and/or factor rainbows to identify the factors of a number. Key Question 5 (Reasoning, Visualization) Students draw all of the possible arrays for the number 24 and explain how the dimensions of the arrays relate to the factors of 24. If students misunderstand Students may not identify all of the factors. (See Extra Support 1.) Students may not connect the numbers of rows and columns with the factors of the number. (See Extra Support 2.) Differentiating Instruction: How you can respond EXTRA SUPPORT 1. Help students understand how they can use a factor rainbow to organize their work. Have students write the numbers 1 to 16 in a row. Tell students to look at each number in the row and use mental math or arrays to decide if it is a factor of 16. If it is a factor, have students circle the number; if it is not a factor, have students cross out the number. Finally, have students draw arches to connect the numbers that can be multiplied together to give a product of 16. For square numbers (16 = 4 4), suggest that students simply draw an arch from the 4 to itself. 2. Have students use grid paper and shade in as many rectangles as possible that have a total area of 24 grid squares. Then have students label each rectangle with the number of rows and the number of columns that are shaded, for example, 4-by-6. Guide students to understand that 4-by-6 means 4 multiplied by 6. Since the area of the rectangle is 24, 4 and 6 are factors of 24. Repeat the exercise, using counters in an array in place of the grid paper, and guide students to connect the numbers of rows and columns in the arrays with the factors of 24. EXTRA CHALLENGE Challenge students to identify the number between 1 and 50 that can be modelled with the greatest number of arrays. Encourage students to develop strategies to help them eliminate some numbers, rather than drawing the arrays for each number. For example, students might eliminate any number that can only be drawn in an array with one row. SUPPORTING DEVELOPMENTAL DIFFERENCES Provide students with an array and have them work together or individually to identify the factors. Then ask students to create another array with the same number of counters. This exercise will give students an opportunity to explore factors and products without identifying all of the factors of a particular number. SUPPORTING LEARNING STYLE DIFFERENCES Kinesthetic learners will benefit from creating their arrays with counters rather than just drawing them. Lesson 1: Identifying Factors 17

21 Chapter 3 2 Identifying Multiples STUDENT BOOK PAGES GOAL Identify multiples to solve problems. PREREQUISITE SKILLS/CONCEPTS Identify factors of whole numbers. Extend a number pattern by multiplying or adding whole numbers. SPECIFIC OUTCOME N3. Demonstrate an understanding of factors and multiples by determining multiples and factors of numbers less than 100 identifying prime and composite numbers solving problems involving multiples. [PS, R, V] Achievement Indicators Identify multiples for a given number and explain the strategy used to identify them. Solve a given problem involving factors or multiples. Math Background In previous grades, students have multiplied factors to calculate a product. In this lesson, students will approach multiplication from a different perspective as they calculate multiples of a number using known multiplication facts and skip counting. Students will multiply a given number by sequential whole numbers to build a list of multiples. For example, to build a list of multiples of 6, students will multiply 6 by 1, 2, 3, 4, to get the multiples 6, 12, 18, 24,. To use skip counting, students will count in units of the given number. For example, to build a list of multiples of 5, students will count by 5s to get the multiples 5, 10, 15, 20, and so on. Students use a number line to help them visualize the pattern in the list of multiples. Students will apply these skills in various problem-solving contexts. Preparation and Planning Pacing (allow 5 min for previous homework) Materials 5 10 min Introduction min Teaching and Learning min Consolidation rulers Optional: counters Masters Number Lines, Masters Booklet p. 33 Optional: Scaffolding for Lesson 2, Question 3 p. 77 Recommended Questions 2, 3, 5, 8, & 9 Practising Questions Key Question Question 5 Extra Practice Mid-Chapter Review Questions 3 & 4 Chapter Review Questions 4 & 5 Workbook p. 18 Mathematical Process Focus Vocabulary/Symbols Nelson Website PS (Problem Solving) and V (Visualization) multiple Visit and follow the links to Nelson Math Focus 6, Chapter 3. Number Lines, Masters Booklet p. 33 Optional: Scaffolding for Lesson 2, Question 3 p Chapter 3: Number Relationships

22 1 Introduction (Whole Class) 5 10 min Briefly review some mental math strategies that students have learned for multiplication. On the board, on a transparency, or on an interactive whiteboard, write the following multiplication expressions: Ask volunteers to share their strategies for calculating each product. Try to elicit a variety of strategies. Sample Discourse How can you calculate the product of 4 and 8? I used doubling. I know 2 8 = 16, so 4 8 = 16 16, which is 32. I used doubling. I know 4 4 = 16, so 4 8 = 16 16, which is 32. How can you calculate the product of 6 and 7? I skip counted up. I know 6 6 = 36, so 6 7 = 36 6, which is 42. I skip counted down. I know 7 7 = 49, so 6 7 = 49 7, which is 42. How can you calculate the product of 8 and 5? I used doubling. I know 2 5 = 10, so 4 5 = 10 10, which is 20, and 8 5 is 20 20, or 40. I skip counted down. I know 10 5 = 50, so 9 5 = 50 5, which is 45, and 8 5 = 45 5, which is Teaching and Learning (Whole Class/Small Groups) min Before reading, remind students that a comet is a small body that orbits the Sun, and it is only visible from Earth at certain points in its orbit. Comets that appear regularly are referred to as periodic comets. Together, read about the comets and then read the central question on Student Book page 74. Have students set up Oleh s List and retrace his steps to show the first multiples of 7. Then direct them to Léa s Number Line. Tell students to use their rulers to draw an open number line with two arrows. Ask them to point out which number Léa starts with on the number line and how she gets to the next number. When students have become comfortable with Léa s method, have them work through Prompts A to C in small groups. You may want to discuss the two methods as a group and have volunteers explain which method they prefer. Sample Discourse Which math operations did Oleh use in his method? How is Oleh s method different from Léa s method? Oleh used multiplication to determine the multiples of 7 and addition to calculate the years the comet would be seen from Earth. Léa only used addition to figure out the years after 2000 the comet would be seen. Lesson 2: Identifying Multiples 19

23 Answers to Reflecting Questions D. For example, you create a multiple of 7 by multiplying 7 by a counting number. So any multiple is 7 times a counting number and 7 must be a factor. E. For example, any factor of 9 has to be 9 or less, so there are only 9 possible numbers. But multiples of 9 are created by continually adding 9s and you can add 9s forever. 3 Consolidation min Checking (Pairs) Draw students attention to the Communication Tip. Ensure that they are comfortable with the notation, which is called an ellipsis. If students require additional guidance, refer them to Oleh s and Léa s methods in the example. You may want to distribute number lines to students; however, students do not need to use scaled number lines; rather, they can sketch empty number lines. Which method is easier for you to use? Explain. Oleh s method is easier because multiplying to determine the multiples is faster than adding, and I only have to replace the last digits of 2000 with the multiples of 7 to get the years. Léa s method is easier because I like adding better than multiplying. Answers to Prompts A. For example, I multiplied 7 by 3 to get 21. B. For example, I added 7 to 2014 to get the year C. For example, I listed the multiples of 7 until I got to 70. I stopped at 70 because I know = 2070 is past , 14, 21, 28, 35, 42, 49, 56, 63, 70, I added these multiples to 2000 to get these years in which the comet will likely be seen from Earth: 2007, 2014, 2021, 2028, 2035, 2042, 2049, 2056, and Reflecting (Whole Class) Here students reflect on the relationship between factors and multiples. Students should recognize that a multiple is the product of a factor and a counting number. Practising (Individual) These questions give students opportunities to practise calculating multiples. Students will also explain connections between factors and multiples. Encourage students to use mental math strategies in their calculations. Encourage students to use number lines as visualization tools. 2. Ensure students understand that the first five multiples can be calculated by multiplying by the first five counting numbers, 1, 2, 3, 4, and 5, or by repeatedly adding the number to itself until five multiples are listed. 3. If extra support is required, guide these students and provide copies of Scaffolding for Lesson 2, Question 3 p Students create lists of multiples of two numbers and then identify the numbers that appear in both lists. In later grades, students will formalize this understanding as they learn about common multiples. Answers to Key Question 5. a) 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 plates b) For example, 10 packages; Pauline needs to buy plates for 80 people, and 80 plates are in 10 packages. c) 12, 24, 36, 48, 60, 72, 84, 96, 108, 120 d) For example, 7 packages; Pauline needs to buy at least 80 glasses, and 6 packages have 72 glasses, which is too little, but 7 packages have 84 glasses, which is enough. 20 Chapter 3: Number Relationships