Multiplication Fact Power and Shortcuts Objectives To discuss multiplication facts and the importance of fact power; and to review fact shortcuts. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Practice multiplication facts. [Operations and Computation Goal 3] Identify patterns in skip counting by 2s, 5s, and 1s. [Patterns, Functions, and Algebra Goal 1] Use arrays to represent turn-around multiplication facts. [Patterns, Functions, and Algebra Goal 4] Use the Commutative Property of Multiplication (the turn-around rule), the Multiplicative Identity, and the Zero Property of Multiplication to generate multiplication facts. [Patterns, Functions, and Algebra Goal 4] Key Activities Children discuss multiplication facts and the importance of fact power. They also review multiplication shortcuts and play Beat the Calculator (Multiplication). Key Vocabulary factor product fact power turn-around shortcut square numbers Materials Math Journal 1, p. 156 Student Reference Book, p. 279 Home Link 4 4 Math Masters, p. 95; p. 446 (optional) transparency of Math Masters, p. 95 (optional) calculator slate per group of 3 children: calculator Advance Preparation Solving Multidigit Subtraction Problems Math Journal 1, p. 88 number grid (optional) base-1 blocks (optional) Children solve multidigit subtraction problems. Math Boxes 4 5 Math Journal 1, p. 89 Children practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 5. [Measurement and Reference Frames Goal 2] Home Link 4 5 Math Masters, p. 96 Children practice and maintain skills through Home Link activities. READINESS Building Facts on a Geoboard Math Masters, p. 97 geoboard rubber bands Children use an array model to find products for basic facts. ENRICHMENT Creating Fact Shortcuts paper Children create and explain their own fact shortcuts. Teacher s Reference Manual, Grades 1 3 pp. 195 198, 199 266 Unit 4 Multiplication and Division
Getting Started Mathematical Practices SMP2, SMP4, SMP5, SMP6, SMP7, SMP8 Content Standards 3.OA.5, 3.OA.7, 3.OA.9, 3.NBT.2 Mental Math and Reflexes Children count from 1 to 3, clapping at the intervals described below. every 5th number 1, 2, 3, 4, 5 (clap), 6, 7, 8, 9, 1 (clap), 11, 12, 13, 14, 15 (clap), and so on to 3 every 3rd number 1, 2, 3 (clap), 4, 5, 6 (clap), 7, 8, 9 (clap), and so on to 3 every 4th number 1, 2, 3, 4 (clap), 5, 6, 7, 8 (clap), 9, 1, 11, 12 (clap), and so on to 3 Math Message 7 4 = 28 is a multiplication fact. Write 5 other multiplication facts. Home Link 4 4 Follow-Up Have partners describe to each other how they solved one of the problems. 1 Teaching the Lesson Math Message Follow-Up DISCUSSION Multiplication facts are products of two 1-digit numbers. Have children call out multiplication facts, with or without the products, while you write them on the board. Use both horizontal formats (with the product to the left and the right) and vertical formats. Tell children that the symbols * and are sometimes used to show multiplication. The numbers being multiplied are factors. The result of multiplication is the product. To support English language learners, label some of the facts written on the board with the words factor and product. Encourage children to use this vocabulary, but do not insist on it. ELL 2 3 = 6 18 = 9 * 2 5 4 = 2 3 8 24 Different multiplication formats Reviewing the Importance of Fact Power DISCUSSION Ask children to name some good and some bad habits. A habit is an automatic behavior that comes from doing something over and over again. Developing basic number-fact reflexes is developing a good habit. In Everyday Mathematics, fact habits are referred to as fact power. Stress to children that they need to learn the multiplication and division facts so that they can solve multidigit multiplication and division problems. Reviewing Shortcuts for Multiplication Facts (Math Masters, p. 95) Algebraic Thinking Tell children they will review shortcuts that will help them develop a good habit knowing the multiplication facts by heart. Lesson 4 5 267
4 3 = 12 and 3 4 = 12 NOTE It is not expected that children use formal terms for the properties of multiplication. It is important for them to know, however, that order does not matter when multiplying factors, that multiplying any number by 1 results in a product that is that number, and that multiplying any number by results in a product of. The Turn-around Shortcut Illustrate the turn-around shortcut (Commutative Property of Multiplication) by writing several fact phrases for small numbers, such as 4 3, on the board. Pair each fact with its turn around, or commutative partner 4 3 and 3 4. The order of the factors does not change the product, so if you know 3 4 = 12, you also know 4 3 = 12. Include square numbers, such as 3 3 and 5 5, and point out that the turn-around fact for a square number is the same as the original fact. Then have children use Math Masters, page 95 to make arrays for the facts you suggest. Children write number models under each array. The arrays that children draw represent the multiplication turn-around facts, similar in principle to the addition turn-around facts. You may wish to use an overhead of the master to model building arrays. Ask: What number models made square arrays? Sample answers: 3 3 = 9 and 5 5 = 25 Ask: What is the pattern of these number models? The factors are the same. Ask: How can you know if a number model will make a square array? If the number model has two identical factors, it will make a square array. Name Date Time Teaching Master 4 3 Grid The Multiplication by 1 Shortcut If 1 is multiplied by any number, or any number is multiplied by 1, the product is that number (Multiplicative Identity). Tell a few times 1 and 1 times number stories, and have children tell some. For example: There are 8 flowers in a pot. You have 1 pot. How many flowers do you have? 1 8 = 8; 8 1 = 8. You have 8 flowers. There is 1 ring in each box. You have 6 boxes. How many rings do you have? 1 6 = 6; 6 1 = 6. You have 6 rings. Adjusting the Activity ELL As you review the number stories, record the number models on the board so that children will be able to see the patterns for the shortcut. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Math Masters, p. 95 268 Unit 4 Multiplication and Division The Multiplication by Shortcut If is multiplied by any number, or any number is multiplied by, the product is (Zero Property of Multiplication). Tell a few times and times number stories, and have children tell some, including some with larger numbers. For example: There are 1 cookies in a box. You have boxes. How many cookies do you have? 1 = ; 1 =. You have cookies.
There are cookies per box. You have 1 boxes. How many cookies do you have? 1 = ; 1 =. You have cookies. The Familiar Facts Shortcut Tell children that the 2s, 5s, and 1s facts are simply counts by 2s, 5s, and 1s remind them of the Mental Math and Reflexes exercise. Have the class recite in unison the 2s, 5s, and 1s facts in order. Point out the following patterns: The products of 2s and 1s facts are always even numbers. (The products of 4s, 6s, and 8s facts are also always even.) The products of 5s facts always end in or 5. The products of 1s facts always end in. Additional Strategies with Familiar Facts 4 times any number can always be decomposed into 2 equal addends. 4 7 can be thought of as 2 7 = 14 and 2 7 = 14. To find the product of 4 7, the two products can be added together. 14 + 14 = 28, so 4 7 = 28. Break one factor into easy addends. For example, 6 7 can be thought of as 6 5 = 3 and 6 2 = 12. The two products, 3 and 12, can be added together to find the product of 6 7. 3 + 12 = 42, so 6 7 = 42. Taking the Fact Power Challenge Divide the class into two groups. Pose a multiplication fact problem. Children in one group use calculators to find the product. Children in the other group find the product without using calculators. Repeat the activity with other multiplication facts. Mix easy and hard facts, so that children in each group can get the product first some of the time. Children are generally surprised to find that fact power can indeed beat the calculator. Student Page Beat the Calculator (Multiplication) Materials number cards 1 1 (4 of each) Beat the Calculator (Multiplication) 1 calculator Players Materials 3 number cards 1 1 (4 of each) 1 calculator Skill Mental multiplication skills Players 3 Object of the game To multiply numbers without a calculator Skill faster Mental than multiplication a player using skills one. Directions Object of the game To multiply numbers without a calculator faster than a player using one. 1. One player is the Caller. A second player is the Calculator. Directions The third player is the Brain. 2. Shuffle 1. One the player cards is the and Caller. place them A second number-side player is down the on the Calculator. table. The third player is the Brain. 3. The 2. Shuffle Caller the draws cards 2 cards and place from them number number-side deck and down asks on for the the table. product of the numbers. 4. The 3. The Calculator Caller draws solves 2 the cards problem from the withnumber a calculator. deck and The asks Brain for solves the product it without of the a calculator. numbers. The Caller decides 4. The Calculator who got the solves answer the first. problem with a calculator. 5. The The Caller Brain continues solves it to without draw 2 cards a calculator. at a time The from Caller the number decides deck who and got ask the for answer the product first. of the numbers. 6. Players 5. The Caller trade continues roles every to 1 draw turns 2 cards or so. at a time from the number deck and ask for the product of the numbers. 6. Players The Caller trade roles draws every a 1 1 and turns a 7. or The so. Caller says, 1 times 7. The Brain and the Calculator 1 each solve The the Caller problem. draws The a 1 Caller and a decides 7. The who Caller got the says, answer 1 first. times 7. The Brain and the Calculator 1 each solve the problem. The Caller decides who got the answer first Student Reference Book, p. 279 Adjusting the Activity Use the minute hand on the clock face to practice the 5s facts. When the minute hand is on the 8, the time is 4 minutes past the hour: 5 8 = 4. When the minute hand is on the 2, the time is 1 minutes past the hour: 2 5 = 1. Pose other clock facts. Games AUDITORY KINESTHETIC TACTILE VISUAL 7 Games 7 NOTE This activity makes the point that for many easy problems, mental calculation is quicker than using a calculator. Over time, children should develop good judgment about when to estimate and when to use calculators, calculate mentally, or use paper and pencil. Playing Beat the Calculator (Multiplication) (Student Reference Book, p. 279; Math Journal 1, p. 156; Math Masters, p. 446) SMALL-GROUP Have children examine the Fact Power Table on journal page 156. The unshaded facts include the s, 1s, and 1s facts as well as the facts from Activity Sheets 1 and 2. The shaded facts are from Activity Sheets 3 and 4. Lesson 4 5 269
Date Time Subtraction Strategies Make a ballpark estimate. Write a number model to show your estimate. Choose at least two problems to solve using the counting-up method. You may choose any method you wish to solve the other problems. 1. Ballpark estimate: 2. Ballpark estimate: 3. Ballpark estimate: 226 134 92 93 47 46 487 129 358 4. Ballpark estimate: 5. Ballpark estimate: 6. Ballpark estimate: 361 248 113 724 396 328 Math Journal 1, p. 88 Student Page 515 367 148 63 192 Estimates vary. Explain to children that they will use the Fact Power Table during Beat the Calculator games to record the facts they get correct while playing the role of the Brain. When children have earned 3 check marks for a fact, they write the product in the table. The page will become a cumulative record of their progress toward multiplication fact power. Divide the class into groups of three to play Beat the Calculator. If necessary, review the directions on page 279 in the Student Reference Book. Have children follow the steps below to record their wins on their Fact Power Table. 1. The Caller selects a problem at random from the Fact Power Table on journal page 156. 2. The Brain finds the product without a calculator. The Calculator finds the product using a calculator. 3. The Caller determines whether the Brain or the Calculator got the product first. 4. If the Brain beats the Calculator and announces the correct product, the Caller makes a check mark in the box for that fact on the Brain s Fact Power Table. Once a fact has three check marks, the Brain writes the product in that box to indicate that the fact has been mastered. 5. You might wish to pass out Beat the Calculator triangles (Math Masters, page 446) to assist with assigning roles to players. NOTE Playing this game frequently will help develop children s automaticity with multiplication facts. Remind children to use their copy of the Fact Power Table on journal page 156 as a record sheet during Beat the Calculator games. Date Math Boxes Student Page Time 2 Ongoing Learning & Practice 1. Use the dots to show a 3 6 array. What is the number model? 3 6 18 3. Solve. Fill in the oval for the best answer. 4 rows of chairs 6 chairs in each row How many chairs in all? 1 chairs 12 chairs 24 chairs 2 chairs 5. Draw a 2 4 rectangle. 2 4 Number model: Area: 8 square units 64 65 66 67 8 Math Journal 1, p. 89 2. Maximum number of points scored: 25 points Minimum number of points scored: 5 points Range of points scored: 2 points 4. Fill in the number grid. 2,946 6. Which is more? $3.45 or $3.9? $.34 or $.9? Points 2,957 2,967 2,976 2,986 2,996 3,6 Game Points 3 A B C D Players $3.45 $.34 $14.55 $14.5 or $14.55? $3.15 or $31.5? $31.5 154 156 36 25 2 15 1 5 79 86 87 7 9 89 Solving Multidigit Subtraction Problems (Math Journal 1, p. 88) Children use counting-up, trade-first, or other methods of their choice to solve multidigit subtraction problems. Encourage children to use the number grid, base-1 blocks, or other manipulatives as needed. Math Boxes 4 5 (Math Journal 1, p. 89) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-7. The skill in Problem 6 previews Unit 5 content. 27 Unit 4 Multiplication and Division
Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 5 Use Math Boxes, Problem 5 to assess children s progress toward finding the area of rectangular shapes. Children are making adequate progress if they are able to draw the rectangle and find the area. Some children might be able to write the number model for the area. [Measurement and Reference Frames Goal 2] Name Date Time HOME LINK Family Note Multiplication-Fact Shortcuts Your child is learning the basic multiplication facts. Listen to your child explain multiplicationfact shortcuts as he or she works the problems. Review some 1s, 2s, 5s, and 1s multiplication facts (facts like 1 3?,? 2 4, 5 5?, and 1 4?). 56 Please return this Home Link to school tomorrow. Tell someone at home about multiplication-fact shortcuts. The turn-around rule: 3 4 12 helps me know 4 3 12. 1. 2 5 and 5 2 15 Home Link Master 1 2. 5 3 and 3 5 2 15 3. 1 2 and 2 1 1 2 Home Link 4 5 (Math Masters, p. 96) If 1 is multiplied by any number, the product is that number. The same is true if any number is multiplied by 1. 9 4. 1 9 and 9 1 9 365 5. 1 9 and 9 1 6. 365 1 and 1 365 9 9 365 Home Connection Children tell someone at home about the multiplication fact shortcuts for turn-around facts and for the s, 1s, 2s, 5s, and 1s facts. 3 Differentiation Options If is multiplied by any number, the product is. The same is true if any number is multiplied by. 7. 12 and 12 8. 99 and 99 9. 9,365 and 9,365 Think about counting by 2s, 5s, and 1s. 1. 1 2 11. 5 4 12. 9 2 13. 2 7 14. 5 3 15. 1 5 2 2 18 14 15 5 Math Masters, p. 96 READINESS Building Facts on a Geoboard (Math Masters, p. 97) 5 15 Min To provide experience with multiplication facts using a concrete model, have children use rubber bands on a geoboard to find products for basic facts. Children record their work on Math Masters, page 97. When they have finished, briefly review which of the facts might be easiest to remember. ENRICHMENT Creating Fact Shortcuts PARTNER 15 3 Min To further explore fact shortcuts, have children write their own fact shortcut for facts they already know. They explain their shortcut on paper and include an example. Possible shortcuts: If I know 2 times a number, adding one more of that number would be 3 times the number. If I know 2 times a number, doubling the answer would be 4 times the number. If I know 1 times a number, I can subtract the number once and have 9 times the number. The children s shortcuts make a wonderful classroom resource. Consider binding them into a class book. Name Date Time Building Facts on a Geoboard For each problem: Use a rubber band to make each rectangle on a geoboard. Draw the rectangle onto this record sheet. Record the number of pins inside the rubber band as the answer to the problem. 9 1. 3 3 pins 6 3. 3 2 pins Make up problems of your own. 5. pins Answers vary. Math Masters, p. 97 Teaching Master Example 8 2 4 pins 12 2. 4 3 pins 16 4. 4 4 pins 6. pins Answers vary. 97 Lesson 4 5 271