p98-205_Gr3ON-NumberUnit2-mulitply-pass2

Unit 2: Multiplication and Division Lesson Content Page 1 Multiplication and Division Introduction 98 2 to 4 The Grapes of Math: An Introduction to Multiplication 102 Representing Multiplication as Equal Groups 111 2 Skip Counting, Repeated Addition, and Multiplication 114 3 Representing Multiplication in Various Ways 117 4 Solving Problems Involving Multiplication 119 5 to 7 Using the Array to Represent Multiplication 121 5 Introducing the Array 123 6 Working with the Array 128 7 Using the Array to Solve Multiplication Problems 132 8 Properties of 0 and 1 in Multiplication 137 9 and 10 Applying Strategies to Solve Multiplication Problems 142 9 Solving for a Better Buy 144 10 Solving More Multiplication Problems 146 11 Scaling Up: Ratios 149 12 to 14 Division as Equal Grouping 152 12 The Grapes of Math: Investigating Division 154 13 Directly Modelling Equal-Sharing Problems (Partitive Division) 157 14 Directly Modelling Equal Group Problems (Quotative Division) 161 15 Guided Math Lesson: What’s Fair? 164 16 and 17 169 16 Division Using the Array 171 17 Investigating the Array for Division 175 18 Solving Division Problems with the Array 178 19 Solving Division Problems Using a Number Line 182 20 Applying Strategies to Solve Division Problems 188 21 Developing Mental Strategies for Multiplication and Division 198 22 Guided Math Lesson: Division Bakes 203 Reinforcement Activities

Multiplication and Division Introduction About the In grade three, students represent multiplication (up to 10 × 10) and division (up to 100 ÷ 10) in various ways, including using arrays. Through these experiences, they also recall multiplication and related division facts of 2, 5, and 10. Having automaticity of these facts helps students understand ratios of 2, 5, and 10 and allows them to scale up quantities in order to solve related problems. While there are many meanings of multiplication and division, students in grade three mainly work with problems involving equal groups. There are three problem structures, defined by the position of the unknown in the part-part-whole relationship: multiplication, partitive division, and quotative division. The following table describes these three structures. Problem Position of Example Structure Unknown Multiplication Whole There are 3 packages of pencils, with unknown 5 pencils in each package. How many pencils are there altogether? (3 × 5 = __) Partitive Division Size of There are 15 pencils altogether. An equal groups number of pencils is in each of the unknown 3 packages. How many pencils are in each? (15 ÷ 3 = __) Quotative Number There are 15 pencils altogether. There Division of groups are 5 pencils in each package. How many unknown packages are there? (15 ÷ 5 = __) Alex Lawson points out that young students typically use “counting and additive reasoning to solve all three problem types. Although additive, this reasoning lays the foundation for a later shift to proportional and multiplicative reasoning for multiplication and division problems in the junior grades” (Lawson, 2015, p. 115). This is important to remember as we continually move students along the developmental trajectory to thinking in multiplicative terms. We need to emphasize the related big ideas such as unitizing—being able to simultaneously see a group of objects as one unit and also as representing several units (e.g., a dime is one coin or unit, yet at the same time represents 10 cents). 98 Number and Financial Literacy

Alex Lawson highlights several strategies that students may use to solve multiplication problems: • D irect Modelling: Students represent the number of equal groups required using concrete materials and then count the objects by 1s or, eventually, skip count to find the total. They may also create one equal group and repeatedly count it, emphasizing the last number in the group, and tracking the number of groups on their fingers. • Counting and Tracking: Without using concrete materials, students may count rhythmically by 1s, emphasizing the last number in each group and tracking the number of groups. Alternatively, they may skip count. Without a model, students need to track the number of equal groups, such as by raising a finger to represent each group counted. • Working with the Numbers: Students use repeated addition or, less frequently, repeated subtraction to represent the equal groups using numbers. They may also use other strategies such as doubling and then counting on. • Proficiency: Students automatically recall the multiplication or division fact that relates to the problem. (Lawson, 2015, p. 110) Lawson stresses that students “may skip some strategies or move back and forth between strategies, depending on the size of the numbers involved and the context of the problem” (Lawson, 2015, p. 114). To move from additive to multiplicative thinking, students need to see and create concrete, pictorial, and symbolic representations of multiplication and division. This includes creating equal groups with concrete materials or drawings and skip counting on number lines to count equal groups. The array, or the rectangular organization of items into rows and columns, is a powerful tool to support multiplicative thinking, since it helps students unitize the equal groups. For example, a 4 × 5 array can be visually interpreted as four rows of an equal number of objects (5), or five columns of an equal number of objects (4). Students can see one row as a single unit, but also as representing five units. The array is effective for learning about multiplication and division and how the operations are related, as well as for developing mental strategies for learning the related number facts. continued on next page Multiplication and Division 99

Students in grade three also focus on the properties of multiplication and division. The emphasis is on understanding and applying them rather than learning their correct names. The properties of multiplication include: – Commutative property (e.g., 3 × 5 = 5 × 3) – Associative property [e.g., (3 × 2) × 5 = 3 × (2 × 5)] – Identity property of whole-number multiplication (e.g., 5 × 1 = 5) – Zero property of multiplication (e.g., 5 × 0 = 0) – Distributive property [e.g., 2 × (4 + 3) = (2 × 4) + (2 × 3)] The properties of division include: – Identity property (e.g., 6 ÷ 1 = 6) Lesson Topic Page 1 The Grapes of Math: An Introduction to Multiplication 102 111 2 to 4 Representing Multiplication as Equal Groups 114 117 2 Skip Counting, Repeated Addition, and Multiplication 119 121 3 Representing Multiplication in Various Ways 123 128 4 Solving Problems Involving Multiplication 132 137 5 to 7 Using the Array to Represent Multiplication 142 144 5 Introducing the Array 146 149 6 Working with the Array 152 154 7 Using the Array to Solve Multiplication Problems 157 8 Properties of 0 and 1 in Multiplication 9 and 10 Applying Strategies to Solve Multiplication Problems 9 Solving for a Better Buy 10 Solving More Multiplication Problems 11 Scaling Up: Ratios 12 to 14 Division as Equal Grouping 12 The Grapes of Math: Investigating Division 13 Directly Modelling Equal-Sharing Problems (Partitive Division) 100 Number and Financial Literacy

Lesson Topic Page 14 Directly Modelling Equal Group Problems (Quotative Division) 161 164 15 Guided Math Lesson: What’s Fair? 169 171 16 and 17 Division Using the Array 175 16 Investigating the Array for Division 178 17 Solving Division Problems with the Array 182 18 Solving Division Problems Using a Number Line 188 19 Applying Strategies to Solve Division Problems 198 20 Developing Mental Strategies for Multiplication and Division 203 21 Guided Math Lesson: Division Bakes 22 Reinforcement Activities Multiplication and Division 101

1Lesson The Grapes of Math: An Introduction to Multiplication Math Number Curriculum Expectations • B2.6 represent multiplication of numbers up to 10 × 10 and division up to Teacher 100 ÷ 10, using a variety of tools and drawings, including arrays Look-Fors • B2.7 represent problems involving multiplication and division, including Previous Experience with Concepts: problems that involve groups of one half, one fourth, and one third, using Students have worked tools and drawings with benchmark numbers of 5 and 10 and can skip • B2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related count by 5s and 10s. division facts • B2.1 use the properties of operations, and the relationships between multiplication and division, to solve problems and check calculations Possible Learning Goals • Mentally composes, decomposes, and recomposes quantities represented visually to create equal groups and find the total number of objects, using repeated addition and multiplication • Connects visual representations of quantity to their numerical representations of repeated addition and multiplication • Identifies the math problem and explains what needs to be solved • Mentally composes, decomposes, and recomposes visual representations of quantities into equal groups • Uses repeated addition and/or multiplication to find the total number of objects • Understands and explains how repeated addition is related to multiplication • Explains how to record repeated addition and multiplication in numerical and symbolic forms and understands how the operations are related About the The Grapes of Math allows students to solve math problems that are embedded in riddles. Students solve the problems by studying visual representations of quantities, which they mentally compose, decompose, and recompose in a variety of ways. Students can use both repeated addition and multiplication to solve the riddles and then record their thinking using equations. Throughout the problem solving, connections can be made between the visual and symbolic representations of numbers so students can also learn to decompose quantities when they are in their numerical form. 102 Number and Financial Literacy

About the Lesson There are 16 riddles in The Grapes of Math—too many for one lesson or session. Therefore, the riddles have been divided among this lesson and its related Math Talks. Four of the riddles are used to introduce the concept of multiplication, and the three accompanying Math Talks use four riddles each to further develop and reinforce the concept. Later in the unit, Lesson 12 revisits some of riddles in The Grapes of Math to introduce the concept of division as repeated subtraction and to connect subtraction to the concept of multiplication. You may decide to further break down the Math Talks and complete only one or two riddles per session. Math Talks with the remaining riddles can be interspersed elsewhere within the unit, as well as used throughout the year so students can spiral back to the concepts of multiplication and division. Assessment Opportunities Observations: Note each student’s ability to: – C ompose, decompose, and recompose visual representations of quantities into equal groups – Use repeated addition and multiplication to find total quantities – E xplain how repeated addition and multiplication are related Materials: Minds On (10 minutes) Written by Greg Tang • Show the cover of the book and read the title and names of the author and Illustrated by Harry Briggs Text Type: Fiction: illustrator. Narrative–Rhyming Verse Riddles: “Fish School,” • Ask what the book may be about. Ask what grapes might have to do with “The Grapes of Math,” “Snail Parade,” and “Ant math. Attack!”; chart paper; marker; class number line • Ask what they think “Mind-Stretching Math Riddles” means. Time: 50 minutes • Draw students’ attention to the illustrations of the grapes. Have them turn and talk to a partner about what math they see. • Discuss how they might count the grapes (e.g., skip counting; counting the top grapes and doubling them to find the total number of grapes). • Ask what addition equations they can make (e.g., top green grapes plus bottom green grapes, 6 + 6 = 12; green and purple grapes on the top, 6 + 9 = 15; all green grapes plus all purple grapes, 12 + 18 = 30). • Ask how they can create a subtraction equation and what it would mean (e.g., how many more purple grapes there are than green grapes, 18 – 12 = 6). Multiplication and Division 103

Working On It (30 minutes) Fish School • R ead the title and have students study the illustration. Ask why the riddle might be called “Fish School.” Ask what a school of fish is. • R ead the first four lines. Ask what the author means by stating that a fish is “smart as smart can be.” • R ead the fifth and sixth lines. Ask students what they are supposed to find. Have them turn and talk to a partner about how they may solve the problem. • D iscuss students’ strategies and the various ways they decomposed the group of fish to find the solution. • R ead the last two lines. If students did not use this strategy, have them use the clue to figure out the problem. Draw a quick sketch of the fish and their configuration on chart paper. Ask how many equal groups there are and how many fish are in each group. Ask how students could record their solution using repeated addition (e.g., 4 + 4 + 4 + 4 = 16). Ask how this would look on the number line and have a student demonstrate the equal jumps, starting from zero, on the class number line. • A sk students if they remember an operation that deals with equal ‘groups of’ objects. Write 4 × 4 = 16. Explain that you are showing the problem using multiplication and that the multiplication sign means ‘groups of.’ Read the problem as “four equal groups of four fish.” Explain that multiplication is a faster way of adding the same number over and over again. The Grapes of Math • R ead the title and the first four lines. Ask why the title suits the riddle. Ask why grapes will soon be raisins. • R ead the fifth and sixth lines. Have students work with a partner to solve the problem. • D iscuss students’ strategies and why they decomposed or recomposed the grapes in their chosen manner. Ask how they could represent the total with an equation. • R ead the last two lines. If students did not solve the problem in this way, have them turn and talk with a partner and use the clues to solve the problem. Discuss what they found and how they can record the solution using repeated addition (10 + 10 + 10 + 10 + 10 = 50). Have a student demonstrate what this would look like on the class number line. • A sk students how they could record the solution using multiplication (5 × 10 = 50). Ask what the multiplication sign means and have students read the equation accordingly, using the words ‘groups of.’ Connect the equation to the visual representation. Snail Parade • R ead the first three lines of the text. Ask what a glade is and have students identify what words or illustrations helped them to make this assumption. Ask what they know about snails and why these animals live in a shell. 104 Number and Financial Literacy

• R ead the fourth line and have students discuss a solution with a different partner. Discuss their strategies as a class. • R ead the last two lines and ask what strategy the clue hints at using. Have students solve the problem using the clue and discuss why visualizing what isn’t there is helpful. Ask how they can record what they visualized using numbers (e.g., 5 + 5 + 5 + 5 + 5 = 25; equal jumps of 5 on a number line). • A sk students how they can use multiplication to represent this problem (5 × 5 = 25). Ask how both equations represent the organization of the snails. Ask how they can finish solving the problem (e.g., subtract the three missing snails, 25 − 3 = 22). • E xplain that the arrangement of the snails in columns and rows is known as an ‘array.’ Explicitly point out which are the columns and which are the rows. Ant Attack! • R ead the first two lines of the text. Ask students if they have ever seen ants, either indoors or outside. Ask why they are often considered a pest. Ask if students have seen ants travel in long rows. Explain that the leader leaves a scent for the others to follow so they have a trail back to their nest. • R ead the third and fourth lines. Ask students how they can use what they learned in the last riddle to help them solve this riddle. • A fter discussing their solutions, read the last two lines. Ask what the clue means and have students explain the strategy. Ask how this is like the strategy used in the previous riddle. (e.g., Both use an array.) • A sk how the solution can be represented using numbers (e.g., 4 + 4 + 4 + 4 = 16 or 4 × 4 = 16, and then 16 + 3 = 19). Ask students whether they think they can make an array out of 19 ants and why they think so. Consolidation (10 minutes) • Discuss how the illustrations helped students find the sums of the items in each problem. Ask how seeing visual representations of the quantities was helpful. • Use an example to connect a numerical representation to a pictorial representation. • Building Social-Emotional Learning Skills: Critical and Creative Thinking: Ask students which strategy they found most interesting. Discuss how we can learn from sharing each other’s ideas and building upon them. Explain that mathematicians often discuss what they have discovered with other mathematicians to see if they can build on each other’s ideas or help each other clarify a concept. Tell students that, with practice, they will find using the strategies easier. Encourage them to look at problems from different perspectives, since this can often lead to discovering creative solutions. Multiplication and Division 105

Materials: Math Talk 1: Riddles: “One Hump or Math Focus: Making equal groups and connecting repeated addition to Two?,” “Sweet Cherries,” multiplication “Doggone It!,” and “Large Pizza to Go!”; Let’s Talk chart paper; marker; class number line Select the prompts that best meet the needs of your students. Teaching Tip • W e are going to read some more riddles from The Grapes of Math. What did you Integrate the math learn from the last time we read this book? How do the clues help you solve the talk moves (see math riddles? page 8) throughout Math Talks to One Hump or Two? maximize student participation and • Listen as I read the first four lines of this riddle. What do you think H2O is? How active listening. do the text and illustration help you to know this? What do you know about camels and their environment? Why do you think they have humps? • Read the fifth and sixth lines. Turn and talk to your partner about how you could count the humps. • What solutions did you find? Listen as I read the rest of the riddle. What is the clue guiding you to do? Where do you see the groups of five? Why are 5s easier to work with than other numbers? How does it help with making equal groups? • How could you represent the number of humps with an equation? (5 + 5 + 5 + 5 + 5 = 25) What operation does this represent? (repeated addition) What other operation can we use to show the number of humps and what would an equation look like? (e.g., multiplication, 5 × 5 = 25) How are both equations represented in the illustration? Sweet Cherries • Read the first two lines. What does the author mean in this rhyme? What are crabby apples referring to? • Read the third and fourth lines. Turn and talk to your partner about how to solve this problem. Try to figure out more than one strategy. What did you find? • Read the last two lines. How is the author suggesting solving the problem? • What equal groups could you make and how can you make them? (e.g., Move cherries around so there are five in each bunch, and then add 5 + 5 + 5 + 5 + 5 + 5 = 30) Show us how you moved the cherries around. How can you show this addition on the number line? • How can this be represented with multiplication and what does the equation mean? (6 × 5 = 30, six groups of five cherries equals thirty cherries) • Did anyone find another way to make equal groups? (e.g., We made three groups of 10 by combining bunches and added 10 + 10 + 10 = 30) How is this represented by multiplication? Why does it make sense that you have half as many bunches when there are 10 in a bunch rather than 5 in a bunch? 106 Number and Financial Literacy

Doggone It! • Read the first four lines. What do you think prairie dogs are? They are rodents that live on the grassy plains of Canada and they live in holes that they make by burrowing into the ground. What do you think burrowing means? Why is the author saying the prairie dogs don’t hunt or bark? • Read the fifth line. Turn and talk to a different partner about how you might solve this problem. What did you find? How did some of the previous riddles help you think of a solution? • Read the last line. What strategy do you think the author wants you to use? What equal groups can you make and how can you represent them using two different equations? (e.g., 9 + 9 + 9 = 27 and then 27 − 4 = 23, or 3 × 9 = 27 and then 27 − 4 = 23) Show where these numbers are in the illustration. How are repeated addition and multiplication equations showing the same thing? Large Pizza to Go! • Read the first two lines. How does the author use the rhyming pattern to make these lines interesting? What other toppings could you have on a pizza? Turn and talk to a partner about how to find out the number of mushrooms. Try to think of two different ways. What did you find? • Read the last four lines. What does the author’s hint mean? How could you count the mushrooms on half of the pizza? (e.g., skip count by 3s) How can you show the skip counting on the number line? How could you solve this using repeated addition? How are skip counting and repeated addition the same? What do you get when you double the mushrooms on half of the pizza? • How could you count the number of mushrooms on only 2 pieces of pizzas to find the total? (e.g., There are 6 mushrooms on one fourth of the pizza, so 6 + 6 + 6 + 6 = 24 or 4 × 6 = 24; you can count 6 mushrooms on one fourth and double 6 to get 12 mushrooms on one half, and then double 12 to get 24 mushrooms on both halves.) Materials: Math Talk 2: Riddles: “Know Dice,” Math Focus: Using visual representations to compose equal groups and represent “Strawberry Seeds,” them with repeated addition and multiplication “Win-doze,” and “It’s a Breeze!”; chart paper; Let’s Talk marker; class number line Select the prompts that best meet the needs of your students. Know Dice • Read the first four lines. What do you think the author means by all of these phrases? These phrases describe different combinations of playing cards. Which continued on next page Multiplication and Division 107

phrases are unfamiliar to you? (e.g., ‘Box cars’ means a pair of 6s and ‘snake eyes’ represents a pair of 1s.) • T urn and talk to your partner about how you can count the number of dots on the dice. Think of two different ways. What did you find? • R ead the last two lines of the riddle. What clue is the author giving? What would this look like in the illustration? • H ow can you represent this using an equation? (e.g., 10 + 10 + 10 + 10 = 40) What does this look like if we use multiplication and what does it mean? (4 × 10 = 40, four groups of 10 equals 40) • Is there a way to use equal groups of 6 and some extra to figure out this total? Where are the groups of 6? What equations can you use? (e.g., 6 + 6 + 6 + 6 + 6 + 6 = 36 or 6 × 6 = 36, and then 36 + 4 = 40) Strawberry Seeds • Read the first four lines. How do strawberries grow, according to the poem? Where are the seeds on a strawberry? How is this different from other fruits? • Read the fifth and sixth lines. Turn and talk to a different partner about how you could solve this problem. Find more than one way. What did you find? • R ead the last two lines. What is the author suggesting you do to solve this problem? Try this strategy with your partner. • How can you represent the groups of 9s using numbers? (e.g., 9 + 9 + 9 = 27 or 3 × 9 = 27) • W hy does the sum of the first and last rows equal the sum of the second and second-last rows? Let’s look on the number line at 2 and 7 from the first and last rows of seeds. Now let’s move to 3 and 6, which are in the second and second-last row. What did you notice about the numbers? (e.g., 2 on the left increases by 1 while 7 on the right decreases by 1, so their sums remain the same.) Let’s see whether this works for other numbers. Let’s try 1 and 9 and then 2 and 8. What did you find? (e.g., Both sums equal 10 because as the number on the left increases by 1, the number on the right decreases by 1.) Win-doze • Read the first two lines of the riddle. What does the author mean by counting windows instead of sheep? Have you ever had trouble going to sleep? What did you do? • Read the third and fourth lines. Turn and talk to your partner and find two ways to solve this problem. What did you find? How did the strategies used in the other riddles help you? • Read the last two lines. What is the author suggesting that you do to solve the problem? How can you represent this using numbers? (e.g., 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35 or 5 × 7 = 35, and then subtract the 7 dark windows, 35 − 7 = 28) Where can you see these equations in the illustrations? It’s a Breeze! • Read the first four lines. What kind of fan is the poem describing? How does the fan work? Have you ever made a fan? Does it keep you cool? 108 Number and Financial Literacy

• R ead the fifth and sixth lines. Turn and talk to your partner about one or two strategies that you could use. What did you find? How can you represent the solutions with numbers? • Read the last two lines. What strategy is the author suggesting that you do? How would you represent this strategy using numbers? (5 + 5 + 5 = 15 or 3 × 5 = 15) The author suggested that you use this strategy instead of seeing groups of 3. If you had solved it by looking at groups of 3, what would this look like in the illustration? What does this solution look like in numbers? (3 + 3 + 3 + 3 + 3 = 15 or 5 × 3 = 15) Look at 3 × 5 and 5 × 3. How is the meaning different and why are the products the same? Materials: Math Talk 3: Riddles: “Scallop Math Focus: Using visual representations to compose equal groups and represent Surprise!,” \"Flying them with repeated addition and multiplication Seeds,” \"It’s a Jungle Out There!,” and “For Let’s Talk the Birds”; chart paper; marker; class number Select the prompts that best meet the needs of your students. line Scallop Surprise! • Read the first four lines. What are scallops and where do they live? (e.g., They are animals related to clams and oysters and live in a shell in the ocean. They can swim fast by opening and closing their shell. The part of the scallop that people eat is the muscle that opens and closes the shell.) Why does the author say “I’d rather see them on my plate?” Why is he talking about scallops swimming in butter sauce? • R ead the fifth and sixth lines. Turn and talk to your partner about how you could find out how many scallops are in the illustration. Try to find more than one way. What did you find? How can you represent your strategies with numbers? • Read the last two lines. What could we do according to the author’s clue? What pattern do you see? How can we represent this pattern using numbers? (11 + 11 + 11 = 33) What equation would show multiplication and what would it mean? (3 × 11 = 33) • C an you solve this problem in another way by finding equal groups? (e.g., You can make 3 squares with 9 scallops in each, and then add in the 3 groups of 2.) How could we show this in numbers? (e.g., 9 + 9 + 9 = 27 and then 27 + 2 + 2 + 2, or (3 × 9) + (3 × 2)) Flying Seeds • R ead the first four lines. Have you ever eaten watermelon before? How do your experiences relate to what the author is saying? What might the author mean when he says, “A watermelon hits the spot.” continued on next page Multiplication and Division 109

• Read the fifth line. Turn and talk to a different partner about how you could find the number of seeds in more than one way. What did you find? How can you represent your strategies using numbers? • R ead the last three lines. What does the author’s clue mean? What does he mean by ‘thrice?’ Try solving the problem according to the clue. (e.g., making equal groups of 11) What does this look like using numbers? (11 + 11 + 11 = 33 or 3 × 11 = 33) How are the two equations related? • Is there another way to solve this using equal groups? (e.g., (3 × 5) + (3 × 6)) Where are these equal groups in the illustration? Which way seems easier for you? It’s a Jungle Out There! • R ead the first four lines. What dangers do bugs face? Who are their enemies? What does the author mean by the phrase, “a sticky tongue right on your back?” What do you think about bugs? • Read the fifth and sixth lines. Turn and talk to a partner and find two different ways to find the total number of beetles. What did you find? How can you represent your strategies using numbers? • R ead the last two lines. How is the author suggesting that you solve this problem? How is this problem similar to other ones in this book? Where are the equal groups? How can you represent the solution using numbers? (e.g., 6 + 6 + 6 + 6 + 6 + 6 = 36 or 6 × 6 = 36, and then 36 – 6 = 30) How are the two equations represented in the illustration? • W hat pattern is there in the first six rows? (e.g., they increase by 1 every time) Let’s look back at “Strawberry Seeds.” Where do you see the same pattern? (e.g., Each row of seeds has 1 more than the previous row.) How can we apply the strategy that we learned in “Strawberry Seeds” to adding up how many bugs are in the first six rows? (e.g., Add the first and sixth rows, the second and fifth rows, and the third and fourth rows.) What did you find? (e.g., Each combination adds to 7, so the total for the first six rows is 7 + 7 + 7 = 21 or 3 × 7 = 21.) For the Birds • Read the first four lines of the riddle. What time of year might this picture be showing? What makes you think that? What does the author mean by the word ‘clan’? What dangers could there be laying eggs on a beach? • Read the fifth and sixth lines. Turn and talk to your partner and find two different ways to find the number of eggs. Before you begin, how many eggs do you estimate there are? Would there be more than 10? Less than 50? • What did you find? How can you represent your strategies using numbers? • Read the last two lines. What clue is the author giving and what does it mean? How can we represent this using numbers? (e.g., 9 equal groups of 4: 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 36 or 9 × 4 = 36) What would the repeated addition look like on the number line? • Is there another way to solve this problem by making equal groups? (e.g., each row has 12 eggs: 12 + 12 + 12 = 36 or 3 × 12 = 36) What would the repeated addition look like on the number line? How is it different from repeated addition using 4s? 110 Number and Financial Literacy

to2 4Lessons Representing Multiplication as Equal Groups Math Number Curriculum Expectations • B2. use knowledge of numbers and operations to solve mathematical Teacher problems encountered in everyday life Look-Fors • B 2.6 represent multiplication of numbers up to 10 × 10 and division up to Previous Experience with Concepts: 100 ÷ 10, using a variety of tools and drawings, including arrays In grade two, students represented and • B 2.7 represent problems involving multiplication and division, including explained multiplication as the combining of problems that involve groups of one half, one fourth, and one third, using equal groups. tools and drawings • B2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to solve problems • B2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related division facts • B 2.1 use the properties of operations, and the relationships between multiplication and division, to solve problems and check calculations Algebra • C 4. apply the process of mathematical modelling to represent, analyse, make predictions, and provide insight into real-life situations Possible Learning Goals • Investigates multiplication as the combining of equal groups using concrete materials, pictures, numbers, and/or words, and makes connections among the representations • Explains their strategies, such as skip counting, repeated addition, and multiplication, and explains how they are connected • Applies concepts to solve problems and selects appropriate tools and strategies • Creates equal groups using concrete materials • Selects an appropriate strategy to total the number of objects (e.g., skip counting, repeated addition, multiplication) • Uses equations to describe their strategies and links them to their concrete representations • Represents multiplicative situations involving equal groups in various ways and explains how they are related • Explains or shows how repeated addition, skip counting, and multiplication are connected • Communicates thinking using appropriate mathematical language • Connects the multiplication symbol (×) to its meaning of ‘groups of’ Multiplication and Division 111

PMraotcheesmseast:ical About the Problem solving, reflecting, It is important for students to initially investigate multiplication by selecting tools and physically creating equal groups using concrete objects and drawings. To rsetrparteesgeinetsi,ncgo, nnecting, find the total number of items, they can start by using familiar strategies communicating, that involve additive thinking, such as skip counting or repeated addition, and proving reasoning and then build on these concepts to bridge to multiplication. “The shift to multiplicative thinking is a substantive challenge for Math Vocabulary: children. Learning to multiply requires that children begin to move away mrmeequpuulelttaiaippttlielyoi,dcnag,atrpidoorduonipd,tsiuocontf, (×), from one-to-one correspondence used in addition and subtraction, to a many-to-one correspondence” (Lawson, 2015, p. 116). In order for students to do this, they must be able to unitize, which is to simultaneously see a group of objects as both one unit and a collection of individual units. For example, students need to understand that a dime is one coin and at the same time represents 10 cents. Although grade three students solve multiplication problems, they do so mostly using additive thinking reflected through their chosen strategies, such as creating equal groups and then using repeated addition or skip counting to find the total. It takes many experiences and time before they are actually thinking multiplicatively in the junior grades and beyond. As students are introduced to multiplication, the term ‘groups of’ can be introduced as a way of describing the models and connecting them to what students know. For example, 4 + 4 + 4 = 12 can be linked to the multiplication equation 3 × 4 = 12, meaning 3 groups of 4 objects, which equals 12 objects. It is critical that students see both the addition and multiplication within the models and make connections between the concrete and symbolic representations. About the Lessons In Lesson 2, students build on the concepts of multiplication introduced in Lesson 1 and further explore skip counting, repeated addition, and the combining of equal groups, using concrete materials. In Lesson 3, students create various representations of multiplication situations and then work as a class to make connections among them. In Lesson 4, students apply what they have learned about multiplication to solve a related problem. As students solve real-life problems, take the opportunity to reinforce the mathematical modelling process and its four components: • Understand the Problem • Analyse the Situation • Create a Model • Analyse and Assess the Model Use an anchor chart to highlight how students move back and forth among the components as they develop and refine a model to represent 112 Number and Financial Literacy

the math. For example, students may first consider the context of a problem (Analyse the Situation) before being presented with the actual problem (Understand the Problem). As they test out their model, they may discover that it is not effective. They may need to revisit the problem (Understand the Problem) or reconsider the context (Analyse the Situation) in order to gather new information and select more appropriate strategies and tools for their model. In Lesson 2, there are suggestions on when and how to reinforce the model, although these will need to be adjusted to be responsive to how your students are progressing through the process. Multiplication and Division 113

2Lesson Skip Counting, Repeated Addition, and Multiplication Materials: Minds On (20 minutes) chart paper, variety • Ask students to brainstorm things in everyday life that come in groups of of concrete materials, BLM 21: KWC Chart two or pairs (e.g., shoes). Record students’ answers on chart paper. Time: 60 minutes • Select one of the students’ examples and draw a representation on the chart paper (e.g., draw five pairs of shoes). • Ask students how many groups they see. (5) Draw a circle around each pair. Have students turn and talk to their partner about how they could figure out how many shoes there are in total. • Discuss their strategies and record them. – S kip Counting: As a group, skip count by 2s and record the pattern, 2, 4, 6, 8, 10. Also show the pattern on the number line, highlighting the regular increments of 2. – R epeated Addition: Ask students how they could use numbers or words to represent the problem using addition (e.g., five groups of two equals ten; 2 + 2 + 2 + 2 + 2 = 10). Ask how they can show this addition on the number line and how it relates to skip counting. – M ultiplication: Ask how five groups of two can be represented with numbers (5 × 2 = 10). • Draw attention to the repeated-addition equation. Ask students how they know from looking at it how many groups there are and how many shoes are in each group. Ask how they can also see this in the drawing. • Connect the repeated-addition equation to the multiplication equation and reinforce that the multiplication symbol means ‘groups of.’ Discuss how multiplication can be a faster way of finding the total number of shoes. • Select another item that comes in pairs. Have partners represent seven pairs of the item using counters. Ask how they could represent their model using skip counting, repeated addition, and multiplication. Together, discuss their responses and continue to make connections among all the representations. Working On It (20 minutes) NOTE: Change the context of the problem so it is meaningful to your students. • Ask students if they have ever seen tennis balls on the bottoms of chair legs and what their purpose may be. Explain that tennis balls are sometimes placed on the bottoms of chair legs to reduce the noise the chairs make when they are moved. (Analyse the Situation) 114 Number and Financial Literacy

• Write the following math problem and read it together. Have students explain what the problem means. (Understand the Problem) – H ow many tennis balls would we need to put on six chairs? • Show the class BLM 21: KWC Chart. Explain what each category means (What do you know? What do you want to know? What are the conditions not explicitly stated in the problem?). Complete the KWC chart for the math problem above. (Analyse the Situation) K W C What Do What Do I Want to What Are the I Know? Conditions? Know? the number of There are 4 legs chairs (6) How many per chair. tennis balls are needed? • Students work in pairs to solve the problem using the concrete materials and/or tools of their choice. Have them transfer their concrete representations onto chart paper using drawings. They can record matching equations to show their thinking. (Create a Model) Differentiation • Change the difficulty level of the problem to suit the needs of your students. Some students may need smaller numbers, yet they can still reason through the problem in context. • For students who need a greater challenge, ask them how many tennis balls are needed if they add two more chairs. Assessment Opportunities Observations: As students solve the problem, note the following: – D o students refer to the KWC chart to ensure they understand the question? – What strategies do students use to solve the problem? (e.g., counting, skip counting, repeated addition) – C an students create equations for their concrete representations and explain what they mean and how they link to the models? Conversations: Use some of the following prompts to further probe student understanding. • W hy are you creating groups of four? What do they represent in the problem? • How could you skip count to find the total number of tennis balls? (by 4s) What tool could help you count? (e.g., hundreds chart, number line) • W hat do these numbers represent in your concrete model? What operation are you using? Why? • How might you check your answer using another strategy? Multiplication and Division 115

Consolidation (20 minutes) • Strategically select two or three solutions, based on the thinking and strategies students used in their solutions (e.g., skip counting, repeated addition, grouping, and/or multiplying). As students explain their work, highlight the relationships and connections among the strategies and thinking. • Make it explicit that representing six groups of four with counters, connecting cubes, or drawings and then skip counting all represent the same thinking, even though they used different tools to represent the same idea. • Highlight how they can move away from counting concrete materials individually by reorganizing the materials into equal groups. This is important so students begin to unitize. • Ask students which strategies they found most meaningful and helpful. Discuss how they may solve a similar problem differently now that they have shared each other’s ideas. (Analyse and Assess the Model) • Reinforce that the multiplication symbol means ‘groups of’ and that it represents a faster way of finding totals than adding. • Connect the written equations to the concrete models and then connect the equations of repeated addition and multiplication to each other. • Have students explain in their own words what multiplication is. • Building Social-Emotional Learning Skills: Healthy Relationship Skills: Students may see other students using math materials they are not yet comfortable using. Partner students who are comfortable with different materials and tools and have them demonstrate and explain to each other how they use the materials or tools. Reinforce the idea that students can learn from each other. Reassure students that with practice, they will get better at using different math materials and tools. Further Practice • Independent Reflection in Math Journals: Ask students to use a real-life example to show how four groups of two is the same as 2 + 2 + 2 + 2. 116 Number and Financial Literacy

3Lesson Representing Multiplication in Various Ways Materials: Minds On (15 minutes) 12–15 books, BLM 22: • Show students three piles of books, with three books in each pile. Puzzle Pieces, chart • Ask how students can determine how many books there are altogether by paper, markers, concrete materials from around the way in which they are organized. As students share their responses, the room record their thinking on a puzzle piece on BLM 22. (e.g., I used repeated addition, 3 + 3 + 3 = 9; I multiplied 3 × 3 = 9; I see 3 equal groups of 3 and I Time: 50–55 minutes know that is 9, I skip counted by 3s.) Discuss how all of the pieces of the puzzle represent the same quantity in different ways. • Change the arrangement of books so there are two piles with six books in each pile. • Ask students how they can determine the total number of books by looking at how they are organized. As students share their responses, record their thinking on another copy of BLM 22 (e.g., I used repeated addition, 6 + 6 = 12; I multiplied 2 × 6 = 12; I see 2 equal groups of 6 and I know that is 12). • Tell students that they will now create their own model with items from around the room, and make a puzzle that shows various representations of the model. Working On It (15 minutes) • Working in pairs, students wander the classroom to collect items to make a model that involves equal groups (e.g., 4 gloves, 3 pairs of shoes, 2 new/full packages of crayons). • Using BLM 22, students record descriptions of their model in the puzzle pieces. Differentiation • Students may require support in deciding which items to use or the number of items to use to be manageable. Assessment Opportunities Observations: Notice if students are able to make connections among the representations of addition, equal grouping, and multiplication. • Do students recognize that the size of the group represents the number you repeatedly add? • Do students recognize that the number of groups represents the number of times you repeatedly add the number in each group? Multiplication and Division 117

Consolidation (20–25 minutes) • Have students participate in a gallery walk. One student in each pair ‘stays’ with the pair’s work to explain what they found, while the other student ‘strays’ to see their peers’ work. Encourage the creators to ask visitors how the information on the puzzle pieces represents the model. Students can then switch roles, with group members returning to their work while the other group member ‘strays.’ • Gather back as a class to discuss what students saw. Using one student example, explicitly highlight how the representations are all related and how the numbers in one representation are reflected in another representation. Further Practice • R epresentation Puzzles: Cut up four or five of the students’ puzzles from the lesson and randomly give one puzzle piece to each student. Have students go for a walk around the room to find others with matching representations. Encourage students to prove to each other that their puzzle pieces match. Extra Challenge: Include some representations that may have the same numbers but in the reverse order (e.g., ‘2 equal groups of 6 is 12’ and ‘6 equal group of 2 is 12’). This can lead to a discussion about the meaning of ‘groups of’ in a word problem and how the context of the problem can affect the order of the numbers. 118 Number and Financial Literacy

4Lesson Solving Problems Involving Multiplication Materials: Minds On (15 minutes) BLM 21: KWC Chart, • Review the connections between skip counting, repeated addition, and chart paper, markers, concrete materials multiplication. Have students explain in their own words and using concrete (e.g., counters) examples how the strategies and operations are related. Time: 55 minutes • Write the following question and read it with students. You can change the context so it has more meaning for your students. – T hree new storage racks are installed at the park. One rack holds 9 bikes, one rack holds 5 skateboards, and the third rack holds 6 tricycles. Which rack holds more wheels? • Together, complete BLM 21: KWC Chart for the question. K W C What Do What Do I Want to What Are the I Know? Conditions? Know? 9 bikes, 5 skateboards, There are 2 wheels on 6 tricycles Which rack holds more a bicycle, 4 wheels on a wheels? skateboard, and 3 wheels on a tricycle. Working On It (20 minutes) • Provide pairs of students with chart paper and markers, and have them select their own concrete materials to solve the problem. Encourage them to transfer their thinking onto chart paper after using the concrete materials. Differentiation • Change the numbers in the problem to meet the needs of your students. This still allows students to reason through the problem and uncover the big ideas, but at a level appropriate for them. • Limit the problem to only two comparisons (e.g., only bikes and skateboards). • For students who need more of a challenge, add a second part to the problem: There are 53 wheels on one rack. There is at least 1 skateboard, 1 tricycle, and 1 bike. What combinations of skateboards, tricycles, and bikes could there be on that rack? Multiplication and Division 119

Assessment Opportunities Observations and Conversations: • O bserve how students begin to solve the problem. Do they use the KWC chart to help identify what they know and what they need to find out? If they are uncertain, draw attention to the chart and pose the following prompts: – What are you finding out? What do you know? (e.g., There are 9 bikes.) – What is important to know about the 9 bikes? (e.g., How many wheels there are altogether.) – How could you represent the number of wheels on 1 bike? On 9 bikes? – L ook at the chart and represent what else you know. • O bserve how students represent the problem and calculate the total number of wheels for each piece of sports equipment. Do they count by 1s, skip count, or mentally add the various amounts? Do they record these methods on their chart paper? If they are not showing their work, pose some of the following prompts: – How many wheels were on the bikes? How did you count them? (e.g., 2, 4, 6, 8…) – H ow could you show that on your chart paper? How can you use a drawing and numbers to help you? Consolidation (20 minutes) • Strategically select two or three pairs of students to share their solutions based on the strategies they used (e.g., counting by 1s, skip counting, repeated addition, grouping, or multiplying). Have students demonstrate with concrete materials, and draw attention to their actions with the materials and how these are represented on chart paper using drawings and numbers. • As students present, ask what equations represent what students did. Record them on the paper (e.g., repeated addition, multiplication). Make connections among them so students can see how they relate to each other and to the visual model. • Highlight how the same thinking can be evident in several representations. For example, students may represent the wheels on the bike using different materials, such as counters, connecting cubes, or a drawing, but if they all make groups of two and then count by 2s to find the total, it is the same thinking and not a different strategy. • Highlight the relationship between the strategies (e.g., counting by 1s, skip counting, repeated addition, multiplication) by showing how each one looks on the number line. 120 Number and Financial Literacy

5 7Lessons to Using the Array to Represent Multiplication Math Number Curriculum Expectations • B2.6 represent multiplication of numbers up to 10 × 10 and division up to Previous Experience 100 ÷ 10, using a variety of tools and drawings, including arrays with Concepts: Students have solved • B2.7 represent problems involving multiplication and division, including problems that involve equal grouping and problems that involve groups of one half, one fourth, and one third, using addition. Students tools and drawings understand how to use concrete objects or • B2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to drawings to support problem solving. It is also solve problems helpful if students have investigated number • B2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related patterns in the hundreds chart and can count by division facts 2s, 3s, 5s, and 10s. • B2.1 use the properties of operations, and the relationships between Teacher Look-Fors multiplication and division, to solve problems and check calculations Possible Learning Goals • Ucosmesmauntaatrirvaeyatnodindviestsrtiigbautteivreepperoatpeedrtaidesdiotfiomnu, mltiuplltiicpaltiicoantion, and the • Applies understanding of arrays to solve multiplication problems • Recognizes arrays as a rectangular arrangement of objects in rows and columns; explains or shows what rows and columns are • Describes arrays in a variety of ways (e.g., as an equation) using appropriate vocabulary (e.g., groups, array) • Connects previous knowledge of equal groups and repeated addition to multiplication • Accurately records the equations that represent relationships within an array and can describe the array using terms such as ‘groups of’ and ‘times’ PMraotcheesmseast:ical About the Problem solving, An array is a rectangular arrangement of items, organized in rows and representing, columns. It can be effectively used to investigate multiplication and division and proving, reasoning and the relationship between the two operations. As students interpret and create arrays, they develop an understanding of the commutative property communicating, of multiplication (e.g., 3 × 5 = 5 × 3). sscteorlanetnceetgicniteginstgo,orlesflaencdting, continued on next page Multiplication and Division 121

Mgarmereraoqrputauuehlyptaa,ispVttrielooooidccnwfa,a,satebt,iduoqucdrnulonia,atl-iruplaoymrrng:oor,nduosuun,pcdts,, It is unnecessary for students to learn the term ‘commutative’—they may ecpqoruomapmteiroutnyta, (toivpetional) refer to the commuted equation as a ‘turn-around equation’ and commuted expressions as ‘turn-around facts.’ It is important for students to realize that the order of the numbers in a multiplication expression doesn’t affect the answer, but can affect the meaning of a problem. For example, if there are 3 nests with 4 birds in each nest, the multiplication equation would be 3 × 4 = 12, meaning 3 groups of 4 equals 12 (the number of birds altogether). There are not 4 groups of 3 birds, even though 4 × 3 would still provide a correct answer. When learning about arrays, students need to know what rows and columns are and identify the equal groups within them. They can describe the embedded operations as repeated addition or subtraction, or as multiplication and division, and make connections between them. As students decompose arrays in different ways, they are also investigating the distributive property. For example, a 4 × 6 array can be decomposed into two 2 × 6 arrays, and the total can be calculated as (2 × 6) + (2 × 6). About the Lessons In Lesson 5, students create mini-multiplication posters for various multiplication equations. In Lesson 6, students study arrays and create equations to describe the operations they use to find out the total number of items. The link between repeated addition and multiplication is explicitly highlighted. In Lesson 7, students create their own arrays and identify the repeated addition and multiplication they see within them. They also decompose and compose arrays in various ways, thereby investigating the distributive property. 122 Number and Financial Literacy

5Lesson Introducing the Array Previous Experience About the Lesson with Concepts: Students have identified In Minds On, students use concrete materials to create arrays that fractions in set models. represent multiplication equations. They’ll describe the arrays in terms of repeated addition and ‘groups of.’ In Working On It, students create mini- multiplication posters for various multiplication equations. Students will practise solving and representing multiplication problems using repeated addition, arrays, ‘groups of’ pictures and statements, and multiplication equations as they create the posters. Materials: Minds On (10–15 minutes) loose parts, chart • Have students sit in a circle on the carpet. Make an array of 20 objects paper, large-sized paper divided into (4 rows of 5). Ask students to describe what they see without giving you the 4 sections, BLM 23: total number of objects (e.g., lines or rows of 5, columns of 4). As they Representing explain, introduce or reinforce the terms ‘columns’ and ‘rows.’ Explain that Multiplication (one per this is called an ‘array’ and that an array has equal groups arranged in rows student, copied and columns. Add these terms to the Math Word Wall. large sized) • Ask students how many objects they see. Have students explain their Time: 45–60 minutes strategies for answering this question. (e.g., I added 5 + 5 + 5 + 5 = 20; I saw 5 groups of 4 so I counted by 5s four times; I saw 2 groups of 5, which is 10 and then another 2 groups of 5, so I doubled 10 to get 20.) • Highlight the 2 groups of 5. Ask what fraction of the whole array it represents. Ask what fraction 1 group of 5 represents. • After students explain their strategies, ask, “What is one repeated addition sentence we can use?” (5 + 5 + 5 + 5 = 20) Record the equation on the board. Ask, “How many groups of 5 is that?” (4) Record 4 groups of 5 = 20. Then ask, “How can we say that in a multiplication equation?” (4 × 5 = 20) Record the equation. Repeat the process for 5 rows of 4. Ask students how the recordings are the same and different (e.g., same answer, the numbers are the same but in a different order, different meanings). • Provide loose parts to students. Record ‘4 × 6’ on the board. Ask students to represent and solve the problem by creating an array with the loose parts. • Discuss and record the repeated-addition equations, ‘groups of’ sentences, and multiplication equations for the array. Multiplication and Division 123

• Co-construct an anchor chart that shows an array along with repeated addition and multiplication equations. Circle the items in columns and rows to show both ways that an array can be read. (See the example below.) Reinforce the concept that an array can represent two different meanings (e.g., 3 × 2 and 2 × 3) but the answer (total number of items) will be the same. You may wish to introduce the term ‘product’ as the answer in a multiplication problem. An array can show repeated addition and multiplication 2+2+2=6 3 groups of 2 = 6 3x2=6 3+3=6 2 groups of 3 = 6 2x3=6 Working On It (20–30 minutes) • Pose the following problem to your students: Close your eyes and visualize 3 tables in the classroom. On each table, there are 4 books. Can you ‘see’ them? How many books are there altogether? Open your eyes. • Have students turn and talk with a partner about what picture they could draw to solve the problem. • Ask a few students to share their ideas. Show students the large paper divided into 4 sections. Tell them that you are going to create a multiplication poster together. Choose two students to come up and draw their pictures (3 groups of 4 and the array) in two separate sections of the poster. If no student describes drawing one of these pictures, ask, “How could we draw this picture in groups?” “How could we draw an array for this picture?” • Label the 2 pictures ‘3 groups of 4’ and ‘Array.’ • Ask students what the repeated-addition equation for this picture would be. Choose a student to answer and write the equation on the poster. • Ask students what the multiplication equation would be: “We said that we have 3 groups of 4, so what would the multiplication equation be?” (3 × 4) Draw a rectangle in the middle of the poster and choose a student to write 3 × 4 = 12 in the rectangle. • Ask students to look at the array. Say, “We can see 3 groups of 4. (Circle the 3 groups of 4 in one colour.) What other equation does this show us? (4 groups of 3 or 4 × 3) Let’s call that the ‘turn-around equation’.” Label the last square “Turn-Around Equation.” Choose a student to record 4 × 3 = 12 in the square. Say, “Now we’ve made a multiplication poster for 3 × 4 = 12.” 124 Number and Financial Literacy

Turn-Around Equation Repeated Addition 4 x 3 = 12 4 + 4 + 4 = 12 3 x 4 = 12 3 groups of 4 Array • Tell students that they are going to be making their own mini-multiplication posters. Write 4 equations on the board that you would like them to create mini- posters for (e.g., 3 x 6 = 18, 7 x 3 = 21, 4 x 5 = 20, 2 x 8 = 16). You may decide to include multiplication facts for 2, 5, and 10 so students have visual references to help them gain automaticity over time. • Provide students with BLM 23: Representing Multiplication and review each step for making the mini-posters. Answer any questions students may have and provide time to complete the task. Differentiation • Work with students who are having difficulty seeing multiplication as ‘groups of’ in a small-group guided math lesson. Tell them contextual story problems and have them use concrete materials to solve them. Have them record both the ‘groups of’ statement and the multiplication equation. Assessment Opportunities Conversations: In one-on-one interviews, ask a student to explain his/her mini-posters. Possible questions include: What does 3 × 6 mean? (3 groups of 6) Which picture shows 3 groups of 6? How many groups are there? How many in each group? How is repeated addition like multiplication? What are the two equations that an array shows? How does the array show 3 × 6? Which are the columns? Rows? When do we use multiplication to solve a problem? (e.g., when there are equal groups) Listen for students to: – Use multiplicative language (‘groups of,’ ‘times’) – Relate repeated addition to multiplication – Understand how to read an array – Explain what 3 × 6 means using a ‘groups of’ statement Multiplication and Division 125

Materials: Consolidation (15 minutes) paper and/or • Have students bring their mini-posters to the carpet. Tell students that you are construction paper, tape, materials with going to give them a clue about a multiplication equation and they have to which to draw or make figure out what the equation is. Explain that the clue will relate to the work arrays (e.g., pencil they did on their mini-posters. crayons, stickers, paper shapes and glue) • Say, “The array for this equation has 3 rows of squares and there are 6 squares in each row. Tell a partner what you think the multiplication equation is.” • After students have solved your problem, invite a few students to give a clue to one of the equations on their mini-posters for the other students to solve. Once students understand the process, have them form an inside/outside circle as follows: – A sk half the class to form a circle. Tell them to bring their mini-posters with them so that they can create clues for the multiplication equations. Ask them to face outwards. – T ell the other half of the class to bring their mini-posters and stand facing a person in the inner circle. This creates the outside circle. – T ell students that they will have a few minutes with their partner to take turns giving clues and guessing the equations. – A fter a few minutes, give a signal to stop sharing. Ask the students in the outside circle to move three persons to the left and to begin sharing with their new partner. Continue for as long as you feel the sharing is productive and engaging. Further Practice • Provide students with experience drawing arrays and identifying both multiplication equations for the array. Give students ‘books’ made of paper or construction paper stapled together and the following directions for making a book of arrays (you can record the steps on chart paper for students to refer to): How to Make a Book of Arrays 1. Draw or make a different array on each page. For example: OOO OOO 126 2. R ecord the 2 multiplication equations for each array. 2 × 3 = 6 and 3×2=6 3. M ake a small paper flap. It must be big enough to cover your equations. 4. C over your equations with the paper flap so we can’t see them. Tape the flap at the top so it can be lifted to show the equations. 5. M ake a title page for your book such as, “My Amazing Arrays” by . 6. E xchange books with a friend and solve their amazing arrays. Number and Financial Literacy

• Place the books of arrays at a math centre for students to solve. Students may solve orally with a partner or record the arrays in their Math Journals and write the corresponding multiplication equations. Once you’ve introduced division, you can ask students to say or record two division equations as well. Differentiation • If you have students who are still having difficulty making the connection between repeated addition and multiplication, have them record a repeated- addition equation for each of the arrays in their book. Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: Inspire your students with inspirational quotations. Consider using them as morning messages. Your students may even be inspired to create their own messages. Here are a few to get started: 1. It does not matter how slowly you go so long as you do not stop. (Confucius) 2. T he greatest mistake you can make in life is to be continually fearing you will make a mistake. (Elbert Hubbard) 3. It’s not that I’m so smart, it’s just that I stay with problems longer. (Albert Einstein) 4. W hether you think you can or think you can’t, you are right. (unknown) 5. M istakes are proof that you are trying. (unknown) 6. A ll things are difficult before they are easy. (Thomas Fuller) Multiplication and Division 127

6Lesson Working with the Array Materials: Minds On (20 minutes) “Cityscape Arrays” (pages • Show the “Cityscape Arrays” pages to students. Have students turn and talk 8–9 in the Number and Financial Literacy big to a partner about what they see and where what is shown in this illustration book and little books), may take place. Ask whether they have been in a large city before and what sticky notes, chart paper, they noticed on the streets. markers Time: 60 minutes • Ask what all of the buildings have in common. (e.g., They all have windows; they all have doors, etc.) Draw attention to the windows. Ask how they are arranged and how they can give clues about the buildings’ features, such as the number of storeys or the number of rooms they have. • Ask how the windows are arranged. Review what an array is (a rectangular arrangement of objects in rows and columns). • Point to the red-brown building with 12 windows (above the Fun Fashions sign). Have students turn and talk to a partner to think of different ways they could find out how many windows there are. Record students’ thinking as they share. Below is a sample conversation: 128 Teacher: How many windows do you see on this building? How do you know the number? Student: There are 12. I counted by 1s, like this. Teacher: [Record 1, 2, 3, …] What would this counting look like on the number line? Is there another way to count the windows? Student: We can count by 2s, like this: 2, 4, 6, 8, 10, 12. Teacher: [Record the number sequence.] How could you show this skip counting on the number line? What are you adding each time? What would this look like as an equation and what would it mean? Student: We are adding 2 six times, so 2 + 2 + 2 + 2 + 2 + 2 = 12, so six groups of two windows. Teacher: [Record the repeated-addition equation.] How would this look on the number line? How could we represent this with multiplication? Student: 6 × 2 = 12, which means six groups of two windows. Teacher: [Record the multiplication equation.] Is there another strategy for finding the number of windows? Student: We can add 4 + 4 + 4, which equals 12. Teacher: How would this look on the number line? How is this different than counting by 1s or 2s? What are we actually adding with the equation 4 + 4 + 4 = 12? Student: 3 groups of 4 windows. Teacher: How could we represent this using a multiplication equation? Student: 3 × 4 = 12 since ‘groups of’ means multiplication. Number and Financial Literacy

Teacher: [Record the multiplication equation.] Is there another way to find the total number of windows? Student: You can total the 2 groups of 4 windows and then add in the last group of 4 windows. Teacher: [Record the equation (2 × 4) + (1 × 4).] So, you can decompose the windows and add them in parts. • Discuss how skip counting, repeated addition, and multiplication are all represented in the array. Working On It (20 minutes) • Students work in pairs or groups of three. Provide each group with a Number and Financial Literacy little book, sticky notes, and chart paper to record their thinking. • Have pairs look for other arrays on the “Cityscape Arrays” pages. Students draw the arrays on their chart paper and then create addition and multiplication equations to reflect how they found the total number of windows in each array. • Students can also record their equations on sticky notes and then place the notes on the windows in the book so they remember which equations match what part of the illustration. Differentiation • For students who may have difficulty recording their work and remembering which equations correspond to the windows in the picture, give them a photocopy of the big book page. They can glue their equations in the middle of their page and annotate around it. • For students who need more of a challenge, encourage them to find smaller arrays within the larger arrays. Assessment Opportunities Observations: Observe how students verbally describe the arrays. Can they differentiate rows from columns? Do they see the equal groups within each row and within each column? Can they differentiate between repeatedly adding columns and repeatedly adding rows? (e.g., for a 3 × 4 array, 4 + 4 + 4 = 12 for rows and 3 + 3 + 3 + 3 = 12 for columns) Conversations: If students are having difficulty linking repeated addition to rows and columns, pose some of the following prompts: – If you are adding up the rows, what are you repeatedly adding? Show me one row. Let’s cover up the other rows with this piece of paper. How many items are you starting with? Let’s uncover the second row. How will I move my paper to do that? (e.g., Move it down.) Look at the second row. How many am I adding to the first row? How many do you think we will be adding when we move down to the third row? How can we record this addition in an equation? continued on next page Multiplication and Division 129

Materials: – L et’s try the same thing for the columns. How will I cover up all but the chart paper, markers first column? How will I move my paper to uncover each column? Predict how we will be adding before I uncover the next column. How can we Teaching Tip record this adding? How is this different from adding the rows? Integrate the math Consolidation (20 minutes) talk moves (see page 8) throughout • Students meet with another group to compare the arrays that they found. Math Talks to • Meet as a class. Have each group share one array and the strategies they used maximize student participation and to find the number of windows. In each case, ask what equation depicts the active listening. strategy that they used. Record the various equations on sticky notes and place them on the matching array in the Number and Financial Literacy big book. Ensure that students understand what each equation represents in relation to the array. • Highlight repeated addition in one array, noting how it differs depending on whether you are looking at the equal groups in the columns or in the rows (e.g., 3 + 3 + 3 + 3 + 3 = 15 and 5 + 5 + 5 = 15). • Highlight an example that shows decomposing the windows into smaller sections and then adding them together. For example, 8 groups of 3 windows (in the grey building) could be decomposed and composed as (4 × 3) + (4 × 3). • Ask how corresponding multiplication equations (e.g., 3 × 8 = 24 and 8 × 3 = 24) relate to the repeated-addition equations and to the array. Ask students if it matters whether they multiply rows by columns or columns by rows if they want to find out how many items there are in total. Tell students they will investigate this more in the Math Talk. Math Talk: Math Focus: Investigating the commutative property of multiplication Let’s Talk Select the prompts that best meet the needs of your students. • L ook at this equation. Do you think it is true or not? Turn and talk to your partner. Record 3 × 5 = 5 × 3. • W hat do you think? (e.g., We think it is true because both 3 × 5 and 5 × 3 equal 15.) Put up your thumb if you agree. Does anyone have an idea to add? • L isten to this problem. “There are 3 plates with 5 cookies on each plate. How many cookies are there?” Do you think this situation is the same as “There are 5 plates with 3 cookies on each plate. How many cookies are there?” Discuss this with a different partner. • W hat do you think? (e.g., We think the two situations look different when you represent them.) Let’s expand on that idea. What do the 3 and 5 represent in the first problem compared to the second problem? (e.g., In the first 130 Number and Financial Literacy

problem, there are 3 plates with 5 cookies on each place, and in the second problem there are 3 cookies on each of 5 plates.) I am going to draw a picture of each situation. • D o the situations affect the total number of cookies? (e.g., No, there are 15 cookies in each case.) • H ow could we represent 3 × 5 and 5 × 3 in arrays? (e.g., 3 rows with 5 counters in each and 5 rows with 3 counters each) How many total items are in each array? Let’s draw both arrays. Draw the two arrays on two separate pieces of paper and label them. Visualize rotating the one array so it is on its side. How does it compare to the other array now? (e.g., The two arrays look the same because the rows are now the columns, and the columns are now the rows.) Let’s actually rotate it to see if this is true. • This is known as the commutative property of multiplication. You don’t need to know the name, but you do need to understand that we can multiply numbers in any order and the product, or total, will be the same. We will record that on our strategies chart. • H ow can this help us learn our multiplication facts? For example, if you know that 2 × 6 = 12, what else do you know? (6 × 2 = 12) Multiplication and Division 131

7Lesson Using the Array to Solve Multiplication Problems Materials: Minds On (10 minutes) “Cityscape Arrays” (pages • Show students “Cityscape Arrays” in the big book. Review what an array is 8–9 in the Number and Financial Literacy big and have students identify the columns and rows. book), chart paper, markers, square tiles, • Using one array of windows as an example, review how students can find the tape Time: 55 minutes total number of windows in a variety of ways (e.g., skip counting, repeated addition, multiplication, decomposing and adding the parts) and the related equations to describe them. Ensure that students can make connections between the numbers and what they represent in the array. Working On It (20 minutes) • Students work in pairs. Give each pair 24 square tiles, chart paper, and markers. • Explain to students that they are going to create different arrays using 24 ‘windows’ (square tiles). Each array must include all 24 tiles, without any tiles left over. • Once students have designed an array, they draw their configuration on chart paper and label it with repeated addition and/or multiplication equations. Differentiation • Adjust the complexity of the task by changing the number of tiles students use to construct their arrays. For example, students who need more support can work with a total of 12 tiles. Other students may be comfortable working with 36 or 48 tiles. Multiple arrays can be made in all these cases. • For students who need more of a challenge after having completed the problem, give them a number of tiles that can produce only a few arrays (e.g., a prime number such as 19). Have students figure out and explain why some numbers can make more arrays than others. Assessment Opportunities Observations: Observe how students start the problem. Do they have a plan or do they use trial and error? Once they have made an array, do they start from scratch for the next one or do they adjust the existing array by moving some of the tiles? Conversations: If students are starting over for each array, pose some of the following prompts: – D escribe this array. How many rows are there and how many tiles in each? (e.g., 6 × 4) 132 Number and Financial Literacy

– Visualize how you could move some tiles to turn this array into a different rectangle. How could you make it different? (e.g., change the number of columns) – T ry moving the tiles in one of the columns into the rows. Does that make a rectangle? – V isualize what tiles you could move this time to make an array. How many more tiles do you need to finish off the rows so the arrangement looks like an array? Which tiles could you move to complete the rectangle? Consolidation (25 minutes) • Discuss the arrays that the students found. As a class, make an organized list of all of the arrays. Highlight the fact that an array with only 1 row or 1 column is still an array. Number of Number of Array Total Number of Rows Columns Windows 1 1 × 24 24 2 24 2 × 12 24 3 12 3×8 24 4 8 4×6 24 6 6 6×4 24 8 4 8×3 24 12 3 12 × 2 24 24 2 24 × 1 24 1 • Ask students what happens to the number of windows in a row when more rows are added. (e.g., As the number of rows increases, the number of windows in each row decreases.) Have a pair of students demonstrate this by making the first array, manipulating the tiles to create the next array, and then manipulating the tiles over and over to create each successive array in the list. Repeat this with the number of columns. • Ask what doesn’t change in each case. (e.g., The total number of windows is always 24.) • Ask students if they see any other patterns in the list. (e.g., There are two arrays for each pair of numbers.) Have another pair of students create a pair of related arrays (e.g., 3 × 8 and 8 × 3). Ask how they are the same and how they are different. • Have students visualize how they could rotate one of the arrays so it is on top of the other one. Draw a 3 × 8 array and rotate it so it is an 8 × 3 array to verify their thinking. Discuss how it is the same array, except it has been turned so the columns are now rows and the rows are now columns. • Discuss how this list shows all of the combinations of numbers that can be multiplied to equal 24. • Draw attention to the 6 × 4 array. Ask students whether they could decompose the windows into smaller parts and then add the parts together. Record their suggestions using equations (e.g., (3 × 4) + (3 × 4); (6 × 2) + (6 × 2)). Multiplication and Division 133

Materials: • B uilding Social-Emotional Learning Skills: Critical and Creative 10-bar rekenreks (one per student or Thinking: It is important that students see that arrays are evident in their one per class) everyday lives and that items are often arranged in arrays for many different reasons. They have seen how windows in a building can be in arrays. Ask what Teaching Tip else in their environment may be arranged in arrays. Challenge them to look for arrays over the next few days and make note of the examples they find. If you do not have Students can also take pictures of the arrays they find (stacked cans or crates 10-bar rekenreks, of fruit in a grocery store, cubby holes for storage, etc.). You can discuss their virtual 10-bar findings a few days later. Discuss why each set of items might be arranged in rekenreks are an array. available at https:// mathies.ca/ Further Practice learningTools.php. • Independent Problem Solving in Math Journals: Have students solve the Teaching Tip following problem: Integrate the math – W hich building has more windows, a building with five rows of four talk moves (see windows or a building with three rows of six windows in each row? page 8) throughout Justify your thinking in more than one way. Math Talks to maximize student Math Talk: participation and active listening. Math Focus: Investigating the distributive property and doubling as strategies for learning the multiplication facts of 5 About the Rekenreks, or arithmetic racks, are excellent tools for representing repeated addition and multiplication in arrays. Beads on a rekenrek are grouped in 5s (5 red and 5 white). This offers visual support for students while using the rekenrek and reinforces anchors of 5 and 10. The colour grouping of the beads also supports subitizing, or instantly recognizing a quantity without counting. If you or your class have not used a rekenrek before, you can demonstrate its use. Beads are ‘out of play’ when they are on the right side of the rekenrek. This is the starting position. Beads grouped on the left are ‘in play.’ Students should be encouraged to slide or push beads from right to left in groups in a single move, rather than counting individual beads (e.g., 8 is created by subitizing 5 and 3 more, then those 8 beads are pushed from right to left in one slide/ motion). The ‘one push’ rule discourages counting by 1s and encourages subitizing 5 and 10. Before a new problem is posed, reset the beads by moving them back to the right side. 134 Number and Financial Literacy

NOTE: Students can work directly with rekenreks throughout the Math Talk, or you may do the Math Talk as a whole-class lesson while projecting one rekenrek for all students to see. Let’s Talk Select the prompts that best meet the needs of your students. • S how students a rekenrek. What do you know about this math tool? • H ow many beads are in one row? How many beads are on the entire rekenrek? All 100 beads are currently out of play when they are on the right side. When I slide them to the left, they are in play. How can I show 7? How can I show 13? Do I need to slide the beads across one at a time? Why? (e.g., No, I can slide 5 across at a time because they are in groups of 5 by colour.) • How can I show 5 + 5 + 5 using only red beads? How could I express this as multiplication? (3 × 5) What does this mean? (e.g., three rows of five) • H ow could I look at these beads differently to create a different multiplication equation? (e.g., Look down at the five columns with three beads each, so 5 × 3 = 15.) What is the same and what is different about the multiplication equations that match our arrangement of beads? • What shape is our representation of 15? (e.g., a rectangle) Where have we seen a representation of multiplication like this before? (arrays) Which are the rows and which are the columns? So, we can make arrays on the rekenrek. • Let’s reset our beads. What is 4 × 5? How might this look on the rekenrek? What equation represents the organization of the beads? (4 × 5 = 20) • W hat represents half of the beads? (e.g., two rows of 5 beads, which is 10 beads, or 2 × 5 = 10) How can you use this to calculate 4 × 5? (double it) So we can say that 2 × 5 plus 2 × 5 is the same as 4 × 5. • Visualize doubling 4 × 5. What would this look like on the rekenrek and how much would this represent? Turn and talk to your partner. • What did you visualize? (8 rows of 5 beads) Let’s show that on the rekenrek. How can we represent this using multiplication? (8 × 5) How many beads are there and how do you know? (40 beads, because we doubled the 20 beads represented by 4 × 5) How can we record this? (4 × 5 plus 4 × 5 = 8 × 5) continued on next page Multiplication and Division 135

• In each of these cases, we have scaled up the number of beads by doubling. • L et’s imagine 5 × 5 on the rekenrek. How many beads are there and how do you know? (e.g., 25, because there are 5 more than 4 × 5, which we already represented; 25 because I skip counted by 5s) Visualize what double that array would look like. How many beads would there be? (10 rows of 5 so 50 beads) How do you know? (e.g., I doubled 25 to get 50) • How can doubling help us learn our ×5 facts? 136 Number and Financial Literacy

8Lesson Properties of 0 and 1 in Multiplication Math Number Curriculum Expectations • B 2.1 use the properties of operations, and the relationships between Teacher multiplication and division, to solve problems and check calculations Look-Fors • B 2.6 represent multiplication of numbers up to 10 × 10 and division up to Previous Experience with Concepts: 100 ÷ 10, using a variety of tools and drawings, including arrays Students learned about the properties of zero in • B2.7 represent problems involving multiplication and division, including addition and subtraction in grade two. In grade problems that involve groups of one half, one fourth, and one third, using three, they have worked tools and drawings with multiplication and have represented • B 2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to problems with equations. solve problems PMraotcheesmseast:ical rrPeerpaorsbeolseneminngtsinoaglnv,dinpgr, oving, Possible Learning Goals communicating, sscteorlanetnceetgicniteginstgo,orlesflaencdting, • Demonstrates an understanding of the zero and identity properties of multiplication • Explains or shows how and why the properties work, using mathematical language and concrete examples • Uses concrete materials or a diagram to represent the zero and identity properties • Uses several examples to prove that the properties work in every circumstance • Explains or shows why the properties work • Applies knowledge of the properties to answer related multiplication problems About the As students explore multiplication, they discover interesting properties or rules that work in all cases and that further their understanding of the operation. Two such properties are • the zero property—any number multiplied by 0 equals 0; and • the identity property—any number multiplied by 1 equals the original number. In order for students to fully understand these properties, they need to investigate the properties using concrete materials and drawings so they continued on next page Multiplication and Division 137

Math Vocabulary: can prove how and why they work. These experiences also help students zperroopeprrtoyp, egrrtoyu, pidseonft,ity create visual images for these properties, which can be mentally retrieved conjecture, at a later time to solve related problems. example About the Lesson In this lesson, students investigate the zero property as it relates to multiplication. They make conjectures and then try to prove or disprove them, using various examples. In the accompanying Math Talk, students investigate the identity property and create examples to prove why it works in all cases. Materials: Minds On (20 minutes) chart paper, markers, • Show students the following equations, one at a time, and discuss what the various concrete materials answers are and how students can prove them with concrete or oral examples. Encourage students to create story problems for each example to Time: 60 minutes explain their reasoning. 40 + 0 = 357 + 0 = 0 + 652 = • Ask students how they know their answers are true and whether adding zero always results in a sum equal to the other number (i.e., Is it a rule?). • Show the following equations. Have students turn and talk to a partner about what rule they can create about subtracting zero. Challenge them to present a convincing argument to support their thinking. Have them create story problems to support their ideas. 27 – 0 = 328 – 0 = 497 – 0 = • Discuss students’ rules and their reasoning for them. Ask whether they are certain that the rule applies to all cases. • Record the properties for adding and subtracting zero on an anchor chart. Zero Property The sum of zero and any number is the original number (20 + 0 = 20). The difference between any number and zero is the original number (20 – 0 = 20). Working On It (20 minutes) • Students work in pairs. Inform students that the zero property applies to addition and subtraction, and they are going to figure out whether the zero property applies to multiplication also. If the same rule does not apply, they are 138 Number and Financial Literacy

going to investigate whether there is another rule that might apply. Discuss what a ‘conjecture’ is (a prediction based on limited evidence; a possible rule) and what needs to be done to prove that a conjecture is true. Emphasize that it only takes one example that doesn’t work to disprove a conjecture. • Give students the following equations to calculate. Have them think about what might be a rule that applies to all situations when we multiply by 0. Inform them that they will need to create other examples to further support their thinking. When they feel they have developed a conjecture (possible rule), have them record it on a piece of chart paper with some supporting examples (e.g., pictures of their concrete models). They can choose their own concrete materials and tools. 3×0= 18 × 0 = 0×9= Differentiation • For students who have difficulty getting started, pose some of the prompts in the Conversations section of the Assessment Opportunities. • For students who need more of a challenge, have them offer proof and examples using other tools, such as number lines. Assessment Opportunities Observations: Pay attention to whether students can create concrete models to represent the given examples. Can they create and articulate a conjecture for their findings? Are they able to produce other examples to further prove their possible rule? Conversations: If students have difficulty representing the examples, pose some of the following prompts: – Let’s think about the problem 3 × 2. How can you describe this expression using words? (e.g., three groups of two, or three groups with two objects in each group) How can you represent this with counters? (e.g., make three groups with two counters in each group, and count up the number of counters) Look at 3 × 0. How is this the same as 3 × 2? (e.g., both have three groups) So, how can you make three groups? How will you finish the representation? (e.g., put zero in each group) What do you find? – L et’s think about 0 × 3. How is this different from 3 × 0? How will you represent it differently? Consolidation (20 minutes) • Have three or four pairs of students share their conjectures, using either a given example or one of their own examples as proof that their conjectures work every time. Ask whether anyone found an example that did not follow the rule they developed. Multiplication and Division 139

Materials: • Clarify how 5 × 0 and 0 × 5 may result in the same product, but look different concrete materials, chart paper, markers when represented with models. Have students create a story problem for each example. Teaching Tip • Co-create a rule that summarizes the zero property for multiplication. This Integrate the math talk moves (see can be added to the anchor chart about the zero property created in the Minds page 8) throughout On. Include some examples, such as drawings, so students can connect the Math Talks to visual representations to the numbers. maximize student participation and Math Talk: active listening. Math Focus: Investigating the identity property as it relates to multiplication and developing a rule Let’s Talk Select the prompts that best meet the needs of your students. • R eview the information about the zero property for multiplication on the anchor chart developed in the lesson. What is a conjecture, and what do we need to do to prove that a conjecture is true? • W e are going to investigate another rule that works in addition and subtraction and determine whether it works every time. • S how students the following equations. What is the same about all of these equations? (e.g., All are addition and 1 is being added in every case.) 6+1= 1 + 15 = 98 + 1 = • Turn and talk to your partner about whether there is a rule for these types of questions. • What did you find? (e.g., The answer is always 1 more than the other number.) How can you prove this? How can you show this on a number line? Record these ideas and examples on chart paper. • L ook at these equations. What is the same about all of them? (e.g., They are all subtraction and 1 is being subtracted in each case.) 9–1= 22 – 1 = 95 – 1 = • Turn and talk to your partner about whether there is a rule for these types of questions. • W hat did you find? (e.g., The answer is always 1 less than the first number.) How can you prove this? How can you show this on a number line? Record these ideas and examples on chart paper. 140 Number and Financial Literacy

• Let’s investigate whether these rules apply to multiplication. Here are three examples for you to use to think of a possible conjecture. 4×1= 12 × 1 = 1×9= • After you have worked on these examples, record your conjecture on chart paper. Then, think of other examples to further prove your conjecture. Select whatever concrete materials or tools you would like to use, and record your examples on your chart paper. • What did you find? What are your conjectures? (e.g., The answer is always the other number.) What examples did you use to prove this? • W hat is a story problem for each of the three examples that we started with? How do they relate to the concrete representations? • Did anyone find an example that didn’t work? Can we consider this a rule? How can we word the rule? (e.g., If you multiply a number by 1, the answer is the number.) What examples can we put with our definition to prove this? Multiplication and Division 141

and9 10Lessons Applying Strategies to Solve Multiplication Problems Math Number Curriculum Expectations • B2.6 represent multiplication of numbers up to 10 × 10 and division up to Teacher 100 ÷ 10, using a variety of tools and drawings, including arrays Look-Fors • B2.7 represent problems involving multiplication and division, including Previous Experience with Concepts: problems that involve groups of one half, one fourth, and one third, using Students have used tools and drawings several strategies to solve multiplication problems. • B2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to solve problems • B2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related division facts • B2.1 use the properties of operations, and the relationships between multiplication and division, to solve problems and check calculations Algebra • C 4. apply the process of mathematical modelling to represent, analyse, make predictions, and provide insight into real-life situations Possible Learning Goals • Applies learned concepts and uses a variety of strategies to solve multi-step multiplication problems embedded in a real-life context • Explains strategies and thinking using mathematical language • Selects one or more strategies to solve the problem • Selects materials and tools to support and represent their thinking • Makes connections between some strategies, such as how repeated addition and multiplication are related • Tries a different strategy if their first attempt doesn’t work as they had anticipated • Works through the multiple steps of the problem, applying the findings in the first step of the problem to solve the second step 142 Number and Financial Literacy

PMraotcheesmseast:ical About the Problem solving, representing, By presenting a realistic problem that requires students to apply what they communicating, have learned about multiplication, you can assess whether they have actually sscrteeoralanestnceoetgnicniitenginsgtgoa,onrledsflapenrcodtviningg, , grasped the concepts. If some students are still uncertain or continue to have misconceptions, it is beneficial to have small-group guided lessons. Math Vocabulary: You can further challenge students who have a firm grasp of the concepts by mgruolutippsli,ceaqtiuoant,ioenqsual providing more problems that may arise in everyday life. For example, students could figure out how many pizzas need to be ordered for a pizza day based on the average number of pieces that each person will consume. Providing realistic problems also reinforces the idea that math plays an integral part in students’ lives and is worth learning. About the Lessons In these two lessons, students apply what they have learned about multiplication to solve problems that require critical thinking and decision making based on real-life contexts. The problems are multi-step and somewhat similar, allowing students to apply what they learn in one step to the next step, and possibly trying to use a strategy that has been discussed but that they have not previously attempted. As students solve real-life problems, it is a good opportunity to reinforce the mathematical modelling process and its four components: • Understand the Problem • Analyse the Situation • Create a Model • Analyse and Assess the Model Use an anchor chart to highlight how students move back and forth among the four components as they develop and refine a model to represent the math. For example, if students are finding their model is not working, they may need to revisit the problem (Understand the Problem) or reconsider some of the parameters surrounding the problem (Analyse the Situation) in order to gather more information and select more-appropriate strategies and tools. There are some suggestions about when and how to reinforce the model in Lesson 9, although these will need to be adjusted so they are responsive to the way your students progress through the process. Multiplication and Division 143

9Lesson Solving for a Better Buy Materials: Minds On (15 minutes) grocery store flyers, • Discuss students’ shopping trips to grocery or department stores. Ask how chart paper, pencils, 2 paper bags (for items are sold (e.g., cereal by the box, produce by weight or quantity, toys demonstration) and books individually). Ask what products come in packages or are sold in multiples (e.g., hot dog buns, bags of potatoes, paper towels, erasers). Ask why Time: 60 minutes it is more convenient to sell some items in packages. (Analyse the Situation) • Show students a grocery store flyer. Have them identify which items are sold individually and which are sold in more than one unit. Have them turn and talk to a partner about which is the better buy (e.g., buying three items for $5 or paying $2 for each item). Ask why they think stores price items in this way. Discuss when it can be a good idea to buy more than one item and when it is not. (Analyse the Situation) Working On It (25 minutes) • Record the following problem on the board or on chart paper, and read it aloud to the students. Ensure that they understand the context, but do not offer input on how to solve it. (Understand the Problem) Pencils are sold in packages of 3 (small) or packages of 5 (large). A package of 3 pencils costs $2. A package of 5 pencils costs $3. We need 24 pencils for our class. 1. How many packages do we need if we buy the small ones? The large ones? 2. Which is the better buy? • Students work in pairs and can select any materials or tools they want to use to solve the problem. Have students record their thinking on chart paper. Encourage them to show all of their work so another person could figure out how the problem was solved just by looking at the chart. (Create a Model) Differentiation • If the multi-step problem is too overwhelming for some students, assign them only the first part in which they find the number of packages needed. They can follow their fellow students’ reasoning and their solutions for the second part of the problem. • For students who need more of a challenge, offer a third purchase option: 10 pencils for $10. You can also change the number of pencils needed to 30 or 40. Assessment Opportunities Observations: Pay attention to how students start to tackle the problem. Can they identify and explain the two parts of the problem in their own words? Do they know which part to solve first? Can they select appropriate materials or tools to represent their thinking? 144 Number and Financial Literacy

Conversations: If students are having difficulty starting, they may need to revisit the problem (Understand the Problem) or consider the conditions surrounding the problem (Analyse the Situation). Pose some of the following prompts: – T ell me in your own words what this problem is about. Visualize the packages of pencils. Let’s use these two paper bags and pencils to show what the different packages of pencils look like. – H ow many parts are there to this problem? How do you know? What do you need to find in the first part? Explain it in your own words. – We have made a concrete example of each of the packages. How can you draw or use tools to represent these different packages? Try using these ideas to find out how many packages would be needed. Consolidation (20 minutes) • Strategically select solutions and strategies that reflect different types of thinking. Keep in mind, for example, that representing 4 groups of 5 with individual counters or connecting cubes and then counting by 1s reflects the same thinking. Different thinking would include counting the groups by 5s, skip counting, or representing the groups as repeated addition. • Have selected groups explain their thinking and strategy for solving the first part of the problem. Annotate any additional information on their chart papers. Name the strategies and thinking that are evident. • Possible strategies and thinking for the first part of the problem include: – S tudents count rhythmically by 1s, emphasizing and tracking every third count on their fingers to count the number of packages, and then repeat this process for the other packages. – S tudents skip count by 3s to 24, tracking the number of counts, and then repeat this for skip counting by 5s. Students need to realize that they can (and may have to) buy more pencils than necessary (e.g., buying 25 rather than exactly 24). – S tudents use repeated addition until they reach 24 or more, first by 3s and then by 5s. They count the number of jumps for each counting series to figure out the number of packages. – S tudents choose numbers to multiply using guess-and-check and then making adjustments; for example, trying 10 packages with 3 pencils each and realizing that will be too much so adjusting. • Have students share and discuss their solutions and strategies for solving the second part of the problem. You may want to select different students. Strategies will most likely be the same as those for the first part, except with different numbers. Discuss which is the better buy and why they think so. • Leave students’ solutions and strategies posted in the classroom for use in the next lesson. • Ask students how they might solve the problem differently after hearing each other’s ideas and strategies. Discuss which strategy made the most sense to them and why. (Analyse and Assess the Model) Multiplication and Division 145

10Lesson Solving More Multiplication Problems Materials: Minds On (15 minutes) concrete materials, • Review the discussion from the previous lesson about how and why stores chart paper, markers sell items in packages. Time: 60 minutes • Review some of the strategies that students used to answer the problem in the previous lesson. Refer to the solutions and strategies on chart paper posted around the room. Working On It (25 minutes) • Write the following problem and then read it over as a class, ensuring everyone understands the context. Sebastian needs stamps to mail out his invitations. He buys 4 books of stamps with 7 stamps in each. He needs to put 3 stamps on each invitation. 1. Will he have enough stamps for 10 invitations? 2. If not, how many more stamps will Sebastian need? How many books of stamps will he need to buy? • Tell students they are going to solve the problem in pairs, but will work with a different partner than in the last lesson. Challenge them to solve the problem in more than one way. Encourage them to try strategies that were discussed in the previous lesson, referring to the posted students’ charts. Differentiation • Students who need more support can solve only the first part of the problem. • To ensure that they are on the right track, before they begin, help students select a possible strategy from the previous day’s solutions, then ask them how they plan to use the strategy. • For students who need more of a challenge, add to the problem: 3. H ow much does Sebastian spend to buy 4 books of stamps if one book costs $3? 4. W hen Sebastian buys his extra stamps, is it better to buy a full book of stamps or to buy individual stamps for $1 each? Explain your reasoning. 146 Number and Financial Literacy