p98-205_Gr3ON-NumberUnit2-mulitply-pass2

Assessment Opportunities Observations: Pay attention to whether students try a different strategy than they used in the previous lesson. Do they verify their work by using another strategy? Are they more intentional in how they approach the problem than in the previous lesson? Conversations: For students who are reluctant to try another strategy or have difficulty choosing one, pose some of the following prompts: – H ow did you solve the problem in the previous lesson? Look at the solutions that we discussed. Where do you see your strategy? Is there another strategy that you understand and that you could use instead, or that you could use to show your thinking a different way? Is there a strategy you don’t understand but would like to try? Let’s go through the steps to see if we can make it clearer. Consolidation (20 minutes) • Strategically select solutions that show more than one way of solving the problem or that use a strategy not discussed in the previous lesson. After each pair of students shares both of the strategies they used, ask the class how they are the same and how they are different. Explicitly make connections between the strategies by showing how one part of the solution is evident in the other one (e.g., connecting skip counting with repeated addition). • For the first part of the problem, students will use various strategies to calculate 4 × 7, such as rhythmical counting, skip counting, repeated addition, or multiplication. The second part can also be seen as being a division problem, which will offer a good transition to studying division in the lessons ahead and to exploring how the two operations are connected. • Possible solutions for the second part of the problem include: – S tudents count rhythmically by 1s, emphasizing every third count and tracking the number of groups on their fingers. – S tudents repeatedly add 3 and keep track of the groups on a number line with their jumps, periodically checking whether they have reached 10 groups. – S tudents may immediately realize that they will need 30 stamps since they know 10 × 3 = 30, so they count back from 30 to 28 to discover that Sebastian needs 2 more stamps. • Tell students that a book of 7 stamps costs $3 and an individual stamp cost $1. Discuss whether it is better for Sebastian to buy another book of 7 stamps, which makes the cost of each stamp cheaper, or to pay more per stamp to get the exact number of stamps he needs. Answers can vary. Some students may think having extra stamps is a waste, while others may think it is better to buy extra at a cheaper price and have them available for future use. Multiplication and Division 147

• Building Social-Emotional Learning Skills: Critical and Creative Thinking: Discuss how many of the decisions that we make in our lives rely on mathematics. Have students brainstorm some ideas (e.g., what we can afford to buy, what size of furniture fits into our homes, how much food we need to make to feed the family, etc.) Discuss how calculations are important, but so are the circumstances of the situation. For example, buying a barbeque that is on sale in the fall may be a good idea to save money, but you still need to have room to store it over the winter. Students need to realize that many factors need to be considered when making wise decisions. 148 Number and Financial Literacy

11Lesson Scaling Up: Ratios 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 • B 2.7 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 represented tools and drawings and solved multiplication problems using a variety • B2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to of strategies. PMraotcheesmseast:ical solve problems rPerporbelseemntsinoglv,ing, • B2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related aracennofdmdlescpmttrriuoanvntgiei,ncggsaiee,tilscneogcn,tinrneegactstoiononglis,ng division facts • B 2.1 use the properties of operations, and the relationships between multiplication and division, to solve problems and check calculations Possible Learning Goal • Scales up numbers by factors of 2, 5, and 10 in order to solve problems • Understands that scaling up by 2 is like multiplying by 2 or doubling • Uses a variety of strategies to scale up values, including multiplication, repeated addition, and using a double number line • Unitizes equal amounts in order to scale up (e.g., repeatedly adds equal groups of 5) • Explains strategy using mathematical language About the Ratios involve thinking in relative rather than absolute terms. The difference is evident in the following problem: One plant grows from 6 cm to 10 cm, while another plant grows from 4 cm to 8 cm. Which plant grew more? From an absolute perspective, it appears that both plants grew the same amount, 4 cm. From a relative perspective, the second plant grew more because it doubled its height, which is a ratio of 1:2. For the first plant to have grown at the same rate, it would have needed to reach a height of 12 cm. As students work with multiplication and division, an important goal is to move them from additive (absolute) to multiplicative (relative) thinking. Giving students realistic situations that involve scaling up (viewing amounts as proportionately larger) allows them to reason continued on next page Multiplication and Division 149

through the process within a meaningful context. For example, if students know what supplies 1 student needs, they can work through how many supplies are needed for 2 or 5 students. In grade three, students work with ratios of 1:2, 1:5, and 1:10, which support their learning of related multiplication facts. One spatial tool that can help students scale up is the double number line, with the first number line showing the original values and the second number line showing the scaled-up values. The following double number line shows a ratio of 1:2, or scaling up by a factor of 2. 0 1 2 3 4 5 6 Math Vocabulary: 0 2 4 6 8 10 12 mgnruuomlutipbpseli,rcealiqtniuoeant,ioenqsu,adl ouble About the Lesson In this lesson, students are presented with a realistic problem that requires them to scale quantities up by factors of 2, 5, and 10. Materials: Minds On (15 minutes) chart paper, concrete • Ask students whether they have put together loot bags for a party or have objects received them. Discuss what items might be inside. Make a list of some of Time: 60 minutes their choices. Ask how many of each item might be in each bag (e.g., 6 gummy worms, 4 balloons, 3 stickers). • As a class, select some items for one loot bag. Ask how students would figure out how many items they would need for 2 loot bags. Working On It (20 minutes) • Students can work independently or in pairs. Each student or pair creates one loot bag by selecting some items and indicating how many of each item will be inside. They can draw their loot bag and materials on chart paper. • Tell them that they are going to figure out how many of each item they will need for 2 people, for 5 people, and for 10 people. Encourage them to try a variety of strategies, which they can record on the chart paper. Differentiation • You may want to limit the number of different items so the task does not become too overwhelming. 150 Number and Financial Literacy

• For students who need more of a challenge, ask how many items they would need for 15 people. Assessment Opportunities Observations: Pay attention to how students solve the problem. • Do they draw the extra bags and then count by 1s to determine how many there are of each item? • Do they use skip counting to scale up their amounts? • Do they use a previous finding to solve one of the other ratios? (e.g., if they know how many they need for 5 people, they just double it for 10 people) • Do they figure out the total using repeated addition or multiplication? Conversations: Pose some of the following prompts if students are counting objects one by one. • How many balloons does each person get? (6) How can you find out how many balloons 10 people get? What would your drawing look like? Is there a way to solve this problem without a drawing? • L et’s start with how many balloons 2 people would get. How did you figure that out? (e.g., I added 6 + 6) What would you do if there were 3 people? (e.g., add another 6) What if there were 4 people? How could knowing the number of balloons 2 people get help you? (e.g., You can double the number of balloons for 2 people, which is 24.) If we doubled 24, how many people would be getting 6 balloons? (8) How do you know? Consolidation (25 minutes) • Have students share their results with another individual or pair. Students need to explain to each other how they solved the problem. • Meet as a class. Selectively choose 2 or 3 student solutions to discuss. Make connections among the solutions (e.g., connect skip counting, repeated addition, and multiplication strategies). • Draw a double number line on chart paper. Discuss how it can be used to scale up numbers by a factor of 5. Ask students how they would extend the number line. Ask how, without extending the number line, they could figure out what the number on the bottom number line would be if the corresponding number on the top number line is 20 (e.g., multiply 20 by 5). 0 1 2 3 4 5 6 0 5 10 15 20 25 30 • Ask students how they could design a double number line for factors of 2 and 10. Co-create the double number lines. Multiplication and Division 151

to12 14Lessons Division as Equal Grouping Math Number Curriculum Expectations • B 2.6 represent multiplication of numbers up to 10 × 10 and division up to 100 ÷ 10, using a variety of tools and drawings, including arrays • B 2.7 represent problems involving multiplication and division, including problems that involve groups of one half, one fourth, and one third, using 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 • 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 PMraotcheesmseast:ical About the rrPeerpaorsbeolseneminngtsinoaglnv,dinpgr, oving, communicating, Multiplication and division are inverse operations, and although students do sscteorlanetnceetgicniteginstgo,orlesflaencdting, not formally investigate this relationship in grade three, they begin to notice it and that it can support them as they learn about division. Alex Lawson explains that students’ “ability to divide may lag behind their ability to multiply, although not necessarily. Children may use their knowledge of multiplication to help them solve division problems” (Lawson, 2015, p. 114). For example, students can apply a multiplication strategy, like knowing how to create equal groups, to division situations, such as sharing a set among a group of people. Lawson points out that few students use repeated subtraction, although some may “count out the total, then remove groups of 4, and repeatedly subtract until they reach zero” (Lawson, 2015, p. 115). It is still important to highlight this strategy to help students recognize that division is the opposite of multiplication, just as repeated subtraction is the opposite of repeated addition. Students benefit from seeing repeated subtraction represented on a number line so they can see how the movement backward matches their action of removing equal groups until zero is reached. When 152 Number and Financial Literacy

Math Vocabulary: solving problems, students may use repeated addition rather than repeated rrmeegeqprpuoueelutaaaippltteeslsid,dchraaaestrdumiiondbnagitt,ri,inaodedcniqvte,iuirosaniol,n, subtraction by continually adding on equal groups to get from zero to the total. As students investigate division, it is also important to expose them to different problem structures, including partitive division and quotative division. It is particularly useful to offer these structures within a meaningful context so students can reason through the problem-solving process based on content. • Partitive Division: The whole and the number of groups are known, and students solve for the number of objects in each group. • Q uotative Division: The whole and the number in each group are known, and students solve for the number of equal groups. About the Lessons In Lesson 12, students revisit the read aloud from Lesson 1, The Grapes of Math, to discover how multiplication and division are related. They are also introduced to repeated subtraction and division equations, as well as the division symbol (÷). In Lesson 13, students solve partitive division problems by directly modelling them. In Lesson 14, students solve quotative division problems by applying a variety of strategies. Multiplication and Division 153

12Lesson The Grapes of Math: Investigating Division Teacher Possible Learning Goals Look-Fors • D ivides wholes into equal groups by decomposing and recomposing, and Previous Experience with Concepts: represents the actions using repeated subtraction and/or division Students have found totals of objects in • Connects the operations of repeated subtraction and division illustrations and • Explains how division is the opposite of multiplication represented the operations of repeated • Identifies words in the text that help them understand the math problem addition and subtraction • D ecomposes and recomposes visual representation of quantities into equal using numbers. groups, and represents actions using repeated subtraction and division • Understands what the division sign means • Understands and explains how repeated addition is related to repeated subtraction • E xplains how to record repeated subtraction and division in numerical and symbolic form, and understands how they are related Assessment Opportunities Observations: Note each student’s ability to: – C ompose, decompose, and recompose visual representations of quantities into equal groups – Divide wholes into equal groups and represent the actions using repeated subtraction and division – Explain how repeated subtraction and division are related Materials: Minds On (15 minutes) Riddles: “Fish School,” • Show the cover of The Grapes of Math. Ask students what they remember “The Grapes of Math,” “Sweet Cherries,” and about the math that was highlighted the last time it was read. Flip through “For the Birds”; counters some of the pages to refresh their memories. Time: 45 minutes • Tell students that they are going to revisit some of the illustrations in the book to learn about another operation. Fish School • Ask students how they had previously solved this problem and what operations they used (e.g., repeated addition and multiplication). Ask what is important to remember when making groups in order to perform these operations. (e.g., There must be an equal number of objects in each group.) Ask how they can make equal groups and how many fish there are in total. 154 Number and Financial Literacy

• Pose the following prompt: “If there are 16 fish in total and there are 4 fish in each group, how many groups can we make?” Have students turn and talk to their partner and solve the problem using counters. • Ask how they divided their counters into equal group of 4 (e.g., put out 16 counters, take out 4 at a time until none are left, and then count the number of groups). Ask how they can represent this action with numbers (16 − 4 − 4 − 4 − 4 = 0). Identify this as repeated subtraction. Ask how it is related to repeated addition. Ask what repeated subtraction would look like on a number line, and have a student demonstrate the movements. • Explain that this can also be represented with division. Introduce the division symbol (÷) and then the equation, 16 ÷ 4 = 4, which means 16 counters divided into equal groups of 4 equals 4 groups. • Look at the fish illustration again and show how multiplication and division ‘undo’ each other: multiplication starts with the parts and finds the whole, and division starts with the whole and a part and finds the other part. Working On It (Whole Group) (20 minutes) NOTE: Select the riddles to focus on according to your students’ needs. The Grapes of Math • Review the solution to this problem students found previously. Record the repeated addition and multiplication equations. • Ask students how many birds could share the grapes if each bird gets 10 grapes of the 50 grapes. Have them visualize each bird taking 10 grapes away, one at a time. Ask how they could represent this action using numbers (50 − 10 − 10 − 10 − 10 − 10 = 0). Ask why there must be none left if it is considered to be an equal or fair share. • Show students how to record this using a division equation, 50 ÷ 10 = 5, meaning 50 grapes divided into equal groups of 10 equals 5 groups. • Ask how the equations relate to repeated addition and multiplication. Sweet Cherries • Ask students how many cherries there are on the page and how they know. Have them work with a partner and use counters to find out how many cherries each person would get if 6 people were sharing. • Once students have finished, discuss how they made the equal groups. Ask how they could record the actions of the problem using repeated subtraction and division. • Ask how many cherries each person would get if there were now 5 people sharing. First ask if each person will get more or less and why they think so. Ask how this scenario can be represented with repeated subtraction and division. • Draw attention to the division equations and ask how they are the same and how they are different (30 ÷ 6 = 5 and 30 ÷ 5 = 6). Multiplication and Division 155

For the Birds • Ask students how they can find the total number of eggs by making equal groups. Record students’ strategies using repeated addition and multiplication. • Ask students how many nests would be needed if there were 6 eggs in each nest. They can solve the problem with their counters. • Record their solutions using repeated subtraction and division. Have student explain how the equations match the actions of solving this problem. Consolidation (10 minutes) • Have students explain what division is in their own words. • Discuss how repeated subtraction and division are related. • Discuss how repeated subtraction is related to repeated addition and how multiplication is related to division. • Co-create an anchor chart that summarizes the discussion. Add the chart to the Math Word Wall. 156 Number and Financial Literacy

13Lesson Directly Modelling Equal-Sharing Problems (Partitive Division) Teacher Possible Learning Goal Look-Fors • S olves problems that involve equally sharing a known number of objects Previous Experience with Concepts: into a specific number of groups, to determine the number of objects in each In grade two, students group have represented and explained division as the • Equally shares a set of objects among a specified number of people sharing of a quantity • Identifies groups as being equal (e.g., each group has the same quantity of equally, through investigation using items) concrete materials and drawings. • Selects and uses appropriate tools (e.g., counters) and models (e.g., array) to help share equally Materials: counters or tiles, chart • Uses an appropriate strategy to share equally (e.g., skip counting by 5s) paper, various quantities • Explains their strategy and why it works of small items in bags or cups, BLM 24: Sharing Minds On (20 minutes) Items Equally Time: 60 minutes NOTE: Select the prompts that best scaffold the learning for your students. • Provide pairs of students with 12 counters or tiles and/or chart paper. Have students solve the following problem with their partner: – T here are 12 cookies and 2 children. How can you share the cookies fairly between the 2 children? The cookies must remain whole and not be broken. • Have students share their strategies for sharing the cookies (e.g., counts out 12 cookies and deals out 1 to each of the 2 people, 1 at a time, until there are no cookies left; deals out the 12 cookies 2 at a time to see if the share is equal and then deals out the remaining cookies, 1 at a time, checking to see if the share is fair; mentally figures out that each person will get 6 cookies and intentionally gives 6 to each child). • Have some students act out each scenario so they can match actions to the descriptions. • Ask what division equation can describe the problem (12 ÷ 2 = 6). Ask what the whole is and what the parts are, and what each number represents in the context of the problem. Multiplication and Division 157

• Ask what would change if the 12 cookies were equally shared among 3 children. Ask students whether each child would get more or less and why they think so. Students can work with their partner to solve the problem. • Have students share their ideas and then have some students act out their solutions. Ask what division equation matches this problem (12 ÷ 3 = 4). • Ask what will change if 4 children share the cookies. Have students solve the problem. Discuss students’ strategies and compare them (e.g., dividing the 12 cookies into 2 groups and then dividing each of the 2 groups into 2 more groups). Ask what division equation matches this problem (12 ÷ 4 = 3), and have students identify the parts and wholes and the meaning of all of the numbers within the context of the problem. • Ask students whether 5 people can equally share the cookies and why they think so. Discuss why it is not an equal share even though all 5 people received 2 cookies. (e.g., There were 2 cookies left over.) Explain that this is known as having a ‘remainder’ of 2 cookies. Ask how they might solve this problem in real life. (e.g., Divide the 2 leftover cookies into 5 pieces each [fifths] and give each person 2 pieces [fifths].) • Discuss what happens when more people share the cookies and why this makes sense. Working On It (20 minutes) • In advance, create collections of various quantities of small items (18 buttons, 35 craft sticks, 40 counters, etc.) in bags or cups. Ensure that there is at least one collection for each pair of students. Place collections at centres so students can share them. • Divide students into groups of four and assign each group to one of the centres. Provide each group with BLM 24: Sharing Items Equally. • Tell students they will work in pairs to discover whether they can equally share collections among 2, 3, 4, 5, and 6 people, with no items left over. They can record their findings on one of the tables on BLM 24, using a different table for each collection. • Once students have shared the items in one collection, they do the same for a different collection at their centre. Encourage students to share at least two collections. • For one collection, have students show their thinking by drawing pictures of how they divided the items. Differentiation • Provide pairs with quantities of items that are most appropriate for their needs. • For students who need the support, fill in the middle column (People Sharing): 2, 3, 4, 5, 6. • For students who need more of a challenge, have them create their own collection using a number of items they choose. 158 Number and Financial Literacy

Assessment Opportunities Observations: Observe how students equally share the objects. • Do they deal out the objects one at a time? Two at a time? • D o they start over every time they share with a different number of people? • D o they apply what they learned in one share to determine whether another share will work? (e.g., Four people can share, so then two will also be able to share.) • D o they apply their understanding of numbers to determine whether there will be a fair share? (e.g., 35 ends in a 5, so five people can share the set.) Conversations: If students seem to be starting over again with each share, pose some of the following prompts: – A mong how many people did you share this set of (e.g., 15) items? (2 people) Did it work? (No) Do you think that 4 people will be able to share? (I don’t know.) – If I share 4 cookies with 2 people, how many does each person get? If 2 more people come along, could the 4 people share the 4 cookies? What would the 2 people have to do? (e.g., Give one of their cookies to a new person.) So, you are dividing the cookies in half again. – If 2 people can’t share two objects, do you think 4 people can? Consolidation (20 minutes) • Selectively choose some of the collections to discuss (e.g., one collection with an even number of items and one collection with an odd number of items; one collection that has double the number of another collection; a collection in which the number of items is a prime number). • Discuss the strategies students used to divide their sets. Make a record of them. • Discuss how the number of items and the number of people (groups) affected the shares. Pose some of the following prompts: – W hat was the most challenging number of people (groups) with which to share items? Why? (e.g., odd number of people; larger number of people) Which numbers was it easier to share with? Why? – W hat was the most challenging number of items to divide? What was the least challenging? Why? (e.g., larger quantities; odd versus even quantities) – W hat strategies did you use to divide the quantities? (e.g., I placed one counter to represent each person and added a counter one at a time to each person until they were all gone.) – H ow did you know you divided using equal sharing? • Review the conditions that need to be in place for a fair share. Multiplication and Division 159

Further Practice • Independent Practice in Math Journals: Have students solve the following problem: – T here are 24 building blocks in a bin. How could you organize them to make it easier for a friend to count? 160 Number and Financial Literacy

14Lesson Directly Modelling Equal Group Problems (Quotative Division) Teacher Possible Learning Goal Look-Fors • Solves problems that involve determining the number of groups that can be Previous Experience with Concepts: made, when the total number of items and the number in each group are In the previous lesson, known students learned about division as equal sharing • Understands they are solving for the number of groups when knowing the and gained the number of items in each group understanding that division involves working • Selects and uses appropriate tools (e.g., counters) and models (e.g., equal with groups that are groups) to solve the problem equal. • Selects and uses an appropriate strategy to solve the problem (e.g., repeated Materials: subtraction to determine the number of groups) large space (e.g., gymnasium, • M akes predictions about the solution (e.g., anticipates how many groups will schoolyard, open space in be created) to reflect on the reasonableness of the answer classroom), concrete materials, chart paper, • Communicates reasoning and mathematical solutions orally and in writing markers Time: 60 minutes Minds On (20 minutes) • Prepare a large space that allows for students to move around and spread out. If weather permits, a large outdoor space would be ideal. Explain to students that they will determine how many groups can be made when a certain number of people must be in each group. • Have 12 students walk around the space. Instruct them to get into groups of 3. Ask the rest of the class how many groups were made. Have the 12 students walk around again. Ask the rest of the class to predict whether there will be more or fewer groups when each group has 4 people in it. Have the 12 students get into groups of 4. Discuss why there are fewer groups. Ask what division equations could describe the two cases that were acted out. Connect the actions to the numbers. • Have the entire class walk around. Call out a number that will not result in an equal division and have students try to group themselves accordingly, with that number in each group. When the grouping doesn’t work out equally, ask students why they can’t always make equal groups. Have the students who couldn’t form a complete group sit down. Ask students what division problem describes the students who are in groups. (e.g., There are 23 students in the class and we made groups with 5 students in each. Three Multiplication and Division 161

students sat down, leaving 20 standing with 5 students in each group, which can be described as 20 ÷ 5 = 4.) Ask why the total number of students does not include the people sitting down. (e.g., They couldn’t be divided equally.) Explain that this is a ‘remainder’ of 3 people. • Repeat this whole-group activity for different numbers of people in each group. When students first get into a group, have them predict whether the number of groups will be more or less than the time before. Record a division equation for each scenario. Working On It (20 minutes) • Students work in pairs to solve the problem below. Change the context to make it meaningful for your students. Read the problem together so that everyone understands it. There are 36 prizes to be shared among a group of students, but they can only be shared if there are no leftover prizes. Can all the prizes be shared in the following situations? If so, how many students can share? Show your work. Record an equation for each situation that works. – 3 prizes per person – 5 prizes per person – 4 prizes per person – 9 prizes per person – 2 prizes per person – 8 prizes per person – 6 prizes per person • Students can select their own concrete materials and tools, and can record their thinking on chart paper. Differentiation • Alter the number of scenarios students solve for, depending on their needs. • For students who need more of a challenge, they can find the maximum number of prizes that could be shared in scenarios that don’t work out equally and create division equations that do not include the remainders (e.g., for 5 prizes per person, 35 of the 36 prizes could be shared equally, 35 ÷ 5 = 7). Assessment Opportunities Observations: Pay attention to how students solve the problem. • Do they make groups with the required number of prizes and then count the groups? • Do they anticipate that some situations will not work? • D o they use their previous findings to help them predict whether a situation will work? 162 Number and Financial Literacy

Conversations: If students start over for each scenario and attempt to solve it without initially thinking about whether it will work, pose some of the following prompts: – You found that 4 prizes per person is an equal sharing of the 36 prizes. Knowing this, what other numbers might work? (e.g., 2 prizes per person) Why? Do you think that more or fewer people would be able to share if each person received 2 prizes? Are there any other numbers that might work, knowing the 4 prizes could be equally shared? – Are there any number of prizes per person that you think cannot be equally shared? Why? Consolidation (20 minutes) • As a class, discuss which numbers of prizes per person could be equally shared and which could not. Ask students whether they could predict which numbers would work. Have them explain their reasoning. (e.g., 36 does not end in a 5 or a 0, so the 36 prizes cannot be equally shared if everyone gets 5 prizes) • Strategically select solutions that offer a variety of strategies and representations. (e.g., concrete materials and drawings that reflect making equal groups, number lines that show repeated addition or subtraction, skip counting, etc.) Connect the thinking in the different representations. • Record the information and division equations in an organized chart. Ask students what patterns they see. (e.g., As people get more prizes, fewer people can share; twice as many people can share when everyone gets 2 prizes rather than 4.) Number of Prizes Number of Prizes: 36 Division Equation Per Person Number of People 36 ÷ 2 = 18 2 Who Can Share 36 ÷ 3 = 12 3 18 36 ÷ 4 = 9 4 12 5 9 — 6 — 36 ÷ 6 = 6 8 6 9 — — 4 36 ÷ 9 = 4 • Discuss what other numbers of prizes it might be possible to share equally, judging from what students found (e.g., 12 prizes per person; 18 prizes per person; 1 prize per person; 36 prizes per person). Highlight how using the commutative property, or turn-around equations, can help students solve this. Multiplication and Division 163

15Lesson Guided Math Lesson: What’s Fair? 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 investigated • B 2.7 represent problems involving multiplication and division, including division as equal sharing using concrete materials problems that involve groups of one half, one fourth, and one third, using and visuals, and have been tools and drawings introduced to recording division using an • B 2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to equation. solve problems Teacher Look-Fors • B2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related PMraotcheesmseast:ical division facts Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, • B 2.1 use the properties of operations, and the relationships between Math Vocabulary: multiplication and division, to solve problems and check calculations dbiyv,idaerr,adyi,vgisrioounp, sd,ivided sharing Possible Learning Goals • Accurately solves division problems • Represents division in a way that clearly shows grouping and an emerging understanding of unitizing • Uses a variety of strategies to solve problems involving division • Explains and represents division as the equal sharing of a quantity • Divides concrete materials into equal groups when dividing • Draws a picture that shows equal groups or arrays • Counts a group as a unit • Says or writes an equation that describes division (e.g., 12 shared by 3 equals 4 or 12 ÷ 3 = 4) • Uses strategies (e.g., trial and error, doubles, multiplication, repeated addition, decomposing) to accurately solve division problems • Orally explains that division involves equal sharing About the Lesson This is an example of a guided math lesson that could be used with the math little book What’s Fair? As students in the small group read through the book with your direction, they will be prompted to solve division problems along with the characters in the story. The problems involve sharing quantities equally. The lesson progresses from modelling division with concrete materials to representing division using pictures, and finally, to the abstract representation of equations. While visual and abstract representations are addressed, it’s most important that students understand that division involves equal sharing and that groups are units that can be counted. 164 Number and Financial Literacy

Materials: Differentiation What’s Fair? little books, • The prompts provided in the lesson are intended to give you ideas about whiteboards or paper and markers, loose parts/ what you may ask and how you can differentiate from group to group. concrete materials Adjust the learning goals to meet the individual needs of each group. Create Time: 20–25 minutes a list of Look-Fors (see the Teacher Look-Fors on page 164 for suggestions), per session and use them to select and formulate prompts to initiate or shape conversations with students. • F or students who need more of a challenge, you can scaffold the learning by having them solve the problems with progressively less support. Assessment Opportunities Observations: • R ecord your observations anecdotally according to the Teacher Look-Fors you have established. • B ased on your observations, determine if students need further opportunities to investigate division using concrete materials and/or visuals. Look for authentic contexts in the classroom that require students to solve a problem involving division. There are often situations that require the sharing of materials or the forming of groups. Take advantage of these opportunities to help students advance their understanding of division. NOTE: Students may be able to solve the division problems and represent them visually by drawing groups or an array but may not yet be able to accurately represent division with an equation. Many more experiences in grades two and three may be necessary for students’ development to progress from concrete to visual to abstract representation. Small-Group Guided Math Lesson • Read the title of the book to the group. Ask, “When do we share in math?” (e.g., when we divide) Prompt if necessary by saying, “What operation involves sharing: addition, subtraction, multiplication, or division?” • Put out 12 loose parts (or another quantity that will divide evenly among the students in your guided group) and say, “How can you share or divide these equally? How many would each of you get?” • Ask one of the students to demonstrate and explain their solution (e.g., she takes the materials and deals them out one at a time to each member of the group, then counts how many each person gets). • Ask, “Were the materials divided equally? How do you know?” (e.g., Each of us has the same number of materials.) • Tell students that the children in this story have trouble sharing materials equally and that there will be some problems in the story for them to solve. Give each student a copy of the math little book. Have paper and markers available for students to record their work. • Read the text aloud together with the students in the small group (or read the text to them). Multiplication and Division 165

Read pages 2 and 3: • Ask, “How many cookies do each of the children have? (e.g., Hassan has 6 and Quinn has 4.) Is that fair? Why or why not? (e.g., No. Hassan has more cookies than Quinn.) How many more cookies does he have? (2 more) How do you know? (e.g., I started at 4 and counted 2 more to get to 6.) Is Hassan’s solution fair? Why or why not? (e.g., Not fair; he’d be able to eat more cookies than Quinn.)” • Ask, “How can they divide the cookies equally?” Tell students to use loose parts to model the problem. Observe their processes as they work and ask questions to determine what strategies they are using to solve the problem. You may want to ask one student to think aloud. • Observe students as they problem solve. Do they start with 10 items and deal them into 2 groups or do they start with 6 and 4 and move one from the 6 over to the 4 pile? What strategies do they use to divide the amount equally? Do they check that both parts are equal by counting? How do they count? How are their materials arranged (i.e., in groups or arrays)? • Ask students to share their answers and explain their solutions. (e.g., Each of them gets 5 cookies; I put 4 and 6 cookies together. That’s 10 cookies. Then I knew that 5 and 5 are 10 so they each get 5 cookies.) Ask, “What is the whole? (10) How many people were sharing? (2) So how many did each get? (5) What 2 parts make up the whole? (5 and 5) How can we use an equation to represent this problem?” (10 shared by 2 equals 5; 10 ÷ 2 = 5) Read pages 4 and 5: • Ask, “Do Hassan and Quinn have the same amount of pencil crayons? (no) How many does Hassan have? How many does Quinn have? How do you know?” • Ask, “How can the students divide the pencil crayons equally? Use the loose parts to model the problem. Then draw a picture to represent your strategy.” Provide students with materials for recording. • Observe students as they problem solve. How do they begin? Do they use the same strategy as they used with the first problem? Do they find the total of 5 and 7 first or make a group of 5 and a group of 7, then move one over from the group of 7 into the group of 5 so that there are 6 and 6. Do they use their doubles facts? Do they use ‘think-addition’? (e.g., I know that 2 × 6 = 12 so I made 2 groups of 6.) Do they form equal groups? • Ask students to share their answers and explain their solutions. Notice the language that students use (shared, divided, made groups, etc.). Expand their use of mathematical terms through modelling and by asking students to use phrases such as ‘divided by.’ Have students explain their pictorial representations. (e.g., I drew 12 circles. Then I circled 2 groups of 6.) Ask, “What does ‘12’ represent? How many pencils does each child get? Why did you circle 2 groups? Which drawing makes it easier to see that there are 2 equal groups without counting all the dots?” Have students notice that an array clearly shows equal groups. Draw an array if none of the students has drawn one. Then ask students to write an equation using the division sign. If 166 Number and Financial Literacy

students are unsure how to do so, model the language ‘12 shared by 2 = 6.’ Then ask students what the division sign means (shared by). Ask them to read the equation using ‘divided by,’ then to write the equation. Model or prompt if necessary. Read pages 6 and 7: • Discuss what Hassan means by ‘put them all in the middle and share.’ Ask students if they think that this is a better or worse solution than dividing the pencil crayons equally. Prompt students to justify their answers, if necessary. Read page 8: • Discuss the problem. Ask, “What is the teacher asking the children to do? (Divide themselves into 5 even groups.) How many children are there? (20) What will the answer represent?” (The number of children in each group.) • Ask students to solve the problem. Tell them that they can use the materials but must also draw a picture and write an equation. Students might ‘accidentally’ read the solution on the next page. That’s okay, remind them that they still have to prove the answer is correct and represent it. • A sk students to share their answers and explain their solutions. Students may misunderstand the meaning of the problem or represent it incorrectly. For example, “I counted out 20 materials. Then I made 4 groups and I counted out items into each group and I had 5 in each group.” Say, “Let’s reread the problem. How many groups did the teacher say to make? (5) Do you have 5 groups? (e.g., No, I have 4 groups.) How can you make 5 equal groups? (e.g., I can take 1 item out of each group and make another group.) Try your strategy and see if it works. How many are in each group now? (4) We’ve discovered something interesting here. What have we learned? (4 groups of 5 is 20 and 5 groups of 4 is 20.) Yes, does it matter how we show it with our materials?” (e.g., Yes, because we have to show what the problem means. Our answer would be wrong.) Have students share their drawings and equations as well. Discuss the advantage of arrays (they’re organized, easy to read, show equal groups, and make it easy to check). Read page 9: • Have students confirm that their solution was correct. Read pages 10 and 11: • Talk about Chanukah and the dreidel game. If you’ve previously explored this Jewish celebration during Social Studies, students may be familiar with the game. If not, you may want to briefly explain it. While a clear understanding of the game is not necessary for solving the problem, the context of playing a game will be useful to students. You may choose to teach students about Chanukah and give them an opportunity to play dreidel at a later time. Discuss how players can spin the dreidel to win chocolate coins (also known as ‘gelt’). Read page 12: • Discuss the problem (e.g., How many chocolate coins does each child get? How many children are sharing the coins? What will the answer tell us?). Multiplication and Division 167

Tell students to solve the problem and represent their solution with a drawing and an equation. Provide students with time to solve the problem. Carefully observe their process and watch for evidence that they understand the problem, divide materials/pictures into equal groups, and can represent their solution with an equation that has the numbers in the correct positions. • Ask students to share their answers and explain their solutions. (e.g., I knew there were 28 chocolate coins and 4 people in a group. So I started by making 4 groups and putting 5 in each group because I know that 4 groups of 5 are 20. Then I saw that I had more left so I gave each person one more. I still had more so I handed out the rest and each person got 7 coins. So I knew that 28 divided by 4 equals 7.) As students share their solutions, have them share their drawings and equations as well. Listen and look for evidence that students understand the concept of division as the equal sharing of a quantity. If you are still unsure, ask them to show and explain what division is. You might ask, “When do we divide? What is a situation that would need division?” Read pages 13 to 15: • Ask students to compare their strategies for solving the problem to how the children in the story solved it. Read page 16: • Discuss how Hassan felt about sharing. Ask students to describe a situation in which they found sharing difficult. • Discuss how sharing relates to division. 168 Number and Financial Literacy

16 17Lessons and Division Using the Array Math Number Curriculum Expectations • B 2.6 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 • B 2.7 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 been tools and drawings introduced to the array and have divided sets of • B 2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to objects into equal groups using concrete materials. solve problems PMraotcheesmseast:ical • B 2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related Problem solving, division facts representing, and proving, reasoning • B2.1 use the properties of operations, and the relationships between communicating, multiplication and division, to solve problems and check calculations sceolnenceticntgintgo,orlesflaencdting, strategies Possible Learning Goals • Investigates arrays as a model for representing division • Uses arrays to solve division problems • Explains what an array is and identifies rows, columns, and equal groups • Interprets and creates arrays using concrete materials and explains what operations are represented in the model • Creates division equations that match the operations represented in the array • Understands division as a way to share a quantity • Applies understanding of arrays to solve division problems About the Students used arrays to represent multiplication involving equal groups. Arrays can also effectively represent division, since they make various ways of equally dividing a set apparent. For example, if students are dividing 24 desks into 4 rows, a 4 × 6 array visually reveals that there are 6 desks in each row. Students can also visually decompose the desks in other ways, such as dividing the 4 × 6 array in half vertically or horizontally to reveal 2 groups of 12, or dividing the set into thirds to reveal 3 groups of 8. In these cases, students are applying their understanding of the distributive property. (Students are expected to understand this property without naming it by its formal term.) continued on next page Multiplication and Division 169

Math Vocabulary: Arrays also highlight the inverse relationship between multiplication and adrirvaidyi,nrgo,wesq, ucaollugmronusp,s division. As students visually recognize both operations within the representation, they can record their findings in multiplication and division equations and make connections among these numerical representations. Understanding the relationship between operations allows students to solve problems in a flexible manner, using whichever one is more appropriate to the context of the problem. It also helps them learn related math facts, such as 4 × 5, 5 × 4, 20 ÷ 5, and 20 ÷ 4. About the Lessons In Lesson 16, students are introduced to the array as a way of representing division. In Lesson 17, students work with arrays to identify different ways to share sets. They record their findings using division equations and explain the meaning of the numerical representations in the context of the problem. 170 Number and Financial Literacy

16Lesson Investigating the Array for Division Materials: About the Lesson In Minds On, students will solve an array using multiplication. Then they’ll be provided with a problem that invites division, and will co-construct the division equation that the problem represents. The class will then co-construct an anchor chart that shows an array and its corresponding multiplication and division equations. The Working On It task involves a series of centres that students visit to solve division story problems as well as to solve and represent arrays. There are 10 story problem cards and 10 arrays. You may choose to have students rotate through some or all of the tasks in partners or groups of three, depending on the number of students you have in your class. It will likely take two or three periods for students to rotate through all of the centres. A reflection sheet is provided for students to reflect on the learning they did in the centres. Digital Slide 12: Arrays, Minds On (20 minutes) chart paper, BLM 25: Division Task Cards (1 • Show students Digital Slide 12: Arrays. Say, “When we were working on copy of each card), BLM 26: Arrays, concrete multiplication, we created multiplication equations to solve these arrays. materials/loose parts, Let’s look at the balls. How many are there? (20 in a 4 × 5 array) What would BLM 27: Reflecting on the multiplication equations be for this? (4 × 5 = 20 and 5 × 4 = 20)” Draw Division Centres, Digital the array using circles to represent the balls. Record both equations on the Slides 13–22: Arrays board beside the array. (optional) • Say, “Yes, we have 20 balls. Now let’s imagine that they are arranged on a Time: 90–125 minutes (over 2–3 bookcase with shelves. Each shelf holds 5 balls. How many shelves would we periods) need to hold all 20 balls?” (4) • Ask a student to explain how he or she solved the problem (e.g., counted the rows). • Then say, “Let’s write a division equation to represent this problem. What could we write?” (20 ÷ 5 = 4) Ask, “What does the ‘20’ represent? (the whole amount, 20 balls) The ‘5’? (the number of balls each shelf holds) The ‘4’? (the number of shelves)” Record the equation beside the array. • Say, “Here’s another problem: There are 20 balls and 4 groups of children that want to play with them. How many balls can we give to each group of children?” (5) • Ask a student to explain how he/she solved the problem (e.g., turn-around equation, counted the columns, divided). Ask, “What division equation represents this problem?” (20 ÷ 4 = 5) Record this equation beside the array. Multiplication and Division 171

Teaching Tip • Say, “Now we have 4 equations for this one array.” • Have students turn and talk with a partner. Ask, “Why do you think we can All of the arrays used below (BLM 26) are write 2 multiplication equations and 2 division equations for an array?” also available on Digital Slides 13–22 • Have students share their ideas. Record a picture of a different array on chart and can be used to conduct a Math Talk paper. Work with the class to co-construct 2 multiplication and 2 division on division, arrays, equations for the array by providing contexts or by asking students to work and the relationship with a partner to create their own contexts. This chart can serve as an between multiplication anchor chart for the following array centres. and division. • Ask students to draw an array that has 3 rows with 5 balls or circles in each Teaching Tip row in their Math Journals and to write 2 multiplication and 2 division You may wish to place equations for the array. the task cards and the arrays in page Working On It (1–3 periods, 20–30 minutes per period) protectors or copy and laminate them for • Use task cards from BLM 25: Division Task Cards as well as arrays from durability. BLM 26: Arrays to set up centres that students will visit in partners. Set the centres up on desks and tables throughout the room and number each centre. Gather the materials needed for the task-card problems and place them with the task cards. Partners may work together to solve the problems but will record their work independently in their Math Journals. Set up as many centres as are appropriate for your class, using the task cards that are most suitable for your students. Students can rotate through the centres in numerical order or you may want them to visit a certain number of centres, moving freely from one to any other that is available. • Explain to students that when they come to a centre with a task card, they should read the card and solve the problem. Materials are provided for direct modelling of the problem (e.g., for making groups or an array). Then students are to record their solutions in their Math Journals by recording the centre number, a picture, and an equation. • Explain that when students visit a centre with an array card, they are to write 4 equations: 2 multiplication and 2 division. Tell students to record the centre number along with a picture of the array, and the equations in their Math Journals. NOTE: Consider holding a mid-point share when students have rotated through a few of the centres. As students share their work, they are exposed to ideas, strategies, and representations that they haven’t yet considered or been ready to try. Listening to and seeing a classmate explain and demonstrate a strategy can often propel a student to take the ‘next step.’ Differentiation • A llow students who are not yet at the abstract level with division to solve and represent the problems in whatever way they can (e.g., drawing and circling the groups, skip counting, repeated addition) but ask questions (e.g., How else could you show 3 + 3 + 3?) and provide prompts to move them forward (e.g., What is the whole? How many people are sharing it? How can you record that?). 172 Number and Financial Literacy

• Challenge students who are thinking multiplicatively to represent the arrays with both multiplication and division equations and/or to create stories to represent the arrays. • If you have students who easily solve division problems by relating them to multiplication and who need a greater challenge, consider doubling or tripling the whole in a task. Using larger numbers will likely lead students to use the commutative (5 × 4 = 4 × 5) and distributive properties [e.g., 9 × 5 = (5 × 5) + (4 × 5)] of multiplication to solve division problems. Just as we encourage students to use ‘think-addition’ to solve subtraction problems, we should encourage students to use multiplication strategies to solve division problems. Assessment Opportunities •O bAssesrtvuadteionntss:work, record your observations of their strategies. Ask them to explain their strategies orally as they work. • Review students’ work in their Math Journals to ensure that they are able to record their solutions and represent their strategies and arrays accurately and effectively. Provide prompts or small-group guided math lessons to support students as necessary. Consolidation (10–15 minutes) • After students have rotated through the centres, meet as a class to discuss the challenges and learning that occurred. • Use the strategy of ‘Mill to Music’ to make the process of sharing more engaging. Play music and tell students to walk around the room without bumping into anyone. Tell them that when the music stops, they must freeze and wait for instructions. Each time you stop the music, tell students to talk to one or two classmates who are nearest to them, and provide one of the following prompts: – Talk about your favourite centre. Tell each other why it was your favourite. – Talk about your greatest challenge. – Tell your partner what you did really well. – Ask your partner a question about the centres. – Tell your partner something that you learned about division. – Tell your partner something that you learned about arrays. – Discuss how multiplication and division are related. – With your partner, solve this problem: I have 12 doughnuts to share with my friend. How many will we each get? – Make up a division problem for 15 divided by 3. Multiplication and Division 173

• After several rounds of Mill to Music sharing, have students complete BLM 27: Reflecting on Division Centres. Have them take out their Math Journals and review their work as they complete the reflection. Differentiation • Have students who are struggling writers reflect orally and scribe their answers on the reflection sheet. 174 Number and Financial Literacy

17Lesson Solving Division Problems with the Array Materials: Minds On (20 minutes) “Bakery Math” (pages • Display “Bakery Math” on pages 10–11 of the big book. Ask students what is 10–11 in the Number and Financial Literacy happening in the scene and where it takes place. Ask whether they have big book and little visited a bakery and what they remember most about their visit. Ask what books), chart paper, happens in a bakery that a customer can see. Ask what happens that a markers, concrete customer does not see. materials (e.g., counters) and tools (e.g., number • Tell students to look at how the baked goods are arranged. Pose the lines), class hundreds chart following story problem: “There are 4 rows of round mini-cakes with 5 cakes in each row. How many cakes are there altogether?” Ask students for the Time: 60 minutes multiplication equation that describes the problem (e.g., 4 × 5 = 20). Record the equation. • Tell students you have another story problem about the cakes. Say, “5 customers come into the bakery and they each buy 4 cakes. There were no cakes left. How many cakes were there?” Ask students how to represent this problem using a multiplication equation. (e.g., 5 × 4 = 20) Record the equation. Ask if it matters which number story or equation they use to solve for how many cakes there are. Why? • Ask students to create a division problem to describe the arrangement of the cakes. (e.g., There are 20 cakes and 4 shelves to put them on. How can we put an equal number of cakes on each shelf?) Ask students to give the division equation for the problem, and record it. (20 ÷ 4 = 5) • Have students imagine that 5 customers come into the bakery. Ask if the customers can equally share the 20 cakes. Ask how the arrangement of the cakes helps them solve the problem. Ask what division equation describes this story and record it. (20 ÷ 5 = 4) • Have students look at the equations you recorded. Ask how the equations are related to the stories and to the arrangement of the cakes. • Ask whether the baker would be able to equally share the cakes among 6 customers. Ask if they can solve this problem just by looking at the arrangement of the cakes. Have them try to visually reorganize the cakes to see if the scenario is possible. • Ask students what numbers of customers can equally share all the cakes (e.g., 20, 10, 2, 1). Ask how they can tell just by looking at the arrangement of the cakes. Ask how they would visually divide the cakes for 2 customers (e.g., 2 shelves or rows of cakes each). • Tell students that they will now work with a partner and figure out how many people can share some of the other baked goods that are shown in the picture. Multiplication and Division 175

Working On It (20 minutes) • Students work in pairs. They can share a copy of the Number and Financial Literacy little book to solve the problem below and then record their findings on chart paper. They can also select their own materials and tools to solve the problem. • Record the problem on chart paper. Read the problem together, ensuring that everyone understands the context and what an equal share involves (e.g., no items left over). Select one of the treats in the picture. Figure out how many of that treat are in the bakery. Then, figure out several ways that you can share all of those treats among some friends. For each way, decide how many friends share and how many treats each friend gets. The shares must be equal, with none of the treats split or left over. Differentiation • For some students, you may limit the openness of the problem by defining how many friends will share. • Students who need more of a challenge can work with remainders. Have them provide a solution that describes the maximum number each person could get and how to deal with the remainders. For example, they might suggest 16 of 18 treats be equally shared among 4 people (16 ÷ 4 = 4), then each of the 2 remaining treats be divided in half and one half given to each of the 4 people. Assessment Opportunities Observations: Observe how students are deciding whether a certain number of people can equally share a treat. • Do they use concrete materials in each case? • Do they visualize rearranging the treats to make an array? • Do they know whether a treat can be equally shared just looking at the numbers? Conversations: If students are using guess-and-check each time rather than predicting which numbers will work, pose some of the following prompts: – How many people are sharing? How are the treats arranged in the picture? Visualize how you could move the treats around so they would be in equal groups for the people sharing. Let’s try acting out what you visualized with some counters. Will it work? – Without moving the counters, do you think one more person will be able to share? Why? What are you visualizing in your mind? – P redict how many people will be able to share and visualize how you would arrange the treats. 176 Number and Financial Literacy

Consolidation (20 minutes) • Strategically select a variety of solutions that feature how different numbers of treats were shared. For each, encourage students to mentally visualize how the treats could be rearranged to show the equal share. Check their predictions using concrete materials. • Discuss connections between the total number of treats and the number of people that could share them, and create some conjectures. Examples include the following: – If there is an even number of treats, then 2 people can share, but an odd number of people cannot share. – If there is an odd number of treats, then an even number of people cannot share. – If the total number of treats ends in a 5 or a 0, then 5 people can share. • For each conjecture, ask students what other numbers they could test to ensure that the conjectures work every time. For some of their suggestions, locate the number on the class hundreds chart and determine if there are number patterns associated with it that would let them predict other numbers that would also work. • Create an anchor chart summarizing some of the conjectures. • B uilding Social-Emotional Learning Skills: Critical and Creative Thinking: Ask students whether they think they have tried enough examples to prove that the conjectures work in every case. Ask what numbers they may want to try in the future to see if the conjectures continue to hold true. Explain that mathematicians spend a great deal of time proving conjectures in order to determine whether they do in fact work in every case. Tell students that the conjectures will be posted in the classroom and they can periodically try other numbers to further test whether their conjectures are true. Further Practice • Independent Problem Solving in Math Journals: Students can choose one of the conjectures and offer 4–5 examples to show whether the conjecture works every time and so can be considered a general rule. Multiplication and Division 177

18Lesson Solving Division Problems Using a Number Line 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 • B 2.7 represent problems involving multiplication and division, including problems that involve groups of one half, one fourth, and one third, using tools and drawings • B 2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to solve problems • B 2.2 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 Possible Learning Goal • Represents repeated subtraction on a number line and makes connections between repeated subtraction and division • Represents repeated subtraction on a number line by showing a decrease in equal lengths • Connects using relational rods to the number line as both representing length • Explains their strategy and why it makes sense, within the context of the problem • Connects repeated subtraction to division • Understands that repeated subtraction is one strategy for solving problems PMraotcheesmseast:ical About the rPerporbelseemntsinoglv,ing, communicating, Linear models can help students visualize and represent division. Rather than sscrteeoralanestnceoetgnicniitenginsgtgoa,onrledsflapenrcodtviningg, , dividing a set of objects into equal groups, they are dividing a length or distance into equal segments. This offers another perspective on equally dividing a whole and on seeing the relationships among the ways in which a quantity can be divided. 178 Number and Financial Literacy

Math Vocabulary: Relational rods are particularly effective tools, because they are all rdeeiqpvuiesaaioltenled, nesgqutuhbastrl,agucrntoiiotusnp,s, proportionally related in length and all rods of the same length have the same colour. This helps students to use spatial reasoning skills to predict how a length may be equally divided. Students can develop mental images of equal segments from their experiences. They can also visually see that, when a length is divided into more segments, the equal segments get smaller. Students can also apply their understanding of the proportional relationships among the rods to determine how a length can be equally divided. For example, when students know that the yellow rod is half the length of the orange rod, they quickly realize that a length of 20 units can be equally divided into two units of 10, as represented by two orange rods, or into four units of 5, as represented by four yellow rods. Once students have had concrete experience with the relational rods, they can extend this understanding to other linear models, such as the number line. The number line visually reveals the equal lengths within a distance. It also reveals the inverse relationship between repeated addition and subtraction, since both operations represent the same distance. This understanding helps students to also view multiplication and division as opposite operations. Materials: Minds On (20 minutes) relational rods, • Provide student pairs with relational rods. Have students imagine that they BLM 28: Activity Cards, BLM 29: Number Lines have a chocolate bar that is 15 units long and they want to share it with five (0–40) people. Time: 60 minutes • Ask students how they can represent the chocolate bar using relational rods. (e.g., one orange rod and one yellow rod; three yellow rods) Tell students to work with a partner to divide a length of 15 units into 5 equal pieces. Ask what rods they will use and how long each rod will be. • When students have finished working, ask them what they found. (e.g., We found that 5 light green rods divide the chocolate bar equally and each piece is 3 units long.) Build a model, such as is shown below. If students have difficulty seeing two or three rods (e.g., one orange and one yellow) as being one chocolate bar, you can have them use a length of 15 cm on a ruler or they can create a strip of paper that is 15 units long. • Move one green rod away at a time as you ask, “If I was giving away the pieces one at a time, what happens to the remaining chocolate bar each time?” (e.g., it gets shorter by 3 units every time) Ask what operation your actions are showing. Ask how you can represent your actions with an Multiplication and Division 179

equation. (e.g., 15 − 3 − 3 − 3 − 3 − 3 = 0) Ask what the 3 represents, and where we see the 5 rods in the equation. Ask why it is important that the equation equals zero. • Explain that the relational rods represent length, much like another tool we have used this year. Ask what tool that is (the number line). Ask how you can show your actions with the rods using the number line. (e.g., repeated subtraction, 15 − 3 − 3 − 3 − 3 − 3 = 0) Record the repeated subtraction on the number line. -3 -3 -3 -3 -3 0 3 6 9 12 15 • Ask what is different about the two representations. (e.g., One has numbers and the other is represented visually with rods.) • Ask in what ways this is similar to repeated addition. Ask what both operations show. (e.g., the length of the chocolate bar, or the distance from 0 to 15, divided into 5 equal segments) • Ask students for a division equation that can be used to describe how the space between 0 and 15 is divided. (15 ÷ 5 = 3) Ask what the equation means in the context of the chocolate bar. (e.g., When a chocolate bar 15 units long is divided into 5 equal pieces, each piece is 3 units long.) • Write 15 ÷ 3 = 5. Ask what this equation means in the context of the chocolate bar. (e.g., When a chocolate bar 15 units long is divided into pieces that are 3 units long, there will be 5 equal pieces.) Ask what this would look like with the relational rods. Ask how this would be represented on the number line and what the numbers represent. Working On It (20 minutes) • P rovide student pairs with activity cards from BLM 28: Activity Cards, relational rods, and BLM 29: Number Lines (0–40). Change the context of the problems to be meaningful to your students. • Have students work in pairs to solve the problems on the activity cards. In each case, they can initially represent their work using relational rods. They can then represent their findings on a number line and create matching repeated subtraction and division equations. Differentiation • Change the numbers in the problems to best suit the needs of your students. • For students who need more of a challenge, offer an open problem. For example: – M r. First had 24 sheets of paper. How many ways can he equally share them among different numbers of groups? 180 Number and Financial Literacy

Assessment Opportunities Observations: Observe whether students are making the connections between the different representations (e.g., concrete, number line, numerical). Do students know what the numbers mean and what they represent within the context of the problem? Conversations: If students cannot create matching equations, or cannot match the numbers to the concrete models and number line representations, pose some of the following prompts: – You have recorded 21 ÷ 3 = 7 and made a number line to solve it. Where is the 21 on your number line? What does it represent? – W hat action did you do on the number line? (e.g., I took 3 away with each jump) What does that represent in the problem? (e.g., giving away 3 mints to one friend at a time) So, the 3 represents mints. What does the 7 in your equation represent? (e.g., the 7 friends) Where is the 7 on your number line? I don’t see a 7. Count your number of jumps. What does each jump represent? (e.g., a friend) Consolidation (20 minutes) • As a class, have students discuss and model how they solved the problems using a number line. Ask what the equal jumps represent and why they need to end on zero to ensure there has been an equal share. Ask how the problem could also be solved using repeated addition. Demonstrate both repeated addition and subtraction on the same number line so students can see the connection. • Make explicit connections between the concrete representations, the number line representations, and the equations. Ensure that students understand what the numbers in the equations mean in the context of the problems. • Write the equation 28 ÷ 7 = 4. Review what it means within the context of the problem. (e.g., 28 sheets of paper, each group gets 7 sheets, so 4 groups get paper) Connect this meaning to a repeated subtraction solution on the number line. Record 28 ÷ 4 = 7. Ask how the problem would change if this was the solution. (e.g., 28 sheets of paper, 4 groups share the paper, so each group gets 7 sheets) Ask how they would solve this on the number line. (e.g., take jumps of 4) Ask what the jumps of 4 represent. (e.g., the 28 sheets divided into piles, one pile for each of the 4 groups) Ask what the length of each jump represents. (e.g., 7 pieces of paper) Further Practice • Independent Problem Solving in Math Journals: Present the following problem verbally and record it on the board for students to solve in their Math Journals. – T he principal has a roll of craft paper that is 30 metres long. She wants to cut it into equal sections so that each class gets 5 metres of craft paper to make a banner. How many classes would she be able to give craft paper to? Use a number line to show your thinking.   Multiplication and Division 181

19Lesson Applying Strategies to Solve Division Problems Math Number Curriculum Expectations • B2.6 represent multiplication of numbers up to 10 × 10 and division up to 100 ÷ 10, using a variety of tools and drawings, including arrays • B2.7 represent problems involving multiplication and division, including problems that involve groups of one half, one fourth, and one third, using tools and drawings • B 2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to solve problems • B 2.2 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 Teacher Possible Learning Goal Look-Fors • Applies understanding of division and multiplication to solve problems and Previous Experience with Concepts: explains their reasoning Students have explored multiplication and • Selects an appropriate strategy to solve the problem division using concrete • Explains how they solved the problem and why their solution makes sense and/or pictorial models • Supports thinking by creating a drawing or concrete model, and/or explains and have created equations that match their mental strategies that they used models. • Connects concrete models to matching equations and explains what the numbers mean within the context of the problem • Finds more than one solution to the problem • Reflects on solutions to see if they are reasonable and make sense • Makes connections between operations, such as multiplication and division or division and repeated subtraction 182 Number and Financial Literacy

PMraotcheesmseast:ical About the Problem solving, Once students have investigated the concept of division, it is important that representing, they can apply their understanding in problem-solving situations. At this and proving, reasoning point in their learning, it is beneficial to present students with open problems that have more than one solution. This offers insight into whether they can communication, view a situation from more than one perspective and flexibly select from the sscteorlanetnceetgicniteginstgo,orlesflaencdting, repertoire of strategies and representations discussed in class to solve the problem in more than one way. Since there are multiple entry points to open MdeasiqaduvuitdbshaititrotiaVionocon,ntc,eisoar,qenburpeuaepllaaesrtayhet:aderding, problems, it is important to let students struggle a bit rather than stepping in to ‘rescue’ them or influencing the way in which they solve the problem. The Consolidation plays a critical role. Students need the opportunity to explain their thinking while their peers listen, question, and add ideas. It is important to explicitly make connections between the strategies and representations and how they link to the big mathematical ideas. Such discussions offer further insight into which students are still uncertain about their thinking and will need further practice and/or feedback, either in small-group or individual guided lessons. About the Lesson In this lesson, students apply their understanding of division to solve an open problem presented with a realistic context. 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 components as they define and refine their model. For example, they may need to analyse the conditions surrounding a problem (Analyse the Situation) before actually being presented with the problem. Students may need to revisit the problem or the situation if they find that their model does not adequately represent the math in the problem. In this way, they can gather more information and select more-appropriate tools and strategies. There are some suggestions on how to reinforce the model in the lesson, although these will need to be adjusted so they are reflective of how your students progress through the process. In the accompanying Math Talk, students explore some coding concepts without the use of a computer. They investigate whether the sequence of step-by-step instructions that involve different operations affects the outcome. This is an excellent way to learn about the difference in how operations make numbers grow. Multiplication and Division 183

Materials: Minds On (15 minutes) Digital Slide 23: • Show Digital Slide 23: Wheels! and discuss how many wheels a bicycle, Wheels!, concrete materials, chart paper, tricycle, and skateboard each have. markers • Show 10 counters and tell students that they represent wheels. Ask how Time: 60 minutes many bicycles 10 wheels would make and if any wheels would be left over. Model students’ suggestions. Repeat for tricycles and skateboards. (Analyse the Situation) Working On It (20 minutes) • R ecord the following problem and read it together, ensuring that all students understand the context and what they are supposed to find. (Understand the Problem) – T here are 30 wheels in the garage. The wheels are on some bicycles, tricycles, and skateboards. How many of each type of sports vehicle could be in the garage? • Students work in pairs. They can select their own tools and materials and record their solutions on chart paper. (Create a Model) • Encourage student to solve the problem in more than one way. Differentiation • Adjust the numbers in the problem to suit the needs of your students. • For some students, you may want to limit the sports vehicles to bicycles and skateboards. • For some students, you may want to make the problem less open. For example, you can establish that there are two tricycles, which reduces the number of options. Assessment Opportunities Observations: • Observe how students start the problem, select tools and strategies, adjust their strategy if one strategy does not work, and think flexibly about different solutions. • Pay attention to whether students are using guess-and-check or are applying their understanding of numbers to develop a system. For example, students could have 4 bikes and convert them into 2 skateboards because they know that 4 is twice as many as 2. Conversations: • L isten to students’ conversations as they begin solving the problem. If students have difficulty getting started, they may need to revisit the problem or its context. (Understand the Problem/Analyse the Situation) – What information do you know? How could you represent this to get you started? 184 Number and Financial Literacy

• If students are not discussing among themselves, you can use prompts such as the following: – Can you explain to your partner what this part of the drawing represents? – E xplain what your partner just said, using your own words. – C an you add on to that? Now you can explain the next step. Consolidation (25 minutes) • Have students meet with another pair to compare solutions. This is important, since they may have found very different solutions. Encourage students to explain their reasoning and the system they used to find more solutions. • Meet as a class. Strategically select solutions that reflect different approaches to the problem, such as the samples below: – W e started with 2 tricycles and 6 skateboards. Then we kept the 2 tricycles and turned the skateboards into 12 bicycles, since the number of wheels on 1 skateboard equals the number of wheels on 2 bicycles. – We started with tricycles and added enough so we had an even number of wheels left. We found if we added 1, 3, or 5 tricycles, we ended up with an odd number that wouldn’t work for skateboards and bicycles. – W e started with 2 tricycles, 1 skateboard, and the rest bicycles. In the next solution, we kept 2 tricycles, made it 2 skateboards, and the rest bicycles. • Select one solution for which to create division equations. For example, below are equations and meanings for 2 tricycles, 6 bicycles, and 3 skateboards. 2 tricycles is 6 wheels; 6 ÷ 3 = 2; 6 wheels, 3 wheels per tricycle, equals 2 tricycles 6 bikes is 12 wheels, 12 ÷ 2 = 6; 12 wheels, 2 wheels per bicycle, equals 6 bicycles 3 skateboards is 12 wheels, 12 ÷ 4 = 3; 12 wheels, 4 wheels per skateboard, equals 3 skateboards • Create a table of some student solutions. You can use the headings below. Discuss any patterns they see in the completed table. Bicycles Bicycle Tricycles Tricycle Skateboards Skateboard Total Number Wheels Wheels Wheels of Wheels • Discuss whether students would solve the problem differently now that they have heard other people’s solutions. Compare strategies and tools and their effectiveness. (Analyse and Assess the Model) Multiplication and Division 185

Materials: Math Talk: Digital Slide 24: Math Focus: Investigating how the sequence of instructions that involve Magical Mix-Up!, operations can affect outcomes BLM 30: Magical Mix-Up!, counters, chart paper Teaching Tip About the Math Talk Integrate the math In this Math Talk, students learn about coding concepts without the use talk moves (see of a computer. They investigate the operations and see how the order of page 8) throughout instructions (code) can affect the outcome. Through this process, they Math Talks to can compare the additive effects of addition and subtraction and the maximize student multiplicative effects of multiplication and division. The sequence of participation and instructions and its effect on the outcome is a critical coding concept active listening. that grade three students are expected to understand. Let’s Talk Select the prompts that best meet the needs of your students. • S how Digital Slide 24: Magical Mix-Up! What do you see on this slide? Who are magicians and what do they do? Have you ever seen a magician? Have you ever seen someone do a magic trick? What was it like? • Why do many magicians have hats and wands? What often happens when something goes into the magician’s hat? (e.g., Something disappears or something different comes out of the hat.) • Look at the order of the pictures. What story do they tell? (e.g., Some balls go into the hat and we don’t know what comes out.) Would there be a different story if the pictures were in a different order? (e.g., Something goes into a hat and some balls come out.) So, order matters since it affects the outcome of the story. • L ook at the instructions on the ripped pieces of paper. What do you think they are and why are they ripped? These are the magical instructions and they are no longer in the correct order. The instructions are like the coding instructions that we give to computers so they carry out the commands and give us the desired outcome. • W hat do you notice that is the same and different about all of the instructions? (e.g., They all involve the number 2, but they each have a different operation.) Do you think the order in which these instructions are carried out will affect the outcome? Why? Partner Investigation • You are going to investigate whether the order of the instructions can result in different outcomes. 186 Number and Financial Literacy

• What happens with a code that creates a repeated event? (e.g., The outcome happens over and over again.) You are also going to investigate whether you can sequence an order that will result in a repeated event. • P rovide copies of BLM 30: Magical Mix-Up! You are going to start with 8 balls going into a hat and then perform each of the operations to find out what the outcome is. You will then change the order of the instructions to see if you get a different outcome. • Print each of the instructions (parts of the code) on separate pieces of paper and then you can adjust the order. You can record the order of your instructions and the outcome on paper. Try to find as many different orders and outcomes as possible. • While you are adjusting the orders, see if you can create a repeating event. Follow-Up Talk • W hat did you find? Let’s record some of the sequences that you used and what the outcomes are. (Below are just a few possibilities.) 8×2+2÷2–2=7 8÷2–2×2+2=6 • W hich order produced a repeating event? How do you know that it is a repeating event? • Why do you think the order of the operations affected the outcomes? Why is it important that we always check the order of our coding instructions to confirm that the code achieves our desired outcome? Multiplication and Division 187

20Lesson Developing Mental Strategies for Multiplication and Division Math Number Curriculum Expectations • B 2.2 recall and demonstrate multiplication facts of 2, 5, and 10, and related Teacher division facts Look-Fors • B2.6 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 have investigated multiplication and • B 2.7 represent problems involving multiplication and division, including division and conceptually understand the two problems that involve groups of one half, one fourth, and one third, using operations and how they tools and drawings are related. • B 2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to solve problems Possible Learning Goals • Selects an appropriate strategy for solving multiplication and division problems and explains why the strategies work • Understands the commutative property, identity property, zero property, and the relationship between multiplication and division and applies them to solve related problems • Applies knowledge of multiplication and division to gain automaticity with the 2, 5, and 10 facts. • Understands the commutative property and applies it to solve problems • Understands the identity and zero properties, and uses them to solve related problems • Selects a strategy for a certain problem and explains why it is suitable and how it works • Applies understanding of the operations and how they can be represented in visual form to recall the 2, 5, and 10 facts of multiplication and division • Understands that multiplication and division ‘undo’ each other and uses this knowledge to solve problems • Solves problems presented in pictorial and numerical form and makes connections between the two representations 188 Number and Financial Literacy

PMraotcheesmseast:ical About the Problem solving, arenfdlescttriantge,gsieelse,cting tools In grade three, students acquire strategies to gain recall and automaticity of ccoomnnmecutninicga,trinegp,resenting, multiplication facts for 2, 5, and 10 up to 10 × 10 and the related division facts. reasoning and This acquisition is aided by students’ experiences with skip counting by 2s, 5s, proving and 10s, and working with 5s and 10s as benchmarks. As students learn strategies and gain automaticity with some facts, they can Mdfafaoaancuctatbehlofl,iaVgnmmougice,inlahyuc,batloepulfcll,uhakhsar,yanco:ldvlnoi,encgks,et, apply their understanding of the properties of operations, such as the strategy commutative property of multiplication. By applying this property, students realize that if 2 × 5 = 10, then 5 × 2 = 10, thereby reducing the number of facts to be learned. It is important to continue using concrete objects, pictorial representations, and tools since they help students formulate mental images to perform calculations in their minds. Strategies for Multiplication Strategies that students may use for multiplication are listed here, some of which are taken from the work of Van de Walle and Lovin (Van de Walle & Lovin, 2006a, pp. 112–113). • Commutative property • Doubling (2× facts) • Decomposing into known facts (distributive property) Strategies for Division Van de Walle and Lovin discuss an interesting question as to whether students are practising division or multiplication when solving problems that involve division facts. They point out that while there is some value in practising division facts, “mastery of multiplication facts and connections between multiplication and division are the key elements of division fact mastery” (Van de Walle & Lovin, 2006a, p. 116). They also recommend posing word problems since the context helps to connect the two operations. It is therefore important to help students learn strategies for multiplication and then explicitly discuss how the inverse relationship between the operations can help them solve division facts and related problems. It must be remembered that students will prefer some strategies, which is entirely acceptable. It is important to expose them to a variety of strategies and discuss the situations where they tend to be effective. This increases students’ repertoire of strategies, allowing them to flexibly choose those that are more effective for particular problems and work best for them. About the Lesson In this lesson, students investigate the inverse relationship between multiplication and division. In the accompanying Math Talks, students investigate various strategies for solving multiplication problems. They continued on next page Multiplication and Division 189

also focus on gaining automaticity with the 2, 5, and 10 facts (multiplication and division). The Math Talks are structured so they move from problems that are presented using concrete or pictorial representations to problems presented with numbers. Each Math Talk can be used several times to reinforce the highlighted concept, changing the numbers in the problems and the related representations in each session. For example: • Session 1: Do the Math Talk with visuals and connect visuals to multiplication equations. • Session 2: Do the Math Talk with multiplication problems represented using numbers. • Session 3: Do the Math Talk with visuals and numbers and also introduce the related division facts. The third Math Talk offers an example of the three sessions that can be applied to all the other Math Talks. Materials: Minds On (20 minutes) Digital Slide 25: • Show Digital Slide 25: Array of Cans for 7 to 10 seconds and then remove it. Array of Cans, chart paper, markers, counters Have students turn and talk to a partner about what they saw and how the objects were organized. Time: 60 minutes • Discuss how many cans they saw and how they know. Show Digital Slide 25 again for a brief time if necessary. • Display Digital Slide 25 a third time and have students think of a word problem to describe what they see. Encourage them to think about the operations of multiplication and division. Ask what equation describes each word problem. Sample problems are shown below. Ensure all four equations are discussed. Record the equations on chart paper. – T here are 3 rows of 5 cans. How many cans are there altogether? (3 × 5 = 15) – T here are 5 columns of 3 cans. How many cans altogether? (5 × 3 = 15) – T here are 15 cans that are divided into 5 equal groups. How many cans are in each group? (15 ÷ 5 = 3) – T here are 15 cans that are divided into groups. Each group has 3 cans. How many groups of cans are there? (15 ÷ 3 = 5) • Discuss how the word problems are different in terms of whether you know the whole or the parts and what you are solving for. Have students explain how each problem relates to the array on Digital Slide 25. • Draw attention to the equations. Ask what is the same about all of them (e.g., the numbers) and how they are all related (e.g., the two operations undo each other). Explain that the four equations make up a ‘fact family,’ 190 Number and Financial Literacy

just like the ones they learned about when studying addition and subtraction. Explain that we call it a fact family because all of the facts are related to each other, just like people in a family are related to each other. • Discuss how knowing one fact in the family can help solve the other facts. Working On It (20 minutes) • Students work in pairs. Provide each pair with a different number of counters. You can include quantities that are divisible by 2, 5, or 10 (e.g., 12, 16, 20, 25, 35, 40, 50, 70, 80, etc.) so they can create arrays with rows or columns of 2, 5, or 10. • Students create as many arrays as possible, using all of their counters in each array. Every time they find an array, they draw the arrangement on chart paper and then record the equations in the related fact family. They can then work with concrete materials to find another array using the same number of counters. Differentiation • Assign numbers so they are most suitable for the individual needs of your students. • Students who need more of a challenge can work with other numbers, such as 24 and 36. Assessment Opportunities Observations: • O bserve how students are creating the arrays. Do they start over with every trial or are they anticipating how they can make a different array and adjust the items accordingly? • O bserve if students can create math facts that relate to the arrays. Can they differentiate between rows and columns? Can they see the turn- around equations within the arrays? Conversations: If students aren’t anticipating before trying a new array, pose some of the following prompts: – Visualize how you could move some of the counters and still keep the arrangement looking like a rectangle. – If you have 4 rows, how could you equally divide them? – W here would you put the counters from the row(s) you are removing? Visualize what the rectangle will look like now. Try moving your counters to see if your prediction is true. Consolidation (20 minutes) • Hang up all student work in ascending order according to the number of items in each array. Have them turn and talk to a different partner about what they see. • Ask why some numbers have more arrays than other numbers. Multiplication and Division 191

Materials: • Discuss how all the arrays on one chart paper represent all of the decks of playing cards multiplication and division facts that equal one particular number. Materials: • Discuss how multiplication and division are opposite operations and actually relational rods ‘undo’ each other. • Discuss how knowing one fact can help to solve the other ones. Ask which operation they are most comfortable with and why. • Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: Explain that it is going to take time to learn all of the multiplication and division facts and that research has found that students usually learn their multiplication facts first and then use them to solve division facts. Suggest that when they are looking at a division problem, they can always turn it into a multiplication problem. With practice and experience, division facts will become easier to solve. Further Practice • 52-Card Challenges: Students use a full deck of playing cards (52) to create arrays according to the criteria given in one of more of the challenges below. In each case, they record matching multiplication and division equations. – Challenge 1: Make arrays with between 10 and 20 cards, with at least 3 equal rows and at least 3 equal columns. – Challenge 2: Make arrays with between 20 and 30 cards, with at least 3 equal rows and at least 3 equal columns. – Challenge 3: Make arrays with between 20 and 30 cards, with at least 4 equal rows and at least 4 equal columns. – Challenge 4: Make arrays with at least 40 cards, with at least 4 equal rows and at least 4 equal columns. – Challenge 5: Make arrays with between 30 and 40 cards, with at least 5 equal rows and at least 4 equal columns. – Challenge 6: Make arrays with between 35 and 50 cards, with at least 6 equal rows and at least 6 equal columns. Math Talk: Math Focus: Using a linear model to investigate fact families of 5 and 10, for multiplication and division Let’s Talk Select the prompts that best meet the needs of your students. • Provide student pairs with a set of relational rods. What do you know about this math tool? If the white rod equals one unit, what rod equals 10 units? 5 units? 192 Number and Financial Literacy

Teaching Tip • H ow are the orange rod (10 units) and the yellow rod (5 units) related? How Integrate the math many yellow rods will evenly fit alongside the orange rod? What is an addition talk moves (see equation to show this? (5 + 5 = 10) How could you represent this using page 8) throughout multiplication and what would the equation mean? (e.g., 2 × 5 = 10 or two Math Talks to groups of five equal ten) We can use the relational rods to show equal groups maximize student for multiplication. participation and active listening. • L ook at the model. What other operation can it represent? (e.g., division, because the length of the orange rod is divided into two equal pieces) • What would the matching equation be? (e.g., 10 ÷ 5 = 2) What does the 5 mean and where do you see it in our model? (5 units) What does the 2 mean? (the number of yellow rods or the number of equal groups) • H ow are 2 × 5 = 10 and 10 ÷ 5 = 2 related to each other? (e.g., They undo each other.) • Is there another way that we could use relational rods to make a model that shows 10 using multiplication and/or division? Work with your partner to find out. • W hat did you find? (e.g., We could lay five red rods on top of the orange rod.) How much are the red rods worth? (2 units) What would the multiplication equation be and what does it mean? (5 × 2 = 10, five rods that are 2 units each equals 10) • W hat is a division equation that describes this and what does it mean? (e.g., 10 ÷ 2 = 5, 10 units divided into 2-unit pieces equals five pieces or rods) • Let’s look at the multiplication and division equations we have made that equal 10: 5 × 2 = 10, 2 × 5 = 10, 10 ÷ 5 = 2, 10 ÷ 2 = 5. What do we call this set of equations? (a fact family) How are these all related? How do they represent the models we made with the relational rods? Partner Investigation • What model have we used before to investigate fact families? (e.g., arrays) In the lesson, we found several fact families. You can work in pairs with a different partner than in the lesson. I will give you a number to represent as a length, using a train of rods. You will find all of the ways to make trains of that same length, with each train made up of only one colour of rod. Draw your trains on chart paper and record the equations that make up the related fact family. Assign numbers to represent the total lengths depending on the facts you are trying to reinforce (e.g., 5, 6, 8, 10, 15, 20, 30). Some pairs may solve for the same number. continued on next page Multiplication and Division 193

• When students are done, post their drawings in ascending order by length. What do you notice? Why do some numbers have more fact families than others? Which number has the most fact families? (12) • L et’s look at the fact family 5 × 6, 6 × 5, 30 ÷ 5, 30 ÷ 6. Where do you see each of these in the relational rod models? What do they mean? • L ook at our chart papers for the lesson and the Math Talk. Many of these include the 2, 5, and 10 multiplication facts that we are going to learn to recall without having to represent them with concrete objects. How can knowing fact families make it easier to learn our 2, 5, and 10 multiplication facts? Materials: Math Talk: BLM 31: Ten Frames, Math Focus: Investigating the doubling strategy for multiplication facts of 2 counters, chart paper, markers, Digital Slides Let’s Talk 26–37: Fast Images, index cards The Math Talk is carried out over three sessions. In each session, select the prompts that best meet the needs of your students. Session 1 • Provide students with ten frames (BLM 31) and counters. Earlier, we investigated what happens when we multiply by 0 and 1. Who can explain in their own words what we found? Let’s make a concrete model of each so we can visualize the rules better. How can you show 2 × 0? 2 × 1? • If I have 4 cookies and you have double that many cookies, how many cookies do you have? Turn and talk to your partner. • What do you think? (e.g., double would be 8 cookies) If another person has double the 8 cookies you have, how many cookies is that? What does doubling mean? (e.g., having two times as many; having a set of cookies like yours and another set just like it) • H ow can we show this doubling of cookies using equations? (e.g., 4 + 4 = 8 and 8 + 8 = 16; 2 × 4 = 8 and 2 × 8 = 16) How are the addition and multiplication equations representing the same thing? (e.g., They both represent 2 groups of 4 or 2 groups of 8.) • H ow can knowing our plus doubles facts help us with our ×2 math facts? • Show Digital Slide 26 for 3–5 seconds. How many dots did you see? How do you know? Show the slide a second time. What multiplication fact describes this fact? Record the multiplication facts on index cards. Repeat for the ten frames on Digital Slides 27 and 28. • L et’s try some more examples together. Show the shapes on Digital Slides 29–31 following the same process as for the ten frames. However, after 194 Number and Financial Literacy

showing each slide a second time, use prompts similar to the following, which are specific to the array of squares on Digital Slide 29: – You said that the first slide represents 2 groups of 3. Can you see another multiplication fact in this picture? (e.g., 3 × 2) What does it represent? (e.g., 3 groups of 2) Where are these groups? What is the product? (6) It is the same as 2 × 3 so these are like turn-around facts. So, what rule can we use if one of the two numbers in a multiplication fact is a 2? – Let’s record this as our doubling strategy on our strategies anchor chart. Session 2 • R epeat the Math Talk using Digital Slides 32–37, which present multiplication facts for 2 and their turn-around partners in numerical form. After removing the number fact, ask students what image they visualize. Then, discuss how students can use the doubling strategy to solve the problem (i.e., find the product). Session 3 • R epeat the Math Talk using problems presented using numbers and pictures (e.g., arrays). Have students look at the pictures and name the related division problems. They can explain where they see the division in the pictures, and how the division is related to the multiplication. Record the division problems on index cards. These can be used in later Math Talks. Math Talk: Math Focus: Investigating strategies for multiplication facts of 5 and 10 Let’s Talk Select the prompts that best meet the needs of your students. • Have 4 students come to the front of the class, raise 1 hand, and then put it behind their backs. How many fingers did you see and how do you know? (e.g., We saw 20 fingers. We counted by 5s.) What would this look like with numbers? (5, 10, 15, 20) How can we represent that with equations? (5 + 5 + 5 + 5 = 20 or 4 × 5 = 20) How can we represent this situation if we know that there are 20 fingers and we want to know how many hands are represented? (20 ÷ 5 = 4) How are the multiplication and division equations related? • Did anyone solve it differently? (e.g., I knew that 2 hands have 10 fingers, so I added 10 + 10.) How can we show this with numbers? (10 + 10 = 20) • Is there another way? (e.g., 2 hands have 10 fingers, so I just doubled 10 and got 20.) How can we show this using an equation? (2 × 10 = 20) What would be a related division equation if we wanted to find out how many students have 20 fingers? (20 ÷ 10 = 2) continued on next page Multiplication and Division 195

• L et’s look at 4 × 5 = 20 and 2 × 10 = 20. How are the equations related? (e.g., We multiply by 4 when there are only 5 fingers but only multiply by 2 when there are 10 or double the number of fingers.) • H ave the 4 students raise both hands and then put them behind their backs. How many fingers did you see this time and how do you know? How could we record these solutions with numbers? (e.g., We counted by 5s [5, 10, 15, 20, 25, 30, 35, 40].) How can you represent this with a multiplication equation? (8 × 5 = 40) • D id anyone solve it differently? (We knew that each person has 10 fingers, so we counted by 10s [10, 20, 30, 40]; twice as many hands were up this time, so we doubled 20 and got 40 [2 × 20 = 40].) How could we represent 4 students with 10 fingers each? (4 × 10 = 40) What would be a related division problem? (40 ÷ 10 = 4) What does this mean? (40 fingers, with each person having 10 fingers, means there are 4 people) • Have one of the 4 students sit down. How many fingers will the remaining students have? How do you know? How can we represent this as an equation? (3 × 10 = 30) What is the related division equation? (30 ÷ 10 = 3) What equation could you use to represent the number of hands? (6 × 5 = 30) • H ow are 6 × 5 and 3 × 10 related? (e.g., We multiply by 6 when there are only 5 fingers and only multiply by 3 when there are double the number of fingers, or 10 fingers in a group.) • H ow can this relationship help us learn our ×5 and ×10 facts? • Make a list of the related facts (e.g., 2 × 10 and 4 × 5, 3 × 10 and 6 × 5, 4 × 10 and 8 × 5, 5 × 10 and 10 × 5). Materials: Math Talk: Digital Slides Math Focus: Investigating for multiplication facts of 5 with odd numbers 38−39: Fast Images Let’s Talk Select the prompts that best meet the needs of your students. • I am going to show you some images for a short period of time. Start with the array of circles on Digital Slide 38. How many shapes did you see and how were they arranged? (e.g., There were 3 rows with 5 shapes in each, so I skip counted and there were 15 circles.) Show the slide again. How can we describe this array with a multiplication equation? (3 × 5 = 15) • Let’s look at the next array. Show Digital Slide 39 for a short time. How many did you see this time and how were they arranged? (e.g., We saw double the number of rows as the last time, so there were 6 rows, so we doubled 15 and got 30.) Show the slide again. Where do you see the first image within the second image? How can we describe the array with a multiplication equation? (6 × 5 = 30) 196 Number and Financial Literacy