Unit2-Add Subtract-p104-175_3rd

Unit 2: Addition and Subtraction to 10 Lesson Content Page Addition and Subtraction to 10 Introduction 104 106 1 and 2 Understanding the Equal Sign 107 109 1 Equality and Balance: Mass 112 115 2 Linking Equality and the Equal Sign 121 124 3 Representing Joining Problems 130 133 4 Investigating Strategies for Joining Problems 138 140 5 Addition: Commutative Property 145 148 6 Addition: Varying the Unknown 155 157 7 Representing Separating Problems 160 165 8 Strategies for Solving Separating Problems 168 173 9 to 11 Part-Part-Whole: Composing Quantities 9 Part-Part-Whole: Composing 5 10 Part-Part-Whole: Composing 6, 7, 8, and 9 11 Part-Part-Whole: Composing 10 12 Whole-Part-Part: Decomposing 10 13 Subtraction as ‘Think Addition’ 14 Compare Problems: Differences 15 Creating Benchmarks Using Mental Strategies 16 Linking Addition and Subtraction 17 Reinforcement Activities

Addition and Subtraction to 10 Introduction Math Vocabulary: About the ebtepshqsaqlueuiluguabasnasntl,,ratcsaiemnoticgadontenk,t,,,enaeamesoaqdqi,twnuduu,qabaasluapy,l,,allsaunminsgt,ciinntey,u,ds, Students in kindergarten decompose and recompose quantities to 10 cdcoeomcmopbmoinspaeot,sioen, s and engage in part-part-whole thinking as they use concrete materials to show ways to make 10. These experiences help to prepare students for understanding addition and subtraction when they enter grade one. Students are exposed to a variety of problem structures, as well as problems that have the unknown in different positions, requiring them to interpret the operations through the context. They apply their abilities to compose and decompose quantities in order to establish benchmark numbers, such as 5 and 10. As they build equations to match the context and operations in the stories, they learn about the symbols, such as plus and minus signs, and gain an understanding of equality as a balance and the equal sign as meaning ‘the same as.’ Throughout the investigations, students discover the relationship between the operations of addition and subtraction. 104 Number and Operations

Lesson Topic Page 1 and 2 Understanding the Equal Sign 106 107 1 Equality and Balance: Mass 109 112 2 Linking Equality and the Equal Sign 115 121 3 Representing Joining Problems 124 130 4 Investigating Strategies for Joining Problems 133 138 5 Addition: Commutative Property 140 145 6 Addition: Varying the Unknown 148 155 7 Representing Separating Problems 157 160 8 Strategies for Solving Separating Problems 165 168 9 to 11 Part-Part-Whole: Composing Quantities 173 9 Part-Part-Whole: Composing 5 10 Part-Part-Whole: Composing 6, 7, 8, and 9 11 Part-Part-Whole: Composing 10 12 Whole-Part-Part: Decomposing 10 13 Subtraction as ‘Think Addition’ 14 Compare Problems: Differences 15 Creating Benchmarks Using Mental Strategies 16 Linking Addition and Subtraction 17 Reinforcement Activities Addition and Subtraction to 10 105

and1 2Lessons Understanding the Equal Sign Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make Previous Experience connections; use technology to explore mathematics with Concepts: It could be beneficial if • Understanding and solving: Develop, demonstrate, and apply students have worked with pan balances before, mathematical understanding through play, inquiry, and problem solving; but not necessary. visualize to explore mathematical concepts Content • M eaning of equality and inequality: Demonstrating and explaining the meaning of equality and inequality; recording equations symbolically, using = and ≠ About the According to an extensive study carried out by Falkner, Levi, and Carpenter, less than 10% of students in any grade from one through six could give the correct response to the question 8 + 4 = — + 5. With greater probing, the researchers discovered that most elementary students have serious misconceptions about what the equal sign means (Carpenter et al., 2003, p. 9). Many interpret it as meaning ‘the answer is.’ According to the British Columbia curriculum, grade one students are expected to demonstrate and explain the meaning of equality and inequality. The research supports the value in taking considerable time to ensure students understand equality as a balance so later misconceptions can be avoided. About the Lessons The following lessons allow students to investigate ‘balanced’ and ‘not balanced’ using mass on a pan balance. Through their investigations, they may use the term ‘weight’ since it is more familiar to them. While mass and weight are different concepts, describing it as weight will be fine at this point in time. The lesson is more about students being able to experience ‘balance’ in a concrete manner. The vocabulary ‘equal’ and ‘not equal’ are introduced. 106 Number and Operations

1Lesson Equality and Balance: Mass Teacher Possible Learning Goals Look-Fors • Investigates ‘balance’ of mass using concrete materials Math Vocabulary: • Explains or shows that ‘equal’ means ‘balanced’ by giving examples and ebpaqaasuln,aanmbl,caaenlsadosn,t,cneleoeq,tvuteabhlla,e,lasnacmeed, even non-examples Materials: • Creates balance on a pan balance using various objects pan balances, items • Recognizes imbalance on a pan balance from around the • Makes reasonable predictions about what might balance an object room, chart paper, • R efines ideas about what might balance an object after several trials markers • E xplains what ‘balance’ and ‘equal’ mean Time: 50 minutes Minds On (15 minutes) • Show a pan balance with the two sides level. Ask students what they notice about the two sides. Highlight the vocabulary that students might use such as ‘even’ and ‘level.’ Introduce the word ‘balanced’ and explain how it is similar to their vocabulary. • Ask students to visualize what will happen if an object (e.g., an eraser) is put on one side. Have students turn and talk with a partner. As a class, discuss their predictions and reasons for them. Put the item on the one side and discuss what happens in relation to their predictions. Ask what they think might happen if another eraser is added to the same side. • First Peoples Perspectives: Use natural objects from the outdoors to create balance. (Characteristics of Aboriginal Worldviews and Perspectives: Engagement with the Land, Nature, and the Outdoors) • Have students discuss with a partner what might be put on the other side to make the two sides balanced again. As a class, try some of their suggestions. Frequently describe the scale in terms of ‘balanced’ and ‘not balanced.’ • Put something very heavy, but small, on one side and show students a collection of smaller objects (e.g., erasers). Have them discuss with a partner how many of the smaller objects need to be put on the other side to create ‘balance.’ An accurate number is not expected, but just the idea that there would be a lot of the smaller objects needed. This helps students gain a general sense of larger numbers when they are estimating. Working On It (20 minutes) • H ave students work in pairs or small groups, depending on how many balances are available in the class. Have them experiment with creating balance. They Addition and Subtraction to 10 107

can select items from around the room. Have them predict what might create balance before actually putting items on the other side. They can record objects that created balance by drawing pictures on chart paper. Differentiation • For some students who may need support with the language, using several terms for one concept, such as ‘level,’ ‘even,’ or ‘balanced,’ can cause confusion. While students are working in small groups, connect the terms. For example, say, “You said that the two sides are even. That means they are balanced. If they are not even, they are not balanced.” Assessment Opportunities Observations: Pay attention to how students create balance. Do they randomly try objects, or do they selectively choose objects based on some kind of reasoning? Do they refine their choices after trying various objects? Are they predicting before putting objects on the balance? Conversations: For students who seem to be working randomly, ask them to visualize and predict what would make the two sides even. Before they place the objects on the scale, ask if their prediction makes sense and why they think so. Consolidation (15 minutes) • Have one pair or group identify an item that they tried to balance. Have the rest of the class visualize and predict what might balance the item. The group members can then reveal their findings. Have some of the other groups present their findings in a similar manner. • Ask students what the pan balance looks like when it is balanced and when it is not balanced. Have them explain the term ‘balance’ in their own words. • Ask students what creates the balance. (e.g., the same mass on each side) Tell students that the masses on each side are equal. Ask what they think ‘equal’ means. (e.g., ‘the same as,’ or ‘balanced’) Use the terms ‘equal’ and ‘not equal’ as you model one example on the pan balance. • Co-create with students an anchor chart with one picture that shows ‘balanced’ and one that shows ‘not balanced.’ Include the labels ‘balanced,’ ‘not balanced,’ ‘equal,’ and ‘not equal.’ Add these terms to the Math Word Wall. Further Practice • Independent Activity in Math Journals: Have students draw one picture that depicts the balance they found between objects, and one picture that depicts objects that are not balanced. They can label their drawings as ‘balanced’ or ‘equal’ and ‘not balanced’ or ‘not equal’ by using the anchor chart as a reference. 108 Number and Operations

2Lesson Linking Equality and the Equal Sign Teacher Possible Learning Goals Look-Fors • Determines, through investigation, how to create ‘balance’ by creating sets of equal quantity • Explains or shows what ‘equal’ means, using concrete or pictorial representations, and starts to connect it to the equal sign • Creates balance on a pan balance in terms of quantity by removing or adding connecting cubes • Creates equations that match their actions • Explains that balance is created by having the same number of cubes on each side • Explains that the equal sign means ‘the same as’ or ‘balanced’ Math Vocabulary: About the ebpaeqaasquln,auanmabl,ctaaienolsadonsn,t,,cnemqeoqu,atuattabhcnlahe,tilitansyng,acmeed, Students have now investigated balance in terms of mass. For students to fully equation understand the meaning of the equal sign in number equations, they need to see equality in terms of quantity. Such experiences help to avoid misconceptions about the equal sign that can last and even grow in the later grades if not addressed. As Van de Walle points out, “after early introductions to the equal sign in grade 1, the assumption is that students understand what you mean by ‘equals’” (Van de Walle & Lovin, 2006, p. 302). About the Lesson In this activity, students create balance between quantities using a pan balance and different coloured connecting cubes. They are also introduced to the equal sign. Materials: Minds On (15 minutes) pan balances, • Review the previous investigations on balance. Ask students what created the connecting cubes, chart paper, class balance. Put three objects with the same size and mass on one side of the anchor chart balance. Ask students what might balance them. Try out some of their suggestions. Use the terms ‘balanced,’ ‘not balanced,’ ‘equal,’ ‘not equal,’ and Time: 45 minutes ‘the same as’ throughout the discussions, clarifying their meaning. • Ask how many of the same objects would need to be put on the other side to create balance. (e.g., 3 objects) Ask why this would work, when in other Addition and Subtraction to 10 109

cases two heavier objects could be balanced by five lighter objects. Discuss how the identical objects all have the same mass, so the same number of identical objects would create a balance. In these terms, it is the number of items that determines equality, since the masses of the individual items are the same. • Put 7 green connecting cubes on the left side of the balance and 5 red connecting cubes on the right side. Have students turn and talk with a partner about how they could make the two sides balanced or equal. (e.g., remove 2 green connecting cubes from the left side or add 2 red connecting cubes on the right side) Discuss their predictions and their reasoning. Ask students how they would describe the two sides now. Working On It (15 minutes) • H ave students work in pairs or small groups. One student can create an imbalance by putting a set of connecting cubes that are the same colour on one side of the pan balance or the other. The other student or students suggest how balance could be created using a combination of connecting cubes of two different colours. For example, one student may put 8 yellow connecting cubes on the right side and the other students put 5 green and 3 red cubes on the left side to create a balance. They can investigate four or five examples, taking turns to put the initial amount on the scale. Differentiation • For students who may need support with the language, when you use terms with the same meaning, such as ‘equal’ or ‘balanced,’ create a gesture with your hands each time so they can connect to a visual reference. • For students who need more of a challenge, have the first student place a number of cubes on the left pan and an unequal number of cubes on the right pan. Their partner then needs to create balance by adding or removing cubes in two different ways. Assessment Opportunities Observations: Pay attention to how students create equality. Do they pick up cubes and place them on one at a time to see what happens, or do they intentionally take a certain number of cubes all at once? In the latter case, they are predicting the change based on their understanding of relationships between the numbers. Conversations: For students working in a random manner, ask them to predict how many cubes they think are necessary and why. Scaffold the learning by asking questions such as, “Do you think it will be more than 5 or less than 5?” This helps students apply their number sense. 110 Number and Operations

Consolidation (15 minutes) • Strategically select two or three groups to share one example each. Have them put cubes on one side and let the rest of the class predict what might go on the other side to create balance. After each example, ask students why the two sides are balanced or equal. (e.g., the number of cubes on each side is the same) Represent some of their examples in drawings. Put one of the drawings on the anchor chart that was started in Lesson 1. • Introduce the equal sign and explain that it means ‘the same as,’ ‘equal,’ or ‘balanced.’ Show how it can go in between the two sets of cubes drawn on the chart paper, reading it as, “7 cubes is the same as, or equal to, 5 cubes and 2 cubes.” Ask what makes them equal. (e.g., the number of cubes) Repeat this for the other examples and the example on the anchor chart. Highlight the equal sign and add a short definition. • Show a combination of cubes that would not be balanced, such as 8 yellow cubes on one side and 2 green cubes and 1 red cube on the other side. Ask whether the equal sign could be placed between the two amounts. Show students the not equal sign (≠) and ask what they think it means. Explain that this symbol is used when two sides are not balanced. Further Practice • Reflecting in Math Journals: Have students draw one example of equality that they discovered using the pan balance and connecting cubes. Have them use words (e.g., balanced, equal) and symbols (=) to show the equality. They can also draw an example of inequality and use the not equal sign to show that the two sides are not balanced. Addition and Subtraction to 10 111

3Lesson Representing Joining Problems Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make connections • Understanding and solving: Develop and use multiple strategies to engage Teacher Look-Fors in problem solving Previous Experience • Communicating and representing: Communicate mathematical thinking with Concepts: Students have been in many ways exposed to the plus and equal signs and know Content what they mean in addition equations, and • Addition and subtraction to 20 (understanding of operation and how an equation matches a story. process): Decomposing 20 into parts; mental math strategies: counting on Possible Learning Goals • Creates, represents, and solves ‘joining’ addition problems by acting them out • Explains or shows the equal sign as representing ‘the same as’ or a balance • Represents a given problem by acting it out and identifies the joining action • C reates a joining problem and acts it out, making the joining action visible • S olves problem by counting on rather than counting three times • M akes an equation that matches the number story and explains what the numbers and signs represent • E xplains that the initial amount usually increases in a joining problem (unless 0 is added) About the Join and separate problems both involve an action that takes place over time. Join problems have the beginning quantity increased, while separate problems have the beginning quantity decreased. Marian Small points out that, “generally, but not always, students find joining and separating situations easiest to deal with” rather than part-part- whole and comparing problems (Small, 2009, p. 107). They are good starting points for addition and subtraction since the actions can be acted out or directly modelled, allowing students to visually see the change in the existing set. The context of the problem and how it is worded are therefore important. 112 Number and Operations

Math Vocabulary: Fosnot and Dolk recommend that the contexts for addition and aeedqqduu,aaplt,ilouensq,,umpallaustscighsniing, gnjo,in, subtraction problems are closely connected to children’s lives and equation are designed to “anticipate and to develop children’s mathematical modeling of the real world” (Fosnot & Dolk, 2001a, p. 24). Stories can be created from recent common experiences, field trips, or discussions that occurred in other subject areas. About the Lesson In this lesson, students represent given problems through direct modelling by acting them out. Students also have an opportunity to create their own addition stories using joining actions in a meaningful context. In the Minds On problems, use your students’ names and change the situations so they relate to common experiences. Materials: Minds On (10 minutes) chart paper, • Present students with the following problem. Modify the names and markers, class anchor chart contexts so they reflect your students and their interests. Time: 35 minutes − 4 children are playing on the swings. 2 more children join them. How many children are playing on the swings now? • Have students act it out. Ask how many children there are now. Explain that the ‘joining’ is like adding or putting together. Show them the addition sign and explain that it means ‘to join or put together.’ Gesture with your hands to make this clear. Have students act out the problem again and record a matching expression (4 + 2). Read it in relation to the problem (e.g., There are 4 children and 2 more come to join them. So 4 and 2 more are put together.). Ask how many children there are now. (You may or may not decide to add ‘= 6.’ The goal is to introduce the concept of adding or joining and linking it to the addition sign.) • Repeat this procedure with a second problem. Ask students how they could represent the problem with numbers and symbols, and explain what they mean in relation to the context. − 1 child is on the soccer field. 7 more children arrive to play soccer. How many children are on the soccer field now? • Have students show, with their hands, what they think the plus sign represents. • First Peoples Perspectives: Include stories that are connected to place, and First Peoples cultural practices. Addition and Subtraction to 10 113

Working On It (10 minutes) • Working in groups of four or five, have students create two or three of their own joining problems and act out the scenarios. Encourage them to be creative, using any materials in the classroom to help tell their stories. • Students may decide to try to represent the stories with numbers and symbols (on chart paper), although it is not necessary at this point in time. As you circulate, you may decide to scribe the number expression (e.g., 3 + 4) as they explain their problem to you, so they get familiar with the addition sign. • While students are working (or during the Consolidation), you could video students’ stories so they can be used later to review some of the adding concepts. Differentiation • For some students who may need support with the language, they can act out a problem and the rest of the students can guess what is happening, offering the vocabulary to match the action. Assessment Opportunities Observations: Pay attention to students who cannot start the problem. Conversations: To scaffold the learning, ask students what they are supposed to do. To get them started, point to two students and ask what activity they could be doing. Have the two students do what classmates suggest. Then ask what might happen next if some students join them. Finally, ask how the story ends. 114 Consolidation (15 minutes) • Groups of students can act out their own problem. The rest of the class interprets what is happening and identifies what the joining action is. • After each group acts out their problem, ask the class what the matching expression (3 + 2) or equation (3 + 2 = 5) would be. Have them explain what it means in their own words. • Refer to the drawing added to the anchor chart in Lesson 2. Review what the drawing of the cubes represent. Ask students where they could put the addition sign to make their drawing complete. Write the complete equation below the drawing, annotating what each number and symbol represents. • B uilding Growth Mindsets: Pose one of the following prompts to evoke some curiosity about adding: – Ask students what they wonder about adding. (e.g., Can you join bigger numbers? Can you join 3 or 4 groups together?) – Ask what teachers or principals may want to add up in the school that might be very large. (e.g., how many students are in all the classrooms) Ask if they think it is possible. Reinforce the idea that when mathematicians are curious about math, they learn new ways of doing things. Number and Operations

4Lesson Investigating Strategies for Joining Problems Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make connections • Understanding and solving: Develop and use multiple strategies to engage Teacher Look-Fors in problem solving Previous Experience • Communicating and representing: Communicate mathematical thinking with Concepts: Students have created in many ways joining problems by acting them out or using Content concrete materials. • Addition and subtraction to 20 (understanding of operation and process): Decomposing 20 into parts; mental math strategies: counting on Possible Learning Goals • Solves addition (joining) problems by using a variety of modelling and counting strategies • Investigates the strategy of counting on • Represents joining problems using concrete materials • Solves a joining problem in at least one way and explains or shows the strategy • Creates a joining problem that reflects the combining action About the Alex Lawson explains that students shift from acting out or drawing themselves in their solutions to representing the problem in ways that are less tied to the context, such as using cubes to represent people. While this is still direct modelling, Lawson stresses that it continues to be valuable since the concrete objects “decrease the amount of information that needs to be held in one’s head” (Lawson, 2016, p. 19). When first solving joining problems, students often count three times. For 5 + 3, for example, they first count out 5 objects, then 3 objects, push the sets together and count the combined set from 1 as their continued on next page Addition and Subtraction to 10 115

Math Vocabulary: third count. With experience, students start to count on from one of the jroeipnr,egsreonutp,sv,issuuamli,ze sets as they include the second set (e.g., 5..., 6, 7, 8). This strategy becomes more evident if the sets remain distinct after being put together (e.g., in two different colours). This helps students reflect on the action after it has taken place so they “can see a relationship between the two parts and the whole” (Van de Walle & Lovin, 2006, p. 73). About the Lesson In this lesson, students solve and create contexts for joining stories, but represent them using concrete materials of two different colours, giving them opportunities to develop different counting strategies such as counting on. Materials: Minds On (15 minutes) class anchor chart, • Refer to the anchor chart started in Lesson 1 and review what the symbols connecting cubes, coloured tiles, two- and numbers represent in the equation under the drawing of the connecting coloured counters, dice cubes. (two per pair of students), chart paper, • Have students turn and talk with a partner and create a story involving coloured markers students in the class that matches the equation represented by the cubes. Time: 45 minutes Share two of the stories. Have students act out one of the scenarios. Ask how the cubes are similar to the people acting out the problem. (e.g., each cube represents one of the people) • Ask students how they could represent the following problem using cubes: 4 red birds are in a tree. 2 yellow birds join them. How many birds are in the tree now? • Have a student model it using red and yellow cubes, while the other students explain how to represent it. Ask students how they might solve the problem. (e.g., count the red out, then count out the yellow, then count all) Ask if there is another way. • Ask how to show the problem in pictures and numbers, like on the anchor chart. On chart paper, draw cubes and record the equation, printing 4 and 2 in matching colours. (If students are still uncertain about creating matching equations, leave this part out so they can focus on solving the problem in several ways.) 116 Working On It (15 minutes) • Students work in pairs. Tell them that they are going to create some problems using colours and numbers. Each student chooses a different colour and rolls a die. Together, they create a problem story using the two colours and two numbers (e.g., 5 blue fish are under a boat. 2 green fish join them. How many fish are under the boat?). They represent and solve it with concrete materials of different colours. Encourage them to solve each problem in more than one way (e.g., While students are working, ask, “Is there another way to solve that problem?”). Number and Operations

• If possible, students can record a matching expression (e.g., 5 + 2) or equation (e.g., 5 + 2 = 7) on chart paper. • Students create and solve one or two stories. Differentiation • For students who have not made the connection between the concrete and symbolic representations, have them just draw their problems on chart paper. As they explain the problem to you, record each number and connect it with the context (e.g., “So these cubes represent 4 blue birds, like this blue 4 here.”). Assessment Opportunities Observations: Pay attention to students’ counting strategies. Do they count all or count on? Do they refine or change their strategies according to the numbers, or use the same strategy every time? Conversations: • F or students who count 3 times, counting out both sets and then counting the combined set starting at 1, use one of the problems that the student has modelled and explained to you (e.g., 4 red birds and 3 yellow birds join them): Teacher: So how many red cubes do you have in this group? Student: 4 (Does the student have to count or just ‘knows’?) Teacher: What are you doing with these yellow cubes? Student: Putting them with the red cubes. Teacher: So you have 4 red cubes…. (Pause and gesture toward the yellow cubes.) • If the student still counts the 4 red cubes one at a time: Teacher: You told me that there are 4 red cubes here, is that right? (Cover the red cubes with a piece of paper and gesture toward the yellow cubes.) • If the student still needs to count the hidden set of 4, he/she may need additional experiences with counting on and more time to be developmentally ready for the concept. Consolidation (15 minutes) • Strategically choose two or three pairs with solutions that reflect various counting strategies that you observed. Have each chosen pair tell one of their stories. Ask the rest of the class to visualize what their problem would look like. Pairs show how they represented their solution and explain how they solved it. • After each problem, ask if other students have another way to solve the problem. Have them demonstrate their strategy and why they chose it. Record each of the strategies that students used (e.g., counting all, counting Addition and Subtraction to 10 117

Materials: on, counting on from the larger set), using pictures on an anchor chart large and small entitled ‘Adding Strategies.’ arithmetic racks (or BLM 5: Blank Ten- • Together, create matching equations for the problems, colour coding the Frames and counters) numbers so they match the cubes. Ask why we might use numbers and Teaching Tip symbols to represent problems rather than words. Integrate the math Further Practice talk moves (see page 6) throughout • Give students more opportunities to explore joining problems in a context Math Talks to maximize student using arithmetic racks. participation and active listening. • Once students have had several experiences with joining problems using concrete materials, they can start to explore different mental strategies for addition, using visual representations. The visual images that students form in their minds can form the basis for developing and applying mental strategies when solving joining problems in numerical form. Math Talk: Math Focus: • Representing joining story problems as part-part-whole relationships • Counting on Let’s Talk Select the prompts that best meet the needs of your students. • V erbally present a series, or string, of word problems (adapt the context to the interests of your students). The first problem is expanded to model some of the dialogue that might unfold. • P ose the problem. There are 3 dogs in the yard. 2 more dogs come to play with them. How many dogs are there now? What kind of action do you hear in the story? (e.g., putting together, adding) Put your thumb up if you agree. Explain to your partner why you agree. With your partner, represent this problem on your arithmetic rack. • What did you do? (e.g., I slid across 3 beads on the top and 2 beads on the bottom.) Can somebody add on to what Zoe said? Put your thumb up if this is how you represented the problem. What do the 3 beads on top represent? (e.g., the first 3 dogs) Do I move them over in one move? Why? (e.g., Yes, because they are in the yard already.) Represent it on the large arithmetic rack. And what do the 2 beads on the bottom represent? (e.g., the 2 dogs that come in later) Do I slide the 2 beads across in one move? (e.g., No, because they come into the yard one at a time.) Does anyone have a different idea? Slide across the 2 beads. • Did anyone represent the story a different way? (e.g., I moved 3 beads across on the top and then slid over 2 more) What do your beads represent? (e.g., It’s the dogs, too, except I showed them altogether in one place.) Do both ways show the same action? Discuss their perspectives, resolving that both ways do represent the same story. 118 Number and Operations

• Now what are we supposed to find out? (e.g., how many dogs are in the yard; I think there are 5 dogs now.) How did you get 5? (e.g., I counted.) How did you count? (e.g., I went 1, 2, 3, 4, 5.) Put your thumb up if you counted this way. Can you show us? Who has another way? (e.g., I know there are 3 on the top, so then I went, 4, 5.) Can someone explain in their own words what Thomas described? Why did you start at 3 rather than at 1? • Is there another way? (e.g., I just knew there were 5.) How did you know? (e.g., I saw 5 red beads together on the top, so I knew.) So you are saying that because the beads are in groups of 5, that the red beads on the top are 5. Is that right? (Yes.) • L et’s solve another problem so we can try out more strategies. You can make an anchor chart of strategies for one of the problems. Partner Investigation • C hallenge students to create more problems so they can find other ways to count. Have them take turns verbally posing a word problem to their partners, who solve it on the arithmetic rack. Together, they discuss whether there are other ways to represent or solve the problem. Follow-Up Talk • M eet after the Investigation to add any more strategies to the anchor chart. Materials: Math Talk: BLM 4: Ten-Frames Math Focus: + 1, + 2 strategies (0–20) using quantities to 10 (or Let’s Talk arithmetic racks), Select the prompts that best meet the needs of your students. counters, class • S elect some of the following prompts to use in your Math Talk. You may number line decide to cover different concepts (e.g., + 1, + 2) on different days. The following dialogue highlights both concepts to serve as an example. • B riefly show students different quantities up to 10 on a ten-frame (or arithmetic rack). Below is a possible dialogue after showing 6 on the ten-frame. • H ow many counters did you see and how did they look? (e.g., I saw 6. I saw 5 and 1 more on the bottom row.) Put your thumb up if you saw the same thing. • S how the ten-frame again. How did you count the 6? (e.g., I counted 1, 2, 3, 4, 5, 6; I counted 5 and then 6.) Why did you start counting at 5? (e.g., I know there are 5 on the top, so I don’t have to count those.) Can anyone explain how Jamie knows? How could we show 5 and 1 more with numbers? (5 + 1) Where are 5 and 6 on the number line? (e.g., right beside each other; one number apart) Show us. continued on next page Addition and Subtraction to 10 119

• V isualize 1 more counter than 6 on the ten-frame. How many would there be, and how do you imagine it? (e.g., I see 6 and then one more added is 7.) How did you know it was 7? (e.g., I counted 6…, 7) Put your thumb up if you counted on like that. What would that look like with numbers? (6 + 1) Where are 6 and 7 on the number line? • D id anyone see 1 more than 6 a different way? (e.g., I saw 5 and 2 more, so I counted 5…, 6, 7.) How could we represent this with numbers? (e.g., 5 + 2) Where are 5 and 7 on the number line? What do you notice? (e.g., They are 2 numbers apart.) • R epeat with other numbers, connecting them to the counting sequence on the number line and to a number expression. Partner Investigation • S tudents take turns showing each other fast images of numbers on ten- frames. Their partners say how many they saw, and what 1 more (or 2 more) would be. They can also say the matching expression. (e.g., 6 plus 1 more equals 7) Follow-Up Talk • S how students some ten-frames (or + 1, + 2 number expressions) and have them figure out the sums. What do you visualize? How can you prove your sum is correct? 120 Number and Operations

5Lesson Addition: Commutative Property Math Curricular Competencies Learning Standards • R easoning and analyzing: Model mathematics in contextualized experiences • Understanding and solving: Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving Content • Addition and subtraction to 20 (understanding of operation and process): Decomposing 20 into parts Teacher Possible Learning Goals Look-Fors • Creates various contexts that represent addition (joining) problems Previous Experience • Explains or shows how the order in which the parts are joined in an adding with Concepts: Students are familiar problem does not affect the sum with addition equations, the addition and equal • Creates stories that represent ‘turning around’ addends signs, and how they • R ecognizes that the order in which two numbers are added does not affect match a story. Students have worked with the total arithmetic racks before (otherwise, they can use • E xplains how ‘turn around’ stories such as 3 + 4 and 4 + 3 can take on connecting cubes). different meanings in a story context, but still result in the same total Math Vocabulary: About the ajmodiadnt,,cpehlqiunusga,le,pqleuuqsautsaioitginonn, , Students may not understand the commutative property (that numbers can be added in any order, a + b = b + a) as they solve addition problems. For example, in joining problems, the action takes place over time and two similar problems (e.g., 2 birds join 3 birds, and 3 birds join 2 birds) can take on different meanings for young children. Carpenter and colleagues point out that, “children may not initially realize that two birds joining three birds gives the same result as three birds joining two birds” (Carpenter et al., 1999, p. 8). Students need several experiences with creating and solving addition problems to understand that 2 + 3 will always result in the same sum as 3 + 2. Addition and Subtraction to 10 121

Materials: Minds On (15 minutes) arithmetic racks or • Students work in pairs with arithmetic racks. (Alternatively, if you do not connecting cubes, dice (two per pair of have arithmetic racks, have students work with connecting cubes of different students), large colours.) arithmetic rack or more connecting cubes, chart • Tell students that they are going to create some joining story problems. Roll paper one die (e.g., roll a 2) and then roll the second die (e.g., roll a 3), so students Time: 4 0–45 minutes can clearly see which number was rolled first. Ask for two or three stories that may match. Choose one to use as a focus (e.g., There are 2 birds. 3 birds join them. How many birds are there now?). • Have students represent the problem on the arithmetic rack (or with connecting cubes). Together, share how they solved the problem (e.g., counting all 3 times, counting on, ‘seeing’ a group of 5). Represent this problem on the large arithmetic rack by moving 2 beads over on the top in one move, and then 3 beads over on the bottom in one move (or with two different coloured sets of connecting cubes). • Ask how the problem would change if the 3 was rolled first rather than the 2 (e.g., There are 3 birds. 2 birds join them. How many birds are there now?). • Have students represent this ‘turn around’ problem with their concrete materials. • Show the first problem that you have represented and record the matching expression (2 + 3) on chart paper, explaining how the context matches the expression. Repeat this for the ‘turn around’ problem (3 + 2). Ask what the sum is for each. • Say, “It is interesting that they both equal the same sum. I wonder if this works with all numbers if we turn them around in an adding story.” Ask for thumbs up if students think so, thumbs down if they don’t think so, and thumbs to the side if they are unsure. • Tell students that they are going to further explore this idea. Explain that mathematicians think of possible rules (conjectures), but they have to find out if they work with all numbers to make them real rules. If they find one example that doesn’t work, then it cannot be a rule. Today, they will be mathematicians to see if the ‘turn around’ rule works with other numbers. Working On It (10–15 minutes) • Students work in pairs. They each roll a die, create a story together, and then represent it on their arithmetic rack or with cubes of two colours. • Next, they change the story so the second number is first, represent it with concrete materials, and then solve it to see if they got the same total as they did in the first problem. • Students repeat this for two or three examples. 122 Number and Operations

Differentiation • For students needing more of a challenge, have them choose their own numbers to try out so they feel they have more input into the investigation. They can also record matching expressions on chart paper to keep track of each group of numbers they have tried. Assessment Opportunities Observations: Pay attention to any students who may find it challenging to make up ‘turn around’ stories or to represent them. They might find it easier working with the cubes, since they have more freedom in how they want to manipulate them. Conversations: Ask students what the two numbers they rolled might represent. (e.g., dogs) Have them act out a scenario with concrete materials as you describe it (e.g., “3 dogs run into the yard [pause while they act it out with cubes] and 2 dogs run into the yard after them [pause]. How many dogs are in the yard?”). Now ask students to act out what would happen if the 2 dogs ran into the yard first. Have them describe the problem in their own words. Challenge them to try creating another problem with other numbers. Consolidation (15 minutes) • Have two or three pairs share one of their stories and explain whether they found the same solutions for their ‘turn around’ stories. • On two arithmetic racks or with cubes, show the two representations of one of their ‘turn around’ problems (e.g., 4 + 3 and 3 + 4). Ask how they are the same and how they are different. • Ask if anyone found an example that did not work. Ask if they think it will work for every number in the world, and whether they have proven the rule. • B uilding Growth Mindsets: Discuss how students were real mathematicians as they worked hard to prove a rule. Tell them that we think the rule works for now, but we won’t know for sure until we try it with even bigger numbers. Reinforce the message that they may not know if the rule always works YET, but with time and hard work they can learn more about it. This supports the First Peoples Principle of Learning that learning involves patience and time. Further Practice • Independent Problem Solving in Math Journals: Give students two matching expressions, such as 2 + 5 and 5 + 2. Have students draw a picture of a scenario that would match each expression. • In future related problem-solving situations, incidentally ask about whether the order in which the parts are added affects the total. Addition and Subtraction to 10 123

6Lesson Addition: Varying the Unknown Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make Teacher connections; develop mental math strategies and abilities to make sense of Look-Fors quantities; model mathematics in contextualized experiences Previous Experience • Understanding and solving: Develop and use multiple strategies to engage with Concepts: Students have had in problem solving experience with joining problems with the two • Communicating and representing: Communicate mathematical thinking parts known and the whole unknown. in many ways; represent mathematical ideas in concrete, pictorial, and symbolic forms Content • Addition and subtraction to 20 (understanding of operation and process): Decomposing 20 into parts Possible Learning Goal • Solves addition (joining) problems with contexts that vary the position of the unknown, using various tools and strategies • Creates and solves a joining problem with either the change or start unknown • Explains or shows how they solved the problem and how they know that their answer makes sense • Explains the action that is taking place in the story and identifies what two parts are being joined together (or separated, if they worked backwards) • Explains their strategy using mathematical language (e.g., join, plus, equals) About the Students are mostly exposed to joining problems that give the parts and the whole is unknown. Problems with different structures, such as the change unknown (e.g., 6 + — = 8) or the start unknown (— + 3 = 7) are often overlooked. Marian Small states that students need exposure to a wide variety of these structures, so they develop “a more complete understanding of addition and subtraction.” She adds, “the different meanings for the operations can be distinguished only through appropriate contexts” (Small, 2009, p. 106). 124 Number and Operations

Math Vocabulary: The order in which the various types of problems are presented helps to aeedqqduu,aaplt,ilouensq,,umpallaustscighsniing, gnjo,in, scaffold the learning. Clements and Sarama highlight how the position of equation the unknown can affect the difficulty of the problems. For example, result unknown problems (5 + 4 = —) tend to be easier than change unknown problems (5 + — = 9), and start unknown problems tend to be the most difficult (— + 4 = 9), with students often using guess and check as the major strategy (Clements & Sarama, 2009, p. 62). About the Lesson In this lesson, students are exposed to joining problems in contexts that describe the unknown in various positions. The focus is on deriving meaning from the stories and clarifying understanding through discussion. Actually creating or understanding the related equations are not the goals of this lesson since this can take a long time to master. The equations are included to clarify the thinking of some students who find the numerical form easier to understand. You do not need to introduce the equations if your students are not ready for them. You may also decide to use the ‘story starter’ frames students are using in the Working On It section instead. (See the sample ‘story starter’ below.) 5 and . and 3 . Now there are 7 . Now there are 8 Materials: Minds On (20 minutes) chart paper, concrete materials • Pose the following story to students. (Change the problems by using your Time: 50 minutes students’ names and familiar situations, so students can connect to the context.) − “2 children are in the gym. 1 more child comes into the gym. How many children are in the gym now?” Ask how they would solve this story and what the matching equation would look like (2 + 1 = 3). Record the matching equation with a simple drawing that depicts the problem. Alternatively, you can record the problem using a ‘story starter.’ • Pose the next problem. − “3 children are in the gym. Some more children enter the gym. Now there are 5 children. How many children entered the gym?” Have them turn and talk with a partner about how this problem is different from the first problem (e.g., We know the final amount, but not how many are being added to make it.). Optional: If you decide to link it to a matching equation (3 + = 5) or a ‘story starter,’ record each part as you say, “3 children AND some more Addition and Subtraction to 10 125

join them. Now there are 5 children.” Connect this to the previous question by covering up the 1 in 2 + 1 = 3. Ask them to put a context to it. (e.g., 2 children are in the gym. Some more join them. Now there are 3 children.) • Have students act out the problem to find the solution. • Pose the next problem. − “There are some dogs in the park. 4 more dogs come to the park. Now there are 6 dogs in the park. How many dogs were in the park at the beginning?” Have some students act it out. Share their strategies (e.g., Some students may work backwards thinking of 6 as the total and 4 as a subset, leaving 2 dogs at the start, or they may use trial and error.). Optional: Ask what the matching equation or ‘story starter’ would be (— + 4 = 6). Cover up the beginning parts of the two previous equations and have students put a context to them so they can connect the same meaning in all three scenarios. • Ask what is the same and different about all three types of problems. (e.g., Sometimes we know the final amount and sometimes we don’t. Sometimes we don’t know how many are at the beginning.) Working On It (15 minutes) • Students can work in pairs. Give students the following ‘story starters’ and have students create stories to complete them. You can print the story starters on chart paper. Students can represent the stories with concrete materials and solve them. • Have students create their own stories and record them using the ‘story starter’ format or pictures. (Students will be verbally sharing their stories, so they just need something to remind them of what they did.) 5 and . and 3 . Now there are 7 . Now there are 8 . Differentiation • You may want to only give ‘change unknown’ problems to students since ‘start unknown,’ as noted by Clements, are much more challenging to solve (Clements & Sarama, 2009). • For some students who are not ready for ‘change unknown’ or ‘start unknown’ problems, they can solve ‘result unknown’ problems. You can offer prompts to further the learning (see Assessment Opportunities). • For students who need more of a challenge, you can have them create the matching equations for their problems. 126 Number and Operations

Assessment Opportunities Observations: Look for students who might just be extracting the two numbers and adding them together without paying attention to the context. Conversations: • Verbally give students a ‘result unknown’ problem to solve using concrete materials (e.g., 5 birds + 2 birds = ). Keep the two parts distinct (e.g., use differently coloured cubes). Teacher: How many birds are there now? How do you know? Student: 7. I pushed the 5 and 2 together and counted them up. Teacher: You say there are 7 birds. Imagine there are only 5 birds (cover up the 2 cubes). What has to happen so there are 7 birds? How do you know? Student: 2 more birds have to join them. They are under your hand. Teacher: So if I told you there are 5 birds and some more joined them and now there are 7 birds, how many birds joined them? Student: That would be 2. Teacher: (cover up the 5 cubes this time) What if there were 2 birds and more joined them, so there are now 7 birds? How many more birds came along? How do you know? • V erbally pose another ‘change unknown’ problem and have students figure it out with concrete materials. There is no need to attach an equation at this point. It is more important that students can use context to help develop new strategies. Teaching Tip Consolidation (15 minutes – 5 minutes for meeting with It may be beneficial another pair and 10 minutes for meeting as a whole class) to have the Consolidation the • Have students meet with another pair. They can take turns verbally sharing the next day to break up the lesson. story that they made up, while the other pair solves it. • Meet as a class. Selectively choose two or three stories that have the unknown in varying positions within the context. Focus on the different ways in which students solved the problems (e.g., Did they work backwards? Did they guess and check?). Optional: As students share, record a matching equation. Ask students how each part of the equation connects to the story. • Ask what is the same about all of the stories. (e.g., They all involve joining two parts to make one whole.) • B uilding Growth Mindsets: Ask students what problems they found hard to solve. Remind them of how much they have learned about adding in the past few days. Reinforce the message that some learning is hard and they may not understand it YET, but by practising hard work, it will get easier. This supports the First Peoples Principles of Learning that learning involves patience and time. Addition and Subtraction to 10 127

Materials: Further Practice large arithmetic rack, small arithmetic • It is important to note that ‘change unknown’ and ‘start unknown’ problems racks for partner use, chart paper, markers, offer much more of a challenge for many students and they will probably not BLM 9: Number Cards understand them after one lesson. Try verbally presenting one simple word (0–10) (one set from problem each day that varies the position of the unknown. For example, “Jen 1–10 per pair of and Amit are at the round table. Some more students join them. Now there students) are 5 students. How many students came?” Students can act out the problem. As they act it out, ask them what the equation would look like and Teaching Tip record it on chart paper. Keep adding equations as more problems are presented. On other days, identify one of the equations and have students Integrate the math make up a story that could match it. This activity can be completed outside talk moves (see of the math lesson when an appropriate context arises, or when there page 6) throughout happens to be five free minutes. Math Talks to maximize student • Independent Problem Solving in Math Journals: From time to time, participation and active listening. have students draw a picture of a given change or start unknown problem that you have posed to the class. Have them record the matching equation or you can help scribe it for them as they describe the action. Math Talk: Math Focus: Solving joining problems with the unknown in varying positions Let’s Talk Select the prompts that best meet the needs of your students. • Pose the following problem using an arithmetic rack, adjusting the context to match the interests of your students: − We want to have a collection of 8 books on 2 shelves. So far, we have 3 books on the top shelf. Slide across 3 beads on the top row. How many books will go on the bottom shelf? • T urn and talk to your partner about how you might solve this. Show your thinking on the arithmetic rack. Put your thumb up when you have a solution. • P ossible strategies to discuss in greater detail: − Count 3 times: Recount the 3 beads, 1, 2, 3, then count on 4, 5, 6, 7, 8, while adding 5 beads to the bottom. Then counting the beads on the bottom, 1, 2, 3, 4, 5. − Same as above, except they ‘see’ the additional group of 5 (subitizing). − Comparing: Slide across 8 beads on the top row and 3 beads on the bottom row, visually match the beads on the top and bottom rows, and count the ‘leftovers’ as 1, 2, 3, 4, 5. • How did you solve your problem? How do you know your answer makes sense? Did anyone solve it in a different way? How are the strategies the same and how are they different? How could I finish off my representation to solve the problem? 128 Number and Operations

• Repeat with other numbers to represent total books and the original collection. • S lide 5 beads across on the top row and cover the entire top row (with a piece of folded cardboard) so students cannot see any of the beads. I have represented a certain number of books and hidden them under the cardboard. A student added 2 more books to the collection. Slide across 2 red beads on the bottom row. Now there are 7 books in the collection. How many books were there at the start? • Turn and talk to your partner and show your thinking on the arithmetic rack. Put up your thumb when you are done. • P ossible strategies: − Count on from the visible set by adding until they reach the total. − Build the total set on one row and the partial set on the other row. See the matching beads on the two rows and count the ‘leftovers’ in the total set. • H ow did you solve the problem? Can anyone show another way? How is your way like Jon’s way? Reveal the beads to confirm their thinking. Partner Investigation • S tudents work in pairs. Student A uses the top row while Student B uses the bottom row. Together, they choose a card from a deck of numbers 1–10. This will represent the total set. The first student slides across any number of beads that are less than the total on his/her row. The second student slides across the number of beads needed to complete the set on his/her row. Students confirm that the number of beads is correct by counting. • S tudents take turns being the first to slide the beads across. Follow-Up Talk • Discuss some of the strategies that students used. Record them on chart paper. Addition and Subtraction to 10 129

7Lesson Representing Separating Problems Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make Teacher connections; develop mental math strategies and abilities to make sense of Look-Fors quantities; model mathematics in contextualized experiences Previous Experience • U nderstanding and solving: Visualize to explore mathematical concepts with Concepts: • Communicating and representing: Communicate mathematical thinking Students have had experience with addition in many ways; represent mathematical ideas in concrete, pictorial, and and have been exposed to symbolic forms separating problems. Content • Addition and subtraction to 20 (understanding of operation and process): Decomposing 20 into parts Possible Learning Goal • Represents ‘separating’ problems by acting them out or using concrete materials • Creates a story problem that involves the action of separation • Selects an appropriate strategy to solve a separating problem and explains or shows how it works • Identifies the whole and the parts in the story • Matches the parts of the equation to the story context • Explains what the minus sign and equal sign mean Msspeeuialqgqubtnuuhst,ra,aaVlttpcaoisoltkcui,ngeas,mnbas,muiwnijlgaoauantirsnyc,y,,,h:eaminqdiugnda,uls, About the equation Addition and subtraction are inverse operations. As Marian Small points out, since they “undo” each other, “any addition situation can also be viewed as a subtraction one, and vice versa” (Small, 2009, p. 107). Alex Lawson adds that while some curricula introduce subtraction well after addition, this does not benefit students. She stresses that “children who have a solid sense of cardinality and who have begun to develop an understanding of the part-whole relationship should be able to learn to subtract” (Lawson, 2016, p. 23). This is especially true when students use a ‘think addition’ or counting up strategy to solve problems, rather than a subtraction strategy. 130 Number and Operations

Materials: Minds On (15 minutes) concrete materials • Propose the following problem, but change the names and the context so it (e.g., connecting cubes or counters), connects to the experiences of your students: 7 children were playing in the chart paper, markers park. 5 had to go home. How many children were left? Time: 45–50 minutes • Have students turn and talk with a partner and visualize the problem. Discuss their visualizations. Have some students act out the problem according to the instructions of the class. Ask how this problem is different from the other problems that they have acted out in the past. (e.g., People are leaving rather than joining the group; they end up with less people rather than more people.) • Introduce the minus sign as representing ‘removing or taking away’ as you create an equation that matches the scenario. Explain that the whole is being broken up into two parts. Have students identify the whole and the two parts in terms of the context. • Offer another problem to act out if students are still uncertain about the ‘removing’ action. Working On It (15 minutes) • Have students work in pairs. Have them create two or three story problems that involve the action of separating the whole. They can choose how they want to represent their problems (e.g., act it out, use concrete materials). Encourage students to be creative and to use materials from around the classroom to help tell their stories. Tell them that they can choose to create a matching equation for their story or wait until later to do so. Differentiation • Some students may have difficulty transitioning from ‘joining’ to ‘separating’ problems presented in a context, not because of the math, but due to the language used in the stories. Offer a familiar but simplistic context using real objects (e.g., books placed on and off of a shelf) to enact an addition problem (e.g., 3 books on a shelf, 2 more added), and then enact a subtraction problem with the same materials (e.g., 3 books on a shelf, 2 taken away). Pair the language (e.g., adding, joining, subtracting, taking away) with the actions. Using the same context, describe a similar problem with different numbers and have them act it out. It is important to simplify the language, but not the math, since students who need support with the language can capably learn with their peers when the language is no longer an obstacle to learning. Assessment Opportunities Observations: Observe whether there are students who continue to make joining problems and do not grasp the separating action. Conversations: If students created a joining problem, ask them to explain what is happening. (e.g., There are 3 students. 2 students sit with them and now there are 5 students.) Ask how many they are starting with in their story and whether they end up with more or less after the story is done. Ask how they could change their story so they end up with less at the end. (e.g., There are 3 students. 1 student leaves. How many students are left?) Reinforce how the action differs. Addition and Subtraction to 10 131

Consolidation (15–20 minutes – 5–10 minutes for meeting with another pair and 10 minutes for whole-class discussion) • Have each pair of students meet with another pair. They take turns proposing their word problems to the other students, who then solve it by acting it out or using concrete materials. • Meet as a class. Since the possible learning goal is about representing separating problems, make connections between students’ representations by acting them out or using concrete materials. Select two or three pairs of students to present their problem to the whole class and explain how they represented it. Ask students how each of the problems can be represented using an equation. Record each equation as students give their explanation, pausing to show how each part relates to the context. Check for understanding by asking what each part represents. Save this list of equations for the next lesson. • Ask why we might use numbers and symbols to represent problems, rather than words. • Building Growth Mindsets: Students are sometimes overwhelmed when they are first introduced to a new concept. They may get quiet and feel embarrassed if they make mistakes or feel confused. Discuss how subtracting or ‘separating’ is a really new concept to understand and they have lots of time in grades one, two, three, and beyond to understand it. Reinforce the message that making mistakes actually helps our brain to grow so we can do things better. • You can also raise students’ curiosity by asking what they wonder about subtracting stories. Encourage wondering since that is what makes mathematicians explore new ideas. Further Practice • Independent Problem Solving in Math Journals: Verbally pose one of the following prompts: – Draw a picture that shows adding or joining. Draw a picture that shows subtracting or taking away. Write a matching equation for each. – Extending Understanding: Draw pictures to match 4 + 3 and 4 – 3. Show how they are the same and how they are different. 132 Number and Operations

8Lesson Strategies for Solving Separating Problems Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make connections; Previous Experience develop mental math strategies and abilities to make sense of quantities with Concepts: Students have had • U nderstanding and solving: Develop and use multiple strategies to engage experience with addition and have been exposed to in problem solving separating problems. • Communicating and representing: Explain and justify mathematical Teacher Look-Fors ideas and decisions Msspeeuialqgqubtnuuhst,ra,aaVlttpcaoisoltkcui,ngeas,mnbas,muiwnijlgaoauantirsnyc,y,,,h:eaminqdiugnda,uls, • Connecting and reflecting: Reflect on mathematical thinking equation Content Materials: concrete materials, • Addition and subtraction to 20 (understanding of operation and BLM 5: Blank Ten- Frames, process): Decomposing 20 into parts; addition and subtraction are related arithmetic racks, chart Possible Learning Goal paper, markers • Solves ‘separating’ problems using a variety of strategies and concrete Time: 40–45 minutes materials • Selects an appropriate strategy for solving separating problems • Explains or shows strategy using concrete materials • Solves the problem in more than one way • Describes the action taking place in separating problems About the Lesson Now that students have had experience with creating and representing ‘separating’ problems, they can focus on investigating various strategies to solve the problems. Minds On (10 minutes) • Review the subtraction equations that were recorded in Lesson 7. Ask what action is happening in the problems. (e.g., taking away) Ask how the equation represents the various parts of the story. • Pose the following problem (change the numbers and context to suit your class): “There are 9 dogs. 4 run away. How many dogs are left?” Addition and Subtraction to 10 133

• Tell students that they are going to solve this problem in as many different ways as they can. They can use various tools to help them solve the problem (e.g., ten-frames, counters, arithmetic rack). Working On It (15 minutes) • Students work in pairs. Have them solve the problem in as many ways as possible, using concrete materials. They can draw a picture of one way they solved the problem so it can be shared with the class. Differentiation • If pairs of students can only find one way of solving the problem, arrange for them to meet with another pair that has a different idea. They can explain their strategy to each other and then continue working, either together, or back in their original pairs. Assessment Opportunities Observations: Pay attention to the various strategies that students use. Ensure that they are different strategies according to the thinking involved and not because they are represented with different concrete materials (e.g., cubes or counters). Students may: – Count three times: Count out a group of 9, remove 4 (counting 1, 2, 3, 4), and count how many are left (1, 2, 3, 4, 5). – Count out a group of 9, and then remove 4 as they count back 8, 7, 6, 5, keeping the 4 counters separate so they can see when they have removed enough (or keeping track of the 4 on their fingers). – Build 9 on a ten-frame, remove the group of 4 all at once, and recognize the remaining counters as a group of 5. Similarly, students may slide across 9 beads on an arithmetic rack in one move (e.g., viewing 9 as 1 less than 10), seeing the group of 4 white beads, and then sliding them back, leaving 5 red beads. – Count out 4 counters, and then add more counters, counting on from 4 as 5, 6, 7, 8, 9, stopping when they reach the count of 9. – Count out 4 counters, add some more, and then recount to see if they have reached 9 yet. Conversations: At this early point, it is best to let students investigate on their own rather than imposing a strategy on them. Encourage them to find other ways or try using different materials or tools (e.g., “Would a ten-frame help you solve this problem?”). They may choose to represent the same thinking with different materials, which is fine. Further input from other students in the Consolidation may help them broaden their thinking. 134 Number and Operations

Materials: Consolidation (15–20 minutes) large and small • Strategically select two to four solutions that reflect the various strategies arithmetic racks (or BLM 5: Blank Ten- that students used (possibilities are outlined in Assessment Opportunities). Frames and counters) • Have the creators of the chosen solutions show and explain one strategy (e.g., either their concrete models or their drawings). Ask the rest of the class if they solved it in the same way (they can show thumbs up), or whether they had another way of solving the problem. If students think they have another strategy, but they solved it in the same way using different concrete materials (e.g., representing it with cubes rather than counters), ask if the thinking is the same or different. Connect how the thinking is the same (e.g., they used counting three times, once with counters and once on the arithmetic rack). • As students share, create an anchor chart of their different strategies. Use drawings and annotate them. Alternatively, take photos of a solution that reflects each of the strategies and add them to the anchor chart. Further Practice • Show students the anchor charts for adding and separating problems. Discuss how addition and subtraction are different (e.g., In addition, we know the parts and find the whole; in subtraction, we know the whole and a part and find another part). NOTE: We cannot conclude that addition always ends up with a larger sum and subtraction always ends up with a smaller difference because this is not true for 0 or when working with integers. • Independent Problem Solving in Math Journals: Verbally pose the following prompt: − Draw pictures to show how 6 + 2 and 6 – 2 are different. Math Talk: Math Focus: • Representing separating problems as whole-part-part relationships • Exploring various strategies (e.g., counting back, counting up) Let’s Talk Select the prompts that best meet the needs of your students. • Pose some of the following prompts. • V isualize the following problem. There are 7 birds in a tree. 2 fly away. What do you visualize? What action is happening? (e.g., birds are going away) Do you think there will more or less birds in the tree? Why? (e.g., There will be less because birds are leaving.) • With your partner, show the problem on the arithmetic rack in at least one way. continued on next page Addition and Subtraction to 10 135

Teaching Tip • Possible solutions they may suggest to probe further: Integrate the math − Slide across 7 beads all at once on the top row, and then slide back 2 talk moves (see white beads (either one at a time or together) and count the remaining page 6) throughout red beads, 1, 2, 3, 4, 5. Math Talks to maximize student − Same as above except they just ‘see’ the group of 5 (subitizing). participation and active listening. − Slide across 7 beads at once on the top row and count back from 7..., 6, 5 as they remove 2 beads (counting back). − Slide across 7 beads on the top and 2 beads on the bottom, match the 2 beads on the top and bottom and count the ‘leftovers’ on the top as 1, 2, 3, 4, 5 (counting up). • How are the solutions the same? (e.g., They all reveal a group of 5 in the end.) How they are different? (e.g., removing versus matching and counting up) • Pose another problem, such as 9 – 5. • Possible solutions they may suggest to probe further: A: Removal Strategies Step 1: Creating 9 − Slide over 5 red and then 4 white beads one at a time on the top row (subitizing 5, then counting on from 5). − Slide over the entire row of 10 on the top and then slide 1 bead back (establishing a benchmark of 10 and seeing 9 as 1 less). − Slide over 9 all at once (above strategy without having to remove 1). Step 2: Once all 9 are on the top row (at the left) − Slide 5 beads back one at a time, and count back 9..., 8, 7, 6, 5, 4, keeping track of the 5 on their fingers. − Slide 4 white beads back all at once (subitizing by colour), and then 1 more, and count up the remaining beads 1, 2, 3, 4. − Slide 5 beads back all at once, and see the 4 remaining (subitizing). − Slide 5 beads across on the bottom row all at once (subitizing), visually match up the 5 beads on the top and the 5 beads on the bottom, and count the beads that don’t match up (1, 2, 3, 4) or subitize the 4 white beads. − Automatically ‘see’ the group of 5 red beads in the set of 9, and subitize the remaining beads, answering “4.” B: ‘Think Addition’ Strategies − Slide 5 beads across on the top row, count on from 5…, 6, 7, 8, 9, adding 1 white bead at a time until reaching the count of 9. • Y ou can create an anchor chart of their solutions or wait until after the partner investigation. 136 Number and Operations

Materials: Partner Investigation BLM 4: Ten-Frames • Give each pair an arithmetic rack. Have them pose subtracting problems to (0–20) using quantities to 10 (or each other to solve. Encourage students to solve them in many ways. arithmetic racks), counters, class number Follow-Up Talk line • A fter the investigation, create the anchor chart of strategies or add them to the chart if it was previously started. Math Talk: Math Focus: – 1, – 2 strategies Let’s Talk Select the prompts that best meet the needs of your students. • S elect some of the following prompts to use in your Math Talk. You may decide to cover different concepts on different days. The following dialogue highlights both concepts to serve as an example. • B riefly show students different quantities up to 10 on a ten-frame. Following is a possible dialogue after showing 9 on the ten-frame: • H ow many counters did you see and how did they look? (e.g., I saw 9. I saw 5 and 4 more on the bottom row.) Put your thumb up if you saw the same thing. • S how the ten-frame again. How did you count the 9? (e.g., I counted 1, 2, 3, 4, 5, 6, 7, 8, 9; I counted 5 and then 6, 7, 8, 9.) Why did you start counting at 5? (e.g., I know there are 5 on the top, so I don’t have to count those.) How many did you add on? (e.g., I added on 4 because I put up a finger for every time I counted on.) Show us how you counted. Let’s all try that. How could we show 5 and that many more with numbers? (5 + 4) • D oes anyone see this quantity in another way? (e.g., I saw 9 like 10 with 1 space empty.) Can someone add to what Maya said? Can you show us? How did you figure out 9? (e.g., I just saw 9 right away; I counted back 10…, 9.) Where are 10 and 9 on the number line? (e.g., They are 1 number apart.) If we first see 10, then how can we represent 9 with numbers? (e.g., 10 – 1) • V isualize taking 1 away from the 9. What do you see? (e.g., I see 8 because it is 1 less than 9.) How can you prove that? Where are 9 and 8 on the number line? (e.g., They are 1 apart.) How can we represent this with numbers? (9 – 1) • D id anyone see it a different way? (e.g., I counted back 10..., 9, 8.). Where are 10 and 8 on the number line? How could we represent this with numbers? (10 – 2) • R epeat with other numbers, connecting them to the counting sequence on the class number line and a number expression. Partner Investigation • Students take turns showing each other either fast images of numbers on ten- frames, or number expressions, and saying what 1 less or 2 less would be. Follow-Up Talk • M ake an anchor chart of students’ strategies. Addition and Subtraction to 10 137

to9 11Lessons Part-Part-Whole: Composing Quantities Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make connections; Previous Experience develop mental math strategies and abilities to make sense of quantities with Concepts: Students have worked • Understanding and solving: Develop, demonstrate, and apply with addition problems and can create matching mathematical understanding through play, inquiry, and problem solving; equations. visualize to explore mathematical concepts • Communicating and representing: Explain and justify mathematical ideas and decisions; represent mathematical concepts in concrete, pictorial, and symbolic forms • Connecting and reflecting: Connect mathematical concepts to each other and to other areas and personal interests Content • W ays to make 10: Decomposing 10 into parts; benchmarks of 10 and 20 • Addition and subtraction to 20 (understanding of operation and process): Mental math strategies: counting on, making 10, doubles; addition and subtraction are related • M eaning of equality and inequality: Demonstrating and explaining the meaning of equality and inequality; recording equations symbolically using = and ≠ 138 Number and Operations

Math Vocabulary: About the ceeoqqmuuaabttiiniooanntssio,,nmbsa,altacnhcineg, cdoemcopmospeo,se Cathy Fosnot emphasizes the importance of students developing an understanding of part-whole relationships in early number sense (Fosnot, 2007, p. 5). In the following three lessons, students are challenged to find all ways in which a quantity can be decomposed into two groups. Composing numbers in various ways helps students see the parts within a whole. These activities also reinforce the concepts of compensation and equivalence (5 + 3 = 4 + 4), which Cathy Fosnot identifies as big ideas. Fosnot states that it is significant when students realize that “if you lose one (from the 5, for example) but gain it (onto the 3), the total stays the same” (Fosnot, 2007, p. 6). About the Lessons In Lesson 9, students investigate combinations of 5 within a meaningful context, since it is a benchmark number for determining other related quantities. While students did similar investigations in kindergarten, it is important to repeat this activity since it creates a bridge to formal addition and subtraction when students use matching equations to match concrete representations. Students may also use more systematic strategies as their understanding increases. In Lessons 10 and 11, students continue to investigate all the combinations of numbers up to 10 and record their related equations. Addition and Subtraction to 10 139

9Lesson Part-Part-Whole: Composing 5 Teacher Possible Learning Goal Look-Fors • Composes and decomposes 5 in a variety of ways, using concrete materials • Creates all different combinations for 5 using two addends, using concrete materials or drawings • Explains how all combinations equal 5 • Creates matching equations for all of the combinations • Recognizes and explains (or shows) how two equations such as 2 + 3 and 3 + 2 result in the same total • Recognizes and explains (or shows) that equations such as 2 + 3 and 3 + 2 can be different when presented in a story context Materials: Minds On (15 minutes) chart paper, markers, • Pose this problem: 3 students can choose between 2 different activities, concrete materials (including connecting either skipping or bean bag toss. What different groupings might occur once cubes), BLM 6: Blank the students make their choices? (3 skipping/0 bean bag; 2 skipping/1 bean Five-Frames bag; 1 skipping/2 bean bag; 0 skipping/3 bean bag) Time: 45–50 minutes • Have three children act out the suggestions given by the rest of the class members. Ask students what the corresponding equations would look like, and record them on chart paper (3 + 0 = 3, 2 + 1 = 3, 1 + 2 = 3, 0 + 3 = 3). • Have students turn and talk with a partner to discuss whether there would be more combinations if there were 4 children. Have them justify their responses to the class. Have students act out the problem of 4 students choosing between the 2 activities. Record the equations that students create. Have students confirm whether there are more or less combinations for 4 students than for 3 students. • Ask students how they could model the question without acting it out. (e.g., Use counters or blocks for people.) Ask how they could use 2 five- frames to represent the 2 activities. Based on students’ suggestions, model how they could split up the 4 students between the 2 five-frames. Emphasize that these are just some possible ways to model the problem. Working On It (10 minutes) • Tell students they are going to work in pairs to investigate how many combinations there will be for 5 students choosing between the 2 activities. They can use any methods and materials they choose to solve the problem. Students can record each combination by making a train using 2 different colours of connecting cubes (they could potentially have 6 different trains). 140 Number and Operations

Differentiation • For students who find this task too challenging, have them work in groups of three or four and act out the combinations for 3 and 4 as was modelled in the Minds On. They can refer to the chart of combinations to help them. Assessment Opportunities Observations: Pay attention to how students are creating their combinations. – Are they randomly making combinations of 5, or are they using a systematic strategy? – Are they clearing off their five-frames each time, or do they build on from one scenario to another? – Do they need to count each time to ensure there are 5, or do they just ‘know’ that there are 5 each time? If students are displaying the latter behaviours, they probably have a good sense of relationships between numbers and are able to apply it in problem-solving situations. Consolidation (20–25 minutes — 10 minutes for Inside/ Outside Circle and 10–15 minutes to discuss as a class) • I nside/Outside Circle: Have each student in the pair take two of the trains they created in Working On It and hold them behind their backs. Have pairs of students stand in a line facing their partner. Bend the line around to make two circles, one inside the other, with partners still facing each other and holding their trains behind their backs. On a signal, have everyone in the outside circle move one person to the right so they are facing a new partner. Students take turns briefly showing their trains, one at a time, and then hiding them again. The partners say what they saw (e.g., “I saw 4 and 1.”). Students can show their trains again if the partner needs more time. After a short time, have the students on the outside move one partner to the right again. (You can signal the time to change with a bell or by playing music.) Students now have a new partner and can play the game again. Repeat this for three or four rotations. • Meet as a class. Have students share their findings and tell whether they saw any combinations they didn’t have when they played the Inside/Outside Circle game. • Ask students how they found their combinations and whether they used a specific strategy. As students share their combinations, record them as addition expressions (e.g., 2 + 3). • Ask students how they could record their findings in an organized list. Ask what happens to the second quantity if the first quantity is decreased by 1, and equality is maintained (5 + 0, 4 + 1, 3 + 2, 2 + 3, 1 + 4, 0 + 5) and rewrite the list according to their suggestions. This is to reinforce the idea of compensation and equivalence. Addition and Subtraction to 10 141

Materials: • Ask whether matching facts such as 3 + 2 and 2 + 3 represent the same thing BLM 6: Blank Five- Frames, counters, in the context of the activities. Ask whether the sums are the same chart paper (commutative property). Teaching Tip • Draw attention to the equation 5 + 0 = 5 and ask what this means. Have Integrate the math them explain why the original number does not change. talk moves (see page 6) throughout Further Practice Math Talks to maximize student • Independent Problem Solving in Math Journals: Have students record participation and active listening. all combinations of 5 in their math journals and show how they are the same and how they are different. Math Talk: Math Focus: Benchmark of 5, combinations of 5 Let’s Talk Select the prompts that best meet the needs of your students. • Show students an empty five-frame. What do you see? Visualize a counter in each square. What do you see now? Discuss their mental images. • P ut 1 counter on an empty five-frame and then quickly cover it up. What did you see and what do you need to add to fill the five-frame? (e.g., I saw 1 counter and I need 4 more counters.) How could we show this with numbers? (e.g., 1 + 4 or 1 + 4 = 5) Repeat this with other combinations, varying the positions of the counters and building matching equations for each. What does the expression/equation mean? (e.g., 1 counter is there and we need 4 more to fill the five-frame.) Partner Investigation • Students can take turns showing their partners up to 5 counters in random positions on the five-frame. The partners say what they see and what they visualize. As an option, they can say the matching equation (e.g., 3 plus 2 equals 5). • Alternatively, students can take turns showing expressions that represent 5, and create them on their five-frame. Follow-Up Talk • S how students the expression 2 + 3. What do you visualize on a five-frame? Show them 3 + 2. What do you visualize now? How are they the same and how are they different? What could you do in your mind to make the one look like the other? (e.g., turn it upside down) • D id you ever show a completely filled or a completely empty five-frame? What equations could show these? (e.g., 5 + 0 or 0 + 5) What do they mean? 142 Number and Operations

Materials: Math Talk: chart paper, Math Focus: Addition and subtraction with 0 markers, counters About the Although 0 is a real number, it can be confusing because the rest of the numbers can be shown with a visual quantity, while the number 0 is represented by an empty set. It is worthwhile investigating the impact that 0 has on addition and subtraction, since it challenges the misconception that addition always results in a larger quantity and subtraction always results in a smaller quantity when working with whole numbers. Let’s Talk Select the prompts that best meet the needs of your students. • Have students work with a partner to represent the following scenarios with concrete materials. (You may decide to do the + 0 and – 0 problems on different days.) • Imagine that you have 4 cookies. I am going to give you 2 more cookies. How many cookies do you have now? Use your counters and act it out with your partner. • W hat action is happening in the story? (e.g., adding, joining) How do you know? (e.g., We ended up with more cookies.) How could I write this as an equation? (4 + 2 = 6) If I give you a story that is exactly the same except for the number of cookies, what do you think will happen at the end of the story? (e.g., There will be more cookies.) Let’s find out. • Act out the following problem with your partner. Imagine that you have 4 cookies. I am going to give you 0 more cookies. How many cookies do you have now? • W hat action is happening in the story? (e.g., It is adding, but it is adding none; there is no action.) How did you represent this with your counters? How is this the same as the first question? How is it different? How can we show this with numbers? (e.g., 4 + 0 = 4) If necessary, pose other adding 0 scenarios and have students act them out. • We created an adding question, but it didn’t end up with more cookies. Does adding always end up being more than what you started with? Why? • V isualize this problem. You have 4 cookies and you eat 2 cookies. How many do you have left? What kind of a question is this? (e.g., It is a subtracting question because cookies are taken away.) How can we show this with numbers? (4 – 2 = 2) I wonder if subtracting questions will always end up with less. You can challenge your partner with some problems and try it out. continued on next page Addition and Subtraction to 10 143

Partner Investigation • H ave students work with a partner to find out if subtraction always ends up with less. Follow-Up Talk • Discuss whether subtraction always ends up with less and how they can prove it. Some people think that addition always makes things bigger and subtraction always makes things smaller. What would you say? How could you prove your thinking? 144 Number and Operations

10Lesson Part-Part-Whole: Composing 6, 7, 8, and 9 Teacher Possible Learning Goal Look-Fors • Composes and decomposes quantities 6 to 9, using concrete materials and a variety of strategies • Finds most or all of the combinations for their number • Explains how all of their combinations represent the same total • Creates matching equations for their combinations • R ecognizes and explains (or shows) that equations such as 4 + 3 and 3 + 4 represent the same quantity • Recognizes that adding 0 to any number results in that number and explains why this is so About the Lesson In this lesson, students are assigned one of the numbers (6, 7, 8, or 9) and asked to find the combinations for only that number. Depending on the level of understanding in your class, you may decide that all students need to find the combinations for all numbers. In this case, you can repeat this lesson over several days. Students can use various methods to find all combinations, such as playing Shake and Spill as described below; modelling a scenario with students choosing between two activities or snacks; or using differently coloured connecting cubes or tiles to represent the problem. Materials: Minds On (15 minutes) two-coloured • Review all of the possible combinations for 3, 4, and 5. Ask students to counters, cup, BLM 15: Shake and Spill Game predict how many combinations there might be for 6 (e.g., will there be Combinations more or will there be less?). Have them justify their responses. Time: 60 minutes • Pose the following problem: There are 6 candies in a bowl. They can be red or yellow. What combinations of candies might there be? Show students 6 two-coloured counters and ask how they might be used to solve this problem. Shake the counters in a cup and spill them out. Ask what the differently coloured counters represent in relation to the context of the story. Ask how it could be recorded into an equation. Colour in the results on a copy of BLM 15: Shake and Spill Game Combinations for 6 and record the matching equation (e.g., 4 + 2 = 6). Tell students that they are going to find Addition and Subtraction to 10 145

all of the different combinations for either 6, 7, 8, or 9 by spilling out counters. Explain that if they spill the same result, they don’t record it again. Working On It (20 minutes) • Have students work in pairs. Assign a number, either 6, 7, 8, or 9, to the pairs and give them the same number of two-sided counters and the related recording sheet. • Have students play the game for about 10 minutes. • Tell students that they may not have gotten all of the combinations while playing the game. Have them figure out if they are missing any combinations and record them on their sheet. Differentiation • You can assign numbers that are the most appropriate for various pairs. Some may need fewer combinations to deal with and can be given 6, while others who require more of a challenge can be given 9. Assessment Opportunities Observations: Look for the same behaviours that were listed in Lesson 9. Pay attention to whether some of the students who were using more random strategies are now refining them. Conversations: • If students are counting the total each time ask, “What number are you trying to find all the combinations for? Can you predict how many combinations there will be before counting?” • If students are finding their combinations in a random fashion, point to one of their combinations on the recording sheet and ask, “Can you visualize a new combination by changing 1 counter in your mind?” Teaching Tip Consolidation (25 minutes – 10 minutes to meet with It may be beneficial another pair and 15 minutes to meet as a whole class) to have the Consolidation the • Each pair of students meets with another who has the same number and next day to break up the lesson. they determine whether they have found all combinations. Together, have them make a list of all of the combinations, either by number combinations or in equation form. • Meet as a class to discuss the numbers, one at a time. For each number, highlight how all of their responses have the same total, but have different combinations of two colours. You may decide to co-create matching equations for all of the combinations. • Ask students what they notice about the number of combinations as the totals increase from 6 to 7 to 8 to 9. • Draw attention to the combinations that involve adding 0. Ask what this means and why the answer is the same as one of the parts. If you have not 146 Number and Operations

Materials: already done so, you may want to do the Math Talk that relates to adding arithmetic racks, and subtracting with 0 (see pages 143–144). chart paper, markers Further Practice Teaching Tip • Independent Problem Solving in Math Journals: Give each student a Integrate the math talk moves (see number that they did not work with during the paired problem solving, and page 6) throughout have them find all combinations for the total using concrete materials and Math Talks to tools of their choice. They can record the matching equations in their journals. maximize student participation and • H ave students show what it means to add 0 to a number. Encourage them to active listening. include at least two examples. Math Talk: Math Focus: Combinations of 6, 7, 8, 9 Let’s Talk Select the prompts that best meet the needs of your students. • Have students work in pairs. Each pair has an arithmetic rack. • Solve this problem on your arithmetic racks. There are 6 books and we want to arrange them on 2 shelves, one above the other. You can put as many books as you want on either shelf, as long as there are no more than 6 books altogether. You can find several ways, but then select one way you like. Put your thumb up when you are finished. • W ho can show one of their arrangements? Put your thumb up if you found the same way. How do you know there are 6 books? Who solved it another way? • Record students’ combinations on chart paper. • We have found several ways. Have we found them all? How do you know? • Is 1 on the top shelf and 5 on the bottom shelf different from 5 on the top shelf and 1 on the bottom shelf? (e.g., Yes, because they look different, but there are still 6 books.) Put your thumb up if you agree. Why? Show one of the scenarios on the arithmetic rack. Visualize what my arithmetic rack would look like if I turned it upside down. Then show students as proof. Partner Investigation • C hallenge students to find all of the combinations for 7, 8, or 9 books (differentiate by changing the numbers for different students, also assign a number they did not do in the previous lesson). They can record their combinations on chart paper. Follow-Up Talk • After the investigation, have students share their combinations as you record them. Probe as to how they found the arrangements (e.g., guess and check, systematically taking one away from the top and putting one more on the bottom). This would be an example of compensation and equivalence (if you take one away from the top, you have to put it on the bottom so there are still 7 books). Addition and Subtraction to 10 147

11Lesson Part-Part-Whole: Composing 10 Teacher Possible Learning Goal Look-Fors • Composes and decomposes the quantity of 10, using a variety of concrete materials • Finds most or all of the combinations for 10 • Explains how all of their combinations represent the same total • Creates matching equations for their combinations • R ecognizes and explains (or shows) that equations such as 4 + 6 and 6 + 4 represent the same quantity • R ecognizes that adding 0 to any number results in that number and explains why this is so Time: 45–50 About the minutes Establishing a benchmark for 10 is especially important to understand and problem solve within our base ten number system. For example, by knowing combinations of 10, students can eventually mentally calculate 7 + 6 + 3 by grouping 7 and 3 into a 10, and then adding in the 6 to make 16. Minds On (5 minutes) • H ave students predict how many combinations they think they can find for the number 10. • Tell them that they are going to find all of the combinations for 10 using no more than 2 parts. They will also create a matching equation for each combination they find. Differentiation • For students who have difficulty with recording full equations, have them record the number combinations instead. 148 Number and Operations

Materials: Working On It (20 minutes) two-coloured counters or different • Following are five different versions of the same activity that deals with coloured connecting cubes finding combinations for 10. One uses a context, while others use different concrete materials. You can choose the activity variation that best suits your Materials: students. You may also decide to have the whole class do two of the activities arithmetic rack to ensure that they get the same results. Alternatively, you may decide to give each one of the activities to a small group and then discern, in the Materials: Consolidation, whether all of the groups have found the same combinations. two-coloured counters Version 1: Materials: • Introduce the following problem to students: There are 10 children and they concrete materials of can choose between two destinations for their field trip: either the apple students’ choice orchard or the zoo. What combinations might there be? Materials: • Students find all of the combinations that could occur, choosing their own relational rods strategies and materials to represent the 10 children. They may decide to use two-coloured counters or two colours of connecting cubes, or to make drawings. Have them record their findings using pictures and matching equations. Version 2: • Students can find all of the combinations for 10 on the arithmetic rack. They can record their findings on chart paper, recording little red and white dots and matching equations or number combinations. Version 3: • Students can play Shake and Spill to find all the combinations for 10. Students can record their findings using pictures of coloured circles and matching equations. Version 4: • Students use concrete materials of their choice to find all of the combinations for 10. Students can record their findings using pictures and matching equations. • First Peoples Perspectives: Students can model their thinking using natural objects from outdoors. Version 5: • Students find all of the combinations for 10 with relational rods (also known as Cuisenaire rods). • Before students begin, show them the staircase made of relational rods. Make sure that students understand that the orange rod is worth 10 and the white rod is worth 1. Ask what they think the values of the other rods are, based on this information. Label each rod with its related number. Addition and Subtraction to 10 149

Note: Version 5 requires experience with • Students work in pairs to find all of the combinations relational rods. If students have not worked with relational rods before, have them engage for 10 by building trains that are the same length as in the following introductory activity: the orange rod (10 units), but they can only use 2 rods in each train. Introductory Activity • They can trace their trains onto chart paper or take a • Give students time to explore and build with the relational rods. If someone built the picture of their work. staircase, draw attention to it and discuss how the rods get progressively longer by the • Ask students to make matching equations for their same amount moving up the staircase. Ask what number the orange rod represents if trains. Display the staircase with the labelled the white rod is 1. relational rods to help students identify the values of the rods. Assessment Opportunities Observe the students’ progress by looking for the same behaviours that were highlighted in previous lessons. Observations: Are students refining their strategies or continuing to use guess and check? Conversations: Teacher: (Show two of their combinations that differ by 1, such as 2 and 8, and 3 and 7.) How can you visualize getting 3 and 7 by moving the 2 and 8 around? (Or, put 2 counters and 8 counters on two ten-frames.) Show how you could move counters so that you have 3 on one ten-frame and 7 on the other. Student: I could move 1 from the 8 and put it with the 2. Teacher: So you moved 1 counter. What happens if you move another counter? Would you get a new combination for 10? Try it. Student: (counts both sets of counters) Now I have 4 and 6. Teacher: Could moving 1 counter be a strategy? Do you think it will work every time? Can you predict what you might get if you tried this strategy again? Observations: Can students transfer strategies from one representation to another? Conversations: Ask, “How is the strategy that you are using with the arithmetic rack like the strategy that you used with the Shake and Spill game?” 150 Number and Operations

Teaching Tip Consolidation (20–25 minutes) It may be beneficial • Hold a gallery walk. Have students show all of their representations for to have the Consolidation the combinations of 10. Give them time to arrange their representations in a way next day to break up that makes sense to them. All students ‘stray’ rather than stay with their the lesson. representations. Remind students that their goal is to look for combinations of 10 to see if they found all of the possibilities. Materials: BLM 5: Blank Ten- • As a class, ask students what they found interesting on their walk, either Frames (or arithmetic racks), with the combinations or how they were arranged. Ask whether there were counters, chart any combinations they hadn’t included. paper • Ask how students found all of their combinations (e.g., Did they work in an orderly fashion, or did they use trial and error?). • Discuss any patterns that they see (e.g., As one rod or group decreases, the partner rod or group gets larger by the same amount.). • Discuss whether the combinations of 2 + 8 and 8 + 2 are linked. Ask if there are any other combinations like that. Highlight that the order does not change the total. • Post a chart of all of the combinations for 10. Further Practice • Independent Problem Solving in Math Journals: Verbally pose one of the following prompts: − Draw how you found all of the combinations for 10. − Draw a picture to show 9 + 0 and 0 + 9. − Show how 6 + 3 and 3 + 6 are the same and how they are different. Math Talk: Math Focus: • Combinations for 10 • Compensation and equivalence Let’s Talk Select the prompts that best meet the needs of your students. • B riefly show students a partially filled ten-frame (e.g., 7 counters). What did you see and what didn’t you see? Turn and talk with your partner. Students can recreate what they saw with counters, or you can just have them visualize. What do the two amounts make together? How do you know? How could we show these two amounts making 10 using numbers? (e.g., 7 + 3) • R epeat this line of questioning for two or three other visual representations. • S how students a ten-frame with 4 counters on the top row. Are there more counters there or more not there? What numbers could show how these amounts equal 10? (Record 4 + 6.) continued on next page Addition and Subtraction to 10 151

Teaching Tip • V isualize how you could change the counters so there is the same number there Integrate the math as not there. (e.g., I could add 1 to the top row.) How does this make the talk moves (see two amounts equal? (e.g., When you add 1 counter, it fills 1 of the 6 empty page 6) throughout spaces, so now there are 5 empty spaces.) How would you represent this with Math Talks to numbers? (Record 5 + 5.) Is there another way to make equal amounts? maximize student (e.g., No, there is only one way.) How do you know? participation and active listening. • H ow could you change the ten-frame so there is 1 more there rather than an equal amount? How can you show that with numbers? (Record 6 + 4.) What do you notice about the 4 + 6, 5 + 5, 6 + 4 combinations? (e.g., If 1 counter is added, there is 1 less empty space.) Why? (e.g., They always have to equal 10 so if you add 1, there is 1 less empty.) Partner Investigation • S tudents can work in pairs to see if adding 1 counter always results in 1 less empty space, and whether adding 2 more (or 3 more) counters always results in 2 less (or 3 less) empty spaces. Further Talk • D id you always have 1 less empty space when you added 1 counter? What did you find out if you added 2 more counters? (e.g., there were 2 less empty spaces) Did it work with 3 counters? How could we make a rule about this? (e.g., When making 10, whatever you add to the ten-frame, there will be that many less empty spaces.) Materials: Math Talk: BLM 16: Dice Math Focus: Mental strategies for solving doubles BLM 1: Dot Configurations (0-6) BLM 1: Dot Configurations (0-6) Doubles or Digital Slides 94–99; BLM 17: Let’s Talk More Dice Doubles or Digital Slides 100–105; Select the prompts that best meet the needs of your students. 10 counters per student; chart paper; (Optional: • B riefly show only half of a fast image of a double on BLM 18: Ten-Frame dice (see BLM 16) and then take it away. The other Doubles (1–5) or Digital Slides 106–110 and half of the fast image is exactly the same as what you BLM 5: Blank Ten- just saw. With your partner, use counters to create what Frames) you saw, and the other half. How many counters are 8 8© 2018 Scholastic Canada Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year)© 2IS0B1N8 S9c7h8-o0l-a0s0t0ic0-C00a0n0a-d0a Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year) ISBN 978-0-0000-0000-0 there? How do you know? How could we show this with numbers? (e.g., 4 + 4) • O ne of you will change the left side of your 4 and 4 image by adding 1 counter and your partner will make the other side of the image look exactly the same. Figure out how many there are altogether. Put your thumb up when you are done. • How did you figure out the total? (e.g., we counted; we had 4 and 4, so making 5 and 5 would be 2 more, 1 for each side, so 8, 9, 10; we could just see the 10 like on two dice) 152 Number and Operations