p106-193-Gr1-ON-Number-Unit2-add-subtract

Teaching Tip • P ossible 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 7) throughout red beads, 1, 2, 3, 4, 5. Math Talks to maximize student participation − Same as above except they just ‘see’ the group of 5 (subitizing). 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 are they different? (e.g., removing versus matching and counting up) • P ose another problem, such as 9 – 5. • P ossible 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 (anchoring 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 right) − 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. • You can create an anchor chart of their solutions or wait until after the partner investigation. continued on next page Addition and Subtraction to 10 155

Partner Investigation • G ive each pair an arithmetic rack. Have them pose subtracting problems to each other to solve. Encourage students to solve them in many ways. Follow-Up Talk • After the investigation, create the anchor chart of strategies or add them to the chart if it was previously started. Materials: Math Talk: BLM 5: Ten Frames (0–20) using Math Focus: – 1, – 2 strategies for recalling subtraction facts quantities to 10 (or arithmetic Let’s Talk racks), counters, Select the prompts that best meet the needs of your students. class number • S elect some of the following prompts to use in your Math Talk. You may line 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: • How 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. • Show 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) • Does 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 one 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 • S tudents 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 • Make an anchor chart of students’ strategies. 156 Number and Financial Literacy

to11 13Lessons Part-Part-Whole: Composing Quantities Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Previous Experience between addition and subtraction, to solve problems and check calculations with Concepts: Students have worked with • B2.2 recall and demonstrate addition facts for numbers up to 10, and related addition problems and can create matching subtraction facts equations. • B2.3 use mental math strategies, including estimation, to add and subtract whole numbers that add up to no more than 20, and explain the strategies used • B2.4 use objects, diagrams, and equations to represent, describe, and solve situations involving addition and subtraction of whole numbers that add up to no more than 50 Algebra • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts PMraotcheesmseast:ical About the tcrrPoeeoorapmolsrsbemolasenuneminndngitcssinaaotrgtlnavi,nditnregepgg,fr,isloeeevsclietnicngtg,in, g 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 11, 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 12 and 13, students continue to investigate all the combinations of numbers up to 10 and record their related equations. Addition and Subtraction to 10 157

11Lesson Part-Part-Whole: Composing 5 Teacher Possible Learning Goal Look-Fors • Composes and decomposes 5 in a variety of ways, using concrete materials • C reates 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 • R ecognizes and explains (or shows) how two equations such as 2 + 3 and 3 + 2 result in the same total • R ecognizes and explains (or shows) that equations such as 2 + 3 and 3 + 2 can be different when presented in a story context Math Vocabulary: Minds On (15 minutes) ceecdoqqoemuumcaabopttiminioooasnpnetsosi,o,s, nmebsaa,ltacnhcine,g • Pose this problem: 3 students can choose between 2 different activities, Materials: chart paper, markers, either skipping or bean bag toss. What different groupings might occur once concrete materials the students make their choices (3 skipping/0 bean bag; 2 skipping/1 bean (including connecting bag; 1 skipping/2 bean bag; 0 skipping/3 bean bag)? cubes), BLM 7: Blank Five Frames • Have three children act out the suggestions given by the rest of the class Time: 45–50 minutes 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 (e.g., they could potentially have 6 different trains). 158 Number and Financial Literacy

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) • Inside/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 10 159

• Ask whether matching facts such as 3 + 2 and 2 + 3 represent the same thing in the context of the activities. Ask whether the sums are the same (commutative property). • Draw attention to the equation 5 + 0 = 5 and ask what this means. Have them explain why the original number does not change. • Building Social-Emotional Learning Skills: Healthy Relationships: Ask students how they felt working with various partners in the “Inside/Outside Circle” activity. Discuss how they can help each other to learn and how they can support classmates if they make a mistake. Explain that we all learn differently and at different times, but working together, we can learn strategies from others that might be helpful. Further Practice • Independent Problem Solving in Math Journals: Have students record all combinations of 5 in their math journals and show how they are the same and how they are different. Math Talk: Materials: Math Focus: Anchoring 5, combinations of 5 to support recall of addition and BLM 7: Blank Five subtraction facts Frames, counters, chart paper Let’s Talk Teaching Tip Select the prompts that best meet the needs of your students. Integrate the math • S how students an empty five frame. What do you see? Visualize a counter in talk moves (see page 7) throughout each square. What do you see now? Discuss their mental images. Math Talks to maximize student participation • Put 1 counter on an empty five frame and then quickly cover it up. What did and active listening. you see and what do you need to add to fill the 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 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. 160 Number and Financial Literacy

Materials: Follow-Up Talk chart paper, markers, counters • 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) • Did 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? Math Talk: Math Focus: Addition and subtraction with 0 to support recall of facts 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. • H ave 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. • What 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. continued on next page Addition and Subtraction to 10 161

• W e 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? • Visualize 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. Partner Investigation • Have students work with a partner to find out if subtraction always ends up with less. Follow-Up Talk • D iscuss 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? 162 Number and Financial Literacy

12Lesson 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 Math Vocabulary: ceecdoqqoemuumcaabopttiminioooasnpnetsosi,o,s, nmebsaa,ltacnhcine,g variety of strategies Materials: • Finds most or all of the combinations for their number two-coloured • E xplains how all of their combinations represent the same total counters, cup, BLM 14: • C reates matching equations for their combinations Shake and Spill Game • R ecognizes and explains (or shows) that equations such as 4 + 3 and 3 + 4 Combinations Time: 60 minutes represent the same quantity • R ecognizes 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. Minds On (15 minutes) • Review all of the possible combinations for 3, 4, and 5. Ask students to predict how many combinations there might be for 6 (e.g., Will there be more or will there be less?). Have them justify their responses. • 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 14: 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 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. Addition and Subtraction to 10 163

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 11 (See Assessment Opportunites on page 159.). 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 another It may be beneficial to pair and 15 minutes to meet as a whole class) have the Consolidation the next day to break • E ach pair of students meets with another who has the same number and 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 already done so, you may want to do the Math Talk that relates to adding and subtracting with 0 (see pages 161-162). 164 Number and Financial Literacy

Further Practice • Independent Problem Solving in Math Journals: Give each student a number that they did not work with during the paired problem solving, and have them find all combinations for the total using concrete materials and tools of their choice. They can record the matching equations in their journals. • H ave students show what it means to add 0 to a number. Encourage them to include at least two examples. Math Talk: Materials: Math Focus: Combinations of 6, 7, 8, 9 to support recall of addition and arithmetic racks, subtraction facts chart paper, markers Let’s Talk Teaching Tip Select the prompts that best meet the needs of your students. Integrate the math talk moves (see • H ave students work in pairs. Each pair has an arithmetic rack. page 7) throughout • S olve this problem on your arithmetic racks. There are 6 books and we want to Math Talks to maximize student participation arrange them on 2 shelves, one above the other. You can put as many books as and active listening. 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. • Who 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 • Challenge 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 10 165

13Lesson Part-Part-Whole: Composing 10 Teacher Possible Learning Goal Look-Fors • Composes and decomposes the quantity of 10, using a variety of concrete Math Vocabulary: ceecdoqqoemuumcaabopttiminioooasnpnetsosi,o,s, nmebsaa,ltacnhcine,g materials Time: 45–50 • Finds most or all of the combinations for 10 minutes • E xplains how all of their combinations represent the same total • C reates 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 About the Anchoring 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. 166 Number and Financial Literacy

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. 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 10 167

Note: Version 5 requires experience with • Students work in pairs to find all of the relational rods. If students have not worked with relational rods before, have them engage in the combinations for 10 by building trains that are the following introductory activity: same length as 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 • Give students time to explore and build with the relational rods. If someone built the staircase, a picture of their work. draw attention to it and discuss how the rods get progressively longer by the same amount • Ask students to make matching equations for their moving up the staircase. Ask what number the orange rod represents if the white rod is 1. trains. Display the staircase with the labelled 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: T eacher: (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?” Teaching Tip Consolidation (20–25 minutes) It may be beneficial to • Building Social-Emotional Learning Skills: Healthy Relationships: Before have the Consolidation the next day to break students engage in the gallery walk (see next page), ask how they can talk up the lesson. about other’s work without hurting feelings. Remind them that they are commenting on the work and not the person. Such reminders help students become more respectful and supportive of one another, and dissociates critiquing mathematical thinking from expressing opinions about classmates. 168 Number and Financial Literacy

Materials: • Hold a gallery walk. Have students show all of their representations for BLM 6: Blank Ten Frames (or combinations of 10. Give them time to arrange their representations in a way arithmetic racks), that makes sense to them. All students ‘stray’ rather than stay with their counters, representations. Remind students that their goal is to look for combinations chart of 10 to see if they found all of the possibilities. paper • As a class, ask students what they found interesting on their walk, either with the combinations or how they were arranged. Ask whether there were any combinations they hadn’t included. • 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) • Repeat 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 10 169

Teaching Tip • Visualize 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 two talk moves (see amounts equal? (e.g., When you add 1 counter, it fills 1 of the 6 empty page 7) throughout spaces, so now there are 5 empty spaces.) How would you represent this with Math Talks to maximize numbers? (Record 5 + 5) Is there another way to make equal amounts? (e.g., student participation No, there is only one way.) How do you know? and active listening. • How 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. Follow-Up Talk • Did 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 15: Dice Doubles or Digital Math Focus: Mental strategies for solving doubles Slides 104–109; BLM 16: More Dice Let’s Talk Doubles or Digital Select the prompts that best meet the needs of your students. Slides 110– 115; 10 • B riefly show only half of a fast image of a double on counters per student; dice (see BLM 15 or Digital Slides 104–109) and then chart paper; take it away. The other half of the fast image is exactly (Optional: BLM 17: Ten the same as what you just saw. With your partner, use Frame Doubles (1–5) or counters to create what you saw, and the other half. Digital Slides 116–120 How many counters are there? How do you know? How and BLM 6: Blank Ten could we show this with numbers? (e.g., 4 + 4) Frames) • 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) Partner Investigation • O ne student makes an amount up to 5 with counters and the partner makes an image that looks exactly the same. Together, they figure out the total number of counters. 170 Number and Financial Literacy

Follow-Up Talk • D iscuss the strategies that students used and make an anchor chart. • Briefly show one of the fast images, showing identical sides, and then take it away. How many dots did you see and how do you know? What is a matching equation? (e.g., 3 plus 3 equals 6) • Repeat these activities another day with other fast images of doubles (see BLM 16 or Digital Slides 110–115). • This activity can also be done with ten frames using BLM 17: Ten Frame Doubles (1–5) or Digital Slides 116–120, 10 counters per student, and BLM 6: Blank Ten Frames. • It can also be done showing identical numbers of beads on the top and bottom rows of the arithmetic rack. Further Investigation • Have students create all of the doubles up to 5 and 5 using connecting cubes of two colours, as double-decker trains (with one row on top of the other rather than as one long train). • S tudents can keep the double-decker trains and flash them to each other like fast images to practise recognizing doubles. They can respond, “I saw 3 plus 3 equals 6.” Addition and Subtraction to 10 171

14Lesson Whole-Part-Part: Decomposing 10 Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Previous Experience between addition and subtraction, to solve problems and check calculations with Concepts: Students have had opportunities to • B2.2 recall and demonstrate addition facts for numbers up to 10, and related create various combinations of the same subtraction facts quantity using concrete materials. • B2.3 use mental math strategies, including estimation, to add and subtract Teacher whole numbers that add up to no more than 20, and explain the strategies used Look-Fors • B2.4 use objects, diagrams, and equations to represent, describe, and solve PMraotcheesmseast:ical acprPenorropndovrnbseienltsercgeamt,nitncetsgoigno,migelsv,mseinrleeugacn, stiiconangtiintnoggo,alsnd situations involving addition and subtraction of whole numbers that add up to no more than 50 McmbdsoaaeuamtlcbtahcobtnrhmiVacnineocap,gtcto,icaosetobeanqmu,ksul,epaaeotraiqysowu:enaa,syt,ions, Algebra • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Possible Learning Goals • Decomposes 10 using concrete materials and various strategies • Connects addition and subtraction as operations that ‘undo’ each other • Finds most or all of the combinations for subtracting from 10 • Explains or shows how the equations match the concrete representations • R ecognizes that the same number pairs are involved in both the addition and subtraction questions for 10 About the This activity gives students more experience with finding all combinations of 10, except from a ‘whole into parts’ context by solving a separating subtraction problem. This offers a good opportunity to link the operations of addition and subtraction and show how they ‘undo’ each other. It also highlights that, unlike addition, the commutative property does not apply for subtraction since the order of subtracting does matter (e.g., 2 brownies can be taken away from 5 brownies, but 5 brownies can’t be taken away from 2 brownies). 172 Number and Financial Literacy

Materials: Minds On (10 minutes) concrete materials, chart paper, markers • Propose the following problem: There are 5 birds in a tree. Some fly away. Time: 45 minutes How many birds could be left in the tree? • Have students turn and talk to a partner to figure out what the solutions might be. Ask whether they think they have found all of the possibilities and how they know. • Ask students what subtraction equations would match their solutions. Record these equations on chart paper. Working On It (15 minutes) • Students work in pairs. Have students solve the following problem: There are 10 birds in a tree. Some fly away. How many could be left? Have students find all of the possibilities using any of the materials they used in the past lessons, and record their results (see Differentiation). Differentiation • Students can record their combinations by drawing simple pictures (e.g., differently coloured dots), by recording number combinations (e.g., 9 and 1, 8 and 2), or by creating subtraction equations (e.g., 10 – 9 = 1). Assessment Opportunities Observations: Are students applying any of the strategies that they used for composing combinations to decompose 10? (e.g., decompose by 1 more each time) Conversations: Probe deeper if students are using guess and check, and not using one combination to form another: Teacher: What did you do here? Student: I had 10, and I took 1 away and there are 9 left. Teacher: Can you visualize what would happen if you took 1 more away? Try it. Student: Now I have 8 and 2. Teacher: Do you still have 10 altogether? Student: (Student may answer ‘yes’ immediately or may need to count.) Teacher: (if they counted again) How many birds should there be altogether? How do you know? Teacher: What strategy did you just use? Student: I took 1 more away. Teacher: Will that work every time? Student: I’m not sure. Teacher: Why don’t you try, and see what happens? Addition and Subtraction to 10 173

Consolidation (20 minutes – 10 minutes to meet with a partner and 10 minutes to meet with the whole class) • Have each pair of students meet with another pair to compare their solutions and to determine if they have all the combinations. • As a class, discuss students’ strategies and record all of the subtraction equations. Ask students how they could put the combinations in a list so they show a pattern. • Show students the chart made in Lesson 13 with all of the adding combinations for 10. Ask what they notice is the same and what is different (e.g., in both operations, 9 and 1 go together when adding to or subtracting from 10; they are different because you can have 9 + 1 = 10, which gives the same sum as 1 + 9 = 10, and you can have 10 – 9 = 1 but not 9 – 10 = 1). • Link addition and subtraction: Addition has part with part to make a whole, while subtraction has the whole broken into parts. Further Practice • Independent Problem Solving in Math Journals: Pose one of the following prompts: – Show how 8 + 2 and 10 – 2 are related. – Make many subtraction stories about 7 cookies. 174 Number and Financial Literacy

15Lesson Subtraction as ‘Think Addition’ Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Teacher between addition and subtraction, to solve problems and check calculations Look-Fors • B2.2 recall and demonstrate addition facts for numbers up to 10, and related Previous Experience with Concepts: Students subtraction facts have had experience with addition and subtraction • B2.3 use mental math strategies, including estimation, to add and subtract activities and have created related equations. whole numbers that add up to no more than 20, and explain the strategies used PMraotcheesmseast:ical • B2.4 use objects, diagrams, and equations to represent, describe, and solve crPeorpmorbemlseuemnnitscinoagltvi,ninrgegf,lecting, situations involving addition and subtraction of whole numbers that add up to no more than 50 Algebra • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Possible Learning Goals • Solves part-part-whole problems using a variety of strategies • Connects addition and subtraction as operations that ‘undo’ each other • C reates a screened activity that can be presented to peers • S elects an appropriate strategy to solve screened activities • E xplains or shows how they solved the problem • C ounts on from total under screen when using ‘think addition’ strategy • E xplains how addition and subtraction are related About the As Marian Small points out, “any addition situation can also be viewed as a subtraction one, and vice versa” since the operations ‘undo’ each other (Small, 2009, p. 107). While addition and subtraction are often taught separately, considerable time needs to be spent on linking the inverse operations. As Van de Walle emphasizes, screened activities that use part-part-whole models can reveal both operations and help students view subtraction as ‘think addition’ (Van de Walle & Lovin, 2006, p. 74). continued on next page Addition and Subtraction to 10 175

Msseecudiaqgqobeutnumtchar,aoapt‘tVmticioohootsnpi,ncnesomsak,,s,bimebanuadulaaldstarci,ntyhmi:coinenin,g,’us In a screened activity, the whole is shown and then a part of the whole is shown as it is removed, while the remaining part is covered. Students determine the part that is covered. These activities encourage a ‘think addition’ strategy such as, “What goes with the part shown to make the whole?” rather than a ‘count what is left’ strategy. Making this connection explicit not only links the operations, but will help students as they learn their number facts. The following Minds On activity was developed by Van de Walle (Van de Walle & Lovin, 2006, p. 74). Materials: Minds On (15 minutes) paper bag, counters or • Pose the following problems: connecting cubes, chart paper, markers − There are 9 cookies in the bag. (Put 9 objects in the bag to represent the cookies.) 3 children came and ate 1 cookie each. (Remove 3 of the Time: 40–45 minutes cookies while students watch.) How many cookies are left in the bag? Ask students how they can be sure that their response is correct. (This is a partially screened task–the total is hidden and students see how many are removed.) − There are 9 cookies in the bag. (Put 9 objects in a bag.) Some children come and take 1 cookie each. (Remove 4 cookies, but do not show students how many were removed.) There are 5 cookies left. How many cookies were taken out of the bag? (This is a fully screened task–the total is hidden and the amount removed is not known.) Have students discuss their solution and the strategy that they used. • Tell students that they will be solving similar problems with a partner. Model the activity with a student. Put out 7 counters on a piece of paper so the student can see. Then have the student cover his/her eyes while you break the 7 counters into two groups (e.g., 3 and 4) and then cover one of the groups. The student uncovers his/her eyes and figures out how many counters are covered, saying, “7 minus 3 equals 4.” Or they may say, “3 plus 4 equals 7.” The covered part can be then revealed to see whether the student is correct. Model how they can record their thinking with matching addition and subtraction equations. Working On It (15 minutes) • Students work in pairs playing the modelled game, switching roles and creating their own problems. After each turn, have students record both the addition and subtraction equations on chart paper. Differentiation • For ELLs, you can create sentence starters that are linked to pictures and/or equations (e.g., minus (–) equals (=) or plus (+) equals (=) ) to help them verbally formulate their ideas. Model how to use them as you act out a problem. 176 Number and Financial Literacy

• If students have difficulty recording matching addition and subtraction equations, they can record the number combinations (e.g., 3, 4, 7). You can help them set up a chart with columns labelled Part/Part/Whole. Assessment Opportunities Observations: As students play the game, observe the strategies that they are using. This can help to assess students’ number sense and whether students see numbers as related to each other or as random quantities. – Are they counting on, are they counting three times, or do they just ‘know’ how many there are? – Do they use their fingers to create both sets or do they use their fingers to track as they count on? – Do they adjust their strategies according to the difficulty of the numbers presented? (e.g., if the difference is one, they may automatically know, while they may need to create sets with concrete materials for numbers that differ by larger amounts) Conversations: Make note of the observations which can be addressed in one-to-one interviews with certain students. For example, if students are still counting three times, make two sets and cover one with a piece of paper so they cannot see how many are underneath. Print the numeral on the top. Tell them, “There are 4 counters under here, so how many are there altogether?” Consolidation (10–15 minutes) • Discuss the strategies that students used to play the game. • Have a discussion of whether these questions are addition or subtraction problems. Highlight the idea that they can be solved using either operation. • Ask how addition and subtraction are related. (e.g., They are the opposite of each other.) Addition and Subtraction to 10 177

16Lesson Compare Problems: Differences Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Teacher between addition and subtraction, to solve problems and check calculations Look-Fors • B2.2 recall and demonstrate addition facts for numbers up to 10, and related Previous Experience with Concepts: Students subtraction facts have had several experiences with adding • B2.3 use mental math strategies, including estimation, to add and subtract and subtracting with concrete materials. whole numbers that add up to no more than 20, and explain the strategies used Students can create matching equations for • B2.4 use objects, diagrams, and equations to represent, describe, and solve models that represent addition or subtraction. situations involving addition and subtraction of whole numbers that add up to no more than 50 PMraotcheesmseast:ical carPeonrpmodrbemplseruoemnvniitncsinaoggtl,vi,cninrogegna,nseocntiinngg, Algebra • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Possible Learning Goal • Compares quantities using a variety of strategies • A ccurately finds the difference between two quantities • Explains or shows how they found the difference • Explains how the problem can be solved with either addition or subtraction About the In comparing problems, the quantities of two sets are being compared, yet they are not subsets of each other like in part-part-whole problems. Instead, the focus is on the difference between the quantities of the two sets. For example, the problem might be, “Jake has 6 cookies. Anna has 4 cookies. How many more does Jake have?” Students can use a variety of strategies using concrete materials to figure out the difference. They may: • C reate a set of 6 cubes and a set of 4 cubes, put them side by side, matching the cubes from each set using one-to-one correspondence, and then count the cubes that are left over. 178 Number and Financial Literacy

Math Vocabulary: • C reate a set of 6 cubes and a set of 4 cubes, and then count on from 4 eectqhqouiumnaakpttiiaoaorndnesds,,,idtmbiiofaafnletacrnehcninec,ge, to 6, tracking the count on their fingers. Materials: connecting cubes, • C reate a set of 4 cubes in one colour, add on different coloured cubes chart paper, markers Time: 45 minutes until a set of 6 is created, and count the cubes in the second colour. • C reate a set of 6 cubes, remove cubes until a set of 4 is left, and count the cubes that were removed. It is important to vary the problems. For example: • A nna has 4 cookies. Jake has 2 more cookies than Anna. How many cookies does Jake have? (compare quantity unknown) • Jake has 6 cookies. He has 2 more cookies than Anna. How many cookies does Anna have? (referent unknown) Minds On (15 minutes) • Have two students stand. Give one student a train made from 5 connecting cubes. Give the second student a train made from 9 connecting cubes. • Ask which train has more cubes and how they know. Ask how they could figure out how many more cubes the second student has than the first student (the strategy is more important than the answer). Students can demonstrate their thinking using the connecting cubes. Through questioning, try to elicit at least two or three strategies. • Pose another problem using a different pair of students. Give one student a train that is 4 cubes long. Give the second student a train that is 7 cubes long, but put it in a bag so the rest of the students cannot see it. Tell them that the second train is 3 cubes longer than the first train. Ask students how long the second train is. Working On It (15 minutes) • Students work in pairs and take turns creating trains and asking the questions. One student looks away, while the other student makes two trains that are different in length. He/she hides one of the trains. The second student looks at the train and the first student poses a question, such as, “The second train is 4 cubes longer. How long is it?” Students can check their answers by revealing the second train. Differentiation • For students who cannot mentally figure out the length of the second train, encourage them to use other cubes to build and support their thinking. • For students who master the activity and need more of a challenge, pose some of the following problems: – The difference between two numbers is 7. What might the numbers be? – The difference between two numbers is at least 5. What might the numbers be? Addition and Subtraction to 10 179

Assessment Opportunities Observations: Pay attention to how students are figuring out the length of the hidden train. – If using mental strategies, are they subitizing the original amount of cubes in the train? Do they need to touch and count each cube? – Can they provide a solution for 1 more/less than problems without counting the length of the train, but by knowing the number sequence? – Are they mentally counting on or back? How do they track the size of the new train? (e.g., on their fingers, nodding their head for each count) – Are they counting on when using concrete materials, or are they creating the original train, adding or removing the difference, and then counting the cubes in the train again? (Ask probing questions highlighted in earlier lessons to get students to count on.) Conversations: • To probe further, ask how students are finding out the difference. Ask what they are counting on their fingers. • If students find 1 more/less than problems difficult, you can carry out some related Math Talks (e.g., 1 more/less than) with small groups or the entire class to reinforce how the differences are related to the number sequence. Consolidation (15 minutes) • Building Social-Emotional Learning Skills: Healthy Relationships: Before beginning the “Inside/Outside Circles” activity, discuss how students can listen attentively to each other. Have them suggest what they might ask if they don’t understand what their partner says, as well as what prompts they can give if their partner hasn’t understood them. This helps students to develop their communication skills. • Inside/Outside Circles: Have each student create two trains of different sizes. Split the class in half and make two circles, one within the other (the inside and outside circles). Students on the inside and outside face each other, with each person in front of a partner. The partners put one of their trains behind their backs so the other students cannot see. They take turns showing their one train, and then posing prompts that compare it to the train behind their backs. After a couple of minutes, have the students on the outside move one person to the right. (You can signal the time to change with a bell or by playing music.) Students now have a new partner and can pose the same or different questions. Repeat this for two or three rotations. • Discuss the mental strategies that students used to figure out the differences between the two trains. • Ask if these are addition or subtraction problems. Through discussion, students should realize that the problems can be solved with either operation. 180 Number and Financial Literacy

Further Practice • Independent Problem Solving in Math Journals: Verbally pose one of the following prompts: – The difference between two numbers is 3. What might the two numbers be? − Show that there are 4 more cookies on one plate than on another plate. Materials: Math Talk: large arithmetic rack, small arithmetic Math Focus: Mental strategies for comparing problems racks (or BLM 6: Blank Ten Frames and Let’s Talk (10–15 minutes) counters, or connecting cubes) Select the prompts that best meet the needs of your students. Teaching Tip • W ith your partner, solve this problem in more than one way on your arithmetic Integrate the math rack. There are 9 birds in the tree and 3 birds on the ground. How many more talk moves (see birds are in the tree than on the ground? page 7) throughout Math Talks to maximize • Possible solutions to discuss: student participation and active listening. Removal: − Slide across 9 beads on the top row (either all at once or 5 red beads and then the 4 white beads); − Slide back 6, one at a time, counting 1, 2, 3, 4, 5, 6, until they can see that there are only 3 left; − Slide back all beads except the 3 (students can subitize this) and then count how many they slid back; − Slide back one at a time counting back from 9, 8, 7, 6, 5, 4, 3, and then counting the beads they slid back. Adding On: − Slide across 3 beads to the right on the bottom row (or they can work on the top row); − Slide across 2 more red beads on the bottom row to create a group of 5, and then 4 white beads all at once to make a group of 9 (2 moves). Then count the beads that they slid over 1, 2, 3, 4, 5, 6; − Mentally count on from 3 to 9, tracking each number on their fingers, and then sliding across a group of 6 counters on the bottom row, since it is easy to subitize the 5 red beads and 1 white bead (1 move). Comparing: − Build 9 on the top and 3 on the bottom. Visually match corresponding beads on the top and bottom and count the ‘extra’ beads on the top, 1, 2, 3, 4, 5, 6. continued on next page Addition and Subtraction to 10 181

• G ive students time to solve the problem. How did you solve this problem? Put your thumb up if you solved it the same way. How did you start? What do those beads represent? What action are you doing? Who can explain in their own words how this strategy works? How is this strategy different from Jon’s strategy? Partner Investigation • Give students another problem to solve to try out some of the strategies that were discussed. Follow-Up Talk • M ake an anchor chart of students’ strategies, illustrating them with red and white dots for beads, and annotating the movement. You can name the strategies after the students who explained them. 182 Number and Financial Literacy

17Lesson Anchoring 5 and 10 Using Mental Strategies Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Previous Experience between addition and subtraction, to solve problems and check calculations with Concepts: Students have had opportunities to • B2.2 recall and demonstrate addition facts for numbers up to 10, and related add and subtract with concrete materials such as subtraction facts ten frames and can read and create matching • B2.3 use mental math strategies, including estimation, to add and subtract equations for the concrete representations. whole numbers that add up to no more than 20, and explain the strategies used Teacher • B2.4 use objects, diagrams, and equations to represent, describe, and solve Look-Fors situations involving addition and subtraction of whole numbers that add up PMraotcheesmseast:ical to no more than 50 cprPeorropnovrnbeienlsecgemt,ninctsgoinomglv,minreugan, sicoantiinngg,and Algebra McrmeoacmatothpcmhoVpisonoecgs,aedeb,equeculqoaaumrtyiaop:tnioossnes,, • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Possible Learning Goals • Solves problems by applying various mental strategies • Investigates benchmarks of 5 and 10 when solving addition and subtraction problems • Uses mental strategies to find sums presented on ten frames • E xplains or shows strategies and how they mentally rearranged the dots on the ten frame • A ccurately recreates the dot images using tens frames and counters, and rearranges them to match their thinking • R eflects on strategy and explains how their rearrangement helped them solve the problem About the Students have had many opportunities to add and subtract with concrete materials, which has helped them to discover the advantages of using 5 and 10 as benchmark numbers. These experiences help students create mental images of these concrete representations, so they can be retrieved later and used to visualize and solve other problems. Addition and Subtraction to 10 183

Materials: Minds On (20 minutes) BLM 18: Fast Images (Set 2) or Digital Slides • Using BLM 18 or Digital Slides 121–126, briefly show Image 1 121–126; BLM 19: Fast Images (Set 3) or Digital and then take it away. Have students explain how they know Slides 127–137; chart how many counters they saw. paper; BLM 6: Blank Ten Frames; counters • Repeat, showing some of the following fast images of ten frames. After each Time: 50 minutes one, ask students for their strategy and an equation that would match their thinking. − Image 2: (5 + 2 = 7) − Image 3: (5 + 4 = 9) − Image 4: (5 + 5 = 10) − Image 5: Ask students what they did in their minds to find out how many counters there were on the 2 ten frames combined. (e.g., Move the sixth counter in the second frame to finish the row of 4 in the first frame.) Optional: Write the expression for the original configuration (4 + 6) and the recomposed configuration (5 + 5). Ask how the first was changed into the second. Ask what the sum is for both and how they know. − Image 6: Ask students how they figured out the total. (e.g., mentally moved 3 counters on the second ten frame to make a row of 5 with the 2 counters on the first ten frame; mentally moved the 2 counters on the first frame to the second ten frame with 8 counters to make a full ten frame) Optional: Write the expression for the original configuration (2 + 8) and the recomposed configuration (5 + 5). Ask how the first was changed into the second. Ask what the sum is for both and how they know. Working On It (15 minutes) • Provide each student with a few ten frame images from BLM 19. Have students work in pairs, taking turns to show a fast image of two ten frames and then taking it away. The other student determines the amount, explains his/her thinking, and then recreates the original image using ten frames and counters. He/she then shows how the arrangement can be recomposed to solve the problem (e.g., decomposing and recomposing the amounts). The two students then compare to see if the re-creation matches the fast image and whether the two sums are the same. Differentiation • For students who need more work with anchoring 5, give them simpler fast images with quantities up to 6 or 7, and images showing 1 or 2 more or less than a group of 5. 184 Number and Financial Literacy

• For students who need more of a challenge, they can create matching equations for the original fast image and the recomposed image. In conversation, ask students how the two equations can still represent the same amount. Ask students to explain how the first equation was changed into the second equation by showing their concrete representations. Assessment Opportunities Observations: Observe what students are doing by paying attention to their actions with the materials. Are they moving counters one at a time to find what may complete a row, or do they move 2 or 3 counters at a time with an intended plan in mind? Conversations: For students who tend to use ‘guess and check,’ ask them to visualize how to rearrange the counters and then make a prediction. This helps them develop visual images and mental strategies. Consolidation (15 minutes) • Using BLM 19 or Digital Slides 127–137, show 1 or 2 of the fast images to the class and discuss the strategies that students used to solve them. • Have ten frames and counters available for each pair of students. Challenge them to create the original fast image and then recompose it to make an easier representation. • For one of the discussed examples, show expressions that represent the original and the recomposed arrangements (e.g., 3 + 7 and 5 + 5, or 9 + 1 and 10 + 0). Ask students to explain how the first arrangement was changed into the second, using concrete representations to support their explanations. • Ask how knowing 5 and 10 can help them figure out other numbers. Further Practice • Independent Problem Solving in Math Journals: Give each student one fast image of two ten frames. Have them use counters and ten frames to rearrange the counters and find the sum. They can use a template of empty ten frames to show what was on the ten frame and how it was rearranged. They can annotate what they did using arrows. For students who need a further challenge, have them create a matching equation for the original and recomposed images. Addition and Subtraction to 10 185

18Lesson Linking Addition and Subtraction Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Previous Experience between addition and subtraction, to solve problems and check calculations with Concepts: Students have worked with various • B2.2 recall and demonstrate addition facts for numbers up to 10, and related problem structures (e.g., joining, separating, subtraction facts part-part-whole, compare problems) as they apply • B2.3 use mental math strategies, including estimation, to add and subtract their understanding of addition and subtraction. whole numbers that add up to no more than 20, and explain the strategies used They have had experiences with • B2.4 use objects, diagrams, and equations to represent, describe, and solve connecting equations to their matching stories. situations involving addition and subtraction of whole numbers that add up to no more than 50 • B1.2 compose and decompose whole numbers up to and including 50, using a variety of tools and strategies, in various contexts Possible Learning Goals • Creates various addition and subtraction problems for a given answer • Investigates the connection between addition and subtraction Teacher • C reates a story that makes sense and matches the given challenges Look-Fors • E xplains how the story either represents addition or subtraction • E xplains how a given matching equation represents their story problem • E xplains or shows how addition and subtraction are related PMraotcheesmseast:ical About the Lesson sscarPteeonrrlapmodetrbceemptlgseirnuoeimegnvnsiitntcsionaogogtl,vil,cnsinrogeagn,an,nsdeocntiinngg, Students have had several experiences working with different structures of addition and subtraction problems using a variety of strategies. This activity reinforces understanding of these structures by offering practice in creating addition and subtraction stories. The key concept is to connect the operations of addition and subtraction as operations that ‘undo’ each other. Time can be spent in the Consolidation making these links, through the stories and also through creating and comparing the matching equations. 186 Number and Financial Literacy

Math Vocabulary: Minds On (15 minutes) eesqquuubaatrttaiioocnntissn,,gmadadtcinhgin, g • Tell students that today they are going to make adding and subtracting Materials: chart paper, markers, problems rather than solve them. concrete materials Time: 50 minutes • Tell them that the answer to the problem is 5 cookies. Have them turn and talk with a partner to discuss what the problem might be. Encourage them to be creative and find at least two different problems. They can use concrete materials to help them devise their problems. • Have two or three pairs share one of their problems with the class (selectively choosing so that at least one represents addition and one represents subtraction). Have the rest of the class put their thumbs up if they had a similar problem. • Ask why the problems can be both adding and subtracting, yet still have the same answer. • Ask what a matching equation would be for each of the problems. Have students explain how each part of the equation relates to the context of the problem. Working On It (15 minutes) • Students can work in pairs or individually. Pose one of the following open-ended prompts to students. They are to create at least two problem stories and include both addition and subtraction. − The answer is 7 penguins. What might be the story? Create as many stories as you can. Make matching equations for each story. − Create many different stories that have 5 and 4 in them. Make matching equations for each story. • Observe whether students are only making addition questions. Ask if they can create a story that includes another operation. Differentiation • For ELLs, the context of penguins, for example, may be unfamiliar. Leaving the question too open can also be overwhelming, since it is difficult to know where to start. Help the students think of a familiar context that makes sense to them. Using real objects, rather than concrete materials representing other objects, can help to give the context more direct meaning. • For students who need more of a challenge, open up the problems. For example, “The answer is 7 penguins” can be opened up by posing “The answer is 7.” Open the second problem by letting students choose the two numbers they wish to use. Addition and Subtraction to 10 187

Assessment Opportunities Observations: This is a good time to check whether students see the connection between addition and subtraction. Conversations: • If students are only creating addition problems, ask whether it is possible to make a subtraction problem with an answer of 7. Ask what action can take place in a subtraction problem. (e.g., taking away or removing) Ask what problem they can create that has a ‘leftover’ of 7. • If students are creating both addition and subtraction problems, ask if they can turn one of their addition stories into a subtraction story. • To help them understand the relationship between addition and subtraction, ask students to investigate the following problem. Record the expressions 5 + 2, and 7 – 2. Ask what would happen if they continually did the first expression, using counters, followed by the second expression, using the same counters, followed again by the first expression. Allow students to investigate and visit them later to ask what they discovered. Consolidation (20 minutes – 10 minutes with another student or pair and 10 minutes with the whole class) • Have individuals or pairs of students meet with another student or pair and pose their problems to each other. Have the other students decide whether it can be solved with adding, subtracting, or both. Together, they can make equations that match the stories. • As a class, have students share some of their solutions. Ask them to explain how addition and subtraction make their stories different. • Show an addition equation that matches one of the students’ stories. Ask how the equation and question could be altered so they represent subtraction. • Ask how addition and subtraction are related. (e.g., One operation undoes the other operation.) Further Practice • Independent Problem Solving in Math Journals: Verbally pose the following prompt: − Create adding and subtracting problems that have an answer of 4. Use pictures, numbers, and/or words to explain. 188 Number and Financial Literacy

Materials: Math Talk: BLM 6: Blank Ten Frames, counters Math Focus: Doubles plus/minus 1 strategies for recalling facts (or arithmetic racks) Let’s Talk Teaching Tip Select the prompts that best meet the needs of your students. Integrate the math talk moves (see • Briefly show 3 on the top row and 3 on the bottom row of a ten frame or page 7) throughout Math Talks to maximize arithmetic rack. How many did you see and how do you know? (e.g., I saw 6 student participation because I know 3 and 3 are 6; I saw 6 and counted 2, 4, 6; I saw 6 because and active listening. they look like the 6 on a die.) • Briefly show 3 on the top row and 4 on the bottom row. How many did you see and how did you know? (e.g., I saw 7 because it is 1 more than 6; I saw 7, 3 and 3 are 6 so 1 more is 7) Where did you see 3 and 3? So knowing the double helped you, is that what you are saying? Did anyone see it differently? (e.g., I saw 4 and 4 is 8, but 1 was missing so it was 7.) Can someone show us where Raj visualized 4 and 4? How did knowing a double help Raj? • C ontinue showing ten frames with 1 more on the top or the bottom. Dot configurations that show 1 more or less dot can also be used. Partner Investigation • Students can add 1 more cube to the top or bottom of their double-decker trains that they previously created (see Lesson 11). They can take turns flashing them to each other. The partners say how many there are, and then state the closest double that they could see. Follow-Up Talk • D iscuss how the doubles and near doubles trains differed. Have a discussion about odd and even numbers. Math Talk: Math Focus: Doubles and near doubles with number sentence sequences to develop strategies for recalling facts NOTE: Once students have had experiences working with doubles and near doubles with concrete and visual representations, they can work through number sentence sequences. Let’s Talk Select the prompts that best meet the needs of your students. • Show some of the following expressions one at a time. For each, ask some of the following prompts. Incorporate the math talk moves to maximize student participation and active listening. What do you visualize? What is the sum? How do you know? How did an earlier problem help you solve this one? How is this problem like the one before it? continued on next page Addition and Subtraction to 10 189

− 3 + 3 (establishing a double) − 3 + 4 (1 more than a double) How does this connect to the first problem? − 4 + 3 (addends in reversed position) How does this connect to the first and second problems? − 4 + 4 (establishing a double) − 4 + 3 (1 less than 4 + 4 or 1 more than 3 + 3) − 5 + 5 (establishing a double) − 5 + 4 (1 less than 5 + 5 or 1 more than 4 + 4) What other problems helped you solve this? How? − 4 + 5 (the same sum as above) How is this like the problem above? Why? Partner Investigation • S tudents can take turns showing each other similar expressions on cards while their partner explains their strategy for finding the sum. Follow-Up Talk • M ake an anchor chart of students’ strategies. 190 Number and Financial Literacy

19Lesson Reinforcement Activities Math • All of the expectations identified for this unit Curriculum Expectations Teacher • S olves word problems using appropriate addition or subtraction strategies Look-Fors • E xplains or shows strategies and justifies how they solve the problem • E xplains the meaning of the addition, subtraction, and equal symbols • E xplains or shows how addition and subtraction are related • E xplains or shows how a given equation can match a story problem PMraotcheesmseast:ical About the Lesson tccrrPoeoeooranpmolsnsrbemoelasencunemitnndinngitcsginaoa, gtlnsvi,nedinrgleepg,fcr,loteivncitgningg, , strategies The following activities can be carried out by the whole class in small groups, or as centres through which students rotate over a few days. They Materials: can also be used throughout the unit any time you decide to offer guided different colours math lessons. For example, you may want to meet with small groups over of connecting a few days and tailor the lessons to meet the needs of the students. While cubes or tiles this can be done while the rest of the students solve the same problem in Time: 20–25 minutes small groups, you may wish to observe how each group works through per activity over a few the same concept. In this case, one group meets with you each day, while days the other groups rotate through some of the following activities. See the Overview Guide for more information on how to manage guided math lessons. Centre 1: Combination Challenges • Students find all of the combinations for 5, using more than 2 groups to make the total (e.g., 2 red connecting cubes, 2 green connecting cubes, and 1 yellow connecting cube). Have them create equations for each combination. Differentiation • For students who need more of a challenge, have them find all the combinations for any other numbers between 6 and 10, using 3 or more addends. They can also choose their own concrete materials to solve the problem. Addition and Subtraction to 10 191

Materials: Centre 2: Creating Compare Problems on the Farm “What Do You See?” • Have students use the “What Do You See?” picture to solve ‘How many (pages 6–7 in the Number and Financial more or less?’ problems. Students create their own problems and verbally Literacy little books), pose them to one another. For example, a student might ask his/her partner chart paper how many more rabbits there are than cats. Materials: Centre 3: Grab Bag Subtraction (from Marilyn Burns, 2000, p. 170) connecting cubes or tiles, • Students work in pairs. They select a number between 5 and 10 and put that paper bag many items in the bag. One student reaches in and removes some of the Materials: items, showing how many have been removed. Both predict how many items connecting cubes, they think are left in the bag. Then they check their predictions, and record chart paper, the matching equation(s). markers Centre 4: Snap It! (from Marilyn Burns, 2000, p. 170) Materials: playing cards • Students work in pairs. Both students decide on a length of train (up to 10), without the face cards, but including and each build one using connecting cubes. Both students put their trains the aces as 1s behind their backs. One student says, “Snap it!” Both students break their trains into two pieces and continue to hold them behind their backs. They take turns showing the cubes in one hand, while the other student figures out how many cubes are behind their back. On the next turn, the other student says, “Snap it!” Have students record some of their equations. Centre 5: What’s the Difference? • Students play in pairs or in small groups. Each player chooses two cards. They take turns figuring out the difference between the two cards and take that many counters. They keep playing until one player has 25 counters. They can keep track of their counters using five or ten frames. Variation: Students roll two dice and figure out the difference between the two amounts showing. They take counters to equal the difference. The first to have 25 counters wins the game. 192 Number and Financial Literacy

Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: Reflect back on the lessons and pose some of the following prompts to help students monitor their learning: – Ask what students’ favourite activities were. (Math is interesting to investigate.) – Ask what they found challenging. (Hard tasks are good and if we keep trying, we can overcome them.) – Ask what they have learned. (Celebrate the accomplishments.) – Ask what they still have to learn. (We may not know it YET, but we will with time.) – Ask how mistakes can help them learn. (Mistakes help us to try new strategies and learn new ways of trying so we can do things better.) Addition and Subtraction to 10 193