p115-198-Gr2-ON-Number_Unit2 AddSub20

8Lesson Applying the Make a Ten Strategy to Addition Using Ten Frames Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationships Previous Experience between addition and multiplication and between subtraction and division, with Concepts: to solve problems and check calculations Reading and analysing graphs and tables is • B 2.2 recall and demonstrate addition facts for numbers up to 20, and recommended but not essential for the related subtraction facts independent task in Working On It. • B 2.3 use mental math strategies, including estimation, to add and subtract whole numbers that add up to no more than 50, and explain the strategies used • B 2.4 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 100 Algebra • C2.2 determine what needs to be added to or subtracted from addition and subtraction expressions to make them equivalent •  C2.3 identify and use equivalent relationships for whole numbers up to 100, in various context Data • D1.5 analyse different sets of data presented in various ways, including in logic diagrams, line plots, and bar graphs by asking and answering questions about the data and drawing conclusions, then make convincing arguments and informed decisions Possible Learning Goals • Applies an understanding of part-part-whole relationships to solve addition problems • Applies the commutative property (5 + 7 = 7 + 5) and the associative property (7 + 5 = [7 + 3] + 2) when adding Addition and Subtraction to 20 163

Teacher • Decomposes a number to make adding more efficient Look-Fors • Uses the make a ten strategy to solve addition problems • Uses 5 or 10 as an anchor PMraotcheesmseast:ical • Uses compensation/near doubles (e.g., when adding 5 + 7, takes one from the Problem solving, rssreeterpalaersteceostgnieniinengtsgit,noacgono,ldnsnpaerncodtviinngg,, 7 and gives it to the 5 to make 6 + 6) communicating About the Math Vocabulary: make a ten, decompose Once students are familiar with the facts that compose 10, they can use this information to add using the make a ten strategy. As Lawson states, when Materials: children use this strategy, they have “constructed the key idea that breaking up the numbers and moving them around still results in the same sum BLM 5: Ten Frames (the commutative and associative properties)” (Lawson, 2015, p. 21). (1–20) or Digital Slides 30–49, BLM 6: While the make a ten strategy is a mental math strategy, it’s necessary for Individual Ten many students to experience it in a concrete way in order to develop a Frame (two conceptual understanding of decomposing in relation to addition. Using per student ten frames allows students to see how the ten is created by breaking up or or pair of decomposing one number and moving it over to join another number to students), make 10. The goal is for students to internalize the ten frame model of counters, make a ten and then visualize it when solving mental math problems. BLM 30: Play Day Problems About the Lesson Time: 55–65 minutes In the Minds On, students will be shown ten frame images that invite them to subitize and/or use the anchors of 5 and 10 to determine the quantity shown. In Working On It, students will use two ten frames to solve addition problems that encourage the use of the make a ten strategy. An independent task will require them to use information in a table to solve a number of addition problems that encourage the use of the make a ten strategy. Minds On (10 minutes) • Use ten frame images from 2–20 to provide students with practice subitizing various combinations that make up ten and more. After briefly showing or ‘flashing’ a card and taking it away, ask one or two students to explain how they determined the number (e.g., 13: I saw ten because the frame was filled and I counted on 3 more). • If students require a greater challenge, flash two ten frames at the same time. This will challenge them to visualize moving over some dots from one ten frame to the other to make a ten and then add on the leftover amount. Working On It (30–40 minutes) • Students can work in partners or independently. Give each student, or pair of students, 2 ten frames and 20 counters. 164 Number and Financial Literacy

• W ork through the questions below using the ten frames. Encourage students to use the make a ten strategy, but accept other strategies as well. Choose a few students to share their solutions after each problem. Always ask one student who has used the make a ten strategy to share his/her solution. • As students share their solutions, represent their thinking on the board. See the example below for the first question. This type of representation shows the decomposing that is involved in the strategy of making a ten. As students become more comfortable with the strategy, you might ask them to do the recording. This will be good practice for when they later use the strategy of decomposing to add or subtract two-digit numbers. 22 62 8+4 8+4 4 + 6 = 10 8 + 2 = 10 10 + 2 = 12 10 + 2 = 12 Question 1: You have 8 gummy bears. I give you 4 more. Now how many do you have? Possible solutions: – Start with 8 in one ten frame and 4 in the other, counting all or counting on to solve the problem. – Start with 8 in one and 4 in the other, then move 2 over to fill the one ten frame, then add on the 2. – Fill one ten frame completely then put 2 in the other ten frame without first starting with 8 in one and 4 in the other. – Start with 8 in one and 4 in the other and move 6 over to the frame with 4, then count on the 2 left. • Possible questions to probe students’ thinking: – How did you solve the problem? – How many counters did you start with in each ten frame? Why? – How many counters did you move? Why? – How many counters do you have in your full ten frame? (10) – How many are in the other ten frame? (2) – Why did that happen? – What strategy were you using? (e.g., make a ten, count on) – Which two numbers add up to ten? – What do you have to do with the 2 that’s left? (count them on) – Let’s compare the two make a ten strategies (8 + 2) + 2 and (6 + 4) + 2. Why did they both work? What’s different and the same about them? Why would you use one or the other? – Why do you think this strategy is called ‘make a ten’? Why is it useful? • Say, “ Let’s try some other problems. Use the make a ten strategy to solve them.” Addition and Subtraction to 20 165

Teaching Tip Question 2: There are 7 books on polar bears and 5 books on grizzly bears on the shelf. How many books are there about bears? The last problem on BLM 30 (Which player Question 3: There were 9 cats and 6 dogs in the yard. How many pets were is the fastest runner?) there? requires students to infer and reason Question 4: For my birthday I got 9 stickers from my friends and 8 stickers (Player C won the from my family. How many stickers did I get? balloon race so he can likely run fast). • Give each student a copy of BLM 30: Play Day Problems and have them work Ask students if we know that it’s Player independently. Read the problem with students, ensuring that they understand C for sure? (No, we the scenario and how to read the table. Provide access to ten frames but do not don’t have enough require their use unless students are struggling with the make a ten strategy. information to be Possible questions include: certain.) Accept all answers as long as – How many players does this table show? students can provide sound reasoning for – How many games are shown for each player? their choice. – What do the numbers across the row mean? (their scores for each game) – Have students Think-Pair-Share: How would you find out what a player’s total score for all 3 games was? – What are you wondering? Differentiation • For students who need more support with the independent task, providing a concrete tool or manipulative such as the ten frame or number line scaffolds the learning for the make a ten strategy as students are able to act out the movements needed. As students consolidate their use of this strategy, the tools will become mental models that students visualize for mental addition and subtraction. • Some students, including ELLs, may have difficulty understanding the questions about the table if they have limited prior experience interpreting data tables. It may help to name the players and talk about the information in the table more extensively if this is the case. Assessment Opportunities Observations: As students work, record observations of how they solve problems using the ten frames. Do they: – Use the ten frame confidently and purposefully to solve problems? – Subitize 5 and 10? – Understand the problem and try a strategy independently? – Apply the make a ten strategy to solve addition problems? – Keep one number whole and decompose another number to make 10, then count on the other number or use a known fact? – Recognize the 2 numbers that add up to 10 when solving the Play Day problems or do they add the numbers in the table from left to right or in random order? – U se a variety of strategies to solve addition problems (e.g., near doubles, count on)? 166 Number and Financial Literacy

Conversations: Converse with individual students as they work independently. For example, a student is working on Question 6 of the Play Day problems and is having difficulty knowing how to approach this problem (finding the difference between total scores): Teacher: I see you’re working on Question 6. What do you have to find out? Student: (Reads the question: What is the difference between the highest and lowest total scores?) Teacher: What does that mean? Student: I don’t know. Teacher: Another way of asking this question is: How much more or less is the highest score than the lowest? Do you know the highest and lowest total scores? Student: The highest is 16. The lowest is 13. Teacher: Are they the same or different? Student: Different. Teacher: Okay. Now we need to find out how much they are different. We need to compare them. Which is higher? Student: 16. Teacher: How much higher? (At this point a student may just count up from 13 to 16 and say 3. If this is the case, you can ask him/her what the difference is between 13 and 16, consolidate the understanding of the terminology ‘find the difference,’ and end the conversation.) Teacher: How could you use your ten frame to help you? Student: I don’t know. Make 16 and 13? Teacher: Try that. Student: (Makes 13 in a ten frame. Teacher stops him/her there.) Teacher: How much do you have in your ten frames? Student: 13. Ten and 3 more. Teacher: Count how many more you’d have to put in to have 16. Student: 3. Teacher: Okay. So you had one ten filled for both 13 and 16 but then there was a difference between how many more ones you needed to add between 13 and 16. What was that difference? Student: 3. Teacher: So what is the difference between 13 and 16? Student: 3. After this conversation, ensure you provide more opportunities for the student to consolidate this concept of finding the difference. Repeat the activity using different contexts and numbers. The ten frame is a good tool for visually and concretely developing the concept. Addition and Subtraction to 20 167

Teaching Tip Consolidation (15 minutes) You may wish to do • The student work will show number sentences but may not reflect the the Consolidation the next day in order strategies students used to solve the problems. Your recorded observations of to analyse students’ students at work, however, will show the strategies and will be useful in work from the deciding which students you’ll choose for sharing with the class. completion of BLM 30: Play Day Problems. • Choose at least two students to explain how they used the make a ten strategy. Choose one student who used the ten frame as a tool and another student who added mentally. It’s not necessary to consolidate each question, but Question 6 (finding the difference) would likely be a good choice as students often have difficulty with this type of question. See the list of possible questions in Working On It and the conversation in Assessment Opportunities for questions and prompts that you might use during the consolidating discussion. • At the end of the sharing, write an equation such as 8 + 4 = on the board. Have students Think-Pair-Share: Explain how to use the make a ten strategy to your partner. Further Practice • Independent Problem Solving in Math Journals: Record and pose questions such as the following: – On Hat Day 8 students wore red hats and 4 students wore green hats. How many students wore hats? – 9 cats and 4 dogs were in the yard. How many pets were there in the yard? – There were 7 robins and 5 bluebirds at the feeder. How many birds were at the feeder? – I took 8 steps and then I took 9 more steps. How many steps did I take in all? Explain to students that they are to imagine that they have a friend who doesn’t know how to use the make a ten strategy. Their job is to show or explain how to use it to solve the problems. They can use words, pictures, and/or numbers to explain their thinking. Scribe for students if necessary. Make ten frames and number lines available to those who may still need to model the strategy before recording. • Make 10 Concentration: This is a game for 2 players. Set-up: Use a deck of cards but remove the tens, jacks, queens, and kings. The aces represent ones. Lay out 16 of the cards in a 4 × 4 array and put the other cards face down in a pile. Directions: – The goal is to turn over 2 cards that add up to 10. – Player 1 turns over 2 cards. If the cards add up to 10, Player 1 takes the cards. If not, Player 1 turns the cards back over and it becomes the next player’s turn. 168 Number and Financial Literacy

– If two cards make a ten and are removed from the array, the cards are replaced with two new cards from the pile. – When all of the cards in the pile have been used and all possible combinations of ten have been made, the game is over. The player with the most cards at the end of the game is the winner. Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: Different tasks pose different challenges for students. For example, some of your students may find the independent task of explaining the make a ten strategy to a friend to be difficult, while others will meet the challenge with ease. When we open up tasks and enable students to approach them in a way that best suits them, they are more willing to persevere. For example, some students may be able to explain the strategy orally, while others may be able to show how to use the strategy through numbers and pictures. Scribing for a student who wants to explain it orally but struggles with writing will address that student’s needs and enable success. Thus differentiating the learning for students helps them develop a self-awareness and sense identity as a mathematician. When we reinforce the message that we all learn and work differently, it frees students to try strategies that work for them, and teaches them to pursue alternatives when they are having difficulties. In other words, it builds resilience. Math Talk: Teaching Tip Math Focus: Make a ten strategy for addition, using the number line Integrate the math Process: Repeat the Math Talk process from Lessons 6 and 7 (see p. 156 and talk moves (see p. 162) but represent student strategies using a number line rather than ten frames. page 7) throughout Math Talks to Possible number strings: maximize student participation and 5+5 2+8+9 active listening. 5+8+5 8+2+7 7+5+5 5+8+2 Let’s Talk Select the prompts that best meet the needs of your students. • Who has a solution for 5 + 5? (5 + 5 = 10) What strategy did you use? (I knew the double. It was a known fact for me.) • D raw a number line. How could I show this on an open number line? (Start at 5 and jump to 10.) Draw 5 on the bottom and show a jump to 10. Label 10 on the bottom. Like this? What do I have to label at the top? (+ 5) Draw a curved line from the 5 to the 10 on top and record + 5. • O kay? Let’s try another problem. Record 5 + 8 + 5 = . (It equals 18. I already knew 5 + 5 = 10 so I just added on 8 more.) Record 18 on the number line and show a jump of 8 from the 10 to 18. So you started with the 10 that we had already figured out and added on 8 more. How did you add 8? continued on next page Addition and Subtraction to 20 169

(I just know that if you have 10 and add another number to it, you just have to change the ones. 1 ten and 8 ones is 18.) • T hank you. Did anyone solve it another way? (I added 5 + 8 and knew that was 13. Then I added on 5 more and that’s 18. I know 3 and 5 are 8 so I know 13 and 5 equals 18.) Draw the jumps on the number line. Does this represent your thinking? (Well I actually started with the 8 and added on 5.) Change the jumps to reflect the student’s thinking. • W hy do both ways work? Turn and talk to a partner. (It doesn’t matter what order you add the numbers in. 5 + 5 + 8 is the same as 5 + 8 + 5.) • Show me if you agree with your thumb up or down. Okay, I see you all agree. Let’s test that. Solve 7 + 5 + 5. (17. I added 5 + 5 to make a 10 and 7 more is 17.) Represent the strategy on the number line. • Did anyone add the numbers in a different order? (I started with 7 and added 5 which equals 12. Then I added 5 more which equals 17.) Represent the strategy. • You both arrived at the same answer but in a different way. Is there another order? (You could add 5 + 7 but you still get 12 and 5 more is 17.) • Okay, so turn and talk to a partner: What rule can we say applies when we’re adding 3 numbers? (You can add 3 numbers in any order and you still get the same answer.) 170 Number and Financial Literacy

9Lesson Applying the Make a Ten Strategy to Addition Using a Number Line Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationships Teacher between addition and multiplication and between subtraction and division, Look-Fors to solve problems and check calculations Previous Experience • B 2.2 recall and demonstrate addition facts for numbers up to 20, and with Concepts: Students should have related subtraction facts prior experience decomposing 10 as in • B 2.3 use mental math strategies, including estimation, to add and subtract Lesson 8 where students use ten frames to model whole numbers that add up to no more than 50, and explain the strategies addition problems using used the make a ten strategy. • B 2.4 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 100 • B 1.1 read, represent, compose, and decompose whole numbers up to and including 200, using a variety of tools and strategies, and describe various ways they are used in everyday life Possible Learning Goals • Uses the make a ten strategy to solve addition problems • Uses a number line to solve addition problems • Represents the make a ten strategy on a number line • Understands the part-part-whole relationship of numbers in addition • Represents and labels their hops accurately on a number line • Keeps one addend whole and decomposes the second addend to create a 10 then adds on the remaining part • Explains the make a ten strategy • Understands the problem and initiates a strategy Addition and Subtraction to 20 171

PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, According to Van de Walle, it is important to model the distance between numbers by drawing ‘hops’ rather than focusing on the dots or numerals on a Math Vocabulary: number line so that students conceptualize the distance between numbers. For example, when modelling 5 + 3 on the number line, draw a hop from 0 to make a ten, 5 and then another hop from 5 to 8. Do not just start at the 5. This clearly number line shows that 8 is made up of 5 and 3. The number line is a good model for helping students understand part-part-whole relationships. It can also be used to show both addition and subtraction depending on which way the arrows are drawn. This may help students develop a deeper understanding of the inverse relationship of addition and subtraction (Van de Walle & Lovin, 2006, p. 73). About the Lesson In Lesson 8, students used the ten frame to develop their understanding of decomposing a number into two parts so that one part could be added to another to make ten. In this lesson, students will solve story problems that invite them to use the make a ten strategy by decomposing to take hops of ten on the number line and then counting on the remaining part. Materials: Minds On (10 minutes) BLM 31: Number • Tell students the following story and ask them to solve it mentally: Some Lines 0–20, BLM 32: Problems for Making students were playing hide and seek on the playground. The person who was a Ten ‘It’ found 7 students and then she found 5 students. How many students did she find in all? Time: 70–75 minutes per period, for two periods • Ask for, and record, students’ solutions. Ask a few students to share their solutions and represent their thinking on a number line using hops. If no one that you have chosen uses the make a ten strategy, ask if anyone used make a ten, or if someone can explain how to use it. For example: Student: I started with the 7 and added 3 more. Teacher: (Draw a hop from 0 to 7 and a hop from 7 to 10; label the hop of 7.) Where did you get 3? Student: I took 3 from the 5 because I know that 3 + 2 = 5. Teacher: You knew that there was a 3 and 2 inside 5. So what did 7 and 3 add to?  172 Number and Financial Literacy

Student: 10. (Label the hop to 10.) Then I added on the 2 that was left. 10 + 2 = 12. (Draw a hop from 10 to 12 and label it.) 10 7 +3 +2 Teaching Tip 0 1 2 3 4 5 6 7 8 9 10 11 12 Contextualize the • Invite students to Think-Pair-Share: Turn to a partner and explain how the problems to suit your class as much as make a ten strategy is represented on the number line. How did I show possible. Lena’s thinking? Working On It (50 minutes per period) • Post and read aloud the following story problems. Tell students to represent their thinking using a number line. (Provide copies of BLM 31: Number Lines 0–20. For students who need more space to work, provide the page with only 3 number lines.) Explain that there are many strategies that can be used to solve the problems, but that today they’re going to practise using the make a ten strategy. After each problem, ask one or two students to share their thinking and number line. As they share, draw a number line that represents their thinking on the board so that all students can clearly see it. At first students will likely have difficulty labelling their hops. Your number lines will provide the model they need to move forward in their representations. As students work, circulate and ask guiding questions to help students who are having difficulty using and/or representing the make a ten strategy. Options for Instruction: 1. Have students work in partners to solve the first problem. Share solutions. Then have students solve the next problem independently in their Math Journals or on BLM 31: Number Lines 0–20. Repeat until all problems have been completed. Do this over 2 days. Consolidate. 2. Have students work in partners to solve every other problem (1, 3, 5) to address the various structures. Consolidate. Then give students the other problems to solve independently in their Math Journals. N OTE: Option 2 provides less scaffolding. If students are unfamiliar with the varying problem structures, choose option 1. Complete this lesson over 2 days. Make copies of BLM 32: Problems for Making a Ten and cut apart the problems if you wish to give students a copy of the problem to attach to their work. Problem 1 (result unknown): The class was growing beans in Science. The first week, the beans grew 8 centimetres. The second week, the beans grew 7 centimetres. How tall were the beans after two weeks? Problem 2 (result unknown): The class was also growing cherry tomatoes. One plant had 4 tomatoes on it. The other plant had 9 tomatoes on it. How many tomatoes were there altogether? Addition and Subtraction to 20 173

Problem 3 (change unknown): The class started to keep track of how many books the teacher read to them. The first week, he read 6 books. By the end of the second week, he had read 11 books. How many books did the teacher read during the second week? Problem 4 (part unknown): There are 18 chairs in the class. Some of them are blue and 4 of them are red. How many blue chairs are there? Problem 5 (start unknown): We had some sets of pencil crayons in our room. Then our teacher bought 7 new sets. Now we have 13 sets. How many sets did we start with? Problem 6 (start unknown): There were some pillows in our classroom but not enough. A parent donated 8 pillows. Now we have 12! How many did we start with? Differentiation • For ELLs, draw visuals beside the words in the problems or refer to concrete examples where possible (e.g., pencil crayons, chairs, pillows). Assessment Opportunities Observations: Analyse the independent work to determine next steps for instruction, including forming small groups to address specific needs in guided math lessons. Take note of which problem structures were most difficult for students and plan to incorporate those structures in future lessons to provide further practice. Conversations: Following is a sample conversation for Problem 3 with a student who is having difficulty understanding and starting to solve the problem. Teacher: What is the problem asking you? Student: How many books the teacher read? Teacher: How many books he read, when? Altogether? Student: I don’t know. Teacher: Let’s read the question part. Student: How many books did the teacher read during the second week? Teacher: Right. Do we know how many he read altogether? Student: Yes, he read 11. Teacher: Okay, what else do we know? Student: He read 6 books the first week. Teacher: Yes. So how can you figure out how many he read the second week? Student: Add from 6 to 11. Teacher: Go ahead and try that. 174 Number and Financial Literacy

Consolidation (10–15 minutes) • Have several students explain their strategies and show how they represented them on their number line. Ensure that each problem structure is addressed. Possible questions include: – What was the problem asking you to find out? – How did you know what number to start with on the number line? – What two numbers add up to 10? – How did you get that number? – How can you label your work so that we can clearly see what hops you took? – What other strategies could you use to solve this problem? (e.g., doubles, count on, near doubles, compensation) How would we show it on the number line? – Turn and explain ’s strategy to your partner. – Does anyone have a question about ’s strategy? – Which problem did you find most difficult to solve? – When does it make sense to use the make a ten strategy? Further Practice • M ake 10 Card Game: This is a game for 2 players. Set-up: Use a deck of cards but remove the jacks, queens, and kings. The aces represent ones. Deal the entire deck out between both players. Each player then turns up 4 of the cards from her/his stack, leaving the rest of the cards in the stack face down. Directions: – The goal is to make combinations of 10 using any of the 4 cards facing up. – Each player tries to make a ten using the 4 upturned cards (a ten can be made from a combination of 1, 2, 3, or 4 cards). – If the player can’t make a ten, she/he takes one of the cards and puts it at the bottom of the stack, replacing it with the top card from the stack. – The game continues until no more tens can be made or when a player runs out of cards. If a player runs out of cards, she/he wins. – If both players run out of cards at the same time, the player with the most tens wins. – Optional: Students can record number sentences for the tens that they made in their Math Journals. Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: According to Carol Dweck, “Every time a student makes a mistake in math, they grow a new synapse” (Boaler, 2016). Let students know that their brains stretch and grow when they make mistakes. Tell them that you give them challenging problems to help them stretch and grow as a Addition and Subtraction to 20 175

mathematician and that their mistakes are how they learn. Your belief in their capability as problem solvers will prompt them to be greater risk takers. You can put this into action by looking at a mistake that a student made (without naming the student) and analysing it together. Say, “Let’s see what we can learn from this great mistake. We can all use it to make our brain grow.” When you put the mistake into such a positive light, the student who made the mistake might even boast that it was his/her work! 176 Number and Financial Literacy

10Lesson Applying the Make a Ten Strategy to Subtraction Using a Number Line Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationships Teacher between addition and multiplication and between subtraction and division, Look-Fors to solve problems and check calculations Previous Experience • B 2.2 recall and demonstrate addition facts for numbers up to 20, and with Concepts: Students should be able related subtraction facts to count back using a number line and should • B 2.3 use mental math strategies, including estimation, to add and subtract have experience solving simple story problems whole numbers that add up to no more than 50, and explain the strategies that involve subtraction. used • B 2.4 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 100 Algebra • C2.2 determine what needs to be added to or subtracted from addition and subtraction expressions to make them equivalent Possible Learning Goals • Solves subtraction problems that have a variety of structures • Uses a variety of strategies and tools to solve subtraction problems • Uses ‘think addition’ to solve subtraction story problems • Uses the make a ten strategy to solve subtraction problems • Uses a number line to solve subtraction problems and represent strategies • Accurately represents jumps on the number line • Understands the problem and initiates a strategy Addition and Subtraction to 20 177

PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, Students often have a more difficult time subtracting, even when working with numbers less than 20, but with practice they can develop efficient strategies. Math Vocabulary: whole, Students may count back by 1s to solve a problem such as 15 – 8 or may nmthuaimnkkbeeaardltdienitnei,o, npart, subtract 5 to get to the benchmark of 10, then subtract 3 more to arrive at 7. Some students might subtract 10 from 15 to get to 5 and then add on 2 more. Once students understand the inverse relationship of addition and subtraction, they may use this relationship to add up and think of 15 – 8 as 8 + = 15. In this case, they can use their ten facts to start at 8, add 2 to get 10, and then add 5 more (8 + 2) + 5 = 15. About the Lesson In the Minds On, students will determine the missing part in an equation using Missing-Part cards and share their reasoning. This activity will provide practice in subitizing and will reinforce the idea that subtraction problems can be solved through ‘think addition’ (adding up). In the Working On It section, students will solve subtraction story problems that have missing parts in various places. The varied question structures will provide a challenge for students. Materials: Minds On (10–15 minutes) BLM 26: • Create a Missing-Part card using BLM 26: Missing-Part Card Template for Missing-Part Card Template, BLM 31: 12. Write 12 in the left section and draw 8 dots in the right section. Finally, Number Lines 0–20, draw 4 dots in the centre section and fold over the flap. class-sized number line • Show students the card and use the language of subtraction by saying, “One Time: 40–60 minutes part is showing (8) but one part has been taken away. What’s the missing part?” • Ask students for answers and record them. Then ask a few students to share their strategies (e.g., known fact, counted back 8 from 12, counted up from 8 to 10 then 2 more to 12, etc.). It’s likely that someone used ‘think addition’ to solve the problem but doesn’t know how to identify the strategy. If a student started at 8 then counted up 4 to 12 or started at 8, made a 10 then added on 2 more, they are using ‘think addition.’ Identify this strategy for the class and record it on your Counting Strategies anchor chart. • Possible prompts to guide discussion include: – What was the whole? (12) – What was the known part? (8) – When I said that ‘one part was taken away’ were you thinking of subtraction or addition? (subtraction) – Could I have just said, “What part is missing?” (yes) 178 Number and Financial Literacy

– I could have. I was just trying to get you to think about subtraction, but many of you added to get the number. Why was that? (e.g., It’s easier, I knew the addition fact.) – That’s interesting, isn’t it? We can actually ‘think addition’ even when we’re being asked to subtract. Write 11 – 9 = on the board. Ask students to solve it. Ask a student to share how he/she used ‘think addition’ to solve it. Ask, “Is this the only way to solve it?” (e.g., No, we could subtract, count back, etc.) – Ask students what the four number facts are for 12, 8, and 4. Reinforce the idea that the facts are related and that we can use addition even when we’re solving subtraction facts. • If desired, create and show a few more Missing-Part cards for practice without having students share their strategies each time. Working On It (20–30 minutes) • Provide each student with a copy of BLM 31: Number Lines 0–20 and a blank piece of paper or their Math Journals and have them solve the following subtraction problems. After each problem, have a few students share their strategies. Represent their thinking on a class-sized number line for all students to see. Accept all solutions, and ensure you discuss and represent the make a ten strategy and ‘think addition.’ Problem 1 (subtraction–separate structure, result unknown): You have 14 stickers. You give a friend 6. How many do you have left? Possible Solutions: Make a ten using ‘think addition’: 6 + = 10, 10 + = 14, 4 + 4 = 8 or Make a ten using subtraction: 14 – = 10; 10 – = 6; Count back from 14 by ones until you get to 6 and keep track of the count; Count back 6 from 14; Double 6 to jump to 12 and add on 2, etc. Problem 2 (subtraction–separate structure, result unknown): The pizza has 16 pieces. 7 of them have bacon. How many do NOT have bacon? Problem 3 (compare structure, difference unknown): Jolin scored 17 points. Sagar scored 9 points. How many more points did Jolin score? Problem 4 (subtraction–separate structure, change unknown): Jenna had 13 marbles. She lost some. Now she has 9 left. How many marbles did Jenna lose? • Record and pose the following problems. Have students work independently to solve the problems using BLM 31: Number Lines 0–20: – There were 16 birds in the tree. Then 7 flew away. How many were left in the tree? – On Monday, I saw 13 stars in the sky. On Tuesday, I saw 7 stars. How many more stars did I see on Monday than on Tuesday? – There were 15 people on the bus. Some people got off the bus at the first stop. Then there were 9 people left on the bus. How many people got off? Addition and Subtraction to 20 179

Differentiation • There may be students who would benefit from working with a partner during the teacher-led problem solving. • Use concrete materials or visuals to support ELLs in understanding the language in the problems. Assessment Opportunities Observations: • Observe students and analyse their work to note any misconceptions students have. Plan to meet with students in small-group guided math lessons to address their difficulties or to provide further practice related to their specific needs. Possible misconceptions or difficulties may include: – Believing that you can subtract digits in any order (as you can in addition) – Difficulty counting back to subtract – Unsure of which direction to move on a number line in relation to the problem – Difficulty interpreting the various problem structures; not knowing whether to add or subtract – Difficulty understanding or determining difference (how much more/less than...) – Needs more practice with combinations for ten in order to use the make a ten strategy – Hasn’t yet anchored 5 and 10 – Confused by open sentences with missing parts and might solve 8 + = 11 by adding 8 + 11 (Have the student explain what the number sentence means and make a story for it.) • There may be students who start with the whole and then show a jump to the known part without thinking about what the problem means. For example, a student may start at 16 and then jump to 7, thinking that the answer is 7. If this is the case, have the student act out the problem as shown below. Conversations: Teacher: Let’s read the problem again and think about what it means: There were 16 birds in the tree. Then 7 flew away. How many were left in the tree? Let’s use concrete materials to act this out. How can you begin? Student: I’ll count out 16 birds, I mean cubes. Teacher: Yes, represent the birds with cubes. Can you organize them so they’re easy to re-count? Student: I can make a group of 10 and a group of 6. 180 Number and Financial Literacy

Materials: Teacher: Great. Now that you have 16 birds let’s read the problem again. What do you think you should do next? BLM 25: Part-Part- Student: Take away 7 birds. I mean cubes. (Takes 7 away by taking the Whole Cards group of 6 away and 1 from the group of 10.) Teacher: That’s right. I like how you took away the 6 to make ten, then took one more away to make 7. You used the make a ten strategy. Now look at your number line. Can you see your error? Student: Yes, I didn’t jump back 7. I just went to 7. Teacher: That’s right. Why don’t you fix it? Student: I’ll jump back from 16 to 10 like I did with the cubes. Then I have to jump back one more to get 9. Teacher: So how many birds were left in the tree? Student: 9. Teacher: Go ahead and show that. Consolidation (10–15 minutes) • Consolidate after students have completed the independent task. Ensure the Consolidation highlights the strategies used, especially the make a ten strategy. Discuss how students decided whether to add or subtract. The third problem may be used to discuss the concept of ‘difference.’ • Possible questions to further probe students’ thinking: – How did you solve the problem? What strategy did you use? – Can you explain how the make a ten strategy works? When would you use it? – How are addition and subtraction related? – How did you decide when to add or subtract? – What is ‘think addition’? – In Question 3 you found the difference between 17 and 9. What do you think difference means? When have we worked with differences before? (e.g., analysing graph data) Further Practice • U se the Part-Part-Whole Cards as a warm-up or as a math centre to reinforce the inverse relationship of addition and subtraction. The cards can also help build automaticity with number facts. • Give students loose parts that they can use to make/act out addition and subtraction stories. Ask students to ‘act out’ a story, and tell the story to a friend. They can solve the problem together and record the number sentence that represents the story. Making up their own stories and ‘acting them out’ Addition and Subtraction to 20 181

Materials: will help students understand the different structures of problems, and will BLM 5: Ten Frames help them understand the different meanings of addition and subtraction. (1–20) For example: Teaching Tip Material: miniature pine cones Integrate the math talk moves (see Problem: The tree had 14 pine cones on it. Some fell off and then there were page 7) throughout only 6 pine cones left on the tree. How many fell off? 14 – = 6 Math Talks to maximize student Building Social-Emotional Skills: Positive Motivation and Perseverance: participation and Many students find it difficult to make mistakes and accept descriptive active listening. feedback. They may think that it means that they’re not good at math. But as Jo Boaler says, when students know that we believe in their capabilities and value their mistakes, they are more likely to develop a growth mindset in math (Boaler, n.d.). Consider this idea: rather than marking student work with checkmarks and Xs and then putting a score on it such as 8/10, simply circle the answers that require a re-do. Have students work with you or a partner to correct the work. Emphasize the learning that occurs when we make and re-think our mistakes. Math Talk: Math Focus: Doubles facts Process: Use finger flashes to practise doubles to 20. Flash a number of fingers and tell students to flash the same number of fingers back to you. As you flash, say, “4 +” and students should finish the fact by flashing and saying, “4 = 8.” Saying the fact provides a little thinking time for students who may not be as quick to recall the facts. After the finger flashes, use ten frame images. Flash a ten frame and ask students to visualize the amount doubled and say the double. For example, if you flash a ten frame with 7 dots, students should visualize and say, “14.” 182 Number and Financial Literacy

11Lesson Doubles Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationships Language between addition and multiplication and between subtraction and division, Curriculum to solve problems and check calculations Expectations • B 2.2 recall and demonstrate addition facts for numbers up to 20, and Teacher Look-Fors related subtraction facts Previous Experience • B 2.3 use mental math strategies, including estimation, to add and subtract with Concepts: Students have had whole numbers that add up to no more than 50, and explain the strategies used experience adding two numbers together. Prior • B 2.4 use objects, diagrams, and equations to represent, describe, and solve exposure to arrays or other organized situations involving addition and subtraction of whole numbers that add up groupings of visuals will to no more than 100 also be helpful when expecting students to Media Literacy represent doubles in a clear, organized way. • 3.2 Produce media texts for specific purposes and audiences, using a few PMraotcheesmseast:ical simple media forms and appropriate conventions and techniques rrtPeeorpfoolrelbesclsetaeimnnngdti,snsocgtlorv,anintnegeg,cisetiesnl,ge,cting communicating Possible Learning Goals • Develops strategies for mental addition and subtraction • Automatically recalls the doubles facts for numbers 2–12 • Explains what a doubles fact is • Represents doubles in pictures clearly and accurately • Recalls doubles facts without having to count About the Doubles facts are generally easy for students to learn, especially if they have a real world visual anchor (e.g., two hands for 5 + 5 = 10). Once learned, students can use doubles to solve other facts, including near doubles (e.g., 8 + 9 can be solved by using a known doubles fact such as 8 + 8 = 16 then adding 1 more to get 17 or by using the known doubles fact of 9 + 9 = 18 then subtracting 1). About the Lesson This lesson should be taught in two parts with each part requiring one period. In Part 1, students will investigate doubles facts by working in partners to generate and record them by writing the doubles fact and drawing a corresponding dot picture. Consolidate the learning before moving on to Part 2. continued on next page Addition and Subtraction to 20 183

Math Vocabulary: In Part 2, students will learn a doubles rap to help them learn the doubles doubles, represent, facts using a rhythmic chant. They’ll also create a poster with visual symmetry anchors for the doubles facts. Students will consolidate this learning through a gallery walk. The more students work with the facts in Math Teaching Tip Talks and problem solving, the more likely they are to achieve automaticity. Being able to recall the facts automatically will make the You may wish to do strategy of near doubles an efficient choice when solving problems such the Math Talk on as 2 + 3, 3 + 4, 4 + 5, 5 + 6, etc. doubles facts (see page 182) before beginning this lesson. Minds On (10 minutes) Materials: • Show students Digital Slide 75. Ask students what they notice about the quantity and how it’s shown (e.g., double, two parts are the same). Ask them what a double is and what double they see. Record the doubles fact (e.g., 4 + 4 = 8) on the board. Ask students what they notice about the numbers (there are 2 of the same number). Digital Slides 75–76: • Show students Digital Slide 76. Ask them what doubles fact they see and tell Doubles Facts, BLM 33: Doubles Rap, BLM 34: them to solve it. Say, “Turn to a partner and tell your partner what the Doubles Rap Recording doubles fact is and what the solution is. Ask your partner how he/she solved Sheet (photocopied on it.” Have a few students share their solutions and strategies with the class. large-sized paper – one Students will use a variety of strategies to solve the problem as most likely per student) don’t yet know the doubles facts. Time: 55–60 minutes (for • Tell students that they are going to investigate doubles facts and that by each of 2 periods) working with them a lot, they will eventually learn them ‘by heart.’ Working On It (30–40 minutes per Part) Part 1: • Have students work in partners. Tell them that their challenge is to find all the doubles facts that add up to 24 or less and to record the facts along with dot pictures in their Math Journals. Tell them that they may use any math tools that they think might help them. Ensure students understand the task before sending them to work by asking questions such as the following: – What is your task? – What will your doubles look like? How will you record them? – What is an example of a double? Come up and write the doubles fact. Teaching Tip – What would a doubles picture look like? Come and draw it. Consolidate following Part 2: each part of the Working On It. • Show students Doubles Rap (BLM 33) by projecting it, showing a chart paper version, or handing out single copies. Read the rap to students with rhythm. Tell students that one strategy for learning something is to put a rhythm or song to it. Explain that the rhythm can help them remember the words, and 184 Number and Financial Literacy

Teaching Tip in this case, the doubles facts. Read it at least twice more and invite students to join in. Carol Thornton suggests these real- • Tell students that another strategy for learning math facts is to use visuals to life visual ideas for doubles: connect a fact to a picture. Tell them that they are going to create a poster that has visuals for each of the doubles facts. Ask students to think of things 3 + 3 a bug with three in real life that would make a good visual for 1 + 1 (e.g., 2 eyes, 2 shoes). legs on each side You may want to brainstorm visuals for each fact. 4 + 4 a spider with 4 • Demonstrate how to draw the visuals so that the double is very clear. For legs on each side example, when drawing a visual for 3 + 3, draw two tricycles side by side 5 + 5 two hands with with all 3 wheels clearly shown. Model an example and a non-example. 5 fingers on each Explain that the sets must be clear for the visual to help you remember the hand fact. You may want to give students scrap paper to practise drawing the visuals before adding them to the poster if you are going to assess this as a 6 + 6 an egg carton media project. If you’re planning to evaluate the poster, you may wish to with two rows of 6 co-construct criteria for it and provide midpoint feedback. 7 + 7 two weeks on a • Provide students with a large-sized copy of BLM 34: Doubles Rap Recording calendar Sheet to record the visuals beside each doubles fact. Alternatively, students 8 + 8 a set of crayons could make a poster on large-sized paper without the doubles rap on it. with 2 rows of 8 in a box Differentiation 9 + 9 an 18-wheeler • The Math Centre in the Further Practice section provides a concrete, visual with two sides of 9 wheels experience for exploring doubles. The doubles rap provides a rhythmic strategy for auditory learners. (Van de Walle & Lovin, 2006, p. 56) • Encourage ELLs to represent the doubles using images that represent their personal cultures. Assessment Opportunities Observations: Observe how students represent the doubles facts. Do they create pictures and groupings that clearly show the double? As students are working, ask individuals a doubles question (e.g., What’s 6 + 6? What’s double 4? Which double has a sum of 14?). Observe whether or not they have automatic recall or need to look at the rap. This can serve as both formative assessment and practice for students. Consolidation (25 minutes — 15 minutes for Part 1 and 10 minutes for Part 2) Part 1: • Have students meet in partners to share their work. Tell students to examine each other’s work to see if they have all of the doubles facts. Then have students share the doubles in order as a class. Record the doubles on the board or on chart paper. Ask students to come up and draw the circles for each double. Reinforce the idea of drawing the visuals in arrays so that the double is clearly shown. Ask students to reflect on their own representations and to consider whether or not they’ve clearly shown the double. Addition and Subtraction to 20 185

Materials: Part 2: miras or mirrors on • Gallery Walk: Have students lay out the posters on their desks. Tell them stands, loose parts, BLM 35: Double Bingo, that they are going to look at each other’s posters to see how everyone dice, BLM 5: Ten Frames represented the doubles facts. Give students a few minutes to mill through (1–20) the room, looking at each other’s posters. Next, have students come to the carpet. Ask them to reflect on the posters. Ask, “What did you notice? What visuals were really helpful for learning the doubles facts? How do you think you might use this poster at home to help you learn the doubles facts?” Further Practice • M ath Centre: Explore doubles with symmetry. Use miras (or mirrors on stands) to visually double a quantity. Provide miras or mirrors and loose parts. Have students put a quantity of loose parts in front of a mirror to see the doubled image and then record the doubles fact. • D ouble Bingo: Provide a copy of BLM 35: Double Bingo and a die to each student. Have students roll the die and double the number. Tell students to cover that square on their game board. The first player to cover the 4 corners and a straight line (vertical, horizontal, or diagonal) wins. Alternative Activity: Cut apart the Double Bingo board and use the various representations to create a concentration game. • Finger Flashes: Use finger flashes to practise doubles to 20 in a kinesthetic way. Flash a number of fingers and tell students to flash the same number of fingers back to you. As you flash, say, “4 +” and students should finish the fact by flashing and saying, “4 = 8.” Saying the fact provides a little thinking time for students who may not be as quick to recall the facts. • Ten Frame Fast Images: Use BLM 5: Ten Frames (1–20). Flash a quantity on a ten frame. Ask students to visualize the amount doubled and to say the double. For example, if you flash a dot card with 7, they should visualize and say, “14.” Once students are familiar with the routine, they can do this in small groups. Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: When students understand that people learn in different ways, it helps them see themselves as capable learners and gives them a sense of belonging. By sharing the strategies that help students learn in an intentional way, such as by explaining that a rap can help some students learn while visuals or hands-on materials help others, students are likely to realize that learning can take different forms. As students move from grade to grade, it’s important that they, and their teachers, are aware of the strategies that help them work successfully in math. Consider co-creating an anchor chart of ‘Strategies That Help Me Learn.’ It can represent learning strategies in math and across all subjects. 186 Number and Financial Literacy

Materials: Math Talk: Digital Slides 77–86: Math Focus: Near doubles, subitizing, building mental models Fast Images (Set 5) Process: Tell students that you are going to show them an image and they are to Teaching Tip find the total. Flash the image and have students share their answers. Record all solutions. Ask a few students to share their strategies. Represent their strategies Integrate the math by circling the dots or drawing them to show how students visualized the parts. Talk Moves (see If no one used near doubles, ask students how this strategy could be used. After page 7) throughout sharing, tell them that you’ll be showing more images that lend themselves to Math Talks to using near doubles and encourage them to try the strategy. maximize student participation and Repeat the process with two or three more digital images. active listening. Let’s Talk Teaching Tip Select the prompts that best meet the needs of your students. Ten frame images or dot configurations • D isplay Digital Slide 82 showing a set of 7 dots and a set of 8 dots side by side can be used to work on near doubles by to represent 7 + 8. What is your solution and strategy? (15. I added 7 + 3 = 10. flashing an image Then I added on 5 more. That equals 15.) and telling students to double the image • You made a ten and then added on the rest. Where did 3 come from? (I in their minds and add one or subtract decomposed 8 into 5 and 3 and added on 3 to 7 so that I made 10.) Circle the one more. This type 7 and 3 dots to show 10. Then circle the 5 dots. Does this represent your of visual subitizing strategy? (Yes.) helps students build mental models • D id someone solve this problem using a different strategy? (I used doubles. I for later use when solving computational added 7 + 7 = 14. Then I added one more. That’s 15.) Draw another set of 7 problems. dots and 8 dots. Circle the 7 dots and 7 of the 8 dots. That’s a good use of near doubles. When numbers are only one apart, we can apply near doubles to help us. • Is there another way we can apply near doubles? (You can add 8 + 8. That’s 16. Then you have to subtract one because you’re really only adding 7 + 8 and that’s 1 less.) Draw a set of 8 dots and a group of 7 dots. So we have a group of 8 and a group of 7. You added one more to the group of 7 so I’ll draw one more using another colour. That’s 16. So now I’ll cross that extra one out. Now there’s 15. Does that make sense? Thumb up or down? • Okay. Let’s try another problem. I’d like you to try using near doubles to solve this next one. Show one or two images from Digital Slides 77–86. • Repeat the Math Talk on subsequent days using other digital images. Addition and Subtraction to 20 187

12Lesson Near Doubles Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationships Teacher between addition and multiplication and between subtraction and division, Look-Fors to solve problems and check calculations Previous Experience • B 2.2 recall and demonstrate addition facts for numbers up to 20, and with Concepts: Students will need to have related subtraction facts knowledge and understanding of the • B 2.3 use mental math strategies, including estimation, to add and subtract doubles facts for addition in order to apply the whole numbers that add up to no more than 50, and explain the strategies strategy of near doubles. used They do not, however, have to have automatic recall of • B 2.4 use objects, diagrams, and equations to represent, describe, and solve all of the doubles facts. situations involving addition and subtraction of whole numbers that add up PMraotcheesmseast:ical to no more than 100 ccssrPooeterrmnalaoensbtmceoeltegncuinmitinenignisgcgst,aoo,artolenrivnelfdislgnpepgracer,ntosidnevginn,tgin, g, Possible Learning Goals • Uses near doubles to solve mental math problems • U nderstands and explains why you have to compensate (give one to one number; take one from the other number) • Recalls doubles facts with automaticity or uses a resource to identify the double • Uses near doubles as a strategy to solve problems where there is a difference of 1 in the addends • R ecognizes the double on the arithmetic rack • Explains and demonstrates how to use near doubles to solve addition problems • F lexibly and accurately uses either the upper or lower double to solve a near- doubles question (e.g., when solving 7 + 8 can use 7 + 7 or 8 + 8 and accurately compensate by adding or subtracting the ‘1’) About the Near doubles are also referred to as the doubles +/– strategy. Near doubles are used when the two numbers being added are only one away from each other (e.g., 1 + 2, 2 + 3, 3 + 4, 4 + 5, 5 + 6, 6 + 7, 7 + 8, 8 + 9). You either double the smaller number and add one more, or double the higher number and subtract one. Once students memorize the doubles facts, near doubles becomes an efficient strategy for them. In addition, near doubles involve compensation which is an important mathematical idea for students to learn. Compensation is a mental math strategy that involves adjusting one of the addends to make 188 Number and Financial Literacy

Math Vocabulary: the equation easier to solve. For example, when using the near doubles strategy daodudbelnedss, ,nseuamr doubles, to solve 8 + 9, you can mentally ‘take’ one from the 9 and ‘give’ it to the 8 so that you are thinking ‘8 + 8 = 16’ and then compensate by adding one more Teaching Tip onto the 8 + 8 double. You may wish to do NOTE: Near doubles can be applied to subtraction as well as addition. For the Math Talk on near example, when solving 16 – 9 students might think ‘16 – 8 = 8 but I’m taking doubles and subitizing away 9 so I have to compensate by taking away one more because 9 is one (see page 187) before more than 8.’ This requires a solid knowledge of the doubles facts and an beginning this lesson. understanding of compensation. Just as many students find it more difficult to subtract than to add, they also find it more difficult to use the near doubles strategy for subtraction than for addition. If you find students are adept at applying near doubles to addition, consider introducing it as a strategy for subtraction. About the Lesson The arithmetic rack is a great tool for introducing and working on near doubles as it has two rows of 10 beads. This provides the opportunity to create clear visual representations of near doubles to 19. In this lesson, students will use the arithmetic rack to investigate near doubles as an addition strategy. Following the investigation, students will work independently on BLM 36: Near Doubles Using the Arithmetic Rack. This will allow you to observe your students’ understanding and application of the near doubles strategy. Materials: Minds On (5–10 minutes) BLM 33: Doubles • Project BLM 33: Doubles Rap (from Lesson 11) and have students say it in a Rap, arithmetic racks (one per pair of students), few of the following ways: paper and markers, BLM 36: Near Doubles Using – Whole class. the Arithmetic Rack – Divide the class in half. Say the chorus together. Group A reads the first Time: 45–65 minutes double rhyme and Group B reads the next double rhyme. Continue alternating until the end of the rap. – Read it in a high voice or a low voice. – Read it quickly or slowly. – Put actions to each double. Working On It (30–40 minutes) • Have students work in partners, sharing an arithmetic rack (Partner A) and paper and marker (Partner B). They can switch roles with each question so both get a chance to work with the arithmetic rack and to practise recording on paper. • Explain that Partner A will use the arithmetic rack and Partner B will record on paper as you record on the board. Write the equation 3 + 4 = on the Addition and Subtraction to 20 189

board and tell Partner B to record it as well. Tell Partner A to push 3 beads on the top and 4 on the bottom. Ask them if they see a double. (3 + 3 = 6) Record the double and have Partner B record as well. Ask how many more they see. (1) Record and say, “You saw a double 3 + 3 = 6. Then you added 1 more to your answer: 6 + 1 = 7.” Record. Tell Partner B to record this as well. Tell students to switch roles and try another question. • Write 5 + 6 = on the board. Tell students to push 5 on top and 6 on the bottom. Ask them what double they see (5 + 5 = 10). Record. Ask how many more are left to add on. (1) Record: 10 + 1 = 11. • Write 7 + 8 = on the board. Tell students to push 7 on the bottom and 8 on the top. Ask them what double they see (7 + 7 = 14). Record it. Ask how many more are left to add on. (1) Write 14 + 1 = 15. Ask them if they see another possible double (8 + 8). Tell students to push over one more bead and say, “Now the double is 8 + 8 = 16.” Record. Then say, “But we used an extra bead to make that fact, so what do we have to do?” (take one away from 16) Record 16 – 1 = 15. Ask students why they think we arrived at the same answer for both methods. • Write 8 + 9 = on the board. Tell students to push 8 on the top and 9 on the bottom. Ask them what double they see (8 + 8 = 16). Record. What is left to add on? (1) Record 16 + 1 = 17. Ask students if we could also record 8 + 8 = 16 + 1. Have a couple of students explain their reasoning. (e.g., 8 + 8 is 16, but 16 + 1 is 17. They’re not the same.) Reinforce that the equal sign means ‘the same amount on both sides.’ Now ask students for the other possible near doubles fact. Have them push one more. Tell partners to work together to record the near doubles strategy for this (9 + 9 = 18; 18 – 1 = 17). Have students hold up their paper for you to check how they recorded. Ask one or two students to explain their strategy. • Give each student a copy of BLM 36: Near Doubles Using the Arithmetic Rack. Explain how to complete the work. Complete the first question together as an example. Have students work individually using red and another colour other than white to represent the beads that are being ‘moved.’ Encourage students to model the question on the arithmetic rack before recording. Differentiation • Provide an anchor chart of the doubles facts to assist students who are still learning these facts. • When explaining the independent task (Near Doubles Using the Arithmetic Rack), check in with how students are feeling about the task (e.g., If you understand and can work independently, show me a thumbs up, if you need to ask a quick question to move on, put your thumb sideways, and if you’d like to work with me in a small group, show me two fingers.). As you send students off to work, answer the quick questions, then work with those students who indicate they need more help in a small-group guided math lesson using arithmetic racks. 190 Number and Financial Literacy

Assessment Opportunities Conversations: When working with a small guided group or an individual who is having difficulty, use low number near doubles such as 2 + 3, 3 + 4, and 4 + 5 to begin so that computational abilities are less likely to be a problem. Use the arithmetic racks to work through a couple of problems together before having students complete the independent task. See the following sample conversation: Teacher: Let’s solve the problem 2 + 3 = . Imagine a bunk bed. There were 2 kids on the top. Push your beads with one push. And there were 3 kids on the bottom. Push your beads with one push. Look at your beads. What double do you see? Student: I don’t see a double. Teacher: Let’s see which beads have a match. That helps us find the double. (Model matching up the first two beads.) Here’s one match and here’s another match. And one is left out. How many matches do you see? Student: Two matches but 4 beads. Teacher: You’re right. There are two matches. So our double is? Student: 2 + 2 = 4. Teacher: Yes, let’s write that double. Now what do we have to do with the extra 1? Student: Add it on. That’s 1, 2, 3, 4, 5. Teacher: Yes. Did you need to count them all again? I think you already figured out 2 + 2? Student: Yes, that’s 4. Teacher: Can you count on from 4? Student: 4, 5. Teacher: Right. What did you add on to 4? Student: 1 more. Teacher: Yes. So what number sentence can we write now to show that? Student: 4 + 1 = 5. Consolidation (10–15 minutes) • Invite students to Think-Pair-Share. Ask them to explain the near doubles strategy to a partner in their own words. Allow them to use the arithmetic rack and/or paper to help them. Have a couple of students share with the class orally with a demonstration on the arithmetic rack and by recording the equations on the paper. • Ask students when they think it is best to use the near doubles strategy (e.g., when numbers are 1 apart and when you know your doubles). Addition and Subtraction to 20 191

Materials: Further Practice BLM 37: Near Doubles Using Ten • H ave students work independently to apply their understanding of near Frames, BLM 38: doubles. Provide one or both of the following BLM activities for students to Double Dip complete: Cones, Digital Slides 12–29: – BLM 37: Near Doubles Using Ten Frames Fast Images (Set 2) – BLM 38: Double Dip Cones Teaching Tip • F ast Images: Flash a dot configuration from Digital Slides 12–29: Fast Integrate the math Images (Set 2). Tell students to double the image in their minds and add one talk moves (see or subtract one more. page 7) throughout Math Talks to Building Social-Emotional Learning Skills: Self-Awareness and Sense of maximize student Identity: Providing students with opportunities to self-assess their participation and understanding of concepts and then enabling them to choose how they access active listening. support is empowering. (See the Differentiation for a strategy that you can use to help students identify their comfort level with a task.) Temporary guided math groupings provide students with a ‘safety net’ when they feel they need extra support. Since the groups are temporary and change frequently, students shouldn’t feel singled out or see themselves as ‘the struggling group’ in math. Ensure that you work with students who are excelling on math challenges in a small group as well, so that students see that everyone needs support sometimes. Working in a small group should be a positive experience that builds mathematical understanding and confidence. Math Talk: Math Focus: Using near doubles to add Process: Record 5 + 6 + 3 = on the board. Ask for student answers and record them. Then ask a few students to share their strategies. Let’s Talk Select the prompts that best meet the needs of your students. • W ho would like to share their solution? (I added 5 + 5 = 10 and then added on 3 more to get 13.) I see one 5 in the equation. Circle it. Where did you get the other 5? (I took one from the 6 so that’s 5.) What strategy were you using? (Near doubles.) • You just missed one step. Can you figure it out? You said you took one from the 6 to get the 5 so I’ll record 6 and show with arrows that you decomposed it into 5 and 1. Then I can record 5 + 5 = 10. (Oh, I have to add on the 1 that’s left from the 6. Oops. Okay then 5 + 5 = 10 and + 1 is 11, and 11 + 3 is 14.) Yes, we have to remember to account for both parts when we decompose a number. • Anna was using near doubles. How else could we name the strategy? (I did the same thing but I knew that 5 and 5 were ten so I was making a ten.) Interesting. Tomas, you went through the same steps but you were thinking about the 10. 192 Number and Financial Literacy

• Did anyone use near doubles in a different way? Jason? (I knew double 6 so I had 12 then I subtracted 1 since I made the 5 go up by 1. That’s 11. Then I knew that 11 + 3 was 14. It’s a known fact for me.) • Turn and talk to a partner. Then we’ll share. How are Anna’s and Jason’s strategies the same and different? (The sum is the same; the addends are the same; they both decomposed; they both used near doubles but one doubled 5 and the other doubled 6; one counted on the last number and the other used a known fact.) • Conduct more Math Talks over a week or two using different equations appropriate for the near doubles strategy. Addition and Subtraction to 20 193

13Lesson Exploring the Calendar Using Addition and Subtraction Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationships Previous Experience between addition and multiplication and between subtraction and division, with Concepts: to solve problems and check calculations Students are familiar with the calendar and its • B 2.2 recall and demonstrate addition facts for numbers up to 20, and basic concepts (e.g., days of the week, month, related subtraction facts number of days in a week) and have had • B 2.3 use mental math strategies, including estimation, to add and subtract experiences reading it. whole numbers that add up to no more than 50, and explain the strategies Teacher used Look-Fors • B 2.4 use objects, diagrams, and equations to represent, describe, and solve PMraotcheesmseast:ical Problem solving, situations involving addition and subtraction of whole numbers that add up ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, to no more than 100 Math Vocabulary: Possible Learning Goals pded(aaevatey.ttgsnee.,t,,rsnwo1,,rescdmetai,knol2esarnn,el dd,mn,alueor3m,snrstdbh,)esr,s • Applies addition and subtraction strategies to solve contextual problems • Recognizes the pattern of counting by 7s and the relationship of days to weeks • Takes leaps (i.e., by 7) rather than counting by 1s • Moves in the appropriate direction when counting on the calendar • Uses ‘think addition’ or adds up to solve subtraction questions About the The calendar provides an authentic context for working with numbers, adding, subtracting, and solving problems. Look for opportunities throughout the year to incorporate math related to the calendar (e.g., countdown to a holiday, recording special events and birthdays, tracking the number of days students are in school, recording the daily temperature for a month, etc.). About the Lesson In the Minds On, students will review calendar skills. In Working On It, students will solve addition and subtraction problems based on the calendar. 194 Number and Financial Literacy

Materials: Minds On (15 minutes) BLM 39: Blank Monthly • Have students sit where they can all see the current month in the calendar. Calendar, BLM 40: Calendar Problems, If you don’t have a large class monthly calendar, create one using BLM 39: paper Blank Monthly Calendar and provide students with a copy of it or project it Time: 40–50 minutes so all can see. Choose a few of the following questions/prompts to discuss the calendar: – What is today’s date? – How many days are there in a week? – How many weeks are there in a month? – How many days are there in this month? – How many days would there be in two weeks? In three weeks? – What patterns do you see in the calendar? (e.g., adding by seven as you go vertically down a column, counting by 1s horizontally, counting by 8s going diagonally down to the right or by 6 going diagonally down to the left) • Record the pattern for counting by 7s on the board (e.g., 7, 14, 21, 28, 35) for students to refer to when discussing the calendar. • Have students Think-Pair-Share: How is using the calendar similar to using the hundreds chart? • Ask a few questions personally related to your calendar, referring to past and upcoming events (e.g., How many days until Pizza Day? How many days ago was it the 4th? How many more days do we have until the last day of the month?). As students answer the questions, have them share their strategies for solving the problems. Ask them to think of a number sentence that would represent their strategies. If students only count on or count back, ask students to think of other ways to solve the problem without counting by 1s. Ask, “Did anyone use a counting pattern or one of our addition/subtraction strategies?” Working On It (15–20 minutes) • Provide each student with a copy of BLM 40: Calendar Problems and paper for working out and recording their solutions. Have students work in partners to solve the first three problems. (The last three problems will be assigned as independent work following the Consolidation.) Differentiation • When doing a whole-class activity such as the Minds On talk, you can differentiate by asking questions that vary in complexity so that all students have an opportunity to participate and move forward in their learning. For example, asking how many days are in a week is less complex than asking how many days there are in 3 weeks. Although students may not be able to count by 7s, they could likely count by 5s, then add on 3 more groups of 2 or they might count by 10s then compensate by subtracting. Addition and Subtraction to 20 195

• Choose students to answer questions that are in their proximal zone of development—the difference between what a learner can do without help, and what they can’t do (Vygotsky, 1978) as we want to support students to move them forward mathematically. When we consistently ask students to answer questions that we already know they can solve, we may be reinforcing their strategies which is beneficial, but it’s not enough to move them forward mathematically. We want to challenge students with questions that require them to think and reason. Knowing what strategies your students are currently using will help you determine what questions will provide just the right amount of challenge. • ELLs may have difficulty with language such as, “If today is the 3rd....” Rephrase this for them and say, “We’re pretending that today is the 3rd. Let’s circle the 3rd.” Assessment Opportunities Observations: In addition to observing the strategies that students use, also note how students record their work (e.g., number sentences, number lines, repeated addition, etc.). Some students may use the calendar in a similar manner to using the hundreds chart. They may circle the starting number and count forward or back by 7s by circling the numbers or by showing jumps as you would on a number line, then count on the remaining ones. This is a good transfer of learning and should be reinforced. Say, “I see you’re thinking as a mathematician and using the number patterns in the calendar like you use the patterns in a hundreds chart to solve the problem.” Conversations: Counting back when numbers are not close together is not an efficient or effective strategy as students often make errors in the count. If you notice that a pair of students are using counting by 1s forward or backwards to solve the problems, adapt the following conversation to encourage taking bigger leaps. Sample question: If today is the 3rd, how many sleeps will you have before you go on the class trip? Teacher: How did you solve the first problem? Student: We started at 3 and counted by 1s to 15. Teacher: Did that work? What answer did you get? Student: 12. Teacher: Let’s try another strategy to check your answer. You started at 3. Could you take a bigger leap or use a number pattern that the calendar has? Student: We could jump down a box. Teacher: What would the jump be? Do you know without counting? Student: 7? (If they don’t know, have them count.) Teacher: Yes, we learned that if we leap down or up a box we’re counting by…? Student: 7. 196 Number and Financial Literacy

Teacher: So now you took a leap of 7 and landed on 10. Now what do you have to do? Student: Count on until we get to the 15. Teacher: Try that. Did you get the same answer as when you counted by 1s? Student: Yes. Teacher: Do you think that you could take leaps rather than counting by 1s next time? NOTE: They could have also taken another leap of 7 to 17 and then counted back 2. Teaching Tip Consolidation (10–15 minutes) You may wish to do • Choose samples of students’ work to be shared that show a variety of the Consolidation the next day in order to strategies from basic (counting by 1s, friendly numbers) to more analyse the students’ sophisticated (e.g., using the 7 pattern then adding/subtracting on or back, work. doubling and adding up to subtract, compensation, make a ten). • Have students Think-Pair-Share: How is ’s strategy the same as or different from yours? • Record the strategies used. Encourage students to choose one of the strategies to apply to the next set of problems they do. • Assign the last three problems on BLM 40: Calendar Problems for students to solve independently. Further Practice • D aily Calendar Problem: Each morning ask a problem related to the calendar in your classroom that students can solve mentally. This can be your Math Talk for the day. • What’s the Difference? Card Game: This is a game for 2 players. Set-up: Use a deck of cards but remove the jacks, queens, and kings (aces represent ones) and counters. Deal the entire deck out between both players. Players keep their stack of cards face down. Directions: – Both players turn over a card showing a number. If both numbers are the same, the cards are discarded and two more cards are turned up. – Each player tries to be the first to determine and say the difference between the numbers or they can work together to find the difference. – The player with the greater number on his/her card takes counters equal to the difference of the two cards. The used cards are then placed in a discard pile. – The game is over when there are no cards left to turn over. The player with the most counters wins the game (Adapted from Lawson, 2015, p. 165) Addition and Subtraction to 20 197

Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: In the Differentiation section, it’s suggested that we challenge our students with questions that are in their zone of proximal development (Vygotsky, 1978) in order to move them forward mathematically. When we challenge students to think and reason, we are sending them a message that we believe in them and are helping them to develop perseverance. Whether or not they can solve the problem is not nearly as important as their willingness to take risks and try to solve it. Reinforce their risk taking and perseverance! 198 Number and Financial Literacy