p115-198-Gr2-ON-Number_Unit2 AddSub20

Unit 2: Addition and Subtraction to 20 Lesson Content Page Introduction to the Addition and Subtraction Units 115 121 Addition and Subtraction to 20 123 130 1 Decomposing Numbers to 20 134 139 2 Decomposing to Solve a Problem 145 150 3 Fact Families Investigation 158 4 Part-Part-Whole Relationships: Determining the Missing Part 163 5 Story Problems Using Addition and Subtraction 171 6 Varying the Unknown in Addition and Subtraction Problems 177 7 Make a Ten: Adding Three Numbers 183 188 8 Applying the Make a Ten Strategy to Addition 194 Using Ten Frames 9 Applying the Make a Ten Strategy to Addition Using a Number Line 10 Applying the Make a Ten Strategy to Subtraction Using a Number Line 11 Doubles 12 Near Doubles 13 Exploring the Calendar Using Addition and Subtraction



Introduction to the Addition and Subtraction Units There are two units that specifically address the conceptual understanding of addition and subtraction in this resource: • Unit 2: Addition and Subtraction to 20 • Unit 4: Addition and Subtraction to 100 Conceptual understanding is critical and should not be overlooked in favour of memorizing the basic facts. Lawson supports this statement with a quote from the National Research Council, “For students in grades K–2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic” (Lawson, 2016, p. 56). In Unit 2, students solve problems that will increase their conceptual understanding of addition and subtraction and the relationship between the two operations. Students will also investigate strategies that will help them develop automaticity with addition and subtraction facts to 20. In Unit 4, which focuses on two-digit addition and subtraction, students will invent various strategies for adding and subtracting and will apply the mental math strategies and computational skills they’ve developed in previous units to add and subtract two-digit numbers. About the As students count, compose, and decompose quantities they learn about number relationships, which supports their understanding of addition and subtraction. Addition is often taught as meaning ‘put together,’ while subtraction is often said to mean ‘take away.’ According to Van de Walle, these definitions are too narrow and can result in a limited understanding of the two operations, and how they are related. Van de Walle explains that, “addition is used to name the whole when the parts of the whole are known (and) subtraction is used to name a part when the whole and the remaining part are known” (Van de Walle & Lovin, 2006, p. 66). Based on numerous research studies, he emphasizes that students can solve problems by “thinking through the structure of the problems rather than by identifying the type of action or key words” (Van de Walle & Lovin, 2006, p. 66). He encourages teachers to have students analyse story problems in order to understand what they mean and to solve them with a focus on the context and what the answer means rather than merely looking for keywords that indicate an operation. Addition and Subtraction 115

Materials Throughout the units, students will work with various materials and tools that will help them develop a conceptual understanding of addition and subtraction. They’ll use concrete objects such as counters, connecting cubes, and ‘loose parts’ (any materials including natural materials such as beans or pine cones), as well as tools such as ten frames, the arithmetic rack, number lines, and the hundreds chart. They’ll also use materials that can be concretely grouped into tens and ones such as connecting cubes as well as base ten materials that show the proportional relationship of ones, tens, and hundreds. These concrete tools will help students form mental models that they can draw upon to solve problems. Structures of Addition and Subtraction Problems There are three basic structures of problems: • Join and separate problems • Part-part-whole problems • Compare problems Join and Separate Problems Join and separate problems both involve an action that takes place over time. Join problems have the beginning quantity increased while separate problems have the beginning quantity decreased. Clements and Sarama highlight how the position of the unknown can affect the difficulty of the problems. For example, result unknown problems (5 + 4 = ) tend to be easier than change unknown problems (5 + = 9), and start unknown problems tend to be the most difficult ( + 4 = 9) (Clements & Sarama, 2009, p. 62). The following table represents the different structures of problems. Action: Join Problems Position of Unknown Example Result Unknown Jesse has 4 balloons. He gets 3 more. How many balloons does he have? 4 + 3 = Change Unknown Jesse has 4 balloons. He is given some more balloons. Now he has 7 balloons. How many balloons was he given? 4 + = 7 Start Unknown Jesse has some balloons. He is given 3 more. Now he has 7 balloons. How many balloons did Jesse have at the start? + 3 = 7 Action: Separate Problems Position of Unknown Example Result Unknown Sana has 7 cookies. She gives 4 away. How many cookies does she have now? 7 – 4 = Change Unknown Sana has 7 cookies. She gives some away. Now she has 3 cookies. How many did she give away? 7 – = 3 Start Unknown Sana has some cookies. She gives 4 away. Now she has 3 cookies. How many cookies did Sana have at the start? – 4 = 3 116 Number and Financial Literacy

Part-Part-Whole Problems There are two parts that comprise a whole, but no action takes place. The two parts remain separate as subsets. What is being highlighted is the relationship between the whole and its parts that remain separated. Unknown Example Whole Unknown There are 4 dogs and 3 cats in the yard. How many pets are in the yard? Part Unknown There are 7 pets in the yard. 4 are dogs and the others are cats. How many cats are there? There are 7 pets in the yard. 3 are cats and the others are dogs. How many dogs are there? Compare 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 them. Unknown Example Difference Unknown Jesse has 4 cookies. Sana has 3 cookies. How many more cookies does Jesse have than Sana? (or How many fewer cookies does Sana have?) Compare Quantity Sana has 3 cookies. Jesse has 1 more cookie than Sana. How many cookies Unknown does Jesse have? (or Jesse has 4 cookies. Sana has 1 less than Jesse. How many cookies does Sana have?) Referent Unknown Jesse has 4 cookies. He has 1 more cookie than Sana. How many cookies does Sana have? (or Sana has 3 cookies. She has 1 less than Jesse. How many cookies does Jesse have?) Adding and Subtracting Strategies Many researchers and educators, including Thomas Carpenter and colleagues, Cathy Fosnot, Doug Clements, Alex Lawson, and John Van de Walle, have studied the strategies that students may use when solving addition and subtraction problems. Following is a synthesis of many of their ideas and findings. More in-depth analysis is available in their books listed in the References section (see pages 507–509). As young children solve problems that are related to addition and subtraction, they intuitively directly model the situation and apply many of their counting skills. With experience, they develop a variety of strategies which become increasingly efficient and more abstract. It is important to note that students will progress in different ways and may or may not engage in all of the strategies. Alex Lawson points out that transition from one strategy to another is not rigid and students may revert to previous and perhaps less efficient strategies, depending on the types of problems that they encounter (Lawson, 2016, p. 18). Carpenter’s research indicates that students generally progress from direct modelling to using various counting strategies to attaining more automatic recall of facts by understanding number relationships, and then to using known facts to derive other calculations (Carpenter, Fennema, Loef Franke, Levi, & Empson, 1999, p. 26). Lawson’s studies reveal that students also use strategies, such as Addition and Subtraction 117

decomposing quantities into groups of 5 and 10 for easier calculation (Lawson, 2016, p. 21). The strategies that students choose are often dependent on the type or structure of the problem and the size of the numbers. For example, students may revert to an earlier strategy as the numbers get larger. Your goal is to provide students with opportunities to develop more effective strategies over time. Direct Modelling Students often begin directly modelling problems by acting them out, using concrete materials, or creating drawings to represent the groups and then recreating the action that takes place. The following problems will be used to illustrate students’ thinking: • Pear Problem: There are 3 pears in a bowl. Jen adds 4 more pears. How many pears are in the bowl now? • B utterfly Problem: There are 7 butterflies in the garden. 3 butterflies fly away. How many butterflies are left? Students use concrete materials to represent the objects and count three times. Pear Problem: • Count to create a group of 3. • Count to create a group of 4. • Mimic the joining action by pushing sets together. • Count the new group as 1, 2, 3, 4, 5, 6, 7. Butterfly Problem: • Count to create a group of 7 objects. • Remove 3 objects, counting 1, 2, 3. • Count the remaining objects as 1, 2, 3, 4. In both cases, the concrete materials directly represent the objects in the problem. Lawson states that the use of concrete objects “decreases the amount of information that needs to be held in one’s head” (Lawson, 2016, p. 19). Counting Strategies A major step occurs when a child no longer sees the need to physically model the situation, but counts in various ways to find a solution. Counting On and Back Pear Problem: • Count on from 3 as 4, 5, 6, 7, raising a finger with each count to track the addition of 4 pears. • Count on from the larger set of 4 as 5, 6, 7, raising a finger with each count to track the smaller set of pears. Butterfly Problem: • Say 7, and then count back 6, 5, 4, raising a finger with each count to track the 3 butterflies that flew away. 118 Number and Financial Literacy

In the counting on strategy, the child counts on from 4 as 5, 6, 7, raising a finger each time. As Lawson points out, raising a finger with each count is tracking the number sequence (Lawson, 2016, p. 20). This strategy is more abstract than directly modelling the problem and more efficient. It takes time for students to transition from using direct modelling to counting strategies, and in many cases they will use one or the other, depending on the problem they are solving. Anchoring Five and Ten Students will often decompose quantities into groups of 5. For example, 4 + 7 can be seen as 5 + 6. As they mentally ‘move’ one unit from the 7 to the 4 to create 5, they are creating equivalence through compensation. As Fosnot explains, if you lose one from one number but gain it in the other number, the total stays the same (Fosnot, 2007, p. 6). Students may also decompose 4 + 7 into 5 + 5 + 1, and then recompose to make 10 + 1, thereby composing a ‘friendly’ group of 10. Composing groups of 10 is important since it aligns with our base ten number system and the concept of place value. For example, 9 + 7 can be seen as 10 + 6, or 1 ten and 6 ones. Once students understand the inverse relationship of addition and subtraction, they can ‘think addition’ when confronted with a subtraction problem. For example, they may see 11 – 5 as 5 + = 11, or they may count on from 5 to 10, and then add one more to get 11. As Lawson highlights, students are “thinking of the difference, not as the result of removal, but as the distance” between the two numbers (Lawson, 2016, p. 21). Automatic Recall of Some Facts and Using Those Facts to Learn Derived Facts Through experience, students can recall certain facts, such as doubles or adding 1 more or less. They can then use these facts to figure out other facts by understanding the relationships between the numbers. For example, they may recall that 4 + 4 = 8 and use this to know that 4 + 5 will be 1 more than 8 in the counting sequence (Carpenter et al., 1999, p. 30). Developing Automaticity with Math Calculations Automaticity is “the ability to use mathematical skills or perform mathematical procedures with little or no mental effort. Automaticity with math facts enables students to engage in critical thinking and problem solving (Ontario Ministry of Education, 2020, p. 33). Cathy Fosnot states, “understanding what it means to add and subtract is necessary before facts can become automatic, but understanding does not necessarily transfer to automaticity” (Fosnot & Dolk, 2001a, p. 97). Alex Lawson adds that this ‘automaticity’ evolves over time by “working with different strategies and construction of key ideas that can be applied across calculations” (Lawson, 2016, p. 22). This is more effective than memorizing facts in isolation since the major focus is on the relationships among numbers and the operations. This takes time and evolves over several years. Addition and Subtraction 119

Caution About Timed Tests and Activities A traditional approach has been to time students while they solve math calculations in order to assess their recall. While we want students to gain proficiency with these calculations, we should question the use of timed tests and activities based on the research. Jo Boaler writes about how “timed tests cause the early onset of math anxiety for students” and is exacerbated over time, leading to “low achievement, math avoidance, and negative experiences of math throughout life” (Boaler, 2014, p. 469). Not only does this affect math achievement, but also self-beliefs. When students underachieve because they cannot access their working memory, they question their own abilities, lose self- confidence, and develop math anxiety. Other Approaches Boaler states that there are many ways to introduce and reinforce mental strategies without the pressure of being timed, such as through discussions about how numbers are related, and applying strategies in games and activities. She also notes that students best internalize math facts when they engage in mentally solving number problems on a daily basis (Boaler, 2014, p. 47). The Development of Mental Strategies Mental math skills “involve the ability to perform mathematical calculations without relying on pencil and paper” (Ontario Ministry of Education, 2020, p. 33). This supports students being able to estimate and judge the reasonableness of their answers. In grade two, students develop mental math strategies to add and subtract numbers to 50. This requires having a strong understanding of numbers and how they can be composed and decomposed, as well as a conceptual understanding of addition and subtraction. Students can also apply their spatial reasoning skills to visualize and mentally manipulate both pictures and numbers that represent addition and subtraction. These spatial activities help students form images in their minds that can be used to develop mental math strategies. 120 Number and Financial Literacy

Addition and Subtraction to 20 PMraotcheesmseast:ical About the crtaPeoonrfomoldelbsmcpltearuiomnnnvgdiic,nssarogtetlr,ivpansirtngeeeg,lgse,eicerntesitna,ingsgo,ning connecting A major goal of the grade two math program is to support students in developing an operational sense which, according to Van de Walle and Math Vocabulary: Lovin, is “a highly integrated understanding of the four operations and the edecfwqdeaqaouichdmumfaofodaeilmbl,tl,eryiisoebn,p,unnaaonpb,cltsuaaiteneomrrn,tua,ncsbmccse,eto,dsbrmfu,easmprctotof,arstyceo,t,t,al, many different but related meanings these operations take on in real contexts” (Van de Walle & Lovin, 2006, p. 65). This unit focuses on developing the varied meanings of addition and subtraction using numbers to 20 so that students are working with numbers they can count and have an understanding of in terms of quantity. Students will solve problems that vary in structure and require them to think about the context of the problem and the meaning of the numbers involved, rather than just focusing on getting an answer. By involving students in discussing problems and explaining their thinking, you have a window into your students’ mathematical development that can be used to plan next steps. In grade two, students learn strategies to support them in recalling addition and subtraction facts to 20. Through investigative tasks, contextual problem solving, and Math Talks students will investigate computational strategies such as make a ten, decomposing, and near doubles which they can apply to gain automaticity of these facts. There is also a focus on decomposing and part-part-whole models in order to support students in developing an understanding of number relationships as a scaffold toward understanding the varied meanings of addition and subtraction and how they relate to one another. Students will use a variety of concrete materials such as counters, arithmetic racks, ten frames, and number lines that will support them in understanding mathematical concepts and contexts as they work through problems and transition from concrete to visual and abstract representations of numbers and problems. It’s important to give students the freedom to use concrete materials to represent and work out their problems. You can encourage students to begin using drawings and abstract symbols (e.g., number sentences) by modelling during Math Talks as well as through partner work and student sharing. Addition and Subtraction to 20 121

Lesson Topic Page 1 Decomposing Numbers to 20 123 130 2 Decomposing to Solve a Problem 134 139 3 Fact Families Investigation 145 4 Part-Part-Whole Relationships: Determining the Missing Part 150 158 5 Story Problems Using Addition and Subtraction 163 6 Varying the Unknown in Addition and Subtraction Problems 7 Make a Ten: Adding Three Numbers 171 8 Applying the Make a Ten Strategy to Addition 177 Using Ten Frames 183 9 Applying the Make a Ten Strategy to Addition 188 Using a Number Line 194 10 Applying the Make a Ten Strategy to Subtraction Using a Number Line 11 Doubles 12 Near Doubles 13 Exploring the Calendar Using Addition and Subtraction 122 Number and Financial Literacy

1Lesson Decomposing Numbers to 20 Math Number Curriculum Expectations • B 2.1 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 have worked with a balance scale and • B 2.2 recall and demonstrate addition facts for numbers up to 20, and concrete materials. related subtraction facts Teacher Look-Fors • 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 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 • B 1.2 compare and order whole numbers up to and including 200, in various contexts Algebra • C 2.2 determine what needs to be added to or subtracted from addition and subtraction expressions to make them equivalent • C 2.3 identify and use equivalent relationships for whole numbers up to 100, in various contexts Possible Learning Goals • Decomposes numbers to 20 • Demonstrates an understanding of equivalence and the equal sign • Explains what the equal sign means (same on both sides, balanced) • Decomposes numbers into two parts • Uses a strategy to decompose in a logical way rather than guessing and checking PMraotcheesmseast:ical About the raaPennrpoddrbesplsetreromanvttieinsngoggil,ev,ssine, glree, cfrlteeinacgstiontnogio,nlgs communicating The equal sign (=) means ‘is the same as.’ However, students often interpret an equal sign as meaning ‘here comes the answer.’ Explicit teaching and contextual experiences help students understand the true meaning of the equal sign. In addition, exploring equations such as 12 + 3 = 8 + 7 can help students understand the idea of equality and the actual meaning of the equal sign. continued on next page Addition and Subtraction to 20 123

Math Vocabulary: About the Lesson edeqequcuaoalmt,iobpnao,lsanenu,cmceodbm,epr ofascet, In the Minds On, students will explore the concept of equivalence using a balance scale and cubes. In Working On It, they will decompose numbers to 20 using a model of the balance scale. Teaching Tip Minds On (15 minutes) You may wish to do • Show students a balance scale. Put 10 connecting cubes (all one colour) on the Math Talk on decomposing and one side of the scale and discuss what they see, for example: equivalence Teacher: What do you notice? (see page 128) before Students: One side is tipping down, they’re not balanced, one side has beginning this lesson. cubes but the other doesn’t, etc. Materials: Teacher: How could I balance the scale so that both sides are the same? balance scale, Students: Put 10 cubes on the other side. connecting cubes, BLM 23: Balance the • Add and count one cube at a time until it balances. Use 3 yellow and 7 green Scales Time: 35–45 minutes cubes. Continue to discuss: Teacher: Why is it balanced now? What does balanced mean? Students: Equal, both the same, they’re even, etc. Teacher: How many cubes are on this side? Students: 10. Teacher: How many cubes are on the other side? Students: 10. Teacher: So they both have the same number. Are the quantities composed the same way? How could I use a number sentence to describe the cubes on this side? Students: 3 + 7 = 10. Teacher: So they both equal 10. Watch how I record this. • Draw a simple balance scale and represent the cubes from the problem on the scale using numbers. Ask students if they can think of any other combinations for 10 that could be used. 37 10 124 Number and Financial Literacy

• Record students’ ideas by writing the combinations below the 3 and 7 on the scale. Ask if there’s a way that we can be sure that we have all the combinations. (e.g., Start with 0 and 10. Continue adding one more to one number and taking one away from the other number each time.) N OTE: Unless students have been exposed to this organized way of determining the facts for 10, they will not suggest this strategy. You will explore this method with students in the Consolidation. • Assign partners. Draw another balance scale on the board. Write 15 on one side and ask students to find as many ways as possible to balance the two sides. Have them record their combinations. Allow access to coloured cubes but some students may choose to do this mentally or may draw pictures. • After a few minutes, have students share their solutions. Possible questions and prompts include: – What do all of your combinations equal? (15) – What does it mean to balance the sides? (e.g., they are equal; both sides are the same; same amounts) – What do you think the equal sign means? • Write an equal sign inside the triangle of the balance scale. Then write the equation: 13 + 2 = 15. • If students do not come to the conclusion that the equal sign means that both sides are equal/same/balanced and still say that the equal sign means ‘the answer is,’ tell them that the equal sign means that the quantities on both sides of the sign are equal. Record 7 + 3 = 12. Ask students if both sides are equal and have them prove why they are not. Record 5 + 3 = 8 and ask students if 5 + 3 is the same as or equal to 8? (Yes.) Ask them to prove it by drawing or modelling. • Reinforce the idea that both sides of the equal sign have to have the same quantity. They can be represented differently, but the quantity must be the same. Working On It (10–15 minutes) • Draw a balance scale and record 16 on one side. Ask students what they notice. (16 on one side but nothing on the other side.) • Ask them what they need to do to balance the sides. (Put in two numbers that equal, or add up to 16.) • Tell students to think of two numbers that will add up to 16. Ask them to put up a finger for each solution they can think of. • Have students share their solutions. Record their combinations below the combinations on the balance scale, showing that there are many ways to decompose 16. • Ask students how they came up with their numbers. Say, “How did you decompose 16 into two parts?” (e.g., some may use an organized strategy such as starting with 0 and 16 and increasing/decreasing the numbers by 1 each time, or they may have randomly chosen a number and counted up to Addition and Subtraction to 20 125

find out the missing part). Tell students that these combinations are called ‘number facts.’ • Record 10 + 6 = 8 + 8 on the board and ask, “Do you think that this is true? Does 10 + 6 equal 8 + 8? Why do you think so? Can you prove it?” Invite students to discuss their ideas with a partner. Then have a few students share their solutions. • Hand out copies of BLM 23: Balance the Scales. Students can work in partners or independently but should each record their work on their own sheets. Instruct students to find at least 4 combinations/number facts to balance each scale. Tell them that they can choose the starting number for the last scale but that the number must have at least 8 number facts. Differentiation • Provide connecting cubes in two colours for students who are having difficulty doing the task mentally so that they can ‘act out’ the problem. Alternatively, provide a number line or hundreds chart. • Model the terms ‘decompose, compose, balance, equal, and equation’ for ELLs often throughout the unit. Create an anchor chart with the terms along with a visual example of each. Add this chart to the Word Wall and prompt students to use the terms during math discussions. • Provide further practice in developing the concept of equality by meeting with a small group to work on the Further Practice Guided Math activity. Assessment Opportunities Observations: Observe how students determine the facts. Are they guessing and checking in a random way or using the strategy of starting with 0 or 1 and using a pattern to proceed? If they are guessing and checking, how are they adding? What strategies are they using? (e.g., using their fingers to count on, using known facts, using 5 as an anchor, making a ten, doubles, near doubles) Consolidation (10–15 minutes) • Draw a balance scale on the board. Choose one of the numbers to focus on for the Consolidation (e.g., 15). As students share their combinations, record them. • Ask, “How can we be sure that we have all the number facts for 15?” Co-construct a T-chart showing the facts in an organized way. Record the first two facts (0 + 15, 1 + 14) and ask students to identify the pattern and predict the next fact. Continue until all of the facts have been recorded. • Invite students to Think-Pair-Share. Say, “Describe the pattern that is occurring in the table. Use the words ‘increasing’ and ‘decreasing’ in your description.” • Review the meaning of equivalence. Write 12 + 3 = 15. Ask, “What does the equal sign mean in this number fact?” (e.g., is the same as; balanced; equal) 126 Number and Financial Literacy

Materials: Ask students why the facts 1 + 14 and 14 + 1 are both included. Ask what they notice about these facts. Determine that they both represent the same BLM 24: Arithmetic amount but could have different meanings in a story (e.g., 1 blue cube and Rack Recording 14 red cubes or 14 blue cubes and 1 red cube). Tell students that these facts Sheet are often called ‘turn around facts.’ • Write 8 + 3 = on the board and ask students to solve it. If all students agree that the answer is ‘11,’ ask if there are any other numbers that we could put after the equal sign to make the equation true. (e.g., 7 + 4, 10 + 1) Ask a few students to explain why this works in their own words. (e.g., The equal sign means ‘the same as’ so 8 + 3 is 11 and 7 + 4 is 11 so they’re the same. They’re equal.) Have a balance scale and connecting cubes available so students can check. • O ptional Exit Pass: Write 3 + 5 = 4 + 4 in your Math Journal. Explain why or why not this equation is true/balanced. Further Practice • Independent Problem Solving in Math Journals: Assign a number between 12 and 20 to each student. Ask students to find and record all of the number facts for that number in their Math Journals. • Partner Work/Math Journals: Assign partners a number/quantity (e.g., 16). Each partner must use 16 connecting cubes to build that number using 2 different colours. Then they compare their creations and record both number sentences showing equality (e.g., 8 + 8 = 9 + 7) in their Math Journals. • Guided Math: Reinforce the concept of equality using the arithmetic rack in a small-group guided math lesson. Ask students to use the arithmetic rack to ‘prove’ whether or not the following equations are balanced. Have them make predictions before they model the equations using the arithmetic rack. Provide copies of BLM 24: Arithmetic Rack Recording Sheet and ask students to colour and label each part of the equation. – Does 6 + 7 = 12 + 1? – Does 3 + 2 = 4 + 1? – Does 4 + 5 = 8 + 2? – Does 8 + 9 = 16 + 1? – Does 7 + 3 = 5 + 4? Building Social-Emotional Learning Skills: Stress Management and Coping: “Teachers should help students develop math facts, not by emphasizing facts for the sake of facts or using ‘timed tests’ but by encouraging students to use, work with and explore numbers. As students work on meaningful number activities they will commit math facts to heart at the same time as understanding numbers and math. They will enjoy and learn important mathematics rather than memorize, dread and fear mathematics” (Boaler, 2015). Ease students’ stress by assuring them that they have time to learn their math facts and that it is more important to understand what they are doing when they add and subtract. Consistently engaging in Math Talks throughout the year (e.g., 3 times per week) will help your students develop math facts and the ability to work flexibly with Addition and Subtraction to 20 127

numbers. Decomposing numbers, as was done in this lesson, also contributes to automaticity. Building a repertoire of strategies for learning facts will help students be resourceful and develop personal resilience when they feel under pressure. Teaching Tip Math Talk: Integrate the math Math Focus: Decomposing, equivalence talk moves (see page 7) throughout Process: Write 7 + 4 = 8 + 2 on the board. Say, “I’ve recorded 7 + 4 = 8 + 2. Your Math Talks to job is to figure out if that’s a true statement.” Have students share their answers. maximize student Record all solutions. Ask a few students to share and justify their thinking. As participation and they share, represent their ideas using an appropriate model (e.g., a number line, active listening. hundreds chart, arithmetic rack, ten frames) so that they can concretely see the lack of equivalence. Let’s Talk Select the prompts that best meet the needs of your students. • D oes 7 + 4 = 8 + 2? (No, because 7 + 4 = 11 but 8 + 2 = 10. That’s not the same.) How do you know they’re not the same or equal? (11 is 1 more than 10.) Can you explain how you added 7 + 4 and 8 + 2? (I know that 7 + 3 equals 10 and then I added one more.) Record the jumps on a number line. +3 +1 7 +3 10 +1 11 • dDiodeysotuhifsigreuprreeoseuntt87y+ou2r?st(rIat twegays?+a(2Ykenso.wI1n0mfaadcet a 10 then jumped one more.) How f1o1r me.) Record the jumps on a number line. 8 +2 10 8 10 • When we look at the two number lines what do you notice, class? (You can see that 11 is one more than 10 so they’re not equal.) • Good observation. How can we change 8 + 2 so that both sides of the equal sign are the same or equal? (Change it to 8 + 3 so that they both equal 11.) 128 Number and Financial Literacy

Materials: Math Talk: large arithmetic Math Focus: Decomposing numbers to 20 using the arithmetic rack rack, construction paper Process: Represent 10 on the arithmetic rack by showing 5 beads on the top and 5 on the bottom. Ask how many beads they see. Have students share their answers and record them on the board. Ask a few students to share their strategies (e.g., I saw 5 on the top and 5 on the bottom and I know 5 + 5 = 10; I doubled 5; I counted all the beads; I subitized; I know an arithmetic rack has 10 and that’s half of 20 which is 10). Repeat the process, showing different combinations for 10 such as 7 on top, 3 on the bottom, 8 on top, 2 on the bottom. Then, using only the top row, show 4 and hide 6. You can create a screen using a folded piece of construction paper. Say, “Let’s think about the top row. We know that the whole is 10. You can see 4. How many am I hiding?” Let’s Talk Select the prompts that best meet the needs of your students. • How many do you think I’m hiding? (6) How do you know? (I knew there were 4 so I counted up to 10 and got 6; I imagined one more bead on top to make 5. Then I knew I needed the same on the bottom. That’s 10.) • C an you explain what you mean by ‘I needed the same on the bottom.’ The same as what? (The same as the 5 I imagined on the top. I know that 5 and 5 are ten. So I needed one more on the top and 5 more on the bottom to get ten. 5 and 1 are 6.) Thank you for explaining that so clearly. • Did anyone use another strategy? (I just did 10 – 4 and got 6.) How did you figure out 10 – 4? (It’s a known fact for me. I know that 6 + 4 = 10 so 10 – 4 must equal 6.) You used fact families to help you. Your strategy shows us that you understand how addition and subtraction are related. • Class, show me with your thumb if you understand the strategy that Zarah used. Note students’ responses. We’ll be doing a lesson soon on fact families that will help everyone understand your strategy. Addition and Subtraction to 20 129

2Lesson Decomposing to Solve a Problem 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 can add related subtraction facts numbers to at least 10 and count to 20. • 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 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 • B 1.2 compare and order whole numbers up to and including 200, in various contexts 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 contexts Possible Learning Goals • Decomposes a number into all of its number facts • Solves problems involving part-part-whole relationships • Understands the commutative property of addition—that the order of the numbers doesn’t matter when adding (4 + 6 = 6 + 4) • Identifies all of the facts for 10 and 20 • Uses a pattern or systematic strategy • Understands the pattern that occurs when decomposing a number into its facts • Represents the solution clearly 130 Number and Financial Literacy

PMraotcheesmseast:ical About the ccaaPoonnrmnoddnbmspeletrcuromtanivtniiecnsggaog,itl,erivnessingpe, grlreee, csfrleteeinnactgstiionntngogi,o,nlgs Composing and decomposing numbers involves additive reasoning and Math Vocabulary: provides students with a strategy for becoming fluent with number cfaocmtsb,indaetcioonmsp, onsuem, ber facts. “Parrish, drawing from Fosnot and Dolk (2001) defines fluency visualize as ‘knowing how a number can be composed and decomposed and using that information to be flexible and efficient with solving problems.’ (Parrish 2014, p 159)” (Boaler, 2015). Research shows that “the best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense” (Boaler, 2015). About the Lesson Ten is such a vital anchor for students that it’s important to provide multiple experiences decomposing it. This lesson builds on Lesson 1 in which students decomposed numbers and discovered how to use a pattern to determine all of the number facts in an organized way. In the Minds On, students will solve a problem that requires them to decompose 10 using cubes of two different colours. Then they will work in partners to solve a story problem that involves decomposing 20. The goal is that students will apply the strategy of systematically using a pattern to determine all of the combinations (a pre-algebraic concept). Materials: Minds On (10 minutes) connecting cubes • Put 10 cubes in a paper bag (some yellow and some blue). Show students the (9 yellow and 9 blue), paper bag bag. Tell them that you have put 10 cubes inside and that some of them are yellow and some of them are blue. Say, “Try to visualize the cubes in my bag. Time: 35–40 minutes How many yellow and blue cubes could be in the bag?” Ask students to share their answers. Record the combinations they give you on the board. As the answers are given, have a student build each combination of cubes as a cube train. • Ask, “How can we be sure that we have all the combinations?” Try out their suggestions. Then say, “Let’s put your ideas in order, starting with 1 yellow and 9 blue.” Continue to order the facts with student input. Rewrite the facts and cross out the ones that have been ordered so that students can see which ones are left in case they can’t yet visualize the pattern. Select a student to order the cube trains as well, so students can visually see the change as it occurs. Ask, “What happens every time I add one more yellow?” (There’s one less blue.) “Does that happen every time?” (Yes.) “Give me a thumbs up if you think you could use that strategy next time to ensure you have all possible combinations for a number I give you.” Addition and Subtraction to 20 131

• After recording the combinations, ask: “Would there be zero yellow or zero blue? Why not?” (I had said there were some of each colour in the bag in my instructions.) Reveal the actual cubes. Differentiation • Have students model or act out the problem. Glue two pieces of construction paper together so that one side is blue and one side is yellow. Cut the paper into 10 squares. Give 10 students each a square. Have them stand in front of the class. As the number combinations are given in order, have students act them out so that students see that as 1 more yellow is added, there is 1 less blue. This provides a concrete example for visual learners. Working On It (15 minutes) • Have students work in pairs to solve the following problem. Have them record their work on a large paper for sharing. Have each partner use a different coloured marker so that you can quickly see their individual contributions toward recording. • Pose the problem: A kindergarten class has 20 students. The class is made up of some students who are in junior kindergarten (JK) and some students who are in senior kindergarten (SK). How many of each might there be? Find all the combinations you can. Record your solution so that we can understand how you solved the problem. Differentiation • Have cubes and other manipulatives available for students who may need to ‘act out’ the problem. • You may need to help ELLs understand the concept of junior and senior kindergarten. If possible, borrow a class picture from a kindergarten class in your school to show and explain the concept. Assessment Opportunities Observations: Notice whether or not students understand that they are working with parts of a whole. Conversations: As students work, ask them to tell you what the whole is and what the two parts are. Ask them what changes when they decompose a number (the parts) and what stays the same (the whole). If students cannot explain or show understanding of this concept, work with them in a small guided group using concrete materials to decompose a smaller number such as 7. Use a story context such as having cookies to share between two children. Have them say and record the parts and the whole using pictures and number sentences. Some students may need many experiences doing this to build their understanding of part-part-whole relationships. You can make it interesting and visual by having students create part-part-whole designs using two kinds of pattern blocks. After they create the design, ask them to describe the two parts and the whole. 132 Number and Financial Literacy

Consolidation (10–15 minutes) • Choose a few pairs with varying strategies to share their thinking and recording. (e.g., built the combinations from cubes, then drew them; drew the combinations as cubes in two colours; used a T-chart; wrote 8 and 12; wrote number facts) • Ask some of the following questions to probe students’ thinking: – What combinations did you find? – How did you find the combinations? – How do you know you have all of the combinations? – What do these numbers represent? – How did you record your thinking? – Do you have any questions about the solution? – How is your solution different and the same as the presenters’ solution? – Do you see a recording strategy that makes the solution very clear and easy to follow? – Which class combinations are most likely to occur in a school? Why? Which are least likely? Why? • Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: Jo Boaler says that “one of the most powerful moves a teacher or parent can make is in changing the messages they give about mistakes and wrong answers in mathematics... when we teach students that mistakes are positive, it has an incredibly liberating effect on them” (Boaler, 2016, p. 15). You can put this idea into practice by sometimes choosing students to share work that has a mistake or misconception in it during a Consolidation. You might say, “this group made a wonderful mistake that will help us all learn.” Check with the group prior to the Consolidation to ensure that they will be comfortable sharing their work. If you approach it positively, students generally love to share their work. Further Practice • Find out how many students are in the JK/SK classes in your school. Pose a part-part-whole problem using the numbers (e.g., There are 25 students in K1. 13 of them are JKs. How many are SKs?). • U sing the number of students in one of the kindergarten classes in your school, have students determine the possible combinations of JK and SK students. Addition and Subtraction to 20 133

3Lesson Fact Families Investigation 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 have had related subtraction facts experience adding numbers to 10 and • B 2.3 use mental math strategies, including estimation, to add and subtract recording number sentences. whole numbers that add up to no more than 50, and explain the strategies used • 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 • B 1.2 compare and order whole numbers up to and including 200, in various contexts 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 contexts Possible Learning Goals • Investigates part-part-whole relationships • Understands the inverse relationship between addition and subtraction • Solves and creates addition and subtraction sentences to represent a situation • Applies the commutative property of addition (a + b = b + a) to find the ‘turn around facts’ • Accurately records the four number sentences involved in a fact family • Knows that for every two addition facts, there are two corresponding subtraction facts in a fact family • Identifies the whole and the parts • Explains or demonstrates an understanding that you add two parts to find the whole (addition) and that if you take one part away from the whole it’s equal to the other part (subtraction) 134 Number and Financial Literacy

PMraotcheesmseast:ical About the ccrPeooramnosnbmoelencumitnninigcsgaoa,tlnrvinedingppgrer,osveinntgin, g, According to Marian Small, a fact family is a set of four number sentences, or Math Vocabulary: equations that can be used to describe the same situation (Small, 2008, pp. faparacortutfsna,dmsfuialmyc,t,sttu,orwtnahlo, le, 109–110). For example, the numbers 2, 3, and 5 have the following subtraction difference and addition relationships: 2 + 3 = 5, 3 + 2 = 5, 5 – 2 = 3, 5 – 3 = 2. Introducing students to fact families helps them build key concepts related to addition and subtraction such as the commutative property of addition (the order in which numbers are added doesn’t change the sum). Fact families help students construct the key idea that addition and subtraction are inversely related. The commutative property leads students to count on more efficiently by starting with the bigger number. For example, once students realize that 3 + 8 = 8 + 3, they will reverse the numbers and start with 8 when adding or counting on. As students construct the commutative property, they are also poised to use the strategy of adding up rather than counting back when subtracting. This strategy is generally more efficient and accurate for young mathematicians. About the Lesson The Minds On introduces students to the concept of fact families. Students will listen to the teacher tell number stories that are related to a fact family. Students will determine and record two addition and two subtraction equations for each fact as the teacher tells the stories. In Working On It, students will work in partners to determine and record the equations for the fact family (5, 3, 2) and then tell each other stories to go with the facts. Subsequently, they’ll choose a fact family, draw a picture to represent it, and write the four related number sentences below the picture. Materials: Minds On (15 minutes) blue and red • Write the numbers 3, 7, and 10 on the board. Draw 3 circles and 7 squares. squares (3 blue and 2 red per student) Say, “I’m going to tell you some number stories and I want you to write down in your Math Journals the number sentences that go with my stories.” Time: 40–50 minutes • Tell students the stories below and ask them to tell you the corresponding number sentence after each one. Record the number sentences on the board for all to see and use to check their own work. 1. A t first I had 3 round stickers for my scrapbook. Then my friend gave me 7 square stickers. How many stickers did I have? (3 + 7 = 10) 2. I glued the 7 square stickers in my scrapbook and then the 3 round stickers. How many stickers did I glue in my scrapbook? (7 + 3 = 10) 3. M y mom gave me 10 stickers. I lost 3 rounds ones. How many stickers did I have left? (10 – 3 = 7) 4. M y friend gave me 10 stickers. I put 7 square stickers in my pocket. How many did I still have in my hand? (10 – 7 = 3) Addition and Subtraction to 20 135

• Ask questions such as the following: – What did you notice about the number sentences you wrote? (e.g., they all use the same numbers) – How many addition sentences are there? What number represents the whole in 3 + 7 = 10? Why did you get 10 with both addition sentences? Does it matter which order you add 3 and 7? (e.g., It matters to the story but the total is the same; you can switch around the numbers) Say, “Yes, some people call them ‘turn around facts’ or ‘switcheroos.’ Tell a partner why those names make sense.” – How many subtraction sentences are there? What is the whole in 10 – 3 = 7? What are the parts? Does the order matter when you subtract? (Yes, you have to start with the whole; If you subtract one number it equals the other number.) – How are these number sentences related? • Invite students to Think-Pair-Share: Ask, “Could we say that 3 + 7 = 7 + 3?” (Yes, they are turn around facts. They both equal 10.) • Have students Think-Pair-Share again. Ask, “Is it true that 10 – 3 = 10 – 7? Why?” (No, they don’t equal the same thing.) • Explain that we call these 3 numbers (10, 7, 3) a fact family. They are related to each other through addition and subtraction. 3 and 7 build up or compose 10 by adding. Then when we subtract, we take apart or decompose 10. Working On It (15–20 minutes) • Give each student 3 blue squares and 2 red squares to glue into their Math Journals. Draw the set on the board and ask: – How many squares do you have in total? (5) Write 5. – How many blue squares do you have? (3) Write 3 (5, 3). – How many red squares do you have? (2) Write 2 (5, 3, 2). • Say, “5, 3, 2 is your fact family. What is the whole? What are the parts? Write two addition and two subtraction facts using only those 3 numbers. After you’ve recorded, tell your partner addition and subtraction stories to go with each fact.” If you find that some students are having difficulty making up stories, have a few students share their stories with the class for inspiration. • Write several fact families on the board. Have students work independently to choose a fact family, draw a picture to represent it, and write the 4 number sentences below the picture. Differentiation • ELLs may have difficulty telling stories related to their pictures and numbers. If there are students in the class with the same first language, pair them and encourage them to use their first language to tell their stories. If this is not the case, support your ELLs by suggesting a simple context that you know they have the oral language to express. 136 Number and Financial Literacy

Materials: • Vary the fact family numbers for students who may need more or less of a BLM 8: Number challenge. Make concrete materials available. Cards (0–20) (1 set of numbers 5–18 per • Have students play Train Wreck or work with the part-part-whole cards on pair of students), BLM 25: Part-Part- BLM 25 (see Further Practice) to consolidate their understanding of fact Whole Cards families and the inverse relationship of addition and subtraction. The Math Talk attached to this lesson will also help students with this concept and can be repeated using different numbers. Assessment Opportunities Conversations: If students are having difficulty, work with them in a small guided group and ask guiding questions to support them. For example, “You have 3 numbers. Which is the whole? How do you know? Build the whole. Pretend it’s a chocolate bar with 5 pieces.” Ensure students understand how the parts are related to the whole by having them tell you stories. You don’t want this to be a procedural task in which they follow a formula. You want them to understand the relationship of the numbers. Have them record the number sentence after each story so that they understand how it represents what they’ve said and done. Consolidation (10–15 minutes) • Have a few students share their fact families after completing the independent task. Ask how they knew what facts to record. • Ask, “What number do you start with when subtracting? When adding? What is the whole? What are the parts? Does it matter what order we add and ? Why not?” • Tell students that the answer in addition is called the ‘sum’ or ‘total’ and the answer in subtraction is called the ‘difference.’ Record these definitions on your Word Wall along with an example of each. • Ask, “Why do you think these are called fact families?” (e.g., the numbers are related; addition and subtraction are related) Then ask, “How can fact families help us when we’re adding or subtracting? For example, how could it help me if I’m working out 10 – 7?” (e.g., If you know that 7 + 3 is 10 then you’d know that 10 – 7 is 3; you just think of the other number in that fact family which is 3, so 10 – 7 = 3; if you’re subtracting, you can still use addition because the numbers are all related and addition is easier.) Further Practice • Train Wreck: Provide partners with a deck of number cards for numbers 5–18 (or use 3 dice). Player 1 takes a card and must make a cube train as long as the card indicates (e.g., 16). Player 1 shows the train to player 2 and says, “My train is 16 cars long.” Then Player 1 puts the train behind his or her back and snaps it into two parts, saying, “Train wreck!” Player 1 then shows one part (e.g., train of 9 cubes) to Player 2 and asks, “How many cars went off the track?” Player 2 figures out the missing part (i.e., 7). Both partners record the 4 fact family number sentences represented by the cube train. Addition and Subtraction to 20 137

Materials: • M ath Centre/partner work: Have students work in partners. Using Digital Slide 72: BLM 25: Part-Part-Whole Cards, one partner covers up one of the numbers Part-Part-Whole with a hand. The other partner has to determine the hidden number. Relationships, BLM 25: Part-Part-Whole Cards Building Social-Emotional Learning Skills: Critical and Creative (family for 8, 5, 3) Thinking: Students develop the belief that they are capable mathematicians when they discover mathematical concepts and big ideas for themselves. When Teaching Tip given open problems to solve, students learn to take the initiative to develop a plan and creatively think of ways to find a reasonable solution. Giving students Integrate the math time to reflect on their solutions allows them to think critically about what talk moves (see they discovered and connect it to what they know, thereby creating page 7) throughout understanding. Math Talks to maximize student As Alex Lawson states, children will learn to check subtraction using addition participation and at some point and it’s important to let them discover this idea through part- active listening. part-whole work themselves, otherwise they’ll never understand why it works (Lawson, 2015, p. 65). By creating fact families, working with the part-part- whole triangle cards, and playing Train Wreck, students are ‘discovering’ the concept of the inverse relationship of addition and subtraction. This prepares them to use the strategy of ‘think addition’ when subtracting. Math Talk: Math Focus: Part-part-whole relationships Process: Show students Digital Slide 72 and point out the visual for 8 (8, 5, 3). Ask students to describe it. You want students to notice that the whole is the same size as the two parts put together and that 5 is shown as a bigger part than 3. Ask students what number sentences could be made with those 3 numbers. Record all 4 facts on the board as students identify them (5 + 3 = 8, 3 + 5 = 8, 8 – 3 = 5, 8 – 5 = 3). Then show students the visual for 12. Ask students to predict what two numbers would be used to complete the fact family (6 + 6). Have students share their answers and justify their thinking. (e.g., The two parts are the same size so the numbers would be the same size. That means it would be a double. 6 + 6 is a double for 12, etc.) NOTE: Copy and cut apart the card for 8 (8, 5, 3) from BLM 25: Part-Part- Whole Cards. Use the parts to act out the 4 facts so that students can see the number being composed and decomposed. 138 Number and Financial Literacy

4Lesson Part-Part-Whole Relationships: Determining the Missing Part 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 have engaged in activities and/or • B 2.2 recall and demonstrate addition facts for numbers up to 20, and discussions about part- whole relationships related subtraction facts (e.g., fact families). • B 2.3 use mental math strategies, including estimation, to add and subtract Teacher Look-Fors 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 • 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 Algebra • C2.2 determine what needs to be added to or subtracted from addition and subtraction expressions to make them equivalent Possible Learning Goals • Applies the concept of part-part-whole relationships to determine the missing part • Subitizes quantities • Deepens the understanding of the inverse relationship between addition and subtraction • Develops the key concept of Hierarchical Inclusion (knowing that numbers nest inside of each other and grow by one each time) • Uses addition and subtraction strategies to determine a missing part • Quickly subitizes a number • Realizes that the two parts add up to make the whole and that if you subtract one part from the whole, it equals the other part Addition and Subtraction to 20 139

PMraotcheesmseast:ical About the Problem solving, cssroetermalaestmceotgniuniinengisgct,aoartoeninlpdsgrepasrnoedvnitningg, , In recent years, brain researchers have found that the students who are most successful with number problems are those who are using different brain Math Vocabulary: pathways—one that is numerical and symbolic and the other that involves part, whole, missing more intuitive and spatial reasoning (Park & Brannon, 2013). Working with part Missing-Part cards involves both pathways as students subitize the dot configurations and decompose the number to determine the missing part. About the Lesson In the Minds On, the teacher will show the class several Missing-Part cards that represent various configurations of the number 6 with a missing part. Students will determine the missing part. In Working On It, students will create their own Missing-Part cards, then work with them in an Inside/Outside Circle activity. Materials: Minds On (10 minutes) BLM 26: • Create a Missing-Part card for 6 (showing the numeral 6 on the left, a dot Missing-Part Card Template (5 card arrangement for 2 in the middle, a dot arrangement for 4 on the right) using templates per student) the blank template. Fold down the flap and show the card to the students. Tell them that the numeral represents the whole and the dots represent part Time: 45–50 minutes of the whole. Say, “Under the flap, there are the missing dots that make up the other part of the whole. On this card the whole is 6. There are 4 dots showing. How many dots are missing?” 6 • • •• •• • Ask students to share their solutions and explain how they determined the missing part. Students may start with the part shown and add up to the whole (4 + 2) or they may subtract the part from the whole (6 – 4). As students share their thinking, help them identify their strategies as subtraction—counting back or as ‘think addition,’ counting on, make a ten, etc. • Repeat with several more cards that show the other fact families for 6 (0, 6; 1, 5; 3, 3). 140 Number and Financial Literacy

Working On It (30 minutes) • Copy BLM 26: Missing-Part Card Template onto cardstock or paper and make enough copies for each student to have 5 cards. Have students make their own Missing-Part cards using the template. • Tell students that they are going to make their own Missing-Part cards. Ask, “What do we need to know before we can make the cards?” (what the whole is and all the facts for the whole) Say, “Let’s try this with the number 6.” Draw a T-chart and elicit the facts for 6. Then show students the template for the Missing-Part cards. Ask them to explain how they would fill it in (write the numeral 6 in one section. Draw a part under the flap, and the other part in the other section). Demonstrate or have students fill in the sections and fold over the flap. Then ask students to turn to a partner and explain the steps of the task. Have one student share back to the class. • Show students several dot configurations and ask them what they notice about how the dots are organized. Reinforce the idea that the dots are arranged in patterns that make it easy to subitize (e.g., straight lines, arrays). Show them a visual of dice or actual dice. Tell them that their dot patterns should be easy to subitize and that they may use the dice or dot configurations to help them make arrangements for their cards. Option 1: Students create at least 5 cards for numbers between 7 and 12. Option 2: Students create at least 5 cards for a specific number. This option reinforces the concept of decomposing a number into its facts. • Inside/Outside Circle: Tell students that they are going to use their Missing- Part cards in a circle with different partners. Have them bring their cards as they create an inside/outside circle. Ask half the class to form a circle. Once they are in the circle, ask them to turn to face the outside. Tell them that they are Partner 1. Then ask the rest of the class to go and stand in front of a partner. They are Partner 2. Tell students that each partner will show their card to the other partner, starting with Partner 1. The other partner must try to figure out what the missing part is. Tell them that you will give them a signal for Partner 2 to move to the right to share with the next partner. Tell them that if they are waiting for the signal, they should wait quietly. Give the signal when most students are quiet. Differentiation • Making Missing-Part cards: Assign the number 10 to students who continue to need practice recognizing facts for 10. • The Missing-Part activity can be used in a small guided group to assist students who are not yet counting on consistently as well as to help students develop part-part-whole relationships. • The visual nature of this task provides support to ELLs who may have more difficulty understanding part-part-whole relationships and subtraction/ addition through story problems. Addition and Subtraction to 20 141

Assessment Opportunities Observations: This lesson allows you to observe whether or not students are constructing the key idea of the part-part-whole relationship of addition and subtraction. Students need to realize that the two parts add together to make the whole. With this understanding in place, they should use an addition or subtraction strategy that involves using one part to determine the missing part. Note the strategies that students use. Possible strategies include: – Counts the known part by 1s. Then tries different numbers to find the missing part (guess and check) and counts on until finding the whole – Subitizes the known part and counts on to the whole – Uses a known fact – Counts back from the whole to the missing part – Subtracts the part from the whole – Adds up from the part to the whole (I see 4. I know 4 + 2 is 6, so 2 is missing.) – Recognizes a double – Uses near doubles – Makes a 10 (the whole is 12; I see 4 dots, 4 + 6 = 10; 10 + 2 = 12 so 6 + 2 = 8; 8 is the missing part) Conversations: • H ave a math conference with students you are wondering about, using a Missing-Part card. Note whether or not the student can quickly subitize the known part. Ask the student to explain his/her strategy. If a student is having difficulty, he/she may count each dot of the known part by pointing at each one. Then he/she may try different numbers to count up to the whole but may lose track of the count or may use his fingers to keep track. If this is the case, the student may have an emergent understanding of part- part-whole relationships as well as difficulty counting on. • Engage the student in tasks where the whole is known (show 7 counters and write the numeral 7 on a sticky note). Then tell the student to close his eyes so that you can take some away (e.g., 4). Ask the student to open his eyes. Have him tell you how many he sees (3). Ask him what the whole was that we started with (7). Tell him to start with 3 and count up to 7. Repeat several times with different numbers. Further practise making fact families by using two colours of cubes may help students to develop their understanding of part-part-whole relationships. 142 Consolidation (5–10 minutes) • After the Inside/Outside Circle activity, have students meet at the carpet. Ask them what strategies they used to determine the missing parts (subitized, added up, subtracted, doubles, counted on, etc.). Record their strategies. Discuss how they used addition and subtraction to find the missing part. Ask them why they think that both operations work. Reinforce the idea that you can add or subtract to find the missing part because addition and subtraction are related as they discovered when making fact families. Number and Financial Literacy

Teaching Tip Further Practice Integrate the math • Math Centre: Create a set of Missing-Part cards for students to practise talk moves (see page 7) throughout solving orally in partners. Alternatively, use the cards that students created Math Talks to in the Working On It task. maximize student participation and Building Social-Emotional Learning Skills: Positive Motivation and active listening. Perseverance: Review the language of bubble gum talk on a regular basis. Refer to the anchor chart you co-constructed with your students. One way to remind students of bubble gum talk is to have them each record a way of showing one of the messages that helps them persevere at a task (e.g., I’ll try again, Mistakes mean my brain is growing, I CAN do it) on big individual speech bubbles. Cut them out and post them around the class. Students love finding them and they are great visual reminders of how we should think and speak. Math Talk: Math Focus: Addition strategies, varying the unknown Process: Write 9 + 4 = on the board and ask students to solve it. Have students share their answers. Record all solutions. Ask a few students to share and defend their thinking. As they share, represent their thinking on a number line. Repeat the process with 7 + = 12 and then + 8 = 11. Record any new strategies on your Counting Strategies anchor chart. Let’s Talk Select the prompts that best meet the needs of your students. • H ow did you figure out 7 + = 12? (I started at 7 and counted on to 12.) What did you count on by? Can you show us? (By ones. 7, 8, 9, 10, 11, 12) I like how you showed a closed fist for 7, then added on the rest of the numbers. What answer did you get? (5. I tracked my numbers by putting up a finger for each number.) So on the number line, I would start at 7 and then jump 5 times to get to 12. • Class, do you think that strategy would work with bigger numbers? Show me what you think with a thumbs up or down. Students indicate their thinking. I see that most of you think that it would work with bigger numbers. Let’s try that tomorrow. We know that counting on works with small numbers. Tomorrow we can test it out on big numbers. • Plan another Math Talk with bigger numbers to show the limitations of counting on and keeping track. • Does anyone else want to share? (I got 5, too. I started at 7 and knew that 7 + 3 was 10. Then I added on 2 more to get to 12. Then I added the 3 and 2 together.) So on the number line, I would start at 7. Jump to 10 (record + 3) and then jump 2 more (record + 2) to land on 12. What is your strategy called? (Make a ten, then count on.) continued on next page Addition and Subtraction to 20 143

• Did anyone use a different strategy? (I guessed and checked. I tried 7 + 7 but it was 14 so I knew it was too much. Then I tried 7 + 3 but it was only 10. Then I knew I could add on 2 more to get to 12.) Let’s see if I can show your thinking on the number line. You started at 7 and jumped 7 more. Did you jump by ones? (Yes.) You landed on 14. Too much. Erase that jump. Then you started at 7 again and jumped 3 to 10. Then 2 more. Is that the same strategy as Josh used? (Yes.) I like how you tried again when one strategy didn’t work. 144 Number and Financial Literacy

5Lesson Story Problems Using Addition and Subtraction 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 have had related subtraction facts experience with joining problems with the two • B 2.3 use mental math strategies, including estimation, to add and subtract parts known and the whole unknown. 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 • 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 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 contexts Possible Learning Goals • Solves addition and subtraction problems that have a variety of structures • Uses a variety of strategies to solve addition and subtraction problems • Represents addition and subtraction problems using number sentences • Explores the reciprocal relationship of addition and subtraction • Understands when a problem requires addition or subtraction • Solves addition and subtraction problems accurately • Uses a variety of strategies to solve addition and subtraction problems • Accurately represents the problem using a number sentence • Clearly explains the strategy used and tells what the numbers represent Addition and Subtraction to 20 145

PMraotcheesmseast:ical About the Problem solving, It’s important to contextualize the operations so that students construct their reasoning and understanding of the part-part-whole relationship among numbers. Solving word selecting tools proving, problems in which the unknown is varied helps students construct this and understanding. According to the National Research Council, “For students in ccsootrmnantmeegcuitneiniscg,a,rterineflgpercetsinegn,ting, grades K–2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic” Math Vocabulary: (Adding It Up, quoted in Lawson, 2015, p. 56). asdedn,tesnucbet,rancutm, nbuemr ber story About the Lesson Teaching Tip This lesson may serve as a review of grade one expectations for addition and subtraction or as a diagnostic lesson to determine if students recognize when a You may wish to do situation calls for addition or subtraction and if they are able to represent the the Math Talk on situation orally, pictorially, and symbolically. The problems in which the addition strategies structure is not simply finding the unknown result will likely be more and varying the challenging for some students. unknown (see page 143) before beginning Using contextual problems also allows you to introduce or reinforce the this lesson. mathematical modelling process as students work back and forth throughout the four components. Refer to the anchor chart you previously developed or Materials: develop one now. The four components are: • Understand the Problem • Analyse the Situation • C reate a Model • A nalyse and Assess the Model In the Minds On, students will solve addition and subtraction problems by acting them out as a class. Students will record the corresponding number sentences on the board. In the Working On It section, students will first work in partners to create and solve addition and subtraction story problems represented by the visuals presented. They’ll orally create addition and subtraction number stories, and then represent them symbolically using number sentences. Students will then solve a few story problems independently that require them to represent their thinking using words, pictures, and numbers. Finally, they’ll create story problems for a partner to solve. Digital Slides 73–74: Minds On (15 minutes) Class Story Problems, BLM 27: Story Problems • Orally pose some addition and subtraction story problems that can be acted Time: 45–50 minutes out by the students in your class. Examples are provided below; however, try to contextualize the problems with scenarios or numbers that make sense for your own class. After posing a problem, have students rephrase the problem in their own words (Understand the Problem). Have students suggest how to act it out. You might choose a student director for each problem. After a problem has been acted out, pose some or all of the possible questions below. Have a student record a number sentence for the problem on the board. 146 Number and Financial Literacy

• Possible questions to use with each problem (Analyse the Situation) include: – How can we act out this problem? – What solution did you find? (Record all responses.) – What strategy did you use? – How can we represent this problem in a number sentence? (Record the number sentence on the board.) – How did you know that this was addition/subtraction? (e.g., there were two groups that we put together; one group joined another; we had a big group then some left, etc.) Problem 1 (addition–join structure, result unknown): 6 students came in before the bell rang this morning. Then 5 more came in after the bell. How many students came to class? Problem 2 (addition–join structure, change unknown): On Monday morning, 4 students signed up for choir. By the end of the day, 12 students had signed up. How many students signed up during the afternoon? Problem 3 (subtraction–separate structure, result unknown): We have [choose a number] students in our class. After the first break, 4 students went home sick. How many were left? Problem 4 (subtraction–part-part-whole structure, part unknown): We have [choose a number] students in our class. 14 of them ordered pizza for lunch. How many did not order pizza? Problem 5 (subtraction–compare structure, difference unknown): 6 students brought bananas for lunch. 10 students brought apples. How many more students brought apples than bananas? Working On It (20 minutes) • Provide partners with paper and markers. Show students the visuals on Digital Slides 73–74: Class Story Problems. Ask them to orally create a number story together for the first picture. Have a few partners share with the class. Then ask students to represent that story using a number sentence on the paper. Observe what students record. Choose pairs that have written different number stories to share, ensuring at least one number story is addition and one is subtraction. Ask students what each number represents. Then ask the class to represent the same picture in a different way (e.g., If they used addition, they should now use subtraction). Record all four options for the number story on the board as students share. • Repeat the process using the second picture, and ask students to represent the picture using at least one addition and one subtraction sentence. • Provide each student with a copy of BLM 27: Story Problems and explain the task. Provide time for students to work individually to solve the problems. Have concrete materials available for students who would like to use them. (Create a Model) Differentiation • Consider working through the problems in the Minds On again in a small- group guided math lesson with students who would benefit from more Addition and Subtraction to 20 147

practice with the various problem structures. You could use the same numbers or alter them. You might use an arithmetic rack to ‘act’ out the problems. • Include ELLs who may need support understanding the language of the problem situations in the small-group guided math lesson. Acting out the problems in the whole class would have provided contextual cues, but a chance to revisit the problems and the problem structures may prove beneficial. Assessment Opportunities Observations: As students share their story problems, observe their understanding of the relationship of the numbers involved in the addition and subtraction stories. Can they tell you what the numbers in their equations represent? Do the actions or situations in their stories reflect the operations they are using? Also note their addition and subtraction strategies. Are they counting on from the bigger number, counting back, counting by 2s or 5s, making a ten, using ‘think addition,’ doubles, etc. Consolidation (10–15 minutes) • After students have completed the independent task using BLM 27: Story Problems, have them share their story problems. Choose a few students to share their problems for each of the three pictures. Ask students to explain why they added or subtracted. Also ask them to identify the parts and the whole. As students share, invite the rest of the class to ask questions or to make connections to the shared problems. Have students explain why addition and subtraction can be used for the same picture. • Discuss the strategies that students used and how they may have changed their approach while solving the problem. Ask how they might solve the problem differently if they were to do it again (Analyse and Assess the Model). Building Social-Emotional Learning Skills: Critical and Creative Thinking: In a recent video series, Jo Boaler talked about the kinds of math tasks that help students develop and maintain a growth mindset. Her belief is that students need to believe that their main job is not to get questions right, but to deepen their learning (Boaler, n.d.). In order to develop critical and creative thinking, provide tasks that encourage a variety of approaches and strategies so that all students have an entry into the problem. Give students a voice by having them share their thinking during consolidations. Focus attention on strategies and representations rather than on just having the correct answer. 148 Further Practice • Provide contextualized problems that are relevant to your students. For example, you might ask students to solve problems related to attendance, lunch orders, lineups, or the calendar (e.g., How many more days until…?, How many days are there in two weeks? How many Tuesdays and Thursdays are there in October?). Have students solve these problems mentally as a Math Talk or in their Math Journals. Have students use a variety of manipulatives (e.g., arithmetic rack, hundreds chart, number line) to help them develop flexibility and so that they can discover which tool will become an effective mental model for them. Number and Financial Literacy

Teaching Tip Math Talk: Integrate the math Math Focus: Addition strategies talk moves (see page 7) throughout Process: Record 7 + = 20 on the board and ask students to solve it. Have Math Talks to students share their answers. Record all solutions. Ask a few students to share maximize student and defend their thinking. As they share, represent their thinking on a number participation and line. active listening. NOTE: This question was chosen because in the previous Math Talk (see p. 143), students used counting on and tracking to solve an addition problem. We want students to realize that counting on is a good strategy when one of the addends is small (less than 5) but that it becomes less efficient with bigger numbers. This question makes it likely that some students will try to count on from 7 to 20. Let’s Talk Select the prompts that best meet the needs of your students. • Who has a solution for the problem? (I got 12. I started at 7 and I counted on until I got to 20.) Did you count on by 1s? Can you show us? (12, 13, 14... 19, 20. Student is trying to keep track with fingers, but gets confused.) • I see you got to 20. How many fingers did you hold up? (I thought it was 12 but this time I got 13.) Let’s try the jumps on the number line. Start at 7 and show jumps of 1 to get to 20. That took a lot of jumps. What is the correct answer? (13) Why do you think you got 12 the first time? (It was hard keeping track. I didn’t have enough fingers.) Do you see a bigger jump that you could have taken? (I don’t know.) • C lass, do you see a bigger jump? (I started at 7 and jumped 3 to 10. Then I doubled 10 to get 20. So it’s 13.) Show the jumps on the number line. +3 +10 7 10 20 • W hy did you jump to 10? (It’s a friendly number. It’s easy to add with.) It only took you two jumps on the number line. That seems very efficient. • Everyone please turn and talk to a partner. When is counting on by 1s an efficient strategy and when would it work better to use a different strategy? • Have students share their thinking and consolidate with the idea that counting on is efficient with small numbers that you can easily keep track of, but that you should try other strategies with bigger numbers. You might decide as a class that if you’re adding 5 or more, you should try a different strategy. It’s possible that some students may remain in the stage of counting on for a while, but you’ll want to create other opportunities to support them in developing other strategies (e.g., make a ten, doubles) that are more efficient through your Math Talks and tasks. Addition and Subtraction to 20 149

6Lesson Varying the Unknown in Addition and Subtraction Problems Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationships between addition and multiplication and between subtraction and division, to solve problems and check calculations • B 2.2 recall and demonstrate addition facts for numbers up to 20, and related subtraction facts • 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 • 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 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 contexts Teacher Possible Learning Goals Look-Fors • Relates numbers to the benchmarks of 5 and 10 Previous Experience • Composes and decomposes numbers to 20 using various combinations with Concepts: • U ses a variety of strategies to solve addition and subtraction problems in Students have had experience decomposing which the unknown is varied numbers to 18 and solving addition problems • Uses the benchmarks of 5 and 10 to add and subtract to 20. Prior experience • Uses a variety of strategies (e.g., make a ten, near doubles, count on, count with the arithmetic rack would be beneficial. back, ‘think addition’) • Represents doubles using the arithmetic rack • Subitizes the coloured groupings of 5 and 10 and pushes them in one movement on the arithmetic rack • Clearly explains strategies using mathematical language (e.g., I knew that 6 was 4 less than 10; I made a ten then counted on, etc.) • Understands what the problem is asking and independently tries a strategy to solve it 150 Number and Financial Literacy

PMraotcheesmseast:ical About the rtPeorpoorlbselseaemnndtsinsotglrvaintegg, iseesl,ecting Ideally, children should be given opportunities to solve contextual problems Math Vocabulary: that relate to their own lives. When we create math problems that build on compose, more than, children’s recent experiences (e.g., building a structure, a field trip experience), less than students connect math to the real world and see it as a subject that makes sense (Van de Walle & Lovin, 2006, p. 71). Although it’s difficult to anticipate Teaching Tip students’ current school experiences in a math resource, story problems can be used to put problems in familiar contexts. Wherever possible, you should You may wish to do substitute an experience/story that best reflects your students’ lives. the Math Talk on addition strategies The arithmetic rack is a visual model that supports students in developing (see page 149) before a variety of strategies for addition and subtraction. The organization of beginning this lesson. the beads into groups of 5 encourages students to build groups and to use the relationships of 5 and 10 when composing numbers, adding, and Materials: subtracting. It is also a useful tool for working on doubles, near doubles, small arithmetic and halving strategies. Using the arithmetic rack to compose and racks (1 per pair of decompose numbers as well as to solve addition and subtraction students), BLM 28: problems provides students with a visual model that can help them build Blank Arithmetic Racks flexibility with numbers and develop number sense. If you do not have Time: 25–40 minutes arithmetic racks, students can carry out the tasks using two ten frames. (depending on how many The can also draw their solutions on the BLM versions of the arithmetic problem sets you do per racks. (For more on the arithmetic rack, or rekenrek, see https:// period) thelearningexchange.ca/videos /building-f luency-with-facts /) About the Lesson Students will become more familiar with how to use the arithmetic rack in the Minds On activity as they explore equivalence, doubles, and the relationship of quantities to 5 and 10. In the Working On It section of the lesson, students will act out and solve problems on the arithmetic rack using the scenario of a carnival game in which players try to hit balloons organized in two rows. The first set of problems will help students develop the relationship of 10 and more and will introduce them to addition and subtraction problems where the unknown is in different parts (start unknown, part unknown, result unknown). It’s important to expose students to a range of problem structures to develop their conceptual understanding of addition and subtraction. Problem sets E and F focus on the strategy of near doubles and may be more challenging for some students. Minds On (10–15 minutes) • Ask students if they’ve ever played games at a carnival or fair. Have a short discussion about the kinds of games they played. Tell students to imagine a game that has two rows of balloons. Each row has 5 red and 5 white balloons in it. Ask, “How many balloons are you imagining?” If students don’t understand the concept of ‘each’ say, “The top row has 5 red and 5 white balloons in it and the bottom row has 5 red and 5 white balloons in it. Draw Addition and Subtraction to 20 151

what you are visualizing to find out how many balloons there are altogether.” Repeat the problem slowly so students can visualize, then draw. Circulate as they draw, taking notice of who is able to visualize and draw the two rows correctly. After a few minutes, have them share their solutions and pictures. Ask some or all of the following questions: – How many balloons did you draw altogether? – How many red? How many white? – How many red in each row? How many white in each row? – Are there more of one colour than another? Are the rows equal? How do you know? • Show students an arithmetic rack and say, “This is an arithmetic rack. It’s just like the picture we drew of the balloon game. We’re going to use it to solve some problems.” If this is the first time using the arithmetic rack, explain how to use it. • Give each set of partners an arithmetic rack. If you don’t have arithmetic racks, they are quite easy to make out of beads and pipe cleaners (see the Teacher’s Website for instructions). Alternatively, you could have students create two rows of two kinds of manipulatives, with five of each kind in each row. • Use the following prompts and questions: – Show 8 on the top row. Try to do it in one push. How many more than 5 is 8? How many less than 10 is 8? Reset by sliding all of the beads back to the right side. – Show 9 using 2 rows. Use only 2 pushes. How did you make/compose 9? How many more than 5 is 9? How many less than 10 is 9? How many more do you need to make 20? How many less than 20 is 9? Reset. – Compose 16. Use only 2 pushes. How did you compose 16? How many more than 10 is 16? How many less than 20 is it? How can you make 16 as a double? (8 in each row) Reset. – Compose a double. How did you make your double? How many are on the top? How many are on the bottom? How many more than 5 is your double? How many more or less than 10? How many less than 20? – Compose a number that is more than 10 but less than 20. What is your number? How did you compose it? How many more than 10 is it? How many less than 20 is it? What are all the possibilities? – Compose a number that is equal to 6 plus 5 using a different combination of beads. What number was I looking for? How did you compose 11? What are all the possibilities? How many more than 10 is 11? How many less than 20? 152 Working On It (10–15 minutes per set of problems, over 2 or more periods) • Below are various problems to solve using the arithmetic rack and the scenario of playing a carnival game. Present these problems over two or more periods. You may wish to work through one set of problems each day by solving one Number and Financial Literacy

problem together as a class and then assigning the second problem in the set for independent work. • For the class work, have students work in partners. Have one partner solve the problem using the arithmetic rack and the other partner record the solution on BLM 28: Blank Arithmetic Racks. Have students switch roles with each new problem. After students work through Problem 1 in each set, have a discussion using the questions provided. After working through all of the problems, determine which structure(s) students find challenging and create similar problems for students to work through. If possible, pose problems using authentic classroom contexts. Problem Set A: Problem 1 (addition–join structure, result unknown): A player broke 8 balloons on the top row and 5 balloons on the bottom row. How many did he break? (13) Questions: How did you solve the problem? (e.g., I pushed 8 on top. Then I pushed 5 on the bottom and counted on; I pushed 8 on top and 5 on the bottom. I know 8 and 2 is 10 and 3 more is 13; I used a known fact; I counted them all; I counted on from 8) Where do you see 10 on your arithmetic rack? How many more than 10 is 13? How did you record? Have students Think-Pair-Share: How is your solution the same or different than your partner’s solution? Problem 2 (addition–join structure, result unknown): A player broke 6 balloons on the top and 9 balloons on the bottom. How many did she break? (15) Questions: How did you solve the problem? How could you use the make a ten strategy to add? Where do you see 10? How many more than 10 is 15? How many less than 15 is 10? Problem Set B: Problem 1 (addition, start unknown): A player broke some balloons on the top and 4 on the bottom. The total number of broken balloons was 11. How many balloons did she break on the top? (7) Questions: How did you solve the problem? What information did we know before solving the problem? What is the whole? What is the part we know? How many more than 10 is 11? How many less than 10 is 4? How many more would she need to hit to break 20? (9) Problem 2 (addition, start unknown): A player broke some balloons on the top. Then he broke 10 on the bottom. He broke 18 balloons altogether. How many did he break on the top? (8) Questions: How did you solve the problem? What is the whole? What part did you already know? How many more than 10 is 18? How many less than 20 is 18? Problem Set C: Problem 1 (subtraction–separate structure, result unknown): There were 20 balloons. Mena broke 14 of them. How many were left unbroken? (6) Addition and Subtraction to 20 153

Questions: How did you solve the problem? What was the whole? (20) How many balloons did Mena break? (14) How many more than 10 is 14? How many less than 20 is 14? Problem 2 (subtraction–separate structure, result unknown): There were 20 balloons. Jehad broke 11 of them. How many were left unbroken? (9) Questions: How did you solve the problem? What was the whole? (20) How many balloons did Jehad break? (11) How many more than 10 is 11? How many less than 20 is 11? P roblem Set D: Problem 1 (compare structure, difference unknown): Sam broke 5 balloons. Maiya broke 12 balloons. How many more balloons did Maiya break than Sam? (7) Questions: How did you solve the problem? (Students may add or subtract to solve this. Bring attention to both strategies.) Model the word ‘difference’ by saying, “What is the difference between Sam and Maiya’s score?” Ask students what they think ‘difference’ means (e.g., how many apart the numbers are). Problem 2 (compare structure, difference unknown): Jessie broke 7 balloons. Abdul won the game by breaking 18 balloons. How many more balloons did Abdul break than Jessie? (11) Questions: How did you solve the problem? What is the difference between Abdul’s and Jessie’s scores? (11) How many did you add to get to 10? How many more to get to 18? Could we subtract to find the answer? How? Problem Set E: Problem 1 (addition, result unknown): A player broke 6 balloons on top and 7 balloons on the bottom. How many were broken? (13) Questions: How did you solve the problem? Where is there a double? (6 + 6 but there’s one more) That’s called a ‘near double.’ Record on the board: 6 + 6 =12; 12 + 1 = 13. Say, “Let’s see if we can use near doubles to solve the next few problems.” Problem 2 (addition, result unknown): A player broke 8 balloons on the top row and 9 balloons on the bottom row. How many were broken? (17) Questions: How did you solve the problem? Where is there a double? (8 + 8) What does 8 + 8 equal? (16) What do we do with the one that is left? (add it to the 16) So 8 + 8 =16; 16 + 1 = 17. Right. So 8 + 9 is near the double 8 + 8. 154 Problem Set F: Problem 1 (parts unknown): A player broke 13 balloons. She broke more balloons on the top than on the bottom. The number on the top and bottom created a near double. How many did she break on the top and the bottom? (7 on top and 6 on the bottom) Questions: How did you solve the problem (e.g., guess and check; thought of all the doubles below 13 and tried them out). Why does ‘7’ have to be on the top? (e.g., You said she broke more on the top and 7 is more than 6.) What is the near double? (7 + 6) So what would the number sentence be to show that we Number and Financial Literacy

used near doubles to solve it? (6 + 6 = 12; 12 + 1 = 13) Why couldn’t we use 7 + 7 – 1 instead? (e.g., It doesn’t show what happened in the story.) Problem 2 (parts unknown): A player broke 19 balloons. She broke less balloons on the top than on the bottom. The number on the top and bottom created a near double. How many did she break on the top and bottom? (9 on top and 10 on the bottom) Questions: How did you solve the problem? Why does 9 have to be on the top? (e.g., You said there were less on the top.) What is the near double? (9 + 10) What would the number sentence be to show that we used near doubles to solve it? (9 + 9 = 18; 18 + 1 = 19) Why couldn’t we use 10 + 10 – 1 instead? (e.g., It doesn’t show what happened in the story.) Differentiation • Assign partners strategically so that one partner has a stronger number sense and can support the other partner if needed by modelling and conferring. • Work on the problems in small-group guided math lessons with students that require more support or further experiences to consolidate the concepts in this lesson. • If students are having difficulty with making ten and near doubles, use Math Talks that build these strategies, revisit the use of ten frames for adding, do dot configuration flashes, and play games that require making tens and doubles. If students are having difficulty with subtraction, work on counting back using the hundreds chart and number line, and work on subtraction as removal using the ten frame or arithmetic rack. If students are not yet counting on but are counting all or counting 3 times, focus on developing counting on through: dice games involving rolling and moving on a game board, using the Missing Part cards from Lesson 4 (see pp. 139–143), and revisiting the Train Wreck activity from Lesson 3 (see p. 137). • Provide a framework for ELLs to create their own problems: There were balloons broken on top and balloons broken on the bottom. How many balloons were broken altogether? Assessment Opportunities Observations: As students work through the problems, observe their problem-solving skills as well as their operational skills. Note which students continue to count all or rely on counting on rather than using strategies such as make a ten or near doubles. Provide more opportunities for these students to develop more efficient strategies through Math Talks, math centres, and/or small-group guided math lessons. Conversations: Engage in individual conferences with students who are having difficulty, and with students for whom you need more data. Have students solve three or four problems using the arithmetic rack. Change the unknown for each problem. Have them orally explain their strategies. Encourage them to use 5 as an anchor and to make a ten when possible, especially if they continually use counting all or counting on/back. Addition and Subtraction to 20 155

Teaching Tip Consolidation (5–10 minutes) Integrate the math • At the end of this series of problem-solving tasks, consolidate by discussing the talk moves (see page 7) throughout strategies that students used (e.g., How did you know when to add or subtract? Math Talks to Which strategy did you find most helpful? Should we use the same strategy for maximize student every problem? Why or why not? How is your strategy the same as or different participation and from [name a student]? When is the make a ten strategy useful? When is the active listening. near doubles strategy helpful?). Ask students to orally reflect on the process as well (e.g., Which problems were the easiest for you to solve? Which were more challenging? What has been your biggest learning? How have you grown as a mathematician by working through these problems?). Further Practice • Math Centre: Have students make up their own story problems using the balloon game or another scenario that would involve the use of the arithmetic rack. Have them record their problems on index cards. Place the story problem cards in a math centre and invite students to solve them. • Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: Reflecting in Math Journals: Have students reflect on the set of lessons in their Math Journal using one of the following prompts: – How has your brain stretched and grown by doing these math problems? Explain your thinking using words, numbers, or pictures. – Did you use a bubble gum brain or brick brain to solve these problems? Explain. – What do you think you still need to work on to achieve your goals? Math Talk: Math Focus: Make a ten Process: Use the following number string to develop the strategy of make a ten. Record 5 + 6 = on the board. Have students share their answers. Record all solutions. Ask a few students to share and defend their thinking. As they share, represent their thinking using ten frames. Repeat the process with: 7 + 4, 8 + 3, 9 + 7. Let’s Talk Select the prompts that best meet the needs of your students. • W ho would like to share their solution? (I started with 5 fingers and I noticed that I could put up my other hand to add another 5. That’s 10. Then I added on 1 more. That’s 11.) Let’s use the ten frame to show that. Here’s 5. Now I draw 5 more dots and the ten frame is filled in so we have 10. I draw one more now we have 11. 156 Number and Financial Literacy

• Did anyone use a different strategy? (I started with 6 and counted on 4 to get 10. Then I had to add one more to get 11.) Model with the ten frames. So does it matter if you start with 5 or 6 when you add? Why? (The numbers can be turned around. It doesn’t matter. They’re the same amount in total. Like our fact families!) • Anyone else? (I used near doubles. I knew that double 5 was 10 and added one more.) • W hat’s another way to use near doubles? (You can double 6 to get 12 and take away 1.) Did you know double 6 or did you count? (I knew the double.) I see. It wouldn’t be as easy if you didn’t know the double, would it? • W hat strategies have we used so far? (make a ten, count on, near doubles) • Let’s try a few more questions. • T his Math Talk can be repeated using different equations. Number strings are great for helping students develop specific strategies such as make a ten, near doubles, decomposing, etc. Addition and Subtraction to 20 157

7Lesson Make a Ten: Adding Three Numbers 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 are able to add related subtraction facts two numbers to 18 and have had experience • B 2.3 use mental math strategies, including estimation, to add and subtract decomposing 10. 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 Possible Learning Goals • Composes and decomposes two-digit numbers in a variety of ways • Uses a variety of strategies to solve addition problems • Applies the commutative property to find all the facts for 10 (6 + 4 = 4 + 6) • Applies the associative property to addition (7 + 5) + 3 = (7 + 3) + 5 • Recognizes combinations for 10 • Uses doubles or near doubles to solve addition problems • Counts on from the higher number when adding • Uses the make a ten strategy when solving problems with three addends • Identifies and explains their strategies 158 Number and Financial Literacy

PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, Students have had practice developing the commutative property of addition (a pair of numbers can be reversed/commuted without changing the sum, i.e., McmoaamtkhpeoVasoect,eadnbe,ucaloadmrdyep:nodsse,, 5 + 3 = 3 + 5) in previous lessons as they worked with fact families and equations Missing-Part cards. When adding, they should realize that it doesn’t matter which addend they begin with, and that they should do what is most efficient Teaching Tip for them (e.g., start with the bigger addend and count on). You may wish to do The associative property is another important property for students to apply the Math Talk on when solving addition problems. According to the associative property, making a ten numbers in an expression can be regrouped without changing the sum. For (see page 156) before example, 8 + (2 + 3) = (8 + 2) + 3. When students decompose numbers to beginning this lesson. work with friendly numbers, or decompose to make a ten for easier addition, they are applying the associative property. Students do not need to know the names of the properties but can gain a conceptual understanding through activities that encourage regrouping quantities. The following lesson will help students recognize pairs of addends that equal 10 so that they can use the make a ten strategy to add more efficiently. For example, if students are adding 8 + 6, they might apply the associative property by mentally decomposing 6 into (2 + 4) in order to add 8 + 2 = 10 and then add on the 4 to get 14. Students are not required to identify and explain the associative property, but it is important that they learn how to apply it when adding. About the Lesson In the Minds On, students will match addition equations with three numbers (two that add up to 10 and an additional number, such as 7 + 3 + 2) to a sum (e.g., 12) on the board and share their strategies. Then they will practise recognizing combinations that make 10 while adding three numbers and recording the steps that are followed when using the strategy of making a ten. This step-by-step process provides a scaffold for applying the make a ten strategy to solve problems such as 5 + 7 and 8 + 4 which students will work on in a subsequent lesson. Materials: Minds On (10–15 minutes) sticky notes, • Write the numbers from 10–18 on the board, leaving some space between BLM 29: Find Ten each number. Leave these numbers displayed for the duration of the lesson. Time: 25–40 minutes • Create addition equations made up of two and three addends that equal the numbers 10–18 on separate sticky notes so that there is one equation for each student. Use combinations for 10 in the equations with three addends. Suggested equations: 8+2 6+4 7+3 5+5 1+9 6+6 7+7 8+8 9+9 8+2+7 6+4+5 8+2+1 5+5+3 1+9+4 8+2+8 6+4+7 7+3+5 5+5+1 7+6+3 5+2+3 Addition and Subtraction to 20 159

• Give each student a sticky note with an equation on it. Draw the class’s attention to the numbers on the board. Tell students that the answer to their equation is one of the numbers on the board. Ask students to solve their equations. • One by one, ask students to share their equations without saying the answers. The student sharing the equation can ask a student (who has a thumb up to show they have an answer) to solve the question. The ‘sharer’ confirms or disagrees with the answer given. If there is a disagreement, both students should explain how they arrived at the answer and determine which answer is correct. Once the answer is agreed upon, the ‘sharer’ should put the sticky note on the number that corresponds to the answer. Assessment Opportunities Observations: Note the strategies that students use to solve the problems. Do they count all, count on, make a ten, use doubles, near doubles, etc.? Working On It (10–15 minutes) • W rite the equation 8 + 2 + 3 = on the board. Tell students to find 2 numbers that add up to 10 (8 + 2). Circle the 8 and 2. Ask a student to explain how he/she knew that 8 + 2 = 10 (e.g., familiar fact, count on from 8). Record 8 + 2 = 10 below the equation. Ask students what they should do next (add on the 3). Record 10 + 3 = 13 below 8 + 2 = 10. Tell students that there are other ways to solve this problem but we are going to focus on making a ten and adding on the leftover number. 8+2 +3= 8 + 2 = 10 10 + 3 = 13 • Give each student a copy of BLM 29: Find Ten. Do the first question together step by step, then have students complete the rest of the questions independently. 6 + 3 + 4= + = 10 10 + = NOTE: Many students make errors when counting on because they count the starting number when they add. When counting on 8 + 3, they might count 8, 9, 10, because they’ve counted the 8. One way to help students count on accurately is by saying, “Put 8 in your hand. Close it. Whisper 8. Now put a finger up for each number you say as you count on 3 more, 9, 10, 11.” You can also show students how to do this using a number line or the hundreds chart. Tell them that they only say a new number as they make a move. Differentiation • Provide access to ten frames, hundreds charts, and arithmetic racks so that students who require a visual model or way to manipulate the numbers can be successful. 160 Number and Financial Literacy

Assessment Opportunities Observations: Observe students as they work on BLM 29: Find Ten. Do they recognize combinations of ten or do they use trial and error and counting all? How do they add on the last number (e.g., known fact, count on, etc.)? Consolidation (5–10 minutes) • After students have completed the ‘Find Ten’ task, write a question on the board such as 7 + 5 + 3 = . Invite students to Think-Pair-Share: Explain to your partner how the make a ten strategy works. Use the question on the board to help you explain it. • Select a few students to share their explanations. Ask, “When would you use this strategy?” (e.g., adding higher numbers, when there are numbers that add up to 10) “When wouldn’t this strategy be a good choice?” (e.g., when there’s a double or near double, when the numbers are less than 10) Further Practice • Give each student five sticky notes. Ask students to create equations with three addends, two of which must add up to 10, for five numbers from 10–18 (see the numbers on the board from the Minds On activity). Ensure they write the answers to their equations and initial their sticky notes. Have students exchange sticky notes with a partner to check for accuracy before attaching the sticky notes to the numbers on the board. Building Social-Emotional Learning Skills: Identification and Management of Emotions; Stress Management and Coping: An important goal for grade two students is to develop flexibility with computational strategies. Exposing them to a variety of strategies through student sharing is a great way to help them develop more efficient strategies. Some students, however, may not move forward without being required to try strategies that they are not yet comfortable with. Ask your students to indicate their comfort level with the make a ten strategy with their thumb (up – I get it; sideways – I sort of get it; and thumb down – I don’t get it yet). Then ask students how they felt while working on this task. Have them show thumbs up, down, or sideways to indicate their feelings. Reassure students that it takes practice to learn how to use a new strategy such as making a ten. Ask them how many practices they think they need to feel comfortable with this new strategy (they can indicate with fingers up). Make a plan together to practise (use Math Talks, games, and learning from next few lessons). Addition and Subtraction to 20 161

Materials: Math Talk: ten frames and differently coloured Math Focus: Make a ten markers Process: Use the following number string to develop and reinforce the make a Teaching Tip ten strategy. Represent student strategies using ten frames. The following number string could be used: Integrate the math talk moves (see 8+2 page 7) throughout Math Talks to 8+2+3 maximize student participation and 8+4+2 active listening. Let’s Talk Select the prompts that best meet the needs of your students. • How did you solve the problem 8 + 2 = ? (I started at 8 and counted on 2 more. 8, 9, 10) Draw 8 dots in one colour in the ten frame and 2 dots in another colour. • D id anyone use a different strategy? (It was a known fact for me.) • L et’s solve this next problem: 8 + 2 + 3 = . Use what you already know to solve this problem. What answer did you get? (13. I added 3 to 10.) Where did you get 10? (I knew that 8 + 2 = 10 and added on 3. 10, 11, 12, 13) • You used the make a ten strategy, then counted on 3. How would I show that in the ten frame? (You already have 10 dots so you just draw 3 more outside of it.) Draw 3 more dots. • D id anyone use a different strategy? (I knew that 2 + 3 = 5. Then I added 8 more.) How did you add 8 to 5? (I knew that 5 + 3 = 8 so I added 5 + 5 = 10 and added on 3 more to equal 13.) • Let’s show that in the ten frame. You said that you knew that 2 + 3 = 5 so let’s draw the 5 dots. Then you said that you split 8 into 5 and 3 and added on 5. So let’s draw 5 more dots in the ten frame. What’s left to add on? (3 more.) So we have 10 made up of two 5s and then we have to add on 3. What’s 10 + 3? (13) • Okay, let’s solve one more problem: 8 + 4 + 2. (That equals 14.) What strategy did you use to solve it? (I made a 10 with 8 + 2. Then I added 4 more. I just know that 10 + 4 is 14 because 14 is 1 ten and 4 ones.) • C lass, what strategy was used to solve all 3 problems? (Make a ten.) Yes. It makes adding bigger numbers easier, especially if we know what 2 numbers add up to 10. Turn to a partner and think of as many combinations of 2 numbers as you can that add up to 10. • Have students share. Record the combinations. 162 Number and Financial Literacy