p256-379-Gr3ON-Number-Unit 4Addition-pass2

Unit 4: Addition and Subtraction Lesson Content Page Addition and Subtraction Introduction 256 261 1 Math Potatoes: Mental Math Strategies 273 275 2 and 3 Solving Addition and Subtraction Problems Using a Variety of Strategies 279 283 2 Solving Problems with Two-Digit Numbers 287 292 3 Solving Problems with Three-Digit Numbers 294 300 4 Mental Strategies that Use the Commutative and Associative Properties 301 304 5 Using Patterns in the Hundreds Chart to Add and Subtract 307 314 6 to 8 Using an Open Number Line to Add and Subtract 320 327 6 Adding and Subtracting Two-Digit Numbers on an Open Number Line 334 7 and 8 Using an Open Number Line to Add and Subtract Three-Digit Numbers 336 7 Adding Three-Digit Numbers on an Open Number Line 339 343 8 Subtracting Three-Digit Numbers on an Open Number Line 349 351 9 Using ‘Think-Addition’ to Subtract Multi-digit Numbers 352 354 10 Investigating Compensation as a Mental Strategy 359 360 11 Using Partial Sums and Differences for Mental Calculations 361 12 Adding and Subtracting Three-Digit Numbers by Applying Known Strategies 366 13 and 14 Directly Modelling Addition and Subtraction of Three-Digit Numbers 368 13 with Regrouping 370 375 Directly Modelling Addition of Three-Digit Numbers with Regrouping 14 Directly Modelling Subtraction of Three-Digit Number with Regrouping 15 Using Partial Sums and Differences Algorithms to Add and Subtract 16 to 21 Using the Standard Algorithm to Add and Subtract Multi-digit Numbers 16 and 17 Using the Standard Algorithm to Add and Subtract (Without Regrouping) 16 Adding and Subtracting Two-Digit Numbers (Without Regrouping) 17 Adding and Subtracting Three-Digit Numbers (Without Regrouping) 18 and 19 Using the Standard Algorithm to Add Multi-digit Numbers (with Regrouping) 18 Adding Two-Digit Numbers (with Regrouping) 19 Adding Three-Digit Numbers (with Regrouping) 20 and 21 Using the Standard Algorithm to Subtract Multi-digit Numbers (with Regrouping) 20 Subtracting Two-Digit Numbers (with Regrouping) 21 Subtracting Three-Digit Numbers (with Regrouping) 22 Solving Real-Life Addition and Subtraction Problems

Addition and Subtraction Introduction In this unit, students build on the skills and conceptual knowledge they developed adding and subtracting two-digit numbers in grade two to add and subtract three- digit numbers. They use their ability to compose and decompose quantities, their understanding of place value and part-part-whole relationships, and the mental math strategies they have previously learned to add and subtract two- and three- digit numbers. Students also learn the standard North American algorithm for addition and subtraction, with an emphasis on understanding how and why it works. While it is a widely accepted convention, it may not be the most effective strategy for them. Ultimately, we want students to be able to choose strategies that work for them. In many of the lessons, students add and subtract with two-digit numbers to review what they learned in grade two and to consolidate the strategies that can then be applied to three-digit numbers. In some cases, lessons are then repeated using three-digit numbers in place of two-digit numbers. This is also true for the lessons on mental math strategies; these lessons and Math Talks are interspersed throughout the unit so students have regular opportunities to reinforce what they have learned. Lessons 16 to 21 build on students’ understanding of addition and subtraction and introduce the North American algorithm for addition and subtraction. Once again, students learn how to use the algorithm with two-digit numbers and then use patterns in the number and place value systems to extend their understanding and apply the algorithm to three-digit numbers. About the Addition is often taught as meaning ‘put together,’ while subtraction is often said to mean ‘take away.’ According to Van de Walle and Lovin, these definitions are too narrow and can result in a limited understanding of the two operations and how they are related. They explain 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, 2006a, p. 66). Based on numerous research studies, they emphasize 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, 2006a, p. 66). Van de Walle and Lovin encourage teachers to have students analyse story problems in order to understand what they mean and to solve them accordingly, rather than merely looking for key words that indicate an operation. 256 Number and Financial Literacy

Problem Structures of Addition (Join) and Subtraction (Separate) Action: Join Problems Position of Unknown Example Result Unknown Jesse has 114 trading cards. He gets 37 more. How many trading cards does he have? 114 + 37 = Change Unknown Jesse has 114 trading cards. He is given some more trading cards. Start Unknown Now he has 151 trading cards. How many trading cards was he given? 114 + = 151 Jesse has some trading cards. He is given 37 more. Now he has 151 trading cards. How many trading cards did Jesse have at the start? + 37 = 151 Action: Separate Problems Position of Unknown Example Result Unknown Jesse has 151 trading cards. He gives 37 away. How many trading cards does he have now? 151 – 37 = Change Unknown Jesse has 151 trading cards. He gives some away. Now he has 114 trading cards. How many did he give away? 151 – = 114 Start Unknown Jesse had some trading cards. He gave 37 away. Now he has 114 trading cards. How many trading cards did Jesse have at the start? – 37 = 114 Materials and Tools Students move developmentally from using concrete to pictorial to symbolic representations throughout the learning. Concrete materials such as base ten blocks and place value mats support students as they develop conceptual understanding of operational skills. Tools such as open number lines are valuable with numbers of greater magnitude, since representing large quantities with concrete materials can be challenging due to limited resources. Students can also use drawings to represent their thinking, which helps them create mental visual images. Once strong mental images are in place, students can progress to working directly with the numbers and developing meaningful ways to record their thinking in numerical form. It is critical that students make connections among the various representations, and understand and explain what they mean. This allows them to flexibly select the representations that work best for them and the problem they are solving. The Development of Mental Strategies In the Ontario Math Curriculum 2020, mental math is described as “the ability to perform mathematical calculations without relying on pencil and paper” (Ontario Ministry of Education, 2020, p. 33). In some cases, students may need to jot down some information if there is too much to retain mentally. It is important to emphasize that the goal is not speed but flexibility and portability, so students can Addition and Subtraction 257

apply skills in real-life situations when tools are not available. In grade three, students develop a variety of mental strategies. Alex Lawson highlights the following strategies that students often develop, build upon, and refine (Lawson, 2015, pp. 85–86): • direct modelling – c ounting three times with concrete materials – c ounting on or back from one of the numbers • counting more efficiently and tracking – c ounting on from the largest number and tracking • working with numbers – s plitting along place value lines and skip counting – s plitting along place value lines and adding partial sums – k eeping one number whole and making jumps of tens and ones (e.g., on a number line and then mentally) In grade three, students are expected to use mental math strategies, including estimation strategies, to add and subtract whole numbers that add up to no more than 1000. Students will draw on their experiences in grade two, when they developed mental strategies involving the addition and subtraction of whole numbers to 50, and developed recall of addition facts to 20 and related subtraction facts. In many cases, they will need a review of these basic facts so they can build on them to learn new mental math strategies. Some of the Math Talks throughout the unit introduce new mental math strategies, extend strategies to work with larger numbers, or offer reinforcement and practice. Students need many experiences to internalize these strategies and to gain fluency and flexibility. It is recommended that the Math Talks be repeated several times throughout the unit and the year to give students regular opportunities to practise. You can tailor the Math Talks to the needs of your students by changing the numbers and offering new challenges. New numbers also allow you to assess whether students can apply what they have learned to new situations and adapt strategies according to the circumstances of the problem. 258 Algorithms In grade three, students add and subtract two- and three-digit numbers using algorithms, with the emphasis on conceptually understanding what each process and procedure means. As students work with larger numbers, algorithms can help them work more efficiently; however, research has shown that “students can have much greater success if they have opportunities early on to develop their own algorithms, generally using concrete models” (Small, 2009, p. 163). Marian Small (2009, p. 162) suggests that students need the following before moving on to use standard or conventional algorithms: • Know their addition and subtraction facts • Understand the basic principles underlying the place value system • Know how to add and subtract multiples of 10 • Understand the basic addition and subtraction principles, as many algorithms are built on those principles Number and Financial Literacy

Strategy or Algorithm? Marilyn Burns distinguishes between a strategy and an algorithm as follows: “In the context of arithmetic, when a strategy can be generalized to a systematic, step- by-step procedure for computing that works for a broad class of problems, then it can be considered an algorithm. Even though not all students’ strategies are algorithms, their strategies are valuable ways for us to assess their understanding and for them to develop, extend, or cement their learning” (Burns, 2015). Why Encourage Invented Strategies and Algorithms? Traditionally, students have been taught to solve three-digit addition and subtraction problems using the North American standard algorithm that involves a number of memorized steps (add the ones column, then the tens column, and then the hundreds column, and ‘borrow’ by crossing out numerals). Many students make errors when using this algorithm because they don’t understand the place value concepts behind it. When students solve problems using rote procedures that they memorize rather than strategies that make sense to them, they are not likely to develop a conceptual understanding of math. If we want our students to develop reasoning skills and to work flexibly with numbers, we need to provide them with many opportunities to develop their own strategies before expecting them to understand and use a standard algorithm in a meaningful way. Reinforcing Mathematical Language As students are learning how to add and subtract three-digit numbers, it is important that we model and reinforce the mathematical vocabulary that best describes the operations. This is especially true as students learn about regrouping. Marian Small states that the terms ‘regroup,’ ‘trade,’ and ‘exchange’ should be used rather than ‘carry’ or ‘borrow,’ because “carrying and borrowing have no real meaning with respect to the operation being performed, but the term ‘regroup’ suitably describes the action the student must take” (Small, 2009, p. 170). In this unit, the term ‘regrouping’ will be used consistently so that students, especially ELLs, understand that it is a process involved in addition and subtraction of three- digit numbers. Students also need to clearly communicate about the value that digits represent in our place value system. For example, if students are adding two numbers in the hundreds column, say 2 and 7, it is important that they express this as adding 2 hundreds (or 200) and 7 hundreds (or 700), to make a total of 9 hundreds (or 900). This can be regularly modelled and reinforced throughout the lessons. Addition and Subtraction 259

Lesson Topic Page 1 Math Potatoes: Mental Math Strategies 261 273 2 and 3 Solving Addition and Subtraction Problems Using a Variety of Strategies 275 279 2 Solving Problems with Two-Digit Numbers 283 287 3 Solving Problems with Three-Digit Numbers 292 294 4 Mental Strategies that Use the Commutative and Associative Properties 300 301 5 Using Patterns in the Hundreds Chart to Add and Subtract 304 307 6 to 8 Using an Open Number Line to Add and Subtract 314 320 6 Adding and Subtracting Two-Digit Numbers on an Open Number Line 327 7 and 8 Using an Open Number Line to Add and Subtract Three-Digit Numbers 334 7 Adding Three-Digit Numbers on an Open Number Line 336 339 8 Subtracting Three-Digit Numbers on an Open Number Line 343 349 9 Using ‘Think-Addition’ to Subtract Multi-digit Numbers 351 352 10 Investigating Compensation as a Mental Strategy 354 359 11 Using Partial Sums and Differences for Mental Calculations 360 361 12 Adding and Subtracting Three-Digit Numbers by Applying Known Strategies 366 368 13 and 14 Directly Modelling Addition and Subtraction of Three-Digit Numbers 370 with Regrouping 375 13 Directly Modelling Addition of Three-Digit Numbers with Regrouping 14 Directly Modelling Subtraction of Three-Digit Number with Regrouping 15 Using Partial Sums and Differences Algorithms to Add and Subtract 16 to 21 Using the Standard Algorithm to Add and Subtract Multi-digit Numbers 16 and 17 Using the Standard Algorithm to Add and Subtract (Without Regrouping) 16 Adding and Subtracting Two-Digit Numbers (Without Regrouping) 17 Adding and Subtracting Three-Digit Numbers (Without Regrouping) 18 and 19 Using the Standard Algorithm to Add Multi-digit Numbers (with Regrouping) 18 Adding Two-Digit Numbers (with Regrouping) 19 Adding Three-Digit Numbers (with Regrouping) 20 and 21 Using the Standard Algorithm to Subtract Multi-digit Numbers (with Regrouping) 20 Subtracting Two-Digit Numbers (with Regrouping) 21 Subtracting Three-Digit Numbers (with Regrouping) 22 Solving Real-Life Addition and Subtraction Problems 260 Number and Financial Literacy

1Lesson Math Potatoes: Mental Math Strategies Math Number Curriculum Expectations • B 2.3 use mental math strategies, including estimation, to add and subtract whole Teacher numbers that add up to no more than 1000, and explain the strategies used Look-Fors • B 2.1 use the properties of operations, and the relationship between Previous Experience with Concepts: multiplication and division, to solve problems and check calculations Students have worked with benchmark numbers Algebra of 5 and 10 and can skip count by 5s, 10s, and 20s. • C 1.4 create and describe patterns to illustrate relationships among whole numbers up to 1000 Possible Learning Goals • Mentally composes, decomposes, and recomposes quantities presented visually in order to add and subtract various amounts • Connects visual representations of quantity to their numerical representations • Identifies and explains the math problem posed in the text • Mentally composes, decomposes, and recomposes visual representations of quantities in a suitable way in order to mentally add or subtract; uses visual patterns do so • Uses anchors of 5, 10, and other friendly numbers to compose and decompose quantities • Flexibly selects strategies that are best suited to the problem, such as the doubling strategy • Mentally adds and subtracts various quantities, using both visual and numerical representations About the Lesson The text allows students to solve math problems embedded in riddles and illustrations. Using visual images rather than numbers allows students to quickly see different ways of organizing quantities. This helps students form powerful mental images of how sets can be composed, decomposed, and recomposed for easier counting, addition, and subtraction. These images can be retrieved later for mentally solving math calculations. Throughout the problem solving, it is important to make connections between the visual and symbolic representations so students can also learn to decompose quantities when they are in numerical form. continued on next page Addition and Subtraction 261

Once students have understood and internalized the mental strategies highlighted in this Read Aloud text and have used them to add numbers to 100, they can apply the strategies to add larger numbers up to 1000. When looking at the solutions in the back of the book, you will notice that the author has highlighted repeated addition as well as multiplication. You may want to revisit this book when you start multiplication to help students make the connection between repeated addition and multiplication. For now, you can focus on the mental addition and subtraction strategies that students will discover as they solve the problems. Math Potatoes has 16 riddles, which is too many for one lesson or session. The lesson addresses the first four riddles and the remainder are considered in three Math Talks (four riddles per Talk). You may decide to further break down the Math Talks and solve only one or two riddles per session. While the related Math Talks all follow this lesson, it is advisable to intersperse them throughout the unit and the year so students get regular practice at mentally composing and decomposing quantities and further developing their mental math skills. Materials: Assessment Opportunities Written by Greg Tang Observations: Note each student’s ability to: Illustrated by Harry – C ompose, decompose, and recompose visual representations of Briggs quantities Text Type: Fiction: – Add and subtract various decomposed parts to find total quantities Narrative–Rhyming – Explain their strategies for composing and decomposing Verse Riddles: “Math-ter Minds On (5 minutes) Cards,” “Sock Hop,” “Vegeta-Bullies,” and • Show the cover of the book and read the title and names of the author and “Shell Shock,” BLM 48: Find the Total, chart illustrator. Ask how the author is making a play on words in the title paper (e.g., math potatoes rather than mashed potatoes). Ask what the book may be Time: 45 minutes about. Ask how the illustration may connect to the title. Ask students what they think “Mind-Stretching Brain Food” means. Working On It (30 minutes) Math-ter Cards • Read the title and have students study the illustration. Ask why the riddle might be called “Math-ter Cards.” Ask what they know about cards and what the numbers, pictures, and shapes represent. • Read the first four lines. Ask what they think Texas Hold’em, 5-Card Draw, 7-Card Stud, and Omaha might be (e.g., they are the names of card games). 262 Number and Financial Literacy

• Read the next two lines. Ask how they might be able to add up the cards without adding each one individually (e.g., grouping cards together to make friendly numbers). Have students work in pairs to add up all of the numbers on the cards. Give them a copy of BLM 48 so they can better see the cards. • Discuss how students solved the problem. For each strategy, encourage students to explain it and why they think it works. Ask how the solutions can be recorded using numbers. Print their solutions on chart paper and explicitly connect the numbers to the visual representations. • One possible strategy is making groups of 10, using the facts that they know (e.g., 6 + 4, 2 + 8, etc.). Students can count the pairs of cards by tens and then add in the extra cards. • Read the last two lines and discuss whether the hinted strategy was used by anyone in the class. Have students work in pairs to try the suggested strategy (e.g., group the first card in each hand to make a group of three, then group the second card from each hand, and continue until they have created five groups of three). Ask what each grouping adds up to. Ask how they can total the cards (e.g., 15 + 15 = 30 and two more groups of 15 will be double that and equal 60. Add in the last group of 15 to total 75.). • Ask students which strategy seems best for them and why they think so. Sock Hop • Read the title. Explain that sock hops were dances that young people used to attend many years ago and were held in school gyms. Students had to remove their shoes and dance in their socks so they wouldn’t damage the floor. • Read the first six lines. Ask what the line “Please excuse the two left feet” might mean. • Have students work in pairs to solve the problem. Give students a copy of the BLM so they have a visual of the problem. • Discuss students’ solutions. Ask how they can record their solutions using numbers. Record their suggestions. Ensure that students see the connection between the numerical and visual representations. • One possible solution is that students count each row as having 10 full socks and then subtract the missing socks (e.g., 10 + 10 + 10 + 10 – 10 = 30). • Read the last two lines. If nobody solved the problem this way, have students do so now. Record their solutions in numerical form and connect the numbers to the visual representation. • Ask which solution they found easier and why they thought so. Vegeta-Bullies • Read the title. Ask how it is a play on words. • Read the first four lines. Ask how the words ‘squash’ and ‘beets’ can each have double meanings. Ask whether onions have ever made them cry. Explain that onions have a substance that irritates the eyes and makes them water. • Read the fifth and sixth lines. Have students work in pairs to solve the problem. Give them a copy of the BLM as a visual. Addition and Subtraction 263

264 • Discuss students’ strategies, record their solutions using numbers and make connections between the numerical and visual representations. • One possible solution is that students assume there are 5 full rows with 12 onions in each. They might mentally add the first two rows as 12 + 12 = 24. They may then double 24 as they add in the third and fourth rows to get 48. Next, they can add in the final row of 12 as 48 + 10 + 2 to total 60. They can then remove the four missing onions by counting backwards or subtracting from 60 (60 – 4 = 56). • Read the last two lines. Have students solve the problem using the author’s suggested strategy if they haven’t already done so. • Ask students which strategy they found worked best for them and why they think so. Shell Shock • Read the title and ask why the author may have chosen it for this visual. Ask what they think are inside the shells. • Read the first four lines. Ask students whether they have ever held up a conch shell to their ears. Explain that it sounds like the waves of the ocean. If possible, bring in a conch so students can experiment with it. Explain that any air that gets into the shell is bounced around the hard surfaces of the shell, which creates a sound much like the waves on the ocean. • Read the fifth and sixth lines. Have students turn and talk with a partner to solve the problem. Give them a copy of the BLM to work with. • Discuss students’ strategies and ask how they could record their solutions using numbers. Make connections between the numerical and visual representations. • One possible solution is that students put shells together so there are the same amount in each group. For example, they may make groups of 5, 10, or 15, skip count to find the number of shells in the like groups, and then add or count in any of the extra shells. If students make groups of 15, you can connect their solution to the one used in Math-ter Cards, which involved adding and doubling with 15. • Read the last two lines. If students didn’t use this strategy, have them work on it now. As they make their squares of 9, they may count by 10s and then subtract one for each group of ten they added to compensate for overcounting by one in each group. Ask students how they can record this solution with numbers. Make connections between the numerical and visual representations. • Ask which solution they found worked best for them and why they think so. Consolidation (10 minutes) • Discuss how the illustrations helped students find the sums of the items in each problem. Ask how seeing visual representations of the quantities was helpful. • Use an example to connect a numerical representation to a pictorial representation. • Explain that students will be revisiting this book again throughout the unit to find other creative ways to group objects and add them together. Number and Financial Literacy

• B uilding Social-Emotional Learning Skills: Critical and Creative Thinking: Ask students which strategy they found most interesting. Discuss how we can learn from sharing each other’s ideas and building upon them. Explain that mathematicians often discuss what they have discovered with other mathematicians to see if they can build on each other’s ideas or help each other clarify a concept. Tell students that, with practice, they will find using the strategies easier. Encourage them to look at problems from different perspectives, since this can often lead to discovering creative solutions. NOTE: The following Math Talks can be interspersed throughout the unit and the year. You may discuss fewer riddles per Math Talk, depending on your focus and the available time. Math Talk 1: Materials: Math Focus: Composing, decomposing, and recomposing visual quantities to make them easier to add and representing the strategies with numbers Riddles: “For Seven’s Sake,” “Nut House,” Let’s Talk “Hanging by a Thread,” “Smart Cookies,” BLM 48: Select the prompts that best meet the needs of your students. Find the Total For Seven’s Sake Teaching Tip • S how the picture of the stars. What do you see in this picture? What do you Integrate the math talk moves (see know about stars? What instrument is being used to look at the stars? Why page 8) throughout do you think it is used? Have you ever looked through a telescope? What was Math Talks to it like? maximize student participation and • R ead the first four lines of text without reading the title. What do you active listening. know about the Milky Way and planets? Why do you think the author says that looking at stars is a “window into yesterday”? • R ead the fifth and sixth lines. What are you supposed to find out? Turn and talk to a partner about how you might solve this. Try and think of one or two different ways. Look at the BLM of the visual in the book to help you. • W hat did you find? (e.g., We found that the second, third, and fourth columns all had 10 stars so we counted, 10, 20, 30. The first row had 12 stars so we added 30 + 12 = 42.) How could we show your solution using numbers? Show us how the numbers connect to the picture with the stars. • R ead the title and the last two lines. Do you think working with sevens is going to be easy or hard? Why? Try making groups of 7 and then use strategies to find the total number of stars. • W hat did you find? (e.g., We used doubling, with 7 + 7 = 14. Then we doubled it again, which accounts for four groups of 7 and equals 28. Then we added another 14 to account for the fifth and sixth groups of 7, which equals 42.) • D id you find it easy to work with groups of seven? Why? • W hat way did you find worked best for you? Why? continued on next page Addition and Subtraction 265

Nut House • S how the picture of the peanuts and read the title. Why might the author have chosen this title for this riddle? Have you ever eaten peanuts before? Have you ever eaten peanuts in the shell? What is the purpose of peanuts growing on a peanut plant? How do the peanuts help the plant? • R ead the first four lines of text. What do you notice about the number of peanuts inside the shells? (e.g., They don’t always have the same number of peanuts.) • R ead the fifth and sixth lines of the riddle. What are you supposed to do? Let’s estimate how many individual peanuts there are first. Remember that an estimate is not an exact amount, but a prediction of ‘about’ how many there are. Do you think there are more or less than 50? Why? Are there more or less then 25? Why do you think so? • T urn and talk to your partner and find a way to solve the problem. You can use the BLM to help you see the visual more closely. • W hat did you find? (e.g., We made groups of five; we made groups of 10; we made groups of 15.) How can we represent your strategies with numbers? How do the numbers connect to the picture? • I am going to read the last two lines of the riddle. What strategy is the author suggesting? Try to find a way to pair the rows so each pair has the same number of peanuts. • W hat did you find? (e.g., If you pair the first and last row, the second and second-last row, and the two middle rows, each pair has 13. You can add the 10s for the three groups of 13 first like 10, 20, 30, and then add in the three remaining peanuts for each pair as 30 + 3 + 3 + 3, which equals 39.) • W hich strategy did you find was better for you? Why? Hanging by a Thread • R ead the title of the riddle. Look at the illustration. Why do you think the title is good for this riddle? • R ead the first two lines. What do you think these lines mean? What does it mean to deceive someone? This is very similar to a famous quote that is over 200 years old. The original quote is “Oh, what a tangled web we weave, when first we practise to deceive.” What lesson can be learned from this quote? (e.g., If you are dishonest or lie, it can create other problems that you can get tangled up in.) How is the spider deceiving bugs and flies? • R ead the next four lines. What are you supposed to find out? Turn and talk to your partner and use a strategy to find the number of spiders. You can use the BLM to help you see the spiders more closely. • W hat did you find? (e.g., We counted the number of spiders in each layer and then added like numbers, 6 + 6 + 6, and 7 + 7 + 7. Then we added 18 + 21 by mentally adding all of the tens and then all of the ones to get 39. Then we added in five more from the layer with five spiders, which totals 44 spiders.) Let’s record your solution using numbers. How do these numbers connect to the visual of the spiders? • R ead the last two lines. What patterns do you see as to how the spiders are organized? (e.g., There are seven layers to the web and each layer has seven 266 Number and Financial Literacy

Materials: places for the spiders to sit.) How can subtraction help us find the number of spiders? How can you apply the strategies we used in “For Seven’s Sake” to find out how many possible places the spiders can sit? (e.g., doubling 7, then doubling 14 to include the first four rows, then adding in one more group of 14 and one group of 7) Does this give us the actual number of spiders? What else do we need to do? (e.g., subtract the missing spiders) • W hich strategy did you find worked best for you? Why? Smart Cookies • S how the illustration. What do you see in this picture? What is the same about all of the cookies? How are the cookies organized? Why is “Smart Cookies” a good title for this riddle? What other title might be suitable? • R ead the first six lines. What problem do you need to solve? Turn and talk to your partner about a possible solution. Try different strategies. You can use the BLM to help you. • W hat did you find? (e.g., We counted the chocolate chips by 2s because it is easier for us than counting by 4s.) • D id anyone have a different strategy? (e.g., We found that five cookies have four chocolate chips each so if we count by 5s... 5, 10, 15, 20, there are 20 chocolate chips in every five cookies. We grouped the cookies into seven groups of five cookies, and then counted those seven groups by 20s… 20, 40, 60, 80, 100, 120, 140. Then we added in eight chocolate chips on the remaining two cookies and got 148 chocolate chips.) How can we show these strategies using numbers? • R ead the last two lines. What strategy is the author suggesting that you try? Work with your partner to find a square. • W hat did you find? (e.g., If you look sideways on the page and focus on the diagonal lines there is a square made up of five rows of cookies and five columns of cookies. We know that five cookies have 20 chocolate chips, so we can count by 20s to get 100 chocolate chips. There are four groups of three cookies still left and each group has 12 chocolate chips. You can find the total of those by doubling 12 for two groups to get 24 and then doubling 24 for two more groups, with a total of 48. Then you can add 100 + 48.) • W hich strategy did you find was the best for you? Why? Riddles: “One-Hit Math Talk 2: Wonder,” “Math Potatoes,” “Flat-Jacks,” Math Focus: Composing, decomposing, and recomposing visual quantities to and “Pearly Whites,” make them easier to add and representing the strategies with numbers BLM 48: Find the Total Let’s Talk Select the prompts that best meet the needs of your students. continued on next page Addition and Subtraction 267

One-Hit Wonder • S how the picture and read the title. Why do you think this riddle is called “One-Hit Wonder”? What do they mean by a ‘hit’? What do you know about musical notes? How do they help musicians know what to play on their instruments? • R ead the first four lines. What do you think ‘repertoire’ means? Have you ever heard of the song called “Chopsticks”? It is often the first song that people learn to play on the piano because it is very easy and you only have to use one finger on each hand. • R ead the fifth and sixth lines. What are you supposed to find? Work with a partner to find a solution? You can use the BLM to help you. • W hat did you find? (e.g., We counted by 2s as we added two notes at a time.) • D id anyone count the notes differently? (We made groups of five and counted by 5s; We made groups of 10 and counted by 10s.) How can we represent these strategies using numbers? • R ead the last two lines. Work with your partner and use the strategy suggested by the author. • W hat did you find? (e.g., There were 15 notes in each column so we doubled 15 to get 30, and then doubled 30 again to account for all four rows, which equals 60.) Let’s show this solution with numbers. • C an you use a similar strategy if you look at the rows rather than the columns? (e.g., The rows have 16, 15, 14, and 15 notes. If you move one note from the first row with 16 to the row with 14, then each row now has 15 notes and you can use the doubling strategy.) • W hich strategy did you find was the best for you? Why? Math Potatoes • S how the picture and read the title. Do you ever eat potatoes? What are different ways that potatoes can be prepared and eaten? Let’s compare your ideas to those of the author. Read the first four lines. • R ead the fifth and sixth lines. Work with your partner to find out how many potatoes there are. You can use the BLM to help you. • W hat did you find? (e.g., We noticed that there is an identical pair of potatoes in each of the rows. We added up the potatoes in the first column, which was 25 potatoes and then we doubled 25 to account for the second column of potatoes to get 50 potatoes.) • H ow did you add the potatoes in the first column? (e.g., There are groups of 5, 6, 3, 4, and 7. We know that 6 + 4 = 10 and 3 + 7 = 10, so that equals 20 potatoes. Then we added in the 5 to get a total of 25 potatoes in the first column.) How can we represent your strategy using numbers? • R ead the final two lines. Try solving the problem using the author’s suggested strategy. • W hy is making groups of 10 an effective strategy? • W hich strategy did you find was the best for you? Why? 268 Number and Financial Literacy

Flat-Jacks • S how the illustration and read the title. Pancakes are often called ‘flapjacks.’ Why do you think this might be a good description? Why do you think the author decided to use ‘flat-jacks’ instead? (e.g., it is a play on words; pancakes are very flat) • R ead the first four lines. Have you ever eaten pancakes? What are they like? Have you ever eaten another food that is flat like a pancake? • R ead the fifth and sixth lines. What are you supposed to find? Work with a partner and try out different strategies. You can use the BLM to help you. • W hat did you find? (e.g., The first row has four plates with five pancakes each so that is 5 + 5 + 5 + 5 = 20. The second row has four plates with four pancakes each so we doubled 4 for the first two plates, which is 8 and then doubled 8 to account for the next two plates, which equals 16. We saw another row of four plates with four pancakes each, so we doubled 16, which equals 32. We did the same doubling strategy for the two rows of four plates with three pancakes each, which totalled 24. Then we added 25 + 32 + 24 by adding the tens first and then the ones.) What did you find when you added the ones? (e.g., We got 11 so we added an extra group of ten to the other tens we already added and then had 1 one left.) How can we record your strategies using numbers? • R ead the last two lines of the riddle. What strategy is the author suggesting? What does ‘rounding’ mean? Try this strategy with your partner. If students are struggling, encourage them to look at the columns rather than the rows. • W hat did you find? (e.g., We found that there were 19 pancakes in each row so we rounded them to 20 and then skip counted by 20s… 20, 40, 60, 80. We found that we added one too many pancake for each column so we subtracted 80 – 4 = 76.) Let’s record your strategy with numbers. • W hy is rounding an effective mental math strategy? • W hich strategy did you find worked the best for you? Why? Pearly Whites • S how the illustration and read the title. What does the author mean by ‘pearly whites’? How are pearls made? Pearls are made inside some types of oysters. An irritating item like a piece of sand gets inside the oyster shell and the oyster tries to coat it with a substance. As the oyster coats the irritating item with more and more layers, a pearl is formed. • R ead the first four lines. What does ‘fake’ mean? What method does the author describe for figuring out whether a pearl is real or fake? • R ead the fifth and sixth lines. Work with a partner and try different strategies to find out how many pearls there are. You can use the BLM to help you. • W hat did you find? (e.g., We made groups of 10 and then skip counted by 10. Then we added in the leftover 6 pearls to get a total of 56 pearls.) What other equal groups could you make? (e.g., We made groups of 5 pearls and continued on next page Addition and Subtraction 269

skip counted by 5s, and then added in the extra 1 pearl.) Let’s show your solutions using numbers. • R ead the last two lines. What solution is the author suggesting? Who can explain the strategy in your own words? (e.g., The pearls are in groups of 7. Double 7 to include two groups, which equals 14. Double 14, which is 28 and accounts for four groups. Then double 28 to account for eight groups, which is double the four groups. There are 56 pearls.) How can the previous riddles help us with this strategy? How can we show this strategy using numbers? • W hich strategy worked the best for you? Why? Materials: Math Talk 3: Riddles: “War of the Math Focus: Composing, decomposing, and recomposing visual quantities to Roses,” “The Emperors’ make them easier to add and representing the strategies with numbers New Clothes,” “In a Pickle,” “Cone Beds,” Let’s Talk BLM 48: Find the Total Select the prompts that best meet the needs of your students. War of the Roses • S how the illustration and read the title. Why do you think the author used this title for the illustration? Many centuries, or hundreds of years ago, there were some wars called the Wars of the Roses. Two sides were fighting over who would be the ruling monarch of England. I am going to read the first four lines. Listen for why the title might be appropriate. Read the first four lines and explain what ‘spouse’ means. What do you think? • R ead the fifth and sixth lines. Solve the problem with a partner. Try out different strategies. You can use the BLM to help organize your thoughts. • W hat did you find? (e.g., We made groups of 5 and counted by 5s.) Did anyone try a different strategy? (e.g., We counted how many flowers there were of each colour and then added them together. There were 19 yellow flowers, 25 red flowers, which we counted by 5s, 16 orange flowers, which we counted by 4s, and 15 pink flowers, which we counted by 2s and then added in the 1 remaining flower.) How can we record your subtotals for the different colours in an equation? (19 + 25 + 16 + 15 = ?) What mental addition strategies can you use? (e.g., Combine friendly numbers, for example, 25 + 15 = 40. For 19 + 16, take one from the 16 and add it to the 19 to make 20 + 15, which equals 35. Then 40 + 35 = 75.) When we take away an amount and give that same amount to another addend we are using the compensation strategy. Why does it work? (e.g., You are not taking away or adding anything, you are just redistributing the quantities.) • R ead the last two lines. What strategy is the author suggesting? Try it with your partner. • W hat did you find? (e.g., You can look at the five diagonal rows. Each row has 15 flowers in it. You can add 15 + 15 for the first two rows, which 270 Number and Financial Literacy

equals 30, and then double 30 to account for the third and fourth rows, which equals 60. Then you can add in the final row of 15, which equals 75.) Let’s record this strategy using numbers. • W hich strategy worked best for you? Why do you think so? The Emperors’ New Clothes • S how the illustration and read the title. Why did the author use this title to describe this illustration? These types of penguins are emperor penguins. They are the largest penguins in the world and they can dive deeper than any other birds and all other penguins. They live in Antarctica and mainly eat fish and some other marine animals. • R ead the first four lines. What do you think a ‘tux’ is? It is a suit that is often worn to formal parties. What makes the penguins look like they are wearing a suit? • R ead the fifth and sixth lines. With your partner, find a strategy to find out how many penguins there are. You can use the BLM to help you. • W hat did you find? (e.g., We took the four penguins in the first two groups and put one penguin in each of the remaining groups. Then we had even rather than odd numbers. Then we added 6 + 8 + 10 + 12 by combining 12 + 8, which equals 20, and then 20 + 10 + 6, which equals 36.) Let’s record your strategy using numbers. • R ead the last two lines of the riddle. What strategy is the author suggesting? Look at how the groups of penguins grow in size from the top to the bottom of the page. What pattern do you see? (e.g., There are two more penguins in each group as you move down the page.) Let’s record the sequence of numbers as 1, 3, 5, 7, 9, 11. How can you make equal groups using these numbers? Turn and talk to your partner. • If we combine the first and last group, how many penguins are there? (12) What happens if you continue this pattern? (e.g., We add the second and second-last group, 3 + 9, which also equals 12, then add the third and third last group, 5 + 7, which also equals 12.) How can we finish totalling the penguins this way? (12 + 12 + 12 = 36) • W hy does this work? (e.g., The second group is 2 more than the first group, and the second-last group is 2 less than the last group, so the 2 more and 2 less cancel each other out.) This is like our compensation strategy. The numbers we are dealing with are evenly spaced. If we add two when we work from the top and we subtract two when we work upwards from the bottom, we are just redistributing the 12 penguins. • W hich strategy worked best for you? Why do you think so? In a Pickle • S how the illustration and read the title. Have you ever eaten pickles before? How would you describe their taste? • R ead the first five lines of the riddle. Work with your partner. Try different strategies to find out how many pickles there are. You can use the BLM to help you. continued on next page Addition and Subtraction 271

• W hat did you find? (e.g., We saw four groups of pickles with either two rows and four columns or four rows and two columns. There is one at the top of the configuration, one at the bottom, and one on either side. So, we doubled 8 to get 16 for the first two groups and then doubled 16 to account for all four groups, which is 32. Then we added in the 12 pickles in the centre that we hadn’t counted yet. So, 32 + 12 = 44.) How can we show your strategy with numbers? • R ead the rest of the riddle. What strategy is the author suggesting? Try it with your partner. • H ow did you total the eight groups of eight? (e.g., We used the doubling strategy. 8 + 8 = 16 and accounted for two groups, then 16 + 16 = 32, which accounted for four groups, then 32 + 32 = 64, which accounted for eight groups. Then we needed to subtract the 20 missing pickles, so 64 – 20 = 44.) Let’s represent this strategy using numbers. • W hich strategy did you prefer? Why? Cone Beds • S how the illustration and read the title. What kind of trees have cones? What are pine cones and what purpose do they have for the trees? • R ead the first five lines. Work with your partner to figure out how many cones there are. Use the BLM to help you. • W hat did you find? (e.g., We counted up all of the green cones and there were 29. Then we counted all of the brown cones and there were 34. Then we added 29 + 34.) How did you mentally add the numbers? (e.g., We took one from 34 and gave it to 29 so we were now adding 30 + 33, which is 63.) Why does this strategy work? (e.g., You are not adding or taking any cones away. You are just reorganizing them by taking one cone from the group of 34 and putting it with the group with 29.) Let’s show your strategy using numbers. • R ead the rest of the riddle. What strategy is the author suggesting? Explain it in your own words. Try using this strategy with your partner. • W hat did you find? (e.g., We moved the cones around and made nine rows with seven cones in each. Then we added 7 + 7 = 14 for the first two rows, doubled 14 to account for the next two rows, which is 28. Then we doubled 28 so we have now accounted for eight of the nine rows, which equals 56 cones. Then we added in the ninth row as 56 + 7 by decomposing the 7. We added 56 + 4 is 60 and then 60 + 3 = 63.) Let’s show your strategy using numbers. • W hich strategy worked best for you? Why? 272 Number and Financial Literacy

and2 3Lessons Solving Addition and Subtraction Problems Using a Variety of Strategies Math Number Curriculum Expectations • B 2.3 use mental math strategies, including estimation, to add and subtract whole Previous Experience numbers that add up to no more than 1000, and explain the strategies used with Concepts: In grade two, students • B 2.5 represent and solve problems involving the addition and subtraction of solved addition and subtraction problems whole numbers that add up to no more than 1000, using various tools and involving two-digit algorithms numbers with and without regrouping, using A bout the concrete materials and various algorithms. They In grade two, students acquired several mental strategies for adding and have developed some subtracting numbers to 50. Although students could apply and adapt several of mental math strategies these strategies to mental calculations with larger numbers, as Alex Lawson for adding and explains, they often revert to simpler and less efficient strategies as numbers subtracting whole increase in magnitude and become less familiar (Lawson, 2015, p. 84). It is numbers to 50. therefore important to support students’ learning with concrete objects and visuals, so they can create mental images they can retrieve later to help carry PMraotcheesmseast:ical out mental math strategies. It is best to find out what students already know so Problem solving, any gaps in understanding or misconceptions can be addressed. arreenfadlessocttnrianintgge,gasieenlsde,cptrionvgintogo, ls ccoomnnmecutninicga,trinegpresenting, About the Lessons Math Vocabulary: In Lesson 2, students solve an addition and subtraction problem involving aspsduaedbrp,ttar,aarwadchtdteioiot,lineoc,,onejm,ossitpnuiam,brtearatec,t, two-digit numbers. In Lesson 3, students extend this understanding to solve addition and subtraction problems involving three-digit numbers. During the Consolidations, they describe their strategies and discuss the connections among them and how they can support mental math calculations. This is a good opportunity to assess what your students know and what concepts and skills need reinforcement. During the lesson, you can reinforce the mathematical modelling process as they develop and refine a model to represent the math in the problems. The four components of the process are: • U nderstand the Problem • A nalyse the Situation • C reate a Model • A nalyse and Assess the Model continued on next page Addition and Subtraction 273

Use an anchor chart to highlight how students move among the four components. For example, if students find that their model does not adequately represent the math, they may need to revisit the problem (Understand the Problem) or reconsider the parameters surrounding the problem (Analyse the Situation) in order to gather new information and select more appropriate tools and strategies. Throughout Lesson 2, there are suggestions about how and when to reinforce these ideas, although these will need to be adapted so they are responsive to the way in which your students are progressing through the process. 274 Number and Financial Literacy

2Lesson Solving Problems with Two-Digit Numbers Teacher Possible Learning Goals Look-Fors • Relates addition and subtraction to appropriate real-life situations • Solves addition and subtraction problems using a variety of tools and strategies • U ses strategies (e.g., composing, decomposing) to accurately add and subtract • Uses concrete materials, diagrams, numbers, and/or words to solve problems • Explains their strategies and why they work Materials: Minds On (10 minutes) concrete materials • Read one of the following problems aloud and display it in written form. and tools (e.g., coloured tiles, counters, ten Change the context so the problems are more meaningful for your students. frames, base ten blocks), Ensure all students understand what is being asked. (Understand the chart paper, markers Problem) – P roblem 1: There are 19 students in the grade two classes, and 47 Time: 60 minutes students in the grade three classes. How many students are in grades two and three? – P roblem 2: There are 63 students in grade four and 47 students in grade three. How many more students are in grade four? • Ask what other information students may need in order to solve this problem. For example, you may ask whether it is important to know how many classes of each grade there are in order to solve the problem. (Analyse the Situation) Working On It (25 minutes) • Students work in pairs to solve the problem. Let students know that they can use any math tools they find helpful. Brainstorm a list of available tools in the classroom. (Create a Model) • Tell students they will need to record how they solved the problem as well as the solution. • Encourage students to solve the problem in more than one way. Differentiation • If students are stuck (e.g., not sure where to start, having difficulty with the numbers), offer a tool that may help them and ask, “How could you use this tool to help you solve the problem?” Addition and Subtraction 275

• Change the numbers in the problems so they are within the understanding of your students. • For students who need more of a challenge, offer an extension to the problem. For example, in the first problem, add “There are 26 students in each of two grade 4 classes. How many more students are there in grades two and three?” Assessment Opportunities Observations: • Observe how students approach the problem. Do they know how to start? Can they select an appropriate tool to help them? They may need to revisit the problem or reanalyse the situation in order to fully understand what they are to do. • Note the strategies students are using. See the Consolidation for possible strategies for each problem. Conversations: Since this is an introductory lesson, minimize intervention unless students are struggling. Pose prompts to ensure they understand the question, such as: – W hat are you being asked to find? – D o you think the solution will be more or less than the numbers in the problem? Why? – W hat operation might you consider using? – W hat tools could help you? What do you think you could try? Consolidation (25 minutes) • Strategically select solutions that reflect different thinking and representations (e.g., concrete materials, pictorial representations, numerical solutions, and mental math strategies). • Have students share and explain their solutions. Ask clarifying questions, such as “Why did you choose to add? What does this number represent? Can you explain this part of your drawing? What part of this solution did you figure out in your mind?” Make connections between the various representations so students can see how the same thinking can be represented in different ways. Possible Strategies for Problem 1: – D irect modelling, adding three times: represents both amounts with concrete materials, either on ten frames or in piles, combines them, and counts again – D irect modelling, counting on from the larger number: puts out 47 objects and counts on, adding one more object at a time until 19 objects have been added, tracking added items on fingers or with tallies – U sing a hundreds chart: starts at either 19 or 47 and then counts on, either by 1s or by taking jumps of 10s and 1s, and tracks jumps at the 276 Number and Financial Literacy

Materials: same time; starts at 47, takes two jumps of 10, and then counts back one chart paper (20 − 1 = 19) – U sing an open number line: starts at 19, takes one jump to 20, then takes Teaching Tip four jumps of 10 to 40, then adds 6 more – U sing mental calculations: Takes one from 47 and gives it to 19 to make Integrate the math 20, then adds 20 + 46 by adding the tens and ones separately and then talk moves (see combining them page 8) throughout Math Talks to Possible Strategies for Problem 2: maximize student – D irect modelling, counting three times: counts out 63 concrete objects, participation and removes 47 objects counting them by 1s, counts the remaining objects active listening. – D irect modelling, counting on (think-addition): counts out 47 objects, adds objects until they reach 63, then counts the number of objects added – U sing a hundreds chart: starts at 63 and counts back to 47; or, starts at 47 and takes jumps of 10s and 1s to get to 63, keeping track of the jumps taken – U sing an open number line: starts at 47 and takes one jump of 10 and then six jumps of 1 to 63, keeping track of the number of jumps – U sing mental calculations: counts on from 47 to 63, tracking the number of jumps on fingers • Make an anchor chart of students’ strategies and highlight the connections among them. For example, show how displaying concrete materials is like drawing items that represent the quantities, and how both can be represented with a hundreds chart or numbers. Highlight any mental decomposing or composing students did and where they see their thinking in drawings and concrete materials. • Ask students how they might solve this problem differently now that they have heard each other’s ideas and strategies. Discuss which approaches were most effective for them and why they think so. (Analyse and Assess the Model) Math Talk: Math Focus: Estimating to judge the reasonableness of a solution Let’s Talk Select the prompts that best meet the needs of your students. • W hat does it mean when we estimate? (e.g., We predict about how much or how many based on what we know.) When might we estimate in our everyday lives? (e.g., when we go to the grocery store and we want to know about how much money we spent) Why is this important? continued on next page Addition and Subtraction 277

Estimating a Sum • L isten to this problem: There were 28 students in the gym. Half an hour later, 44 students joined them. About how many students are in the gym now? Write the problem on chart paper. • W hen the question asks “about how many,” what kind of answer do we need to find? Turn and talk to your partner about how you would estimate the number of students in the gym. Be prepared to explain your reasoning. As students share, record their estimation strategies on chart paper to create an anchor chart. • W hat do you think? (e.g., We estimated about 70 students are in the gym. We thought that 28 was close to 30 and that 44 was close to 40, so 30 + 40 = 70.) Put up your thumb if that estimate sounds reasonable. Why do you think 70 is a friendly number for estimating? • D id anyone have a different estimate? (e.g., We estimated about 75 students were in the gym. We thought that 28 was close to 30 and that 44 was close to 45, so 30 + 45 = 75.) Put up your thumb if you think this estimate is rea- sonable. Why did you round to 75? (e.g., We think it is a friendly number since we often count by 25s, like when we count quarters when dealing with money.) Estimating a Difference • L isten to this problem: There were 97 students in the gym during lunch. 43 of the students returned to their classes after lunch. About how many students were left in the gym? Write the problem on chart paper. Turn and talk to your partner to find an estimate. • W hat did you find? (e.g., We thought there would be about 50 students, because 97 is close to 100 and then 43 is close to 50, so 100 − 50 = 50.) Put up your thumb if this estimate seems reasonable. • D id anyone estimate differently? (e.g., We estimated 50 too, but we did it differently. We thought that 97 was close to 95 and 43 was close to 45. We rounded 97 down, so we thought we would round 43 up to balance it. Then we had 95 − 45, which is 50.) Put up your thumb if this makes sense. So, you found the same estimate, but in different ways. • D id anyone have a different estimate? (e.g., We thought 60 was a good estimate. We thought that 97 was close to 100, but 43 was closer to 40 than 50, so we subtracted 100 − 40, which equals 60.) Does this estimation seem reasonable? Why? (e.g., We rounded up 97, so we thought we could round down 43 to make up for it.) • In all cases, what statement could we make to give people a general idea of how many students are left in the gym in relation to the original amount? (e.g., about half) • W hy is it a good idea to estimate before we figure out all problems? (e.g., to see if our answer is close to our estimate and makes sense) 278 Number and Financial Literacy

3Lesson Solving Problems with Three-Digit Numbers Teacher Possible Learning Goals Look-Fors • Solves word problems involving the addition and subtraction of three-digit numbers using a variety of strategies, materials, and tools • E xplains their strategies and why they work, using mathematical language • U nderstands the problem and identifies an operation and strategy to solve it • Reads three-digit numbers, and understands how much a digit represents according to its place value position in the number • Represents three-digit numbers using concrete materials or drawings • Accurately adds amounts by combining and counting them, using counting methods such as skip counting by 100s, 10s, and 1s • Composes three-digit numbers into groups of hundreds, tens, and ones • Estimates to judge the reasonableness of their solution Materials: Minds On (10 minutes) concrete materials • Present one or both of the following problems, ensuring that all students and tools (e.g., base ten blocks, place value understand the content and what is being asked. You can change the context mats [see BLM 2], of the problems to make them more meaningful for your students. hundreds charts [see BLMs 3 and 44], number – P roblem 1: There are 122 people in the gym when the bell rings. After lines), chart paper the bell, 345 students join them for an assembly. How many students are in the gym? Time: 55 minutes – P roblem 2: On another day, 465 students attend an assembly. After it is over, 213 students return to their classrooms. How many students are still in the gym? • Inform students that they can use any materials or tools that are available. Brainstorm some of the choices that they have (e.g., base ten blocks, place value mats, hundreds charts, counters, number lines). • Encourage students to solve the problem(s) in more than one way. Working On It (20 minutes) • Students work in pairs or in groups of three. Give students chart paper and markers to record their solutions. They can select any other materials and tools they need. Encourage them to solve the problem(s) in more than one way and to clearly show their work so their peers will understand their strategies. Addition and Subtraction 279

Differentiation • C hange the numbers so the level of difficulty is appropriate for your students, but keep the context the same. • If students are stuck, offer materials or tools such as base ten blocks or number lines, and ask students how they might use the materials to help them solve the problem. Try to refrain from directing them what to do. • For students who need more of a challenge, vary the structure of the word problem (e.g., 345 + ___ = 756). • For students who need more of a challenge, have them create their own word problem to solve. They can exchange problems with another pair and solve, and then compare their solutions. Assessment Opportunities Observations: • N ote how students approach the problem. Do they discuss the problem first? Do they have a way to begin? Do they select a tool to help them? • If a group is completely off track, use questioning to get them back on track, but avoid leading them in a specific direction. • A s you observe students at work, note what strategies they use. Students often use a combination of possible strategies, including: – d irect modelling and counting all – c ounting on or back – c ounting on or back from the larger number – s kip counting forward or backward by 10s and 100s – d ecomposing and adding or subtracting partial sums – k eeping one number whole and taking jumps of 100, 10, and 1 forward or backward – compensation (e.g., jumping too far to a friendly number and then adjusting) Conversations: As students work, you may need to ask a few questions to understand their strategies. Sometimes their thinking on paper is not clear, but when explained orally, their ideas make sense. Use prompts such as: – E xplain your strategy. Did you do something before this representation? How could you show what you did? – W hat does this part of your diagram represent? What could you add to make it clearer to understand? – H ow could you show what operation you are using in your solution? Consolidation (25 minutes) • Read the problem again and have students estimate the answer. Remind them that their estimation is not the actual amount, but ‘about’ how much the answer will be. Discuss their estimation strategies. 280 Number and Financial Literacy

• Strategically select solutions so you have a variety of strategies and representations to discuss, including solutions that use concrete materials, pictorial representations, and numbers. Possible Solutions for Problem 1: – R epresenting both amounts with base ten blocks, putting them together and counting three times – R epresenting the larger quantity numerically and then counting on by adding base ten blocks, then counting the materials that were added – R epresenting the amounts with pictures that unitize the numbers as hundreds, tens, and ones (e.g., a square with 100 written on it) rather than depict them individually (e.g., 100 dots), then counting up or adding the two amounts – U sing a number line to count jumps of 100, 10, and 1 – U sing an invented algorithm and decomposing the actual numbers along place value lines – U sing an invented or standard algorithm to work directly with the numbers Possible Solutions for Problem 2: – R epresenting the whole amount with base ten blocks, removing the blocks that represent the amount being taken away, counting up the remaining base ten blocks – R epresenting the partial amount with base ten blocks, adding blocks to reach the whole amount, counting the blocks that were added – R epresenting the amounts with pictures that unitize the various place value groups; either representing the whole and removing a part, or representing a part and then adding up to the whole – U sing a number line to count jumps of 100, 10, and 1, either by counting back or adding on – U sing an invented or standard algorithm to work directly with the numbers • Since considerable time is devoted to learning about number lines and algorithms later in the unit, focus the discussion around the use of concrete materials and drawings and how they represent the problem. Make some connections to the other strategies (e.g., How is adding a tens rod like skip counting by 10s?). Make a list of the strategies discussed in class and inform students they will explore them in upcoming lessons. • Have students compare their answers to their estimations. Discuss how an estimation can help them determine whether their solution is reasonable. • Building Social-Emotional Learning Skills: Healthy Relationship Skills: Discuss how students felt when they may have seen their peers using strategies that they don’t understand or tools that they have never used before. Reassure them that they do not have to be concerned, because they are going to have plenty of time in the lessons ahead to try out the tools and materials and to better understand the strategies. Pair up students who use different tools and/ Addition and Subtraction 281

or strategies. They can take turns explaining their strategies and how they used their tools. Before meeting, discuss how they can explain their ideas and how they can help their partners understand if they don’t immediately catch on to the ideas. Highlight how we can learn from each other, especially if we take the time to listen to each other and ask questions. 282 Number and Financial Literacy

4Lesson Mental Strategies that Use the Commutative and Associative Properties Math Number Curriculum Expectations • B 2.1 use the properties of operations, and the relationships between Teacher multiplication and division, to solve problems and check calculations Look-Fors • B 2.3 use mental math strategies, including estimation, to add and subtract whole Previous Experience with Concepts: numbers that add up to no more than 1000, and explain the strategies used Students investigated the commutative property in Possible Learning Goal grade two. • Applies the commutative and associative properties to mentally add two or more addends by reordering or composing and decomposing the numbers • U nderstands that the context of the problems requires adding • Selects and uses a reasonable strategy and explains how it works • Adds numbers in various orders and explains why the order does not change the sum • Composes and decomposes numbers to create benchmark or friendly numbers • Explains their strategy and why it makes addition easier PMraotcheesmseast:ical About the Problem solving, csscroteeormalanestnmceoetgniucniintengiinsgct,gaoar,toenirnlpedsgrfelapesnrcoedtvninitnigng,g, , Applying the commutative and associative properties can help students work more efficiently when mentally adding and subtracting two-digit numbers. With the commutative property, numbers can be added in any order and the sum will be the same. Students often use this principle when adding two numbers, by starting with the larger number even when it is the second number in a problem. They can then count or add on the quantity represented by the first number. For example, for 23 + 165, students may start with 165, count on by 10s two times, and then count on by 1s three times. continued on next page Addition and Subtraction 283

Math Vocabulary: In grade three, students study the associative property, which involves the idea dcapeonrcmodopmmaesruptstyooasc(toeiiva,pettciivopoemrnoappl)oesrtey, that three or more numbers can be regrouped in different ways and added and the sum will be the same. As Marian Small explains “as a consequence of this property, you can take away from one number and add what you took away to the other number without changing the sum” (Small, 2009, p. 109). For example, when adding 59 + 24 + 11, students can subtract 1 from the 11 and add it to 59, making 60 + 24 + 10. They can then add 60 + 10, and then finally add 24. This allows students to make friendly numbers by decomposing and recomposing. It is important that students understand and can apply the properties, although they do not need to know their formal names. About the Lessons In this lesson, students review the commutative property and are introduced to the associative property. During the Working On It, students investigate the properties by mentally solving problems. For further practice, repeat this lesson using three-digit numbers in order to reinforce mental math strategies for adding and subtracting numbers to 1000. Materials: Minds On (15 minutes) chart paper, markers, • Pose this problem: BLM 49: Addition Challenges – T here are 12 cans of tomato sauce on the shelf. The storeowner adds 36 cans of soup beside them. How many cans are there now? Time: 60 minutes • Ask how this information could be recorded using an equation (e.g., 12 + 36 = __). Discuss whether students think it matters if we add the numbers in a different order and why they think so. Review the commutative property. Ask why you might want to add the 36 first if you are calculating this in your mind. (e.g., Starting with the larger amount can be easier.) Record the meaning of the commutative property on an anchor chart. Explain that this is a proven rule and that it works every time. • Repeat this line of questioning with a problem that involves three quantities, perhaps by adding more cans to the problem above (e.g., 12 + 36 + 10). Ask students if they think it makes a difference to the sum if they regroup the numbers differently (e.g., group together and add 12 + 36 and then add 10, or group together and add 36 + 10 and then add 12). Work through the problem together to prove that the sum does not change. Inform students that this is the associative property. Create a definition and add it to the anchor chart. • Inform students that they are going to solve some problems and they can use the commutative and associative properties to make the calculations in the solutions easier. 284 Number and Financial Literacy

Working On It (20 minutes) • Students work in pairs. Provide them with chart paper, markers, and BLM 49: Addition Challenges. Read the challenges together to ensure everyone understands them. • Have students solve the challenges, which include creating their own problem (Challenge #5). Differentiation • Change the quantities in the problems to suit the needs of your students. • Limit the number of challenges to one or two if completing them all is overwhelming for any students. Assessment Opportunities Observations: Pay attention to the strategies students are using. • D o they apply the commutative property? • A re they decomposing and recomposing numbers so they are easier to work with? • A re they regrouping any numbers to make calculations easier? • A re they drawing pictures to help them with their thinking, or are they using mental math strategies without concrete materials or pictures? Conversations: If students are adding numbers in only one way, pose some of the following prompts: – H ow did you add your numbers? Did your partner add them the same way? – Is there another way that you could add the numbers? – W hat friendly number could you make from one of the numbers? How could you do that? Consolidation (25 minutes) • Have student pairs meet with another pair. Pairs can compare solutions and strategies and see if they found the same sums for Challenges 1–4. Have the pairs exchange the problems they created (Challenge 5) and solve them. Students can compare their solutions to see if they found the same answer. • As a class, discuss how the two properties (commutative and associative) helped them with their calculations. • Focus on one problem and strategically select and display as many different student strategies for solving it as are available. For each, have the students who used the strategy explain it. Addition and Subtraction 285

Possible Strategies for Challenge #4: – A dd the 3 from 13 to 27 to make 10 + 30, which equals 40; add 1 from 22 to 39 to make 21 + 40; now add the two 40s, which equals 80; finally, add in the 21 to equal 101. 13 + 39 + 22 + 27 39 + 1 + 21 10 + 3 + 27 40 + 21 10 + 30 40 40 + 40 + 21 80 + 21 101 – A dd all the tens, which is 8 tens or 80; for the ones, add 3 + 7 and decompose 9 + 2 into 9 + 1 + 1, which recomposes into 10 + 10 + 1; add 80 + 20 and then 1. 13 + 39 + 22 + 27 10 + 30 + 20 + 20 3 + 9 + 2 + 7 3+7 9+1+1 80 10 + 10 + 1 80 + 20 + 1 100 + 1 101 • Discuss with students which solution makes the most sense for them. Further Practice • Repeat the lesson using three-digit numbers in order to reinforce mental math strategies for adding and subtracting numbers to 1000. 286 Number and Financial Literacy

5Lesson Using Patterns in the Hundreds Chart to Add and Subtract Math Number Curriculum Expectations • B 1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and Teacher strategies Look-Fors • B 2.3 use mental math strategies, including estimation, to add and subtract whole Previous Experience with Concepts: numbers that add up to no more than 1000, and explain the strategies used Students have had experiences adding and • B 2.5 represent and solve problems involving the addition and subtraction of subtracting one-digit numbers and two-digit whole numbers that add up to no more than 1000, using various tools and numbers. They should also algorithms have an understanding of the place value of digits in Algebra a number. • C 1.4 create and describe patterns to illustrate relationships among whole numbers up to 1000 Possible Learning Goals • Uses hundreds chart patterns (i.e., base ten patterns) to add and subtract • Mentally decomposes numbers into tens and ones to add and subtract • S elects appropriate strategies to add and subtract on a hundreds chart (e.g., keeps one number whole and takes leaps by 10s, then counts the 1s) • Uses information from the hundreds chart to determine how to move in the appropriate direction when adding and subtracting tens and ones (e.g., notices number patterns such as numbers increasing by 1s left to right in a row) • Explains movements and what they mean when describing their thinking (e.g., I moved down 5 tens to add 50 and left 2 ones to subtract 2.) • Starts from any number and adds on or subtracts any two-digit number, using the hundreds chart or mental strategies • Begins to apply the patterns to mentally solve addition and subtraction problems Addition and Subtraction 287

PMraotcheesmseast:ical About the Problem solving, rssreeterpalaersteceostgnieniinengtsgit,noacgono,ldnsnpaerncodtviinngg,, Recognizing the patterns of tens and ones in our base ten number system communicating allows students to use the strategy of counting by leaps of 10 and 1 when adding and subtracting. This is a flexible strategy that doesn’t require Math Vocabulary: regrouping, as do some traditional algorithms. In grade two, students learned hounnedsr,eodpsecrahtaiortn,,taednds,, to skip count by 20s, 25s, and 50s to 200. Most students have no problem subtract counting forward by 10s or adding on a ten when beginning with a ‘decade number’ (e.g., 20, 30, 40). Initially, however, they may find it more challenging to count forward or backward by 10s when adding on to, or subtracting from, a number that is not a decade number (e.g., 42, 56). Students need time and experiences to develop the key mathematical idea that, in our place value system, the tens digit increases by one and the ones digit stays the same with each ten that is added. This understanding extends beyond reading off a pattern in the hundreds chart for 1–100 (Lawson, 2015, p. 98). Students can then extend these patterns to add and subtract three- digit numbers using other hundreds charts (101–200, 201–300, etc.). The hundreds chart offers an effective visual that can help students identify the tens and ones patterns. Students can begin by physically using the hundreds chart as a tool to add and subtract. An eventual goal is that they internalize the process and visualize the patterns so they can mentally count by tens and ones, without having to refer to a hundreds chart. About the Lesson In this lesson, students use the patterns in the hundreds chart to add and subtract. They are also encouraged to visualize the hundreds chart so they can internalize the process and start solving related problems using mental strategies. Materials: Minds On (20 minutes) Digital Slide 1: • Show Digital Slide 1. Ask students where they see groups of 10 represented in Hundreds Chart (1–100), BLM 3: Hundreds Charts the hundreds chart (e.g., 10 rows, 10 columns, 10 numbers in each row, 10 to 500, two-sided numbers in each column, 10 numbers in the centre diagonals). counters, other hundreds charts (see Digital Slides • Ask how the numbers change horizontally in the hundreds chart. (e.g., As 2–5 and/or BLM 44). you move from left to right, the ones get bigger and the tens stay the same; Time: 60 minutes as you move from right to left [starting in the second-last column] the ones get smaller and the tens stay the same.) • Ask how the numbers change vertically. (e.g., As you move down, the tens get bigger and the ones stay the same; as you move up, the tens get smaller and the ones stay the same.) • As students respond, highlight observations on the hundreds chart. Ask what operation is represented when you move to the right or move down on a hundreds chart and why they think so. (e.g., It represents addition because numbers increase.) Ask what operation is represented when moving to the left or up and why. (e.g., It is subtraction, because numbers are decreasing.) 288 Number and Financial Literacy

• Provide each student with a counter and a hundreds chart (1–100, from BLM 3). Tell students they will be adding and subtracting using the hundreds chart. Read the following instructions below, one line at a time, as students move their counter on their hundreds chart. After each instruction, check to see whether students have placed their counter on the correct number. Ask how the movement changed the number. (e.g., When moving straight down one row/line, the ones stay the same and the tens increase by 1 ten.) Instructions: Start at 20. (20) Add 30. (50) Subtract 5. (45) Add 30. (75) Subtract 9. (66) Add 21. (87) • Write the problem 20 + __ = 87. Explain that this represents the movement from the first number to the last number. Have students turn and talk with a partner about how they could solve this problem in fewer moves. Encourage them to find two or more ways. • Discuss their strategies and trace the moves on a large hundreds chart as students explain their solutions. In each case, ask how the number changed from one move to the next. • Show students other hundreds charts such as 101–200 and 201–300. Ask how adding and subtracting will be the same and different when using these charts with larger numbers. Working On It (20 minutes) • Students work in pairs. Give students problem to solve using hundreds charts with larger numbers. Some examples are given below: 137 + 46 = __ 257+ 39 = __ 372 − 67 = __ 287 + 138 = __ 326 − 147 = __ • Students can use the various hundreds charts and a counter and record their moves directly on the hundreds charts using different colours. Differentiation • Adjust the numbers so they are most appropriate for your students. Addition and Subtraction 289

Assessment Opportunities Observations: Pay attention to whether students see how the patterns in the 1–100 chart continue in the other hundreds charts. • C an they transfer knowledge of the patterns to add and subtract with three-digit numbers? • Can they carry out the mental calculations over the hundreds, from one chart to the next? Conversations: You can use the following prompts to help students recognize the patterns in the counting system: – How can you add 47 + 39 on the hundreds chart? How can you count by tens and then compensate? (e.g., count forward by 10s four times (40) and then count back one space) – Let’s change the problem to 147 + 39. Where is 147 on the 101–200 chart? What do you notice about its position compared to 47? How could you use the same strategy to add on 39? – H ow would this work when adding 347 + 39? What patterns do you see? Consolidation (20 minutes) • Have students meet with another pair to compare solutions. • Meet as a class. Discuss students’ strategies for the given problems. • Highlight how students either added or subtracted using more than one hundreds chart. • Discuss how the patterns continue from hundreds chart to hundreds chart. Math Talk: Teaching Tip Math Focus: Adding and subtracting multi-digit numbers by visualizing a hundreds chart Integrate the math talk moves (see Let’s Talk page 8) throughout Math Talks to maximize Select the prompts that best meet the needs of your students. student participation and active listening. • W e have been adding and subtracting using the hundreds chart to help us. In real life, it’s not always possible to have a hundreds chart in our pockets. Imagine you were out shopping and you wanted to add up the prices. You wouldn’t have a hundreds chart with you, but you can imagine or visualize the hundreds chart to help you. Let’s try it together. What does 49 + 21 look like? Close your eyes and try to solve the problem by visualizing the hundreds chart. Put your thumb up when you have a solution. Open your eyes. • Wspahcaet did you visualize first? (e.g., I started in the middle right of the chart, one before 50.) Everyone try and visualize what was described. 290 Number and Financial Literacy

• Wdohwant do you visualize doing next? (e.g., I can visualize taking two jumps of 10 the chart from 49 to 59 to 69.) Everyone visualize those moves. You can make the gestures with your finger. • tWo h7a0.t)dVidisuyoaulizveistuhaaltizaenndesxhto?w(e.tgh.e, I moved one space over to the right from 69 movement with your finger. • Dmiodvainngyotone5d0.o) it a different way? (e.g., I first added the 1 from 21 to 49 by Let’s visualize and gesture with our finger how we moved. • 2W0.h)aDt iddidyoyuoudodoitninexot?ne(eo.gr.,twI owemnotvdeos?w(ne.gth.,eIcuhsaerdt from 50 to 70 to add in the one move, because I know that 50 + 20 equals 70.) Let’s show that move with our fingers. • H ow would your strategy change if you were adding 149 + 121? What would be the same and what would be different? Addition and Subtraction 291

6 8Lessonsto Using an Open Number Line to Add and Subtract Math Number Curriculum Expectations • B 1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 2.3 use mental math strategies, including estimation, to add and subtract whole numbers that add up to no more than 1000, and explain the strategies used • B 2.5 represent and solve problems involving the addition and subtraction of whole numbers that add up to no more than 1000, using various tools and algorithms About the The open number line is a valuable tool for performing math calculations. It must be explicitly introduced, since “number-line related strategies will only PMraotcheesmseast:ical appear in classrooms that foster them” (Lawson, 2015, p. 89). The number line cctPoooromnolnbsmeleacumntnidnicsgsao,ttlrrvianeintgpegrge, isseeesnl,eticntgin,g is a tool that supports splitting, or decomposing, numbers into groups like hundreds, tens, and ones, and using ‘jumps’ to add and subtract. For example, if a student is adding 49 + 32, she/he might start by recording 49, then saying and showing, “I started at 49, then I jumped 3 tens to 79 because 32 has 3 tens in it. Then, I jumped 2 more because there were 2 extra ones remaining from the 32.” + 10 + 10 + 10 +1 +1 49 59 69 79 80 81 In this example, the strategy involves keeping one addend whole (49), decomposing the other addend (32) into tens and ones, and taking leaps of 10s and then 1s. Strategies that students use may include •• mkeoevpiinngg one number whole and then counting on or back; number, forward or backward to the nearest decade or hundred • mthoenvitnagkfionrgwaadrdditoiornbaalcjkuwmaprds;to other friendly numbers and then taking • ajudmdipnsginorfasumbitlriaacrtiinncgremmoernetos;r less than needed and then compensating by • aaddddiinngg or subtracting the extra amount; or subtracting the same amount from each number to keep the • addifdfeinregnlciekebeptlwaceeenvatlhueemgrcoounpssta(en.tg;.,ahnudndreds first, then tens, and then ones). 292 Number and Financial Literacy

Math Vocabulary: The number line can also help students visually see that subtraction involves ohpuenndrneudms bchear rltin, eju, mp finding the difference between two numbers, and related problems can be solved using either addition or subtraction strategies. When students use ‘think- addition’ to find differences, it enhances their understanding of the inverse relationship between addition and subtraction. About the Lessons In Lesson 6, students are introduced to using a number line to add and subtract two-digit numbers, with problems that traditionally do or do not require regrouping. In the Minds On, students are introduced to the open number line and how they can use it to record their strategies as they add. In the Working On It, students solve problems that require addition, drawing their own open number lines to find and represent their solutions. In the accompanying Math Talk, students mentally solve subtraction questions and use the number line to record their strategies. In Lessons 7 and 8, students add and subtract three-digit numbers using the open number line. Addition and Subtraction 293

6Lesson Adding and Subtracting Two- Digit Numbers on an Open Number Line Teacher Possible Learning Goal Look-Fors • Solves two-digit addition and subtraction problems using an open number line • Uses the number line effectively to solve problems • Decomposes numbers to make friendly numbers • Takes leaps of ten and more on a number line and orally counts the movements • Explains mathematical ideas and solutions orally with peers using appropriate vocabulary • Represents the strategy used by drawing the jumps, and recording the numbers and operations the jumps represent Materials: Minds On (20 minutes) BLM 8: Open Number Lines, BLM 3: Hundreds • Show students the following problem: 23 + 58. Ask how they could solve it in Charts to 500 Time: 60 minutes their minds. Have them turn and talk to a partner to discuss their ideas. +10 • Explain that you are going to show them how they can use an open number line to record their thinking. Draw five identical open number lines stacked vertically (one above the other) and write “0” at the far left on each. Explain that, unlike a closed number line that has all the numbers and increments recorded, an open number line has only the numbers needed to solve a problem and represent the thinking. • Record the jumps involved in different solutions on different number lines, so the solutions can be compared. Some sample solutions, prompts, and recordings are given below. – W hat did you do in your mind? (e.g., I started at 23 and skip counted by 10s five times to 73, and then I counted by 1 eight times.) Which direction will we move on the number line? How do you know? Where do we start? (0) +10 +10 +10 +10 +1 +1 +1 +1 +1 +1 +1 +1 0 23 33 43 53 63 73 74 75 76 77 78 79 80 81 294 Number and Financial Literacy

– H ow would our thinking look different if we added 50 to 23 all at once? (e.g., one curved arrow with +50 above) How is the thinking represented on the two number lines the same/different? +50 +1 +1 +1 +1 +1 +1 +1 +1 0 23 73 74 75 76 77 78 79 80 81 +20 – W hat is another way to solve the problem? (e.g., I added all the tens first and then all the ones, so I jumped from 0 to 20, and then added 50 by skip counting 30, 40, 50, 60, 70, then I did 3 counts of one jump each and then 8 counts of one jump each.) +10 +10 +10 +10 +10 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 0 10 20 30 40 50 60 70 71 72 73 74 75 76 77 78 79 80 81 – D id anyone do it slightly differently than that? (e.g., I did the same with the tens, but added the ones (3 and 8) differently. I knew that 8 + 2 equals 10 so I added on 10 more in one jump from 70 to 80, and then one more single jump for the leftover 1 from the 3.) How will that look on the number line? +20 +10 +10 +10 +10 +10 +10 +1 0 10 20 30 40 50 60 70 80 81 – Is there another way? (e.g., I started with 58 because it is the larger number and then added 2 ones from the 23 to make 60. Then to add on the 20 in the remaining 22, I skip counted from 60 as 70, 80, and then I added 1 more left over from the 3 in 23.) How would I record that thinking? +2 +10 +10 +1 0 58 60 70 80 81 • Compare the thinking on the number lines. Highlight how the numbers were decomposed. Working On It (20 minutes) • Students work in pairs. Provide each pair with two copies of BLM 8: Open Number Lines. Tell students they are going to mentally solve two problems in as many different ways as possible and record their thinking on the open number lines (one BLM per problem). They can refer to the number lines completed as a class in the Minds On for examples of how to record their thinking. Addition and Subtraction 295

• Give students these problems: 45 + 28 19 + 76 Differentiation • Change the numbers so they meet the needs of your students. For example, if some students need to work with decade numbers, provide questions such as 50 + 30 or 54 + 30. • You’ll likely have some students who have difficulty using the open number line and/or recording the jumps accurately. Students with fine motor difficulties may find the recording particularly challenging as it requires some precision in the spacing of the numbers. Others may confuse the numbers on the number line with the numbers they are adding. Work interactively with these students in small-group guided math lessons to solve a few problems using the number line. Begin with problems that require only jumps of 10 and progress to problems that require jumps of 10 and more. • Have students who need more of a challenge create their own problem. You could also have them solve a problem with three addends (e.g., 41 + 27 + 39). . Assessment Opportunities Observations: Observe how students use the number line. • Do they know what number to record as a starting point? (Choosing the larger number may make adding easier, but is not necessary.) • Do they take and record jumps of 10 or are they counting only by 1s? • Do they realize that the jumps represent the amount that they are adding? • Are they connecting this process to decomposing? • A re they moving in the appropriate direction to represent addition and subtraction? Conversations: As students are working, ask them to explain their jumps on the number line. As they talk through their process, ask questions to clarify their thinking and prompt the use of mathematical language. For example, say: – I see you have drawn three jumps. What do they represent? – H ow can you show that your jumps mean adding tens? Where should you record ‘+10’? How did you know how many jumps of 10 to take? Consolidation (20 minutes) • Strategically select different solutions based on the strategies students used. Ensure you have an example of keeping one addend whole, and of taking jumps of 10 and more. Have students model and explain their solutions. • Discuss how we choose the numbers for an open number line and how we record to show the numbers that we’re adding. 296 Number and Financial Literacy

• Ask students how this strategy of taking jumps of 10 and more connects to decomposing and place value. You might ask, “How does understanding place value help you use the number line to take jumps?” (e.g., If you know what the tens and ones are, you can take jumps of 10 then jumps of 1; you can decompose the number into tens and ones and jump by tens first, and then ones.) • Ask students to visualize an open number line in their minds and solve the problem 34 + 21. Discuss what they saw in their minds. • Building Social-Emotional Learning Skills: Healthy Relationship Skills: Carol Dweck emphasizes that while effort is important for developing a growth mindset, there are other parameters to consider. She explains that “students need to try new strategies and seek input from others when they’re stuck. They need this repertoire of approaches—not just sheer effort—to learn and improve” (Dweck, 2015, p. 21). Discuss what students have learned from each other by sharing their strategies and thinking. Provide students with a small piece of paper and ask them to reflect on one strategy they would like to improve upon or try. Ask them to write one ‘next step’ they will take to grow in this area. Collect their reflections and return them after students have had time to solve more problems with their chosen strategy. They can then revise their goal or add other another one. Math Talk: Materials: Math Focus: Developing mental strategies for two-digit subtraction, including chart paper decomposing to make friendly numbers Process: Work through the following equations one at a time. Record students’ Teaching Tip strategies on open number lines on chart paper. Integrate the math 100 − 25 = talk moves (see 75 − 49 = page 8) throughout 73 − 25 = Math Talks to maximize student participation Let’s Talk and active listening. Select the prompts that best meet the needs of your students. • Who has a solution for 100 − 25 = ? (I think it’s 75. I counted back from 100 by 10s—90, 80—and then counted back 5 more to 75.) Let’s represent your thinking on the number line. • Did anyone have another way? (e.g., When you count by 25s, it is 25, 50, 75, 100, so 25 less than 100 is 75.) So, you used counting by 25s to help you. Let’s represent those counts on the number line. Which way are we moving on the number line? How do we mark counting backward by 10? (− 10) • Is there another way? (e.g., I know that 100 is like 100 cents or 4 quarters, and if you take 25 cents away, you have 3 quarters or 75 cents.) Your way is like counting by 25s, except you are thinking in terms of money. continued on next page Addition and Subtraction 297

• Let’s look at the second problem, 75 − 49. (e.g. I counted back by 10s from 75 to take away the 40 in 49, like this, “65, 55, 45, 35,” and then I counted back nine times by 1s and got to 26.) Let’s record this on the number line. Is there another way of counting back? (e.g., I counted back by 10s from 75 five times—65, 55, 45, 35, 25—to take 50 away. But I took away 1 too many, since 49 is 1 less than 50, so I had to add 1 more to 25, so the answer is 26.) Let’s record this on the number line and compare it to the previous strategy. How are these strategies the same and how are they different? Where can you see the 49 being subtracted in each case? 75 − 49 +1 +1 +1 +1 +1 +1 +1 +1 +1 +10 +10 +10 +10 26 27 28 29 30 31 32 33 34 35 45 55 65 75 –10 –10 –10 –10 –10 +1 45 55 65 75 25 26 35 • Let’s look at the final problem, 73 − 25. How could you solve this in more than one way? Turn and talk to your partner. Record students’ solutions on a number line. Possible solutions and prompts include: – I counted back from 73 by 10s two times—63, 53—and then counted back by 1s five times to get 48. How did you track the 25 that you took away? (e.g., I put up a finger on one hand for each of the two groups of 10, and then put a finger up on my other hand to show each of the 5 ones.) – I counted up from 25 by 10s—35, 45, 55, 65—and then counted up by 1s eight times to 73. How did you track your counts? (e.g., I used my fingers too.) – I counted up from 25 to 75 by 25s—50, 75—but that is 2 too many since 73 is 2 less than 75, so I took away 2 from 50, which is 48. Did you need to track your count? (e.g., No, I could do it all in my head.) – I added 2 to 73 to make 75, like 75¢, then I took 50 away, like taking away 2 quarters, to get to 25¢, then I took away the 2 that I added at the beginning from 50, which ends up at 48. How is your way like the previous one that was presented? • How did solving the first two problems help you solve the last problem? (e.g., It made me think in groups of 25, so I used the amounts 25, 50, and 75 as friendly numbers.) • Which way seemed easier for you? Why? • Why can you use both addition and subtraction strategies to solve this problem? 298 Number and Financial Literacy

Further Practice • Independent Problem Solving in Math Journals: Pose the following problem and have students record their thinking on an open number line: – A t 7 o’clock in the morning, there were 36 cars in the parking lot. Two hours later, 24 more cars had parked. At the end of the day, 43 of the cars had left the parking lot. How many cars were still in the parking lot at the end of the day? Addition and Subtraction 299

and7 8Lessons Using an Open Number Line to Add and Subtract Three-Digit Numbers Possible Learning Goals • Solves addition and subtraction problems involving three-digit numbers using an open number line • R ecords their thinking on an open number line with numbers and arrows, and explains what they mean • Decomposes three-digit numbers in a variety of ways to add and subtract Teacher • Applies understanding of place value to decompose numbers into hundreds, Look-Fors tens, and ones Previous Experience • Applies understanding of known facts to decompose numbers into friendly with Concepts: numbers (e.g., decomposing to create a decade number) Students have used an open number line to add • Takes leaps of hundreds, tens, and ones, either forward or backwards, on the and subtract two-digit number line and counts accordingly numbers. • Clearly explains strategy using mathematical language • Represents strategy by drawing jumps on an open number line and recording the numbers and operations that the jumps represent (e.g., + 5) 300 Number and Financial Literacy

7Lesson Adding Three-Digit Numbers on an Open Number Line Materials: Minds On (20 minutes) chart paper, marker Time: 60 minutes • Ask what an open number line is and how it is different from the classroom number line. (e.g., It only includes the numbers that are necessary to solve the problem; the jumps from one number to another are labelled with the operation and size.) • Pose the following problem: – T here are 29 books on the lower shelf in the library and 43 books on the upper shelf. How many books are there altogether? • Have students turn and talk to their partner about how they could solve this using an open number line. • Draw several number lines on chart paper, one underneath the other, and record students’ strategies. Potential strategies include the following: – S tart at 29, take four jumps of 10 to 69, and take three single jumps to 72. – S tart at 29, take one single jump to 30, take four jumps of 10 to 70, and take two single jumps to 72. – S tart at 43, take two jumps of 10 to 63, and take nine single jumps to 72. – S tart at 43, take seven single jumps to 50, take two jumps of 10 to 70, and take two single jumps to 72. – S tart at 43, take three jumps of 10 to 73, and take one single jump back to compensate for the one extra jump (30 rather than 29). – S tart at 40, take two jumps of 10 to 60, take a jump of 9 (from the 29) to 69, and take three single jumps to 72 (to represent the 3 from the 43). • Discuss how the strategies are the same and how they are different. (e.g., Some keep one number whole and then add on by decomposing the second number; the last example decomposes both numbers according to place value groups and adds them individually.) Ask students which strategy they think works best for them and why they think so. • Ask students whether they can solve addition questions for three-digit problems using the open number line. Ask what size of jumps they may take (e.g., 100, 10, 5). Working On It (20 minutes) • Have students work in pairs. Read the following problem together, ensuring all students understand the context. You may decide to change the context so the problem is more meaningful for your students. 301 Addition and Subtraction

– A t the aquarium, there were 178 tiny blue fish in one tank and 432 tiny yellow fish in the other tank. How many fish are in the two tanks? • Students can draw their own open number lines on chart paper and record their thinking. Encourage them to find at least two different ways of solving the problem. • Before they begin, have students estimate about how many fish there are and discuss their estimation strategies. Differentiation • Some students may find it difficult recording bigger jumps, such as 100, on the number line. Work interactively with these students in a small-group guided lesson. Begin with problems that need only jumps of 100. Once students can evenly space these, have them work with adding 100s and then a 10. Ask how they will adjust the size of the jumps to reflect their magnitude. • For students who need more of a challenge, have them create their own problem. They can record their problem on a separate piece of paper, exchange problems with another pair, and then solve each other’s problems and discuss their solutions. Assessment Opportunities Observations: Observe how students use the number line. • Do they know what number to record as a starting point? (Choosing the larger number may make adding easier but is not necessary.) • D o they take jumps of 100 and then adjust the size of the jumps when adding 10s? Can they accurately record the magnitude and operation of the jumps? Conversations: As students are working, ask them to explain their jumps on the number line. As they talk through their process, ask questions to clarify their thinking and prompt the use of mathematical language. For example: – I see you have drawn three jumps. How much do they represent? What operation are you using? How can you show that they mean adding hundreds? – T hese next jumps look the same size as your first jumps. Do they represent the same amount? How could you adjust the size of your jumps? How can you record numbers so people know how much you are adding? Consolidation (20 minutes) • Select various students’ strategies that reflect different ways of decomposing the numbers. As students share, record their solutions on several number lines. The possible solutions shown below are grouped into three categories. 302 Number and Financial Literacy

Keep one number whole and add on – S tart at 178, add 4 hundreds one at a time to 578, add three jumps of 10 to 608, then add two single jumps to 610. – S tart at 432, add 1 hundred to 532, add eight single jumps to 540, then add 7 tens one at a time to 610. Add by place value groups – S tart at 100 (the 100 from 178), add 4 hundreds (from 432) all at once to 500, add 7 tens (from 178) all at once to 570, add 3 tens (from 432) all at once to 600, add 8 ones (from 178) in one jump, and add two more single jumps (from 432) to 610. – S tart at 400 (from 432), add 100 (from 178) to 500, add 1 more hundred (the tens, 70 + 30, from the two numbers) to 600, then add 1 ten (the ones, 8 + 2, from both numbers) to 610. Use friendly numbers and compensate – S tart at 432, add 8 ones in one jump to 440, add 2 hundreds (rounding up from 178) in one jump to 640, then subtract 3 tens all at once to compensate for the extra 30 that was added with the 200 to 610. • Have students compare the different strategies. Create categories for them, such as those given above, and name the strategies. Record this information on the same chart paper as the number line solutions to create an anchor chart. • Discuss which solution works best for them and why they think so. • Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: When students make a mistake in their work or in an explanation, it is important that the class culture values mistakes and uses these opportunities for learning. According to Jo Boaler, research indicates that “the brain grows when we make mistakes, even if we are not aware of it because it is a time of struggle; and the times when we struggle and are challenged are the best times for brain growth.” Research further found that, “there was greater brain activity and growth when people had a growth mindset than when people had a fixed mindset” (Boaler, 2017). As a class, you can celebrate a mistake and discuss what students learned from it, emphasizing that if they hadn’t made the mistake, the learning may not have happened. Further Practice • Independent Problem Solving in Math Journals: Pose problems for students to solve independently on an open number line, to check for understanding. Often, students can solve a problem when working with a partner but experience more difficulty when working on their own. You can then identify students who need more input and have a guided lesson with them. Addition and Subtraction 303

8Lesson Subtracting Three-Digit Numbers on an Open Number Line Materials: Minds On (15 minutes) chart paper, markers, • Ask students how they used the open number line to help them add three- BLM 3: Hundreds Charts to 500 and BLM 44: digit numbers. Review what is important to include when recording Hundreds Charts to 1000 thinking on the open number line (e.g., the starting point, the operations, (optional) the magnitude of the jumps, the points of landing on the number line). Time: 60 minutes • Read the following problem to students, changing the context if needed to make it more meaningful to your students. – T here are 531 students in the school. One day, 276 students went on a field trip. How many students were still in the school? • Ask students whether the solution will be greater or less than the number of students in the school and why they think so. Have them estimate about how many students will be left in the school. Discuss their estimation strategies. Working On It (20 minutes) • Students work with a partner to solve the problem from the Minds On using an open number line. Encourage them to do so using at least two strategies. They can draw their own number lines on chart paper and record their thinking. Differentiation • Adjust the numbers to be suitable for the needs of your students. • Students who have difficulty counting back by 10s or 1s can use hundreds charts (BLMs 3 and 44). For example, if they are counting back from 462 by 10s, the chart can help them see the pattern (52, 42, 32, 22, and so on). • For students who need more of a challenge, vary the problem structure. (e.g., There are 531 students in the school. Some went on a field trip. There were 255 students left at the school. How many students went on the field trip?) • For students who need more of a challenge, have them create their own problem, solve it, and then exchange their problem with another pair. They can solve each other’s problems and then compare their strategies. 304 Number and Financial Literacy