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

Assessment Opportunities Observations: Notice how students approach the problem. • Do they use subtraction and/or think-addition? • W hen recording numbers on the open number line, do they record the numerals in the appropriate places? • Do they jump forward or backward and recognize which operation each direction represents? • Do they take reasonable jumps, such as 100s, 10s, and 1s, or other jumps that create friendly numbers? • What other strategies are they using? (e.g., compensation) Conversations: If students are not clearly recording their solutions, ask them to explain their strategy. As they do, pose some of the following prompts so they can add more written detail to their solutions: – What number did you start at? How can you show that on your number line? – W hat does 30 represent? What number did you decompose to get 30? Are you adding or subtracting 30? How can you show the operation on your number line? – How are you keeping track of all of the amounts between the two numbers? Consolidation (25 minutes) • Strategically select three or four solutions that reflect a variety of strategies. As students are explaining them, record the solutions on separate number lines positioned one below the other. Some possible strategies are given below. Keep one number whole and subtract parts of the other number – S tart at 531, take away 2 hundreds (from 276) in two jumps to 331, take away 1 one (from the remaining 76) to 330, take away 3 tens all at once (from 75) to 300, take away 4 tens one at a time (from the remaining 45) to 260, and take away the remaining 5 to 255. – S tart at 531, take away 6 ones (from 276) one at a time to 525, take away 7 tens (from the remaining 270) one at a time from 525 to 455, and take away 2 hundreds (from the remaining 200) one at a time to 255. Keep one number whole and add/count up to the larger number – S tart at 276, add 2 hundreds in one jump to 476, add 5 tens one at a time to 526, add 5 ones one at a time to 531, and total the amount added (200 + 50 + 5 = 255). – S tart at 276, add 4 ones all at once to 280, add 2 tens all at once to 300, add 2 hundreds all at once to 500, add 3 tens all at once to 530, add 1 to land on 531, and total the amount added (255). Addition and Subtraction 305

Use constant differences – C hange the question by adjusting both numbers down 1 unit so the numbers are now 275 and 530. Solve in one of the following ways: • S tart at 530, subtract 3 tens (from 275) all at once to 500, subtract 2 hundreds (from the remaining 245) all at once to 300, subtract 4 tens (from the remaining 45) to 260, and subtract 5 ones all at once to 255. • S tart at 275, add 25 to 300, add 2 hundreds to 500, and add 30 all at once to 530. • Have students compare the strategies. Categorize them into groups as shown above and name the strategies. Record this information on the same chart paper as the number line solutions to create an anchor chart. 306 Number and Financial Literacy

9Lesson Using ‘Think-Addition’ to Subtract Multi-digit Numbers 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 solved problems using the • B 2.5 represent and solve problems involving the addition and subtraction of ‘counting on’ strategy. whole numbers that add up to no more than 1000, using various tools and algorithms Possible Learning Goals • Solves problems involving two-digit subtraction with and without regrouping • Uses strategies, including ‘think-addition,’ to solve subtraction problems • Demonstrates an understanding of the inverse relationship between addition and subtraction by explaining why ‘think-addition’ can be used to solve subtraction problems • Selects and uses appropriate tools to solve the problem (e.g., number lines, hundreds chart, place value mats, base ten blocks) • Selects and uses appropriate strategies to solve the problem (e.g., adds on friendly numbers) • Clearly explains that ‘think-addition’ works to solve subtraction problems because addition and subtraction are inversely related • Uses another strategy to assess the reasonableness of their solution (i.e., uses think-addition after using subtraction) • Shows all steps when solving a problem (e.g., keeps track of the count when adding on) Addition and Subtraction 307

PMraotcheesmseast:ical About the arrPeenrfadolesbsocltetnrimanintgge,sgoasileenvlsdien,cpgtr,ionvgintogo, ls ccoomnnmecutninicga,trinegpresenting, Any subtraction problem can be solved with addition by counting up from the lesser number to the greater number and finding the difference. Van de Math Vocabulary: Walle and Lovin call counting up or ‘think-addition’ an “amazingly tdshiuifnbfektrr-aeancdtcd,eita,idoondp,eration, powerful way to subtract” (Van de Walle & Lovin, 2006a, p. 167). Students who have had successful experiences counting up when solving questions involving the basic facts can apply the strategy to subtracting with larger numbers. This requires a good understanding of number order and flexibility in skip counting by different quantities from various starting points. Think-addition highlights the inverse relationship of addition and subtraction. Problems that have a missing part encourage students to use think-addition. For example, 45 − __ = 30 invites students to start with 30, add 10 to get to 40, and then add 5 more to arrive at 45. When using think- addition to subtract two-digit numbers, students initially may need to record the parts they are adding (e.g., the 10 and 5) so that they don’t forget them. About the Lesson In this lesson, students work in pairs or groups of three to solve word problems using the think-addition strategy with two-digit numbers. For further practice, repeat this lesson or parts of it using three-digit numbers. Materials: Minds On (20 minutes) concrete objects, • Write 81 − __ = 65 and draw an open number line. Ask students what we are chart paper, markers supposed to find in the equation. Have someone make a story about the Time: 60 minutes equation to give it context. (e.g., Ana had 81 baseball cards. She gave some away. Now she has 65 cards. How many cards did she give away?) • Ask which numbers are important to record on the number line and where they should go. Ask students what they are trying to find as far as the two numbers given are concerned (e.g., the distance between the two numbers). Have students turn and talk to a partner about how they could solve the problem. • Discuss and then record two or three student strategies on the number line, including some that are similar to the following two strategies: – S tart at 81, jump back by 1 ten to 71, jump back by 1 one to 70, jump back by 5 to 65. Add up the jumps as 10 + 1 + 5 = 16. – S tart at 65, jump forward 1 ten to 75, jump forward 5 to 80, and jump forward 1 one to 81; 10 + 5 + 1 = 16. • Discuss what students notice about the two strategies. (e.g., The distance is the same, both strategies result in finding the same difference.) 308 Number and Financial Literacy

• Record 81 − 16 = 65 and 65 + 16 = 81 to represent the subtraction and addition strategies. Have students identify the parts and the whole in each equation. Discuss how in the given problem, they knew the whole and a part and used this information to find the other part. Reinforce part-part-whole relationships by discussing that we were trying to find the same missing part in both equations. • Reinforce that counting on from one part to get to the whole is known as ‘think-addition,’ since we are adding on to find the difference. Use the number lines created above to discuss how addition and subtraction ‘undo’ each other. Working On It (20 minutes) • Have students work in partners or groups of three. Present the following problem (orally and in writing): – A nna has a set of 63 collector cards. She wants to keep 47 cards and trade the others. How many cards will she trade from her collection? • Encourage students to find more than one way to solve the problem. Suggest that they try the ‘think-addition’ strategy as one way. Provide access to tools that may help them (e.g., number lines, hundreds chart, place value mats, base ten blocks). • Have students record their work on a piece of chart paper. Differentiation • If the numbers in the problem are too challenging, use multiples of 10 instead (e.g., 60 and 20) as a parallel task. • For an additional challenge, have students visualize their solution before working directly on the number line. • The term ‘think-addition’ might be confusing for some students, including ELLs. Use the term at the same time as you model the direction of movement on the number line so students link the term with a visual. Assessment Opportunities Observations: • L ook for evidence that students understand the problem (e.g., they start right away, they try a strategy, they choose an appropriate tool). • O bserve what strategies they use (e.g., concrete materials, counting backward or forward, addition or subtraction, mental math). Did they try think- addition as suggested? Do they use more than one strategy? • N ote how they are representing their work. Do they use concrete materials, drawings, hundreds charts, number lines, mental math? Can they articulate their mental math strategies, even though they may have difficulty recording them? continued on next page Addition and Subtraction 309

Conversations: If a student is unable to solve this problem using think- addition, pose a similar problem with simpler numbers (e.g., Sara has 17 books. 12 of the books are fiction and the rest are non-fiction. How many are non- fiction?). Ask some of the following prompts. • W hat are you trying to find? Will the solution be greater than or less than the number of books that Sara has? • V isualize the problem. Imagine Sara has sorted her books into two piles. What do you see in your mind? Let’s put out 12 counters to represent the fiction books. What would the other pile look like? How could you count on to find out? • N ow let’s show what you did with the counters on the number line. You started at 12 and then how did you move? Were you adding or subtracting to find the total number of books? Which direction would you move on the number line? Could you work back from 17 to get to 12? What operation is that? Notice how it is the distance between the two numbers that you are finding in both cases. Consolidation (20 minutes) • Strategically select pairs or groups to share their solutions with the class so that a variety of strategies are shown. Sample strategies include: – P ut 47 at the left end of the number line and 63 on the right. Start at 47 and jump up to 50, adding 3. Jump by 10 to 60, adding 10 more, and then jump to 63, adding 3 more. 3 + 10 + 3 = 16. – U se think-addition on place value mats with base ten blocks. Start with 4 rods and 7 units. Add one more rod to make 57. Add 6 more ones to get to 63. 10 + 6 = 16. – O n a hundreds chart, start at 63, jump down a row to take away 10, count back 6 more numbers to 47 to take away 16 altogether, 63 − 10 − 6 = 47. • Ask why the problem can be solved with both subtraction and addition. Discuss how addition combines the parts to make the whole, and subtraction starts with the whole and takes part away to find the other part. Since the difference between the numbers remains the same, you can work back from the whole, which is subtraction, or you can work toward the whole from the part, which is addition. • Have students visualize the following problem: – T here are 70 pets in the store. 45 are dogs and the rest are cats. How many are cats? Ask what they are solving (e.g., the difference between the number of dogs and the total number of pets). Have students visualize how they might solve this problem on a number line. Discuss what they saw in their minds. Demonstrate some of their ideas on the number line so students can start to make the link between visuals and their mental images. 310 Number and Financial Literacy

Teaching Tip Further Practice Integrate the math • Repeat this lesson, or parts of it, using three-digit numbers. talk moves (see page 8) throughout Math Talk: Math Talks to maximize student Math Focus: Using rounding to the nearest ten as a mental math strategy for participation and subtracting two-digit numbers active listening. Let’s Talk Select the prompts that best meet the needs of your students. • Think about how you can solve this problem: 29 − __ = 12. • W ho has a solution to share? (I got 17. I started with 12 and added 10 to get to 22, and then counted by 1s to get to 29.) How did you keep track of your counts by 1s? (e.g., I counted them on my fingers.) I’ll keep track of your jumps on the number line. This is especially helpful when you work with larger numbers and it is harder to keep track of what you are adding or subtracting. You might even visualize a number line in your head to help you. What strategy did you use to solve this problem? (e.g., think-addition) • Let’s try solving another problem in your minds. You may decide to visualize a number line or hundreds chart in your head or mentally plan out the moves by looking at our classroom tools. The problem is 61 − 37. • W hat did you find? (e.g., I counted back by 10s, like 51, 41, and then back by 1s, 40, 39, 38, 37.) How did you track how many times you counted back by 1s? (e.g., I counted on my fingers.) Let’s look at how this solution would look on the number line. • Are there any friendly numbers between 61 and 37 that you could use to help you? (e.g., Count back 1 to 60, then back by 10s, like 50, 40, and then back three times by 1.) So, you used 60 as a friendly number. How did you keep track of the 1s? (e.g., I just remembered there was 1 and then 3 more.) Let’s show the jumps on the number line. • Are there any other friendly numbers? (e.g., Count by 1s three times to 40, then by 10s like 50, 60, and then 1 more.) You used 40 as a friendly number. Why? • W hy does it help to use one of the decade numbers as a friendly number? We call this ‘rounding’ up or down to the nearest ten. • Try visualizing a number line in your head. What friendly number could you use with 83? (e.g., You could use 80.) Why? What friendly number might you use with 28? • W hat do you do if you are working with 35? (e.g., You could use 30 or 40, because it is right in the middle.) You are correct. Often in math, we use a rule that we round up to the next ten when a number ends in a 5. That doesn’t mean that rounding down isn’t right. It is generally that people agree to round up. It continued on next page Addition and Subtraction 311

can also depend on the situation. For example, let’s say that you need 43 pencils and they come in packages of 20. Would you round up or down to find out how many packages you need to buy? (e.g., You would round up to 60 rather than down to 40, so you have enough pencils.) • Let’s summarize how we can use rounding to help us find friendly numbers when we are adding and subtracting two-digit numbers in our minds. Materials: Math Talk: chart paper, Math Focus: Investigating the constant difference strategy to mentally add and classroom number subtract two-digit numbers line with numbers to at least 50 Let’s Talk Select the prompts that best meet the needs of your students. • Write the equation 29 – 14 = ___ so all students can see it. What is 29 − 14? Share your thinking with a partner. • W hat did you find? (e.g., It is 15.) How did you find this out? (e.g., We counted back by 10s from 29 to 19 and then we counted back by 1s five times to get to 14.) How did you keep track of your counts backward? (e.g., We remembered the one group of 10 and then counted the single jumps on our fingers.) • W hat does the 15 represent? (e.g., It is the difference between the two numbers.) What does this look like on the number line? Let’s show these two numbers on our classroom number line that has all of the numbers and count the difference. • Write the expression 30 − 15. What is the difference between 30 and 15? (e.g., It is 15.) How did you solve this problem? (e.g., We counted back by 5s; we counted back by 10 and then 5; we knew that 15 + 15 = 30, so the answer had to be 15.) • Look at the numbers in the two problems. Why do you think the answer is the same for both of them? (e.g., 30 is 1 more than 29, and 15 is 1 more than 14, so the difference between the numbers is the same.) • Let’s look at this on the classroom number line. Mara, please point to 29, and Devon, please point to 14. Show us how you move to the new numbers of 30 and 15. Class, what did they do? (e.g., They each moved one number to the right.) Why is the difference still 15? (e.g., They both moved the same amount to a new number.) • This is interesting. Do you think this strategy of adding or subtracting the same from both numbers would always end up with the new numbers having the same distance between them? This is a conjecture, or prediction, that we could further explore. 312 Number and Financial Literacy

• We added 1 to each of our numbers. What else could we do to both numbers, 29 and 14? (e.g., Subtract 5 from each of them.) Mara and Devon, what do you need to do to subtract 5 from your numbers? Show us. What are the new numbers? (24 and 9) How can you figure out the difference between the two numbers? (e.g., I can start at 9 and take a jump of 10 to 19, and then five jumps of 1 to 24.) Is the difference still 15? • Repeat with other examples suggested by the students. • W hy does it make sense that the difference in all cases is 15? (e.g., If you change the two numbers by the same amount, the difference or distance between then is the same.) Let’s record this as a possible rule for now. • How can this rule make adding and subtracting easier? Look at our examples. (e.g., You can make friendlier numbers by adding or subtracting the same to both numbers and then finding the difference easier, like 30 − 15 rather than 29 − 14.) • Let’s remember this as a mental math strategy that we can use. Addition and Subtraction 313

10Lesson Investigating Compensation as a Mental Strategy 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 In grade two, students added and subtracted • B 2.5 represent and solve problems involving the addition and subtraction of two-digit numbers using concrete materials, whole numbers that add up to no more than 1000, using various tools and student-generated algorithms algorithms, and standard algorithms. Possible Learning Goal PMraotcheesmseast:ical • Investigates the compensation strategy to determine if it works in all cases cctPoooromnolnbsmeleacumntnidnicsgsao,ttlrrvianeintgpegrge, isseeesnl,eticntgin,g • Understands and explains what a conjecture is • Explains or shows what the compensation strategy is • Explains why the compensation strategy works • Investigates whether the compensation strategy works every time by selecting and trying several examples About the In grade two, students used their understanding of basic addition and subtraction facts to develop mental strategies to add single-digit numbers. One effective strategy that grade three students can apply to adding and subtracting two-digit numbers is making combinations of 10. For example, by knowing that 4 + 6 = 10, students can decompose expressions such as 24 + 36 into more friendly decade numbers, such as 30 + 30 or 20 + 40. Friendly numbers can also be composed by applying the concept of equivalence through compensation. If you lose 1 from one number but gain 1 in the other number, the total stays the same. For example, if 10 + 20 = 30, then the sum of 11 + 19 will also be 30, since one is taken away from the 20 and added to the 10. Before students can efficiently apply this strategy, they need to understand what compensation is and why it works. 314 Number and Financial Literacy

McoantjhecVtuorcea, bcuomlarpye:nsation About the Lesson strategy In the Minds On, students learn about the compensation strategy. In the Working On It, they test a conjecture to determine whether the compensation strategy works with all numbers. They apply this knowledge to create friendly numbers when adding two-digit numbers. As students develop a model to prove or disprove a conjecture, it is a good opportunity to reinforce the mathematical modelling process and its four components: • Understand the Problem • Analyse the Situation • Create a Model • Analyse and Assess the Model Use an anchor chart to highlight how students move among the four components as they develop and refine their model. There are some suggestions on how to do this throughout the lesson, although they will need to be adjusted so they are responsive to how your students move through the process. Materials: Minds On (20 minutes) Digital Slide 44: • Tell the story of Carl Friedrich Gauss, a famous mathematician who lived in Gauss’s Strategy, classroom number line Germany in the late 1700s. When Gauss was 11 years old, his teacher asked the from 1 to 100, chart class to determine the sum of all the numbers from 1 to 100; that is, to add paper, markers, 1 + 2 + 3 + 4 and so on, up to 100. A few minutes later, Gauss approached the calculators teacher with the solution, amazing his instructor by solving the problem so quickly. Time: 60 minutes • Show students Digital Slide 44: Gauss’s Strategy. Have students turn and talk to a partner about what Gauss did to solve the problem. (Analyse the Situation) • If students are unable to figure out the strategy from the visual, pose a few prompts to clarify its meaning, such as, “What numbers do you think go in between the 4 and 97? Why do you think the number pairs are connected? What operation could you use to figure out the strategy? What do you notice about the sums of the number pairs that are shown? (e.g., They all add up to 101.) Do you think all of the number pairs from 1 to 100 add up to 101? Why?” Addition and Subtraction 315

• Have one student point to the number 1 on the classroom number line (from 1 to 100), while another student points to the 100. Ask what the sum of the numbers is. Have the student pointing to 1 move to the next number to the right (2), and have the student pointing to 100 move to the next number to the left (99). Ask what the sum of these two numbers is. Have the two students keep moving inward, one number at a time, while the rest of the class find the sum. Once students are convinced that all the totals will be 101, ask why this pattern works. (e.g., As the number on the left increases by 1, the number on the right decreases by 1, so the total remains the same.) • Have students turn and talk to their partner about how this could help them find the sum of all numbers from 1 to 100. • Ask how many number pairs there are for the numbers 1 to 100 and why they think so. (e.g., There are 50 pairs because half of 100 is 50.) Ask what they could do to find the sum of the numbers from 1 to 100. (e.g., Add 101 fifty times.) Ask what other operation they could use, since they are dealing with equal groups (e.g., multiplication, so 50 × 101 to represent 50 groups of 101). Explain that these numbers are bigger than the ones we multiply in grade three. Model using a calculator to find the total. Explain that you are typing in 50 × 101, which means 50 equal groups of 101. Share that the total is 5050. Working On It (20 minutes) • Ask students whether they think the strategy that Gauss used would work with any string of numbers in the counting sequence. Tell them that this will be the conjecture they will try to prove or disprove. (Understand the Problem) Explain that a conjecture is like a prediction. Tell them that a conjecture can only be considered a rule if it works every time, and it takes only one example that doesn’t work to disprove the conjecture. Explain that mathematicians often try to prove conjectures to find out whether they can be considered rules. (Analyse the Situation) • Students work in pairs. Together, they discover whether this pattern would work with other numbers. They select any 10 consecutive numbers in the counting sequence (e.g., 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 or 26, 27, 28, 29, 30, 31, 32, 33, 34, 35). After writing out their number sequence, students make number pairs, like Gauss did, and find the sums to see if they are all the same. (Create a Model) • Have students try this with two or three different sets of numbers. Differentiation • Give calculators to students who find mentally adding the numbers time consuming. • You may want to assign lower number sequences to some students, so the numbers are more manageable. • Have students who need more of a challenge select a counting sequence of 20 numbers. They could also choose 10 numbers in a skip-counting sequence, such as 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. 316 Number and Financial Literacy

Assessment Opportunities Observations: • D o students understand that one number is increasing by 1 while the other is decreasing by 1? • Can they explain why the sums of the number pairs are all the same? Conversations: If students are just following the procedure but not really understanding the idea of compensation to maintain equivalence, pose some of the following prompts: – Y ou have 10 and 19 paired up and 11 and 18 paired up. What is the sum of each pair? (29) Why are they the same? (e.g., I am not sure.) – Let’s represent the two numbers 10 and 19 on either side of your desk using counters. You can put the counters in ten frames. Can you see the 29? – Now, rearrange your 29 counters so there are 11 counters on the left and 18 counters on the right. How many do you see now? Did you add any more counters? What did you do to rearrange them? – Now, rearrange the 29 counters again so there are 12 on the left and 17 on the right. How do you know there are still 29 counters? (e.g., I didn’t add or take any away.) What did you do to rearrange the counters? Did you have to make the amounts by starting over? (e.g., No, I moved one from the right pile over to the left pile.) How would you make the next number pair? Explain to me in your own words why all of the number pairs add up to 29. Consolidation (20 minutes) • Meet as a class. Ask students whether their sums always equalled the same amount. Strategically share some of the number sequences to show that this is true with a variety of numbers. • Ask whether we can conclude that our conjecture could be a rule. • Explain that this rule is known as ‘compensation,’ which means that you are taking an amount from one number and giving that amount to the other number, but the sum remains the same. Show students how this works using concrete materials (see the Assessment Opportunities above). • Ask whether compensation will work if they take 2 away from one number and give the 2 to the other number. Students can turn and talk to their partner. If any students worked with a number sequence representing skip counting by 2s, have them share their results. If necessary, use concrete materials to prove that taking 2 away from one number and adding 2 to the other number keeps equality. Ask whether the compensation will work if 5 counters are taken away and given to the other number. Through discussion, highlight that compensation will work as long as the amount taken away from one number is the same as the amount given to the other number. • Create an anchor chart that summarizes the compensation strategy. Addition and Subtraction 317

• Ask students whether they are convinced that the conjecture will work with all numbers and what they could do to be certain that it works in every case. Ask students how they might try to prove or disprove this conjecture differently if they were to do it again. (Analyse and Assess the Model) Materials: Math Talk: chart paper, markers Math Focus: Using compensation to create friendly numbers when adding Teaching Tip multi-digit numbers Integrate the math Let’s Talk talk moves (see page 8) throughout Select the prompts that best meet the needs of your students. Math Talks to maximize student • L et’s look at the anchor chart that we created on the strategy of compensation. participation and active listening. Tell me in your own words what compensation is. What would be an example? Why does it work? • W e are going to use compensation as we add some two-digit numbers in our minds. On chart paper, record 22 + 21 in a horizontal number sentence. How could we solve this problem using compensation? Turn and talk to your partner. • W hat did you find? (e.g., We subtracted 1 from the 21 and added it to 22, so we now had 23 + 20, which equals 43.) Record this on the chart paper, indicating what was added and what was subtracted from the two numbers. Why does this work? • W ho has another way? (e.g., We subtracted 2 from the 22 and added it to the 21, so we had 20 + 23, which equals 43.) Which of the two ways did you find easier? Why? How are the two ways the same and how are they different? Record the strategy. • L et’s try another example. Write 48 + 33. Turn and talk to your partner about how you could use compensation to solve this problem. (e.g., We found that you can subtract 3 from 33 and add it to 48 to make 51 + 30, which equals 81.) How did you know 48 + 3 is 51? (e.g., We counted on from 48 in our heads three times.) • D id anyone have a different way? (e.g., We wanted to make 50 from 48 so we needed 2 more, so we subtracted 2 from 33 and got 50 + 31, which equals 81.) Which of the two ways did you find easier? Why? • D oes it matter how much we subtract from one number? (e.g., No, as long as we add the same amount to the other number.) When does it make the most sense to use compensation? (e.g., When one of the numbers is close to a decade number; when we might have to regroup to add.) • R epeat this Math Talk using three-digit numbers. Make connections between the strategies that students use for two- and three-digit numbers. 318 Number and Financial Literacy

Materials: Math Talk: chart paper, markers Math Focus: Using compensation with the associative property to mentally add two-digit numbers Let’s Talk Select the prompts that best meet the needs of your students. • W e are going to use compensation to solve some more addition problems. Explain in your own words what compensation is. • W rite 33 + 19 + 27 on chart paper. Turn and talk to your partner about how you could use compensation to solve this problem. • W hat did you find? (e.g., We took 3 from 33 and added it to 27 to make 30. Now we have 30 + 19 + 30. Then we added 30 + 30, which equals 60, and added in 19, which equals 79.) Why can you add 30 + 30 before adding 19? (e.g., We found in another lesson that is doesn’t matter what order you add numbers in. You will get the same sum.) Yes, this is known as the associative property, which can make adding in our minds much easier. • D id anyone use a different strategy? (e.g., We gave 1 from 33 to 19, so our number sentence is now 32 + 20 + 27. We know that 32 + 20 equals 52, because we counted by tens two times from 32. Then we added the tens as 30 + 20 + 20, which equals 70, and added the ones as 7 + 2, and combined the tens and ones as 70 + 9, which equals 79.) Explain why you can change 33 + 19 into 32 + 20 and it doesn’t change the sum. • R epeat this Math Talk frequently throughout this unit and throughout the year, changing the numbers so students can practise applying the compensation strategy and associative property as they develop their mental math skills. Addition and Subtraction 319

11Lesson Using Partial Sums and Differences for Mental Calculations 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 In grade two, students added and subtracted • B 2.5 represent and solve problems involving the addition and subtraction of two-digit numbers with student-generated whole numbers that add up to no more than 1000, using various tools and algorithms and standard algorithms algorithms. PMraotcheesmseast:ical Possible Learning Goals Problem solving, • Uses partial sums to mentally solve two-digit addition problems arreenfadlessocttnrianintgge,gasieenlsde,cptrionvgintogo, ls • Uses partial differences to mentally solve two-digit subtraction problems ccoomnnmecutninicga,trinegpresenting, • Develops flexibility in applying addition and subtraction strategies • Mentally decomposes numbers into tens and ones to make operations easier • Adds and subtracts by working with tens and ones separately and then recomposing the amounts • Understands and can explain the partial parts strategies • Demonstrates a conceptual understanding of the process by explaining their strategies About the In grade two, students used partial parts strategies with concrete materials to add and subtract two-digit numbers. They can now apply these strategies to mentally solve problems. Partial parts strategies involve adding and subtracting numbers in parts, such as working with tens and then ones and then combining the results. If students have had multiple experiences decomposing numbers and have a good understanding of part-whole relationships, these strategies may prove useful in mentally performing calculations. 320 Number and Financial Literacy

The partial sums strategy involves decomposing numbers into parts using place values and adding the parts together. Students may choose to add the tens first and then the ones, or they may add the ones first and then the tens. They should be encouraged to do what works best for them. For example: 32 + 27 30 + 20 2+7 50 + 9 = 59 The partial differences strategy involves finding partial differences. Students subtract in parts that are easiest or friendliest for them to work with, which are often along place value lines. Students decompose and use strategies such as compensation to make the calculations as easy as possible. For example: 57 – 23 50 – 20 7–3 30 + 4 = 34 Students can also use partial differences for subtraction questions that typically require regrouping, such as for 74 − 38 = ___, as shown here: Keep one number 74 – 38 + 2 to make friendly number whole and subtract 74 – 30 8 + 2 the tens Subtract 2 more than necessary 44 – 10 34 + 2 + 2 to compensate for the extra 2 subtracted above About the Lesson In this whole-group lesson, students perform mental calculations using partial parts strategies and learn how to represent their thinking using the numerical form of the quantities. The lesson includes accompanying Math Talks focused on variations of the partial parts strategies. Addition and Subtraction 321

Materials: Minds On (10 minutes) chart paper, markers • A sk students what we mean when we talk about mental math strategies. Time: 50 minutes • A sk students what 5 + 6 is. Discuss how they solved this problem in their minds. Ask what they visualized to do the task (e.g., ten frames, dice, hundreds chart) and what strategies they used (e.g., doubles plus one). Tell students that they are going to solve problems in their minds using larger numbers. Working On It (30 minutes) • Show students the number sentence 32 + 27. Have students turn and talk to their partner about how they can mentally solve the problem. Students can use tools such as hundreds charts and open number lines if necessary. • Share some students’ strategies. In each case, show how the thinking can be represented with numbers. Two possible examples: 1. K eep 32 whole. Take two jumps of 10 to represent the 20 in 27, which equals 52. Then take single jumps to 79. Show this solution on a number line. Then, demonstrate how the strategy can be recorded with numbers, ensuring that students see the connection between the two representations. 32 + 20 = 52 52 + 7 = 59 2. A dd the tens (30 + 20), add the ones (2 + 7), and then add the tens and ones together. 32 + 27 30 + 20 2 + 7 50 + 9 = 59 Show this on the number line by starting at 30, taking two jumps of 10 to 50, and then taking single jumps to 59. Demonstrate how this can be represented with numbers. Make the connection between the two representations. Ask students whether they could add the ones first and then the tens. Show how it is much the same except for the order in which the adding takes place. • Repeat this with other students’ strategies. You may also want to repeat this procedure for another problem. Differentiation • You may want to make connections to the hundreds chart as well, so students have more examples of how visuals connect to numerical representations. 322 Number and Financial Literacy

Teaching Tip Assessment Opportunities Integrate the math Observations: Pay attention to whether students see the connection talk moves (see between the visual representations and the numerical representations. You page 8) throughout may want to do a think aloud through the steps to make it explicit what the Math Talks to numbers represent. maximize student participation and Consolidation (10 minutes) active listening. • Summarize how to record the various strategies using numbers in an anchor chart. Post it in the classroom so students can refer to it during the upcoming Math Talks. Math Talk: Math Focus: Mentally adding two-digit numbers with regrouping, using partial parts strategies Let’s Talk Select the prompts that best meet the needs of your students. • L ook at this number sentence: 23 + 68. Turn and talk to your partner about how you can solve it using mental math strategies. • W ho would like to share their strategy? (e.g., I added 2 tens and 6 tens and that equalled 8 tens.) What number is composed of 8 tens? (80. Then I added the ones, 3 + 8 = 11, so I knew I had 1 ten and 1 one. I added 1 more ten to 80 to equal 90 and had 1 one to add to 90. That equals 91.) What do we call the strategy you used? (I decomposed.) • Y es, you decomposed into tens and ones and added in parts. Let’s explore how you can record this using numbers. You can refer to our anchor chart from the lesson. Record student thinking as it is shared. 23 + 68 20 + 60 3 + 8 80 + 11 80 + 10 0 + 1 90 + 1 = 91 • W hat was your first step? (I added the tens.) I’m going to record that as 20 + 60. What did you do next? (I added the ones.) I’ll record that as 3 + 8. What is the next step you took? (I added the tens and ones, but I regrouped.) So far, you added two parts: the tens and the ones. Then you continued on next page Addition and Subtraction 323

added the two parts together. Let’s record that as 80 + 11 and break that down into 80 + 10 and 0 + 1, and then 90 + 1 = 91. Does that capture what you did? • W ho can explain in their own words what each of these steps means? Put your thumb up if you agree. Does anyone have anything to add to that explanation? • C ould we use this same strategy, but add the ones first and then the tens? Explain how this would be different and similar. So it doesn’t matter whether you start with the ones or tens, as long as you account for them all and add the parts together accurately. • L et’s try another example together. Repeat with one or two other addition number sentences to ensure that all students understand what the steps mean and how they represent their mental strategies. Math Talk: Math Focus: Mentally subtracting two-digit numbers without regrouping, using partial parts strategies Let’s Talk Select the prompts that best meet the needs of your students. • L ast time, we used partial parts to add two-digit numbers. Let’s try using this strategy for subtracting by looking for groups of tens and ones. Turn and talk to your partner about how you can use mental strategies to solve this number sentence: 87 − 52. • W ho would like to share their strategy? (I subtracted the tens in the numbers, so 80 − 50 = 30.) Put up your thumb if this makes sense. What did you do next? (I subtracted the ones, so 7 − 2 = 5. Then I added the parts, 30 + 5, which equals 35.) • If this is a subtraction question, why did you add the parts? (e.g., I subtracted in parts, but I have to put those amounts together so I know how much I subtracted altogether.) • L ook at our anchor chart for recording our mental math strategies. How can I show what we did in our minds with numbers? Record student thinking as it is shared. 87 – 52 80 – 50 7 – 2 30 + 5 = 35 324 Number and Financial Literacy

• C ould we subtract the ones first and then the tens? Turn and talk to your partner. We agree that it seems to work in this case. We will try other cases in the days ahead to see if it works every time. • W ho can explain in their own words why we are adding in the final step? • H ow could we solve this problem using think-addition? Turn and talk to your partner. • Is there a number sentence that we could record to help us? (e.g., 52 + ___ = 87) What does it mean in your own words? (e.g., 52 plus some amount equals 87) • W hat could you do? (e.g., We need to think what we can add to 50 to get 80, which would be 30 and what we can add to 2 to get 7, which would be 5. Then we add those two parts together, 30 + 5 = 35.) Let’s record that using numbers. 50 + 52 + = 87 =7 = 80 2+ 30 + 5 = 35 • W ho can explain in their own words what we did? How does this connect to the subtraction strategy we used? • L et’s try another example. Repeat this process with other subtraction number sentences that do not require regrouping. Materials: Math Talk: base ten blocks Math Focus: Mentally subtracting two-digit numbers that require regrouping, using partial parts strategies Let’s Talk Select the prompts that best meet the needs of your students. • T urn and talk to your partner about how you can use mental strategies to solve this number sentence: 54 − 38. • W ho would like to share their strategy? (I started with 54 and then I subtracted 30, which equals 24. Then I subtracted 4 to get to 20. Then I subtracted 4 more by counting back from 20 to 16.) Record the student’s process on an open number line. • W hy did you subtract 4 and then 4 more? (I had to subtract the 8 ones that are in 38, so I split 8 into 4 and 4 so I could take 4 from 34 easily to continued on next page Addition and Subtraction 325

get 30. Then I had to subtract the 4 that was left.) So you decomposed and subtracted in parts. • E xplain your steps again and I will record your ideas using numbers. (I kept one part whole and subtracted the tens from the other part.) Record 54 − 30 = 24. (Then I subtracted the ones. I split the ones up into parts though.) Then I’ll record both parts to show that. Record 24 − 4 = 20 and 20 − 4 = 16. • H ow would we solve 54 − 38 using base ten blocks? (e.g., We would exchange one of the 5 tens for 10 ones and have 14 ones, and then take away 8.) Have students demonstrate their solution using base ten blocks. How does subtracting the ones in parts make it unnecessary to regroup? (e.g., We subtract enough to make a friendly number and then subtract the rest by knowing our math facts. I know that 10 − 4 equals 6, so 20 − 4 = 16.) So knowing your math facts helps you solve this problem. • L et’s try another example together. 326 Number and Financial Literacy

12Lesson Adding and Subtracting Three- Digit Numbers by Applying Known Strategies 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 used an open number line to add • B 2.5 represent and solve problems involving the addition and subtraction of and subtract two-digit and three-digit numbers. whole numbers that add up to no more than 1000, using various tools and algorithms Possible Learning Goals • Uses and applies various strategies to add and subtract three-digit numbers • Records thinking, and clearly explains strategies and why they work • Selects a reasonable strategy to solve a problem and explains how it works • D ecomposes three-digit numbers in various ways, including into hundreds, tens, and ones • Adds and/or subtracts like place value groupings • R ecords thinking and verbally explains it • M akes connections between their strategy and other strategies PMraotcheesmseast:ical About the Problem solving, arreenfadlessocttnrianintgge,gasieenlsde,cptrionvgintogo, ls Students in grade three are expected to add and subtract three-digit ccoomnnmecutninicga,trinegpresenting, numbers using concrete materials and using student-generated and standard algorithms. While all students may not generate their own algorithms or methods per se, they may adapt what they have learned continued on next page Addition and Subtraction 327

Math Vocabulary: previously and apply it to new situations. For example, to calculate with dhoeupcneodnmrenpduosms, ebt,eencrosl,imnoepnoesse,, three-digit numbers, students may apply compensation or constant differences strategies they have used to mentally add and subtract two- digit numbers. They may also take a decomposing strategy that they used with a visual tool and adapt it so it can be used with quantities in numerical form. For instance, students who added by taking jumps of 100s, 10s, and 1s on an open number line may use the same thinking, but directly decompose the quantities in numerical form instead. Throughout this unit to date, students have been exposed to a variety of strategies, both through you (explicit teaching) and through the sharing of ideas with peers. This exposure allows students to develop the ability to select their own tools and strategies, depending on the problems they are solving. Strategies include: • Applying the inverse relationship of addition and subtraction and ••• UuAUsdssiiindnngiggncgtcohoamennscpdtoeamsnnustmbadtturiiofatfnacettritievnonegmcaeunassdiinn/togaripnalsaescqoeucviiavataliulveeengpcreroouppesrties The most important aspect of using different strategies is that students understand and can explain what they are doing. About the Lesson In this lesson and accompanying Math Talks, students use and adapt some of the strategies they have learned for mentally calculating two- digit numbers to adding and subtracting three-digit numbers. The goal is to expose students to a variety of strategies, so they can eventually select the strategy that best matches the requirements of the problem they are solving. In the lesson, they adapt adding place value groups on a number line to adding with numbers. In the Math Talks, they investigate the strategies of constant differences and of applying the inverse relationship of addition and subtraction to maintain equivalence. Materials: Minds On (10 minutes) chart paper, base ten • Show students the following problem with one possible solution on a number blocks line (record this information on chart paper before the lesson). Time: 50 minutes 378 + 145 0 300 400 470 510 518 523 • Explain that there are many ways to solve this problem, including the one on the chart paper. Have students turn and talk to their partner about what strategy was used in the example. 328 Number and Financial Literacy

• Ask what they notice about the size of the jumps. (e.g., They keep getting smaller.) Ask what this means. (e.g., Smaller and smaller amounts are added to the 300.) Ask what numbers need to be filled in above the arrows. Fill them in as students give explanations. 378 + 145 + 300 + 100 + 70 +40 + 8 + 5 0 300 400 470 510 518 523 • Ask how the problem was solved. (e.g., The numbers were added along place value lines, adding hundreds, then tens, and then ones.) • Ask students how they could represent this solution without the number line, using just the numbers. Ask how the numbers would be decomposed. As student explain, record the following steps. They can use concrete materials to decompose, if necessary. 378 + 145 300 + 70 + 8 100 + 40 + 5 300 + 100 70 + 40 8+5 400 + 110 + 13 510 + 13 523 • Make connections between the solution on the number line and the numerical solution so they know how the thinking is the same. Ask how this solution can help them add (e.g., dealing with friendlier numbers). Working On It (20 minutes) • Have students work in pairs. Have them solve some of the following problems by decomposing the numbers along place value lines. 526 + 249 375 + 125 473 + 252 691 + 219 • Students can record their solution on chart paper. They can use the solution from the Minds On as an example or create their own method of recording. • Encourage students to find another way to decompose the numbers (e.g., creating friendly numbers). • Students can use concrete materials to help them solve the problem. Addition and Subtraction 329

Differentiation • Select the problems that best suit the needs of your students. • For students who need more support, start with problems that do not typically require regrouping. Then, provide a problem that requires regrouping of only ones to tens. • For students who need more of a challenge, provide a problem with three addends. Assessment Opportunities Observations: Pay attention to whether students can decompose the numbers along place value lines and then combine the like groups. Can they record their thinking and explain what it means? Can they decompose the numbers in a different way to recompose the numbers? Conversations: If students are having difficulty adding some of the subgroups (e.g., 70 + 40), pose some of the following prompts: – W hat are you adding together? (e.g., the tens, 70 + 40) How could you decompose 40 so you can make 100 from the 70? Think of your number facts that equal 10. Let’s look at the number line to help us. (7 + 3 equals 10, so 70 + 30 = 100) What is left to add from the 40? What is your total now? (110) – How could you add the remaining 13 to 510? Think about how you could reach another decade number. (e.g., Add 10 and then add 3.) 330 Consolidation (20 minutes) • Strategically select various solutions. Since the highlighted strategy is adding along place value lines, students should have followed this method but may have recorded their thinking differently. Have the students explain how their recorded work connects to their explanations. Make connections among all of the recordings. • You may want to connect the written form of one example to a concrete example of the solution so students see the connection explicitly. • Ask students whether they found it helpful to add along place value lines. • Discuss some of the alternative ways that students decomposed the numbers. Below is an example of a possibility for each problem: 526 + 249: Decompose 526 as (525 + 1) and add 1 to 249 so it equals 250, then compose 525 + 250. 375 + 125: Decompose 125 as (100 + 25) and add 25 to 375 so it equals 400, then compose 400 + 100. 473 + 252: Decompose 252 as (250 + 2) and add 2 to 473 so it equals 475, then compose 475 + 250 either by adding along place value lines or decomposing further as 500 + 225. 691 + 219: Decompose 219 as (210 + 9) and add 9 to 691 so it equals 700, then compose 700 + 210. • Highlight how the above strategies are used to decompose into friendly numbers that are multiples of 25. Number and Financial Literacy

Materials: • Building Social-Emotional Learning Skills: Healthy Relationship chart paper, markers, concrete materials Skills: It is important for students to understand that learning is a process that involves thinking and effort and math is something you work at to get Teaching Tip better. Through whole-class discussions, students discover that the community can work together and learn from one another. Highlight Integrate the math situations when students offer an idea and other students build upon it or talk moves (see when an idea evokes a new way of looking at the problem. Role-play some page 8) throughout situations so students can experience listening and being responsive to each Math Talks to other. Include prompts such as: maximize student participation and – H ow can you express her idea in your own words? active listening. – Is there something that you can add on to that idea? – D o you agree with his idea? Why? – C an you give a real-life example of what she is talking about? Through these experiences, students can become more active listeners and not always look to the teacher when offering a response. Math Talk: Math Focus: Applying the inverse relationship of addition and subtraction to maintain equivalence through compensation About the Students can apply their understanding that addition and subtraction are inversely related, or ‘undo’ each other, to make calculations with three-digit numbers easier. For example, when carrying out a subtraction problem, they may decide to subtract more than is necessary and then add it back at the end to ‘undo’ the extra that was originally subtracted. While this is a powerful mental strategy, students can also record their thinking with paper and pencil when dealing with three-digit numbers so they remember what they did. Let’s Talk Select the prompts that best meet the needs of your students. • R ecord this equation horizontally, as shown: 623 − 397 = __ . • T urn and talk to your partner about how you might solve this problem. You can use concrete materials, draw a picture, or use the numbers. • W hat did you find? Have some students show their representations using concrete materials and drawings. • H ow can we solve this problem by only looking at the numbers? How could we round 397 so it is a friendlier number? (400) What do we have to add to it? (3) continued on next page Addition and Subtraction 331

Materials: • R ecord 397 + 3 = 400. Let’s circle the 3, because that is how much we added. • R ecord 623 − 400 =___. How could you solve this problem? (e.g., This chart paper, markers would be 223. I can count back by hundreds from 623.) • Is this the answer? Why? (e.g., No, because we took away 3 too many.) What could we do now to compensate for this? (e.g., We could add back the 3, 223 + 3 = 226.) Record the solution in the following manner. I am going to record our thinking for this problem. Explain the steps that I am recording. 623 − 397 397 + 3 = 400 623 − 400 = 223 223 + 3 = 226 • L et’s try another question. Look for friendly numbers that can help us make addition easier. Record 307 + 496. Turn and talk to your partner about how you could solve this. • W hat did you find? (e.g., We took away 7 from 307 and then added 300 + 496, which equals 796. Then we added the 7 that we took away. We added 4 of the 7 to equal 800 and then added the remaining 3, which equals 803.) Why can you do this? (e.g., We took away 7 at the beginning, so we had to add it back at the end.) Record their solution. • Did anyone have a different way? (e.g., We added 4 to 496, which equals 500, and then we added 307 + 500, which equals 807. Then we took away the 4 that we added at the beginning, which equals 803.) Why can you do this? Record their solution. • W e are learning that if we add too much to a number so it is easier to calculate, we can then subtract that amount at the end. Who can explain what we do if we subtract too much to make the number easier to work with? (e.g., We add it back at the end.) Math Talk: Math Focus: Subtracting three-digit numbers using constant differences About the The constant differences strategy is based on the fact that the difference between two numbers remains the same if you add or subtract the same amount from each number. For example, envision two numbers, 23 and 39, on the number line. If you add 1 to each number, they each move one space to the right on the number line to 24 and 40, thereby keeping a constant difference between them. This strategy can be especially helpful for creating friendlier numbers to use in calculations. 332 Number and Financial Literacy

Let’s Talk Select the prompts that best meet the needs of your students. • L ook at the problem 13 − 6. What are we trying to find? (e.g., the difference between the two numbers) Look at the number line. What does the difference between 13 and 6 look like? (e.g., the space between 13 and 6 on the number line) • W e can call this the distance between the two numbers. How could you calculate this distance? (e.g., We could count back from 13 to 6 or count up from 6 to 13.) Does it matter if we count back or up from one of the numbers? (No.) Why? (Because the distance remains the same.) • Let’s explore another strategy. Rather than dealing with 13, what would be a friendly number? (e.g., 10) What do we have to do to change 13 into 10? (e.g., Subtract 3.) What would that look like on the number line? (e.g., Move three spaces to the left.) How could we keep the distance between 13 and 6 the same if we take 3 away from 13? (e.g., Take 3 away from 6.) Let’s try that on the number line. If we move three spaces down from 13 at the same time as we move three spaces down from 6, what happens to the distance between the two new numbers? (e.g., It is the same.) So, do you agree that 13 − 6 = 10 − 3? • L et’s see if we can use this strategy with larger numbers. Look at this problem: 441 − 204. What are we trying to find in this problem? (e.g., the difference or the distance between the two numbers) • H ow could we apply our ‘same distance’ strategy to solve 441 − 204? Turn and talk to your partner. • W hat did you find? (e.g., We could add 9 to both numbers and now work with 450 − 213, then count back by 200, which equals 250, and then subtract 10 and then 3 more, which equals 237.) Let’s show what you did on an open number line. • D id anyone find another way? (e.g., We subtracted 1 from 441 and 1 from 204 to have a new problem of 440 − 203, and then we subtracted 200 from 440, which equals 240, then subtracted 3 more, which equals 237.) • D id anyone adjust the numbers and then add on to the higher number? (e.g., We subtracted 4 from 204 and 4 from 441 and had a new problem of 437 − 200. Then we counted up from 200 by 2 hundreds to 400, and then added 37. We added 237 altogether, so that is the distance between the numbers.) • Did we get the same solution when we adjusted the numbers different ways? Why? Did we get the same solution if we counted back or counted on from one number to the next? Why? Addition and Subtraction 333

13 14Lessonsand Directly Modelling Addition and Subtraction of Three-Digit Numbers with Regrouping 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 used base ten blocks to represent addition • B 2.5 represent and solve problems involving the addition and subtraction of and subtraction of two- digit numbers. whole numbers that add up to no more than 1000, using various tools and algorithms Possible Learning Goals • Solves addition and subtraction problems that involve regrouping three-digit numbers using concrete materials, and explains their strategy • Decomposes three-digit numbers along place value groups using concrete materials and recomposes them by exchanging quantities for equivalent amounts • Applies understanding of place value to decompose numbers into hundreds, tens, and ones • Exchanges equivalent amounts of base ten blocks and regroups in order to add and subtract • Identifies the quantity that a digit represents, based on its position in the number • Explains how each place value group is 10 times greater than the place value to the right • Clearly explains strategy and their actions with the materials, using mathematical language 334 Number and Financial Literacy

PMraotcheesmseast:ical About the Problem solving, arreenfadlessocttnrianintgge,gasieenlsde,cptrionvgintogo, ls Alex Lawson points out how students will often revert back to simpler ccoomnnmecutninicga,trinegpresenting, strategies when they are dealing with larger numbers that are less familiar to them (Lawson, 2015, p. 84). This is why it is important that grade three Math Vocabulary: students have several opportunities to solve addition and subtraction tptdheleraneccseoe,-mdovinapgeloiutsse,ne,cu,homgumrnbopdeuorrpess,,des, , problems with three-digit numbers using concrete materials. These regroup experiences help them to internalize the processes of the operations by physically acting them out with materials. While some students may be ready to represent their thinking with drawings or numbers, it is beneficial for all students to see the connections to concrete materials. Direct modelling involves using “manipulatives or drawings along with counting to represent directly the meaning of an operation or story problem” (Van de Walle & Lovin, 2006a, p. 158). With addition, for example, students count out both quantities that represent the addends, combine them, and then count a third time to find the sum. Direct modelling becomes challenging as students work with larger numbers, since there is often a shortage of materials to represent such large quantities. It can also be inefficient and time consuming to count out concrete materials to solve a problem such as 137 + 258. Base ten blocks are valuable since they reflect the proportional relationships within our number system, with each representation being 10 times greater than the previous one. Since students cannot break the models apart, they must make exchanges to acquire smaller, equivalent units. These physical actions help students understand when regrouping needs to take place, what the equivalent amounts look like, and how the process works. About the Lesson In the following two lessons, students add and subtract three-digit numbers using base ten blocks and place value mats. In the first lesson, they explore how to exchange equivalent blocks and regroup them in order to add three-digit numbers. Similarly, in the second lesson, students investigate how to regroup with the materials in order to subtract. Addition and Subtraction 335

13Lesson Directly Modelling Addition of Three- Digit Numbers with Regrouping Materials: Minds On (15 minutes) base ten blocks, • Read the following problem aloud: BLM 2: Place Value Mat, chart paper, markers – T here are 68 soccer balls and 57 basketballs in the storage room. How many balls are in the storage room? Time: 55 minutes • Have students estimate the solution. Pose some of the following prompts: – W ill the answer be less than or greater than the numbers in the problem? Why? – D o you think there are more than 100 balls in the storage room? Why? Do you think there are more than 500? more than 200? • Ask students how they could represent the problem using base ten blocks and a place value mat (BLM 2). They can turn and talk to their partner. • Discuss solutions as a class, with different students explaining their strategies using the base ten blocks. Once the two amounts are created, ask what the problem is with the unit blocks in the ones section of the place value mat and how this can be resolved. Have another student regroup according to the students’ instructions. Ask what the solution is to the problem. • Students should notice that there are now more than 10 groups of tens. Ask how this can be resolved. Discuss how the flat represents 10 rods. Lay 10 rods on top of a flat to prove equivalency. Have a student demonstrate how to regroup. Ask what the solution is. • Review how there can only be nine units in each place value section because there cannot be a number represented by two digits in one column. Working On It (20 minutes) • S tudents work in small groups of two to four students. Provide each group with base ten blocks and BLM 2: Place Value Mat. Post the following problem, read it together so all students understand the context, and challenge students to solve it in at least one way. You can change the context so it is more meaningful for your students. – T here are 176 fiction books and 238 non-fiction books in the library. How many books are there altogether? 336 Number and Financial Literacy

• S tudents solve the problem and leave the base ten blocks for one of their solutions in place so they can be shared in the Consolidation. Students can record their alternative solutions on chart paper using diagrams of the base ten blocks. Differentiation • Change the numbers to suit the needs of your students. For example, rather than having them regroup two times in one problem (e.g., regrouping ones and then regrouping tens), you may use numbers that require regrouping only one of the place value groups. • For students who need more of a challenge, adjust the numbers so that no tens will be left after regrouping (e.g., 166 + 238). Students can figure out how they can use a zero as a placeholder for one of the place value groups. • For students who need more of a challenge, add a third element to the problem, such as “There are also 64 magazines in the library. How many pieces of reading material are there?” Assessment Opportunities Observations: Although the problem is a closed question with only one solution, students are challenged with figuring out how to regroup twice in the same problem. If students run into difficulties, can they rethink the problem and try a new strategy? Can they apply the concepts they learned with two-digit numbers to three-digit numbers? (e.g., as soon as there are 10 units in a column [place value group], students need to regroup) Can they interpret the concrete representation and express the sum as a number with digits? Conversations: It is best to let students struggle a bit with the problem so they can work as a group and try to reason through the problem. If they are stuck, ask a few prompts to get them started. • What are you being asked to do? • What operation do you think you are doing? • What could you do to get started? (e.g., represent the two numbers) • How could the base ten blocks help you? Consolidation (20 minutes) • Strategically pick two or three groups to explain how they solved the problem. If possible, pick a group that started with a strategy that didn’t work and so had to try a new strategy. You can move as a class to their concrete representations so they can focus their explanation on what they did with the materials. • With each solution, record students’ strategies and what they did with the concrete materials using pictures and numbers. Explicitly connect the mathematical language and the numerical representations to the concrete materials. For example, if you record that 200 + 100 = 300, ensure that Addition and Subtraction 337

students see that the equation connects to 3 flats, formed by combining 2 flats and 1 flat. This recording can become an anchor chart for adding with regrouping. Information can be added to the anchor chart in later lessons as students develop more strategies.  Possible solutions include: – S tudents represent both addends on their place value mats, decomposing each into hundreds, tens, and ones. They combine the two groups and count the parts, starting with the hundreds, then the tens, and then the ones. They may regroup at the end or as they proceed. – S tudents represent one addend and add the quantity of the second addend in parts, first adding the ones and regrouping if necessary, then adding the tens and regrouping if necessary, and finally adding the hundreds. • Have other students explain the steps that their peers described, using their own words. Refer to the anchor chart so students continually make connections between the representations and verbal explanations. Further Practice • Offer many opportunities for your students to add and regroup with concrete materials such as base ten blocks. If necessary, scaffold the learning so they are dealing with regrouping only one place value group before moving to regrouping twice in the same question. Have students verbally explain their strategies to you and to others so they can internalize the process. 338 Number and Financial Literacy

14Lesson Directly Modelling Subtraction of Three-Digit Numbers with Regrouping Materials: Minds On (15 minutes) base ten blocks, • Review the anchor chart for directly modelling addition of three-digit BLM 2: Place Value Mat, chart paper, markers numbers with regrouping from the previous lesson. Have students explain one or two of the solutions in their own words. Reinforce the mathematical Time: 55–65 minutes language, such as adding 2 tens and 7 tens rather than adding 2 and 7. Include terms such as ‘regrouping,’ ‘trading,’ and ‘exchanging.’ • Show students the following problem and read it over together, then ensure everyone understands it. You can change the context so it is more meaningful for your students. – Jesse has 426 trading cards. He gives 167 to his little brother. How many cards does Jesse have left? Working On It (2–3 periods, 20–30 minutes per period) • Have students work in pairs, triads, or groups of four, depending on how many base ten blocks are available. Challenge them to solve the problem in at least two ways using the base ten blocks and a place value mat (see BLM 2). They can leave their base ten representations out, displaying one of their solutions. They can record any other solutions on chart paper, using diagrams and numbers to represent the base ten blocks. Differentiation • Change the numbers so they suit the needs of your students. For example, rather than having them regroup two times in one problem (e.g., regrouping ones and then regrouping tens), use numbers that require regrouping only one of the place value groups. • For students who need more of a challenge, adjust the numbers so they need to regroup when a zero is used to represent one of the place value columns (e.g., 410 − 167). Students can figure out how they need to regroup in order to take the ones away. Addition and Subtraction 339

Assessment Opportunities Observations: Pay attention to how students deal with having to regroup twice in the same problem. • If students run into difficulties, can they rethink the problem and try a new strategy? • C an they apply the concepts they learned with two-digit numbers to three-digit numbers? (e.g., If there are not enough units to take away in a place value column, they need to exchange and regroup from the column to the left.) • Do they try to subtract the smaller quantity from the larger quantity, even though the smaller quantity is part of the whole? (e.g., when subtracting the ones in 426 − 167, they take 6 away from 7) Conversations: It is best to let students struggle a bit with the problem so they can work as a group and try to reason through the problem. If they are ‘stuck,’ ask a few prompts to get them started. • What are you being asked to do? • What operation do you think you are doing? • What would be something you could do to start? (e.g., show the amount of cards Jesse has) Represent that number and then we will read the problem again. Will there be more or fewer cards at the end of the problem? How does this help you know what to do? Consolidation (20 minutes) • Strategically pick two or three groups to explain how they solved the problem. If possible, pick a group that started with a strategy that didn’t work at first and had to try a new strategy. As a class, move to the groups’ concrete representations of the solution. On chart paper, record their strategies, using pictures of the base ten blocks (e.g., squares, sticks, and dots) and numbers. Explicitly connect the mathematical language and the numerical representations to the concrete materials. For example, if you record 400 − 100, ensure that students see the connection between the equation, the representation, and the action of 1 flat being removed from 4 flats. • The recorded student strategies can become the anchor chart for subtracting three-digit numbers with regrouping. Information can be added to it in later lessons when students develop further strategies. Possible solutions include: – Students represent the whole on their place value mats, decomposing it into hundreds, tens, and ones. They remove the amount being taken away in parts, starting with the hundreds; regrouping once they move to the tens and realize there are not enough tens to take away; then regrouping again when they move to the ones, and realize there are not enough ones to take away. – S tudents represent the whole, decomposing it into hundreds, tens, and ones, They remove the amount being taken away in parts, starting with 340 Number and Financial Literacy

Materials: the ones and then regrouping as necessary as they proceed from right to left. base ten blocks, – S tudents represent the part amount in the problem (e.g., 167), BLM 2: Place Value decomposing it into hundreds, tens, and ones. They then add parts Mat (think-addition), starting with the hundreds, then the tens, and then the ones, until the whole amount is reached. Students regroup where Time: 60 minutes necessary and then count the base ten blocks they added. • Have other students explain the steps that their peers described in their own words. Refer to the anchor chart so students continually make connections between the representations and explanations. • Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: As teachers, we set goals for our students in accordance with the curriculum and their personal readiness. It is also important that they are involved in the goal-setting and monitoring process. Only then will students truly appreciate how much they are growing and progressing. A very simple self- or peer-reflection that students might use is “Two stars and a wish.” If self-assessing, students record two things that they’ve done well or learned (e.g., I persevered to solve the problem, or I subtracted two-digit numbers correctly) and one wish (or goal) such as “I’ll try another way when I get stuck, or I’ll practise rounding numbers using the hundreds chart to help me.” Further Practice • Many of your students will need several opportunities to practise subtracting with regrouping before the concept is internalized. Offer many experiences for them to subtract using base ten blocks, adjusting the numbers to reinforce the concepts they may find difficult. Scaffold the learning so they are first regrouping one place value group at a time before moving to regrouping two times in one question. Reinforce the relevant language and encourage students to verbally explain their strategies so they become internalized. Math Talk: Math Focus: Subtracting with regrouping that involves zero Let’s Talk Select the prompts that best meet the needs of your students. • Look at this problem: 402 − 193. What are you being asked to do? Estimate what the answer will be. Why do you think so? • Work with your partner to solve the problem using concrete materials or drawings. • What did you do first? (e.g., We represented 402.) Show us with your materials what that looks like. How is your representation different from numbers we have represented in other problems? (e.g., There weren’t any tens rods in the tens column.) Why does this make sense if you are looking at the digits in the number? continued on next page Addition and Subtraction 341

Teaching Tip • How did you start subtracting 193? (e.g., We had to subtract 3 ones from 2 Integrate the math ones.) How do you know that you are not subtracting 2 ones from 3 ones? talk moves (see page 8) throughout • What did you do? (e.g. We found we couldn’t subtract 3 ones from 2 ones, Math Talks to maximize student so we thought we could exchange a ten for 10 unit blocks, but we couldn’t participation and because there aren’t any tens.) active listening. • What could you exchange? (e.g., We could exchange a hundreds flat.) Let’s visualize this scenario without using our base ten blocks. How many ones would you get if you exchanged a flat, and where would you put them? (e.g., You would get 100 ones and you would put them in the ones column.) How many ones would there be now? (102 ones) • Visualize how many ones would be left after you subtracted 3 ones. (99) Can you have 99 ones in the ones column? What would you need to do? (e.g., We would need to make groups of 10.) How many rods of ten do you think you could make? (9 rods) Where would you put them? (e.g., in the tens column) Draw a picture of the 3 flats, 9 rods, and 9 ones to help them visualize. So, what would you have now and what do you still have to subtract? (e.g., We have 3 flats, 9 tens, and 9 ones, so 399 and we still need to subtract 9 tens and 1 flat.) Visualize what you would have left. (2 flats, no rods, and 9 ones) On the drawing, put an X through these amounts. • Why is it not very practical to exchange flats for ones? (e.g., We don’t have enough in our class for everyone to have 100.) • What could we do instead, so we don’t need so many ones? (e.g., We could exchange the flat for 10 rods.) Let’s do that with our base ten blocks. What do we have now? (3 flats, 10 rods, and 2 ones) What can we do so we can subtract 3 ones? (e.g., Exchange 1 rod for 10 ones.) Let’s do that. Where do we put the 10 ones? How many ones are there now? (12) What are we subtracting? (3) How many do we have left? (9) • What do we do now? (e.g., We have 9 rods and we need to subtract 9 tens.) What do we have left? (0 rods) • Is there any more to subtract? (e.g., We need to subtract 1 flat.) What do we have left now? (2 flats, 0 rods, and 9 ones) Is that the same as if we had exchanged 1 flat for 100 ones? (yes) • What is the answer to our problem? How do we read it? (two hundred nine) How do we write that number? Can we write it as 29? Why? What do we need to do to make sure the number is written correctly? (e.g., Put a 0 in the tens column.) How do you see that 0 represented in the problem? (e.g., There are no rods in the tens place.) • Zero plays an important role when we write numbers. It lets us know that there are none of those groupings, but it holds the place open so we don’t read the number incorrectly. • Visualize what 430 would look like using base ten blocks. Explain what the zero tells us in this number. 342 Number and Financial Literacy

15Lesson Using Partial Sums and Differences Algorithms 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 Possible Learning Goals • Uses partial sums and partial differences to add and subtract three-digit numbers • Represents their solutions using numbers and makes connections to the concrete and/or pictorial representations • Explains strategies using appropriate mathematical language (e.g., adding 7 tens [or 70] and 2 tens [or 20], rather than adding 7 and 2) Teacher • Decomposes numbers into hundreds, tens, and ones to make calculations Look-Fors easier Previous Experience • D ecomposes numbers in alternate ways to make calculations easier with Concepts: • Adds or subtracts like place value groups and identifies the partial sums or Students have had experiences adding and differences subtracting two-digit numbers using a variety of • Explains how they calculated the partial parts and how the parts connect to strategies and decomposing their concrete or pictorial representation three-digit numbers in a variety of ways. • Explains why they add partial sums or partial differences together to find the solution (e.g., to find the total of the partial parts) • Explains how their written solution connects to their concrete or pictorial representations and the actions taken Addition and Subtraction 343

PMraotcheesmseast:ical About the Problem solving, rrreeepfalreseocstneininngtgi,nacgon,ndnpercotviningg, , The partial parts algorithms can be used for both addition and subtraction communicating and can be very useful for students who have a good understanding of part- whole relationships. Partial Sums: The partial sums algorithm involves adding various parts of the numbers separately, and then adding the partial sums together. This is often, but not always, done along place value lines. In the example below, numbers are decomposed into hundreds, tens, and ones, and then added together, either adding hundreds first and proceeding to the right or adding ones first and proceeding to the left. It is important to connect the numerical representation to the concrete or pictorial representations so students develop a better understanding of what the numbers mean. Example: 245 + 615 245 + 615 800 (2 hundreds + 6 hundreds) 50 (4 tens + 1 ten) + 10 (5 ones + 5 ones) 860 Partial Differences: The partial differences algorithm involves subtracting parts of the numbers separately and then adding the subtracted parts together to find out how much was subtracted in total. Students may also decompose the number being subtracted into friendly numbers and then progressively subtract them from the whole (see the example below). The algorithm can be connected to concrete or pictorial representations or to counting backward on the number line. Example: 615 − 245 615 − 200 415 −5 410 − 10 400 − 30 370 When using partial sums and partial differences, the order in which steps are carried out is flexible. For example, students can start with the greatest place value position and move left to right, or begin with the lesser place value position and move right to left. Students can also decompose and then order steps in ways that simplify calculations. For example, they can decompose numbers and sequence the operations in a way that avoids regrouping. 344 Number and Financial Literacy

Math Vocabulary: About the Lesson vpdeairfrtftieicaraelln,scuhemosrsiz,opnatartli,al In this lesson, students explore partial parts strategies and use various Materials: recording methods, including numerical representations that are chart paper, marker, oriented vertically and horizontally. base ten blocks, BLM 2: Place Value Minds On (20 minutes) Mat Time: 60 minutes • Write the number 345 on chart paper, and ask students what they know about the number. (e.g., it has three digits; it has 3 hundreds, 4 tens, and 5 ones) • Have a student compose the number using base ten blocks on a place value mat (BLM 2). • Point out the labels on the columns above the number 345 (Hundreds, Tens, Ones). • Pose the following prompts to reinforce the value of digits when they are in the various columns: – W hat digit would change if 2 tens were added? – W hat digit would change if 4 ones were subtracted? – W hat digit would change if 6 hundreds were added? – W hat digit would change if 30 was subtracted? • Record the following problem vertically: 345 + 124 • Have students turn and talk to their partner about how they may solve it. • Have one or two students demonstrate to the class how to solve the problem using base ten blocks. • Discuss and record the other strategies students had and connect them to the concrete example. Include the example below. Have students explain the strategy while you record and explain the procedure. 345 + 124 400 (300 + 100) add the hundreds 60 (40 + 20) add the tens + 9 (5 + 4) add the ones 469 • Record the problem horizontally and ask what the solution may look like. Make connections between the two recordings and to the concrete example. 345 + 124 = 400 + 60 + 9 = 469 Addition and Subtraction 345

• Repeat using an example of a student’s strategy that adds the ones first, then the tens, and then the hundreds. • Present and record the following problem. Have students talk to a partner about how to solve it. Encourage them to adapt some of the strategies from the addition problem. 345 − 135 • Ask students what the differences would be if they subtracted the columns along place value lines. They might start either with the hundreds or the ones (see example recording of starting with the ones below). Accept and record both ways. Have a student demonstrate what this would look like using concrete materials. Connect the written partial differences to the concrete example. Ask how they calculate the final answer. Ask why they add when this is a subtraction problem. (e.g., We subtracted in parts and now we have to find out how much we subtracted altogether.) 345 − 135 0 (5 − 5) 10 (40 − 30) + 200 (300 − 100) 210 345 − 135 = 200 + 10 + 0 = 210 • Show students an example that requires regrouping. 345 + 165 • Ask how many hundreds, tens, and ones will result when these numbers are added together (see example recording below). Ask how adding 4 tens and 6 tens results in 100. Have a student demonstrate using concrete materials. 345 + 165 400 (300 + 100) 100 (40 + 60) 10 (5 + 5) 510 • Have students show how the partial sums connect to the concrete example. Ask what they think they do next and why. 346 Number and Financial Literacy

Working On It (20 minutes) • Students work in pairs to solve the problems below. Students can choose to solve using concrete materials such as base ten blocks and a place value mat (BLM 2), or by drawing their solutions using squares, sticks, and dots. They can then represent their concrete or pictorial representations using numbers. Although many students may use place value groupings to add and subtract, accept other decomposition strategies as well. • Read the problems aloud. Change the contexts as needed so they are more meaningful for your students. – P roblem 1: Georgie and Manny counted the new books in the library. They counted 286 fiction books and 267 non-fiction books. How many new books are there altogether? – P roblem 2: There are 432 vehicles in the arena parking lot before the hockey game and 149 vehicles in the parking lot after the game. How many vehicles left right after the game? • As a class, have students estimate what the solutions might be before they begin. Share their estimation strategies. Differentiation • Change the numbers to meet the needs of your students. You may want to have questions with no regrouping while students are learning the process of recording, or you may want to limit the regrouping to one place value group. • For students who need more of a challenge, you could revise the problem to offer a different problem structure. (e.g., There are some fiction books in the library and 267 non-fiction books. Altogether, there are 453 books. How many fiction books are there?) Assessment Opportunities Observations: Pay attention to how students are decomposing the numbers and creating partial sums/differences. • Can they explain how the new groupings make calculations easier? • Can they record their steps using numbers and link the numbers to their concrete or pictorial representations? • Can they explain where the partial sums/differences come from? Conversations: If students are having difficulty understanding what the partial sums/differences represent, pose some of the following prompts: – You recorded that as 4 tens plus 7 tens is 11 tens. What is 11 tens? (e.g. I am not sure.) Look at your base ten blocks. Where are the 4 tens? What is 4 tens? Try counting them. (40) Where are the 7 tens in your base ten blocks? Count to find out how much 7 tens is. (70) What action did you do to get 11 tens? (e.g., I combined all the rods.) – How can you count the 4 tens and 7 tens? (e.g., skip count by 10s) What do you get? (110) How can you record that as a partial sum? What could you do with the 11 rods? (e.g., Exchange 10 rods for 1 flat.) What do you have now? (e.g., 1 hundred and 1 ten) Which columns will you record these numbers in? Addition and Subtraction 347

Consolidation (20 minutes) • Students meet with another pair. One student in the first pair explains what they did with the concrete materials or drawings for Problem 1. Then the second student explains the process using numbers and shows how the numbers connect to the concrete example. Encourage the other pair to ask questions if explanations are unclear. Pairs can switch roles and the second pair can explain how they solved Problem 2. • Strategically select solutions that show different ways of decomposing the numbers or different ways of recording the partial parts. Ask why the partial parts make calculations easier. • Add any new information to the chart started in the Minds On. 348 Number and Financial Literacy

to16 21Lessons Using the Standard Algorithm to Add and Subtract Multi-digit Numbers Math Number Curriculum Expectations • B 1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and Previous Experience strategies with Concepts: Students have used base • B 2.3 use mental math strategies, including estimation, to add and subtract whole ten materials and place value mats to compose and numbers that add up to no more than 1000, and explain the strategies used decompose three-digit numbers and have added • B 2.5 represent and solve problems involving the addition and subtraction of and subtracted three-digit numbers using a variety of whole numbers that add up to no more than 1000, using various tools and strategies. algorithms PMraotcheesmseast:ical About the Problem solving, rrreeepfalreseocstneininngtgi,nacgon,ndnpercotviningg, , The standard North American algorithm, or process, for multi-digit communicating addition and subtraction is based on place value. For example, with addition, it requires students to add the ones and regroup if necessary, add the tens and regroup if necessary, and then add the hundreds. This may be opposite to what students have been doing when using invented strategies to add multi-digit numbers. Often, students decompose and add the hundreds first, followed by the tens, and then the ones. The standard algorithm may require them to shift their practice. It’s important that students know that this standard algorithm isn’t necessarily the best algorithm nor is it the ‘right’ way to add. Although you are intentionally teaching students the standard algorithm, they should be encouraged to use strategies or algorithms that make sense to them outside of these specific lessons. When introducing a standard algorithm, it’s good practice to have students work through the steps using concrete objects like base ten materials. Marian Small points out that no matter what algorithm students use, they should be able to explain what they are doing and why when following the steps (Small, 2009, p. 163). NOTE: The term ‘algorithm’ does not need to be introduced. Instead, students can view this process as another strategy that they can add to their repertoire for solving two-digit addition problems. You may wish to call it a ‘place value strategy.’ continued on next page Addition and Subtraction 349

Math Vocabulary: About the Lessons rromeeppgaerrtero,asurtepoion,wntp,sal,mal ccpooeldaluvecmale,lunvesa,lue The following lessons introduce students to a standard algorithm for adding and subtracting two-digit numbers. Once students understand and have mastered the process, they can apply and extend it to adding and subtracting three-digit numbers. It is important to explicitly highlight how the process for two-digit numbers extends to three-digit numbers due to the patterns built in to our place value system. Two anchor charts (‘Place Value Addition’ and ‘Place Value Subtraction’) begun in Lesson 16 will be completed and referred to throughout the lessons. 350 Number and Financial Literacy

and16 17LessonsUsing the Standard Algorithm to Add and Subtract (Without Regrouping) Teacher Possible Learning Goals Look-Fors • Uses a standard algorithm involving place value to add and subtract two- and three-digit numbers, using concrete materials, drawings, and numbers, and explains the process • Composes and decomposes numbers by place value groups in a variety of ways • F ollows the steps of the algorithm to accurately add and subtract two- and three-digit numbers that do not require regrouping • Explains the process of adding using the standard algorithm as a strategy • Explains the process of subtracting using the standard algorithm as a strategy • Accurately represents the standard algorithm using concrete materials and/or drawings and solves problem • Accurately records the processes using numbers organized in a vertical format Addition and Subtraction 351

16Lesson Adding and Subtracting Two-Digit Numbers (Without Regrouping) Materials: Minds On (15 minutes) BLM 50: Operational • Have students work in pairs. Give each pair base ten blocks and BLM 50: Place Value Mat, base ten blocks Operational Place Value Mat. Time: 55 minutes • Have students model the number 35 in different ways (e.g., 35 ones, 2 tens and 15 ones, etc.), and count their representations. • Ask students how they can model 35 with the smallest number of blocks. • Have students represent 2 tens and 13 ones. Ask how they can regroup this using the smallest number of blocks. Highlight the term ‘regrouping’ as representing an equivalent quantity in a different way. Add this term to the Math Word Wall. Working On It (20 minutes) • Pose the following problem. Change the context so it is relevant to your students. – T here are 42 pencil crayons and 36 markers. How many writing utensils are there? • E xplain that while there are many strategies to solve this problem that they are going to use a specific strategy that is accepted and used in many countries. • H ave students represent 42 in the first row of the mat and 36 in the second row using the fewest base ten blocks. Explain that they first add the ones and record the sum below (in the third row), and then add the tens. • Pose a new problem and challenge students to solve it using the same strategy of starting with the ones and then moving to the tens. – T here are 58 pencils and 34 erasers. How many more pencils are there than erasers? • R eview students’ strategies and solutions. • G ive students another addition problem and a subtraction problem that do not require regrouping to solve in their pairs. 352 Differentiation • Adjust the numbers in the problems so they meet the needs of your students. • Work with students in a small-group guided lesson to support those who are having difficulty with the process. You can use multiples of ten (e.g., 20 + 30) to scaffold the process. Number and Financial Literacy

Assessment Opportunities Observations: Observe the process that students use to solve the problem. Do they understand the steps in the algorithm? Can they explain the process as they perform it? Consolidation (20 minutes) • Review the process that students used to solve the problems. • C o-create two anchor charts entitled ‘Place Value Addition’ and ‘Place Value Subtraction.’ Record the steps in each process, but leave space to add the regrouping steps and the ‘add/subtract the hundreds’ step (highlighted below) in future lessons. Place Value Addition – M odel each number using place value (hundreds, tens, ones) – R egroup if you have more than 9 ones – A dd the ones – R egroup if you have more than 9 tens – A dd the tens – A dd the hundreds – R ecord your answer Place Value Subtraction – M odel each number using place value (hundred, tens, ones) – If there aren’t enough ones to subtract, regroup – S ubtract the ones – If there aren’t enough tens to subtract, regroup – S ubtract the tens – S ubtract the hundreds – R ecord your answer Addition and Subtraction 353

17Lesson Adding and Subtracting Three-Digit Numbers (Without Regrouping) Materials: Minds On (20 minutes) BLM 50: Operational • As a class, have students work in pairs using a shared operational place value Place Value Mat, base ten blocks mat (BLM 50) and base ten blocks. Present the problem below (addition, part-part-whole, whole unknown). Time: 60 minutes – T here were 234 cars and 542 trucks in the parking lot at the mall. How many vehicles were there altogether? • Work through this problem interactively as you show students how to use the standard algorithm and operational place value mat. A sample script follows: 1. W hile there are many strategies to solve this problem, we’re going to use our operational place value mats and base ten blocks. 2. W hat operation do we need to use to determine the number of vehicles? [Write these numbers and an addition symbol stacked horizontally on the left.] How do I represent these numbers using base ten blocks? Hundreds Tens Ones 234 +5 4 2 354 Number and Financial Literacy