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3. U sing this strategy, we start adding right to left. Which column do we start with? What do we do? (e.g., Add 4 ones and 2 ones, then add 3 tens and 4 tens, then add 2 hundreds and 5 hundreds.) How would I record this using numbers? (6 in the ones column, 7 in the tens column because there are 7 tens [70], and 7 in the hundreds column because there are 7 hundreds [700]) How do we read the answer? Hundreds Tens Ones 234 +5 4 2 776 7 7 6 • Tell students that now they will work in pairs to solve a problem on their own using their operational place value mats. Working On It (20 minutes) • H ave students work in pairs. Read the problem below aloud and ask students what operation they could use. – A t lunch time, there were 687 vehicles in the mall parking lot. At 5 o’clock in the afternoon, there were 255 vehicles remaining. How many vehicles left the parking lot? • E xplain to students that they are going to use subtraction to solve the problem, and they should try to follow the same strategy they used in the Minds On. • W hen students are done, they can create their own word problem involving addition, solve it, and then record it on a separate sheet of paper. Differentiation • Depending on the needs of your students, you may decide to break this lesson into two, dealing only with addition in the first lesson and subtraction in the second. Addition and Subtraction 355

• Adjust the numbers in the problems to create a greater or lesser challenge. • If students need more support prior to solving problems independently, solve a few more problems interactively. • To support those students who are having difficulty with the process, carry out a small-group guided math lesson. Ensure students understand why they are following the steps and can explain their process as they model it. Use problems involving multiples of ten to make operations easier if necessary (e.g., 600 − 400). Assessment Opportunities Observations: Observe the process students use to solve the problem. • Do students understand the steps involved in using the algorithm? • Can they explain the process as they perform it? Conversations: If students are having difficulty explaining the process, pose some of the following prompts: – Why did you start with removing the ones blocks? How do you know where you will move next on the place mat? – You just removed some rods from your place value mat. Why did you do this? How does this link to the problem? – Y ou have the number 3 under the hundreds column. What does the 3 represent? – W hat does your answer represent? Consolidation (20 minutes) • Meet as a class. Discuss what was different about how they solved the addition problem(s) compared to the subtraction problem(s). • Selectively choose one pair to demonstrate their strategy for the given Working On It subtraction problem using concrete materials. Periodically check with the rest of the class about whether they used the same process. Discuss any differences. Explain that, while there may be other ways to solve the problem, this time they needed to follow the steps outlined in the Minds On. • Have another pair of students explain how they recorded their solution using numbers. Compare how the process is the same and different for when they recorded the standard addition strategy. • Add the new second-last step (‘add/subtract the hundreds’) for three-digit numbers to the two anchor charts (‘Place Value Addition’ and ‘Place Value Subtraction’) started in the previous lesson, as shown below. The highlighted steps are still to come. Place Value Addition – M odel each number using ones, tens, and hundreds – A dd the ones 356 Number and Financial Literacy

Materials: – R egroup if you have more than 9 ones chart paper – A dd the tens – R egroup if you have more than 9 tens Teaching Tip – A dd the hundreds – R ecord your answer Integrate the math talk moves (see Place Value Subtraction page 8) throughout – M odel the whole number using ones, tens, and hundreds Math Talks to – If there aren’t enough ones to subtract, regroup maximize student – S ubtract the ones participation and – If there aren’t enough tens to subtract, regroup active listening. – S ubtract the tens – S ubtract the hundreds – R ecord your answer • B uilding Social-Emotional Learning Skills: Critical and Creative Thinking: Following a procedure can seem meaningless for students, especially if they are accustomed to using their own strategies to solve problems. Explain that there are many ways to solve problems and that this way is just one that is generally accepted by many people. People record it in the same way so it can be understood by anyone who may see the solution. It is standard, just like a metre is standard and is the same anywhere in the world. Tell students that this is known as a ‘convention’ that has been agreed upon by many people. Mathematicians have conventions so they can share and understand each other’s thinking. Explain that students will sometimes be expected to carry out and record their work using this strategy so everyone understands what is happening, while other times they can use their own strategies to solve problems. Math Talk: Math Focus: Adding three-digit numbers without regrouping; recording and reading the question in a vertical format Let’s Talk Select the prompts that best meet the needs of your students. • R ecord 231 + 426 = on the board or on chart paper. • I often record number sentences horizontally like in this example, but we can also record them vertically as well.  231 + 426 continued on next page Addition and Subtraction 357

Materials: • What is different about the two representations? (e.g., There is no equal sign base ten blocks, in the vertical representation.) In the vertical format, the answer goes under BLM 50: Operational the line. Place Value Mat • How are the numbers lined up in the vertical format? (by hundreds, tens, and ones) • Using the algorithm, what do we add first? Where do you think you put the total number of ones? (e.g., under the ones column) • What are we adding when we are working in the tens column? (e.g., 3 tens and 2 tens, which equals 5 tens, or 30 plus 20, which equals 50) How will we record the sum? (e.g., put a 5 under the line in the tens column) What does the 5 represent? (50) • What are we adding in the hundreds column? (200 + 400, which equals 600) How do we represent the 600? (e.g., Put a 6 under the line in the hundreds column.) • What is our final answer? (657) • What do you think is important when adding or subtracting numbers in the vertical format? (e.g., to accurately line up the numbers in the correct place value columns) Further Practice • Students can exchange the addition problem they created in the Working On It with another group, solve each other’s problems, and then compare their solutions. • Students solve the following problems using BLM 50: Operational Place Value Mat and base ten blocks, and show their solutions in their Math Journals. – F ind two three-digit numbers that have a sum between 700 and 800. – F ind two three-digit numbers that have a difference between 100 and 200. 358 Number and Financial Literacy

and18 19Lessons Using the Standard Algorithm to Add Multi-digit Numbers (with Regrouping) Teacher Possible Learning Goals Look-Fors • Uses a standard algorithm to add two- and three-digit numbers that require Previous Experience with Concepts: regrouping Students have had experience adding and • Understands and models equivalence by composing and decomposing subtracting two- and three-digit numbers using numbers in a variety of ways the standard algorithm without regrouping. • Represents two- and three-digit addition with regrouping using concrete materials, visuals, and/or numerals • Follows the steps of the algorithm to accurately add two- and three-digit numbers with regrouping • Accurately represents addition of three-digit numbers using concrete materials, drawings, and/or numbers • Composes and decomposes numbers into hundreds, tens, and ones • Understands and can explain the value of the digits in a three-digit number • Explains the process of adding and subtracting with regrouping • Accurately records the process using numbers organized in a vertical format About the Marian Small states that the terms ‘regroup,’ ‘trade,’ and ‘exchange’ should be used rather than ‘carry’ or ‘borrow’ because “carrying and borrowing have no real meaning with respect to the operation being performed, but the term ‘regroup’ suitably describes the action the student must take” (Small, 2009, p. 170). About the Lessons In Lesson 18, students learn about regrouping when adding two-digit numbers using the standard North American algorithm. In Lesson 19, students extend their understanding to adding three-digit numbers. Addition and Subtraction 359

18Lesson Adding Two-Digit Numbers (with Regrouping) Materials: Minds On (15 minutes) base ten blocks, • Students work in pairs. Have them represent 38 in at least two ways. BLM 50: Operational • As a class, discuss students’ representations. Have them prove that all of the Place Value Mat, chart paper representations are equivalent. Use and review the term ‘regrouping.’ Time: 50 minutes Working On It (20 minutes) • R eview the anchor chart for Place Value Addition from the previous lessons. Tell students that they are going to use this strategy to solve a problem. Explain that they may need to make some adjustments, but they still need to follow the general steps of adding the ones and then adding the tens. • Present the following problem and change the context as needed so it is meaningful to your students. – T here are 28 students in grade two and 34 students in grade three. How many students are in grades two and three altogether? • Students work in pairs. Provide them with base ten blocks, BLM 50: Operational Place Value Mat, and chart paper. They can record their solution on the chart paper, representing the base ten blocks as sticks and dots. Differentiation • Adjust the numbers so they are most suitable for your students. • Some students may benefit from working with you in a small-group guided lesson. Assessment Opportunities Observations and Conversations: As students work on solving the problem, observe how they manipulate the base ten blocks. Ask them to explain their process of regrouping. Listen for the use of place value terms, such as “I regrouped the ones.” Also note how students represent regrouping visually on their papers. Do they clearly show the regrouping? Do they add the ones and then the tens? Ask students to refer to the steps in the anchor chart, if necessary. 360 Consolidation (15 minutes) • Choose several students to share their work. Have students demonstrate the regrouping physically, using the base ten blocks. Connect the physical representations to the drawings of sticks and dots on the chart paper. • Reinforce correct place value terms. Refer to the steps in the anchor chart as students explain the process. Ask what step needs to be added to the chart to account for any necessary regrouping. (“Regroup if you have more than 9 ones.”) Number and Financial Literacy

19Lesson Adding Three-Digit Numbers (with Regrouping) Materials: Minds On (15 minutes) base ten blocks, • Review the anchor chart of the steps in the standard algorithm for addition. BLM 50: Operational Place Value Mat, chart Solve an addition problem that does not require regrouping (e.g., 314 + 271) paper together if necessary. Time: 55 minutes • Display the following problem: 395 + 236. You may decide to express this as a word problem to provide a context. Have students turn and talk to their partner about how they could estimate the sum. Discuss their estimation strategies. • Have students work in partners. Give each pair BLM 50: Operational Place Value Mat and base ten blocks. Tell students they are going to solve the problem using the ‘place value strategy’ (standard algorithm). Have students work on the problem without intervening (e.g., do not tell them how to regroup). Observe their strategies and solutions. • Discuss students’ strategies and solutions. Here is a sample discussion: Teacher: How did you solve the problem? Students: (Students show their concrete materials as they explain.) We made 395 by making 3 hundreds, 9 tens, and 5 ones and placed them in the correct columns. Then we made 236 by making 2 hundreds, 3 tens, and 6 ones. Teacher: What did you do next as you followed the steps in the anchor chart? Students: We added up the ones. We found that there were 11 ones, so we exchanged 10 ones for 1 rod and put it in the tens column. Teacher: Let’s put it above the other tens rods so we can keep it separate. [Add this step to the anchor chart if you haven’t already: “Regroup if you have more than 9 ones.”] What did you do next? Students: We added up the tens. Including the one we exchanged, we had 13 tens, which represents 130, and is more than 9 tens, so we exchanged 10 rods for a flat and put the flat in the hundreds column. Teacher: Let’s put the flat above the other flats so it is separate. We can add this step to our anchor chart: “Regroup if you have more than 9 tens.” How did you complete the question? Addition and Subtraction 361

Students: We added the flats which equals 6 hundreds. Our final answer is 631. Teacher: Does the answer seem reasonable compared to our estimate? Teacher: Let’s review what we did, and I will show you how we record this process using numbers as you explain it. [Record the main points on the anchor chart.] – W e record the numbers vertically, like the columns in our place value mat. – W e add the ones and there were 11. We put a 1 in the ones column to represent the remaining 1 once we regrouped the other 10 and exchanged them for a rod. Then we put a 1 above the other numbers in the tens column. What does it represent? (1 ten) – W e add the tens, including the one we regrouped. There are 13 tens, so we put a 3 in the tens column to represent the 30 that remain after regrouping. Then we put a 1 at the top of the hundreds column. What does it represent? (1 hundred) – N ext, we add the hundreds, including the one we regrouped, and record 6 to represent the 6 hundreds. Working On It (20 minutes) • G ive students the following problem: – T omas made a model out of 178 building blocks. His sister made a model out of 337 building blocks. How many blocks did they use altogether? • Students work in pairs with BLM 50: Operational Place Value Mat and base ten blocks to solve the problem using the standard algorithm. Then, they record their steps in writing as indicated on the anchor chart created in the Minds On. • W hen they are done, have students explain to another pair how they solved the problem and what the numbers they recorded mean. Differentiation • Some students may benefit from a small-group guided math lesson to review regrouping. Focus on the process rather than on recording the algorithm using numbers. • Some students may need more reinforcement with the recording process. Have them explain the problem to you as you do a ‘think aloud’ while recording the steps. Then ask students to explain the steps in their own words and why they are doing each step. • For students who need more of a challenge, give them two numbers that result in no tens left over after regrouping (e.g., 395 + 206). Ask how they will represent the number with no tens in the column. • For students who need more of a challenge, have them add 3 three-digit numbers using the standard algorithm and correctly recording the process. 362 Number and Financial Literacy

Materials: Assessment Opportunities base ten blocks, Observations: Observe how students work through the problem using the BLM 50: Operational algorithm. Place Value Mat, chart paper • Do they represent the numbers accurately with hundreds, tens, and ones? Do they regroup when necessary? • Do they add the ones, then the tens, followed by the hundreds accurately? • D o they make errors by forgetting to add or subtract a regrouped ten or hundred? • C an they correctly record the quantities that have been regrouped? Can they explain what the small numbers represent? • C an they link the written numbers to what they did with the concrete materials? Conversations: If students are simply following the steps in the anchor chart to record their calculations with numbers, check their understanding by posing some of the following prompts: – How many ones did you find when you added them using concrete materials? Show me where all of the ones are represented in the numbers. – W hat does this 1 above the tens column represent? Where did this group of ten come from? Show me with the base ten blocks. What must you do with this 1 when you are adding up the tens column? Consolidation (20 minutes) • Select one pair to show how they worked through the problem. Have the students demonstrate the regrouping using the base ten blocks. Ask the class how they would record the actions with numbers and where the numbers go. Continually link the numbers to the concrete representations. Regularly refer to the anchor chart as the process unfolds to reinforce what each step means. • Reinforce the correct use of place value terms. (e.g., “I regrouped 10 rods for one flat because they both equal 100, and I put the flat in the hundreds column.”) • Add the new step to the addition anchor chart if you haven’t already. (“Regroup if you have more than 9 tens.”) Math Talk: Math Focus: Recording the standard algorithm for addition problems requiring regrouping Let’s Talk Select the prompts that best meet the needs of your students. • T oday, we are going to learn how to record addition with regrouping using only numerals. continued on next page Addition and Subtraction 363

Teaching Tip • Work in pairs to solve the following problem using base ten blocks and Integrate the math operational place mats. This is the problem: talk moves (see page 8) throughout - T here are 24 vanilla cupcakes and 38 chocolate cupcakes for sale at the Math Talks to bakery. How many cupcakes are there altogether? maximize student participation and • What did you find? Explain your strategy. Did you need to regroup? Why? active listening. • How can we record the problem in vertical form? What do we need to remember about the place value columns? • Let’s look at the ones column. You said that 8 + 4 = 12 but I can’t put ‘12’ under the ones place. I can only record one numeral in a column. I will record ‘2’ to represent the 2 ones. I’ll show the ten we regrouped by writing a small ‘1’ in the tens column by the top of the ‘2.’ It’s in the tens column, so what does the 1 represent? (10) • Now we can add the tens. How many are there? Don’t forget to add in the regrouped ‘1’ or 1 ten. (e.g., there are 6 tens or 60) Where do you think we record the 6 tens? (in the tens column) • What is our final answer? (62) • L et’s write what our recording means. 24 + 38 12 (4 + 8) 50 (20 + 30) 62 • Let’s try a question with three digits. How can I record 369 + 274 in vertical form? • Let’s represent the problem together on the operational place value mat. Create an operational place value mat on chart paper and select students to do various parts of the problem. • How will we record the addition of the ones, 9 + 4? (e.g., It totals 13, so we put the 3 that represents the ones in 13 in the ones column.) How do we represent the 1 ten? What does the little 1 at the top of the tens column represent? (1 ten) • Now what do we do? (e.g., Add up the tens, which equal 14 tens including the ten from the regrouping.) What do the 14 tens represent? (140) What problem do we have? (e.g., We can’t record two digits in one column.) Which number will go in the tens column? (e.g., the 4 because it represents 4 tens, or 40, in 140) • What does the 1 represent? (the 100 in 140) Where do you think we will put the 1? (in the hundreds column) What do we do now? (e.g., Add up the hundreds, including the little one that represents the regrouped hundred.) • W hat is our final answer? Let’s record what it means. 369 + 274 13 (9 + 4) 130 (60 + 70) 500 (300 + 200) 643 • How do you think the pattern for recording would continue with larger numbers? 364 Number and Financial Literacy

Materials: Further Practice BLM 51: Comparing • Independent Problem Solving in Math Journals: Provide copies of BLM Solutions 51: Comparing Solutions. Have students record their responses in their Math Journals. Addition and Subtraction 365

20 21LessonsandUsing the Standard Algorithm to Subtract Multi-digit Numbers (with Regrouping) Teacher Possible Learning Goals Look-Fors • Uses the standard algorithm to subtract two- and three-digit numbers and can accurately record the process vertically • Decomposes numbers into hundreds, tens, and ones and regroups when necessary • Explains the steps in the standard algorithm and how they connect to concrete and pictorial models • F ollows the steps of the standard algorithm to accurately subtract two- and three-digit numbers that require regrouping • Regroups hundreds, tens, and ones when needed • Explains the process involved in using the standard algorithm while modelling the steps with concrete materials • Represents the steps involved in subtracting with regrouping using the standard written vertical convention About the According to Marian Small, “The traditional North American algorithm for subtraction is not always the easiest for students. One of the reasons is that it proceeds in a left-to-right direction to regroup but in a right-to- left direction to subtract. The algorithm is built around focusing on one place value position at a time and therefore, often involves several states of regrouping” (Small, 2009, p. 167). This means that students need to decide whether regrouping is necessary and then return to the process of subtraction in each step of the process. Building a strong conceptual understanding of place value and why this standard algorithm works is important. It’s not enough that students are able to follow the steps. They must be able to explain the process and demonstrate an understanding of why it works. 366 Number and Financial Literacy

About the Lessons In Lesson 20, students learn about the standard algorithm for subtracting two-digit numbers when regrouping is necessary and how to record the steps involved in the process. In Lesson 21, students extend this knowledge to subtract three-digit numbers. Addition and Subtraction 367

20Lesson Subtracting Two-Digit Numbers (with Regrouping) Materials: Minds On (15 minutes) base ten blocks, • Have students work in pairs. Provide students with base ten blocks, BLM 50: BLM 50: Operational Place Value Mat, chart Operational Place Value Mat, and chart paper. Review the anchor chart paper entitled ‘Place Value Subtraction.’ Tell students that while there are many ways to solve the following problem, they are going to use the strategy Time: 50 minutes highlighted in the chart. • Pose the following problem: – T here were 35 birds in the tree. 14 flew away. How many birds are left? • Students can solve the problem using concrete materials and then record their answer on chart paper using sticks and dots to represent the base ten blocks. • Discuss their solutions and have a student model how to represent the problem visually with sticks and dots on a large operational place value mat. Review their solution and connect it to the steps in the anchor chart. Working On It (20 minutes) • Students continue to work in pairs to solve the following problem using the strategy outlined in the anchor chart. They can use concrete materials and then record their answer on chart paper using sticks and dots. – T here are 62 books on the library shelf. 28 of the books got signed out. How many books are left on the shelf? Differentiation • Adjust the numbers in the problem so they are most suitable for your students. Assessment Opportunities Observations and Conversations: • A s students work on the problem, observe how they model the numbers using tens and ones and how they follow the steps of the algorithm. As they work, you may want to ask them to explain what they are doing and why. • Note whether students see the need to regroup in order to subtract. Can they regroup using base ten blocks and explain what they are doing? 368 Number and Financial Literacy

Consolidation (15 minutes) • Discuss students’ strategies for solving the problem. Have a student represent the problem using base ten blocks and another student demonstrate how to regroup and subtract. Reinforce why it was necessary to regroup. • Check the steps that students demonstrated against the steps in the anchor chart. Ask what needs to be added to the chart in order to accommodate problems that require regrouping when subtracting. Add “If there aren’t enough ones to subtract, regroup” to the anchor chart. Addition and Subtraction 369

21Lesson Subtracting Three-Digit Numbers (with Regrouping) Materials: Minds On (20 minutes) base ten blocks, • Review the anchor chart on using the standard algorithm for subtraction BLM 50: Operational Place Value Mat, chart (Place Value Subtraction). Provide the problem below and tell students they paper, markers are going to use this procedure to solve it. Time: 60 minutes – S amara is reading a book that is 534 pages long. She has read 145 pages. How many more pages does she need to read to complete the book? • Read the problem and have students estimate what the answer might be. Discuss their estimation strategies. • Students can work in pairs using BLM 50: Operational Place Value Mat and base ten blocks. They can also solve it by drawing squares, sticks, and dots to represent the various base ten blocks. Encourage students to use the anchor chart to help them with the steps. • Discuss students’ solutions and have one student model how to represent the problem with concrete materials as the class discussion progresses. On chart paper, record their thinking using squares, sticks, and dots, and then the written standard vertical format beside it. Below is a sample conversation: Teacher: We are going to record the problem in a vertical way, using a standard procedure. I will record it on the side of my chart paper with my drawing so you can see how they connect. What must I be careful to do? (e.g., Line up the columns.) How did you solve the problem? Student: We represented 534, putting out 5 hundreds, 3 tens, and 4 ones. [Draw this representation with squares, sticks, and dots.] Now we need to take away 145, starting with the ones. We are supposed to take 5 ones away, but there are only 4 ones so we need to regroup by exchanging 1 rod for 10 unit cubes. Teacher: I am going to strike out one of the ten sticks that I drew and then add 10 more dots to the ones section. How could I record this step using numbers? Student or Teacher: Strike out the 3 and put a 2 beside it, and then put a 1 in front of the 4 ones to represent 14. Teacher: What do these little numbers represent? Can you subtract now? Student: Yes, we found that when we subtracted 5 from 14, we had 9 ones left. 370 Number and Financial Literacy

Teacher: We can record that using numbers by putting a 9 in the ones column, under the line that divides the question from the answer. What did you do next? Student: We looked at the tens. We are supposed to take 4 tens away, but there are only 2 tens so we regrouped by exchanging a flat for 10 rods and putting the 10 rods with the 2 rods. Teacher: So in my drawing, I will strike out one of the flats (square) and add 10 sticks beside the other 2 sticks. How can I write this with numbers? Student or Teacher: Strike out the 5 and put a 4 since there are now 4 hundreds left, and put a 1 in front of the 2 tens, since there are now 12 tens. Teacher: Can you subtract now? Student: Yes, we can subtract 4 tens from the 12 tens, which equals 8 tens. Teacher: We can record 8 under the line in the tens column. What did you do next? Student: We had 4 hundreds left and we subtracted 1 hundred from it, leaving 3 hundreds. Teacher: How can we record this using numbers? Student: You can print a 3 under the line in the hundreds column. Teacher: What is the final answer and where do we see it in the drawing and in the numbers? Does the answer seem reasonable compared to our estimation? Teacher: Let’s take turns explaining what the number version of the solution means and how all of the numbers and symbols represent what we did with our base ten blocks. Working On It (20 minutes) • Provide the following problem. Read it with the students and ensure everyone understands. Change the context so it is more meaningful for your students. – A nna’s family was driving to visit friends who lived 724 kilometres away. By the time they stopped for lunch, they had travelled a long distance, but they still had 257 kilometres to go. How far had they travelled before lunch? • H ave students estimate what they think the answer will be. Discuss their estimation strategies. • Students solve the problem in pairs by drawing their solutions using squares, sticks, and dots. They also record their solution using numbers, referring to the anchor chart as necessary. Differentiation • If needed, work with a small group of students in a guided math lesson to reinforce the way in which they regroup and then record their thinking using numbers. Addition and Subtraction 371

• For students who need more of a challenge, have them create their own problem and solve it. They can then exchange their problem with another pair and solve each other’s problems. Assessment Opportunities Observations: • Observe how students work through the problem using the algorithm. Do they understand the problem and begin with the whole? Do they represent the numbers accurately with hundreds, tens, and ones? Do they recognize when regrouping is necessary, or do they try to subtract the top number from the bottom number when the bottom number is larger? When subtracting the tens, do they remember that a ten was regrouped and do they account for it? • Observe how students record the subtraction vertically. Do they align the digits correctly, or do errors in alignment result in subtraction errors? Do they reverse digits or record them in the wrong column? Do they show the regrouped hundred or ten in the column to the right and can they explain the quantity that it represents? Conversations: Ask students to explain how to subtract three-digit numbers with regrouping using the standard algorithm. If students are simply reading or retelling the steps from the anchor chart, check their understanding through questioning, using the following prompts: – H ow do you know when to regroup? – What does this number represent in the problem? Where can you see this quantity in your drawing? – What does this little ‘1’ represent [point to a regrouped 10]? Why is it there? Where can you see this regrouped ten in your drawing? – W hat did you do after finishing subtracting in the tens column? Why? Consolidation (20 minutes) • Have students meet with another pair. One pair explains how they subtracted with their drawing, while the other pair explains how the numerical version represents what was done in the drawing. Encourage them to explain without looking at the anchor chart. They can refer to the chart after their explanations to check whether they left anything out. • As a class, review the steps to the solution, posing questions about what the numbers mean and how they link to the drawings and concrete representations. • Remind students that this is only one of many ways to subtract three-digit numbers and that, in many situations, they can choose which way works best for them. The most important thing is that they understand what they are doing as they use any strategy. 372 Number and Financial Literacy

Materials: Math Talk: base ten blocks Math Focus: Recording numerically when using a standard algorithm for Teaching Tip subtraction 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 • Today, we are going to focus on how we can record the strategy we have participation and active listening. been using for subtraction using numbers. Consider the problem we did in a previous lesson (Lesson 20): There are 62 books on the shelf and 28 of them were signed out. How many books are left on the shelf? • As a group, let’s represent the problem using base ten blocks. We will solve the problem with the blocks as we record the solution. • How can I record this problem in vertical format? What is important to remember? (e.g., Line up the place value columns accurately.) • According to the anchor chart, what do we do first? (subtract the ones) Can we do this? (e.g., no, because there aren’t enough ones to subtract from) What can we do with the blocks? (e.g., regroup by exchanging 1 tens rod for 10 unit cubes) • How many tens are left? (5) So, I can cross out the 6 and replace it with a little 5. What does the little 5 represent? (5 tens or 50) • How many ones do I have now after regrouping? (12) I am going to record a ‘1’ at the top by the ‘2’ in the ones column. The ‘1’ means 1 ten so now I have 12 ones, just like we did with the base ten blocks. • Can I subtract the ones now? Where do you think I will put the answer? (below the line in the ones column) • What do we do next? (e.g., subtract the 2 tens from the 5 remaining tens, which is 3 tens) Where do you think we will record the 3 tens? We can put a 3 in the tens column, which represents 3 tens or 30. • What is our final answer? • Let’s try another problem. Record 425 − 147 in vertical format. Have a student demonstrate how to represent the problem using base ten blocks. Another student can solve the problem, showing any regrouping that is necessary. • If we look at the problem recorded in vertical format, what do we do first? As we regroup so there are enough ones, we cross out the 2 tens and put in 1 ten instead. • What do you think I put in front of the 5 in the ones column and why do I do it? (e.g., a 1, which represents 1 ten) What action did we do with our base ten blocks that represents this? (regrouping) Can we subtract the ones now? Where do I put the answer? continued on next page Addition and Subtraction 373

• Let’s look at the tens column. Do we have enough tens to subtract now? How can we represent the regrouping that we did with the blocks? (stroke out the 4 hundreds and replace it with a 3 that represents 300 and put a one beside the 1 at the top of the tens column, which represents 10 tens or 100) Can we subtract the tens now? Where do we put the answer? • How can we finish off the question? (subtract the hundreds and record the number of hundreds left in the hundreds column) • Let’s write down what our recording means beside our work. 425 (300 + 110 + 15) – 147 (100 + 40 + 7) 278 (200 + 70 + 8) Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: It is important to give students an opportunity to self-assess their learning by reflecting back on their work and achievements. This helps students realize that they are growing and progressing. This is a good time to give students the self-reflection tool known as ‘two stars and a wish.’ Students record two things that they have done well or learned, and one wish, which is a goal of how they may proceed in order to master something that they are less certain about. This reinforces the message that learning does not stop at the end of a unit, and that there will be more opportunities to improve through related tasks throughout the year. As teachers, it is important for us to regularly spiral back to the concepts that students have previously learned and link them to their new learning. It is also beneficial to reinforce these concepts and skills in other subject areas and in their everyday lives. The more we can make learning relevant to students, the more they will value their learning and strive to improve. Further Practice • S tudents will continue to need practice adding and subtracting using the standard algorithm. Give them problems to solve, initially with concrete materials or drawings, and then by recording the steps in numerical form. 374 Number and Financial Literacy

22Lesson Solving Real-Life Addition and Subtraction Problems 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 added and subtracted three-digit • B 2.5 represent and solve problems involving the addition and subtraction of numbers using a variety of strategies and two-digit whole numbers that add up to no more than 1000, using various tools and numbers using a variety of algorithms mental strategies. Spatial Sense • E2. compare, estimate, and determine measurements in various contexts Possible Learning Goals • Applies addition and subtraction strategies to compare statistical data and make meaning of the quantities • Explains what the statistics mean by making connections to other data, familiar benchmarks, or to facts about themselves • U nderstands and distinguishes among measurable attributes, such as length, height, and distance • Understands how long centimetres and metres are and how they relate to each other • Accurately uses a metre stick or metre tape to measure lengths • Adds and subtracts two-digit numbers using a variety of mental strategies • Adds and subtracts three-digit numbers using a variety of strategies and tools • Makes meaningful comparisons among statistics • Explains what their findings mean and how they relate to the provided context • Explains their strategies and offers reasons why their solutions make sense Addition and Subtraction 375

PMraotcheesmseast:ical About the rrrPeeerpfaolresbeoclsetneimninngtgi,snoacglonv,ndinnpger,cotviningg, , communicating It is important that students get opportunities to apply the addition and subtraction strategies they have learned to solve real-life problems. One Math Vocabulary: engaging way of accomplishing this is by presenting students with dlmeifnefgetrtrehes,n,chceee,ignshtuitmm, de, ittsrotetasanl,ce, interesting facts and statistics and having them make sense of the data by quantitatively comparing the numbers. Working with real data can also raise students’ curiosity about the world around them, motivate them to further investigate, and to think critically about what their findings mean within a context. About the Lesson In this lesson, which takes place over two days, students analyse statistics about various lengths and heights of animals. They make comparisons among the animals as well as to themselves so they can make meaning of the numbers. Students are encouraged to select the tools and strategies that best match the problem and make the most sense to them. There is no Consolidation on Day 1, only on Day 2. Day 1 Materials: Minds On (15 minutes) “Comparing Lengths” • Show “Comparing Lengths” from the Number and Financial Literacy big (pages 6–7 in the Number and Financial book and read the title. Ask students what they think the pictures are about Literacy big book and and what they have in common. little books), metre stick, painter’s tape, • Have students turn and talk to a partner about what features of the animals base ten blocks, number lines, chart can be measured. Ask which units they could use. Students can share copies paper of the Number and Financial Literacy little books to get a closer look at the Time: 45 minutes pictures. • Show a metre stick. Discuss metres and centimetres and the relationship between the two units. Tell students that they will be working in centimetres. Working On It (Whole Group) (30 minutes) • D raw attention to the elephant on page 7 and read some of the related statistics aloud. Clarify terms such as ‘length’ and ‘height.’ Ask students how they could use the metre stick to get an understanding of the size of the animal. Together, create a visual of the size of the elephant by marking out certain measures using painter’s tape. • C ompare some of the measurements to those of students, such as the elephant’s height to a student’s height. Have students estimate how many of the student’s heights would equal the elephant’s height. As a class, figure out the actual difference through computations involving either addition or subtraction. • H ave students look at the other animals shown on pages 6–7, and ask how they could compare the elephant’s measurements to those of other animals. 376 Number and Financial Literacy

Materials: • D o some examples together. For example, compare the length of the cheetah “Comparing Lengths” (head and body) to that of the elephant. Ask about how many cheetahs could (pages 6–7 in the line up beside the elephant. Ask students how they could find out how much Number and Financial longer the elephant is. (e.g., Subtract the cheetah’s length from the elephant’s Literacy big book and length; add on from the cheetah’s length to the elephant’s length.) Have students little books), chart work with a partner to carry out the calculations. They can use any tools or paper, base ten blocks, concrete materials they choose, and they can record their solutions on chart number lines, sticky paper. notes Time: 60 minutes • D isplay and discuss students’ solutions and make comparisons among them. (e.g., Counting up on a number line is finding the difference between two numbers, just like subtracting the two numbers using an algorithm.) • R ead the statistic about the length of the cheetah’s stride. Ask what they think ‘stride’ means. Show the distance on the floor using painter’s tape. Have students look at the other animals and ask how they could compare lengths of strides or jumps. Ask what operations they could use to find the answer. • Select one student to make a jump from the starting point of the cheetah’s stride and have other students measure the length of the jump. Ask about how many of the student’s jumps would equal the stride of the cheetah. Ask students how they could calculate the total of the suggested number of student’s jumps to see how close it gets to the length of the cheetah’s stride. Have students solve the problem. You may want to encourage them to solve it in a different way than they did the previous problem. They can record their solutions on chart paper. • D isplay and discuss students’ solutions and make connections among them. Day 2 Minds On (5 minutes) • Revisit “Comparing Lengths” in the Number and Financial Literacy big book. Ask students what other comparisons they could make among the animals. • Review some of the solutions to the problems that were solved the previous day. Working On It (25 minutes) • Students work in pairs. • Tell students that they are going to make as many comparisons as they can among the animals. Ask whether it would make sense to compare the length of an animal to the height of another animal. Establish that they should be comparing the same measurable attributes. They can also compare the various lengths, heights, and strides of the animals to their own lengths, heights, and strides. • E xplain that they can use any strategies they want for adding and subtracting in order to make the comparisons. They can record their comparisons and calculations on chart paper. Addition and Subtraction 377

Differentiation • You may decide to list the comparisons that students could make so the task is less open. • For some students, you may decide to have them compare only two or three animals so the task is more focused. • Some students may need to use a calculator if they are having difficulties with the mechanics of the calculations. Assessment Opportunities Observations: • Are students able to compare the same measurable attributes (e.g., heights to heights and strides to strides)? • D o students pick an appropriate operation to carry out the calculations? Can they explain what they are doing with the calculations? • Can they explain their answer in terms of the context? Conversations: If students are having difficulty finding appropriate data to compare, pose some of the following prompts: – Let’s look at these two animals and read the statistics together. What features do they share? – W ould it make sense to compare the length of one animal to the distance that another animal can jump? – If we want to know which animal is longer, what number should we look at? – If we want to know which animal can jump further, what numbers can we look at? Scan the page and find numbers that you could use to compare how far animals can jump. Consolidation (30 minutes) • Display students’ solutions and have a gallery walk. One student in each pair ‘strays’ to visit, while the other student ‘stays’ with the work and explains what they found to the visitors. • Encourage students to ask questions of each other. For example, the creators of the work could have the visitors predict what an estimation of the answer might be. • Give each student a sticky note. At the end of their visit, they can copy down the comparison that they found most interesting. • Students can switch roles. • As a class, share comments from students’ sticky notes. List some of the interesting comparisons they found. Ask students what surprised them about their findings. 378 Number and Financial Literacy

• For some comparisons, highlight the proportional relationships between data. For example, have students estimate how many of their jumps would equal one kangaroo jump. Express their estimations in terms of the kangaroo being able to jump four times or five times further than a student. • Highlight some of the addition and subtraction strategies students used. • B uilding Social-Emotional Learning Skills: Critical and Creative Thinking: Ask students what they wonder about now that they have learned facts about the animals featured on the page. Discuss how we often have more questions once we start learning about a topic, which can lead to more investigations. Explain that this often happens to mathematicians and scientists who are exploring a topic. Record some of their wonders. At a later time, you can have students research some of their questions and make new comparisons. Addition and Subtraction 379