p206-255-Gr3ON-Number-Unit3-counting

Unit 3: Quantities and Counting to 1000 Lesson Content Page 1 and 2 Quantities and Counting to 1000 Introduction 206 1 Large Numbers in Our Lives 208 2 Investigating Distances 210 3 Representing Large Numbers 212 4 Composing Quantities to 1000 216 5 Composing Numbers to 1000 Using Base Ten Blocks 221 6 Equivalent Representations of Quantities 225 7 Skip Counting by 25s, 50s, 100s, and 200s 229 8 Counting by Place Value Amounts (1s, 10s, 100s) 232 Skip Counting Using a Calculator 237 9 and 10 Comparing Quantities to 1000 241 9 Investigating Quantities that are More Than or Less Than 243 10 Comparing Numbers on a Number Line 246 11 Estimating Large Quantities (to 1000) 248 12 Applying Strategies to Solve Real-Life Problems 252

Quantities and Counting to 1000 Introduction About the In this unit, students spiral back to revisit concepts addressed in Unit 1 (Quantities and Counting to 500) and extend them to understand quantities up to 1000. They initially explore these larger numbers in their lives and establish meaningful benchmarks to judge how much these numbers represent. Students also use smaller subgroups of sets to help them estimate the magnitude of larger quantities. Students continue to use concrete materials such as base ten blocks, tools such as open number lines, and the numerical forms of quantities to represent, compose, and decompose three-digit numbers. Connections are made between the various representations so students can create mental images from the concrete models and link them to more abstract ways of representing the numbers. This is important since it becomes increasingly challenging to have enough concrete resources on hand to directly model such large quantities. Digital slides and/or blackline masters for base ten blocks and other tools are included and recur throughout the unit. Students also explore the proportional relationships between 1, 10, 100, and 1000. This furthers their understanding of our place value system and how to decompose and recompose quantities based on place value positions. Students will decompose and represent numbers in a variety of ways. For example, they may represent 783 as 7 hundreds, 8 tens, and 3 ones, as 78 tens and 3 ones, or as 7 hundreds and 83 ones. This flexibility in composing and decomposing helps prepare students for adding and subtracting three- digit numbers, using student-generated as well as standard algorithms. In grade three, counting continues to be important as students further develop flexible strategies for more complex tasks. For example, students need to count by place value amounts, switching their count as they move from hundreds, to tens, to ones. They also benefit from recognizing important connections, such as the link between skip counting and the operations of addition, subtraction, multiplication, and division, which helps them better understand these operations. Tools such as concrete materials, hundreds charts, and number lines offer visual supports for understanding counting and the patterns inherent in our number system. 206 Number and Financial Literacy

Lesson Topic Page 1 and 2 Large Numbers in Our Lives 208 210 1 Investigating Distances 212 216 2 Representing Large Numbers 221 225 3 Composing Quantities to 1000 229 232 4 Composing Numbers to 1000 Using Base Ten Blocks 237 241 5 Equivalent Representations of Quantities 243 246 6 Skip Counting by 25s, 50s, 100s, and 200s 248 252 7 Counting by Place Value Amounts (1s, 10s, 100s) 8 Skip Counting Using a Calculator 9 and 10 Comparing Quantities to 1000 9 Investigating Quantities that are More Than or Less Than 10 Comparing Numbers on a Number Line 11 Estimating Large Quantities (to 1000) 12 Applying Strategies to Solve Real-Life Problems Quantities and Counting to 1000 207

1 2Lessons and Large Numbers in Our Lives Math Number Curriculum Expectations • B1.1 read, represent, compose, and decompose whole numbers up to and including 1000, using a variety of tools and strategies, and describe various ways they are used in everyday life • B1.2 compare and order whole numbers up to and including 1000, in various contexts • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Possible Learning Goals • Develops an understanding of the role that large quantities play in their lives • Creates concrete models of quantities greater than 500 and explains how they connect to their numerical representations Teacher • Explains how larger numbers play a role in their lives Look-Fors • Correctly reads three-digit numbers in numerical form • Creates concrete models of numbers represented in numerical form, using Previous Experience with Concepts: metre sticks and base ten blocks Students have represented and compared numbers • Locates values on an open number line using benchmark numbers and to 500, and composed explains their reasoning for the numbers’ positions and decomposed numbers along place • Decomposes numbers into place value amounts value lines (see Unit 1). • Creates reasonable true/false statements about quantities and explains why they are true or false PMraotcheesmseast:ical About the Problem solving, cccsrooeoemnamlensmcpoetuncuinttinanignitgcgitoao,antonrianeldslgpspraetrnrosadevtinentggiin,egs,, As students explore numbers to 1000, it is important that they see how these numbers are relevant in their lives. Identifying quantities in their environment helps create meaning for larger numbers. Students can also establish familiar benchmarks that they can use to make comparisons among quantities. 208 Number and Financial Literacy

MphbcluaaaontsmchdeeprVetvaeaodrnelcsu,,abeetbl,eosutncthliaskmo,rsuy,oas:ntaeensd, s, While students may be able to sequence numbers from least to greatest, they also need to get a sense of how large quantities are in relation to each other. For example, 568 and 962 are both larger than 500, but 962 is almost double the size of 500. Initially using concrete materials to represent the quantities helps students create mental images, which can then be linked to more abstract representations, such as comparing quantities on an open number line. Students also need to understand the proportional relationships between the place value positions and how they affect the values of the digits. About the Lessons In Lesson 1, students compare distances to certain Canadian destinations by concretely representing them using metre sticks. In Lesson 2, students investigate the relationships between 1, 10, 100, and 1000 in order to compare the populations of cities and towns. They also represent the populations using base ten blocks and then make comparisons among the models. Quantities and Counting to 1000 209

1Lesson Investigating Distances Materials: Minds On (15 minutes) Digital Slide 41: • Show students Digital Slide 41: Road Sign. Draw their attention to the photo Road Sign, Digital Slide 42: The James Bay and discuss the wilderness terrain through which the road travels. Ask where Road, metre sticks, they think this road might be and what it would be like to travel on. Ask markers, chart paper students if they have ever seen a sign similar to this before. Ask where they think the sign is located, what its purpose is, and what information it is Time: 60 minutes conveying. Have them turn and talk to their partner. • Ask what the names and numbers represent (names of cities/towns and distances to them). Ask what units are used to represent the numbers (kilometres). Ask what they notice about the numbers (e.g., they are all greater than 100 and less than 700). • Explain that this is a sign at Matagami, Quebec, which is at the beginning of the James Bay Road that extends into northern Quebec. Show the map of the road on Digital Slide 42: The James Bay Road. Ask students what they think the road might be like and why it might be challenging to travel such distances (e.g., finding places to stop for food, gas; weather conditions; large animals crossing the road). • Display Digital Slide 41 again and provide some true/false statements for students to consider (e.g., Eastmain is further away from this location than Radisson; Radisson is closer to this location than Chisasibi; The distance between Eastmain and Nemiscau is greater than the distance between Eastmain and Wemindji; The distance between Wemindji and Radisson is less than the distance between Eastmain and Nemiscau.). Briefly discuss each point. The statements will be further addressed later in the lesson. 210 Working On It (20 minutes) • Show students a metre stick. Ask what they know about it, what the markings on it mean, and what units are represented on it. Ask how many centimetres are in one metre. • Tell students that they are going to work in groups. Each group will create a model on chart paper of one of the distances on the road sign so the class can compare them. Ask students how they could use centimetres to represent kilometres. Establish that 1 centimetre will represent 1 kilometre in their models. • Ask students what they need to remember if they want to compare the distances (e.g., they need a common baseline; they need to measure accurately). Establish a common starting point. • The groups create their distances, including significant benchmarks along the way (e.g., marking off 100 cm, or 500 cm). Students also figure out a way to count the units in the distance to prove that their measurements are accurate. Number and Financial Literacy

Differentiation • You may want to choose different cities and distances so they are more relevant to your students and so the numbers are within their abilities. • You may want to include some shorter distances that are expressed in three- digit numbers, or if necessary in two-digit numbers, for students who may still be struggling with three-digit numbers. Assessment Opportunities Observations: Since this is an introductory lesson, observe what students know and what misconceptions they may have. Pay attention to how students measure the distance and choose the benchmarks. • Do students iterate one metre stick or do they use two metre sticks to iterate the distance? • How do they use the metre stick to measure the last part of the distance that does not end at a full metre? • Are they being accurate in their measurement, ensuring there are no gaps or overlaps as they iterate their measuring tool? • Do they mark benchmarks along the distance as they go or do they do it after the full distance is measured? • Can they explain why they chose certain benchmarks? Consolidation (25 minutes) • Meet as a class. Have groups explain how they measured their distances and why they chose certain benchmarks. Have them count out the distance to confirm that it is accurate. • Have students study the distances and make comparisons among the different lengths. Have them differentiate which distances are much farther apart from those distances that are relatively close. • Revisit the true/false statements from Minds On and confirm the validity of each, using the concrete model as proof. • D raw an open number line on chart paper. Together, create endpoints and benchmarks that would be appropriate for showing the distances to the various towns. Discuss where each distance should be placed and why students think so. • Together, create a list of words and phrases that students used to compare the distances (e.g., more than, less than, greater than, fewer than, least, greatest). Quantities and Counting to 1000 211

2Lesson Representing Large Numbers Materials: Minds On (15 minutes) base ten blocks, • Show students the following base ten blocks: a unit cube, a tens rod, a chart paper, Digital Slide 43: hundreds flat. Ask how they are related to each other. Discuss how, since Population there are 10 unit cubes in one rod and 10 rods in one flat, each larger block is ten times larger than the previous one. Have students predict what quantity Time: 60 minutes 10 flats represent. • As a class, stack 10 flats, one at a time, and count by 100s to 1000. Introduce the thousands cube and exchange it for the stack of hundreds flats. • Print the numerals 1, 10, 100, and 1000 on chart paper. Ask students how they can see the ‘ten times more’ relationship represented in the numbers and how this relates to the models. • Tell students that they are going to be looking at numbers up to 1000 in something from everyday life: the populations of cities and towns. Discuss what the word ‘population’ means. Ask students whether they think a city or town with 1000 people is large or small and why they think so. • Record some of the populations of cities in your area. Include examples of cities with populations of about: 1000 people, 10 000 people, 100 000 people, and/or 1 000 000 people. Ask how the numbers are different as far as how many place value positions there are. Create an extended place value chart to show the ones-tens-hundreds patterns in our number system. For example, highlight how the ones, tens, and hundreds are also evident in the one thousands, ten thousands, and hundred thousands, as well as the one millions, ten millions, and hundred millions. This helps students realize that there are numbers much larger than what they are studying and that these large numbers follow a pattern that is already familiar to them. Hundred Ten Millions Hundred Ten Thousands Hundreds Tens Ones Millions Millions Thousands Thousands 100 000 000 10 000 000 1 000 000 100 000 10 000 1 000 100 10 1 • Have students visualize how much 10 000 would look like in terms of thousands cubes. Repeat this with 100 000 and 1 000 000, highlighting the ten-times-more increase in amounts. Although students only need to work with numbers to 1000, it is beneficial for them to learn that there are much larger numbers and many cities with populations larger than 1000. • Having learned about local populations, ask again whether they think a city or town with 1000 people is large or small. 212 Number and Financial Literacy

Working On It (20 minutes) • Show the population signs on Digital Slide 43. Locate these towns on a map of Canada. Ask students what they know about the sizes of the towns and how they compare in general terms. • Tell students that they are going to make concrete models of each of these populations using base ten blocks. Each unit block represents one person and they can connect the blocks in any way to create a ‘building’ that will house all of the citizens. • Have students work in groups of four. Each group creates a concrete model of one population using base ten blocks. They also establish a method for counting the blocks to confirm that their model of the population is accurate. When students are finished, they create another model representing a benchmark number that will help to make comparisons (e.g., if their population is 735, they may make a benchmark model that represents 100 and then show how their population model is about 7 times greater than 100). Students can practise how they will explain their model in the Consolidation. Differentiation • Replace the given towns with local towns/cities or with towns whose population numbers are within your students’ ability levels. You may want to include two-digit numbers if some students are struggling with three-digit numbers. • Add more towns or cities so each group can have a different population to work with. Assessment Opportunities Observations: Pay attention to how students build with the blocks to represent the numbers. • Do students use hundreds flats as their base or do they choose a different block (or combination of blocks) to represent the various ‘floors’ in their model? • Can they flexibly count the base ten blocks, adjusting the count for their different values? • Do they select or create an appropriate benchmark to use as a comparison? Consolidation (25 minutes) • Meet as a class. Have each group present and explain their model. Before the group begins, have the other students estimate the number of people represented in each model and why they think so. Have students explain how they can use the group’s benchmark models to make comparisons. Quantities and Counting to 1000 213

Materials: • C ompare all of the models to each other. Explain what may make it hard to compare them (e.g., some buildings may be tall and narrow, while others are shorter but more spread out). • As a class, decide on a design that would make it easier to compare the populations. Have the groups adjust their buildings to reflect the new design. • Compare the buildings again, noting which amounts are much more or just a little bit more than others. • Show the open number line that was used with the distances to towns on the James Bay Highway (see Lesson 1). Ask students where they would place the populations on the number line. Have students explain their reasoning. • Together, create summary statements about the populations of the towns. Ensure that some of them reflect proportional relationships. For example, include statements such as, the population of Norman Wells is more than double the population of Alameda, or the population of Hare Bay is almost triple the population of Alameda. • B uilding Social-Emotional Learning Skills: Healthy Relationship Skills: Explain that while students only learn about numbers up to 1000 in grade three, there are numbers much larger than 1000, as was discussed in the Minds On. Brainstorm things in the environment that can be described by larger numbers. • Explain that numbers go on forever and the counting sequence never ends. Mathematicians continue to explore increasingly larger numbers as they make new discoveries, such as other galaxies, which require larger and larger numbers to describe. Show students a place value chart that extends up to one billion and have them look for patterns in how the numbers grow. Highlight the ones-tens-hundreds patterns (e.g., one thousand, ten thousand, and one hundred thousand are followed by one million, ten million, and one hundred million). • Ask what students wonder about larger numbers. Students can add to these questions throughout the year. As a class, you can periodically take time to find answers for some of these questions. Inspire students to be curious, wonder, and ask questions to help them develop a positive attitude toward math and the role it plays in their lives. “Let’s Math Talk: Estimate” and “Estimating Larger Math Focus: Looking at visual representations of sets to estimate Quantities” (pages 4–5 and 20 in the Number Let’s Talk and Financial Literacy big book and little Select the prompts that best meet the needs of your students. books), small jar filled with grains of rice or • W hat does it mean when we estimate? If I said that it took me about an hour other very small items to drive to the store, how long could the trip have actually taken? What would not be realistic based on my estimate? 214 Number and Financial Literacy

Teaching Tip • Show students a jar filled with rice. Look at this jar. What do you see inside? Integrate the math About how many grains of rice might there be in the jar? Turn and talk to talk moves (see your partner. page 8) throughout Math Talks to • Put your thumb up if you think there are more than 100? 200? 500? Why do maximize student participation and you think so? What are your estimations? Let’s record some of them. active listening. • Show the big book page “Let’s Estimate.” Let’s look at jelly beans in the jar on page 4. We estimated the number of jelly beans in a previous lesson (Unit 1, Lesson 3). What helped us make a reasonable estimation? (e.g., finding out how many are in a smaller group and then visualizing how many smaller groups are in the larger set) • Do you think finding a smaller group like 10 is going to be helpful with a large set like this jar of rice? How could we adjust our strategy? (e.g., create larger subgroups, like 25, 50, or 100) • Show big book page 20, “Estimating Larger Quantities.” Let’s try our estimation strategies with these sets. What do you see in the set at the bottom of the page? (diced/chopped vegetables) What makes this set challenging to estimate? (e.g., You can’t see all of the pieces of vegetable; they are overlapping.) Do you think there are more than 50? 100? 500? Why? • Estimate how many pieces of vegetable you think there are. You can share the little book versions of the big book to get a closer look. • What are your estimates? What strategies did you use? How many pieces did you use for your subgroup? Now that we have shared our estimates, does anyone want to change their prediction? Which estimates seem most reasonable? • Let’s look at the picture of the stones. What do you notice about them that may make it more challenging to estimate? (e.g., they are all different sizes and colour, they are overlapping) • Turn and talk to your partner about the number of stones. What strategies did you use to estimate the number? Which estimates seem most reasonable? Why? • Look at our jar of rice. Do you want to change your original estimate? Which of our estimates seem the most reasonable? Do we need to know exactly how many grains of rice are in this jar? Why or why not? • W hy does it make sense to estimate the items that are in these pictures rather than knowing the exact number? How would you know how much of the vegetables to use when you are cooking? How do we usually measure ingredients when cooking or baking? (e.g., We use a cup or tablespoon. We use a scale. We count the wholes instead of the pieces, e.g., 1 carrot, 3 bananas.) Why does this make more sense? Quantities and Counting to 1000 215

3Lesson Composing Quantities to 1000 Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and PMraotcheesmseast:ical including 1000, using a variety of tools and strategies, and describe various ways Problem solving, they are used in everyday life ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, • B 1.2 compare and order whole numbers up to and including 1000, in various contexts • B 1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Possible Learning Goals • Flexibly composes and decomposes 1000 in different ways, using a variety of strategies • Uses mental strategies to find compatible pairs that combine to make a target number Teacher • Uses appropriate strategies for reaching 1000 from a starting number Look-Fors (e.g., adding on to 1000, determining the distance to 1000) Previous Experience • Uses a variety of tools to find numbers that total 1000 (e.g., base ten with Concepts: materials, number lines) Students have composed and decomposed values • Decomposes and recomposes quantities along place value lines up to 500 using concrete • Recognizes and explains how decomposing along place value lines can be materials and have exchanged base ten used to identify and determine compatible numbers (e.g., the digits in the ones blocks for equivalent position should add to 10) amounts. • Explains strategies for composing smaller numbers into larger numbers About the As students further develop their number sense, it is essential that they can flexibly compose and decompose quantities. As researcher Lauren Resnick explains, “probably the major conceptual achievement of the early school years is the interpretation of numbers in terms of part and 216 Number and Financial Literacy

Math Vocabulary: whole relationships. With the application of a Part-Whole schema to cdttoheemoncuosps,maaotpnnibodelssese,,,phbpaualinsra,decdreeigdviast,,lue, quantity, it becomes possible for children to think about numbers as ten compositions of other numbers” (Van de Walle & Lovin, 2006b, p. 42). Students need experiences composing and decomposing larger numbers in various ways, especially along place value lines, which prepares them for performing operations such as subtraction that requires regrouping. Grade three students can also apply their composing and decomposing skills to develop mental strategies for adding and subtracting multi-digit numbers. They can use compatible numbers, which John Van de Walle and LouAnn Lovin refer to as “numbers that go together easily to make nice numbers” (Van de Walle & Lovin, 2006b, p. 55). These are often numbers that combine to make tens, hundreds, or thousands or that end in 5 or multiples of 25. Students benefit from both identifying pairs of compatible numbers and mentally building on from one compatible number to a friendly whole number. About the Lesson During the lesson, students decompose 1000 into various parts and find missing parts from a quantity that will combine to make 1000. In the Math Talk, they work at identifying compatible numbers that add to 1000 by applying various mental strategies. Materials: Minds On (15 minutes) base ten blocks, • Show students a thousands cube. Have them visualize breaking it into two BLM 40: 0–9 Spinner, paper clips and pencils, parts so one part is small and the other part is much larger. Ask what it BLM 41: Create a Pair would look like and about how many hundreds would be in each part. Recording Sheet, chart Students can turn and talk to their partner. paper, markers, other tools as needed • Discuss their responses and record the number pairs (e.g., 100 and 900, 300 (e.g., number lines, place value mats, BLM 12: and 700). Have some students represent these amounts using the base ten Base Ten Blocks) blocks to confirm that one amount is much larger than the other amount. Represent the number pairs on an open number line as two jumps that Time: 60 minutes together reach 1000. • Repeat this process by asking students how they might decompose the cube so one part is only slightly larger than the other part. Ask how they might talk in terms of hundreds and tens to describe the two amounts (e.g., 510 and 490). Have some students represent one or two of the suggested amounts using base ten blocks to confirm that one amount is slightly larger than the other amount. • Ask what the other part might look like if one part of a decomposed thousands cube has 318 units in it. Ask about how many units it may have. Have a student create 318 using base ten blocks. Ask how they could add to those blocks so the total equals 1000. Have a student demonstrate while the Quantities and Counting to 1000 217

other students explain what to do. Ask students how they would count on from 318 to 1000 and record the count on the open number line. Working On It (20 minutes) • Students work in pairs. They take turns creating a three-digit number by spinning a spinner three times. (See BLM 40: 0–9 Spinner.) Together the partners create a second number such that the two numbers together equal 1000. Students can use base ten blocks, open number lines, place value mats, or other tools and materials to represent their thinking. If there are not enough base ten blocks for all students, they can work with paper base ten blocks from BLM 12. • Students can record their numbers and thinking on BLM 41: Create a Pair Recording Sheet. For two of their pairs, they can cut out pictures of base ten blocks and glue them on to chart paper to show how the two quantities equal 1000. Alternatively, students can draw squares, sticks, and dots to represent their work. Differentiation • For students who need more support, they can work with friendlier numbers, e.g., numbers with 0 ones. Have them spin the spinner two times to determine the numbers for the hundreds and tens places and they can record a zero in the ones place. • For students who need more of a challenge, have them decompose each of the numbers/quantities into two parts so they now have four numbers that add to 1000. Assessment Opportunities Observations: • Are there any numbers/quantities that give students particular difficulty or that reveal any misconceptions? Selectively choose such numbers to use in the Consolidation. You may also want to choose some solutions represented with base ten blocks on chart paper that would be good for discussing how to count on from one value to another by adjusting the count to the size of blocks being used. • Pay attention to how students are finding the second number: – D o they create the first number with base ten blocks and then add on until they create the thousands cube? – Do students add on from the first number and only use the base ten blocks for the second number? – C an students flexibly adjust their count to account for the different values of the base ten blocks? – Are students using open number lines to help them solve the problem? 218 Number and Financial Literacy

Materials: – A re students using mental math to solve the problem? Do they start by making 10 ones and then progressively move to the tens and chart paper to record hundreds or do they start with the hundreds? open number lines Conversations: If students are having difficulty finding the second number, pose some of the following prompts: – Y ou have built 327 on your place value mat, using base ten blocks. About how much more do you think you will need to make a 1000 cube? Look at the 7 ones. How many ones does 1000 have? (0) How many ones do you have if you add one more? Two more? Three more? Can you have 10 ones? What can you do? (e.g., You can exchange them for a rod.) So when you add 3 ones, are there any ones left? What is important for you to do? (e.g., You create 10 ones so they can be exchanged.) Think of how you can do that for the tens column. Consolidation (25 minutes) • Have two pairs meet together. The pairs take turns showing one number from a pair represented with base ten blocks on chart paper, while the other pair of students estimates about how much the other number will be. The creators then explain how they solved the problem. • Meet as a class. Discuss some of the strategies that students used. Show student examples of each strategy while the creators explain what they did to solve the problem. • Record the strategies that students used. • Using one example, show the base ten representations and then ask how one of the quantities could be shown on an open number line. Ask students how they could solve for the other quantity (to make 1000) by just looking at the number line. Connect the numbers to the concrete representations. Math Talk: Math Focus: Mentally creating compatible pairs that add to 1000 Let’s Talk Select the prompts that best meet the needs of your students. • In the lesson, we found numbers that add to 1000. What is an example with one number being much larger than the other number? How can you add on from one number to the other in your mind? How can we show this on an open number line? • Let’s try some more examples. What number could we add to 875 so the two numbers equal 1000? Turn and talk to your partner. continued on next page Quantities and Counting to 1000 219

Teaching Tip • W hat did you find? How did you add on from one number to the other? Integrate the math (e.g., We added 25 and then 100 and counted: 875, 900, 1000—so we added talk moves (see 125.) Let’s show the addition on the number line. Is there another way? page 8) throughout (e.g., We added 5, then 10, then 10, and then 100 and counted: 875, 880, 890, Math Talks to 900, 1000.) Let’s put that amount on the number line. maximize student participation and • W hat can you add on to 730 to create 1000? About how much will it be? active listening. (e.g., 300) Will it be more or less than 300? How do you know? Materials: BLM 42: Compatible • W hat did you find? (e.g., add 20 to 730 so it equals 750, add 50, which equals Pairs Memory Game 800, add 200, which equals 1000, so we added 270) How did you keep track of the numbers that you added? (e.g., I kept track of the tens that I added, 20 and 50, on my fingers and then added 200 to the 70.) How can the number line help us keep track? Let’s show the jumps and their amounts on the number line. • For both 875 and 730, what strategy are we using? (e.g., First adding to get to the nearest hundred and then adding by hundreds.) • What can you add to 137 to equal 1000? About how much will it be? Will it be more or less than 700? How do you know? • What did you find? (e.g., We added 3 to 137 to get to 140, then added 10 to get to 150, then 50 to get to 200, then 300 to get to 500, then 500 to get to 1000.) Let’s record the jumps and those amounts on the number line. • What strategy did you use this time? (e.g., We counted up to the nearest 10 and then added to the benchmark of the nearest 50.) • Let’s record our mental strategies for adding on to 1000. Further Practice • Math Game: Compatible Pairs Memory Players: 2–4 Directions: – E ach group needs one page of cards from BLM 42: Compatible Pairs Memory Game. The compatible pairs of three-digit numbers differ in their ones digits: Numbers on page 1 have 0 ones, numbers on page 2 have 0 or 5 ones, and numbers on page 3 have all possible numbers of ones. – P layers cut out the cards, shuffle them and then place them face down in an array. – P layers take turns flipping over two cards to see if they add to 1000. If the cards match, the player removes them and takes another turn. If there is no match, the player returns the cards to their original positions, face down, and the next player takes a turn. The game end when all the cards are taken. The player with the most cards wins. 220 Number and Financial Literacy

4Lesson Composing Numbers to 1000 Using Base Ten Blocks Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life • B 1.2 compare and order whole numbers up to and including 1000, in various contexts • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Possible Learning Goals • Creates various three-digit numbers to 1000, using a given number of base ten blocks and given parameters • Accurately counts the number of items in their representations • Creates examples of what the numbers cannot be and explains their reasoning • Creates more than one quantity using a given number of base ten blocks • Understands and explains the relationships between the place value amounts represented by the blocks • Accurately counts the blocks, using and switching between appropriate skip- counting methods to match the value of the blocks • Records concrete representations in numerical form • Explains how their representations meet the parameters in the problem • Gives an example of a number that does not meet the criteria and explains why it does not belong • Makes generalizations about what the numbers can and cannot be and explains their reasoning Quantities and Counting to 1000 221

PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, According to Marian Small, “as students get older, the numbers they deal with in their everyday lives become more complex. Students need strategies MphbgluaaaentscnhdeeerrVetvaeaodlnilcszu,aabetbtl,eiouotnchlnaskos,rsuy,os:naensd, s, for representing and making sense of these greater numbers” (Small, 2009, p. 138). For example, students may realize that 900 is larger than 300 but not be able to visualize that it is three times larger. They need visual representations to make such connections, yet it is challenging to have such large numbers of concrete materials on hand, and to work with them if you do. Base ten materials can effectively help students create mental images of larger quantities and how they relate to each other. It is also important that students can make generalizations about numbers by extending the patterns in our counting system up to 1000. For example, any number that ends in a 3 is odd, and any number that is greater than 799 will have at least 8 groups of hundreds. Students can make assumptions based on such patterns rather than having to analyse each number individually to understand its meaning. About the Lesson In Unit 1, students built various three-digit numbers up to 500 using different combinations of base ten blocks. In this lesson, students create numbers up to 1000 by following some given parameters. They are using more base ten blocks than in the previous lesson, allowing for more possible combinations. Students also make an equivalent representation for some of the numbers and create examples of what the numbers cannot be, given the parameters. Minds On (20 minutes) Materials: • Students work in pairs. Tell students that they can choose any 4 base ten base ten blocks, blocks with which to create a number, and they must use all 4 blocks in the BLM 12: Base Ten number. Before students begin the task, have them turn and talk to their Blocks (optional), BLM partner about what the largest and smallest numbers could be. Discuss their 43: Creating Large predictions and have them explain their reasoning. Numbers and/or chart paper • Have students solve the problem. Record some of their findings using Time: 60 minutes squares, sticks, and dots (e.g., 400, 310, 301, 220, 211, 202, 121, 130, 112, 103, 40, 31, 22, 13, 4). Have students count each representation. Ask what all of the numbers have in common (e.g., all of the digits in each number add to 4). Ask why this is true. • Ask students which numbers could still be included in the list of possible answers if the numbers need to be between 100 and 299. Have students explain their reasoning. • Ask students which numbers cannot be included. Ask whether they need to look at each number individually or can make some generalizations (e.g., it can’t be any number that has no hundreds flats; it can’t be any combination with 3 hundreds flats). 222 Number and Financial Literacy

Working On It (20 minutes) NOTE: If you have limited quantities of base ten blocks, you can use paper versions, which students can cut out of BLM 12. • Students work in pairs to solve one or more of the following problems: – Choose exactly 8 blocks. Create numbers between 500 and 700. – Choose exactly 11 blocks. Create numbers between 750 and 1000. – Choose exactly 12 blocks. Create numbers between 625 and 850. • In each case, students record their numbers and a drawing of the base ten blocks (a representation of the number), using squares, sticks, and dots. Students also record an example of a number that doesn’t meet the criteria and include a drawing that justifies their answer. They also record any generalizations, such as “any number with 2 hundreds flats will not work.” They can use BLM 43: Creating Large Numbers or chart paper. Differentiation • Select the number of problems to be solved and the number of representations to be created based on students’ needs and abilities. • You can simplify the task by not requiring students to provide a non- example. • For students who need more of a challenge, have them create their own math problem. They can exchange it with another pair and students can solve each other’s problems. Assessment Opportunities Observations: • Can students find more than one solution? • Can they explain why their solutions meet the criteria? • Can students flexibly count the collections of blocks? Do they change their way of counting to match the value of the blocks? • Can they give examples of numbers that do not work? Can they justify why they do not meet the criteria in the problem? • Can they generalize what they have found to other numbers? Conversations: If students are having difficulty making generalizations, pose some of the following prompts: – Y ou just told me that you can’t make 411 with 5 blocks. Can you make 412 with 5 blocks? Why? Can you make 420? 450? Why? What is the largest number you can make with 5 blocks that is under 500? Why? What rule can you make? Quantities and Counting to 1000 223

Consolidation (20 minutes) • Meet as a class. Select one of the problems to discuss. • Discuss the possible numbers that students created for a problem and record them using numerals, and squares, sticks, and dots. In each case, have students explain how the number meets the criteria in the problem. • Discuss any systems students had for finding all of the solutions or whether they just randomly tried various combinations. • Discuss what students notice about all of the numbers that meet the criteria (e.g., their digits add to the number of blocks used). • Have students share their non-examples and explain how they do not meet the criteria. Look at all of the non-examples and ask students if they can make any generalizations about which numbers qualify and which do not. • Draw attention to the problem that invited students to work with 11 blocks. Ask what numbers they could create that are less than 150. Highlight the fact that they can make numbers such as 11 from 11 ones (instead of 1 ten and 1 one) or 110 from 11 tens. • Draw attention to the examples that do meet the criteria. Change the criteria so more examples are eliminated. Ask students how they know which numbers cannot be included and whether they can make generalizations to rule out several of the numbers (e.g., any numbers with 6 hundreds cannot meet the criteria). 224 Number and Financial Literacy

5Lesson Equivalent Representations of Quantities Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life • B 1.2 compare and order whole numbers up to and including 1000, in various contexts • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Possible Learning Goals • Composes and decomposes numbers to 1000 in a variety of ways • Understands and explains the purpose of zero as a placeholder for quantities expressed in numerical form • Describes the value of each digit in a three-digit number • Flexibly decomposes and recomposes three-digit numbers (e.g., 345 is 345 ones; 3 hundreds, 4 tens, and 5 ones; or 2 hundreds, 14 tens, and 5 ones) and renames the numbers • Applies their understanding of regrouping to compose numbers with no more than 9 base ten blocks in each place value position • Gives examples and non-examples of numbers that meet given criteria • Explains the purpose of zero as a placeholder for quantities expressed in numerical form PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, “Being able to recognize and generate equivalent representations of the same number is the part of number sense that will serve students well during tasks that require estimation, comparison, or computation” (Van de Walle & Lovin, 2006b, p. 43). For example, when solving 423 − 278, continued on next page Quantities and Counting to 1000 225

MphbluaaantschdeerVetveaodnlcsu,abetbl,eoutnchlasko,rsuyos:naensd, s, students can regroup the 2 tens and 3 ones in 423 into 1 ten and 13 ones in order subtract the 8 ones in 278. They can further regroup the 4 hundreds and 1 ten into 3 hundreds and 13 tens in order to complete the calculation. While this concept is covered in the addition and subtraction unit, it is beneficial if students have several experiences creating equivalent quantities using concrete materials. These hands-on activities help students form the mental images that will support them as they regroup amounts without the use of concrete materials. As students move away from concrete representations to working solely with numbers, they also need to have a firm understanding of zero as a placeholder. While they can clearly see 5 flats and 3 units as 5 hundreds, 0 tens, and 3 ones (or 503) on a place value mat, they may record the representation as 53 in numerical form. Understanding zero’s role is also important as students create an equivalent amount for a number like 503 so they can subtract 382. They must recognize that 4 hundreds and 10 tens are equivalent to 5 hundreds and 0 tens. About the Lesson In this lesson, students create quantities with base ten blocks that meet certain criteria. They can then regroup and rename their quantities using an equivalent representation. Materials: Minds On (20 minutes) base ten blocks, • Students work in pairs. Show students 5 hundreds flats and ask how much BLM 12: Base Ten Blocks (optional), chart they represent. Tell them that they are going to add on exactly 11 base ten paper, teacher-made blocks—tens rods or unit cubes, in any combination—to make numbers that chart (see Consolidation) are greater than 600. Which blocks will they add and why? Time: 60 minutes • Record students’ responses using numerals and squares, sticks, and dots (e.g., 11 tens and 0 ones to make 610, or 10 tens and 1 one to make 601). Ask students what they had to do in order to create a number more than 600 (e.g., there needed to be at least 10 tens to create another group of 100). • Have a student demonstrate how they needed to regroup the base ten blocks and how this action renames the number. • Ask what the zeros represent in each case. Ask why zeros are needed when writing the quantities in numerical form. Discuss the importance of zero as a placeholder. 226 Number and Financial Literacy

Working On It (20 minutes) • Pose the following problem: – Start with 7 hundreds. Choose exactly 13 more base ten blocks, either tens rods or unit cubes, in any combination. Use all of your blocks to create numbers that are a) between 700 and 750 or b) greater than 800. Differentiation • Change the quantities and criteria in the problem so they meet the needs of your students. • For some students who are struggling with regrouping to create new numbers, have them create any numbers using 13 blocks. They can see an example of the regrouping when it is discussed in the Consolidation. Assessment Opportunities Observations: Pay attention to how students solve the problem. • Do they work within the parameters? • Can they create more than one number within the parameters? • Can they count the base ten blocks and name the number? • Can they give an example of a number that will not work? • Do they have a system for creating the numbers or do they randomly choose blocks to see if they work? • Can they make generalizations? (e.g., If this number is too large, then all other numbers greater than that number will be too large too.) Conversations: If students are randomly choosing blocks and not seeing patterns, pose some of the following prompts: – What number did you make with the 7 hundreds, 10 tens, and 3 ones? (803) What would happen if you used 11 tens and 2 ones? Would the number be more or less than 803? How do you know? (e.g., It would be more because 11 tens is worth more than 10 tens.) What do you think will happen if you use more tens than 11? (e.g., The numbers will keep getting larger.) So what generalization can you make? (e.g., Using any amount of tens over 10 tens will make a number that is greater than 803.) Quantities and Counting to 1000 227

Consolidation (20 minutes) • Meet as a class. Record students’ findings in a chart. Ask what they notice about the number of tens rods and the number of unit cubes (e.g., They both add up to 13. As the number of tens decreases by one, the number of ones increases by one.). Hundreds Tens Ones Regrouped Drawing Number 7 13 0 830 7 12 1 821 7 11 2 812 7 10 3 803 7 0 13 713 7 1 12 722 7 2 11 731 7 3 10 740 7 4 9 749 • Ask students what they needed to do in order to make numbers that met the criteria, in most cases (e.g., they needed to choose more than 9 tens or 9 ones). Ask why this wasn’t necessary for 749. Discuss how they needed to regroup and how it changed the number of blocks in each place value column. Have the class name the regrouped number as a student demonstrates how to regroup the concrete models. • Draw attention to the numbers 830 and 803. Ask what is the same and what is different about them. Ask what the zeros represent. Ask why 830 can’t be written without the zero even though there are no ones, and why 803 can’t be written without the zero even though there are no tens. Discuss how it is easy to see from the concrete representations that there are no tens or ones, but harder to realize this when just the numbers are written. It is important to have zeros as placeholders to indicate the absence of any amounts in the place value columns. • Ask what the zeros mean in the number 1000. Ask how to write the number that is one more than 1000, the number that is 10 more than 1000, and the number that is 100 more than 1000. Record students’ responses and discuss the importance of the zeros in order for us to be able to differentiate the numerals and their matching quantities. 228 Number and Financial Literacy

6Lesson Skip Counting by 25s, 50s, 100s, and 200s Math Number Curriculum Expectations • B1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life Previous Experience • B 1.2 compare and order whole numbers up to and including 1000, in with Concepts: Students have skip various contexts counted forward by 25s. • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Possible Learning Goals • Accurately counts by 25s, 50s, 100s, and/or 200s (to 1000) and applies knowledge to solve related problems • Describes the patterns that result from counting by various amounts • Accurately counts by 25s, 50s, 100s, and/or 200s (to 1000) • Describes the patterns that emerge when counting • Selects and uses tools and strategies to help solve problems • Represents counting using a variety of tools (e.g., hundreds chart, number line, base ten materials) • Clearly explains the strategies they used to solve the problems PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, Counting serves the purpose of finding out the number of objects in a set. As students work with larger quantities, it is important that they can count accurately and efficiently. Students in grade three learn to skip count by 50s, 100s, and 200s (to 1000) and apply this knowledge to solve problems. This involves unitizing, or simultaneously seeing a group as representing one unit and several units. For example, they learn that a group of ten can be perceived as one unit of ten or ten individual units at the same time. Quantities and Counting to 1000 229

Math Vocabulary: About the Lesson ophoplunaenecndesr,nevbuadamlsus,beete,etrtnehlnsion, ues, ands, In the Minds On, students review skip counting by 25s and discuss the patterns that evolve. They extend this knowledge to count by other amounts. In the Working On It, they apply this knowledge to solve problems. Materials: Minds On (15 minutes) quarters (or coin • Show students a set of 13 quarters that are randomly organized. Review the manipulatives), BLM 44: Hundreds name and the value of the coins. Charts to 1000, concrete objects, markers, chart • Have students turn and talk to their partner and estimate the number of paper, calculator (optional) quarters and their total value. Time: 55–60 minutes • Together, count the quarters by 25s. Highlight the pattern in the numbers using hundreds charts 1–100, 101–200, 201–300, and 301–400. • Explain that they investigated skip counting by 25s up to 500 in an earlier unit. Ask whether they think this pattern will continue up to 1000 and why they think so. Working On It (20 minutes) • Tell students that they are going to apply their understanding of skip counting to solve one or more problems (see below). Explain that the problem can be solved in other ways, but today they will solve it using skip counting. After, they can confirm their answer by solving the problem with a different strategy. • Students work in pairs. They can solve the problem(s) and record their thinking on chart paper. – P roblem 1: Anna has 735 cookies. She wants to package them in groups of 25. How many packages can Anna make? Are there any cookies left over? – Problem 2: At the candy factory, workers put 50 candies in each package. How many packages can workers make with 954 candies? – Problem 3: There are 800 marbles. In what different ways could they be packaged so there are the same number in each package? Each way must have at least 20 marbles per package. • Let students know that they can use base ten blocks, hundreds charts, or number lines to support their problem solving. Differentiation • Adapt the quantities so they meet the needs of your students. • For students who find it difficult to skip count, encourage them to use the hundreds charts. They can also use a calculator. 230 Number and Financial Literacy

• For students who need more of a challenge, add the following to Problem 1: – How many packages can you make if there are 75 cookies per package? Will there be any cookies remaining (left over)? – W hich packaging method results in the least number of leftover cookies? Assessment Opportunities Observations: • Do students understand the counting pattern? • Can they extend this pattern to larger numbers up to 1000? • How do they keep track of the number of groups as they are skip counting? • Do they understand what the number of groups represents in terms of the context of the problem? Consolidation (20–25 minutes) • Meet as a class. Discuss the solutions to the problems. Highlight the patterns in their skip counting. Draw attention to the ways in which students kept track of the number of counts. • Discuss how students dealt with the remaining cookies in Problem 1. Explain that in real-life situations, items often don’t come in the exact amount that we want to group them in so we need to make accommodations. • Ask students if they found another way to check that their solutions were correct. For Problem 1, for example, discuss how students could have counted 4 packages for every group of 100 since there are 4 groups of 25 in 100. Compare this to grouping 4 quarters together into $1 or 100¢. • If students did not solve the addition to Problem 1 given in the Differentiation section (having 75 cookies per package) pose the problem now. Discuss how knowing how many packages there are with 25 cookies in each can help them solve this problem. Ask which way of packaging results in the least number of leftover cookies. • Discuss how students would deal with any remaining (leftover) quantities in their solutions. This reinforces the idea that context can affect how we deal with real-life situations when things may not work out evenly in terms of the mathematics. • B uilding Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: Conclude the discussion by asking students to look back at their work for 5 to 10 minutes. Have them share with the class one or two things they might do differently if asked to solve this problem again. Reinforce the importance of looking back at their work to reflect on the strategies used in order to do better in the future. It is through reflection and sharing that we all can improve. Quantities and Counting to 1000 231

7Lesson Counting by Place Value Amounts (1s, 10s, 100s) Math Number Curriculum Expectations • B1.1 read, represent, compose, and decompose whole numbers up to and Previous Experience including 1000, using a variety of tools and strategies, and describe various with Concepts: ways they are used in everyday life Students have experience counting by 1s and 10s • B 1.2 compare and order whole numbers up to and including 1000, in on a hundreds chart. They have worked on various contexts tasks that require them to decompose and represent • B 1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools 100s and 10s. and strategies Teacher Look-Fors • B1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Possible Learning Goals • Counts forward and backwards by 1s, 10s, and 100s from various starting points • Understands how the digits in a number relate to each place value position • Recognizes how the direction of skip counting (forward or backwards) impacts the magnitude of the number • Counts by 1s, 10s, and 100s from various starting points • Explains the patterns that emerge from skip counting from various starting points PMraotcheesmseast:ical About the ccssrPooeterrmnalaoensbtmceoeltegncuinmitinenignisgcgst,aoo,artolenrivnelfdislgnpepgracer,ntosidnevginn,tgin, g, As Marian Small explains, “place value work depends on students’ ability to count using more than one grouping in the same situation” (Small, 2009, p. 144). For example, when counting a value like 347, students first must count three times by 100s, then switch to counting by 10s for four counts, and finally continue counting on by 1s seven times. To do this, students need to understand the relationships among the digits in our number system. 232 Number and Financial Literacy

Math Vocabulary: John Van de Walle and LouAnn Lovin caution about the language we use to skip count, forward, describe place value groupings. Referring to numbers as having “4 tens and backward 3 ones” can be very confusing. They suggest initially using phrases such as “groups of tens and leftovers,” or “bunches of tens and singles” until students clearly understand what “4 tens and 3 ones” means (Van de Walle & Lovin, 2006a, pp. 132–133). The hundreds chart is a useful tool for helping students recognize number patterns that emerge from counting by 10s, and 1s. Several hundreds charts can be taped or shown together to create a thousands chart (see BLM 44: Hundreds Charts to 1000). This allows students to see the patterns that emerge when counting by 100s. About the Lesson In the Minds On, students review skip counting from various starting points. In the Working On It, they play a game that reinforces counting by place value amounts from various starting points, adjusting their counting pattern to match the size of the increments. Materials: Minds On (20 minutes) base ten blocks, • Ask students when they might skip count and why it makes sense to do so. BLM 44: Hundreds Charts to 1000, counters Discuss how skip counting requires having equal groups and is beneficial in different colours, when counting large quantities. BLM 45: Base Ten Spinner, paper clips and • Have one student build a thousands cube by adding hundreds flats, one at a pencils, number cube (1–6), markers, time, as the rest of the students skip count by 100s to 1000. Have another chart paper student remove the hundreds flats, one at a time, while students skip count backwards. Ask how the two counting patterns are the same and how they Time: 55 minutes are different (e.g., they both have the same numbers, but it is the direction of counting that is different). • Next, ask students to pick any two-digit number. For example, they might pick 23. Ask how they would count on from 23 by 100s. Continue until the first number past one thousand is reached—in this case, 1023. Record the counting sequence and discuss the patterns that students see (e.g., which numbers stay the same and which numbers change). • Have students pick a large three-digit number and count down by 100s from that number. Record numbers as students count. Ask how this pattern is similar to counting forward by 100s. • For the next counting-by-100s exercise, have students select a one-digit number (e.g., 9) to use as a starting point, to assess whether they understand the role of zero as a placeholder in the tens column. Ask how each number in the counting pattern would be recorded. Record all the numbers to just past 1000 (e.g.,. 109, 209, 309, …, 1009). Discuss why there is a zero in the tens column. Ask why there are two zeros as placeholders in the number 1009. Quantities and Counting to 1000 233

Working On It (20 minutes) • Show students hundreds charts taped or arranged together to form a thousands chart (see BLM 44: Hundreds Charts to 1000). Put a counter on 3. Ask, “If I were to move forward by 100, where would I end up?” (103) Ask where they would end up if they add another 100. • Put the counter on 156 on the hundreds chart. Ask, “If I move forward by 100, where would I end up?” (256) “If I moved backwards by 100 from 156, where would I end up?” (56) • Put a counter on 396. Ask, “If I move forward by 10 three times, where would I end up?” • Repeat until students understand this movement on the charts. • Students work in pairs and play the game Race to 1000: – G ive each pair a copy of the thousands charts (BLM 44), BLM 45: Base Ten Spinner, a paper clip and pencil to use on the spinner, and a number cube. – B oth players put their marker on the top left corner of the thousands chart. The goal is to be the first player to get to 1000. – P layers take turns spinning the spinner, which indicates the amount that students will move, and rolling the number cube, which indicates how many times students will move that amount. For example, if a player is at 25 on the ‘board’ (the thousands chart) and rolls a 5 on the number cube and a 10 on the spinner, the resulting move is 5 jumps of 10, which will take them to 75. Players count their movements aloud while their partner confirms that their count is correct. – The first person to reach 1000 wins. Differentiation • Play the game in a guided math group for students who still require practice skip counting from various starting points. • For students who need a greater challenge, they can play the game backwards, starting at 1000 and ending at 1. Assessment Opportunities Observations: Pay attention to students’ counting as they play the game. • Listen for students who may be less confident in properly saying the names of the numbers. You may want to have a guided lesson on how the numbers are spoken (e.g., 107 is “one hundred seven” and not “one hundred and seven”). • Do students need further support in counting by 1s, 10s, and/or 100s from various starting points? • Can they adjust their count according to the magnitude of the place value amounts? 234 Number and Financial Literacy

• Can they anticipate where their turn will end or do they need to count it out? • Can they correctly say the names of the numbers as they count on the hundreds chart? Consolidation (15 minutes) • Meet as a class and discuss any of the problems or misconceptions students may have experienced as they played the game. Pose some of the following prompts: – What were the most challenging counting situations? What made them challenging? (e.g., counting backwards by 10s from an odd number like 137; forward counting by 1s across decade numbers) – What amounts were the easiest to count by? Why? – How did your counting change when you moved from one hundreds chart to another? • B uilding Social-Emotional Learning Skills: Healthy Relationship Skills: Ask students to think/pair/share to answer: – Why do you think we play math games? (to learn new things, to have fun, etc.) – Explain to students that when we play games we are trying to be effective game partners. In order to learn how to be effective game partners, we need to learn what an effective game partner is like. – Create an anchor chart entitled “Effective Game Partner.” Divide the chart into four sections and write “Them Winning” at the top left and “You Winning” at the bottom left. On the right side, write “Them Losing” at the top and “You Losing” at the bottom. – Have students think/pair/share about how an effective game partner would react to each scenario. Write their ideas down under the appropriate headings. Further Practice • Math Game: 1000 Snap! Directions: – P lay this game as a class. The goal is to count forward by 25s, 50s, or 100s, without being the student who reaches 1000. – S tudents stand in a circle. One student starts the count at 0 and each student continues the count using their choice of 25, 50, or 100 as increments. The student who lands on 1000 says “Snap!” and sits down. – T he remaining students play the game again. Play continues until there is only one student left. That person is the winner! Quantities and Counting to 1000 235

• Variations: – A llow students to say 1 or 2 numbers at a time. – S tudents can play this game as a smaller group independently for further practice. – H ave the count start from various starting points, rather than from 0. 236 Number and Financial Literacy

8Lesson Skip Counting Using a Calculator Math Number Curriculum Expectations • B1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life Previous Experience • B 1.2 compare and order whole numbers up to and including 1000, in with Concepts: Students have skip various contexts counted by 25s, 50s, and 100s to 500. • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Algebra • C1.4 create and describe patterns to illustrate relationships among whole numbers up to 1000 Possible Learning Goals • E xplores skip-counting patterns using a calculator • A pplies understanding of skip counting to solve a problem • Uses a calculator to count forward by 50s and 100s (to 1000) • Recognizes and describes the patterns that emerge in the skip-counting sequence • Predicts what the next numbers will be before using the calculator • Explains emerging number patterns and why they occur • Understands what is being asked in a given problem • Applies understanding of skip counting to solve a problem • Explains the solution within the context of the problem and why it makes sense PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, Skip counting and predicting numbers in a growing pattern develop students’ understanding of number relationships. Calculators allow students to predict how familiar number patterns continue across larger numbers. Students can also generate a series of numbers using the repeat continued on next page Quantities and Counting to 1000 237

Math Vocabulary: function on the calculator and then study the patterns within those srekpipeacot ubnutt,tocnalculator, numbers. They can also compare the patterns that emerge when skip counting by different amounts. For example, they can skip count by 5s from a number that is not a multiple of 5 and then compare the pattern created to the those generated when counting by 10s and 25s from the same starting point. Students can apply their knowledge of number patterns to solve problems. By using a calculator, they can work with larger and less friendly numbers. The focus can be on the patterns that emerge rather than on performing each calculation with concrete materials or using mental strategies. About the Lesson During the Minds On, students review the different counting strategies and numbers used so far. The Working On It is a guided lesson in the use of a calculator, before students use calculators in pairs to skip count and extend number patterns. Materials: Minds On (15 minutes) calculators (one per • Review skip-counting patterns on a hundreds chart. In each case ask how the pair of students), markers, chart paper, patterns continue past 100. base ten blocks • If students do not remember how to skip count using the calculator (see Unit Time: 55 minutes 1, Lesson 7), review it by providing the following instructions while students work in pairs on calculators. 1. Press the On/C button. What happens when you press it? (You see 0.) 2. Press + then press 50. Now press =. What do you see? (50) 3. Press = again. Now what do you see? (100) 4. What is happening? (It’s counting up, adding.) 5. Let’s keep pressing the = sign and read the numbers that appear. [Record the pattern on the board as students say the numbers.] 6. What number pattern is that? (counting up by 50s) 7. P redict what the pattern will be if you start skip counting from 1 instead of 0. Then confirm your prediction on the calculator. [Press 1 + 50 = = = etc.] 8. Predict what the pattern will be if you start skip counting from 292 for 10 numbers. Then check it out on your calculator. 238 Number and Financial Literacy

Working On It (20 minutes) • Students work in pairs. Pose the following problem and ensure all students understand the context. Explain that it can be solved in different ways, but today they will use a calculator. Have them record their thinking and what they did on the calculator on chart paper. – The city is building a skyscraper that will hold 943 offices. They want to have 25 offices per floor. The remaining offices will be equally split between the top two floors of the building. How many floors will have 25 offices? How many offices will be on the top two floors? Differentiation • Adjust the numbers so they meet the needs of individual students. • For students who need more of a challenge, have them solve the problem with different numbers of offices per floor. You may want to have 50 or 75 offices per floor so they can predict the pattern while thinking of the multiples of 25. You may also challenge them to find the number of floors if there are 40 offices per floor. Ask them how they could accommodate the extra offices on upper floors that have fewer than 25 offices each. Assessment Opportunities Observations: • Can students use the calculator independently? • Are they able to predict the next numbers in the pattern? • Do they have a strategy for keeping track of their counts? • Do they understand what each group of 25 represents within the context? • Do they understand what each count represents within the context of the problem? • Do they have a plan to deal with the remaining offices? Conversations: If students have difficulty visualizing the problem, pose the following problem and have them create a model using base ten blocks. • Let’s imagine that there are 550 offices with 100 offices per floor. How could we show one floor using base ten blocks? How many flats will you need to show all of the offices? Build the skyscraper with the base ten blocks and then count the number of offices. How many floors have you created? Have you accounted for all of the offices? What can you do? (e.g., create a floor with only 50 offices) Imagine each one of these flats is worth 25 offices. Will there be more or fewer floors? How could you start building the skyscraper in the problem? Quantities and Counting to 1000 239

Consolidation (20 minutes) • Meet as a class. Strategically select two or three pairs to explain how they solved the problem using the calculator. Discuss how they kept track of the number of times they skip counted. Ask what each count represents within the context of the problem. Record the skip-counting pattern on an open number line and connect this representation to the series of numbers generated on the calculator. • Discuss how students figured out how many offices would be on the two top floors. On the number line, highlight how the count continues from 925 with a jump of 9 to 934, and another jump of 9 from 934 to 943. Ask whether the offices on the top floor would be bigger or smaller than the offices on the other floors and how they know. Ask whether they would be more or less than double the size. • Discuss other ways students could have solved this problem (e.g., figuring out the number of hundreds and then knowing there are 4 floors per hundred). Further Practice • Independent Task in Math Journals: Have students complete the following actions: – Pick a starting number, an operation (addition or subtraction), and a number to count by (25, 50, or 100). – Use a calculator to skip count by that number and record the series of numbers you find. – Count again using the same starting number and operation, but a different number for skip counting. – Compare the two patterns that emerge. 240 Number and Financial Literacy

and9 10Lessons Comparing Quantities to 1000 Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life Previous Experience • B1.2 compare and order whole numbers up to and including 1000, in with Concepts: Students have used various contexts various tools to skip count by 50s, 100s, and • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools 200s up to 1000. and strategies • B1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Possible Learning Goals • Identifies quantities that are more or less than a given amount and locates them on a number line • Creates comparison statements for given numbers and locates them on the number line with reasonable accuracy • C reates meaningful benchmarks on a number line to best accommodate the numbers being used • Selects an appropriate strategy to find an amount that is more than or less than a given quantity • Selects reasonable endpoints for a number line, given the numbers being represented • Selects and locates meaningful benchmarks for a number line and explains their reasoning • Locates given numbers on a number line with reasonable accuracy • Explains how the benchmarks help to locate the position of the quantities on the number line • M akes both general and more precise statements when comparing three numbers in numerical form Quantities and Counting to 1000 241

PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, As students extend their understanding of quantity to 1000, it is important that they can compare and order numbers according to their relative Math Vocabulary: magnitude, which is “the size relationship one number has with another—is more than, less than it much larger, much smaller, close, or about the same?” (Van de Walle & Lovin, 2006b, p. 45) On number lines with endpoints 0 and 1000, more general comparisons can be made, while students can make more precise comparisons when the endpoints are closer together, such as 700 and 800. In either case, students need to realize that the relative distance between the numbers remains the same, no matter what the endpoints are. Students also benefit from ordering the same quantities on number lines with different endpoints. This requires flexible thinking and being able to change the benchmarks that are most meaningful, depending on the endpoints and the numbers being used within the problem. About the Lessons In Lesson 9, students use number lines to compare quantities. They establish meaningful benchmarks and then play a game to practise identifying and locating quantities that are more or less than a given amount. In Lesson 10, students compare three numbers on the number line and select the best benchmarks and endpoints for their number line. 242 Number and Financial Literacy

9Lesson Investigating Quantities that are More Than or Less Than Materials: Minds On (20 minutes) BLM 46: More • Hang a string across the room within students’ reach. Use index cards and Than/Less Than Spinner, BLM 47: More paper clips to label the endpoints of the string 0 and 1000. Ask what numbers Than/Less Than Cards, could be on the number line and which numbers could not. Ask what string, paper clips, index benchmarks would be helpful to have on the number line. Discuss different cards, markers, chart alternatives (e.g., the hundreds; 250, 500, 750). paper, different-coloured counters, BLM 44: • Discuss what strategy students could use to add the benchmarks. Discuss Hundreds Charts to 1000 (optional) how these need to be equally spaced across the number line. Time: 60 minutes • Ask what would change and what would be the same if students put the benchmarks on a shorter string. (e.g., the numbers would all be closer but they would still be evenly spaced) • Show students a three-digit number (e.g., 372). Discuss where they would place it on the number line and why they would place it there. • Ask which number would be 200 more and where it would be placed on the number line. Ask which number would be 300 less and where it would be positioned. Ask what number would be 70 more and what number would be 40 less than the original number. Discuss where to place the numbers on the number line. Working On It (20 minutes) • Students work in pairs. Give each pair a long strip of chart paper on which to draw a number line. They should label the ends 0 and 1000, select benchmarks, and add them to the number line. • Each student then gets a different-coloured counter and puts it on/below 500. Students take turns spinning the more than/less than spinner (BLM 46), choosing a card from BLM 47: More Than/Less Than Cards, and moving their counter forward or backwards along the number line accordingly. They also record the new number on the number line. • Students play the game until someone reaches 1000 or is closest to 1000 when a set time has expired. Quantities and Counting to 1000 243

Differentiation • Some students may benefit from working on hundreds charts taped together to make a thousands chart (see BLM 44). You may also limit the range of the numbers that certain students are given for their number line. • For students who need more of a challenge, have them work with numbers that are less friendly. Assessment Opportunities Observations: Pay attention to how students are locating numbers. • How do they place the benchmarks? Are they evenly spaced? Do they make adjustments? Do they fold the number line to help them find the midpoint, etc.? • D o they use the benchmarks to place the numbers in the game? Do they place the numbers relative to their values or do they just put them in between two benchmarks? • D o they accurately record their new position each time? Conversations: If some students are having difficulty figuring out how much more or less a number is, try the following activity with a small group of students. Frame the Number: Give each student a hundreds chart (e.g., 701– 800) and some counters and ask students to carry out the following steps: – Put a counter on 20 less than 765 (745). – Put a counter on 2 less than 758 (756). – Put a counter on 5 more than 731 (736). – Put a counter on 5 more than 742 (747). – What number is framed (or surrounded) by all of the numbers that you just covered? (746). Consolidation (20 minutes) • Meet as a class. Have some students share their number lines and their strategies for figuring out the new number after each move. Highlight how they may have decomposed the numbers in order to move to the new number (e.g., move 70 in two jumps of 50 and 20). • Select some numbers that students may have found difficult to accurately place on the number line. Discuss where to locate them on the class number line. • Change the endpoints of the class number line (e.g., 100–800). Ask which numbers can no longer be on the number line and where the other numbers need to be located now. Ask whether the numbers are closer together or further apart than on the previous number line. 244 Number and Financial Literacy

• Discuss how the numbers are further apart on a number line covering a smaller range, but the relative distances are still the same. • Ask students when they may want to solve a problem on a number line from 0–1000 and when they might prefer a number line that has closer endpoints, such as 500 and 700. • B uilding Social-Emotional Learning Skills: Healthy Relationship Skills: Review the chart created about effective game partners (Lesson 7, above). Discuss how students can help a partner in a math game, even when both partners are trying to win the game. Further Practice • Independent Reflection in Math Journals: Give students a hundreds chart and have them create their own Frame the Number clues for another student to solve (see Assessment Opportunities, Conversations, above). Quantities and Counting to 1000 245

10Lesson Comparing Numbers on a Number Line Materials: Minds On (15 minutes) string, index cards, • Use the class number line made of string from the previous Lesson, with no markers, chart paper numbers on it and no endpoints. Time: 55 minutes • Show students three numbers (e.g., 242, 354, 837). Have students turn and talk to their partner about all the comparing statements they could make about the numbers. • Record students’ ideas on chart paper as they share them. Prompt them to create general statements (e.g., 354 is larger than 242 but much smaller than 837; 242 and 837 are both to the left of the midpoint of the nearest hundreds [250 and 850], but 354 is to the right of the midpoint [350]) as well as more precise statements (e.g., 354 is 112 more than 242). • Ask what endpoints they would put on the number line to accommodate all three numbers. • Discuss what benchmarks they could use and why they would be helpful. Together, decide where the benchmarks should go. • Have students discuss where to place the three numbers. Revisit the comparisons they made and prove that each is correct by looking at the number line. Working On It (15 minutes) • Students work in pairs. They choose three numbers to compare. They can draw a number line on chart paper, choose the best endpoints and benchmarks to accommodate their numbers, and then mark the locations of their numbers. • They can create comparing statements and record them under their number line. Differentiation • You may decide to choose the three numbers for some students if they need reinforcement with certain numbers or increments. • For students who need more of a challenge, have them choose four or five numbers to compare. 246 Number and Financial Literacy

Assessment Opportunities Observations: • C an students choose appropriate endpoints and benchmarks for their chosen numbers? • Are students placing their numbers with reasonable accuracy, taking into account the relative positions of the numbers? Conversations: If students are just placing a number between benchmarks and are not considering its relative position between those benchmarks, pose some of the following prompts. – W here does number 469 go? (e.g., between 400 and 500) If you could divide the space between 400 and 500 in half, what number would be at the halfway mark? Why? Show me where that would be on the number line? Would 469 be between 400 and 450 or between 450 and 500? Would 469 be closer to 450 or 500? How do you know? Is it a lot closer or just a little closer? What would be the midpoint between 450 and 500? How can that help you place the number 469? Consolidation (25 minutes) • Students meet with another pair. They take turns sharing their numbers. Together, they can create statements that compare the two groups’ numbers to each other or compare their numbers to a third number of their choice. (e.g., Which of our numbers is closer to 158?) Students can then share their comparison statements and number lines. • Meet as a class. Strategically choose two or three pieces of students’ work to share and discuss. Highlight why the endpoints and benchmarks used are most appropriate for the numbers. Have the creators of the work explain their comparison statements and their reasoning for them. • In each case, ask how the position of the numbers would change if different endpoints were used. Reinforce the idea that the closer the endpoints are in distance, the further apart the numbers are spread out, yet the relative distances remain the same. Quantities and Counting to 1000 247

11Lesson Estimating Large Quantities (to 1000) Math Number Curriculum Expectations • B1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life Previous Experience • B1.2 compare and order whole numbers up to and including 1000, in with Concepts: Students have estimated various contexts large quantities by looking at visual Possible Learning Goals representations of sets. • Connects numbers to benchmark numbers on a number line • Understands that benchmark numbers are ‘friendly’ numbers close to numbers we are working with • Demonstrates the ability to make accurate estimates that indicate the relative size of a number • Uses estimates to make predictions and assess reasonableness of answers • Selects appropriate tools and strategies to estimate (e.g., hundreds charts, number lines, benchmark numbers) • Identifies key vocabulary words to determine how to estimate (i.e., ‘less than’ versus ‘more than’) • Explains and justifies their estimates using appropriate vocabulary (e.g., about, nearly, almost) PMraotcheesmseast:ical About the ccaaPoonnrmnoddnbmspeletrcuromtanivtniiecsngagog,itl,erivnesisngpe,grlree,ecsrfelteeianncstgtioinntngogi,no, gls Estimation strategies are “mental math strategies used to obtain an approximate answer. Students estimate when an exact answer is not required and when they are checking the reasonableness of their mathematics work” (Ontario Ministry of Education, 2020). According to Marian Small the context in which we estimate determines: • if an estimate or an exact answer is needed; • how close the estimate needs to be to the actual value; • whether a high or low estimate is needed (Small, 2009, p. 160). Small uses the example of using a high estimate to find the cost of two items to ensure you have enough money, and using a low estimate when you are figuring out how much you can afford to buy. Estimates can have a lot of room for variance, therefore the context might determine whether the estimate needs to be to the nearest thousand, hundred, ten, or other meaningful benchmark. 248 Number and Financial Literacy

Math Vocabulary: About the Lesson enmnsuetemiamarbslayeu,trre,ae,c,lmbeheneontisigcmthhetm,traaebr,koruutle, r, During the Minds On, students are presented with various estimation statements and they determine what the actual numbers might reasonably be. In the Working On It, students work with estimates of animals’ lengths and heights and place them on an open number line. Materials: Minds On (15 minutes) “Comparing Lengths” • Review what an estimate is, emphasizing that it is an educated prediction or (pages 6–7 in the Number and Financial guess based on what is known or can be seen. Discuss why estimation is Literacy big book and valuable in many circumstances (e.g., estimating to closest dollar amounts little books), metre sticks, when shopping, estimating the time it takes to drive a long distance). chart paper, markers Time: 55 minutes • Discuss what types of numbers we often use for estimation, highlighting how friendly numbers are easier to visualize and to use in calculations. • Read different estimation statements to students one at a time. Personalize some of the following examples so they are meaningful to your students. Discuss what the actual amounts could be based on the estimates (e.g., If Jesse is ‘nearly 200 cm tall,’ he might be 190 cm tall, or 196 cm tall.) and what words in the statements indicate why their suggestions are reasonable. – There are just over 550 students at our school. – Jesse is nearly 200 cm tall. – Simon can run much farther than 1000 m without stopping. – There are a lot fewer than 800 candies in the jar. – We have raised slightly more than $600 through our fundraiser. • With each example, mark the estimated value and the suggested actual values on an open number line. Ask whether there could be other numbers used as the estimates. (e.g., If Jesse is 192 cm tall, we could say he is ‘just over 190 cm tall.’) Working On It (25 minutes) • Show students the big book pages titled “Comparing Lengths.” • Review what length is and what attributes of the animals on the page have a length that could be measured (e.g., body, tail, trunk). • Show students a metre stick and review the units of centimetres and metres and the relationship between them. • Students work in pairs to formulate reasonable estimates for some of the length statistics of the animals. In each case, they record their estimate and the actual amount on an open number line, selecting the best endpoints for each open number line. They record estimation statements under each number line. Quantities and Counting to 1000 249

• Co-create an anchor chart of some of the words and phrases that students could use in their estimation statements. (e.g., about, nearly, slightly more than, almost, close to, a little more than, a little less than, quite a bit more than) Add the chart to the Math Word Wall. Differentiation • Limit the number of estimation problems to reduce the scope of the task. • Select the numbers that students are estimating so they are within their understanding. • For ELLs or any students who need language support, you may want to have a guided lesson to explain and demonstrate the terms that can be used in the estimation statements. • For students who need more of a challenge, have them find two or three measurements that can be estimated using the same number but a different estimation statement (e.g., the length of a cheetah’s head and body is ‘almost 150 cm’ and the length of the a red kangaroo’s head and body is ‘just over 150 cm’). Assessment Opportunities Observations: • A re students selecting reasonable benchmark numbers? • D o they adjust their benchmarks from question to question? • Can they formulate matching estimation statements using the appropriate language? Conversations: Use some of the following prompts to further probe student thinking. • What does this estimation phrase on the anchor chart mean to you? Does it tell you whether the actual amount is more or less than the benchmark number? How can you use that language in your estimation statement? • Let’s measure out the length using metre sticks, and measure 1 metre beyond the actual measurement. Which metre is the actual amount closest to? What estimation statement can you make using words from the anchor chart? • If we want an estimation amount that is closer than a full metre, what could we use? (e.g., a half metre—rather than 600 or 700 use 650) Consolidation (15 minutes) • Strategically select some student work to share with the class. Discuss why their estimates are reasonable and whether the estimation statements are helpful for choosing the most appropriate benchmark numbers. • Discuss how they chose the key words for their estimation statements. Ask if there are any others they could add to the chart. 250 Number and Financial Literacy

• Ask how one number could have different estimates and estimation statements, depending on the circumstances of the problem. Further Practice • Independent Reflection in Math Journals: Use a ruler to measure five things in the classroom to the nearest centimetre. Write estimation statements about each one. Have a partner guess the actual measurement of the object. Quantities and Counting to 1000 251

12Lesson Applying Strategies to Solve Real-Life Problems Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and including 1000, using a variety of tools and strategies, and describe various ways they are used in everyday life • B 1.2 compare and order whole numbers up to and including 1000, in various contexts • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Algebra • C 4. apply the process of mathematical modelling to represent, analyse, make predictions, and provide insight into real-life situations Possible Learning Goal • Applies understanding of numbers to 1000 to solve related problems and explain their strategies and thinking Teacher • Understands the problem and what needs to be solved Look-Fors • Selects appropriate tools and strategies to solve the problem • Explains their strategies and thinking using appropriate math vocabulary Previous Experience • Explains their solution in terms of the context and why it is reasonable with Concepts: • Flexibly composes and decomposes quantities Students have represented • Counts quantities by flexibly adjusting counting strategies to the place value quantities to 1000 in various ways, including amounts being represented using place value amounts. 252 Number and Financial Literacy

PMraotcheesmseast:ical About the Problem solving, ccssrooetermnalaenstmceoetgncuinitinenignisgcgt,ao,artoenrinelfdslgpepracerntosidnevginn,tgin, g, In this unit, students have had several experiences with representing and comparing numbers to 1000. It is important to offer them opportunities to Math Vocabulary: apply what they have learned to solve real-life problems. This helps them snkuimp bcoeur nlitnineg, open make meaning of the numbers and understand why these large quantities are important in their lives. Modify the context of any problems, if/as needed, to make it more meaningful and relevant to students. About the Lesson In this lesson, students solve real-life problems involving quantities to 1000. There are a number of problems from which to choose. You may decide to repeat the process of this lesson two or three times using a different problem each time. Alternatively, you may assign different problems to different groups, according to the needs of your students. This 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 Throughout the lesson, use an anchor chart to highlight how students move back and forth among the four components. For example, as students test out their model, they may need to revisit the problem (Understand the Problem) or the conditions surrounding the problem (Analyse the Situation) to develop and refine strategies and select more appropriate tools. There are suggestions in the lesson about when and how to reinforce the process, although these will need to be altered so they are responsive to how your students progress throughout the lesson. Materials: Minds On (10 minutes) a variety of concrete • Display one of the problems and read it together so all students understand materials and tools, such as base ten blocks, the context and what they are supposed to solve. (Understand the Problem) number lines, hundreds charts, chart paper • Discuss what other information students may need in order to solve the Time: 55 minutes problem. For example, they may need to understand other parameters that are evident in the context. (Analyse the Situation) • Reinforce that students can solve the problem in any way they choose and use any types of tools or concrete materials to design and refine their model. (Create a Model) • Encourage students to solve the problem in another way to confirm that their thinking makes sense. Quantities and Counting to 1000 253

Problems – Jack has run 372 metres in his 1000-metre race. What do you know? How much farther does he have to run? If he wanted to run the rest of the race in three intervals that are about the same distance, what might they be? – M akwa has 4 groups of beads in different colours that add up to 1000 beads. Two groups have the same amount of beads, one has fewer beads than the two equivalent groups, and the other group is greater than the two equivalent groups. How many beads might there be in each group? – B uild a structure using base ten blocks that is worth $1000, knowing that the unit cube is worth $1. How could you make the structure taller by not changing its dollar value? – T he answer is 786. What might the problem be? How can you solve it in more than one way? – U se 16 base ten blocks to represent some numbers. Have some be very large, some be very small, and some be in between. What numbers can you not make? Why? Working On It (20 minutes) • Students work in pairs to solve the problem. Have them record their solutions on chart paper. (Create a Model) Differentiation • Select the problems that best meet the needs of your students. Change the context so the problems have more relevance. • For students who need more of a challenge, they can create and solve their own problem. Assessment Opportunities Observations: Pay attention to whether students know how to start their problem. What tools and concrete materials are they selecting? Can they switch to a new strategy if the one they are using does not work? They may need to revisit the problem (Understand the Problem) or the conditions surrounding it (Analyse the Situation) in order to select appropriate tools and strategies to refine their model. Conversations: If students are having difficulty starting the problem, read it over with students and ensure they understand what is being asked. Ask which tools might help them start the problem. Encourage them to represent one of the quantities and then read the problem again to see how they can progress from there. 254 Number and Financial Literacy