p27-97-Gr3ON-Number-Unit1-count

Unit 1: Quantities and Counting to 500 Lesson Content Page Quantities and Counting to 500 Introduction 27 30 1 Read Aloud: 365 Penguins: First Reading 35 40 2 365 Penguins: Second Reading 44 3 Estimating Large Quantities (to 500) 48 4 Rounding Two-Digit Numbers to the Nearest Ten or Hundred 52 57 5 Extending Patterns in Two-Digit Numbers to Three-Digit Numbers 62 (to 500) 68 69 6 Skip Counting by 50s and 100s 73 76 7 Skip Counting Using a Calculator 78 80 8 Using Base Ten Blocks to Compose Numbers 84 87 9 and 10 Composing Numbers to 500 in Various Ways 90 95 9 Representing and Renaming Equivalent Quantities 10 Composing Quantities to 500 through Problem Solving 11 and 12 Using a Number Line to Represent and Compare Quantities 11 Ordering Numbers on a Number Line 12 Comparing and Ordering Numbers on a Number Line 13 Problem Solving with Quantities to 500 14 Representing Quantities Using a Concept Circle 15 365 Penguins: Revisiting the Math 16 Reinforcement Activities



Quantities and Counting to 500 Introduction About the In this unit, students represent, compare, and order quantities to 500 using concrete materials, drawings, various tools, words, and symbolic representations. Making connections between the various representations is critical for helping students make sense of the magnitude of these increasingly large quantities. Students also need to develop meaningful benchmarks, such as perceiving 200 as the number of students in the school. Creating benchmarks, like a set of 100 paper clips, can also help students form a visual image that can be used to estimate larger quantities. Composing and Decomposing Numbers Students develop a sense of quantity by composing and decomposing different numbers in various ways. The ability to look at a quantity in relation to its parts is described as a part-part-whole relationship. John Van de Walle and LouAnn Lovin explain that “to conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers” (Van de Walle & Lovin, 2006a, p. 43). For example, knowing that the number 8 can be broken into 5 and 3 allows students to solve an expression like 5 + 8, by decomposing it into 5 + 5 + 3, thereby creating a group of ten. Counting As grade three students represent, compare, and order whole numbers to 1000, counting continues to play an integral role. Counting larger quantities requires more efficient strategies such as organizing sets into equal groups and skip counting. Such experiences reinforce the ability to unitize—to see a group as one unit but simultaneously representing several units. For example, when using base ten blocks, a hundreds flat represents one unit of one hundred as well as one hundred individual units. Flats can be counted by 1s and interpreted as the number of hundreds, or counted by 100s to represent the number of individual units. Grade three students also skip count by 50s, 100s, and 200s, using a variety of tools and strategies. Tools such as hundreds charts, number lines, and coins can help students count in a meaningful way. Place Value Place value is a key concept explored in this unit. Grade three students become more aware of the relationship among ones, tens, hundreds, and thousands and how digits in a number represent ten times more continued on next page Quantities and Counting to 500 27

moving right to left in the place value positions. The following place value principles (Small, 2009, p. 139) will be reinforced in the unit: 1. Y ou group in tens for convenience so that you need only 10 digits (0–9) to represent all numbers. 2. P atterns are inherent in our number system because each place value is 10 times the value of the place to the right. 3. A number has many different ‘forms.’ For example, 123 is 1 hundred, 2 tens, 3 ones and also 12 tens and 3 ones. 4. A place value system requires a symbol for a placeholder. For example, the ‘0’ in 304 is a placeholder. It pushes the digit ‘3’ over to show that it represents 300 instead of 30. 5. N umbers can be compared when written in standard or symbolic form. About the Lessons It will be useful to have a large hundreds chart (1–100), number line, and place value chart/mat posted in your classroom. Introduce each tool the first time it is used and invite students to refer to the classroom versions as needed throughout this and future units. These tools are also available as digital slides and blackline masters and can be used for any lesson. Helpful and Frequently Used Tools BLM Digital Slide Hundreds charts to 500 BLM 3 Slides 1–5 Hundreds charts to 1000 BLM 44 — Place value mats BLM 2 — Number lines (labelled and open) BLMs 4, 8, 29 Slide 6 Base ten blocks (templates) BLM 12 — Base ten blocks are perhaps the most commonly used concrete tool in this and future units. You will want to teach students the quick and easy way to draw the blocks—as a square (for the hundreds flat), a stick (for the tens rod), and a dot (for the ones unit/unit cube). Place Value Pre-assessment Interview Materials: Consider using the following pre-assessment to determine your students’ current understanding of place value (adapted from Van de Walle & Lovin, connecting cubes, 2006a, p. 123). Base your choice of lessons in this unit on the specific needs of base ten blocks your students. Conduct this assessment as a one-to-one interview. 1. A sk the student to count out 47 connecting cubes and note how this is done. (Does the student count by 1s into a pile or group the cubes in some way?) 28 Number and Financial Literacy

2. A sk the student to write the number 47. (Does the student write 47 or a reversal, 74?) 3. N ext, ask the student to write the number that is 10 more/less than the number he/she just wrote. (Does the student count, keep track of the count on fingers, or just know the answer right away based on using the tens pattern?) 4. S how the student 8 tens rods and 6 unit cubes and have them count the blocks. Observe whether they can count by 10s and then switch to counting by 1s. a. H ave the student make a quantity that is 10 more and ask how many there are. (Does the student need to count or do they just know?) b. A dd 6 more unit cubes and have the student count again. (Can the student count over the hundred? Do they exchange 10 unit cubes for a tens rod?) Lesson Topic Page 1 Read Aloud: 365 Penguins: First Reading 30 35 2 365 Penguins: Second Reading 40 44 3 Estimating Large Quantities (to 500) 48 4 Rounding Two-Digit Numbers to the Nearest Ten or Hundred 52 5 Extending Patterns in Two-Digit Numbers to Three-Digit Numbers 57 (to 500) 62 68 6 Skip Counting by 50s and 100s 69 73 7 Skip Counting Using a Calculator 76 78 8 Using Base Ten Blocks to Compose Numbers 80 84 9 and 10 Composing Numbers to 500 in Various Ways 87 90 9 Representing and Renaming Equivalent Quantities 95 10 Composing Quantities to 500 through Problem Solving 11 and 12 Using a Number Line to Represent and Compare Quantities 11 Ordering Numbers on a Number Line 12 Comparing and Ordering Numbers on a Number Line 13 Problem Solving with Quantities to 500 14 Representing Quantities Using a Concept Circle 15 365 Penguins: Revisiting the Math 16 Reinforcement Activities Quantities and Counting to 500 29

1Lesson 3 65 Penguins: First Reading Language Introduction to the Read Aloud Curriculum Expectations This humorous and engaging story contains some interesting mathematical Social Studies problems that are embedded within the context. During the first reading of 365 Curriculum Penguins, students apply literacy strategies such as predicting, making Expectations connections, inferring, and evaluating to understand and enjoy an adventure about how a family deals with receiving penguins in the mail from a mystery sender. Students also discuss land use and its impact on climate change and habitat destruction, which aligns with specific grade three Social Studies curriculum expectations in the People and Environments strand. During the second reading of the text, students act as mathematicians and apply the mathematical processes to solve problems that arise from having to deal with an ever-increasing number of penguins in the house. Students compose and decompose quantities in various ways and are introduced to some of the operations that are used with larger quantities. Oral Communication • 1 .3 identify a variety of listening comprehension strategies and use them appropriately before, during, and after listening in order to understand and clarify the meaning of oral texts • 1 .5 distinguish between stated and implied ideas in oral texts • 1 .6 extend understanding of oral texts by connecting the ideas in them to their own knowledge and experience; to other familiar texts, including print and visual texts; and to the world around them • 1 .7 identify and explain the importance of significant ideas and information in oral texts People and Environments • B 2.1 formulate questions to guide investigations into some of the short- and/or long-term effects on the environment of different types of land and/ or resource use in two or more municipal regions of Ontario • B 2.5 evaluate evidence and draw conclusions about some of the short- and long-term effects on the environment of different types of land use in municipal regions of Ontario and about key measures to reduce the negative impact of that use Visual Literacy • S peech bubbles • Mathematical clues written on letters, boxes, etc. • Answers written upside down 30 Number and Financial Literacy

Assessment Opportunities Observations: Note each student’s ability to: – Use their prior knowledge to make and support their predictions (I think ... because ...) – Demonstrate literal understanding based on text and illustrations – Make inferences by connecting to their prior knowledge and using clues from the words and pictures – Make inferences about the purpose of text features and how they help the reader understand the text Materials: Read Aloud: 365 Penguins Written by Jean-Luc Summary: 365 Penguins is a story about a family that receives a penguin a day Fromental for 365 days. As the penguin population increases in the house, the narrator Illustrated by Joëlle (one of the children) describes the many perplexing problems that arise as the Jolivet family tries to feed, house, and organize their new guests. At the end of a long Text Type: Fiction: year, the mystery sender arrives at the house to explain the reason for sending Narrative–Humorous the family so many penguins. Story NOTE: Select and use the prompts that best suit the needs and interests of your Time: 20–30 students. Unless otherwise noted, the During Reading prompts are meant to be minutes used after you have read the pages(s) in question aloud. Differentiation • For English Language Learners (ELLs), preview vocabulary that will recur throughout the story. Show photos of penguins in their natural habitats. Possible vocabulary or concepts to preview: penguins, 365 days in a year, organize, anonymous, sender, housing. Before Reading Predicting/making Activating and Building On Prior Knowledge connections • Show students the front cover and read the title and names of the author and Predicting/analysing illustrator. Ask whether they think the story is fiction or non-fiction and why they think so. Ask what is significant about the number 365. Review that there are 365 days in one year. Ask students what they think the story will be about. • Setting a Purpose: Say, “We have now made our predictions. Let’s read the story to find out what the problem is and how it is solved.” During Reading Inferring/predicting “On New Year's Day...” • A sk what is special about New Year’s Day and why people celebrate it. Have students predict when the story might end. Quantities and Counting to 500 31

• Ask where this story might take place. • Have students predict what the man might be delivering. Inferring/making “I opened the box…” connections • Ask who they think the narrator is and why they think so. Ask who they think may have sent the penguin. • H ave students explain how each of the family members may be feeling. Have them offer evidence from the text and from the illustrations. Ask how they would be feeling if they received such a mysterious package. Text features/inferring “The next morning…” • A sk students why the author chose to write “nothing” in larger text. Look back at previous pages to see where else there is larger text and discuss why the author made that choice. Draw attention to the words “Ding dong!” and explain that this is known as an exclamation bubble. Ask why the author may have used this text feature. • A sk what they think “anonymous” means. Inferring/making “And on it went…” connections • Ask why the family decided to name the penguins after the first week. Ask students how they choose names for pets or stuffed animals. Inferring/making “…we’re going to keep them!” connections • Ask why they think the family has decided to keep the penguins. Ask what other alternative the family might have. • Ask how they think their family members would feel if they had 31 penguins in the house. Ask what might be challenging about keeping penguins at home. Text features/inferring “February has only…” • Ask how the author uses the size of the text to show that the problem is becoming bigger and bigger. Inferring “Give them away…” and “Plus one” • Ask why the mother may be changing her mind about keeping the penguins. Ask whether they think organizing the penguins will help solve the problem. Inferring/predicting “On April the tenth exactly…” • Ask what they think the mother means when she says, “This weird sense of humor reminds me of someone.” • H ave students predict what problems the family might have now that there are 100 penguins in the house. Record some of their ideas on chart paper. Inferring/activating prior “1) Feeding the penguins” knowledge • After reading the opening sentence, ask students what they think penguins eat. Ask where the family might be able to get so many fish. 32 Number and Financial Literacy

• R ead the rest of the page and then explain that they will discuss the math during the second reading. Ask whether they think the family can afford to pay for all of the food the penguins need, and how they might find the money to do so. Inferring/predicting/ “2) Taking care of the penguins” making connections • After reading the opening sentence, ask students what the family would need to do to take care of the penguins. Ask what they have to do to take care of any pets they may have. • R ead the rest of the page. Ask what “It’s time to ditch them” means. Ask why the family may be changing their mind about keeping the penguins. Inferring “3) Housing the penguins” and “The days went by…” • Ask what they think the expression “desperate times call for desperate measures” means. Ask whether they think storing the penguins in boxes is practical. Inferring/predicting/ “When the summer arrived…” and “c) Let’s not talk…” activating prior knowledge • After reading the first sentence, ask what the word “complications” means. Have students predict what the problems might be now that summer has arrived. • R ead the rest of the pages. Ask how the family could solve these problems. Inferring “On the fourth of August…” and “Once you’ve reached…” • Ask what they think “the point of no return” means. Ask why the narrator says “You become penguin.” Have students study the illustration to show how the family members have become like the penguins. Inferring/predicting “Before we could say…” • Ask why the family is celebrating New Year’s Eve on the front yard even though it is cold outside. Ask who the 365 guests in dinner jackets are. • A sk who they think is at the door now that all of the penguins have arrived. Inferring/making “364… 365…” and “Then Uncle Victor explained…” connections/analysing/ activating prior knowledge • Have students explain in their own words why Uncle Victor sent all of the penguins. • A sk what the phrase “endangered species” means. Ask what other animals are endangered and the possible reasons for it. • Show students a map. Ask where the South Pole and the North Pole are. Ask why Uncle Victor thinks the North Pole might be a good environment for the penguins. • A sk why they think the family may decide to take care of Chilly. Quantities and Counting to 500 33

Inferring/predicting “And life started…” • Have students predict who—or what!—might be at the door and why they think so. Inferring/predicting/ “I am number 1” analysing • Have students predict what might happen now. Ask who they think sent the polar bear. Ask why the polar bears may be more of a problem than the penguins. After Reading Inferring/evaluating • Ask students what the author’s purpose might have been for writing this book. Evaluating • Ask, “Do you think that Uncle Victor’s solution of sending the family all the penguins is a good one? Is it a solution for climate change?” • S ocial Studies Focus: Discuss how climate change can endanger the habitats of penguins in Antarctica and polar bears in the Arctic and in northern Canada. Your discussions can tie in with your social studies investigations into land use in Ontario municipalities. Discuss the short- and long-term effects of land use on Ontario’s animal populations. Have students propose or consider specific solutions, such as establishing parks for endangered species or building structures like bridges or fences that prevent animals from being killed on busy highways. Discuss what can be done to reduce air and water pollution. 34 Number and Financial Literacy

2Lesson 365 Penguins: Second Reading Math Number Curriculum Expectations • B1.1 read, represent, compose, and decompose whole numbers up to and PMraotcheesmseast:ical including 1000, using a variety of tools and strategies, and describe various Problem solving, ways they are used in everyday life creoamsmonuinngicaatnindgp, roving, trcoeoopnlrsneesacentndintigsnt,grra,etsfeelgeleicectsitningg, • B 1.2 compare and order whole numbers up to and including 1000, in Teacher various contexts Look-Fors • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of Previous Experience with Concepts: tools and strategies Students had experiences representing and ordering • B1.5 use place value when describing and representing multi-digit numbers numbers to 200 in grade two. in a variety of ways, including with base ten materials Possible Learning Goals • Reads, represents, and compares numbers to 500 and has an understanding of their relative magnitudes • Composes and decomposes numbers in various ways including using place value • Identifies quantities in numerical form and explains/shows how each digit’s place value position relates to the whole number • Identifies and explains some patterns in our number system • Uses base ten blocks to compose and decompose numbers from the text • Explains how larger numbers are made up of smaller units About the As students extend their understanding of quantity to larger numbers, it is important to review and reinforce what they learned in grade two about numbers to 200 so they can extend the patterns inherent in our number system. For example, students can represent two- and three- digit numbers by decomposing them in various ways using concrete materials. Recognizing that 163 can be represented as 1 hundred, 6 tens, and 3 ones, or as 16 tens and 3 ones, helps students flexibly think about how to decompose numbers greater than 200. In addition, reinforcing the grade two concept of skip counting to 200 using a variety of groupings helps students extend these patterns to count to 500 and beyond. The more students understand the patterns of our number system, the easier it will be for them to learn about larger quantities. Quantities and Counting to 500 35

Math Vocabulary: About the Lesson pmbuclaanoasciuttee,nchttvuieunabnngleudb)eor,le,nordpe,csglaka,rdscotdeeu(finnplvasgiant,l,gouor,nenoeds, , Students revisit the text to explore the math embedded in the context. You can use the lesson as an assessment of where your students are and which students may need more review of two-digit numbers before beginning with larger quantities. Materials: Assessment Opportunities Observations: • D o students know how to read two- and three-digit numbers using proper mathematical vocabulary? • C an students explain/show what each digit of a number represents? • C an students represent numbers using base ten blocks? • C an students decompose numbers using place value? 365 Penguins, base NOTE: You can read or paraphrase some of the pages as you revisit the book ten blocks, BLM 2: and then focus on the pages for which prompts are provided below to explore Place Value Mat, BLM 3: the math. As students analyse the math, they can use base ten blocks and Hundreds Charts to BLM 2: Place Value Mat to represent various quantities. Students can also use 500, connecting cubes BLM 3: Hundreds Charts to 500 to show counting sequences. Select from or centimetre cubes, among the many prompts those that best suit the needs and interests of your calculator students. Time: 30–40 minutes Before Reading • Ask students, “What do you know about the number 365?” Record student responses in a brainstorming web. Ask when (in what situations) it might be considered a large number and when it might be considered a small number, depending on the context. For example, 365 pairs of shoes in your house is a lot, but 365 pairs of shoes in a shoe store or factory is not a lot. • Review the names and values of the various base ten blocks and how they can be used to represent whole numbers on a place value mat. Allow students time to use the blocks to represent some numbers (e.g., 1, 14, 25, 76, 100, 137, 182). • Tell students that they are going to revisit parts of the story, but this time as detectives uncovering the mathematical ideas embedded in the text and illustrations. 36 Number and Financial Literacy

During Reading Reasoning and proving “The next morning…” • W hen the third penguin arrives, the note says, “There is always a 3 after a 2.” What does this mean? Can we also say “there is always a 13 after a 12, or a 93 after a 92?” Why? Let’s look at these patterns on the hundreds chart. What would be another example of this pattern using numbers between 100 and 200? Connecting “…we’re going to keep them” Representing/reasoning • How do we know there are exactly 31 penguins by the last day in January? and proving • Using the base ten blocks, how could you represent the penguins at the end of January in more than one way? How would you count each way? How are the representations the same and how are they different? Problem solving/ “February has only…” representing • If there are 31 penguins at the end of January and 28 more delivered in Representing/ communicating February, how many more penguins arrived in January than in February? How can you show this on a number line? • How many penguins are there altogether? How can you use the base ten blocks to show this? What operation are we doing? • How can you calculate the total number of penguins in your mind? (e.g., take away the 1 in 31 and add it to 28 so you are adding 30 + 29) Why does this strategy work? What does this look like using the base ten blocks? • Now there are 60 penguins. How can you represent 60 in more than one way using the base ten blocks? How can you count each way? Representing/ “Give them away…” communicating • T he family organizes the penguins into groups of 15. What is the pattern they are using to build each group? What would the next row look like if you put another row on the bottom? How many would be in the group now? Use connecting cubes or centimetre cubes to build and extend this pattern. • If the 60 penguins were organized into groups of 2 or 5, how would you count them? What does each way look like on the hundreds chart? Which way of counting seems easier? Why? Representing “On April the tenth exactly…” Reasoning and proving • Using the base ten blocks, represent the penguins that are at the home on April 10 in different ways. How can you count each way? How can you prove that 10 rods equal 1 flat? • Why do the authors say that the family’s problems really began after the three- digit numbers? Quantities and Counting to 500 37

Problem solving “1) Feeding the penguins” • The number 2.5 represents two and one half, so each penguin eats two and a half pounds of fish per day. How many pounds of fish would one pair of penguins eat? How many pounds of fish would 10 pairs of penguins eat? How can counting by 5s help to solve this problem? What would this counting sequence look like on a hundreds chart? Estimate how many pounds of fish 20 pairs of penguins would eat. Show your solution using the hundreds chart. Representing “3) Housing the penguins” Problem solving • The family is organizing the penguins into dozens. What is a dozen? How are the compartments they are making like an egg carton? How many sets of compartments do they need to make? Is it more than 10? How do you know? • How many more penguins will they be able to accommodate with the 144 compartments than they are using right now? On what day will they fill the compartments? How can you find this out? Reasoning and proving “The days went by…” • On May 24, there are 144 penguins. What is the value of the digits 1, 4, and 4 in the number 144? Prove this by representing the number using base ten blocks. How else can you represent this number? (e.g., 14 tens and 4 ones) Reasoning and proving “On the fourth of August…” and “Once you’ve reached…” Representing/reasoning • There are now 216 penguins. The text says, “Once you’ve reached the point of and proving no return, one penguin more or one penguin less each day doesn’t make much difference anymore.” If there are 216 penguins, how many penguins would there be if there were one more? One less? What if there were two more? Two less? Ten more or ten less? One hundred more or one hundred less? • What do the authors mean by organizing the penguins like a cube? What do we know about cubes? Let’s build a cube like the one in the book together using centimetre cubes, starting with the base. Look at the illustration. How long is the base of the cube? How wide will the cube be? How do we finish creating the base? How many cubes have we used now? (36) How tall will the cube be? How do we finish the cube? How many layers are there now and how many are in each layer? (6 layers with 36 cubes each) Let’s use a calculator to see if our total adds to 216. “364… 365…” Reasoning and proving • By the end of the year, there are 365 penguins. Uncle Victor counts, “364... 365.” What method is he using to count the penguins? (counting by 1s) Is it an efficient way to count a large number of penguins? What would be a faster way? How would you organize the penguins to count in your way? Representing/reasoning • How can you represent 365 using base ten blocks? and proving • W hy can’t Uncle Victor divide all of the penguins into couples? Could he divide them into couples if there were 243 penguins? Why? How many more penguins would he need to do this? What rule can we make about making couples with different numbers of penguins? What do we call these different types of numbers? (even and odd) 38 Number and Financial Literacy

After Reading Reflecting • As a class, create a list of questions that students have about the math in the text. Pose some prompts to help them get started. For example, ask how the number of penguins would change if they were only delivered on weekdays. Revisit these questions at the end of the Quantities and Counting to 500 unit. • Discuss any/all of the following questions: – W hich strategies help us when we are counting larger quantities? – H ow do you think the counting patterns to 100 continue to 500? – H ow can base ten blocks help us represent two- and three-digit numbers? – H ow many ones (unit cubes) are needed to make one rod? How many rods are needed to make one flat? How many flats do you think you will need to make 1000? What pattern do you notice? • Building Social-Emotional Learning Skills: Critical and Creative Thinking: Conclude the discussion by asking students what they wonder about in terms of counting to 500 and beyond. Ask how far they think they can count. Ask how far people need to be able to count and why they think so. Ask what tools they could use to help them count larger numbers. Ask how scientists may count the stars in the sky and whether they end up with an estimation or a precise amount. Make a list of some of the students’ queries and revisit them to initiate inquiries. In this way, students develop a curiosity and wonder about math that motivates them to learn more. Quantities and Counting to 500 39

3Lesson E stimating Large Quantities (to 500) Math Number Curriculum Expectations • B1.1 read, represent, compose, and decompose whole numbers up to and PMraotcheesmseast:ical including 1000, using a variety of tools and strategies, and describe various caPonrmodbmpleruomnviincsoaglt,vinrineggp, ,rereseanstoinnign,g ways they are used in everyday life sceolnenceticntgintgo,orlesflaencdting, strategies • B 1.2 compare and order whole numbers up to and including 1000, in Teacher various contexts Look-Fors • B1.3 round whole numbers to the nearest ten or hundred, in various Previous Experience with Concepts: contexts Students estimated quantities up to 100 in • B 1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools grade two. 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 • Reasonably estimates large quantities of items in a picture using various strategies, including visualizing smaller, familiar groups within larger sets • Explains estimating strategies • U nderstands that estimation is a reasonable prediction of about how much is in a pictured quantity • Identifies smaller groups within a larger collection of objects • Uses a benchmark quantity to visualize and estimate • Accurately skip counts according to the groups they have created • Estimates the quantity in a larger group by visualizing how many smaller, equal groups it contains • Explains estimation strategies used • Explains why their estimation is reasonable About the According to Marian Small, “one particularly important application of visualization is the strategy of estimating by comparing one amount with another” (Small, 2005, p. 146). When students learn about numbers to 100, they develop mental images of how much the quantities 40 Number and Financial Literacy

Math Vocabulary: represent by creating sets using concrete materials. Tools, such as ten epsretidmicatt,eh, urenadsreodnsa,ble, frames, help students organize the materials so they do not need to be tens, ones individually counted, and students can create mental images of amounts like 5 and 10. Students can use these images to estimate how many of the smaller sets are in a larger set of objects. Students can apply this strategy to estimate larger quantities by establishing mental images of groups such as 25, 50, or 100. For example, if students have a mental image of what 100 beans look like, they can visualize 1000 as 10 piles of 100 beans. About the Lesson In this lesson, students analyse pictures of large sets and use several strategies to estimate how many items make up the larger set. Throughout the lesson, students visualize and explain their reasoning. Materials: Minds On (15 minutes) jar of 200–300 small • Ask students what ‘estimation’ is. Reinforce the idea that it is a reasonable items (e.g., beans, popcorn kernels, beads), prediction of a solution to a problem. Record this term on the Math Word “Let’s Estimate!” (pages Wall. 4–5 in the Number and Financial Literacy big • Show students a jar filled with 200 to 300 items in it. Have them estimate book and little books); chart paper; about 10 how many objects might be in the jar and why they think so. Take out 10 of collections of sets the objects and count them together. Ask if they would like to refine their containing 50 to 500 estimates. Count out a total of 50 objects and ask whether they would like to items, placed in clear further refine their estimates. bags • Show the picture of jelly beans spilling out of a container on page 4 of the big Time: 6 0 minutes book. Ask students what they see and how the objects are organized. Students can think/pair/share about their estimation for the number of jelly beans. They can use the little book versions of the big book to get a closer look at the picture. Record some of their estimations, and in each case have the students justify their responses. • Ask whether they can see a smaller group of items within the larger group. Discuss how they can identify a smaller group and then visualize several of those groups within the entire collection. Ask how they count and keep track of the items. Working On It (20 minutes) • Students work in groups of two or three, with each group sharing a little book version of the big book. Together, they make estimates of the rest of the objects shown on the spread. They can share their estimates on chart paper and explain their estimation strategies. • G ive each pair of students a clear bag that has a collection of 50 to 500 items in it (e.g., beans, toothpicks, grains of rice). Have them estimate how many items Quantities and Counting to 500 41

they think there are before removing the items from the bag. After their estimates, they can work with the concrete materials to further refine their predictions. For example, they may make a group of 10 or 100 to estimate the quantity in the entire set. Encourage them not to count each item, because an estimate is all that is needed. When they are finished, they write their estimate on a small piece of paper. • A fter making their estimates, students can then count the number of items to find out how many items there actually are. They can record this on the small piece of paper too. Differentiation • Give groups different amounts of materials, depending on their individual needs. • For students who need more of a challenge, have them take a handful of items that will approximate part of the set. For example, if there are 250 items in the set and they have made a group of 25 to help them estimate, challenge them to pick up a handful that will have 75 items in it. Assessment Opportunities Observations: Observe how students are identifying smaller groups within the larger group, either in the pictures or with the concrete materials. • What size of group are they making? Do they adjust the size of the subgroup according to how many items may be in the set and the size of the individual items? • C an they accurately count the subgroups by skip counting by the size of their groups? • Do they have a way of tracking the number of groups and figuring out the total? Conversations: Pose some of the following prompts to further uncover student thinking. • W hy did you decide to count out that many items in your smaller set? About how much of the set does it take up? Is it less than one half? One fourth of the set? Can you visualize more than 10 groups within the larger set? More than 20 groups? • Could you make a larger group that might make it easier to estimate? How much would you have if you doubled the size of the group you just made? Consolidation (25 minutes) • Have students meet with another pair to compare their estimates for the pictures in the big book. Students can take turns explaining their estimation techniques. Pairs exchange their sets in the clear plastic bag and have them estimate how many items there are. The students can then compare their estimates. 42 Number and Financial Literacy

• Meet as a class and discuss their estimates for each picture. Have them justify their reasoning. For two or three of the pictures in the big book, record and compare all students’ estimations. Discuss the range of estimates and why they are reasonable or not. • Discuss what size of subgroups they found most helpful to create in order to estimate. Ask how they adjusted the size according to the size and number of objects. Highlight how they skip counted and kept track of the number of subgroups within the entire set. • Reinforce the idea that an estimate is a reasonable prediction and does not have to be a precise amount. Discuss why you would not need to know the precise amounts of the items pictured in the big book (e.g., you don’t need to know the exact number of grains of rice when you are cooking dinner). Ask how we may measure these items (e.g., using a cup or a spoon). • Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: Accept students’ estimations without judgement, even if they seem unreasonable. Ask them to justify their answers. In doing so, you will make students feel that their ideas are valued. Encourage students to listen to the estimation strategies of others. Provide prompts, such as “Do you think the quantity is greater than 100? Why? Do you think it is less than 500? Why?” This will help students develop more reasonable estimation skills based on analysing the situation rather than just making a guess. Further Practice • S et up an Estimation Centre with more sets of 50 to 500 items. Prepare a simple T-chart for each set so students can record their estimates when they visit the centres. After a few days, compare and discuss the estimates recorded on the charts. • Frequently use estimation bags or jars in the classroom. (See Estimation Jars, Embedding Number Concepts Throughout the School Day, page 13.) Quantities and Counting to 500 43

4Lesson Rounding Two-Digit Numbers to the Nearest Ten or Hundred 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 In grade two, students have worked with • B1.2 compare and order whole numbers up to and including 1000, in numbers to 200. various contexts Teacher Look-Fors • B 1.3 round whole numbers to the nearest ten or hundred, in various contexts Possible Learning Goal • Rhuonudnrdesdnwuimthbearnsdpwreistehnotuetdainnuremabl-elrifelincoentexts to the nearest ten and • Identifies multiples of 10 (to 100) and multiples of 100 (to 500) • Identifies the two tens or hundreds that a number is in between • Places given numbers on a number line with reasonable accuracy • Explains how to round a number to the nearest ten or hundred • Uses a number line to round numbers to the nearest ten or hundred PMraotcheesmseast:ical About the Problem solving, representing, Estimating is a practical skill that students will use throughout their lives, communicating, since an estimation is often all that is necessary when solving many real-life preroflveicntgin, gc,ornenaescotniningg and problems. For example, you may assume that it will take you half an hour to travel to the store rather than using the exact time of 27 minutes. One estimation method is rounding. Van de Walle and Lovin say, “to round a number simply means to substitute a nice number that is close so that some computation can be done more easily. The close number … should be whatever makes the computation or estimation easier (Van de Walle & Lovin, 2006a, p. 145). For example, when buying groceries, we might round up the costs of items to the nearest dollar to ensure we have enough money. In other cases, we may round down the amount of money we have so we do not spend beyond our means. 44 Number and Financial Literacy

Math Vocabulary: In grade three, students round whole numbers to the nearest ten or the reosutnimdiantgin, gn,ecaornevset ntetino,n nearest hundred, in a variety of contexts. It is generally accepted that we round down to the nearest ten if the number being rounded ends in 4 or less and round up to the nearest ten if the number ends in 5, 6, 7, 8, or 9. Similarly, we round down to the nearest hundred if the number of tens and ones combined is 49 or less and round up to the nearest hundred if the number of tens and ones combined is 50 or greater. Using a number line to round quantities helps students make sense of these conventions since they can visually see which ten or hundred is closest to the number being considered. It is important that students understand that this is a convention generally accepted by many people, but there are circumstances when another way of rounding may be used. About the Lesson In this lesson, students round amounts that are described in real-life contexts. They also use the number line to help them determine which ten is closer to the number being rounded. Materials: Minds On (15 minutes) BLM 4: Number Lines to 100 • Review what ‘estimate’ means (e.g., making a reasonable prediction of how Time: 55 minutes much, based on our prior knowledge; about how many; not exact; close to). Review what numbers students used to estimate when looking at the big book pictures in the previous lesson and why ‘friendly’ numbers are convenient. • Pose some real-life estimation problems that are relevant to their lives, such as: – A bout how many students are in our school? – A bout how many books are on the shelf? About how many more books are on the lower shelf than on the middle shelf? – A bout how many crayons are in the bin? About how many of the crayons are red? • After students estimate, find the answers together to confirm their predictions. This helps students refine their estimation skills. Working On It (Whole Group) (20 minutes) • E xplain to students that they are going to learn a new estimating skill called ‘rounding to the nearest ten.’ • D raw an open number line and label the endpoints 0 and 100. Have students skip count by tens and ask how students could place these numbers on the number line. Together, add these benchmark numbers to the line and reflect on how it is important to space them out evenly. Quantities and Counting to 500 45

• Tell students that these ‘decade’ numbers will be the benchmarks for rounding, which means deciding which ten a given number is closest to. • E xplain that the school needs to order 57 basketballs. Ask about how many basketballs that is. Have them locate the number 57 on the number line. Ask which two decade numbers it is in between and which ten is closer to 57. • A sk about how many basketballs there are if 52 balls need to be ordered. Have students show where 52 is on the number line and which ten it is closest to. Explain that when we are following the rounding rule, we would round down to 50 because 50 is closer to 52 than 60 is. • A sk students how they would round if basketballs came in packages of 10 and they needed 52 balls. Explain that in some real-life situations, you need to change the rounding rules so they suit the circumstances. In this case, students would need to round to 60 in order to get at least 52 balls. • A sk students how they would round if they needed 55 basketballs. Explain that the general rule or convention is that when a number ends in 5, it is rounded up to the nearest ten. • A sk students what they think the general convention is for numbers that end in 0 to 4 and numbers that end in 6 to 9. Record the rule for students to access as they work. Independent Work (10 minutes) • Give students some two-digit numbers in a meaningful context (e.g., 28, 42, 76, 18, 31, 64). Have them round the numbers to the nearest ten. Provide students with copies BLM 4: Number Lines to 100 as needed. They can locate the numbers on a number line and then draw an arrow from each number to the nearest ten. Differentiation • Some students may benefit from using a number line with all of the numbers marked on it so they don’t have to locate the numbers on their own. • For students who need more of a challenge, have them create their own number lines. Assessment Opportunities Observations: Observe students as they work on the independent task to see if they can locate the numbers on the number line and then round to the nearest ten. If there are any numbers that consistently seem to give students problems, discuss them during the Consolidation. Conversations: If students are having difficulty rounding, pose some prompts to see if the problem is with locating the numbers or actually rounding to the nearest ten: – Where do you think 73 would go on the number line? Which two tens is it in between? What number would be halfway between 70 and 80? Is 73 more or less than 75? Do you want to adjust where the number would be? 46 Number and Financial Literacy

Materials: Consolidation (10 minutes) BLM 3: Hundreds Charts • Students briefly meet with another pair to compare their answers. to 500 and/or Digital • Meet as a class. Review the rounding conventions and co-create an anchor Slides 1–5 chart that summarizes them, including examples for each. Teaching Tip • Review when rounding by the conventions may not be appropriate or practical. Integrate the math talk moves (see It is important for students to understand that these are generally accepted page 8) throughout ways of estimating, but they may need to be altered in real-life situations so Math Talks to they pertain to the context of the problem. maximize student participation and Math Talk: active listening. Math Focus: Rounding to the nearest hundred Let’s Talk Select the prompts that best meet the needs of your students. • What were we doing in the lesson when we were rounding to the nearest ten? What were the benchmarks that we used to round? (tens) • Let’s imagine that we are working with larger numbers. What other benchmarks might we want to use to round? • We are going to round to the nearest hundred. What would be the numbers that we round to? Let’s count by 100s to be sure. • Show hundreds charts that extend up to 500. We are going to use the hundreds chart as a tool for rounding to the nearest hundred. • Locate the number 237 on one of the hundreds charts. What are the two hundreds that it is in between? (200 and 300) Turn and talk to your partner about which hundred you would round it to. Be prepared to explain why. • What did you find? (e.g., We would round down to 200 because it is closer than it is to 300.) Do we agree on that? How is this different than rounding to the nearest ten? How would we round 237 to the nearest ten? (round to 240) In what circumstances might you want to round to hundreds rather than tens? (e.g., if you don’t need such an exact estimation) • Let’s try the number 367. How would you round it to the nearest hundred? How would you round it to the nearest ten? • How would you round the number 450 to the nearest hundred? Why? (e.g., We would round it up to 500 because 450 is right in the middle of 400 and 500. When we rounded to the nearest ten, we rounded up when the number was in the middle. So, we think we do the same thing.) • That is the convention that we normally use. Sometimes, there are circumstances when we may not follow the conventions. If we are rounding money amounts, we may always round up so we make sure that we don’t underestimate how much money we have to spend. Partner Investigation • Give students some other numbers to round to the nearest ten and the nearest hundred. Quantities and Counting to 500 47

5Lesson Extending Patterns in Two-Digit Numbers to Three-Digit Numbers (to 500) 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 In grade two, students worked with numbers • B 1.2 compare and order whole numbers up to and including 1000, in to 100 on the hundreds chart and skip counted by various contexts 20s, 25s, and 50s to 200. • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools Teacher Look-Fors 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 • Recognizes various patterns in the hundreds charts and explains how they work • A pplies understanding of patterns to predict the missing numbers in a pattern • Skip counts by various amounts from different starting points • Locates a target number on a hundreds chart and predicts other numbers surrounding it • Explains patterns in the hundreds chart and how they can be extended • Accurately skip counts forward and backwards from a given number, with and without a hundreds chart • Generalizes patterns in the 0–100 chart to larger numbers PMraotcheesmseast:ical About the Problem solving, rrreeepfalreseocstneininngtgi,nacgon,ndnpercotviningg, , Students can gain greater understanding of our number system by tcooomlsmaunndicsatrtaintge,gsieeslecting recognizing patterns in numbers 0 to 100 and then extending those patterns to larger numbers. This allows students to compare and order larger numbers with greater accuracy, and to expand their skip counting abilities. Tools such as hundreds charts are 48 Number and Financial Literacy

Math Vocabulary: particularly effective for helping students create visual images of how hmuonrder,etdesnclheassrt,, ten numbers are organized and how they compare in magnitude. At first, patterns students can use hundreds charts for larger numbers, such as 101–200 or 201–300, but eventually they can internalize the patterns and apply them to understanding larger numbers. Skip counting is also useful for carrying out operations when students can flexibly count on from various starting points and identify the patterns the numbers make. For example, when adding 127 + 52, students can count on from 127 by 10 five times (137, 147, 157, 167, 177) to add the 50 in 52, and then by 1 two times to add the 2 ones. About the Lesson During the Minds On, students review what they know about patterns in the hundreds chart and extend this knowledge to larger numbers. In the Working On It section, students solve for missing numbers in jigsaw puzzle pieces from various hundreds charts. Students can also create their own jigsaw pieces, to further reinforce their understanding of patterns in larger numbers. Materials: Minds On (15–20 minutes) hundreds charts to • Display a hundreds chart for numbers 1–100. 300 (Digital Slides 1–3), • Ask students what happens to the number of penguins throughout the story chart paper, BLM 5: Jigsaw Hundreds, BLM 3: 365 Penguins. Ask how many penguins there were at the end of January (31). Hundreds Charts to 500, Have them locate 31 on the hundreds chart. Ask how they can use the chart BLM 6: Big Jigsaw, BLM to find out how many penguins there were the day before, the day after, 10 7: Blank Jigsaw days before, and 10 days after. Discuss how the numbers increase and Hundreds decrease by 1s and 10s and the patterns they make on the hundreds chart. Time: 6 0–65 minutes • Ask students how they could find the number of penguins by the end of February. Together, count from 31, first by 2 tens (to 51) and then by 8 ones (to 59), highlighting the patterns in the chart. • Ask how many penguins there would be 100 days after January 31 and then 100 days after February 28. Ask how the numbers 31 and 59 change and how they stay the same (e.g., they increase by 100 but the number of tens and ones remains the same). • Show students the next hundreds chart, for numbers 101–200. Have them locate 131 and 159. Ask what they notice about the position of the numbers 31 and 131 and 59 and 159 in the two charts. Discuss how the patterns in the charts are the same. • Remind students that at one point in the story there are 144 penguins. Have them predict how many more penguins there will be the day before, the day after, 10 days before, and then 10 days after. Confirm their predictions by highlighting the patterns in the chart. • Pose some other problems such as how many penguins there are 9 or 11 days before or after a given number. Highlight how they can still count vertically Quantities and Counting to 500 49

up or down for 10 and then move one space backward or forward to create the 9 or 11 moves on the chart. • Ask how many penguins there would be 100 days after there are 144 penguins. Have them predict what the next hundreds chart would look like (e.g., numbers 201–300, with the same patterns). • Reproduce the following image on a piece of chart paper. Tell students that this is a “puzzle piece” from a hundreds chart. Can they figure out which hundreds chart it comes from—what the smallest and largest numbers on the chart are? Ask them to explain how they know, and to fill in the missing numbers on the puzzle piece. 173 • Ask students what other numbers surround 173 and how they know. Extend the puzzle piece (by adding squares) to include some of their suggestions. Confirm their predictions by showing the 101–200 hundreds chart. Working On It (20 minutes) • Have students work in pairs. They will work on three separate tasks. 1. P airs fill in the missing numbers in the puzzle pieces on BLM 5: Jigsaw Hundreds. Give them a copy of the hundreds chart for 1–100 (see BLM 3) to help them find patterns and then extend those patterns to the larger numbers. 2. G ive each pair one puzzle piece from BLM 6: Big Jigsaw to complete. Explain that all of the pieces make one hundreds chart and that they will put it together in the Consolidation. Students can fill in the numbers and keep this piece separate from their other puzzle pieces. 3. P airs select one template from BLM 7: Blank Jigsaw Hundreds and create their own puzzle. They make two copies: one with just one number on it and the other with all the numbers filled in (the answer key). Differentiation • Assigns students numbers that suit their needs. You can also change the numbers in the existing puzzles or create your own puzzles using the templates on BLM 7. • For students who need more support, give them copies of all of the hundreds charts up to 500 (BLM 3). • For students who need more of a challenge, they can work without a copy of the hundreds chart. • For students who need more of a challenge, have them design their own puzzle piece on a blank sheet of paper, without using a template. 50 Number and Financial Literacy

Assessment Opportunities Observations: Observe how students fill in the chart. • Do they need to refer to the hundreds chart? • Can they extend the patterns in the 1–100 chart to create patterns for the larger numbers? • Do they just match and copy the numbers from the hundreds chart or do they understand the patterns? Conversations: If students seem to be copying the numbers without paying attention to the patterns, pose some of the following prompts: – W hy did you put the number 234 above 244? How are the two numbers the same and how are they different? What does the 3 represent in 234? (30) What does the 4 represent in 244? (40) Let’s look at the hundreds chart. How far apart are these numbers? (10 spaces) What number would be directly above 234? Why? What number would be 9 spaces away from 234? How can you figure that out without counting back 9 spaces? What number would be 10 less? Then what number would be 9 less? Consolidation (25 minutes) • Students meet with another pair to compare the puzzle pieces they completed (Task 1). They also exchange the puzzle piece they created (Task 3) and complete each other’s pieces. They can check their solutions against the answer keys they created. • Meet as a class to complete Task 2. Have one pair of students lay down their puzzle piece from the big jigsaw puzzle. Have other pairs come up, one at a time, and place their piece where they think it belongs. After each piece is added, discuss whether it seems to be in the right place or whether it’s position needs to be adjusted. • Review some of the patterns that apply to all of the charts (e.g., move vertically up or down to find a number that is 10 less or 10 more). Create an anchor hundreds chart to visually show the patterns. Further Practice • Independent Problem Solving in Math Journals: – On a hundreds chart, a mystery number is two rows above 245 and one square to the right. What number could the mystery number be? – Explain how you could use a hundreds chart to show that 256 is smaller than 266. Quantities and Counting to 500 51

6Lesson Skip Counting by 50s and 100s 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 had various contexts experience using open number lines. • 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 Algebra • C 1.4 create and describe patterns to illustrate relationships among whole numbers to 1000 Possible Learning Goal • Skip counts by different amounts from any starting point • Understands that skip counting involves adding or subtracting an equal number with each count • Counts by 50s and 100s from any starting point • Accurately continues the counting pattern over decade numbers (e.g., 20, 30) and the hundred numbers (e.g., 100, 200) • Records the count on an open number line, using equal jumps to represent the increments • Explains the patterns that are in the counting sequences and how they can help to predict the next numbers PMraotcheesmseast:ical About the Problem solving, ssacrteenorlpamdetrceepmtgsrinoueiegvnnsiitntciongaog,til,rnsecgafo,lnenrdcneteaincsgtoi,nnign,g Students can learn to skip count to larger quantities by following the patterns inherent in our number sequence. It is also important that they attach meaning to these larger numbers so they have an idea of how much they represent; students should thus link visual or concrete 52 Number and Financial Literacy

Math Vocabulary: representations of the quantities to their numerical representations. skip counting, number When using base ten blocks, for example, students can see the 100 patterns individual units that make up a hundreds flat as they count by 100s, as well as the equal increase in quantity with each count. Money can also act as a valuable tool, especially as grade three students learn to estimate, represent, and count the value of a collection of coins and bills. In grade two, students skip counted forward by 20s, 25s, and 50s to 200. In grade three, students skip count by 50s, 100s, and 200s, which is an important skill they will apply as they learn about various operations, such as when students use mental strategies to add and subtract multi- digit numbers. For example, if students are adding 153 + 48, they can start with 153 and add 50, which totals 203. Students can then subtract 2 to compensate for the extra 2 that were added when adding on 50 rather than 48. About the Lesson In this lesson, students solve problems using skip counting as one possible strategy. They can use concrete materials and open number lines to represent their skip-counting strategies. In the accompanying Math Talk, students solve problems that involve money, using skip counting as one possible strategy. Materials: Minds On (20 minutes) hundreds charts • yPoouser the following problem. Change the context so it is more meaningful to 1–100 and 101–200 students. (Digital Slides 1–2), base ten blocks, open – Jesse has 253 trading cards. He gets 6 new packages with 50 trading number lines (Digital cards in each. How many trading cards does he have now? Slide 6 and BLM 8) • iDsitshceusssamsoem. Sehoofwthtehier solutions. Highlight how the amount added each time Time: 60 minutes counting sequence on an open number line. • A(es.gk.,ssttuadretnwtsithho2w53thaenydcaodudldorneepfrleastenfotrsekvipercyo2ungtrionugpussoinf g50b,acsoeutnentinbgloocnk,s 303, 353…). Refer to hundreds charts if/as needed. • sAtuskdehnotwdtehmeocnosutrnattewtohuilsdocnhathnegenuifmJebsesre started with 353 cards. Have a line or with base ten blocks. Show the counting sequence on a separate open number line. Have students compare the patterns on the two number lines. • Aweskreh1o0w0 the counting sequence would change in the two scenarios if there trading cards in each package. Show the counting sequences on the same number lines as the ones that represent skip counting by 50. Quantities and Counting to 500 53

Working On It (20 minutes) • H ave students work in pairs to answer the following problems. Tell them that there are many ways to solve the problems, but today they are going to use skip counting as at least one strategy. They can support their counting using concrete materials and then record their findings on open number lines (available on BLM 8). – P roblem 1: Jayden uses 193 chocolate chips to bake muffins. He wants to bake 6 more muffins and each muffin has 50 chocolate chips. How many chocolate chips will he need altogether? – P roblem 2: Anna uses 284 blocks in her tower. She is going to add 4 more layers. Each layer has 50 blocks. How many blocks will she use to build the whole tower? – P roblem 3: Andrea is reading a book with 500 pages and she is on page 137. She has 8 more reading sessions left and wants to know whether she should read 50 or 100 pages per session. What would be the better choice? How many pages will she have finished at the end of each reading session? Differentiation • Csthuadnengetst.hTehcios nhteelxptsstohfetmhecopmropbrleemhesnsdotthheemy aargenmituodree meaningful to your are using. of the numbers they • Csthuadnengetst.he numbers in the word problems to best meet the needs of your • pFroorbslteumdeunstisnwg haosenceoenddmstorraeteogfya, challenge, encourage them to solve the counting. to confirm their findings with the skip Assessment Opportunities Observations: Observe how students solve the problems using skip counting. • Can they begin the count from different starting points? • C an they continue the counting patterns over the decade and hundred numbers? • C an they record their sequences on a number line? Do they use equal- sized jumps to represent the count? Do they adjust the size of the jumps when they are counting by a lesser or greater amount? Conversations: If students are having difficulty beginning at different starting points, pose some of the following prompts: – Y ou are starting at 167. What are you skip counting by? (by 50s) How much are you adding each time? Let’s look at the hundreds chart. Count on 50 spaces from 167. Remember that you count the jumps you are making from number to number and not the numbers. What pattern do you notice? 54 Number and Financial Literacy

– Y ou stopped at 97 and you need to continue to count by 50. How would this count continue past 100? We can use this hundreds chart from 101–200 to help us. What pattern do you notice? Without looking at the hundreds chart, what do you think the next two numbers will be? Let’s confirm it on the hundreds chart. Consolidation (20 minutes) • Dthiesmcuossntohpeesnkinpu-cmobuenrtilningesst.rYaoteugimesayfoarlesoacwhaonfttthoeapsrkosbtluemdesnatsnhdorweperlesseent they could have solved each problem so they realize that skip counting is not the only strategy. • Hpaatvteersntsudcoennttsinluoeokovfoerr the patterns in the various counts. Highlight how the the decade and hundred numbers. • pAosiknststu. Hdeingthslihgohwt tthhaetythweounuldmcboeurnotfbhyu1n0d0rseidfsthcheyanbgeegsa,nbuatt various starting the number of tens and ones remains the same. Ask what is the same and different in the number sequence of counting by 100s and counting by 50s. Materials: Math Talk: coin manipulatives (quarters) Math Focus: Using quarters to skip count in various ways Teaching Tip Let’s Talk Integrate the math Select the prompts that best meet the needs of your students. talk moves (see page 8) throughout • Show a set of 8 quarters that are randomly organized. Look at this set of Math Talks to maximize student quarters. How many do you think there are? If you look closely, you can see participation and that there are 8 quarters. Do you know what the value of a quarter is? How do active listening. you know? Let’s count the quarters up to $1. (25¢, 50¢, 75¢, 100¢) How do you think we can continue counting in the rest of the quarters if we count them in cents? (125¢, 150¢, 175¢, 200¢) Record this pattern on an open number line. What pattern do you see in the numbers? (e.g., the numbers end in 25, 50, 75, or 00) • If we added another 8 quarters how would we count on to add them in, using cents? I have recorded these numbers on our number line. Does the pattern continue? • How many quarters do we have now? (16) How else could we group the quarters so we could count them in a different way? (e.g., count two quarters at a time and count by 50s) • Let’s imagine that we already have 30¢. Let’s count in the quarters, two quarters at a time. (80, 130, 180, ...) Record the count on an open number line. What patterns do you see? continued on next page Quantities and Counting to 500 55

• How else could we group the quarters? (e.g., put four together to make 100¢) Imagine that we have 80¢. Let’s count in the 16 quarters by adding in four at a time. Record the count on an open number line. What patterns do you see? • How would you count in the quarters if we added 8 quarters at a time? Let’s imagine that we have 25¢. Let’s count in the quarters 8 at a time. • Why does it make sense to group quarters when you are counting a large number of them? 56 Number and Financial Literacy

7Lesson S kip 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: In grade two, students various contexts skip counted by 20s, 25s, and 50s to 200. • B 1.4 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 Algebra • C1.4 create and describe patterns to illustrate relationships among whole numbers to 1000 Possible Learning Goals • Explores skip-counting patterns using a calculator • Counts forward and backwards in a variety of ways, using a calculator • Counts by 50s and 100s, both forward and backwards, using the calculator and other tools like the hundreds chart • Clearly explains patterns that emerge from using the calculator or hundreds chart • Predicts how a given counting pattern will extend and gives some examples • Recognizes and represents patterns on an open number line • Uses a calculator to extend number patterns PMraotcheesmseast:ical About the Problem solving, sacreenolpmdercepmtsrinouegvnniitntciongaog,til,rnsecgafo,lnenrdcneteaincsgtoi,nnign,g Students need many experiences skip counting, predicting numbers in a strategies sequence, and exploring patterns that involve large numbers. Using the calculator to explore counting patterns engages students and enables them to work with larger numbers that may be beyond their continued on next page Quantities and Counting to 500 57

Math Vocabulary: counting ability. Students can connect to the idea that patterns repeat sppkarietptdecircontu,sn,ctarienlcpgue,laanttu,omr ber through the use of the repeat function. By investigating various ways of counting on the calculator and recording the results, students can identify emerging patterns and use them to predict other numbers in the counting sequence. About the Lesson Students first review skip counting using tools such as the hundreds chart and the open number line. Next, they predict, extend, and create skip-counting patterns using a calculator. They record their pattern explorations on the open number line. In the accompanying Math Talks, students use the calculator to skip count forward and to skip count back from 500. Materials: Minds On (15 minutes) calculators, open • Review skip counting by 50s on the hundreds charts to 500, starting from 0. number lines (Digital Slide 6 and BLM 8), Have students extend this understanding and count on from larger amounts, hundreds charts to 500 such as from 200 to 500 (200, 250, 300, 350…). (Digital Slides 1–5 and BLM 3), paper • Demonstrate how these patterns can be transferred onto an open number Time: 55 minutes line. • Ask students how they could count by 10s starting from 7. Have them show the pattern that emerges using the hundreds chart. Choose a larger number, such as 418, as a starting point. • Repeat this line of questioning for counting by 20s from various starting points. Working On It (20 minutes) • H ave students work in partners with a calculator and a piece of paper for recording. Project an online calculator to demonstrate as you give instructions (for example, see www.online-calculator.com). Review the keys that are on a calculator if this is new learning. • T ell students that you are going to give them instructions on how to use the calculator to skip count. Say: 1. P ress the On/C button. C means clear. What happens when you press it? (You see 0.) 2. P ress + then press 20. Now press =. What do you see? (20) 3. P ress = again. Now what do you see? (40) 4. W hat is happening? (It’s counting up by 20s.) 5. If we keep pressing the = sign, what numbers do you think will appear? 6. N ow start at a different number, any number less than 200, and press + then 20 at least 5 times. After each press, predict and record the next 3 numbers in the sequence. Take turns being the calculator and the recorder. 58 Number and Financial Literacy

Materials: 7. W ork with your partner to start at different numbers that don’t end in 0 and are less than 200 (e.g., 43, 101). For each number, count on by 20s. In calculators, paper, each case, first predict what the next 5 numbers will be. Then use the open number line calculator to confirm what you predicted. Record your sequences of numbers on a piece of paper. [Demonstrate this process so all students know what to do.] • H ave students repeat this process and count by 50s. Differentiation • There is a great deal of receptive language (listening and following directions) involved in this lesson. Pair students strategically so that at least one partner has a strength in this area. Assessment Opportunities Observations: Are students able to follow your directions? Can they predict the number patterns before using the calculator? Are they able to describe and explain how a pattern repeats? Are they able to use the calculator efficiently or do they require a guided lesson to support the use of this math tool? Consolidation (20 minutes) • Have some students share their patterns. They can offer the first three numbers and then ask the rest of the students to predict the rest of the pattern. As they share the patterns, represent them on an open number line to help them see the connections between the tools. They can draw their own number lines or use BLM 8: Open Number Lines. • Pose some of the following prompts: – H ow did you predict the next numbers in the pattern? – H ow did the count change if you went over a hundred? – W hy do we skip count, especially with larger numbers? – H ow can the calculator help us problem solve? – If you didn’t use a calculator, what other tool could you use to solve a problem that involves skip counting? Math Talk: Math Focus: Using the calculator to skip count Let’s Talk Select the prompts that best meet the needs of your students. • How can we use the calculator to skip count? Which way of skip counting (i.e., skip counting by which number) made the number pattern grow or shrink the quickest? Why? continued on next page Quantities and Counting to 500 59

Teaching Tip • How can you use the calculator to count forward by 100s? Choose any number Integrate the math between 1 and 100 and figure out how to skip count by 100s. Before beginning, talk moves (see predict the first 5 numbers and then confirm your predictions using the calculator. page 8) throughout Math Talks to • What did you find? What part of the number changes in the pattern each maximize student participation and time? active listening. • Imagine you counted forward by 50s from the same number. How would the pattern change? Check it out on your calculator. Why is every second number the same as in the sequence created when counting by 100? Show both patterns on an open number line. • What is one half of 50? How would this change the counting pattern? • Work with your partner. Start at 0 and count up to 500 by 25s. Predict what the next number is before using the calculator. Record the number sequence on a piece of paper. • Let’s record your findings on the open number line. What pattern do you notice in the number sequence? (e.g., All of the numbers end in 25, 50, 75, or 00). • Knowing this pattern, how would you count on by 25s from 175? Let’s check your predictions using the calculator. Materials: Math Talk: calculators, paper, Math Focus: Using the calculator to uncover patterns when skip counting open number line, backwards hundreds charts Let’s Talk Select the prompts that best meet the needs of your students. • Explain how we used the calculator to skip count forward. What buttons made the pattern repeat? • Imagine you have 500 candies and you give 50 candies away to one friend and then another friend, and you continue doing this until you have no candies left. How many friends would get candies? How could we solve this problem using a calculator? • Turn and talk to your partner and experiment with your calculator to show how you could show this pattern on the calculator. • What did you find? So instead of pressing the plus (+) key, we press the take away or minus (−) key and we can skip count backwards. • Pick any number between 400 and 500. Predict what the pattern will be for counting backwards by 50. Then confirm your predictions by counting back on the calculator. Take turns using the calculator while your partner records the counting sequence. Do at least two patterns each. • What did you find? Let’s look at these patterns on the hundreds chart. • Now try counting back by 100s. What is the same about the numbers in each sequence? 60 Number and Financial Literacy

• Did you have any remainders or did you always end up at 0? Why might this be? • Let’s record one example of each of these counts on the open number line. What do you notice about the relationship between counting by 50s and 100s? • What do you predict would happen if we did the same ways of counting backwards but started at 492? Would we end up at 0? Why do you think so? Would any of the counts end up at 0? Why? Quantities and Counting to 500 61

8Lesson Using Base Ten Blocks to Compose Numbers Math Number Curriculum Expectations • B 1.1 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 In grade two, students represented numbers • B 1.2 compare and order whole numbers up to and including 1000, in to 200. They should be familiar with various contexts representing two-digit and some three-digit • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools numbers using base ten materials and have and strategies had prior experiences counting and grouping • B1.5 use place value when describing and representing multi-digit numbers objects into groups of hundred, tens, and ones. in a variety of ways, including with base ten materials Possible Learning Goals • Flexibly composes, decomposes, and represents numbers using groups of hundreds, tens, ones and accurately counts their representations • Reads and compares numbers represented numerically and concretely/ visually Teacher • Decomposes a three-digit number in a variety of ways Look-Fors • Creates the largest number by arranging digits • Identifies the hundreds digit and tens digit as representing groups/units of hundreds or tens (e.g., the 3 in 305 means 300 or 3 hundreds blocks) • Explains why one number is greater than, less than, or equal to another number • Adds on multiples of hundreds or tens using the tens pattern PMraotcheesmseast:ical About the Problem solving, creoamsmonuinngicaatnindgp, roving, In our base ten system, the position that a digit occupies determines its trcoeoopnlrsneesacentndintigsnt,grra,etsfeelgeleicectsitningg, value. This concept can be very abstract when dealing with numerical representations of quantities. For example, students may not recognize that the digit ‘4’ in the number 342 represents 40. It is therefore important that students have several opportunities to create 62 Number and Financial Literacy

quantities with concrete materials so they can form mental images of each place value position and how much a digit in each position represents. Base ten blocks effectively represent whole numbers. Each type of base ten block—the ones unit (or unit cube), the tens rod, the hundreds flat— is ten times larger in size and value. Students can now unitize each block, simultaneously seeing it as one unit and as representing many units. As Cathy Fosnot and Maarten Dolk describe, this is a complex shift in understanding for many students, since their foundational understanding of number is based on a one-to-one correspondence (Fosnot & Dolk, 2001, p. 11) Math Vocabulary: About the Lesson hpmulanactde,rebvdaasslu,eetet,enpnsla,bcoloencevksas,lue In this lesson, students make the largest number possible from three Materials: digits and then represent the number using concrete materials. base ten blocks, Many activities for further practice are provided. paper and pencils, Digital Slide 7: Which Minds On (20 minutes) One Doesn’t Belong?, BLM 9: Numeral Cards • Show students Digital Slide 7: Which One Doesn’t Belong? Have students turn (1–4), BLM 10: Numeral Cards (0–9), BLM 11: and talk to a partner about which of the four quantities pictured doesn’t Base Ten Recording belong with the others and why they think so. Explain that there are no Sheet, BLM 2: Place incorrect responses as long as they can justify their thinking. Students can Value Mat work with base ten blocks if they want to represent the amounts as they solve the problem. Time: 60 minutes • As a class, create base ten block representations and numerical representations for all of the quantities shown. Draw attention to how the position of the digits in the number determines how many of each block is needed. • Ask what all of the numbers have in common (e.g., they all have a 5 digit and a 1 digit). • Ask how each number is different from the other numbers. Record students’ ideas as they share. Some possible responses: – 5 1 does not belong because it has no hundreds or it is the only two-digit number. Quantities and Counting to 500 63

– 5 01 does not belong because it is the only number worth more than 200. – 1 05 does not belong because it is the only number that has 5 ones. – 1 50 does not belong because it is the only number that has 0 ones. • Pose some of the following prompts: – W hat representation helped you compare the numbers more easily? Why? – W hat makes the quantities different from one another? (e.g., the number of hundreds, tens, and ones; their size) – H ow can all of the numbers have digits in common but be so different in value? – W hat do we need to pay attention to when we determine how much a number represents? Working On It (20 minutes) • H ave students play the following game in pairs. Each pair needs a set of large and small numeral cards from BLMs 9 and 10, along with base ten blocks, a place value mat (BLM 2), and BLM 11: Base Ten Recording Sheet. 1. S huffle both decks of cards and place them face down in two separate piles. 2. P layer A draws 1 large card, which will represent the number of hundreds, and then draws 2 smaller cards, which will represent the other two digits in a three-digit number. Player A builds the greatest number possible using the digits and then represents it on the place value mat using base ten blocks. 3. P layer B counts the representation to confirm that it is correct. Player A records the number on the recording sheet. 4. P layer A returns his/her number cards to the decks, shuffles them, and turns them over. Players B takes a turn drawing cards and making a number (steps 2 and 3). 5. P layers A and B look at their numbers and decide which number is larger. The player with the larger number gets 1 point. Play continues until time runs out or their recording sheet is filled. Differentiation • For students who are not yet ready to deal with three-digit numbers, have them create two-digit numbers using only the small numeral cards. • For students who need a greater challenge, have them draw all three cards from only the small deck (0–9), thereby allowing them to create numbers up to 1000. 64 Number and Financial Literacy

Assessment Opportunities Observations: • Can students create the largest number possible using their cards? • Can they accurately represent their numbers using base ten blocks? • C an they correctly count the blocks in their representations? Are they able to count by hundreds and then flexibly count on by tens and then ones? Conversations: If students are having difficulty counting the representations, pose some of the following prompts: – How many hundreds do you have? (e.g., 3) How can you count them? (e.g., 100, 200, 300) Now look at the tens. How many tens rods do you have? (e.g., 4) How do you count them? (e.g., 10, 20, 30, 40) Let’s put the two together. What comes after 300 if you are counting by 10s? Can it be 400? Why? If you have 300 and 10 more, it is three hundred ten. How can you count in the other tens? Consolidation (20 minutes) • Have students think/pair/share to discuss their strategies for winning the game. Pose some of the following prompts: – W hat was the best strategy for ordering your numeral cards to win a point? – What is the best place to put a “0” numeral card in order to win a point? – What numbers were challenging to make? Why? – H ow would you change your strategy if you were making the smallest possible number? • Review how students have represented numbers in two forms, with concrete materials and with digits. Explain that there are other ways to represent numbers. • Ask students how they can represent 234 using base ten blocks and have some students build the representation. Show them how they can record the digits only on a place value mat: Hundreds Tens Ones 2 3 4 • Ask students how they could show adding the base ten blocks using numbers. Record the expanded form as 200 + 30 + 4. • Ask how the addition sentence would change if students added each block one at a time. Record this as 100 + 100 + 10 + 10 + 10 + 1 + 1 + 1 + 1. Quantities and Counting to 500 65

• Record the various representations on an anchor chart entitled Representing Numbers: Base Ten Blocks Expanded Form Repeated Addition Numbers and Place Value 200 + 30 + 4 234 100 + 100 + 10 + 10 + 10 + 1 + 1 + 1 + 1 Materials: Further Practice base ten blocks, • M issing Digit Problems: Give students ‘missing digit’ problems to solve. small paper/cloth bags They will have to use their knowledge of the place value system to reason and to justify their answers. For example: – P rove that 3_9 is greater than 307, no matter what number is in the blank. – P rove that 5_9 can be larger than 590. – A mystery number is 2_ _. What digits can be added to make this number close to 300? To make it about halfway between 200 and 300? • Independent Problem Solving in Math Journals: Pose one or both of the following problems: – Alex says that 390 is larger than 309. Is he correct? Explain. – Arrange the digits 4, 7, and 0 to make: a) the largest number possible (740) b) the smallest number possible (407) c) a number in between _____ and _____(e.g., 470 or 704) • M ath Game: Base Ten Mystery Bags: Demonstrate the game before having students play in partners or groups. The game can also be played with the entire class. Directions: Pick 5–10 different base ten blocks (e.g., 2 hundreds flats, 4 tens rods, and 1 ones unit for the number 241) and put them in a bag so that students cannot see. Present the clues below, one at a time and have students guess what the mystery number is. Record the clues so students remember them. After each clue, ask what the number could not be and why students think so. Ask what they do know about the number and what they still need to know. 66 Number and Financial Literacy

– There is at least one of each type of base ten block in the bag. – There are twice as many tens as hundreds. – There is only one unit block. – The number is less than 300 and more than 200. – The numbers 3, 5, 6, 7 and 9 are not in the number. Once students have guessed the number, reveal the base ten blocks to confirm the correct response. • Math Game: Secret Number: One person picks a number between 100 and 500 and asks the others to guess what it is. The other students ask questions that can be answered with only ‘yes’ or ‘no’ responses such as: – Is it bigger than 200? – Is it smaller than 300? – Is it even/odd? – D oes it have a zero? – C an I count by 10s from 0 to get to the number? – Is the hundreds digit bigger than the tens digit? Quantities and Counting to 500 67

9 10 ComposingLessonsand Numbers to 500 in Various Ways 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 • B1.2 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 • B1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials PMraotcheesmseast:ical About the rPerporbelseemntsinoglv,ing, communicating, According to Marian Small, “the ability to rename numbers is ssrceteoralanestnceoetgnicniietngisngtgoa,onrldesfpalerncodtviinngg,, fundamental to many of the algorithms involving addition, subtraction, multiplication, and division that students will learn” (Small, 2009, p. 142). Math Vocabulary: For example, it is important to know that 13 tens is equivalent to 1 rtmeegnarsto,, ubopnainesegs,,tphelunancderevdaslu, e hundred and 3 tens in order to subtract when regrouping is required. This blocks requires students to think flexibly, transform quantities into an alternate form, and rename them. Students also need to be able to count their representations of different quantities, changing their counting patterns as they move from one place value amount to another. About the Lessons In the Minds On of Lesson 9, students analyse and compare two representations of the same quantity to determine how they are similar and how they are different. During the Working On It session, students build a concrete representation of a number and then create an equivalent but different representation by regrouping and renaming the quantity. In Lesson 10, students create various numbers by following given parameters about how many base ten blocks they can use. There are several solutions to the problems, allowing students to be creative in their thinking. Students also look at other ways to represent the same number, with fewer blocks or more blocks. 68 Number and Financial Literacy

9Lesson Representing and Renaming Equivalent Quantities Teacher Possible Learning Goals Look-Fors • Represents and compares whole numbers using base ten materials Previous Experience • Composes and decomposes three-digit numbers in a variety of ways with Concepts: In grade two, students • Describes the value of each digit in a two-digit number regrouped two-digit • Flexibly decomposes and represents two-digit numbers (e.g., 34 is 34 ones, numbers into tens and ones. 3 tens and 4 ones, 2 tens and 14 ones, 1 ten and 24 ones) Materials: • Compares two numbers to determine which is larger and explains their Digital Slide 8: Base reasoning Ten Representations, base ten blocks and/or • Applies understanding of regrouping to compose numbers with no more than BLM 12: Base Ten 9 base ten blocks in each place value position Blocks, pencils, BLM 2: Place Value Mat, BLM Minds On (15 minutes) 13: Numeral Cards (11– 21), BLM 14: Base Ten • Show students the base ten representations on Digital Slide 8: Base Ten Regrouping Recording Sheet Representations. Have students turn and talk to a partner and discuss what is similar and/or different in the two representations. Record their thinking as Time: 50 minutes they share. • Prompt with the following questions: – W hat number(s) are represented? – H ow are they represented differently? – If more base ten blocks are used, does it mean the number is larger? – W hich concrete representation is easier to interpret as a number? Why? – H ow else could you represent this amount? Working On It (20 minutes) • H ave students play the following game in pairs. Give each pair two sets of the numeral cards from BLM 13 (shuffled together), a pencil, base ten blocks, BLM 2: Place Value Mat, and BLM 14: Base Ten Regrouping Recording Sheet. You can also provide them with paper base ten blocks (see BLM 12) if you have a limited supply of the actual concrete materials. 1. P layer A draws two number cards and places one in the ones column and the other in the tens column on the place value mat. Player A represents the number with base ten blocks (rods and unit cubes only) and then records the number of tens and ones on the recording sheet. (e.g., 13 tens and 11 ones) Quantities and Counting to 500 69

2. P layer A regroups the base ten blocks by making equivalent exchanges (e.g., 10 rods for 1 flat). Player B checks the work, ensuring that there are no more than 9 blocks in each place value position and that blocks were traded properly. Player A records the new, regrouped number on the recording sheet. 3. N umber cards are returned to the deck and the deck is shuffled. Player B takes a turn drawing cards, building a number, and regrouping (Steps 1 and 2). 4. P layers A and B analyse both numbers and decide which one is larger. The player with the larger number gets a point. Partners continue playing until they either run out of time or fill the sheet. Differentiation • Students who are having difficulty regrouping can use only one numeral card to put in the tens or ones column and a number cube to generate the other number. This will reduce the amount of regrouping the student needs to do. • Students who need help visually seeing the regrouping might benefit from base ten blocks that click together to form the larger blocks. Instead of exchanging blocks of one kind for another, the student can merely regroup (put together) blocks and move the newly formed block over. • For students who require more of a challenge, they can select any number from 1 to 4 and place it in the hundreds column. This allows them to work with larger numbers. • For students who need more of a challenge, have them regroup the base ten blocks again in another way, thereby creating another different representation for the number. Assessment Opportunities Observations: Notice how students are forming the numbers. Do they: – P hysically need to join blocks before seeing how to regroup? – Recognize when regrouping is necessary and make appropriate exchanges? – U nderstand how one number can have more than one name? – P erceive quantities in different ways. (e.g., 34 is 34 ones, 3 tens and 4 ones, 2 tens and 14 ones, 1 ten and 24 ones)? Conversations: Probe whether students understand that blocks can be exchanged. • How many blocks can we have in each section? Why? • How many units make up this rod? How might knowing this help you? • What is the largest digit we can have in each place value position? Consolidation (15 minutes) • Have students think/pair/share about some of the following prompts: – W hy can numbers be represented in more than one way and have more than one name? – W hen is it helpful to have numbers represented in certain ways? 70 Number and Financial Literacy

Materials: – W hat strategies helped you regroup numbers? Explain. Digital Slides 9–11: – H ow can one number be read in more than one way? Quick Images 1–3 • Create an anchor chart that captures students’ thinking and add it to the Math Teaching Tip Word Wall. Some ideas include: Integrate the math – N umbers can be decomposed/grouped into hundreds, tens, and ones in talk moves (see different ways. page 8) throughout – A hundreds flat can be exchanged for 10 tens rods or 100 ones. Math Talks to – T en can be a tens rod or ten ones. maximize student – A ny base ten block can be exchanged for smaller blocks. participation and – A ny group of 10 blocks can be exchanged for 1 larger block. active listening. – A ll numbers can be written in more than one way and the quantities are equivalent. Math Talk: Math Focus: Investigating the importance of zero as a placeholder Let’s Talk Select the prompts that best meet the needs of your students. • Briefly show students Digital Slide 9 and remove it from sight. How much did you see (what quantity) and how did you see it? Turn and talk to your partner. • What did you see? (e.g., I saw 2 flats and some tens rods and some unit cubes) What do we know about the number? (e.g., It is over 200) Can the quantity be more than 300? How do you know? Is the number a lot more than 200 or just a little more? Why do you think so? • I am going to show you the base ten blocks again so you can gain more information. Show Digital Slide 9 briefly a second time. What more did you learn about this number? (e.g., It has 3 ones.) So how could we write what we know about this number in numerical form? (e.g., 2__3) Is there anything else you know? (e.g., I think there are at least 5 rods, so the number is greater than 253) Let’s look at the image again. How can we complete writing the number? • Look carefully as I show you another image. Briefly show Digital Slide 10. How much did you see? Turn and talk to your partner. • What did you see and what do you know about the number? (e.g., I saw 4 flats so the number is over 400) Is the number a lot more than 400 or just a little more? How do you know? (e.g., It is a lot more because I saw lots of tens rods.) Did you see any unit cubes? What does this tell us about the number? How could we write the number so far? (4___0) What does the zero represent? • I am going to show the representation again without taking it away. How can we complete writing the number? (490) If there are no unit cubes, why can’t we write the number as 49? (e.g., That would be forty-nine.) What purpose does the zero continued on next page Quantities and Counting to 500 71

have and what does it tell us? (e.g., It tells us there are no ones. It holds the place for the ones.) • I am going to show you a third set of base ten blocks. Briefly show Digital Slide 11. Turn and talk to your partner about what you saw (the quantity, the number and type of blocks). • What did you see? (e.g., I saw 3 flats so the number is over 300.) Is it a lot more than 300 or just a little more? Why do you think so? (e.g., It is just a little more because I only saw unit cubes and no tens rods.) • I am going to show the representation again and leave it for you to study. What new information do we have? (e.g., There are 7 unit cubes.) How can we write the number that is represented? (307) What does the zero mean? (e.g., There are no tens rods.) Why can’t we write the number as 37? (e.g., That would be thirty- seven and it would make you think there are 3 tens and 7 ones.) So the zero is there so we know there are no tens, but there are hundreds and there are ones. • Zero plays a very important role in numbers. It can be a ‘placeholder,’ which means it holds open a place for one of the place value columns. • Let’s read some of the following numbers. What base ten blocks do you visualize in your head with each number? Show numbers that include at least one zero (e.g., 402, 160, 390, 208, and 300). Ensure students are naming the numbers properly (e.g., four hundred two) and discuss what base ten blocks they visualize. 72 Number and Financial Literacy

10Lesson Composing Quantities to 500 through Problem Solving Teacher Possible Learning Goals Look-Fors • C reates various three-digit numbers to 500, using a given number of base ten Materials: base ten blocks, blocks and given parameters paper, BLM 15: My Representations, • A ccurately counts and explains the representation of a quantity hundreds charts (optional) • Creates more than one quantity using a given number of base ten blocks Time: 60 minutes • 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 • Creates the same amount in more than one way • Explains how their representations meet the parameters in the problem Minds On (20 minutes) • Students work in pairs. Have students choose 3 base ten blocks, in any combination, from the tens rods and the unit cubes and identify the number they make. Have them choose 3 different base ten blocks and make a different number. Students continue doing this until they think they have found all the numbers possible (3, 12, 21, 30). Have students record their numbers on a piece of paper. Highlight how they counted the blocks and changed their counting pattern when they added in different blocks. • Record the numbers along with drawings of the corresponding base ten blocks, using sticks and dots. Ask what all of the numbers have in common (e.g., the digits in all of the numbers add to 3). Ask why this makes sense. • Tell students that this time, they can also choose hundreds flats when they select their 3 blocks. Have them predict how many possibilities there might be. Record the numbers and accompanying drawings, including squares to represent the hundreds flats (111, 102, 120, 201, 210). Highlight how students counted the blocks. Discuss why the digits in these numbers also add to 3. • Present the following challenge: – U se any combination of 5 base ten blocks to create a number that is between 100 and 200. What numbers can you make? Quantities and Counting to 500 73

• Have students predict what the digits in the numbers will add to and why they think so. • Ask what block must be included (1 hundreds flat) and what combinations of blocks cannot be included (e.g., 2 hundreds flats). Have students explain their reasoning. • Record some of their responses using squares, sticks, and dots. As students count the combinations, draw attention to how they change their counting with the various place values. Working On It (20 minutes) • Students work in pairs to solve one or more of the following problems: – C hoose exactly 5 blocks. Create quantities between 0 and 225. – C hoose exactly 6 blocks. Create quantities between 200 and 400. – C hoose exactly 8 blocks. Create quantities between 300 and 500. • In each case, students record their numbers and make a drawing of the blocks, using squares, sticks, and dots. Students also represent each number in a different way using other base ten blocks. In this situation, students may use as many rods as the want (e.g., If they make 123 using 1 flat, 2 rods, and 3 units, they can also represent it as 1 flat, 1 rod, and 13 units). Students can record their representations on BLM 15: My Representations (they will need at least 1 copy of the BLM for each problem). Differentiation • Select or adapt the problems so they meet the needs of individual students. • For students who need more of a challenge, have them create their own 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 combination that meets the criteria? C an they give examples of numbers that do not work? Can they explain • or show how they do not meet the criteria in the problem? • Can students flexibly count the collections of blocks? Do they change their count to match the new blocks? • C an they represent the same number in more than one way? Conversations: • sIfamsteudneunmtsbearr,ephoasveisnogmdeifoffictuhletyfoflilonwdiinngg another way to represent the prompts: – H ow can you count the number that you made with the base ten blocks? What exchanges could you make with your blocks? Try exchanging one block. How can you count your number now? 74 Number and Financial Literacy