• aIfnsottuhdeern, ytsohuacvaendpifofsiceuslotymcehoafntghinegfotlhloewcionugnpt rforommptosn: e type of block to – You have counted your flats as 100, 200, 300. What blocks are you counting now? (e.g., the rods) How much are they worth? (10) Can you count from 300 to 400? Why? What is 10 more than 300? Let’s look at the hundreds chart (301–400). How will you count in 10 more? How will you count in the remaining rods? What do you need to do when you add in the unit cubes? Look at the hundreds chart to help you. Consolidation (20 minutes) • Have students meet with another pair that solved the same problem(s). They can compare the numbers that they found and explain how they meet the criteria given. They can also explain how their second representation of each number is equivalent to their first representation. • Meet as a class. Select one of the problems to discuss. • Discuss students’ solutions and record them using squares, sticks, and dots. Have students justify their solutions and how they meet the criteria. • Ask students how they found all of the combinations and whether they had a system for doing so. • Have students explain and justify why some numbers do not meet the criteria in the problem. • Have students share and justify some of their equivalent representations. Focus on how they counted the blocks in each representation and record the counting on an open number line. Ask when we represent numbers in the different ways (e.g., when we are regrouping to add or subtract). Quantities and Counting to 500 75
and11 12Lessons Using a Number Line to Represent and Compare Quantities 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 working with number lines. They have • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools worked with quantities to 500 using base ten blocks. 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 • R epresents, compares, and orders benchmark numbers on number lines that have various endpoints • U ses benchmarks and endpoints to locate other quantities on the number line • Uses endpoints to determine significant benchmarks on a number line • Adjusts the position of numbers as the endpoints change so the numbers are still correctly positioned relative to each other • Uses number relationships (greater than, less than) to determine the placement of numbers on a number line • Orders numbers with reasonable accurately on the number line • Communicates their reasoning for strategically placing numbers at certain points on the number line • Applies their understanding of the number line to place numbers that have smaller differences closer together and larger differences farther apart • Locates the positions of numbers that are greater than or less than a number by a specific amount 76 Number and Financial Literacy
PMraotcheesmseast:ical About the Problem solving, careonpmdrempsrueonnviitcninaggt,i,ncrgoe,nanseocntiinngg, Ordering numbers requires a solid understanding of the counting sequence arenfdlescttriantge,gsieelsecting tools and the patterns within it. It is also important that students understand how the numbers are relatively positioned. For example, students may be Math Vocabulary: able to sequence numbers 257, 265, 398, and 418 in ascending order, but nnebsuuseemmtnqimcbubheeeamrnrtitlolaiiiannnrlkee, ,,noupmenbers, may not understand that in relative terms, 265 is much closer to 257 than it is to 398. Students benefit from seeing visuals, such as the number line, that reveal the distance between numbers. Doug Clements and Julie Sarama state that the number line is a valuable geometric/spatial model for ordering numbers since each point on the line is uniquely identified with one and only one number (Clements & Sarama, 2009, p. 44). This allows students to compare the magnitudes of numbers by visually seeing the distance between them. For example, 138 is 10 units larger than 128. Comparisons can also be made in relative terms since the proportional relationships between numbers remain, regardless of the endpoints of the number line. For example, 50 will always be halfway between 0 and 100. To help students understand the proportional relationships between numbers, it is valuable to have them order the same numbers on number lines with different endpoints. This requires them to make adjustments based on the relative distances between numbers. Working with the number line also offers students practice at establishing meaningful benchmarks, such as halfway points, and using them to judge the relative positions of numbers. About the Lessons In Lesson 11, students work as a whole group to compare and order numbers to 500 on the number line. They also identify significant benchmarks that can be used and how they can change as the endpoints change. In Lesson 12, students locate numbers on number lines and investigate how their relative positions change when the number line has different endpoints. Quantities and Counting to 500 77
11Lesson Ordering Numbers on a Number Line Materials: Minds On (20 minutes) string, blank index • Hang up a string across a portion of the room at a height that is within cards, marker, paper clips, index cards with students’ reach. Using paper clips, attach the endpoints 0 and 100 (on index various numbers from cards) at either end. Ask for examples of numbers that can and cannot be on 0–500 the number line. Have students support their reasoning. Time: 60 minutes • Ask students what the midpoint is and how they know. Ask where 50 would go, write it on an index card, and hang it on the number line. • Ask what some other meaningful benchmarks could be and where they would be located on the number line. Students can turn and talk to their partner. • Together, discuss and place their suggested benchmarks on the number line. • Change the endpoints on the number line so they are now 0 and 200. Ask what the midpoint is now. Ask students how they would move the existing numbers so they are in the right positions on the new number line. • Ask what other benchmarks would be beneficial to add, considering the new endpoints. Working On It (Whole Group) (25 minutes) • Remove all numbers and change the endpoints to 0 and 500. Ask students where they think 100 and 200 should go on the new number line. Ask what the new midpoint would be and place it on the number line. • Give each student an index card with a number between 0 and 500 printed on it. Have them turn and talk to a partner about where they think the number should be placed. • Have two or three students place their numbers on the number line. The rest of the students can analyse the position of the numbers and make suggestions about whether any of the numbers should be moved and why they think so. • Continue having students add their numbers and check in with the class periodically about whether there need to be any adjustments. • Pose some of the following prompts to elicit discussion. – A re all the numbers in sequential order? How do you know? – H ow did you determine how far apart those two numbers should be? – W hy are these two numbers closer together than the first two numbers we looked at? – H ow did the benchmarks help you place your numbers? 78 Number and Financial Literacy
Assessment Opportunities Observations: Observe how students determine the position of their numbers. • Do students consider the number line in its entirety or just focus on the benchmarks close to their number? • D o they suggest adjustments throughout the process as more and more numbers are added? • Do they place the numbers sequentially between two numbers or do they also consider whether the number should be closer to one number than another? Consolidation (15 minutes) • Once all numbers are placed, have students analyse the number line and determine if they need to refine the location of any of the numbers. • Discuss which numbers would be the most beneficial as benchmarks and why students think so. • Highlight three numbers that are beside each other and within the same hundred on the number line (e.g., 307, 322, 393 are all between 300 and 400). Ask if they are in the correct sequence, from smallest to largest. Ask whether they are correctly spaced in terms of their values. Discuss why they are or are not evenly spaced. • Use another string to create a second number line with a narrower range (e.g., endpoints 300 and 400). Ask what benchmarks it would be good to establish (e.g., the midpoint, 325, 375) and place them on the number line. Ask students where the three numbers highlighted on the first number line should be located on this new number line. Discuss how the numbers are further apart than on the first number line because of the different endpoints, yet the distances between the numbers are still relative to each other (e.g., 307 and 322 are closer to each other than 322 and 393 are). • Discuss how the endpoints affect where numbers are located on the number line. Quantities and Counting to 500 79
12Lesson Comparing and Ordering Numbers on a Number Line Materials: Minds On (15 minutes) string for number line, • Hang up the string that was used in the previous lesson and put on the index cards, markers, paper clips, long strips of endpoints of 110 and 210. chart paper, BLM 3: Hundreds Charts to 500 • Ask students what the midpoint is and why they think so. Ask what other (optional) benchmark numbers might be helpful to have on the number line. Ask what Time: 55 minutes would be the best way to locate these benchmarks on the number line and why they think so (e.g., locate the midpoint and work out from there in either direction; locate numbers closest to the endpoints and work in to the middle). Together, place the benchmarks on the number line. Discuss whether any of the positions need to be adjusted. • Show students two or three numbers (e.g., 117, 135, 178) one at a time, and have them turn and talk to their partner about where they should go on the number line. Discuss their responses and have students place the numbers, justifying their reasoning as they do so. • Change the endpoints of the number line so they are now 150 and 400. Ask which numbers, if any, need to be removed from the number line. Discuss how the remaining numbers need to be readjusted. Ask what other benchmarks it would be helpful to have on the number line. Working On It (20 minutes) • Have students work in pairs. Give each pair different endpoints, anywhere between 0 and 500, with which to create their own number line on a long strip of chart paper. Each pair identifies and records at least 3 benchmark numbers on their number line. On a separate piece of paper, they record 3 or 4 other numbers that fall between the same endpoints. • Pairs exchange their number lines and other numbers with another pair, and add those numbers to the number line. Together, the pairs discuss the placement of the numbers and whether there need to be any adjustments. • Students can repeat this again with different number lines. Differentiation • Select endpoints that are within the abilities of your individual students. • Some students may benefit from working with number lines that have the endpoints and midpoint marked on them. 80 Number and Financial Literacy
• For students who need extra support, provide access to hundreds charts up to 500 (BLM 3) to help them with the order of the numbers. • For a greater challenge, have students create their own endpoints. Assessment Opportunities Observations: • Pay attention to how students select their benchmark numbers. Can they explain their choices to each other? Can they equally partition the number line? • O bserve how students place the numbers. Do they use the benchmark numbers? Can they explain their thinking? • D o they recognize 10 as a friendly number? Are they able to use 10 as a friendly number or do they need practice counting by 10s from various starting points (forward and backwards)? Conversations: If students have difficulty selecting appropriate benchmarks, show them a hundreds chart and pose some of the following prompts: – Where are the two endpoints from your number line on the hundreds chart? What number would be about in the middle? How could you find it using the hundreds chart? – What other numbers could you put on your number line that are between the midpoint and the endpoints and are friendly numbers? Can you see a pattern made by friendly numbers? How far apart are they? (e.g., they are all 10 spaces apart) How could you add them to your number line so they are equally spaced? Consolidation (20 minutes) • Strategically select students’ work to share and discuss as a class. For example, you may choose one number line with endpoints that are close to each other and another number line that has endpoints that are farther apart. In this way, you can focus on how certain numbers need to be more spread out when the endpoints are closer together. Pose some of the following prompts: – W hy did you choose those benchmarks? How did you decide where they would go on the number line? Why is it important to have benchmarks like 320, 330, 340—benchmarks that increase by 10 each time—equally spaced? – W hat helped you identify benchmark numbers? – H ow did the benchmark numbers help you place the other numbers? Did you ever feel that you needed to readjust the benchmark numbers? Why? – W hy are these two numbers (e.g., 350 and 375) so close together on this number line, yet so far apart on the other number line? Quantities and Counting to 500 81
Teaching Tip Math Talk: Integrate the math NOTE: This Math Talk is adapted from Teaching Student-Centered talk moves (see Mathematics: Grades 3–5 (Van de Walle & Lovin, 2006b, pp. 45–46) and can be page 8) throughout repeated frequently throughout the year using different numbers. Math Talks to maximize student Math Focus: Comparing numbers by their relative sizes participation and active listening. Let’s Talk Select the prompts that best meet the needs of your students. • W rite the following three numbers on the board: 475, 498, 377. • W hat numbers do you see? Which numbers can be described as being less than the others? • W hich numbers can be described are being greater than the others? How do you know? • W hich two numbers are closest together? How can you prove that? Which number is closest to 480? • W hat is a number that is in between 475 and 498? 377 and 475? Which of the two numbers is it closer to? • H ow far apart are 377 and 250? What is a number that is a lot more than 250 but a little bit less than 377? • W hat is a number that is in between 377 and 475 but a little bit closer to 377? What is a number that is a lot closer to 377? • W hat is a number that is about half the size of 498? • W hat benchmark numbers could you use for locating 377? Why? Materials: Math Talk: string, index cards, Math Focus: Investigating relative distances on the number line marker, paper clips Let’s Talk Select the prompts that best meet the needs of your students. • H ang the string that was used as a number line in the lesson. Put on the endpoints 150 and 300. • W hat do you know about this number line? What numbers can be on it and what numbers cannot be on it? • W hat number do you think is midway between the two endpoints? Turn and talk to your partner. What do you think? What strategies did you use to find it? (e.g., jump forward 50 from 150 and jump backwards 50 from 300, and then find the midpoint between 200 and 250). How can you prove that it is the midpoint? 82 Number and Financial Literacy
(e.g., The distance between 150 and 225 is the same as the distance between 225 and 300.) Let’s mark the midpoint on the number line. • W hat number is 70 less than 300? How can you use the midpoint to help you? (e.g., 300 is 75 away from the midpoint so add on 5 to the midpoint to get 230) Let’s place 230 on the number line. How do you know where it will go? • W hat number is 70 less than 230? What benchmarks can you use this time to help you? (e.g., count back 30 to 200 and then count back 40 more to 160) Add 160 to the number line. What do you know about the distance between 160 and 230 and 230 and 300? • W hat number is 87 more than 150? How can the midpoint help you? Turn and talk to your partner. What did you find? (e.g., It is 75 from 150 to the midpoint, 225, so jump 10 more to 235, 5 more to 240, and then 2 more to 242.) Let’s place 242 on the number line. • H ow much greater is 242 than 230? How do you know? • L et’s change the endpoints to 150 and 500. Which of our numbers can still be on the number line and which ones cannot? Can the numbers that are still on the number line remain where they are? Why? What do you know about the spacing of some of the numbers? • Is 225 still the midpoint? What is the new midpoint? • W hat do you know about the distances between 160 and 230 and 230 and 300? (e.g., the distances will be equal to each other but they will be smaller now because the numbers are closer together) • W hat benchmarks could we place on the number line? • W hat numbers will be 125 more than 300 and 125 less than 300? What do you know about these distances? Let’s place these numbers on our number line. • W hat happens to distances on the number line when we change the endpoints? Quantities and Counting to 500 83
13Lesson Problem Solving with Quantities to 500 Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and including 1000, using a variety of tools and strategies, and describe various ways they are used in everyday life • B 1.2 compare and order whole numbers up to and including 1000, in various contexts • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and strategies • B 1.5 use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials Algebra • C 4. apply the process of mathematical modelling to represent, analyse, make predictions, and provide insight into real-life situations Teacher Possible Learning Goal Look-Fors • Applies understanding of numbers to 500 to solve related problems and Previous Experience with Concepts: explain their strategies and thinking Students have represented quantities to 500 in • Understands what the problem asks various ways, including • Selects appropriate tools and strategies to solve the problem using place value • Explains their strategies and thinking using appropriate math vocabulary amounts. • Explains their solution in terms of the context and why it is reasonable • Flexibly composes and decomposes quantities • Counts quantities by flexibly adjusting counting strategies to the place value amounts being represented PMraotcheesmseast:ical About the Problem solving, creoamsmonuinngicaatnindgp, roving, In this unit to date, students have had several experiences with representing arreenfpdleresctstrieanntget,ignsigee,lseccotninngectotionlgs, and comparing numbers to 500. It is important to offer them opportunities to apply what they have learned to solve real-life problems set in contexts that are familiar and meaningful. This helps students make meaning of the numbers and understand why these large quantities are important in their lives. 84 Number and Financial Literacy
Math Vocabulary: About the Lesson quantities, hundreds, tens, ones In this lesson, students solve real-life problems involving quantities to 500. There are a number of problems from which to choose. You may decide to repeat the process of this lesson two or three times using a different problem each time. Alternatively, you may assign different problems to different students according to their needs. Adjust the context of each problem if/as needed. This is a good opportunity to reinforce the mathematical modelling process and its four components: • Understand the Problem • Analyse the Situation • Create a Model • Analyse and Assess the Model Throughout the lesson, use an anchor chart to highlight how students move back and forth among the four components. For example, as students test out their model, they may need to revisit the problem (Understand the Problem) or the conditions given in the problem (Analyse the Situation) to develop and refine strategies and select more-appropriate tools. There are suggestions in the lesson about when and how to reinforce the process, although these will need to be altered so they are responsive to how your students progress throughout the lesson. Materials: Minds On (10 minutes) a variety of concrete • Display one of the problems below, read it together, and discuss what materials and tools such as base ten blocks, students think they are supposed to solve for. (Understand the Problem) number lines, hundreds chart, chart paper • Ask what other information students may need to solve the problem. For Time: 5 5 minutes example, for Problem 1, they need to know how to convert between different units of time so they are meaningful. (Analyse the Situation) • Reinforce the fact that students can solve the problem in any way they choose and use any types of tools or concrete objects. (Create a Model) • Encourage students to solve the problem in another way to confirm that their thinking makes sense. Problems: 1. H ow old is a person who has lived for 500 days? 500 months? 2. B uild two creatures out of base ten blocks. The two creatures should be made of different blocks but have the same value: 273. 3. Jesse built three different-sized towers using a total of 500 blocks. The middle-sized tower has 187 blocks. How many blocks could the other two towers have? 4. A nna and her family drove 500 kilometres over 4 days. On the first two days, they drove the same distance. On the third day, they drove 87 Quantities and Counting to 500 85
kilometres, which was the smallest distance driven on any day. On the last day, they drove the farthest distance. How much could they have travelled each day? Working On It (20 minutes) • Students work in pairs to solve their problem. Have them record their solutions on chart paper. Differentiation • Select the problem(s) that best meet the needs of your students. Change the context so the problems have more relevance. • For students who need more of a challenge, they can create and solve their own problem. Assessment Opportunities Observations: Pay attention to whether students know how to start solving their problem. What tools and concrete materials are they selecting? Can they switch to a new strategy if the one they are using does not work? They may need to revisit the problem or reanalyse the situation in order to develop a more effective model. Conversations: If students are having difficulty getting started, read the problem over with them and ensure they understand what is being asked. Ask which tools might help them start the problem. Encourage them to represent one of the quantities and then read the problem again to see how they can progress from there. Consolidation (25 minutes) • Have students meet with another pair who solved the same problem. They can share and discuss their solutions. If some students created their own problem, they can exchange it with another group to solve. • Select one of the problems to discuss with the entire class. Connect the thinking in the different strategies. • Discuss other circumstances when students would be solving problems involving numbers to 500. • Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: Have students reflect on the mathematical modelling process and how they progressed through it. (Analyse and Assess the Model) Ask students how they changed their plans if their initial strategy did not work. Discuss how it is important to be flexible when thinking about problems and to look at them from different perspectives. Mathematicians often don’t solve a problem on their first try and may have to use several different strategies before they find one that works. They also benefit from talking to teach other and sharing ideas. Often, hearing another person’s ideas will evoke a new idea that may work. 86 Number and Financial Literacy
14Lesson Representing Quantities Using a Concept Circle Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life Previous Experience • B1.5 use place value when describing and representing multi-digit numbers with Concepts: Students have previous in a variety of ways, including with base ten materials experience using base ten blocks, place value mats, Possible Learning Goal composing and decomposing numbers, • Composes, decomposes, and recomposes three-digit numbers up to 500 in a and renaming numbers. variety of ways and explains how various representations of the same PMraotcheesmseast:ical number are equivalent Problem solving, • Represents a three-digit number in multiple ways (e.g., words, expanded reasoning and notation, place value mat, base ten materials, trading tens, trading hundreds, representing, proving, drawings) communicating, • Understands that any three-digit number can have equivalent representations ssreterflaeletcectgtiniineggs,tocoolnsnaencdting, (e.g., 13 tens are equal to “one hundred and thirty”) • Explains or shows how two representations of the same number are equivalent About the Now that students have had several experiences representing numbers to 500 in several ways, it is valuable to give them a culminating task that allows them to show what they have learned. The culminating task in this lesson involves concept circles. Dr. Cathy Marks Krpan describes the concept circle as a graphic organizer designed to encourage students to think critically and make connections. It is also effective for evoking meaningful math talk and debate (Krpan, 2013, p. 118). The organizer is a circle divided into sections. Students write the concept in the middle of the circle and represent it in several ways in the sections, using drawings, numbers, and symbols. continued on next page Quantities and Counting to 500 87
Math Vocabulary: About the Lesson dpfoelarcmcoem, vepaxolpusaeen,, drseetagdnroduaprd, notation In the Minds On, students are introduced to the concept circle and brainstorm ideas to cooperatively fill in an example with various representations. During the Working On It, students independently complete a concept circle that shows multiple representations of a three-digit number up to 500 of their choosing. During the Consolidation, students share and explain their concept circles in a gallery walk. Materials: Minds On (15 minutes) a variety of concrete • Draw a concept circle with 6–8 sections on chart paper or a whiteboard. Put materials and tools (e.g., base ten blocks, the number 75 in the middle. Explain that a concept circle helps people relational rods, number understand an idea by showing it in different ways; in this case, by showing lines, money different representations and examples of the same number. The examples manipulatives), sticky can be made using pictures, numbers, or words. notes, chart paper • Have students turn and talk to a partner about how they could represent 75 Time: 60 minutes in several ways. • Discuss their various representations. As they describe their ideas, record them in the sections around the concept circle. Here are just some of the ways students could represent the number 75: – 3 quarters (25¢, 25¢, 25¢) – 7 tens rods and 5 unit cubes – 7 orange relational rods and 1 yellow relational rod, or 15 yellow relational rods – o n a number line – 2 0 + 20 + 20 + 10 + 5 or 100 – 25 – the age of their grandfather – the number on their house • Pose prompts to elicit a variety of answers. Working On It (20 minutes) • Have students work independently to create their own concept circle on chart paper for a three-digit number of their choosing. They print the symbolic form of the number (e.g., 188) in the middle of the circle and then represent it in the surrounding sections. Differentiation • You could give students who require more support a concept circle with tools or strategies written in some of the segments (e.g., base ten blocks, drawing, addition sentence). Remind students of the various tools and concrete objects 88 Number and Financial Literacy
that they might want to use (e.g., base ten blocks, relational rods, number lines, money manipulatives). • If writing is a challenge for some students, scribe their ideas for them. • For students who need more of a challenge, have them choose two numbers to compare in several ways. Assessment Opportunities Observations: • C an students accurately represent their number in more than one way? • Can students flexibly decompose their number, using concrete materials and/or the symbolic representation of the number (e.g., ‘75’) alone? • C an they adequately explain their representations? Conversations: If students are having difficulty representing the number, pose some of the following prompts: – W hat tools or concrete materials could you use to show this number? – Look at the concept circle we made together (in the Minds On). Can you adapt any of those ideas for your number? – H ow could you use addition or subtraction to show your number? – H ow could you change your base ten blocks so you are showing the same number, but in a different way? Consolidation (25 minutes) • Post all of the concept circles and hold a gallery walk. Have half of the students ‘stray’ to look at the work of the students who ‘stay’ with their creations. Encourage the creators to explain some of their representations and the visitors to ask questions. Students can then switch roles. • Afterwards, you may decide to do a quick walk around as a class and note any interesting ideas that were represented on the concept circles. Quantities and Counting to 500 89
15Lesson 365 Penguins: Revisiting the Math Math Number Curriculum Expectations • B 1.1 read, represent, compose, and decompose whole numbers up to and Teacher including 1000, using a variety of tools and strategies, and describe various Look-Fors ways they are used in everyday life Previous Experience • B 1.2 compare and order whole numbers up to and including 1000, in with Concepts: Students listened to 365 various contexts Penguins read aloud at the beginning of the unit • B1.4 count to 1000, including by 50s, 100s, and 200s, using a variety of tools and deconstructed the text as both readers and and strategies mathematicians. They asked questions about the • B 1.5 use place value when describing and representing multi-digit numbers numbers and mathematics in the text. in a variety of ways, including with base ten materials PMraotcheesmseast:ical Possible Learning Goals crPeoramosbmolenuminngicsaoaltnvindingpg, r,oving, tcroeoopnlrnseesacentnidntigsn,tgrsa,etreleegfclieteicsntging, • Decomposes quantities in a variety of ways • Accurately skip counts to solve various problems • Selects one or two appropriate strategies for decomposing the numbers • Identifies the whole number and explains/shows how each place value position relates to the whole number • Explains how larger numbers can be broken down into smaller units About the Throughout this unit, students have had experience representing, comparing, and ordering numbers to 500, counting in several ways, and flexibly composing and decomposing quantities. By revisiting the same text used to introduce the unit, students can reflect on how much they have learned throughout the lessons, which can reinforce the idea that their efforts have been worthwhile. Seeing their own growth in understanding can also help increase confidence in their math abilities. In Lessons 1 and 2, students also generated questions that they had about numbers and larger quantities. They can now review these questions and possibly answer many of them based on what they have learned. Such reflection is important for nurturing positive attitudes toward math and building social-emotional learning skills. Further problems can also be posed, so students can apply what they have learned throughout the unit. Some of the problems can deal with time relationships, such as the number of days in a week or the number of weeks in a year. 90 Number and Financial Literacy
Mdttheeaonctuoshs,maVopnonodcessase,,b,hbupualalnsardecyre:etedvnas,,lue, About the Lesson doubling In this whole-group lesson, students revisit the book 365 Penguins and Materials: solve various problems that are embedded in the text and illustrations. They also revisit and answer questions that they generated when they 365 Penguins, concrete initially read the book at the beginning of this unit. materials and tools (e.g., base ten blocks, Assessment Opportunities yearly calendar, open number line), markers, Observations: chart paper, Digital Slide 3: Hundreds Chart (201– • D o students know how to read numbers using proper mathematical 300) Time: 30–40 minutes vocabulary? • Can students decompose numbers and accurately skip count to find the value of their representations? • A re students able to see the relationship between skip counting and grouping? • A re students using what they know about skip-counting patterns to determine how they can and cannot group numbers? (e.g., 161 cannot be grouped by 2s because it ends in a 1) • D o students understand the relationship between days and weeks, and weeks and years? Before Reading • Ask students what they remember about the book 365 Penguins. Discuss the problem that the family had and how it was solved in the end. Ask what new problem transpires at the end of the story. Ask what numbers students remember as being significant in the story. • Tell students that they are going to revisit the book, after having learned much more about numbers to 500, and solve some new problems that are embedded in the story. During Reading NOTE: You may decide to reread the entire story or have students paraphrase what happens based on the illustrations. Stop at some of the following pages to further explore the math. Select the problems that best meet the needs and ability levels of your students. Problem solving “And on it went…” • R eview how many penguins the family has after one week. Ask how many penguins there would be after 2 weeks, after 4 weeks, and then after 8 weeks. Record the number of weeks and the number of penguins after each week. Quantities and Counting to 500 91
Discuss the doubling patterns (e.g., 1, 2, 4, 8 and 7, 14, 28, 56). Have students predict how many weeks it would be before there are about 100 penguins. • A sk students how they could find out the number of penguins after 7 weeks (e.g., 7 × 7; 7 + 7 + 7 + 7 + 7 + 7 + 7; if there are 56 penguins after 8 weeks then subtract 7 penguins to find out how many penguins there are after 7 weeks). • Ask how many weeks are in a year. Ask how students can use the information in the story to figure out how many penguins there are after 50 weeks (e.g., There are 365 penguins after 52 weeks, 358 penguins after 51 weeks [365 − 7], and 351 penguins after 50 weeks [358 − 7]). Problem solving “…we’re going to keep them!” • At the end of January there are 31 penguins. Ask how many penguins there would be at the end of April. Tell students you would like them to estimate the number before calculating it. Have students turn and talk about how they would solve this problem. Together, total up the days by referring to a yearly calendar, which indicates the number of days in each month. • H ave students figure out about many penguins there will be at the end of August. They may add up the days for another four months, or they may double the amount at the end of April (4 months) to get an estimation of how many there are at the end of August (8 months). Representing/ “Give them away…” communicating • Ask students what would be another way to organize 60 penguins Communicating/ reflecting/reasoning and (e.g., 6 groups of 10, 2 groups of 30, 12 groups of 5). Ask how they can use skip counting to find the total. proving • If there are 60 penguins, ask how many times more there would be if there were 600 penguins. Ask whether it would take more or less than two years to get 600 penguins if one penguin arrived each day. Ask how they might organize 600 penguins. Have them represent 600 penguins in more than one way, using base ten blocks. • Ask students how they could count backwards to represent the 60 penguins being taken out of the house 10 penguins at a time. Show the count on an open number line. Ask how they could count backwards if the 60 penguins are taken out 5 penguins at a time. Represent this count on the number line and have students compare it to the sequence of counting back by 10s. Ask what patterns they see. Reflecting/representing “On April the tenth exactly…” • Ask students how they could count 100 penguins by skip counting. Demonstrate the skip counting on the number line. Ask how they would continue the counting pattern up to 500. • H ave students imagine that the 100 penguins are taken out of the house in smaller, equal groups until no penguins are left. Ask what size the groups could be. Have students skip count back from 100 in several ways. 92 Number and Financial Literacy
Reasoning and proving/ “1) Feeding the penguins” representing/ • Discuss how feeding the penguins is a problem. Ask many pounds of fish are communicating needed to feed the penguins for 2 days, 4 days, and 8 days. Discuss strategies students can use to solve this (e.g., doubling). • Ask how students could use the same strategies to figure out how much it would cost to feed the fish for 2 days, 4 days, and then 8 days. Problem solving/ “The days went by…” representing • Ask students how they can represent 144 penguins with base ten blocks in more than one way. Ask how many penguins there would be if 60 more penguins arrived all in one day. Ask how they could represent the new amount in more than one way. Have them prove that the multiple representations equal the same quantity. Problem solving/ “On the fourth of August…” communicating • Ask students how they can represent 216 in more than one way, using base ten blocks. • Ask how they can count from 216 to about 300 if 5 penguins are added at a time. Ask how the count would change if 10 penguins are added each time. Show the count on Digital Slide 3: Hundreds Chart (201–300). Problem solving/ “Before we could say…” connecting/representing • Ask how many weeks have passed since the penguins started to arrive. Ask how many months have passed. • Ask students how they could count up by 10s from the last total in the book, which was 217. First, have them predict whether they will finish counting at exactly 365 and how they know. Have them count up by 10s from 217. Ask how they would continue the count after getting as close to 365 as they can counting by 10s. • A sk whether they can get closer if they count by 5s and why they think so. Ask what number they would end up on before having to count by 1s. Have students count up by 5s from 217. • A sk how they can represent 365 with base ten blocks in more than one way. • Ask how many penguins there would have been at the end of November. Have students solve the problem in more than one way. After Reading • Pose some of the following prompts: – W hich strategy do you think makes it easiest to group and count? Why? – W hat math tools/concrete materials help you the most to group and count? – H ow do you keep track of your count? – W hich counting strategy are you not as comfortable with? Why? Quantities and Counting to 500 93
– W hen is skip counting a good/efficient strategy to use? – W hen is counting by 10s a good/efficient strategy to use? • B uilding Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: As a class, revisit the list of questions that students generated at the end of Lesson 2 about the text and quantities to 500. (e.g., How might the number of penguins delivered have changed if they were only delivered on weekdays?) Have students work in pairs or small groups to answer one or more questions. Discuss questions and answers as a class. Emphasize how much students have learned and improved as mathematicians after completing this unit. Discuss what they feel they still need to practise in order to keep improving and to get more comfortable with numbers to 500. 94 Number and Financial Literacy
16Lesson Reinforcement Activities Math • All expectations covered in this unit Curriculum Expectations Teacher Possible Learning Goals Look-Fors • Practises representing, comparing, composing, and decomposing quantities to 500 • Counts in various ways to 500 • Applies understanding of skip counting to accurately count during the activities • Composes and decomposes three-digit numbers in more than one way • Explains how two representations of the same number are the same • Compares numbers by looking at their digits and how they are positioned (place value) • Uses a variety of concrete materials and tools, including base ten blocks, to represent quantities About the Lesson The following activities are games that can be played by the whole class in small groups, or at centres through which students rotate over a few days. The activities can also be used throughout the unit any time you decide to offer guided math lessons. For example, you may want to meet with small groups over a few days and tailor the lessons to meet the needs of each group. Although this can be done while the rest of the students solve the same problem in small groups, you may wish to observe how each group works through the same concept. In this case, one group meets with you each day, while the other groups rotate through some of the following activities. See the Overview Guide for more information on how to manage guided math lessons. Quantities and Counting to 500 95
Materials: Activity 1: Memory (Players: 2–4) BLM 16: Memory Game Cards Directions: Time: 20–25 minutes • Shuffle the deck of 40 cards and place them face down in a 4-by-10 array. • Players take turns flipping over two cards. If the two cards match, the player keeps them and can take another turn. If there is no match, the cards are returned to their places, face down, and the next player takes a turn. • The game is over when all the cards are matched. The winner is the person with the most cards. Materials: Activity 2: Build the Largest Number (Players: 2) BLM 10: Numeral Directions: Cards (0–9), three copies of each number • The goal of the game is to make the largest three-digit number. in one deck of playing • Shuffle the numeral cards and place them face down. Players take turns cards, BLM 2: Place Value Mat (one per drawing a numeral card and placing it in the ones, tens, or hundreds column player), BLM 17: Build on their place value mat. Once the number is in place, it cannot be moved. the Largest Number Recording Sheet • Players draw a second card and place it in one of the remaining columns, Time: 20–25 minutes and finally draw a third card to complete their three-digit numbers. • Players compare their numbers and determine which is larger and by how much. They record their numbers and proof of which number is larger on BLM 17: Build the Largest Number Recording Sheet. The player with the larger number records a point in their Points column. • Play continues until one player reaches 5 points. • NOTE: To play in larger groups (3–4 players), students will need two sets of numeral cards and a modified recording sheet (columns added for each additional player). Variation: • Students can change the goal of the game so the player with the smallest three-digit number wins each round. 96 Number and Financial Literacy
Materials: Activity 3: Hundreds Chart Race (Players: 2) BLM 3: Hundreds Directions: Charts to 500, BLM 18: Skip-Count • Players choose one hundreds chart to play with. Both players place their Spinner, pencil and paper clip, number cube counters on the first number of their chosen hundreds chart. or die, two counters in different colours • Players take turns spinning the spinner and rolling the number cube. They Time: 20–25 minutes move their counter the number of spaces indicated on the cube by the number and direction indicated on the spinner. For example, if a player rolls a 3 and spins ‘5s forward,’ they count forward by 5s three times from their start number. Players verbally say the numbers as they skip count as well as read the number that they land on. • The player who reaches the end of the hundreds chart first is the winner. Materials: Activity 4: Make 500 (Players: 2) base ten blocks, Directions: BLM 19: 0–5 Spinner, pencil and paper clip, • Players take turns spinning the spinner three times to create a three-digit BLM 2: Place Value Mat, BLM 20: Make 500 number, recording the first number in the hundreds column, the second Recording Sheet number in the tens column, and the third number in the ones column of their place value mat. If they spin a 5 for the hundreds, they must spin again Time: 20–25 minutes for a smaller number. • Players record their number on the recording sheet and create the number using base ten blocks. The other player then adds base ten blocks to the original number until they have made the number 500 and records the number of blocks used and the number on the recording sheet. The activity continues for 5 to 10 rounds. Quantities and Counting to 500 97
Search
