Materials: Consolidation (25 minutes – 10 minutes to meet with masking tape or painter’s tape, another pair and 15 minutes for group discussion) personal benchmarks for 100 cm and • Have pairs meet with another pair to compare their results. If their results 1m are not the same, challenge them to figure out why they may be different. Materials: sidewalk chalk, • As a group, discuss whether all students found the same order of paths, from various measuring tools shortest to longest. If a group worked with the big book, have them compare their order to that of the other students. Discuss why their order is the same. • Discuss the units that students chose (e.g., centimetres) and the tools they used to measure the paths. Ask whether the students who measured in the little books should get the same results if they all measured in centimetres. Discuss why the lengths in the big book are different from the lengths in the little books. (e.g., the scale of the paths is different) • Discuss how students compared the lengths. Strategically choose two or three different methods and have pairs that used them demonstrate doing so. Explicitly connect the methods, showing how they all reveal the differences in the lengths of the paths. • Ask what is important when sharing the differences in the lengths of the paths. (e.g., Students need to report a number and the unit.) Further Practice Comparing More Paths: • Create two to three different paths on the classroom floor using masking tape or painter’s tape, all under 1 m long. • Allow students to investigate each path. • Have each student predict which path is the shortest. Use a tally system to record students’ choices. • Have students estimate the length of each path in centimetres. You may want to remind students to use a personal benchmark for 100 cm or 1 m to help with making a reasonable estimate • As a whole group, measure the lengths of the paths. Then allow students to reflect on their estimates. If they were close, ask what helped them make a good estimate. If they were not close, ask what knowledge or information they could use to make a better estimate. Chalk Paths: • Students work outdoors in pairs. Each pair draws two curved or zigzag lines on the pavement that they feel will be about the same length. • Students determine two ways to measure their chalk paths (e.g., footsteps, standard units). Students measure and then compare their results to determine if the paths are in fact about the same length. Measurement 195
Materials: Stick Person: paper or small • Give each student a piece of paper or a small whiteboard and a tool with whiteboards, straight edges (for which to draw straight lines (but not a tool with which they can measure drawing but not lines). measuring straight lines), rulers • Call out instructions, one at a time, for creating a stick person. Students will estimate the lengths of the lines they draw as they follow the instructions. – D raw a straight line in the centre of their page that is about 10 cm long and label it. (We want students to visualize personal benchmarks while creating this line.) – A dd lines near the top of the previous line for the arms. Make each arm 4 cm long. – A dd 2 legs near the bottom of the first line, for the legs. Make each leg 6 cm long. • Using a standard ruler, students can measure each line to see how accurate their estimates were. 196 Spatial Sense
10Lesson Measuring with a ‘Broken’ Ruler Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make First Peoples connections; model mathematics in contextualized experiences Principles of Learning • Understanding and solving: Develop, demonstrate, and apply Teacher mathematical understanding through play, inquiry, and problem solving; Look-Fors visualize to explore mathematical concepts; develop and use multiple strategies to engage in problem solving Previous Experience with Concepts: • Communicating and representing: Communicate mathematical thinking Students have measured with standard rulers that in many ways; use mathematical vocabulary and language to contribute to are divided into mathematical discussions; explain and justify mathematical ideas and centimetres. decisions • Connecting and reflecting: Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts Content • D irect linear measurement, introducing standard metric units: Centimetres and metres; estimating length; measuring and recording length, height, and width, using standard units • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) Possible Learning Goal • Accurately measures using a “broken” ruler that does not begin at 0 by counting the iterated spaces • Recognizes that the broken ruler is made up of centimetres, a standard unit • Explains or shows how the broken ruler is different from a regular ruler (it does not begin at 0) • Lines up the length being measured with the beginning of one unit on the broken ruler • Counts the number of iterated units (the spaces) to the end of the length • Reports the length using a number and the standard unit (centimetres) Measurement 197
Math Vocabulary: About the centimetres, ruler In order for students to understand how to properly use measuring tools comprised of standard units, such as centimetres, they must first understand how the standard units are represented on tools. The unit is the space that appears between any two numbers. On most tools, the unit starts at 0 and then is iterated to the end of the tool. For example, a ruler begins at 0 and ends at 30 cm, resulting in 30 uniform, standard units on a ruler. It is critical for students to understand that when measuring with a ruler, they are counting these spaces and not the hash marks that mark where one space ends and the next one begins. Using the hash marks can lead to counting the beginning position—the 0—as 1, thereby throwing off the count. An excellent way to help students learn to count the spaces is to have them use a “broken” ruler, or a ruler that does not start at 0. Rather than relying on the number where the length ends or on counting the hash marks, such a ruler forces students to count the actual spaces that are iterated along the tool when it is lined up with the length of the object. About the Lesson In the lesson, students measure lengths with “broken” rulers that do not begin with 0. As a result, they must count the spaces that represent the iterated centimetres. NOTE: To make broken rulers, cut up a measuring tape into segments, each of which begins at a different whole number greater than 0. Alternatively, photocopy a measuring tape and cut out paper segments. Materials: Minds On (15 minutes) “broken” rulers (see • Say, “I thought that we could share the measuring tape as we measure today. NOTE, one per pair); objects with uniform So each of you are getting a piece of the measuring tape. What do you notice length (e.g., unused about your ‘broken’ ruler? What do you wonder about?” (e.g., Each piece of erasers or pencils, one measuring tape has numbers on it as well as spaces between the numbers per pair); standard representing the units. No one’s measuring tape begins with the number 0.) rulers (30 cm); BLM 16: Measuring with a Broken • Give each pair an unused eraser or pencil. Say, “Measure these objects using Ruler your broken ruler.” Time: 55 minutes • Ask, “What did you notice? What made the task challenging? What length did you get? Put your thumb up if you got the same length. Why should we all get the same length? How did you use your broken ruler to measure?” (e.g., We lined up one number with the eraser and counted the spaces on the tool until we reached the end of the eraser.) “Did you use the numbers on the broken ruler? Why?” 198 Spatial Sense
Working On It (20 minutes) • Pose the following challenge: “Select five objects and measure their lengths using your broken ruler. Record the name of each object and the measurement. Then, measure the length of the same objects with a standard ruler, and record those measurements, too.” • Students can record their findings in the table on BLM 16. Differentiation • Students who find measuring with the broken ruler difficult can be given, or asked to measure, objects with lengths that are shorter than their ruler (so the ruler does not need to be iterated). • Students who need more of a challenge can be given, or asked to measure, at least one object that is longer than their broken ruler. Assessment Opportunities Observations: Pay attention to whether students can use the broken ruler by counting the spaces or are confused by the numbers printed on them (e.g., They interpret the last number lined up to the item as the length.). Pose some of the following prompts. Conversations: • You said that this item is 9 cm long. Why do you think so? (e.g., The number at the end of my pencil is 9.) • What units are on the ruler? (centimetres) Show me 1 cm. You are pointing to the number 5. Is that point on the ruler a centimetre? Show me with your fingers how long a centimetre is or show me a relational rod that is 1 cm long. How is this different from the point on this ruler? Put the centimetre cube beside the number 5. Where does it stretch to? (the number 6) So the centimetre is the space between 5 and 6. Show me another centimetre beside that centimetre. (the space from 6 to 7) You have lined up your pencil at the number 3. Count how many centimetres (spaces) there are from the 3 to the 9. How many centimetres long is your pencil? Consolidation (25 minutes) • As a class, discuss how students measured their items. Pose some of the following prompts: – H ow did you measure with your broken ruler? What were you counting to find the length? Did the numbers help you? Why or why not? – What did you notice about the two measurements of each object—the measurement with a broken ruler and the measurement with a standard ruler? Why did you get the same number of units with the different tools? What do we know about the units on each ruler? (e.g., They are the same length. Both rulers use centimetres, which are standard units.) Measurement 199
– W hat are we counting when we are measuring with a standard ruler or a broken ruler? (e.g., We are counting the spaces between the numbers.) How can the numbers help you on a ruler that is not broken? (e.g., The numbers can tell you how many spaces long something is, but only if you start at 0— if the object is lined up with the 0.) – H ow did you measure if your broken ruler didn’t stretch across the entire length? (e.g., We knew that our ruler was 8 units long so we put it end to end two times and then a little bit more. We added 8 + 8 and then counted another 3 units to get 19 units.) Did you use the numbers printed on the ruler this time? • If students are still struggling with this concept, give them more opportunities to measure using both the broken and standard ruler and to compare their findings. 200 Spatial Sense
11Lesson Guided Math Lesson: Cool Facts!: Bugs Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make connections; estimate reasonably; model mathematics in contextualized experiences • Understanding and solving: Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving; visualize to explore mathematical concepts; develop and use multiple strategies to engage in problem solving • Communicating and representing: Communicate mathematical thinking in many ways; use mathematical vocabulary and language to contribute to mathematical discussions; explain and justify mathematical ideas and decisions; represent mathematical ideas in concrete, pictorial, and symbolic forms • Connecting and reflecting: Reflect on mathematical thinking; connect mathematical concepts to each other and to other areas and personal interests; incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts Content • D irect linear measurement, introducing standard metric units: Centimetres and metres; estimating length; measuring and recording length, height, and width, using standard units Science Curricular Competencies Learning Standards • Questioning and predicting: Demonstrate curiosity and a sense of wonder First Peoples Principles of about the world; make simple predictions about familiar objects and events Learning • Planning and conducting: Make and record simple measurements using informal or non-standard methods Content • M etamorphic and non-metamorphic life cycles of different organisms • Similarities and differences between offspring and parent • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) Measurement 201
Teacher Possible Learning Goals Look-Fors • Estimates, measures, and records various lengths using standard units • Recognizes linear measurements such as length, width, height, depth, and distance • Explains the various linear measurements and shows examples of each • Compares different linear measurements and describes how they are the same and how they differ • Estimates a length and then measures it in standard units using various strategies (e.g., iterates multiple or single centimetre cubes, uses a ruler or string) • Estimates the length, width, etc. of a given object and then measures to confirm the estimate • Draws a given length and compares it to other lengths • Estimates how much longer or shorter a length is compared to another length • Records lengths with a number and a unit (e.g., 6 cm, 8 m) • Explains why a unit is appropriate for a given length Math Vocabulary: About the ldcemeenepngttttrhihem,,sedw,tismirdett,ashn,e,cschetmie,m,igahtte,, measure Students have had several experiences measuring linear attributes in standard units using a variety of strategies and tools. It is important to give them opportunities to apply their conceptual understanding and to practise and further reinforce their measuring skills. This includes allowing them to choose the tools that they think are most appropriate for what is being measured and to justify their selections. About the Lesson This is a guided math lesson for the little book Cool Facts!: Bugs. You can modify the lesson to meet the specific needs of students in each group. As you carry out the lesson with small groups over two or three days, the rest of the class can engage in activities you have set up as centres. You may have students rotate through the centres or allow them to freely visit the centres. Select the way that best suits your class. Remember that the purpose of the little book is to provide context for the math. The reading is not meant to be a barrier to the math, nor is the goal to have students independently read the text) although this would be a welcome secondary outcome). If students are struggling with the text, read it to them so they can focus on solving the problems and applying their mathematical thinking skills. 202 Spatial Sense
Materials: Differentiation Cool Facts!: Bugs • The major purpose of a guided math lesson is to be able to differentiate little books, various measuring tools (e.g., instruction and assess students’ math abilities and thinking. There are more string, centimetre problems within the little book than it is feasible for students to solve in one cubes, rulers), chart session. Select the ones that best meet students’ needs. You may decide to do paper each selected problem with the group, or you may do two or three problems Time: 20–25 together and have students work through the rest independently or in pairs minutes once the problems have been explained. • You may decide to have students select the way in which they measure, or you may suggest a method so you can assess their mastery and understanding of that method. For example, you may want students to use a single centimetre cube to iterate across a length or you may want them to show how they can measure using a ruler. As a further challenge, you may want students to measure with a “broken” ruler that does not begin at 0, thus requiring them to count the spaces between the numbers rather than rely on reading the numbers at the end of the object being measured. Guided Math Lesson NOTE: There are more prompts than can be covered in one lesson. Select the prompts that best meet the needs of the students in each group. Front Cover • Read the title and ask students what they think the book will be about. Ask how the book might involve measurement. Ask whether they think the book is fiction or non-fiction and why they think so. Pages 2 and 3 • Read page 2. Ask students which bugs they recognize and where they have seen them. Ask whether they think all of the bugs are insects and why they think so. • Read page 3. Have students paraphrase in their own words why these bugs are amazing. • Tell students that the bugs on these pages are shown at the size they are in real life. Prompt students to make comparisons. For example: • O rder the bugs on page 2 from shortest to longest. • H ow much longer than the white grub is the goliath beetle? • F ind the two longest and the two shortest bugs on pages 2 and 3. Pages 4 and 5 • Read both pages and discuss why calling these bugs walking sticks is appropriate. Have students estimate how long the nymph that is hatching on page 4 might be if the egg is 1 cm long. • Have students measure out the approximate maximum length of a Chan’s megastick (56 cm). Have them also measure out the length of the longest Measurement 203
megastick to show the difference. Ask what the length of the longest walking stick is in centimetres. • Have students estimate how much a walking stick grows from being a nymph to being an adult. Ask whether the appearance of the walking stick changes from nymph to adult or whether it just gets larger. • Students can measure their arm to answer the question at the end of page 5 (to compare the length of their arm to the length of the walking sticks). Pages 6 and 7 • Read both pages. Ask students what the difference between ‘length’ and ‘width’ is. Have them show the lengths and widths of the bugs pictured. • A sk students how wide the wingspan of the atlas moth is. Ask how much wider the wingspan of the atlas moth is compared with the wingspan of the tarantula hawk wasp. • H ave students estimate with their hands how long and wide each bug is. Have them measure out the estimated lengths and widths, either on paper (by drawing lines) or using concrete materials (e.g., connecting cubes or centimetre cubes). You might get each student to represent one of the bugs and then compare them as a group. Pages 8 and 9 • Read page 8. Have students show how long half a centimetre is. Ask how long a line of 8 peacock spiders would be if they lined up end to end. • Read page 9. Have students visualize a person jumping over a skyscraper. Have students visualize and estimate, using personal benchmarks, how high the flea and froghopper can jump. Have them measure out the length of a froghopper and the height it can jump. Ask whether the jump is closer to 1 m or half a metre. Repeat this for the flea’s length and jump height. • A sk how much higher than the flea the froghopper can jump. Pages 10 and 11 • Read page 10. Have students use personal benchmarks to visualize and show the length of the grasshopper and the distance it can jump. Students can then measure out the length and distance to confirm their estimations. • Read page 11. Have students visualize what it would be like to jump more than half the length of a hockey rink in one jump. • H ave students use personal benchmarks to visualize and show the length of a trap-jaw ant and the distance it can jump. Have them show the approximate length and distance with their hands. Then have them measure out both measurements with a suitable measuring tool. Pages 12 and 13 • Read page 12. Have students select a personal benchmark for 1 m and use it to visualize 10 m. Tell them that the ants can travel that distance 20 times in a row. • H ave students show the lengths of the three types of ants using personal benchmarks. Have them figure out how many of the giant Amazonian ants 204 Spatial Sense
would fit on the trail in the book. Ask whether more or fewer giant bull ants could fit on the same trail and why they think so. Have them estimate and then measure to find out how many giant bull ants would fit. Repeat this line of questioning for the bullet ant. • Read page 13. Discuss the length of the giant burrowing cockroach in relation to the size of the hand that is holding it. Have students measure their own hands to see if it would fit there. Have students visualize and then show how deep into the ground the cockroach can burrow. • H ave students show how deep in the ground the cicada nymph lives. Discuss how the appearance of the cicada changes from when it is a nymph to when it is an adult. Pages 14 and 15 • Read both pages. Ask students what ‘life-size’ means. Students can measure different dimensions of the bugs pictured (e.g., length of body, wingspan, length of tail). Pose some prompts to encourage comparisons. For example, how much longer than the dragonfly is the titan beetle? Page 16 • Read the last page. Discuss how the monarch butterfly changes throughout its life cycle. Discuss how its appearance changes throughout the metamorphosis. • H ave students show the length of the caterpillar in each instar (stage), either using concrete materials or drawing lines. Ask when the caterpillar seems to grow the most. Ask how much it grows from the first to the last instar. • D iscuss other bugs that experience metamorphosis. For extended learning, you could research the changes that they go through and how much they grow from stage to stage. Measurement 205
12Lesson Measurement Reinforcement Activities Math Curricular Competencies Learning Standards • All learning standards identified in this unit Materials: Foot Figuring (Marilyn Burns, 2000, p. 53) butcher paper • Students work in pairs. Each student traces their left foot, cuts it out, and or chart paper records their name on it. This will be their personal measuring tool. • Students can estimate and then measure each other’s heights in “feet.” • They can jump across the floor and then estimate and measure how many “feet” long their jump was. • Students can pick three other objects to measure with their “foot.” Materials: Not Enough Units uniform units (e.g., pipe • Provide students with 3 uniform units (e.g., 3 pipe cleaners). Students select cleaners, coloured tiles) 5 different objects from around the room to measure with their 3 units. For example, they might measure the length of their leg or the bookcase, but they can only use their 3 units. Students will need to iterate the units in some way to find the lengths. Materials: Estimating Lengths relational rods • Propose the following challenges: Using relational rods as benchmarks, measure and record an object in the classroom that is about the length of 3 orange rods/5 white rods/6 red rods/7 yellow rods, etc. Compare your findings with those of a classmate. How many centimetres long are your objects? Materials: Body Measurements uniform concrete • Give students a variety of tools to measure parts of their body. With a materials to measure with (e.g., pipe partner, students measure their fingers, hands, arms, legs, and feet as well as cleaners, identical the distance around their head and waist. Students later reflect on what other lengths of string), things they can measure about their bodies (e.g., their height, the length of rulers, tape measures their hair). When students know the length of an arm, the length of a finger, etc., they begin to build personal benchmarks that help them understand the size of a standard unit. They can use these body measurements to help them estimate other measurements. 206 Spatial Sense
Materials: Broken Ruler long, thin objects, • Set up a centre with a wide variety of long, thin objects. Students work in “broken” rulers (see Lesson 10) pairs. They each have a “broken” ruler with a different starting point. They first estimate and then measure the objects with their broken ruler and record their answer with a number and a unit. The students in the group then compare their findings to see if they reached the same results. Measurement 207
References British Columbia Ministry of Education. (2015). Aboriginal worldviews and perspectives in the classroom. Victoria, BC: Queen’s Printer for British Columbia. Burns, M. (2000). About teaching mathematics: A K-8 resource, Second Edition. Sausalito, CA: Math Solutions Publications. Chapin, S. H., O’Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn, grades K–6, Second Edition. Sausalito, CA: Math Solutions. Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge. McGrath, C. (2010). Supporting early mathematical development: Practical approaches to play-based learning. New York, NY: Routledge. Moss, J., Bruce, C. D., Caswell, B., Flynn, T., & Hawes, Z. (2016). Taking shape: Activities to develop geometric and spatial thinking, grades K–2. Don Mills, ON: Pearson. Newcombe, N. S. (2010). Picture this: Increasing math and science learning by improving spatial thinking. American Educator, Summer 2010, 29–43. Newcombe, N. S., & Frick, A. (2010). Early education for spatial intelligence: Why, what, and how? Mind, Brain, and Education, 4(3), 102–111. Small, M. (2007). PRIME: Geometry: Background and strategies. Toronto, ON: Nelson. Small, M. (2009). Making math meaningful to Canadian students, K–8. Toronto, ON: Nelson Education Ltd. Small, M. (2010). PRIME: Measurement: Background and strategies. Toronto, ON: Nelson. Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K–3, Volume One. Boston, MA: Pearson. Wai, J., Lubinski, D., & Benbow, C. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101, 817–835. 208 Spatial Sense
Spatial Sense Part of Math Place BC Grade 2 Lead Author: Diane Stang, National Math Consultant for Scholastic Education Math Reviewers: Jennifer Carter, Vernon School District #22 Debbie Nelson, Comox Valley School District #7 Indigenous Consultant: Diane Jubinville, Delta School District #3 Director of Publishing: Molly Falconer Project Manager: Jenny Armstrong Editor: Dimitra Chronopoulos Proofreader: James Gladstone Art Director: Kimberly Kimpton Designer: Dennis Boyes, Kimberly Kimpton Production Specialist: Pauline Galkowski-Zileff Copyright © 2021 Scholastic Canada Ltd. 175 Hillmount Road, Markham, Ontario, Canada, L6C 1Z7 Photo, p. 156: © Yuliya Vadi/Dreamstime ISBN: 978-1-4430-5408-9 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, scanning, recording or otherwise, without the prior written consent of the publisher or a license from The Canadian Copyright Licensing Agency (Access Copyright). For an Access Copyright license, call toll free to 1-800-893-5777.
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