GR2_unit2-measurement-p146-210_2nd

Unit 2: Measurement Lesson Content Page 1 Measurement Introduction 146 2 What We Measure and Why: Math Counts: Length 150 3 How We Measure 157 4 Measuring Length and Height with Non-standard Units 162 5 to 7 Measuring Distance with Non-standard Units 169 5 Transitioning to Standard Units 173 6 The Centimetre 175 7 The Metre 178 8 The Centimetre–Metre Relationship 182 9 Using Benchmarks to Estimate and Measure 187 10 Using Benchmarks to Make Comparisons 192 11 Measuring with a ‘Broken’ Ruler 197 12 Guided Math Lesson: Cool Facts!: Bugs 201 Measurement Reinforcement Activities 206

Measurement Introduction About the According to Marian Small, measurement involves “assigning a numerical value to an attribute of an object, shape, or event—such as an object’s height, the amount of space a 3D shape occupies, or the duration of an event—in order to make comparisons” (Small, 2010, p. 2). She adds that measurement allows us to make decisions about our everyday lives by quantifying and comparing things in our environment. The study of linear measurement can be particularly engaging for students since it can relate directly to the world around them. For example, students may naturally ask, “How tall am I? How high is my desk? How wide is that window? How long is my foot? How far did we walk?” Students require many experiences that allow them to explore measurement within real and personally meaningful contexts. Conceptual Understanding Understanding the attributes being measured is the first and most critical component in developing a conceptual understanding of measurement (Van de Walle & Lovin, 2006, p. 225). Direct comparison, such as lining up two objects to compare their heights, is effective because it puts the focus on the attribute, without the need to measure or produce a number (Van de Walle & Lovin, 2006, p. 226). Vocabulary such as ‘longer’ or ‘shorter’ (comparative adjectives) can be used to make the comparisons. Van de Walle and Lovin identify the second component in developing a conceptual understanding as figuring out how “filling, covering, matching, or making other comparisons of an attribute with measuring units produces a number called a measure” (Van de Walle & Lovin, 2006, p. 225). When students work with familiar non-standard units, it is easier to focus on the attribute. It is also easier to keep numbers reasonable by selecting measures that students can count and understand (Van de Walle & Lovin, 2006, p. 227). When students use non-uniform units, such as hand spans, they begin to understand the value of uniform standard units. These experiences help grade two students transition from using non-standard to standard units. Basic Skills In addition to conceptually understanding measurement, students need to master the basic skills of making measurements. These include using a common baseline when directly comparing lengths and leaving no 146 Spatial Sense

gaps or overlaps when iterating multiple units or a single unit. Many basic measuring skills are related to foundational concepts in number and operations, such as being able to unitize (to see units within other units). Students also need to know that they are counting the multiple units that haven’t been iterated and not the markings between units on a measuring tool, a misconception which often arises when students measure with standard rulers (see Lesson 10, Measuring with a “Broken” Ruler). Linear Measurement with Standard Units Students in grade two focus on direct linear measurement with standard units. According to John Van de Walle and LouAnn Lovin, length is usually the first attribute students learn to measure (Van de Walle & Lovin, 2006, p. 228). Length is the distance between two points, or the measurement of something from end to end. For grade two students, linear measurement includes length, width, and height presented in various contexts. Students can often get confused about which of these attributes they are measuring. All three are linear measurements, but length is often a horizontal measurement while height is often a vertical measurement.  In grade two, students transition from measuring with non-standard units to measuring with standard units. It is important that they understand that it makes sense to use standard units because these are the same everywhere; they are a reliable measure over time and place. As students learn about centimetres and metres, they explore the relationship between the two units and establish benchmarks to help them make estimates. They also investigate which units are most appropriate for measuring certain objects, as well as the tools that it is most practical to use. Measurement Involves Approximation Marilyn Burns points out another important measurement idea: “that the physical act of measuring produces, at best, an approximate measure” (Burns, 2000, p. 8). Grade two students can begin to understand, through their investigations, that a measurement is never exact, and there is always a way to make the measurement more precise by further subdividing the units being used. As a result, measurement often involves some estimation, or rounding up or down to one unit or another. This is important as students will often find that the length they are measuring does not perfectly line up with the units being used and so they must approximate which unit the length is closest to. Measurement 147

Embedding Measurement Throughout the School Day • W ord Wall: Measurement involves a great deal of new vocabulary, including the names of attributes (height, length, width), related comparative terms (e.g., longer than, shorter than), and the names of standard units (e.g., centimetres, metres). Add new vocabulary to your Math Word Wall. Give a definition and have students locate the matching word. Ask for an example in the classroom. Alternatively, give the word and have students explain its meaning and give a real-life example. The more you do this, the better students become at recognizing the words, knowing their meanings, locating them on the Word Wall when they want to use them or be reminded of what the words mean. • H ave students line up according to different criteria (e.g., from shortest to tallest, from the person with the shortest hair to the person with the longest hair). • M easure the height of all students at the beginning of the year by marking how tall they are in centimetres on a long piece of chart paper. Revisit the chart three or four times during the year to track how much each person has grown. • Do a daily sort of your students according to various measurable attributes. Sort them according to a secret rule (e.g., students with long hair and students with short hair) by having them stand on one side of the room or the other. Once they are all sorted, have them look at each other and figure out what the sorting rule is. • Grow a plant in the classroom such as an amaryllis or a bean plant. Measure how much the plant grows on a daily or weekly basis, using standard units. • W ill It Fit? Show students a box and then point to an object that is on the other side of the room. Ask them if they think it will fit in the box and why they think so. Ask how they could measure before they try to put the item in the box. Vary the box size or use different types of containers. • E stimate: Have students estimate how many ‘footprints’ it would take to get from the door to the bookshelf. Then measure the distance with students’ feet. They can follow this up by measuring in centimetres or metres. The more students do this kind of estimation, the more reasonable their predictions will be. • Draw attention to two objects on either side of the room. Ask which one is longer or taller. Then measure using non-standard and standard units to find out. • P lay ‘I Spy’ using measurement clues, such as, “I spy something that is taller than the door, but shorter than the window.” • Q uick Scavenger Hunt: Give students quick challenges, such as, “Bring something to the meeting spot that is longer than 30 cm.” Measure the items collected and order them from shortest to longest. • P ose a daily riddle. For example: “I am shorter than 2 m but longer than 1 m. What can I be?” 148 Spatial Sense

Lesson Topic Page 1 What We Measure and Why: Math Counts: Length 150 2 How We Measure 157 3 Measuring Length and Height with Non-standard Units 162 4 Measuring Distance with Non-standard Units 169 5 to 7 Transitioning to Standard Units 173 5 The Centimetre 175 6 The Metre 178 7 The Centimetre–Metre Relationship 182 8 Using Benchmarks to Estimate and Measure 187 9 Using Benchmarks to Make Comparisons 192 10 Measuring with a ‘Broken’ Ruler 197 11 Guided Math Lesson: Cool Facts!: Bugs 201 12 Measurement Reinforcement Activities 206 Measurement 149

1Lesson What We Measure and Why: Math Counts: Length English Introduction to the Read Aloud Language Arts Learning Standards The Read Aloud text used in this lesson introduces math concepts in a meaningful context. Throughout the reading, students will apply their literacy Math strategies, such as making connections, inferring, and synthesizing Learning information, to understand the relevance of measurement in people’s lives. Standards They will also apply their math thinking skills to understand the characteristics of length, how length is measured, and the units that are used. First Peoples In the Math Talks, the attribute of width is introduced and students look for Principles of examples in the classroom. Learning 150 Spatial Sense Curricular Competencies • C omprehend and connect: Use developmentally appropriate reading, listening, and viewing strategies to make meaning; use personal experience and knowledge to connect to stories and other texts to make meaning • Create and communicate: Exchange ideas and perspectives to build shared understanding Curricular Competencies • 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 • 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 • C onnecting and reflecting: Reflect on mathematical thinking; connect mathematical concepts to each other and to other areas and personal interests • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) Possible Learning Goals • Begins to understand the concept of length and the related linear measurements of width, height, and distance in realistic contexts • Makes connections to measurement in their lives and explains why it is important to everyday living • Begins to understand the need for, and advantages of, standard units

Teacher • Explains or shows what length, distance, height, and width are and gives Look-Fors examples of each • Explains how measuring length is important in their lives • Begins to understand the difference between non-standard and standard units and can give examples of each • Gives reasons for selecting one standard unit depending on what is being measured Assessment Opportunities Observations: Note each student’s ability to: – M ake connections to measurement in their everyday lives – D istinguish between length, height, and distance – Discuss the difference between standard and non-standard units and give examples of each Math Vocabulary: About the mmmmdwieneieiisladatltMirstsamehuuanserrtc(,eethienr,c,e,tTeslrsahoont,laendtkkinigu:mgi,dlchWoealetemtr,hdrndaetegatsttlr,hle,,s, It is important to establish the relevance of measurement to students’ lives Is Width?) before students engage in measuring activities so they realize that measurement serves a purpose and that many decisions depend on the measurements made. For example, the size of the furniture a person might buy depends on the size of the room in which it will be used. As students learn about linear measurements (length, distance, height, width), they may not be able to distinguish among them. Regularly and incidentally providing realistic examples, and using the appropriate vocabulary, will help students refine their definitions and understanding. The Read Aloud provides excellent real-life examples of where, how, and why we measure length. Length and distance are similar, with length being “how far it is between the endpoints of the objects” and distance being “how far it is between any two points in space” (Clements & Sarama, 2009, p. 164). It is important to introduce distance as distinct from length, even though both are measured in much the same way. Students also need to learn related vocabulary, such as ‘near’ and ‘far,’ and to realize that these descriptions are relative. Giving students a reference point helps them to realize that the same distance can be far away from one perspective and close from another perspective. The Read Aloud touches on all of the above. As students transition from using non-standard units to using standard units, they need to realize that standard units are always the same, regardless of where or when they are used. In many countries, including Canada, the system of standard units used is the metric system. The Read Aloud introduces the metric system but also lets students know that there are other standard measurement systems used in the world. Measurement 151

Materials: Read Aloud: Math Counts: Length Summary: This non-fiction book introduces length and two related linear measurements, height and distance. Real-world examples are given, along with some ways in which the various linear measurements are made. Math Counts: Length by NOTE: There are more prompts than it is feasible to use in one session. Select Henry Pluckrose those that best suit your students’ needs and interests. You can revisit some of the pages and do more investigation, either in a second session or in a Math Talk. Text type: Non-Fiction: Description – Explanation Before Reading ball of string, rulers, metre stick or tape measure, picture of a speedometer Time: 25–30 minutes Connecting and Activating and Building On Prior Knowledge reflecting • Show the cover of the book. Ask students what they think the title that is Predicting/using prior knowledge written in cursive writing says. Ask what they think the word is made from. Ask what they think ‘length’ means and have them give some examples. Ask what else on the front cover has length. Ask which piece of yarn is the longest/shortest and how they know. • Setting a Purpose: Ask students what they think the book will be about and whether they think the book will be fiction or non-fiction. Have them justify their responses. During Reading Reasoning and Pages 4 and 5 analyzing (estimate • After reading the question on page 4, ask students about how long they reasonably) Understanding and think the string may be. Ask whether they think it would be longer than a pencil, a metre stick, or the length of the classroom. solving • Show students a small ball of string. Ask them to estimate how long they think Reasoning and analyzing (estimate the string is. Unravel the string until you have a piece that is as long as a desk. Ask how many more desks long students think it might be. Unravel more of the reasonably) string so it is now three desks long. Ask students if they would like to adjust their estimations. Keep doing this until the string is completely unravelled. • Ask how long students think the truck on page 5 is in comparison with the length of your classroom. Ask why they think it is important to know exactly how long the truck is. Communicating and Pages 6 and 7 representing • Read the text. Ask students to explain what length is in their own words. 152 Spatial Sense Have students show what the length of the pool is in the photograph. Ask why it is important to know the exact length of the pool for the race that is

Connecting and pictured. Ask how they would figure out the total length of a race if the reflecting/making competitors had to swim three lengths of a 100-m pool. connections • Ask students how they would measure the length of a race that is run on a Reasoning and curved track. analyzing • Ask students if they have ever watched a swimming or running race and Reasoning and analyzing (estimate what they noticed about how the winner is determined. reasonably) Pages 8 and 9 Understanding and • After reading the text, ask how the length of a highway and the length of a solving piece of fabric would be different. Ask students whether they would use the Communicating and same tool to measure the highway and the fabric and why. representing Pages 10 and 11 Making connections • After reading the text, ask why measuring with hand spans may not give Reasoning and analyzing exact results. • Have students estimate how many large hand spans long the table in the picture is and then how many small hand spans long it is. Together, measure by iterating the lengths of the two hands to confirm students’ predictions. Pages 12 and 13 • Read the text, then ask students which person will probably have the most exact measurement and why they think so. (e.g., The woman because she is putting one foot in front of the other with no space.) Ask which person will have the most number of paces when they measure the length of the lawn. Discuss why people’s paces may not be the best way to measure length. Pages 14 and 15 • Read both pages. Have students explain in their own words what a standard measure is. Ask what they think a non-standard measure is. Have them compare the two using examples from this book (e.g., the metres versus hand spans). Ask what standard measures we use in Canada. Tell them that we mainly use metres and centimetres. • Ask students what they think a surveyor does and what he is measuring in the picture. Ask if they have ever seen a surveyor with a tripod. • Ask what tools the architect might use to draw with exact measurements. Pages 16 and 17 • Ask what the tailor is measuring in the picture. Ask why it is important for his measurements to be exact. Ask whether the tailor is measuring in centimetres or metres. Show students a metre stick or tape measure. Ask how many centimetres are in a metre. Have students estimate how long their arms are in centimetres. Measure the arms of one or two students to confirm their estimates. • After reading page 17, provide students with rulers. Ask students what they notice and what they wonder. Draw attention to the distance of one centimetre and the little lines that subdivide it. Explain that the distances between the little lines are millimetres, and these are used to measure things smaller than Measurement 153

1 cm. Ask students what things might be smaller than 1 cm. Students can hunt around the room for some examples. Understanding and Pages 18 and 19 solving • Ask students whether they have seen road signs that show distances to other Making connections places. Explain that distance is just the same as length in that both are the length between two points. • Explain that the sign on page 18 is in a country that uses kilometres to measure distances. Discuss how 20 km in that country would be exactly the same distance as 20 km in Canada because the kilometre is another standard measure, like the metre and the centimetre. • Together, use a metre stick to measure out 10 m on the floor. Explain that students would need to do this 99 more times to make 1 km. Ask what might be 1 km away from the school. • Show students a picture of a speedometer and point out that it shows both kilometres per hour and miles per hour. Explain that miles are units of measurement used in the United States. If you cross the border, it is important to change your car’s instruments so they read in miles rather than kilometres. This is important so you know how fast you are driving and can stick to the posted speed limits. Communicating and Pages 20 and 21 representing • Read the text. Have students explain what height is in their own words. Ask what other things they could measure in the classroom that have height. Ask what is important to remember when you are measuring a person’s height so it is exact. • Ask students how they know that the basketball players were once the same height as all of the students in the class. Making connections Pages 22 and 23 Connecting and • Ask what tall buildings the students have seen and whether they have been reflecting inside them. Ask how they got to the top of the building. • Ask students what other things they have seen that are tall. • Ask students whether they would measure the heights of the building and the cliff in centimetres, metres, or kilometres and to give a reason for their choice. Reasoning and Pages 24 and 25 analyzing • Ask students why they think sea level is a good base or ‘starting line’ for Making connections measuring a mountain. (e.g., Sea level is the same all over the world so it is good for comparing heights above it.) • Explain that the highest mountain in Canada is Mount Logan and is located in the Yukon. It is 6000 m tall. • Explain that large planes can fly over 10 km above sea level. They fly so high because they get less air resistance there and therefore don’t need to use as much fuel. • Ask students if they have ever seen a mountain and what they thought about it. 154 Spatial Sense

• Ask students if they have ever flown on a plane. Ask how the cities and landscape look from that high above the ground. Connecting and Pages 26 and 27 reflecting • Discuss how the way we see the heights of other things often depends on our own height. Use the plants in the book as an example. (e.g., The plants appear short to the girl, but very tall to the lizard.) Ask students what is tall to them and short to them. Ask whether those items would be considered tall or short to an ant, a giraffe, or an adult. Reasoning and Pages 28 and 29 analyzing • Read both pages. Ask students why the lumberjack is measuring the length rather than the height of the tree. Discuss whether the measurement of the height and length would be different and why they think so. Communicating and Pages 30 and 31 representing • Have students explain, in their own words, the difference between length and height. After Reading Synthesizing • Together, make a list of all the linear measurements that were mentioned in the book (length, distance, height). Create a short definition for each and draw a visual to support the definition. • B uilding Growth Mindsets: Explain that over the next several lessons, students are going to study measurement in more detail. This year, in grade two, they are going to study linear measurements, but in years to come, they will also learn about other measurements such as mass, capacity, volume, and time. They may not know all there is to know about measurement yet, but with time and effort, they will learn to measure many things that are in their daily lives. Materials: Math Talk:What Is Width? desk Math Focus: Introducing width Teaching Tip Let’s Talk Integrate the math talk moves (see Select the prompts that best meet the needs of your students. page 7) throughout Math Talks to • W e just finished this book on length. What are we measuring with length? (e.g., maximize student participation and how long something is, how high, how far) active listening. • W hat are some questions that you might ask during the day that would make you want to measure the length of something? (e.g., How long does this ribbon need to be to wrap this present? How long is my foot so I can get new shoes? How tall am I? How far is it to the store? How many books will fit across my desk?) Brainstorm and record as many ideas as possible. Let’s decide what continued on next page Measurement 155

Materials: w e would be measuring in each of these cases: length, height, or distance. Print Optional: picture ‘length,’ ‘height,’ or ‘distance’ beside each example. illustrating scenario • If we wanted to cover the top of a desk with paper, what would we need to 156 Spatial Sense know? (e.g., the length of the desk) This is a good start. We need to cut a piece of paper so it is the same length as the desk. Who can show us the length of the desk? What else do we need to know if we want the paper to fit the desk exactly? (e.g., how wide the desk is) This measurement is known as the width. When we measure the longer side of the desk, we measure the length, and when we measure the shorter side of the desk, we are measuring the width. Would knowing the height of the desk help us to make our cover? Why? So we can make our cover if we know the length and the width. • W hat other things in the room do you see that have width? Let’s add ‘width’ to the list of linear measurements we started during our lesson and create a definition for it. Math Talk:What Do We Need to Measure? Math Focus: Distinguishing among linear attributes/measurements Let’s Talk Select the prompts that best meet the needs of your students. • P resent the following scenario or vary it as needed to suit the configuration of your classroom: I want to put a bookcase between the two windows. What do we need to know to find out if the bookcase will fit? Turn and talk to your partner. Alternatively, you could illustrate the scenario with a picture. • W hat do we need to know? What do we need to measure? (e.g., You need to measure the distance between the windows; you need to measure the bookcase.) • W hat do we need to know (measure) first? Students can discuss determining the distance between the windows and then comparing that distance to the length (or width) of the bookcase. • W hat attributes of the bookcase do we need to measure to see if it fits? (e.g., You need to measure the length of the bookcase. You could measure the height of the bookcase to be sure it won’t cover the windows.) • Is there anything else I should measure to be sure the bookcase fits? What measurement(s) of the bookcase are not important to know? (e.g., depth, mass) Why? • L et’s measure our bookcase and the distance between our windows. First, let’s predict whether it will fit.

2Lesson H ow We Measure Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make First Peoples connections; estimate reasonably; model mathematics in contextualized Principles of experiences Learning • U nderstanding and solving: Develop, demonstrate, and apply Teacher Look-Fors mathematical understanding through play, inquiry, and problem solving; visualize to explore mathematical concepts • Communicating and representing: Communicate mathematical thinking in many ways; explain and justify mathematical ideas and decisions • Connecting and reflecting: 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 • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) Possible Learning Goal • Explores the concept of linear measurement, what can be measured in their everyday lives, and the tools used to measure • Identifies some tools that are used to make linear measurements Previous Experience About the with Concepts: Students have discussed Now that students have been introduced to and explored some measurable measurable attributes attributes (Lesson 1), it is important that they investigate the tools used to such as length, height, measure. John Van de Walle and LouAnn Lovin describe measuring as and distance and why ‘filling’ or ‘covering’ or ‘matching with a unit of measure with the same measurement is attribute’ and say that “measuring instruments such as rulers, scales, important. protractors, and clocks are devices that make the filling, covering, or matching process easier” (Van de Walle & Lovin, 2006, p. 224). They give the example of a ruler that lines up the units of length and numbers them so they do not need to be counted. It is important to highlight some of the technological advances that have been made in measurement over the years. For example, students may have seen step counters or a GPS, but they may not know about laser measuring tapes, which are especially used in construction. continued on next page Measurement 157

Math Vocabulary: It must be noted that even though students may be able to read a ldcesionhsmgotatrphnt,a,crseheh,e,oielgroshtnettig,rm,,watlaoitdelnlt,,ghe, r, measurement on a tool, it does not mean they understand what the taller, measurement means. It is important to help students connect the skill of measuring tool measuring with a conceptual understanding of measurement. About the Lesson In this lesson, students explore the concept of linear measurement, the attributes that can be measured, and the tools that are used in measurement. Through discussion, students will recognize that measurement is an important part of their everyday lives. Materials: Minds On (15 minutes) “How We Measure” • Review what measurement is and why we measure. Ask students what they (pages 12–13 in the Spatial Sense big book); learned about measuring length from the read aloud book in the last lesson concrete materials for (Length). selected pictures (details below) • Show students the “How We Measure” pages in the Spatial Sense big book. Time: 50–55 Read the title. Pose some of the following questions to launch the discussion: minutes per – W hat do all of these objects have to do with measurement? (e.g., They all session measure something.) Are they things that are being measured or things that are used to measure? Why do you think so? Materials: paper clips, – W hat measurement tools are familiar to you? Where have you seen them? pencil – H ave you used that measurement tool before? 158 Spatial Sense – W hat did you measure when you used it? Working On It (Whole Group) (20–25 minutes) NOTE: There are more photographs and prompts than can be used in one session. Select the ones that best suit the needs and interests of your students. You may decide to discuss two or three photographs only and revisit others in another session. Paper Clips (page 12) • Look at the paper clips. How can they be used to measure something? How could you use paper clips to measure a pencil? Let’s try it with this pencil and real paper clips. What is important when we are measuring with paper clips? (e.g., start at the beginning of the pencil and finish at the end; use paper clips that are the same size; place the paper clips end to end and don’t leave any gaps) What are we measuring when we measure the pencil this way? (e.g., length; how long it is) What would happen if someone measured the same pencil with longer paper clips? What else could we use to measure the pencil? (e.g., a ruler, erasers) What other lengths could we measure with paper clips? What lengths might not be a good idea to measure with paper clips? Why? What else on this page could you use to measure the length of a pencil and always get the same answer?

Materials: Girl with Arms Outstretched (page 13) ruler, pencil • How could this girl use her body as a measuring tool? (e.g., She could measure with her arm span, with her hands, by taking footsteps.) What might she measure with her arms outstretched? (e.g., the width of a window) What part of a wall could she measure? (e.g., She can measure across the wall.) What do we call that measure? (e.g., width) How is measuring width different from measuring height? Would it be practical for her to measure the height of the wall with her arms? Why? How would the measurement of the width of the wall be different if another person measured with his/her arm span? (e.g., The two people would get different measurements because their arm spans would be different.) Wooden Ruler (page 13) • What do you notice about this ruler? What units is it divided into? They are called centimetres. How would you use this ruler to measure the length of a pencil? (e.g., line up one end of the pencil at the 0 and then find out at which number the other end of the pencil lines up with or is closest to; count the spaces from 0 to the end of the pencil) Why must we start at 0 and not at the other end of the ruler (15)? Let’s try it with a real ruler and pencil. Our rulers have a small space between the end of the ruler and 0. What do we need to do? (e.g., place the end of the pencil at 0) What would the length be if you measured the pencil with another ruler that is in centimetres? Why do you think so? Measuring Tape and Pants (page 13) • What is being measured in this picture? (e.g., the width of a pant leg) How is this different from measuring the length of the pant leg? What is the tool being used to measure? What do the numbers on the measuring tape mean? What else besides width can you measure with a measuring tape? Yarn (page 12) • How can a ball of yarn be used to measure length? What kinds of things could you measure the length of? When would it be better to measure with yarn instead of with a plastic or wooden ruler? (e.g., when measuring things that are curved, coiled up, or not straight) Explain how you would use the yarn to measure the width of this desk compared to the width of my desk. (e.g., Stretch the yarn the length of the teacher’s desk and then see if that piece [or length] of yarn is longer than the student’s desk.) What would we do if we wanted to know how much longer the width of my desk is? (Line up the length of the yarn to something that can be counted, like paper clips, or to a measuring tape and find out how many units long each desk is.) Laser Measuring Tool (page 12) • How do you think this measuring tool works? It is known as a laser measuring tool. It sends a laser beam straight to a target and, by using how much time it takes for the beam to be reflected back, it figures out the distance between the tool and the target. It is good for measuring distances from far away. What other ways do you see a laser measuring tool being Measurement 159

helpful? What can the laser measuring tool not measure? (e.g., It relies on straight laser beams so it can only measure distances that are straight and not distances or lengths that are bent, or curled up.) Girl Measuring Her Height Against a Picture (page 12) • What is the girl trying to do? What is she measuring? • How is she measuring? What do we know about the girl’s height by looking at the picture? (e.g., She is shorter than the drawing of the giraffe.) • If we really wanted to know the height of the girl compared to the height of a real giraffe, what would this picture have to look like? (e.g., The drawing of the giraffe would have to be at the actual size of a real giraffe; the girl and the drawing would have to be standing on the same level. Or, we would have to see the girl standing next to a real giraffe.) Do you think that the girl would be shorter than she appears in this picture if she were standing beside a real giraffe? Why? • What other measuring tool is the girl using? (e.g., lines that are evenly spaced) Do we know how much space is between each line? (e.g., No, so they might just be ‘made-up’ units.) How could we know how many lines tall she is? (e.g., We would need to count the lines from the bottom.) Where would the lines have to start? What makes counting the lines a little easier? (e.g., Every fifth line is longer, so you could count them by 5s.) What would make it even easier to measure the girl’s height? (e.g., having numbers beside the lines) Doctor Measuring Child’s Height (page 13) • What is the doctor doing? Have you ever had your height measured at the doctor’s office? What was it like? How do you think the tool that the doctor is using works? What do you notice on the measuring tool? (e.g., There are little lines and numbers.) What might the units of measure be? How exact do you think this measurement tool is? Why? How does this compare to the way the girl with the drawing of a giraffe is measuring herself? How is this tool easier to use? (e.g., You don’t have to count every line to find out the height because there are little numbers beside the lines.) Step Counter (page 12) • What do you think this instrument measures? Have you ever seen one of these instruments before? Where? Why do people use it? This is a big number [read the number to students]. It is much bigger than one hundred. To take that many steps, you would have to take 100 steps and then do that over 100 more times! Do you think you would reach the principal’s office with over eleven thousand steps? Do you think you would go off school property? We are measuring the length travelled with this tool, but what do we often call this measurement? (distance) When do we want to know distance? Do you think that if two people took the exact same number of steps in the same direction they would end up in the same place? Why? GPS (page 13) • What is this instrument? Have you seen it before? Where? Who uses it and why? What does the instrument do? (e.g., It gives you directions; it tells you 160 Spatial Sense

how to get to different places.) Why is measuring distance important when giving directions? (e.g., You need to know how far to go in one direction before you turn in another direction.) What units does the instrument use when it gives distances? (e.g., metres, kilometres) We can see that the GPS says to go 200 m before turning left. Why is it important to give the distance in metres rather than footsteps? Differentiation • You may want to group and discuss the tools pictured according to how they are used. (e.g., tools that measure length/width, tools that measure distance) If so, keep in mind that some tools are more versatile than others and will fit in more than one group. (e.g., the step counter can measure only steps/ distance, the ruler can measure length, width, and height) Assessment Opportunities Observations: Since this is an introductory lesson, it is best to listen to what students are saying to determine what they know, what vocabulary they use, and any misconceptions they may hold. Use what you learn to plan next steps forward and to identify students who may need extra input and reinforcement. Consolidation (15 minutes) • Make a list of tools that can be used to measure length and the units each tool uses. Emphasize that a measurement that makes sense needs to include a number and the kind of units used (e.g., 5 steps, 3 m, 12 small paper clips, 25 cm). • Discuss which tools might produce different results, depending on the size of the tool (e.g., paper clips, arm spans). • Ask students what length or object might be 10 shoes long. Ask whether students would need more or fewer shoes to measure the same length/object if the shoes are larger. Make note of any students who understand the relationship between the size of the measuring unit and the number that are needed to measure a length (i.e., the larger the unit, the fewer are needed). • B uilding Growth Mindsets: Ask students what else they wonder about when it comes to measuring and whether there are other measuring tools that they have seen and are curious to know more about. Explain that they will be studying measurement over the next couple of weeks and have lots to learn, and that being curious about things helps them learn. Further Practice • Throughout the year, point out other tools that are used to measure different attributes or characteristics of objects. For example, you may show a scale when students are working in science. Although students don’t learn about mass until grade three, it is good for them to become aware of other attributes that can be measured. Measurement 161

3Lesson Measuring Length and Height with Non-standard Units Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make First Peoples connections; estimate reasonably; model mathematics in contextualized Principles of experiences Learning • Understanding and solving: Develop, demonstrate, and apply Teacher Look-Fors mathematical understanding through play, inquiry, and problem solving; visualize to explore mathematical concepts 162 Spatial Sense • Communicating and representing: Communicate mathematical thinking in many ways; explain and justify mathematical ideas and decisions • Connecting and reflecting: 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 • Direct 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 • Measures the length and height of objects using two different non-standard units to investigate the relationship between the number of units needed and the size of the units • Explains the similarities and differences between height and length and how they can be measured • Chooses two appropriate non-standard units for the objects being measured • Measures by laying units end to end with no gaps or overlaps to produce a reasonably accurate measure • Understands that a measurement includes a number of units and a type of unit • Uses suitable language to express approximate measurements, e.g., in situations where the non-standard unit they are using to measure an item doesn’t line up perfectly with the end of the item, but is a little long or a little short • Explains or shows that the smaller the unit, the more units are needed to measure the same length • Accurately measures by iterating a single unit and keeping track of the number of units (Math Talk)

Math Vocabulary: About the hlteoeasnillgtgeihemr,tr,,attlaleoellnn,eggmsetthe,s,atu,lsnoutinatr,gell,, Now that students have been introduced to linear measurements and measuring tools, it is essential that they actually engage in a variety of measurement activities so they develop skills alongside conceptual understanding. According to Clements and Sarama, students measure length by “identifying a unit of measure and subdividing (mentally and physically) the object by that unit, placing that unit end to end (iterating) alongside the object” (Clements & Sarama, 2009, p. 164). They also need to keep track of the number of units used. They are now connecting a numerical value and a unit to their measurements. It is important to begin by measuring with non-standard units so students understand the process of putting multiple uniform units end to end with no gaps or overlaps as well as accurately iterating a single unit across a length. Measuring in this way helps students understand that each unit takes up space alongside the object, which helps them to realize, when they transition to using rulers, that it is the spaces between the numbers that are counted and not the actual numbers. Using non-standard units also helps students grasp the big idea that the larger the unit, the fewer of them you need to measure the same length; if you have a smaller unit, you will need more of them. Clements and Sarama caution that early measuring experiences with several different units can confuse children (Clements & Sarama, 2009, p. 166). While children are grappling with the concept of measurement, it is important to use a few non-standard units that clearly mimic or reflect the attribute being measured (e.g., using toothpicks to measure length and not square tiles). About the Lesson In Minds On, students review and compare the attributes of length and height and differentiate them from other measurements that are shown in a picture. In Working On It, students select objects to measure the lengths and heights of using two different non-standard units. Measurement 163

Materials: Minds On (20 minutes) “What Is Tall? What Is Show students the “What Is Tall? What Is Long?” pages from the Spatial Sense Long?” (pages 14–15 in big book. Use some of the following questions to engage students in discussion. the Spatial Sense big book), a rectangular • How do we decide how long something is? sheet of paper, a few • Which item in this picture looks long? What might look short even though it materials to use as measuring is really long? Why? (e.g., Looks can be deceiving—we can’t tell how long tools (e.g., something is if it isn’t straightened out, or if we see it at an angle.) Can you paper clips, think of something else that is long but doesn’t look long? (e.g., The skipping toothpicks, rope can be long but it might not look long because it is curled up.) straws), BLM 12: • Would you call the table on page 14 long? (e.g., It is long compared to the Estimates and length of the sign on the tree but not long if we compare it to the length of Measurements the fence.) Time: 60 minutes • How do you decide if something is longer than something else? (e.g., line 164 Spatial Sense them up end to end and see which is longest; measure each with a non- standard or standard unit of measurement) • Do we usually say “the tree is long?” (e.g., No, the tree is tall; we use the attribute of height when we think about the measurement of a tree.) How would you describe the height of the scarecrow compared to the height of the tree on page 14? How would you describe the height of the pine trees compared to the height of the skyscrapers behind them on page 15? • When do we want to measure something very carefully and exactly? (e.g., when you are measuring your body for clothes such as pants, so the pants are the right length for your legs) When is an approximate measure okay to have? (e.g., if we want to know whether a desk will fit through a door) • What is the difference between length and height? (e.g., Discuss how length is often a horizontal measurement and height is often a vertical measurement.) What is the difference in the way we measure them and the tools we use? • [Show students a rectangular sheet of paper with the longer side placed horizontally.] Let’s measure the length of this paper. What is the length? What tools could we use? (e.g., paper clips) Can we use different sizes of paper clips? Why? How can we place the paper clips to measure the length? What do you notice about how I lined the paper clips up? (e.g., There are no gaps or overlaps.) What is the length? (e.g., 15 paper clips) • What if we want to measure the height of this desk? What is the height— show me with your hands. What could we use to measure it? (e.g., our hand spans) Why might hand spans be a better tool than paper clips? (e.g., The paper clips would fall down unless we taped them or strung them together; we would need a lot of them.) What do you notice about how this student measured the height of the desk? Working On It (Whole Group) (20–25 minutes) • Students work in pairs to estimate and measure the lengths of two objects (with different lengths) and the heights of two objects (with different heights). Invite pairs to choose four suitable objects from around the

classroom. Then have them select two different non-standard measuring units to measure with. • Students should first estimate and then measure the lengths with one unit, and then estimate and measure the lengths with the other unit. They do the same for the heights of their other two objects. • Students can record their estimates and measurements in the tables on BLM 12. Differentiation • For students who find it challenging to choose appropriate non-standard units, consider assigning the two units they use. Be sure the units are different in size, for example, paper clips (smaller unit) and straws (longer unit). • Some students may benefit from having the attribute in question (length or height) highlighted physically on the object they are measuring, for example, by placing masking tape along the height of a toy or the length of box. • In light of the research (see About the Math, page 163), you may want to restrict the available units to those that mirror linear measurement (e.g., toothpicks, straws, pencils, or paper clips, but not square tiles or buttons). • You can give students who need more of a challenge a limited number of units (e.g., only three paper clips) and have them devise a way to measure without having enough units to cover the entire length or height. Assessment Opportunities Observations: Pay attention to how students use units of measure. Are they using the same type of unit consistently each time? Are they placing the unit so there are no gaps or overlaps to ensure accuracy? Are they assigning a number of units to the length of each object? Are they using approximate language (‘about’) when they measure the length of an item and the last unit falls just a little short or a little long of the end point (some measurements do not result in a whole number of units). Conversations: While observing students as they measure, consider asking some of the following prompts to elicit thinking: – What was the length of your first item? – W hat unit did you use to measure with? Why did you choose this unit? – Which item is longest? How do you know? – W hat changed when you used a different unit? What stayed the same? – Is there a way to find the length of your items with only one of the units you measured with (instead of a lot of them)? Show me what you might do to find the length. (This begins the conversation on unit iteration.) continued on next page Measurement 165

– W hy is it important to use the same unit when we measure? (e.g., trying to be as accurate as possible in our measurement, wanting to compare lengths using the same unit) – W hat part of the item are you measuring? (e.g., Be sure students are focused on the length of one side of the item.) – What was the difference when you measured with the first unit rather than the second unit? – H ow do the items compare in length? – I noticed your last unit went over/past the length of the item. What did you do about that? – Would you say your measurement is accurate? Explain. – What if I wanted to measure the [width or height] of this object? Where would I measure? Consolidation (20 minutes) • Have pairs meet with another pair to discuss their findings. They take turns showing their items to the other pair so the students can estimate how long or tall they think the items are. They can measure each other’s items to confirm their measurements. • Meet as a class. Strategically select one pair to share with the class how they measured their items. If they iterated multiple units, discuss the importance of laying the items end to end with no gaps or overlaps. • Strategically select one group of two pairs to discuss how accurate their estimates were and whether their estimate using one measurements unit affected their estimate using the second measurement unit. • Strategically select a pair that measured with two units that varied significantly in size. Discuss whether they needed more or fewer of a unit if it was longer/shorter than the first unit they used. Highlight the big idea that the shorter the unit, the more units are needed, and the longer the unit, the fewer units are needed. • Discuss whether the two units that pairs chose were practical for measuring all of their chosen items. (e.g., The paper clips worked for the book, but they were too little to measure the length of the carpet.) • Ask, “How can you report to another group the length or height of your item?” Emphasize that their response needs both a number and the name of the unit. (e.g., The pencil is about 7 paper clips long.) 166 Spatial Sense

Materials: Math Talk: Measuring with a Single Unit one paper clip for each pair (half Math Focus: Iterating with one unit receive a small paper clip, half receive a large Let’s Talk paper clip); little book versions of the Spatial Select the prompts that best meet the needs of your students. Sense big book; sticky notes • W hat were some of the tools or units that we measured with in the lesson? How did we use our tools, let’s say paper clips, to measure carefully? (e.g., We put them end to end with no gaps and no overlaps; we counted the number of paper clips carefully.) If we work carefully, we should get an accurate measurement. Accurate means that the measure is correct and fairly exact. It is hard to get perfectly exact measurements, but we want to get as close as possible. • W hat if we only had one paper clip? Would we still be able to measure? Think about how you might do that. Partner Investigation • W ith your partner, you are going to measure the length and width of the cover of the little books. You will also measure the length of the book when it is opened up. You can measure with one paper clip. I am going to give half of you one size of paper clip, and the other half a different-sized paper clip. Make sure that you estimate first! You can record your estimates and measurements on a sticky note. Then meet with another pair that used the same size of paper clip to compare your results. Follow-Up Talk • H ow did you measure to make sure you got an accurate measurement? (e.g., We put our finger to mark where the paper clip ended so we knew where to put it down again; we put a little pencil mark on the sticky note so we knew where it would end; we flipped our paper clip over and over.) • H ow did you keep track of the number of paper clips? (e.g., We counted them as we went; we put down a mark on the sticky note for each paper clip.) • H ow did you count the leftover space at the end? • D id you get the same answer as the other pair? • L et’s look at the two sizes of paper clips. Which one would produce a larger answer? Why? Further Practice • D esk Measure: Students can measure the length of their desks using two different units. They can measure using paper clips and new pencils, and then compare the number of paper clips and pencils needed to measure the length of the same desk. • Top 3: Students work with a partner. Together, they choose three objects to measure that are very different in length. Students discuss which unit it Measurement 167

would be best to measure each object with, explaining their reasoning. First they estimate the length of each object, then they measure, and finally they compare their measurements with their estimates. While observing students as they work, ask some of the following questions to further their thinking; – W hich unit did you choose and why? – H ow did you estimate the length of each object? – W hen you measured each object, what did you notice? Were the objects very different in length? – Were your estimates close to your actual measurements? Why or why not? – Consider having students record their estimates in marker or pen so that they can think strategically about their estimate and not change it once they have chosen it. This will reinforce to students the importance of estimation, and that it is in fact a strategic choice and not a random guess. • Step Ahead: Students work with a partner. Each student takes turns taking one step forward. Students measure the length of the step (or stride), from the tip of the toe on their back foot to the back of their front foot. Students choose an appropriate non-standard unit to measure the length of each other’s footsteps and to compare them. Next, they estimate how long they think a giant step would be using the same unit of measure. Students can then take a giant step and repeat the same process (measure the lengths of their steps and compare them). 168 Spatial Sense

4Lesson Measuring Distance with Non-standard Units Math Curricular Competencies Learning Standards • R easoning and analyzing: Use reasoning to explore and make First Peoples connections; estimate reasonably; model mathematics in contextualized Principles of experiences Learning • U nderstanding and solving: Develop, demonstrate, and apply Teacher Look-Fors mathematical understanding through play, inquiry, and problem solving; visualize to explore mathematical concepts Previous Experience with Concepts: • Communicating and representing: Communicate mathematical thinking Students have a basic understanding of the in many ways; explain and justify mathematical ideas and decisions attributes of length, width, and height and • Connecting and reflecting: Connect mathematical concepts to each other have measured in non-standard units. 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 • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) Possible Learning Goals • Accurately measures a distance using body parts and records the distance with a number and a unit • Begins to understand the need for standard units in order to reproduce results in other situations • Selects a body part as an appropriate unit for the chosen distance • Uses a consistent method for measuring, with no gaps between units and no overlapping units • Accurately keeps track of the number of units used • Expresses the distance using a number and a unit (e.g., 7 arm spans) • Describes or shows the method for measuring and proves that it gives a reasonable measure of the distance • Compares measurements made by the group and gives reasons for differences among them • Explains or shows why standard units can be better than non-standard units like body parts Measurement 169

Math Vocabulary: About the ldeinsgtathn,cew,idnteha, r, far, close to While a distance from point A to point B remains constant, students may interpret the distance from their own perspective only. For example, one student may describe the teacher’s desk as being a short distance away, while another student at the other end of the room perceives the desk as being far away. It is therefore important to give students a reference point to work from and to include those reference points in their verbal descriptions. Students often find it convenient to use body parts to measure because they are readily available and because students can easily iterate two of the same measure, such as feet or hands, across a distance. This is a good opportunity to discuss how these particular units can vary from person to person and to thus establish the need for standard units (to be able to reproduce results). About the Lesson Students work in groups of three or four to measure using body parts the distance to a chosen object in the classroom. All students in the group will measure with the same body parts (e.g., feet) so their findings can be compared. This offers an opportunity to assess students’ techniques for measuring and whether they are accurately able to iterate units across a distance. In the Consolidation, students discuss the need for standard units in order to reproduce the same results in other situations and over time. The lesson can also be done in a larger indoor space (e.g., gymnasium) or outdoors (e.g., schoolyard, playground). Materials: Minds On (15 minutes) objects from around the room, half • Ask students to look for something that is ‘very close’ to them in the piece of chart paper for recording classroom. They can discuss their choices with a partner before discussions Time: 50 minutes with the class. Ask how they know the object is very close. Ask whether their chosen object is very close to all students in the classroom, or whether it is 170 Spatial Sense closer to some than to others. • Repeat this line of questioning by having students locate something that is ‘far away’ from them in the classroom. • Ask how they might measure the distance between themselves and these objects. Ask what body parts they could use to measure the distance to the very close object and what body parts would be suitable for measuring the distance to the object that is far away. Working On It (15 minutes) • Students work in groups of three or four. Have them choose an object in the room and then decide on a body part as their unit of measure. Each student in the group takes a turn measuring the distance to that object with the same body part.

• Give students half a piece of chart paper so they can record their findings and how they carried out their measurements. Differentiation • Students with limited verbal counting ability may be encouraged to select an object that is not too far away. • Students who need more of a challenge can measure the distance more than once, using a different body part each time. Assessment Opportunities Observations: This is a good opportunity to observe rather than carry out conversations. From your observations, you can discern which students may need more experiences measuring with non-standard units before moving to standard units. Pay attention to students’ ability to select a suitable body part as a unit, to accurately measure with that unit, and to keep track of the number of units in the distance. – Are they using a body part that is appropriate for the measurement? – Are they accurately iterating the body part with no gaps? – How are they keeping track of the number of body parts? – Can they measure the distance in a reasonably straight line? – How are they handling any distance that is “left over” (i.e., if the last body part iterated doesn’t quite reach the object)? – A re they recording their distances with a number and a unit? – W hat reasons do they give for any discrepancy in the measurements from person to person? Consolidation (20 minutes) • Ask what each group’s distance was and the body part(s) the group members used to measure it, without discussing the actual results. Have students justify why they chose to use that particular body part as a unit of measurement. • Discuss some of the strategies students used to ensure their measurement was accurate. Ask how they dealt with any leftover distance that did not exactly line up with the end of the last unit. • Select a group that measured a short distance and have them share their results. Ask whether each student in the group got the same results and why they think they did or did not. Ask which student in the group produced the measurement with the most units and which student produced the measurements with the least units and how they think this might have happened. • Select a group that chose a long distance and have them share their results, including a comparison of their results. Measurement 171

• Ask the class why the results for each group member were not the same. Ask how this could be a problem. Give an example of giving someone directions in the class to locate a specific item. • Explain that this is why people have developed standard units that are the same all over the world, and that don’t change from place to place or from time to time. Tell students that these units are known as standard units and that other units, like feet or hands, are known as non-standard units because they are different from person to person. • B uilding Growth Mindsets: Ask what other non-standard units students have used to measure. (e.g., paper clips, new pencils) Have them brainstorm other appropriate units. Highlight how many different ideas they can come up with when they work as a group. Explain that mathematicians often work together because they often come up with new ideas when they share their thinking. • Tell students that there is lots to learn about measurement and with work and time they are going to learn more about standard units. Explain that, like non-standard units, there are lots of different standard units and they will learn about new ones all through their years at school and beyond. 172 Spatial Sense

5 7Lessonsto Transitioning to Standard Units Math Curricular Competencies Learning Standards • R easoning and analyzing: Use reasoning to explore and make Previous Experience connections; estimate reasonably; model mathematics in contextualized with Concepts: experiences Students have measured lengths/distances using • U nderstanding and solving: Develop, demonstrate, and apply non-standard units, either by iterating mathematical understanding through play, inquiry, and problem solving; multiple units or a single visualize to explore mathematical concepts unit, and have recorded their answers with a • Communicating and representing: Communicate mathematical thinking number and a unit. in many ways; use mathematical vocabulary and language to contribute to First Peoples mathematical discussions; explain and justify mathematical ideas and Principles of decisions Learning • C onnecting and reflecting: 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 • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) About the Van de Walle and Lovin stress that the use of non-standard units early in the study of measurement is beneficial at all grade levels because it provides a good rationale for using standard units when, for instance, a class has measured the same objects with their own non-standard units and arrived at different and sometimes confusing answers (Van de Walle & Lovin, 2006, p. 227). Van de Walle and Lovin further add that “the move to standard units should be guided by how well you believe your students are developing an understanding of the measurement of that attribute” (Van de Walle & Lovin, 2006, p. 227). continued on next page Measurement 173

Math Vocabulary: Centimetres or Metres? Where to Start cvlseoomishqnmueougaptr,artlael,ilzor,iseennch,,gelomeeenrsrntet,teigaimlmrtos,henuas,tgrthreeeeo,,srtt,est, Marian Small discusses whether metres or centimetres should be introduced first when delving into linear measurement with standard units. Introducing the metre first can be advantageous “because measuring in metres results in fewer errors due to problems with small motor coordination” (Small, 2009, p. 376). She adds that educators may choose to introduce centimetres first since measuring smaller objects can be more convenient for getting all students working on measurement activities at the same time. It is a matter of choice depending on the materials available and the needs of your students. While this resource introduces the centimetre first, you may decide to begin with the metre. If so, you can do Lesson 6 before Lesson 5. Small describes how measuring with centimetre cubes can help students transition from non-standard to standard units since the cubes, like non- standard units, can be lined up along the length of the object, but the measurement can be reported in standard units, centimetres (Small, 2009, p. 376). It is important that students can iterate multiple uniform units, as well as a single unit, along a length. You can decide which approach to focus on in the following lessons, depending on which strategy requires more reinforcement. Alternatively, you can have students select their own strategy. Conservation of Measure As students transition from non-standard units, they need to be sure that a standard measure such as a metre will be the same, whenever and wherever it is used. Similarly, the lengths of objects remain the same. This involves understanding conservation of measure, which Marian Small describes as being able to “recognize that the measurement attributes of an object do not change, even if the object is moved or parts of the objects are separated and rearranged” (Small, 2010, p. 42). It takes time and several experiences to fully understand this concept. Small cautions that it can be difficult to discern whether students have acquired mastery since they can participate in related tasks and get correct answers without realizing or knowing that the same results would be obtained with the same standard unit at a later time. It is beneficial to have students predict whether they would achieve the same results the next day if they used the same standard measure. 174 Spatial Sense

5Lesson The Centimetre Teacher Possible Learning Goal Look-Fors • Transitions from non-standard units to a standard unit, centimetres, by measuring length and width using centimetre cubes • Explains or shows how centimetre cubes are standard units and compares them to non-standard units such as paper clips or hand spans • Gives a reasonable estimate for the measurement of a print, using centimetres as the unit, and explains their reasoning • Accurately measures the animal print with centimetre cubes by lining them up end to end • Accurately counts the centimetre cubes, either as they line them up or afterwards • Reports the measurement using a number and a unit • Differentiates between length and width by showing examples of each About the Lesson In this lesson, students first estimate and then measure the length and width of animal prints using centimetre cubes and then report their findings in centimetres. Materials: NOTE: Since this is a long lesson, you may decide to do the Consolidation the next day. “Animal Prints” (pages 16–17 in the Spatial Minds On (20 minutes) Sense big book and little books), small • Show the big book pages full size and ask, “What do you see? What do you paper clips, ruler, centimetre cubes, think made these prints? Have you ever seen animal prints before? Have you sticky notes ever made tracks in the snow with your boots or on pavement with wet feet/ or BLM 13: shoes? What can you learn about an animal from looking at its prints?” Animal (e.g., how big it is; where it is going, where it has been; where it may live, eat, Prints or get water; whether it is following another animal) Time: • Reduce the whole image to be about 43 cm high and explain to students that 60–65 minutes the prints are shown to scale—they are the size that they would be in real life. If a real red fox, house mouse, or giant panda stepped in some mud and then stepped on a piece of paper, this would be the size of its paw print. • Ask students which prints, if any, they recognize. Ask which prints look the most interesting and what makes them interesting. Which prints do students wonder most about? Measurement 175

• Ask, “If we wanted to compare and order the prints by size, what could we measure? (e.g., length and width) What tools could we use?” • Have students choose two prints to compare, and ask them what tool they would use to measure the prints and why they think that tool is a good choice. Suggest using small paper clips. Ask students to estimate how many paper clips would be needed to measure the length of the two prints, and have them share their thinking with a partner. • Record some of the students’ estimates, then invite one or two students to come up to the whiteboard and measure the first print. After they have done so, ask the other students: “What do you notice about how these students measured? Do you think we would get the same answer if other students measured with the same paper clips? So it is important that when we compare size that we use the same tools and use them in the same way.” Repeat with the second print and compare the two prints. • Ask, “Do you think students in another classroom would get the same answers? Why or why not? (e.g., They wouldn’t if they used different- sized paper clips.) What could students in any classroom anywhere in the world do so that when we measure these prints we would get the same answers?” (e.g., We could use centimetres.) Say, “We have talked about centimetres. Look at these little cubes.” Hold up some centimetre cubes and continue: “They are 1 cm long. I can line them up beside a ruler to show you. [Do so.] It is divided into centimetres and you can see how the cubes fit to match them. Today we are going to measure the animal prints using centimetre cubes.” • Distribute copies of the little book, and ask students to compare the prints in the little book to the prints in the big book. What do students notice? (e.g., They are prints made by the same animals but the whole page has been shrunk.) “So these prints in the little book are not the actual size, but since all of the prints have been shrunk, will the biggest print still be the biggest? Why?” Working On It (Whole Group) (20–25 minutes) • Students work in pairs to measure the lengths or widths of three to five prints using centimetre cubes. Then they order the measurements from smallest to largest. They can record their measurements on sticky notes (one per animal), order the sticky notes from smallest to largest, and then number them from 1 (so the largest print gets the largest number). Differentiation • You may provide size-as prints of familiar local animals for students to measure and sequence. • You may decide to have all students measure the same attribute, length, to allow for a full class discussion. • You may decide to have students record their findings in a table (see BLM 13) rather than on sticky notes. 176 Spatial Sense

Assessment Opportunities Observations: Use the Teacher Look-Fors as you observe students at work. Conversations: Use any of the following prompts as needed: – W hat attribute are you measuring? Is it length or width? How do you know? – H ow did you estimate the length of this print? Why do you think this is reasonable? Estimate how long you think this print is. How can knowing length of the other print help you with estimating? Put down two cubes. Do you want to change your estimate? – How are you keeping track of how many centimetre cubes you are using? – Y ou have put down 4 as your length. Is that 4 paper clips or 4 pencils? How can you make sure others know what the 4 is? Consolidation (20 minutes) • Have two or three pairs show which prints they measured and compared. For each pair, the class can estimate and predict the order of the prints from smallest to largest, as well as how many centimetres long they are. Then the pair that measured can share their results. Ask: – “ What do we need to remember when we tell other people what our measurements are?” (e.g., We need to say a number and what the unit is.) – “Do you think you would have gotten the same order if you measured with a different unit? Why?” – “Do you think you would get the same order if you measured your prints again tomorrow? Why?” – “Why is it a good idea to measure in centimetres?” Measurement 177

6Lesson The Metre Teacher Possible Learning Goal Look-Fors • Uses the benchmark unit of 1 m to estimate, compare, and order the lengths Previous Experience with Concepts: of various objects Students have measured objects with non- • Understands that the length of an object does not change regardless of the standard units and have tool used to measure the length been introduced to the centimetre. • Has a basic understanding of what centimetres and metres are • Reasonably estimates a length using the metre as a benchmark • Uses benchmark to measure and confirm, refine, or disprove estimations • Explains what to do when the end of the object doesn’t line up exactly with the end of the measuring tool About the Lesson In this lesson, students use a string 1 m long as a benchmark to estimate, measure, compare, and order the lengths of various objects. In the accompanying Math Talks, students further explore the metre as they estimate height. Materials: Minds On (15 minutes) metre stick, string, • Hold up a metre stick and ask, “What is this? Where have you seen this scissors, clipboards and copies of BLM 14: before? What do people use this for? This is a metre stick and it is exactly Measuring with Metres 1 m long. The interesting thing about a metre stick is that it is exactly the Time: 50 minutes same length anywhere in the world. All countries that use the metric system, like Canada, have metre sticks that are exactly this long. Are you taller or 178 Spatial Sense shorter than one metre? How would you find out?” • Ask students to look around the room for objects that are about as long as the metre stick. They can discuss their ideas with a partner. • Have students share their choices with the class. (e.g., We think the window is about 1 m long.) Ask the students listening to put their thumb up if they agree or down if they don’t agree, and invite some students to explain their reasoning. • Now, compare some of the objects chosen to the metre stick by lining them up side by side. Ask students what you should do to make sure that you are comparing carefully. (e.g., The bottom/end of the metre stick has to be even with the bottom/end of the object.) • Say, “Let’s do a different comparison. Find an object that is just a little shorter than the metre stick. Let’s use the metre stick to confirm our estimations.”

• Next, ask students to find an object that is about 2 m long, and again, use the metre stick to confirm their estimations. • Ask students how you could create a piece of string that is 1 m long. Cut enough pieces of string (or have pieces pre-cut) so that each pair can have one to measure with. Working On It (Whole Group) (20 minutes) • Students work in pairs, preferably outdoors so that they can spread out and have more objects to measure. Each pair can have a copy of BLM 14 on a clipboard. • Ask students what kinds of objects they could measure. (e.g., height of bush, width of door, height of fence, length of an object in the playground) • Ask students to find five objects, each with a different length: – less than 1 m – e xactly 1 m – about 2 m – longer than 3 m but less than 5 m – longer than 5 m but less than 8 m • Have pairs look for objects that they estimate to meet the criteria. Once they have estimated, they can compare the objects to their ‘metre string’ to see if their estimations were accurate. They can record their findings in the table on BLM 14. Differentiation • It may be more appropriate for some students to find objects that are “longer than 1 m,” “about the same as 1 m,” and “shorter than 1 m.” • You could challenge students to find objects that are as close to 3 m, 4 m, and 5 m as possible. Assessment Opportunities Observations: Listen to how students determine their estimations. Are they just guesses or do they seem reasonable based on their benchmark of 1 m? Do students adjust future estimations based on previous estimations and measurements? As students measure, observe how they iterate the string and whether they pull it tightly and leave no gaps or overlaps. Consolidation (15 minutes) • Have students meet with another pair after they have found and measured their objects. They can compare their results and then confirm whether their measurements are accurate. • Look at one or two of the students’ findings as a group. How close were their estimations? How did they compare the object to the metre stick? Measurement 179

Materials: • Ask students to describe what they recorded if the end of the string didn’t “Animal Prints” reach the end of an object or went past it. How did they decide which (pages 16–17 in the measurement the length was closest to? Spatial Sense big book), metre stick • Ask students to show the length of a metre with their arms. • B uilding Growth Mindsets: Ask students what they did when an object Teaching Tip that they thought was, say more than 3 m long, was actually less then 3 m Integrate the math (much shorter). How did this change their thinking as they looked for an talk moves (see object that fit the criteria? (e.g., I need something that is longer than what I page 7) throughout first picked.) Math Talks to maximize student • Explain that when we are not right the first time, our first answer can help participation and active listening. us get better at our task. We can learn from our first tries and get better at what we are doing. We can also learn from our mistakes by trying 180 Spatial Sense something different that might work. Math Talk: Using the Metre Stick to Compare Heights Math Focus: Comparing linear measurements using visual images and the metre stick as a benchmark Let’s Talk Select the prompts that best meet the needs of your students. • L et’s look at the animal prints again. • W hen we measured the last time, which prints were the largest? The smallest? Can the prints always tell you how big an animal is? Let’s investigate. • D raw attention to the print of the Northern giraffe and a second print of your choice. Which animal do you think would stand the tallest? Why? What animal do you think made these prints? One animal is a giraffe. Do you want to change your prediction of which animal is taller? A baby giraffe is about 2 m tall when it is born. Let’s measure how tall that is using this metre stick. What is special about the metre stick? (e.g., It is the same anywhere in the world and is used to measure lengths and distances.) Measure 2 m up the wall, either by using two metre sticks or iterating one. Would a baby giraffe fit in our classroom? • A n adult giraffe can grow to be 6 m tall. Do you think an adult giraffe would fit in the classroom? Why? Turn and talk to you partner. What are your predictions? Let’s measure as far up the wall as possible. • N ow let’s predict how tall the other animal is. Repeat the line of questioning for the second animal print that you chose. You will need to look up the animal’s height. • W hat do you notice about the two heights? About how much taller is the giraffe? Mark the heights and names of the animals on masking tape and put the markings on the wall. • A n animal’s prints may not always tell us how tall the animal is. Let’s predict which animal might have a height in between our two animals. Look up the

heights of some of the other animals that students suggest and add their heights to the measurement wall using the same prompts. Follow-Up Talk • Y ou can continue to add animal heights to the measurement wall as more animals are discussed. Use the metre stick as a benchmark. You can also add a small picture of the animal beside its height as a visual label. • H ave students suggest some local animals. Research their heights and add them to the measurement wall using the same prompts. Math Talk:What Measures About One Metre? Math Focus: Creating a personal benchmark for 1 m Let’s Talk • P resent the following prompt to students: Can you find something in this room that has the same length as the distance from the bottom of the door to the doorknob? (NOTE: This distance is about 1 m.) • T ogether, stand by the classroom door. Give students time to take turns standing beside it, to familiarize themselves with the height of the doorknob in relation to themselves. Then, have students roam around the classroom looking for two to three items that meet this criteria. • A fter roaming for a few minutes, have students pair up to show each other the items they were thinking about and explain the reasons for their choices. You may wish to have students switch partners a few times so that they begin to see more lengths within the classroom. • Gather as a class to create a list of the items that students believe are about the same length as the distance from the bottom of the door to the doorknob (or the height of the doorknob). • W hile standing away from the door, show students a metre stick. Ask students whether they think the distance from the bottom of the door to the doorknob is more than, less than, or about the same as 1 m. Confirm by measuring the distance with a metre stick. Label the list of items that students found “Benchmarks for About One Metre.” • N ow that we know this, how can we use this information to help us make good estimates? We want students to understand that we can use the benchmark of a metre to make many estimates. For example: The seat of the chair is about half of a metre tall. I know this because I can visualize the doorknob and the chair is about half the height of the doorknob. Measurement 181

7Lesson The Centimetre– Metre Relationship Teacher Possible Learning Goal Look-Fors • Investigates the relationship between centimetres and metres and the tools Previous Experience with Concepts: that are used to measure in these units Students have been introduced to centimetres • Understands and shows that 100 cm make up, or are equivalent to, 1 m and metres and have • Establishes other benchmarks for centimetres (e.g., 1 orange relational rod measured with each unit. equals 10 cm) and proves them using concrete examples • Shows other relationships between centimetres and metres (e.g., 10 orange rods equals 1 m; 20 yellow rods equals 1 m; 5 red rods equal 1 m) • Explains or shows how more smaller units are necessary to measure the same distance • Correctly uses a measuring tool by lining it up at one end of the object and iterating it if necessary • Uses an appropriate counting strategy to total smaller lengths or units (e.g., skip-counting by 2s, 5s, or 10s) About the Marian Small explains that “part of learning about metric units is learning about the relationships between one unit and another” (Small, 2009, p. 378). Students in grade two need to know that there are 100 cm in 1 m. Small also emphasizes the need for students to actually see 100 centimetre cubes lined up beside a metre stick. This helps them create a powerful mental image that can later be retrieved to remember the relationship. As students start measuring with more precision, it is helpful for them to have other benchmarks besides the metre and centimetre. For example, they can create a benchmark for 10 cm that they can physically and mentally use as they estimate lengths. This also helps students establish other metric relationships, such as that there are 10 lengths of 10 cm in a metre. Such experiences help students to think proportionately. 182 Spatial Sense

Materials: Relational rods are excellent concrete materials for creating benchmarks since the ten different-coloured rods increase in length from 1 cm to metre stick, 10 cm in increments of 1 cm. In finding relationships between the rods, relational rods, students find relationships among lengths in centimetres. centimetre cubes 1 Time: 60 minutes 2 3 4 5 6 7 8 9 10 About the Lesson In this lesson, students use concrete materials to see and learn that there are 100 cm in 1 m. They also explore other benchmarks for centimetres, such as that an orange relational rod represents a length of 10 cm. Minds On (15 minutes) • Hold up a metre stick and ask students what they can tell you about it. Do the same with a centimetre cube. • Ask, “How many centimetre cubes are there in a metre? How could we prove your estimation? How could we prove that there are 100 cm in a metre?” Students may know that there are 100 cm but they need to prove it. • If students suggest lining the centimetre cubes alongside the metre stick, ask how you could group the centimetre cubes together so they are easy to count. (e.g., in groups of 10) Have each pair of students make a centimetre- cube train 10 cubes long. • Say, “Now line the 10-cube trains up alongside the metre stick and count together as we go. How many centimetres are in a metre? Does this ever change?” Measurement 183

Working On It (Whole Group) (15 minutes) • Hold up an orange relational rod and ask students to estimate how many centimetres long they think it is. Ask how they could measure to find out. (e.g., Line up centimetre cubes alongside it. Place it alongside the metre stick and see how far it goes.) • Place one orange rod alongside the metre stick with its end lined up with the end of the metre stick. Ask, “How many more orange rods would we need to line up end to end to make a train the same length as the metre stick? Turn and talk to your partner. Let’s add some more orange rods. [Add two or three rods, then pause.] Do you want to change your estimation? Let’s line up orange rods until we reach the end. How can we count the rods to be sure of our answer?” • State the answer: “So there are 10 orange rods in 1 m. Put up your thumb if you agree.” • Hold up a yellow rod, and ask, “Do you think we will need more or fewer yellow rods than orange rods to make a train 1 m long? Why? How many do you think we will need? How many centimetres long is the yellow rod? How can you prove it? (e.g., Put 5 centimetre cubes on top of it.) So how many yellow rods will we need? How can we count them? How many more yellow rods did we use than orange rods? (e.g., We used 10 more. We used twice as many.) Small Group (10 minutes) • Students work in pairs to determine how many red rods make a train 1 m long. • First, ask students to estimate based on their experience: “Will we need more or fewer red rods than yellow rods? Why do you think that? Will we need more or fewer red rods than white rods/centimetre cubes? Why do you think that?” • Explain how this investigation will differ from what students did in Minds On: “We don’t have enough red rods for everyone to line them up against their own metre stick, so we will have to think of another way to discover how many red rods make up a metre. Each group can have five or six red rods and any other colour rods you would like to help you. Differentiation • You may decide to do the red rod investigation as a whole group. Alternatively, you may simply discuss with the class whether they would need more or fewer of the red rods than the orange and yellow rods rather than figuring out exactly out how many. In this case, reinforce the idea that many more red rods are needed because they are smaller than the orange and yellow rods. 184 Spatial Sense

Assessment Opportunities Observations: Observe students rather than probing their thinking. Arriving at a solution—determining how many red rods make a metre—is not as important as experimenting with various strategies to solve the problem. Do students realize that they will need more red rods than yellow or orange rods? Do they think it is only a few more or many more? Do they try to figure out how the red rod relates to rods of other colours? Do they use their skip-counting skills to tally up numbers of rods in a line or to determine how many of one colour rod fit alongside another colour? Consolidation (15 minutes) • Ask, “How long is the red rod? (2 cm) What strategies did you use to find out how many red rods make 1 m?” Strategies: – L ay red rods across one orange rod. There are 5 red rods in an orange rod, so we would need 5 red rods for every orange rod. How could we count that? Count 5 for every orange rod alongside the metre stick. – L ay centimetre cubes across the red rod. The red rod is 2 cm long, so if we put red rods alongside the metre stick we have to count by 2s. – T he red rods are twice as long as the centimetre cubes, so instead of 100 centimetre cubes lined up with the metre stick, we would need only half as many red rods, and half of 100 is 50. • Ask students to close their eyes and to visualize a metre stick with orange rods lined up beside it. How many rods do students see? How many centimetre cubes do they see lined up alongside the metre stick? • Make a list with students of everything they now know about centimetres and metres and how these standard units are related. Materials: Math Talk: Measuring with a Ruler standard rulers Math Focus: Introducing the standard ruler; linking measurement with (30 cm), centimetre centimetre cubes and measurement with a standard ruler cubes, students’ shoes, chart paper About the It is important that students discuss how measuring with actual unit models, like centimetre cubes lined up along the length of an object, compares to measuring using a single instrument, such as a ruler, that is divided into standard units. Without discussion, students may not understand that these two methods are essentially the same (Van de Walle & Lovin, 2006, p. 233). By measuring the same length with both centimetre cubes and a centimetre ruler, students can make connections between concrete materials and measurement tools. continued on next page Measurement 185

Teaching Tip Let’s Talk Integrate the math Have three students give one of their shoes for the investigation. Strategically talk moves (see choose the three students in advance of the lesson: one student who has a page 7) throughout smaller shoe and two students whose shoe length might be close to one Math Talks to another. Select the prompts that best meet the needs of your students. maximize student participation and • H ow can we represent the lengths of these shoes on this chart paper? (e.g., make active listening. a line as long as the shoe) 186 Spatial Sense • I am representing the length of each shoe by drawing a line from the toe to the heel. Put each shoe on the chart paper and hold a ruler alongside it to draw the line as described. Place the line segments in various locations on the chart paper and not lined up next to each other, to prevent students from directly comparing their lengths.  • W hich shoe is the longest? How do you know? Whose shoe is the shortest? How do you know? How can we measure the length of the three shoes? What could we use to measure the length of the shoes? Would it be best to measure in non- standard or standard units? Why? Would it make more sense to measure in centimetres or metres? Why? • S how me with your fingers about how long a centimetre is. Here are some centimetre cubes to give you an idea. Estimate how many centimetres long the first shoe is. Turn and talk to your partner. • Have students estimate how many centimetres long the first shoe is and record some of their estimates. Do your estimates seem reasonable? Put the chart paper on the floor and invite two students to measure the line that represents the first shoe using centimetre cubes. Do you think the students were accurate in measuring? How do you know? How close were our estimates to the measurement? • H ere is a ruler that is divided into centimetres. What do you notice about it? How do you think we can use it to measure the first line? What is important to do? (e.g., make sure the end of the line lines up with the 0) What measurement did we get? How does it compare to our measurement with the centimetre cubes? Why do we get the same results? • W hat can we do if the centimetres don’t line up exactly with the end of the line? (e.g., round to the nearest centimetre, use the number closest to the end of the line) • H ow are the two ways of measuring the same and how are they different? Which way did you find easier? Why? • L et’s look at the second line. Do you think it is longer or shorter than the first line? Why? How can the measurement of the first line help you estimate the length of the second line? I am writing down your estimations. Are any estimates too high? Are any estimates too low? How do you know? Measure the shoe with centimetre cubes and a ruler to confirm that the results are the same. • R epeat the same line of questioning with the last line. • H ow can we order the shoes from shortest to longest? • H ow is measuring with centimetre cubes like measuring with a ruler?

8Lesson Using Benchmarks to Estimate and Measure Math Curricular Competencies Learning Standards • R easoning and analyzing: Use reasoning to explore and make First Peoples connections; estimate reasonably; model mathematics in contextualized Principles of experiences Learning • Understanding and solving: Develop, demonstrate, and apply Teacher Look-Fors 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 • 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 • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) Possible Learning Goal • Establishes and uses benchmarks for standard units to estimate and measure various lengths • Reasonably estimates the length of objects using various benchmarks for standard units • Shows benchmarks for 1 cm, 10 cm, and 1 m and proves that they are the actual length • Uses benchmarks to describe the “how muchness” of 1 cm, 10 cm, and 1 m • Uses a combination of 10 cm benchmarks and 1 cm benchmarks to estimate and measure lengths • Counts combinations of relational rods according to the number of units (centimetres) in each (e.g., counts the orange rods by 10s and then the white rods by counting on by 1s) Measurement 187

Mlnseceotanesantgntnh-ittsmdhimtVa,aaorehntdcetedra,iueagbcn,rhuodimttlm,,auerdpntyiaris:ter,te,ance, About the Marian Small states that an important big idea in measurement is that “there is always value in estimating a measurement, sometimes because an estimate is all you need, and sometimes because an estimate is a useful check on the reasonableness of a more precise measurement” (Small, 2009, p. 364). Developing and becoming familiar with personal benchmarks for standard units will help students to make reasonable and sound estimates throughout all of their measurement investigations. For example, they can physically use a benchmark to iterate partway across a length and use this to estimate the entire length. With time and experience, students can mentally iterate a benchmark across a length to produce their estimates. About the Lesson In this lesson, students use benchmarks for standard units to estimate length. They also investigate the relationships between the benchmarks. Materials: Minds On (15 minutes) orange relational rods, white relational • Show students an orange rod. Reinforce ideas introduced in the previous rods or centimetre cubes, standard rulers lesson by asking how many centimetres long one orange rod is, how many (30 cm), BLM 15: More orange rods are in 1 m, and how many centimetres are in 1 m. Show Estimates and students a standard ruler 30 cm long without the numbers showing and an Measurements orange rod. Ask them to visualize and estimate how many orange rods Time: 50 minutes would line up along the ruler. Together, verify that the ruler is 30 cm long because it is the same length as three orange rods. 188 Spatial Sense • Draw a line that is 37 cm long. Have students visualize and estimate about how many orange rods long it is. Together, line up orange rods along the line. Once you’ve seen that a fourth rod extends (or would extend) beyond the end of the line, ask students what they can do to find the leftover length. Together, measure the remaining length with white rods (centimetre cubes). Skip count, by 10s and then by 1s, to find the total length of the line. • Ask students how they could use the standard ruler, knowing that it is 30 cm long, to help them measure the line without using orange rods. Working On It (20 minutes) • Give each pair of students a copy of BLM 15, 1 orange rod, 10 white rods (centimetre cubes), and a standard ruler. • Students find three objects of different lengths in the room: one between 40 and 50 cm long, one between 60 and 70 cm long, and one between 80 and 90 cm long. Students first estimate exactly how long their objects are and then measure them in various ways using their measuring tools. Students record their estimates and measurements on their tracking sheet.

Differentiation • For students with limited counting abilities, you may want to assign lengths that are under 30 cm. • You can encourage students who need more of a challenge to find two additional objects of their choice, estimate their lengths, and then measure them in two different ways. Assessment Opportunities Observations: Pay attention to students’ use of benchmarks to make their estimates. – D o they visualize iterating the benchmark along the length, or do they hold up one copy of the benchmark to get an idea of how much of the length it occupies? – Are they able to use more than one benchmark to measure a length? (e.g., a ruler, then one orange rod, then six centimetre cubes or white rods) Can they count their combinations of measuring tools appropriately by switching to the unit being represented? – A re they making connections between the length of the orange rod and the length of the ruler? When they are estimating, are they using the language of approximation (e.g., ‘about’)? – Do they make reasonable estimates using the standard units of measure? (e.g., The notepad is about 12 cm wide because it is just a little wider than the orange relational rod which is about 10 cm long.) Do they adjust any of their estimations using, or according to, the length of objects that they measured previously? – H ow are they dealing with any leftover length when they measure? Do they round to the nearest unit (e.g., orange rod) or do they fill in the difference with centimetre cubes? Conversations: While observing students at work, consider asking some of the following questions to elicit thinking: – W hy do you think your estimate is reasonable? Which benchmark do you find the most helpful to use? Why? – H ow did finding an object between 40 and 50 cm long help you find an object that is between 60 and 70 cm long? – How did you count up your total number of centimetres? Is there another way to count them? – Is there another way to measure the same length? Which strategy seems easiest to use? – (To encourage students to use one measurement to make other estimates) How many centimetres do you think would describe the length of this object? How can you compare that to the length of this other object? Measurement 189

Teaching Tip Consolidation (15 minutes) Analyze student • Select two or three pairs of students to share the findings from their solutions and choose which strategies you investigation. Ask them to explain how they made their estimates. Ask them would like to have why they chose the measuring tools they used for certain objects. Students shared. can describe how they used them. Materials: • Discuss how the benchmarks helped students to estimate lengths. paper or string, • Ask students how they could estimate the length of a table. Ask which metre stick benchmark would be most helpful: the centimetre, the 10-cm orange rod, or Materials: the metre. Put the suggested unit at one end of the table. Have students sidewalk chalk, visualize iterating the unit across the table to estimate the length. Keep measuring tape or adding more of the unit, one at a time, and pause periodically to ask whether metre stick they want to change their estimate. When you get near the end, ask students to predict whether the last unit will line up exactly with the end of the table, will fall just short of the end of the table, or will extend beyond the end of the table. Ask students how they can adjust their estimate to take the leftover amount into account. Further Practice • A Benchmark for 50 Centimetres: Give students a strip of paper or string that measures 50 cm. Together, compare it to the length of a metre stick so they can see that 50 cm is half a metre long. Have students find three objects in the classroom that are about the same length. Using their previous experiences with benchmarking 1 cm, 10 cm, and 30 cm, they should estimate about how long their objects are in comparison to the 50-cm strip. As a class, share some students’ benchmarks. Periodically ask them to visualize one of their benchmarks to estimate another length in the classroom so they internalize this new benchmark. • M ath Journal or Oral Reflection: Provide one or both of the following prompts for students to respond to in their journals or to discuss orally with a math partner. Consider using a recording device to capture their oral responses to assess their learning. – W hat personal benchmarks do you have for 1 cm, 10 cm, half a metre, and 1 m? Explain or show how these help you to estimate the length or height of objects without actually measuring them. – D escribe a situation when you would use centimetres rather than metres to measure the length of something. Explain your thinking. • M easuring Shadows: Students work with a math partner to measure their “heights” as seen in their shadows. Using sidewalk chalk, partners mark the top and bottom of their shadows. Students then estimate, in standard units, the length of their shadows on the ground. Students measure each shadow with a measuring tape or metre stick and record their findings. Lastly, students estimate whether or not their actual height is longer or shorter than the “height” of their shadow. Partners take turns measuring their height with the same measuring tool to compare the length of the shadow to their actual height. 190 Spatial Sense

Math Talk:Will It Fit Through the Door? Math Focus: Directly and indirectly comparing linear measurements Materials: About the benchmarks for standard units used Presenting this measurement task in a problem-solving context helps students to in the lesson (e.g., see that measurement is purposeful. Discussions about possible ways to measure string, relational rods, objects allow students to think about measurable attributes and the different standard rulers) kinds of units needed to measure different attributes. To further develop students’ concepts about length, they can also compare objects indirectly when Teaching Tip a direct comparison is impossible or difficult. Students then use a tool, such as a string, to measure both objects and then compare the lengths. Integrate the math talk moves (see Let’s Talk page 7) throughout Math Talks to Select the prompts that best meet the needs of your students. maximize student participation and • P resent the following scenario: Our class has to place all the furniture from our active listening. classroom into the hallway so the custodian can clean the floors. Can we slide the tables and the desks through the doorway? • W ill the desks/tables slide through the doorway? How do you know? Invite students to stand up and discuss their responses and reasoning with a partner (Stand and Talk). • W hat information do we need to decide if the desks and tables can slide through the doorway? (Possible prompts: How wide is the desk/table? How wide is the doorway?) • W hat benchmarks that we have for standard units could we use to help us estimate? • H ow can we make sure that the desks/tables will slide through the door without actually moving them? What tools could we use? Partner Investigation • W orking in pairs, have students determine whether the desk/table will slide through the door without actually sliding it through the door. They can use some of the benchmarks they have developed. Follow-Up Talk • D id we need to actually measure the desk or doorway with standard units to decide if it can fit through the door? Why? • W hat other tools can we use? (e.g., take a string that is the length of the desk/ table and then hold it across the doorway to see if it is longer or shorter) Measurement 191

9Lesson Using Benchmarks to Make Comparisons Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make First Peoples connections; estimate reasonably; model mathematics in contextualized Principles of experiences Learning • U nderstanding and solving: Develop, demonstrate, and apply Teacher Look-Fors mathematical understanding through play, inquiry, and problem solving; visualize to explore mathematical concepts; develop and use multiple Previous Experience strategies to engage in problem solving with Concepts: Students have an • Communicating and representing: Communicate mathematical understanding of a metre and a centimetre and the thinking in many ways; use mathematical vocabulary and language to relationship between the contribute to mathematical discussions; explain and justify mathematical two units. ideas and decisions • 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 • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) Possible Learning Goal • Estimates, measures, and compares lengths/distances using benchmarks for standard units • Offers a reasonable estimate for a length and justifies their reasoning • Selects an appropriate benchmark with which to measure a certain length and justifies their choice • Places benchmarks end to end along a length, with no gaps or overlaps, to produce an accurate measurement • Uses a system to accurately count and/or track the units • Understands that a measurement includes a number of units and a type of unit • Measures and compares the lengths of paths that are not straight • Determines how much longer certain paths are than other paths using a suitable strategy 192 Spatial Sense

Math Vocabulary: About the lhceceoneimgngtthphimta,,redeetis,sretstia,mtanmancteedet,a,rerd unit, Now that students are familiar with some benchmarks for standard units, Materials: they can use these benchmarks to compare various lengths. This involves selecting a benchmark that is suitable for all the lengths being compared, accurately measuring each length using a consistent method, and then comparing the results. Students may initially order the distances from shortest to longest and use comparative language such as ‘longest,’ ‘longer than,’ or ‘shortest’ to describe their results. We eventually want students to quantify the difference using a number and a unit—to describe how much more or less a length is. When comparing lengths that are all straight (linear), students may physically line them up along a common starting point and then concretely ‘add on’ units from one length to the other to find the difference. Comparing becomes more challenging when the lengths in question are not straight. Students may measure with flexible tools, such as pieces of string, by laying them along each curvy length and then directly comparing the lengths of the straightened strings. They may measure with short benchmarks, such as the red relational rod (which represents 2 cm) by placing them along each curvy length and then putting the same numbers of rods in trains/rows to directly compare their lengths. Alternatively, they may compare lengths by using various counting strategies to find the difference in lengths in their symbolic form (e.g., this path is 10 red rods long and the other is 12 red rods long, so the second path is 2 rods longer). It is important to make connections among their various strategies to show how they are all related. About the Lesson In this lesson, students explore the lengths of paths that are not straight (linear). Students select an appropriate unit as well as suitable measuring tools to estimate, measure, and then compare the lengths of the paths. “Park Paths” (pages Minds On (15 minutes) 18–19 in the Spatial Sense big book and • Display the big book and allow students time to look over and analyze the little books), 30-cm lengths of string, paper image on pages 18–19.  clips, pipe cleaners, connecting cubes, • Tell students that this is an overhead or bird’s eye view of a park and there relational rods, measuring tape or ruler are three pathways (in purple) to different equipment or activities in the park.  Time: 60 minutes • Ask, “What is challenging about measuring the lengths of these paths? Which path do you think is the shortest, and how do you know?” • Allow time for students to visualize and think about how they would solve this problem. Encourage discussion in pairs before whole-group sharing. Measurement 193

• Ask, “Which path is the longest? How do you know? If we want to know how much longer one path is than another, what would we need to do?” (e.g., measure and record the measurement with a number and a unit) • Draw attention to the same page that is in the little book versions of the big book. Ask, “Do you think that the longest path in the big book will also be the longest path in the little book? Why?” • Ask, “What standard unit would be most appropriate, centimetres or metres? (centimetres) Why? What other tools might you use?” (e.g., relational rods, string, rulers) Working On It (20 minutes) • Students can work in groups of two or three, with each group having a copy of the little book version of the big book. Students start by predicting which path is longest and estimating the length of all three paths using their chosen unit of measurement (likely centimetres). Then, they measure the paths in any way they think is appropriate, using whichever tool(s) they wish. Students need to report the length of each path as well as how much longer or shorter it is compared to the others. Students can use a blank piece of paper to record their findings and the way in which they figured the problem out. Differentiation • One group can work with the big book to see if they get the same order of paths (shortest to longest) as the students working with the little books. When discussing this group’s results in the Consolidation, you can talk about why their measuring tool, which is probably longer, is more appropriate for the paths in the big book. The rest of the students in the class can also estimate the lengths of the paths in the big book before the students report their findings. Assessment Opportunities Observations and Conversations: • Observe students as they measure each path. Ask, “What measuring tool did you select? Why that one?”(We want students to understand that we select a tool that is appropriate for what is being measured and is not chosen randomly.) • Observe whether students have a consistent strategy for measuring each path. Ask them to explain and demonstrate their strategy. Ask whether they can use a different method to measure the other paths and whether it matters. • Observe how students compare the lengths of the paths. Are they using concrete materials or are they finding the differences in length by comparing the numbers in their symbolic form? Ask, “How did you determine how much longer this path was than the other one?” (We want to understand how students used the information from measuring to draw conclusions about each path.) 194 Spatial Sense