Big Idea: Measurement relationships

Expectations:

• E2.2 solve problems that involve converting larger metric units into smaller ones, and describe the base ten relationships among metric units

• D1.2 collect data, using appropriate sampling techniques as needed, to answer questions of interest about a population, and organize the data in relative-frequency tables

teacher background

Students should be familiar with selecting the most appropriate standard units in problem-solving contexts, and with justifying their choices. Students also need previous experience in displaying information on graphs. In particular, familiarity using broken-line graphs will allow students to focus on interpreting, drawing conclusions, and comparing their data with related sets.

KEY CONCEPTS:

• Conversions within the metric system rely on understanding the relative size of the metric units (see E2.1) and the multiplicative relationships in the place-value system (see Number, B1.1).

• Because both place value and the metric system are based on a system of tens, metric conversions can be visualized as a shifting of digits to the left or right of the decimal point a certain number of places. The amount of shift depends on the relative size of the units being converted. For example, since 1 km is 1000 times as long as 1 m, 28.5 km becomes 28 500 m when the digits shift three places to the left.

• There is an inverse relationship between the size of a unit and the count of units: the smaller the unit, the greater the count. Remembering this principle is important for estimating whether a conversion will result in more or fewer units.

Although this expectation focuses on converting from larger to smaller units, it is important that students understand that conversions can also move from smaller to larger units using decimals. Exposure to decimal measurements is appropriate for Grade 5 students.

Canada is world renowned for its natural beauty and vast expanses of wilderness. These areas afford Canadians opportunities to enjoy a wide variety of outdoor activities. In Ontario, the Bruce Trail, extending from Niagara to Tobermory, provides an ideal setting for hiking adventures. In this learning activity, students explore measurement problems and relationships in the context of a Bruce Trail hike. Working with conversions, students will be required to recognize and apply the relationship between kilometres and metres. They will use their understanding of elapsed time in dynamic ways to determine distances covered over time.

RATES OF TRAVEL If a hiker travels 9 km in 3 hours, the rate of travel is 3 km/h. This is equivalent to 3000 m in 1 hour, or 750 m every 15 minutes, or 500 m every 10 minutes, or 50 m every minute.

The Bruce Trail, extending from Niagara to Tobermory, provides an ideal setting for hiking adventures. At nearly 800 km, it is Ontario’s longest trail.

Describe the following scenario to the class:

“An Ontario hiker has just completed a three-hour hike along this trail. At a number of picturesque locations, the hiker stopped to take photographs. The digital camera recorded the time each photograph was taken. The hiker wants to use these recorded times to pinpoint the locations on a map for a hiking club’s website. Over the three-hour period the hiker travelled nine kilometres and took 5 photographs. The hiker left the trailhead at 9:00 a.m. and took photographs at the following times:

• Photograph 1 - 9:15 a.m.

• Photograph 2 - 9:20 a.m.

• Photograph 3 - 10:25 a.m.

• Photograph 4 - 11:20 a.m.

• Photograph 5 - 11:56 a.m.

“How far is each photograph location from the trailhead, assuming the hiker walked at a steady pace?”

SHARED READING OF THE PROBLEM: Shared reading is an effective instructional approach that can be applied in this context as the students read and interpret the problem together as a class. They will benefit from explicit reading instruction regarding the format and features of mathematics text. A shared-reading approach will support students as they learn to isolate the key pieces of information they need to solve this problem. Through a skilfully led discussion, you can prompt students to justify the reasoning behind their choices. Modelling the “skimming and scanning” strategy during the initial read will help students to recognize an effective reading approach for such math problem formats. On subsequent readings, shift the focus to locating pertinent facts. You may wish to use a highlighter to facilitate this process. The shared-reading format also provides opportunities for students to engage in mathematical talk and to clarify their understanding of the task.

CREATING A VISUAL REPRESENTATION: At this point in the learning task, it may be helpful if each triad of students used chart paper to create a visual representation (such as a time line or a hiking route) of the key information identified during the shared reading of the math problem. The reading comprehension strategy of visualization encourages students to represent key information to synthesize what they know. This visual representation also provides a personally relevant referent from which each student can work.

CALCULATING THE DISTANCE FOR EACH PHOTOGRAPH

Have groups of 3 determine the distance from the trailhead of each photograph location. While students are engaged in this task, circulate and encourage mathematical talk. Ask:

• “How are you using the information in the problem to determine the hiker’s rate of travel?”

• “How could the information on the time each photograph was taken help you to determine the distance from the trailhead of each photograph location?”

• “How will you decide which units of measure to use in your calculations?”

This task provides a rich opportunity for students to reason mathematically as they determine relationships. If the rate of travel is 3 km/h, students may use this information to determine distances travelled over time. For example, the knowledge that a hiker travels 3 km/h allows students to determine that the hiker will travel 1.5 km per half hour and 0.75 km or 750 m in 15 minutes. This line of reasoning will help them identify distances travelled over smaller increments of time. If 750 m can be travelled in 15 minutes, then 250 m can be travelled in 5 minutes, and 50 m can be travelled in 1 minute.

Ensure that students are able to recognize the relationships between rate, time, and distance. An integral part of these calculations will involve working flexibly with conversions from kilometres to metres. It may be necessary to engage in small- or whole-group mathe- matics instruction. Alternatively, you may invite students to share and discuss their strategies for determining rate and distance.

Once groups have determined the distance from the trailhead of each photograph location, reconvene the class and have students communicate their findings. As they share their work, record (or have a student record) accurate distances related to time on a class T-chart.

Ask students to summarize the information on the T-chart, using a ine graph that displays time on the horizontal axis and distance on the vertical axis. The graph will be a straight line, since the distance/time rate is constant (3 km/h). (Students can use the Pages app to create a line graph).

While students are engaged in the reading, interpretation, and analysis of their data, ask:

• “What conclusions can you draw about the relationship between time and distance by examining your graph?”

Ongoing assessment opportunities are embedded throughout this learning activity. Suggested prompts and questions have been provided in the Minds On and Action sections. Focus your observations in order to assess how well students:

• express their understanding of measurement relationships (time, distance, rate);

• work flexibly with conversions (kilometres to metres);

• select and compare units of measure and justify their reasoning;

• draw upon their understanding of quantity and fractional relationships with respect to time; • apply reasoning and logical thinking;

• communicate and justify their solutions.

Assessment Checklist

SEL Self-Assessments (English) and Teacher Rubric