Big Idea:

The emphasis in this learning activity is on interpreting the problem

situation, applying meaningful procedures rather than simply using an algorithm, and making sense of the solution.

Expectations:

B2.8 represent and solve problems involving the division of three-digit whole numbers by decimal tenths, using appropriate tools, strategies, and algorithms, and expressing remainders as appropriate

Ask students: “How long do you think it would take you to bike 100 km?”

Have a few students estimate the time it might take, and ask them to explain how they made their estimates.

Continue the discussion by asking: “What information would help you answer the question more accurately?”

Students might suggest, for example, that it would be helpful to know an actual 100 km distance or an average biking speed (e.g., kilometres per hour). Explain to students that you will provide some information to help them refine their estimates.

Display the following statements on the board or on chart paper. The distance from the school to the [local site] is 5 km. • It takes about an hour for a typical recreational cyclist to bike 15 km to 20 km.

Divide the class into groups of two or three students. Instruct students to work in their groups to determine the length of time it would take to bike 100 km. Invite them to use information (from the displayed statements or based on their own knowledge) to determine a solution.

Provide each group of students with a sheet of paper on which they can record their work. Ask them to record their solution and to be prepared to share it with the class. As students work on the problem, examine the various strategies they are using. For example, students might:

• refer to a familiar 100 km distance (e.g., the distance between two nearby towns) and estimate the time it would take to bike the distance;

• estimate the time it takes to bike 5 km and multiply this time by 20;

• consider the time it takes a recreational cyclist to bike 20 km and multiply this time by 5.

When students have finished recording their solutions, ask different groups to present their strategies and solutions to the class. Attempt to include groups who used a variety of strategies. Discuss the variety of approaches by asking questions such as the following:

• “Which strategies are similar? How are they alike?”

• “Why do the solutions differ? Is it possible to have an exact time for the solution?”

• “Which solutions seem reasonable? Why do you think they are reasonable?”

• “Which strategy, do you think, provides the most accurate solution? Why?”

• “What variables or factors might affect the time it takes to bike 100 km?”

For the last question, students might respond that factors such as the terrain, the kind of bike and condition of the bike, the physical condition of the rider, and the weather will influence the amount of time it would take to bike 100 km.

STRATEGIES STUDENTS MIGHT USE

USING REPEATED SUBTRACTION: Students might begin with 1550 and repeatedly subtract 95 until they reach a remainder of 30. They then count the number of times 95 was subtracted.

USING DOUBLING: Students might double 95 to calculate the distance travelled in 2 days, and then continue to double the distance and the number of days until they reach a distance close to 1550.

95 + 95 =190 (2 days)

190 + 190 = 380 (4 days)

380 + 380 = 760 (8 days)

760 + 760 =1520 (16 days)

1550 – 1520 =30 (30 kilometres more to travel)

USING “CHUNKING”: Students might subtract “chunks” (multiples of 95) from 1550.

USING AN ALGORITHM: Students might use an algorithm to divide 1550 by 95.

Note: If students attempt to use an algorithm that they have learned in previous grades, encourage them to think about the meaning of each procedural step.

When students have solved the problem, provide each group with markers and a sheet of chart paper or large sheet of newsprint. Ask students to record their strategies and solutions on the paper and to clearly demonstrate how they solved the problem.

Make a note of the various strategies used by students, and consider which groups might present their strategies during Reflecting and Connecting. Aim to include a variety of strategies (e.g., using repeated subtraction, using doubling, using “chunking”, using an algorithm).

After students have finished solving the problem and recording their solutions, bring the class together to share their work. Ask a few groups of students to explain their strategies and solutions to the class. Pose guiding questions to help students explain their procedures:

• “What strategy did you use to solve the problem? Why did you use this strategy?”

• “How did you know that you were on the right track?”

• “Did you alter your strategy as you worked on the problem?”

• “What is your solution to the problem?”

• “Is the solution to the problem reasonable? How do you know?”

• “What did you do with the remainder?”

It is important that students have an opportunity to examine and discuss various strategies and evaluate their efficiency in terms of ease of use and effectiveness, in order to provide an accurate and meaningful solution. The purpose of this evaluation is not to have the class make definitive conclusions about which strategies are best, but to allow students, individually, to make decisions about which strategies make sense to them.

Observe students as they solve the problem to assess how well they:

• understand the problem;

• use an appropriate problem-solving strategy;

• judge the efficiency and accuracy of their strategy;

• solve the problem;

• explain the meaning of the remainder within the context of the problem and their solutions;

• explain their strategies and solutions clearly and concisely, using mathematical language;

• determine whether the solution is reasonable.

Provide an additional assessment opportunity by having students solve an additional problem.

Provide students with copies of Div6.BLM2: Detour to Edmonton, and discuss the problem.

“If Ben and Jen take the Yellowhead Highway from Winnipeg to Edmonton and then travel south to Lake Louise, the total distance is 1910 km. If Ben and Jen travel at a more leisurely pace of 85 km a day, how many days will it take them to complete the trip?”

Encourage students to think about the various strategies that the class used to solve the previous problem, and to apply one that would work well to solve this problem. Remind students to show their strategy and solution clearly so that others will know what they are thinking. Observe students’ completed work and assess how well they apply an appropriate strategy, solve the problem, and explain their strategy and solution.

SEL Self-Assessments (English) and Teacher Rubric

Simplify the problem for students who experience difficulties because of the size of numbers in the problem (e.g., “How many days will it take Ben and Jen to complete a trip of 260 km if they travel 65 km each day?”). It may be necessary to demonstrate a simple strategy, such as repeated subtraction, or to pair students with classmates who can explain a simple problem-solving method. For students who require a challenge, ask them to solve the following problems:

• If Ben and Jen were to cycle 45 km in 3 1/2 hours, about how many kilometres would they cycle in 8 hours?

• If a 4-day cycle trip costs approximately $635, about how much would Ben and Jen spend on their trip from Winnipeg to Lake Louise?

Encourage students to consider the effectiveness and efficiency of each strategy by asking the following questions after each presentation:

• “Was it easy to find a solution using your strategy?”

• “What are the advantages of this method? What are the disadvantages?”

• “How would you change your strategy if you solved the problem again?”

Conduct a think-pair-share activity. Provide 30 seconds for students to think about the different strategies they observed and to choose the strategy that they think worked best to solve the problem. Next, have them share their thoughts with a partner.

Ask a few students to share their thoughts about effective strategies with the class. Pose the following questions:

• “In your opinion, which strategy worked well?”

• “Why is the strategy effective in solving this kind of problem?”

• “How would you explain this strategy to someone who has never used it?”