options: Recognize and represent proportional relationships between quantities.

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
problem_activity: Optional activity
In this activity, students build a trundle wheel and use it to measure distances in the classroom. The trundle wheels need to be stored in the classroom for use in the next lesson. If it is not feasible to store a trundle wheel from every group of 3–4 students, have them combine to form larger groups before building the wheels. Each student should still get a chance to practice measuring with the trundle wheel.
It is suggested that students build their trundle wheels using a paper plate as the wheel, two rulers taped together end to end as the handle, a metal paper fastener, and an index card taped to the wheel to produce an audible “click” when it hits the handle. There are many other ways to build a trundle wheel. If this fits into the culture of the class, students can use other designs and materials, e.g., students could use a bike with a playing card in the spokes to count rotations.
Launch
Keep students in the same groups, or if necessary, combine them to form larger groups. Discuss how to build the trundle wheel and distribute the supplies. Give students 15 minutes of group work time to build and try out their trundle wheels, followed by whole-class discussion.
Student Facing
Your teacher will give you some supplies. Construct a trundle wheel and use it to measure the length of the classroom. Record:
the diameter of your trundle wheel
the number of clicks across classroom
the length of the classroom (Be prepared to explain your reasoning.)
Student Response
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Anticipated Misconceptions
Students may need some trial and error in building working trundle wheels, in particular for placing the clicking device. Encourage them to try out their design and then revise it as necessary.
Activity Synthesis
The goal of this discussion is for students to double-check that their trundle wheels work correctly and their results are reasonable. Invite one group that used each size of paper plate to demonstrate how their wheel works by walking across the classroom and counting the “clicks.” Record the data for all to see and discuss how these groups’ calculations for the length of the classroom compare, which may include the following points:
The group with the smaller plate had a larger number of clicks and vice versa.
Each group needs to multiply their number of clicks by the circumference of their wheel.
The circumference of the wheel can be found by multiplying the diameter times
\(\pi\)
or the radius times
\(2\pi\)
.
Should we report our answer in terms of
\(\pi\)
or use an approximation?
Are the groups’ answers for the length of the classroom pretty close or very different?
Is it possible to count half rotations of the wheel?
Representation: Develop Language and Symbols.
Display or provide charts with important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding and memory of: formula for circumference, rotation and the relationship between circumference of a wheel and the distance traveled.
Supports accessibility for: Conceptual processing; Memory
Representing, Conversing: MLR7 Compare and Connect.
Rearrange groups of students so that they are seating with another group that used the same size paper plate to build their trundle wheel. Invite groups to share their solutions and reasoning for the length of the classroom with each other. Ask, “What is the same and what is different?” about how they used their trundle wheel to measure the classroom. This will help students to consider precision when measuring and use mathematical language as they converse.
Design Principle(s): Maximize meta-awareness; Cultivate conversation


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