Proportions Lesson 5 (Teachers)

Forming a Unit Ratio

Kate and Christopher extend their use of diagrams to form a unit ratio in a speed context.

Episode 1: Making Sense

Kate and Christopher compare two different ways of reasoning – multiplicative reasoning and additive reasoning.

Episode 2: Exploring

Christopher and Kate figure out how many miles a car should travel in one minute so it goes the same speed as a car traveling 10 miles in 4 minutes. This results in a unit ratio.

Episode 3: Reflecting

Kate and Christopher use diagrams to demonstrate why additive reasoning does not work to find a unit ratio.

Mathematics in this Lesson

Common Core Math Standards

CCSS.M.7.RP.A.2.b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

In this lesson, the students use two strategies to identify a unit rate for a car that is traveling 10 miles in 4 minutes.  First, they find the unit rate by a numerical operation. They divide both the number of miles and minutes by 4. They state that this car is going at a speed of 2.5 miles in 1 minute. Secondly, the students also create a diagram to determine the unit rate of this car. They partition the diagram representing a car going 10 miles in 4 minutes into four identical little trips of 2.5 miles in 1 minutes.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP4Model with mathematics.

According to the Common Core’s description of Math Practice 4, mathematically proficient students “identify important quantities in a practical situation and map their relationships using such tools as diagrams.” In this lesson, Kate and Christopher, use diagrams in two productive ways. First, they use a diagram to show why traveling 2.5 miles in 1 minute is the same speed as traveling 10 miles in 4 minutes by iterating identical small journeys of 2.5 miles in 1 minute to make up the larger journey of 10 miles in 4 minutes [2:39 in Episode 2]. Later, they identify that a journey of 10 miles in 4 minutes can be partitioned into four identical trips of 2.5 miles in 1 minute [6:02 in Episode 3].