# First Rate

In the problem First Rate, students use measurement, rates of change, and algebraic thinking to solve problems involving proportional relationships, metrics, and multiplicative relationships. The mathematical topics that underlie this problem are repeated addition, multiplication, division, percents, linear measurement, proportional reasoning, rates, distance-time-velocity, change, functions, algebraic reasoning, and related rates.In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K
In this task, students are asked to determine who wins a race between two brothers jumping up the stairway, where one jumps farther than the other.

LEVEL A
In this level, students are presented with a measurement problem. In the problem, students are asked to determine who wins a race between two brothers jumping up the stairway, where one jumps farther than the other. The students have to reason between number of jumps and length of jumps.

In this level, students count to answer “how many” questions (K.CC.B.5) to identify the number of jumps each student must make; they also answer questions to compare the number of jumps the brothers must make (K.CC.C.6). Students also use some logic to identify which brother will win the race up the stairs.

LEVEL B
In this level, students are challenged with a problem involving two runners who run different rates. They will need to reason about the relationship between the two rates to determine who is faster.

In this level, students solve a problem using a unit rate (6.RP.A.3b) by comparing the rates at which two runners travel to determine which runner will reach the finish line first. To solve the problem, students must calculate the unit rate of each runner (6.RP.A.2).

LEVEL C
In this level, the students are presented with the challenge of determining how two students different rates can complete at ask. The students are asked to determine how fast the task can be completed when the two students work together.

This level supports 7.RP.A.3 as students must reason through a problem situation using rates. One approach to solving the problem involves writing and solving an equation (7.EE.B.4a).

LEVEL D
In this level, students analyze track races where runners are running at different rates. The students investigate when a runner needs to make a change in order to overtake a runner in front of him.

This level supports 7.RP.A.2b and 7.RP.A3 as students solve a multi-step problem involving ratios and rates. They determine various unit rates of runners and adjust these rates given different distances and times needed to win a race.

LEVEL E
In this level, students are presented with a situation that involves related rates in the context of a football game. Students are asked to determine how far a receiver is from the quarterback at a given moment and the rate at which the distance between them is changing.

This level supports G-SRT.C.8 as students are asked to find side lengths of a right triangle using the Pythagorean Theorem in an applied problem. Students also find the average rate of change of a quantity over an interval (F-IF.B.6) to determine the rate at which the distance between the quarterback and receiver is changing. Finally, students generalize this relationship by examining data and writing an expression that gives the change in distance between the quarterback and receiver (F-BF.A.1a). This level builds students’ intuition around related rates, a key concept in calculus.

PROBLEM OF THE MONTH

You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide.  Download the guide as iBook with embedded videos or Download as PDF without embedded videos.

SOLUTIONS
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