Linear Functions: Stacking Cups

By Darice Gammon (inspired by/stolen from Dan Meyer. Google him and read his fantastic blog)

Teacher Overview

Students will discover a linear relationship by collecting data on stacked cups. This may include using tables, graphs, patterns, and equations. The idea is for the students to complete the activity and select students to present their findings to the whole class at the end. The presentations will be selected in order of sophistication, leading up to an equation be presented last to tie it all together. This will lead into a discussion of the teachable questions (listed below) or other questions that could arise. The goal of the lesson is for students to discover that the cups may be represented by a linear equation, with the slope representing the lip of each cup and the y-intercept representing the base of the cup. (1-2 days)

The frameworks addressed by this activity are:

8.F.2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

8.F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Design Rationale/Problem Scenario

How many Styrofoam cups would you have to stack to reach the top of your math teacher's head?

Teachable questions/strategies to watch for:

  • Discuss strategies students used to figure out an answer (table, proportion, guess/check, equation, graph). How does each strategy highlight the slope and the y-intercept in relation to the cup?
  • What was demonstrated through this activity? The goal is for students to see the parts of the cup in a linear function. The lip of the cup represents slope and the base of the cup represents the y-intercept.
  • Which part of the cup matters the most over the long run? The lip of the cup, i.e. slope. The rate of change is more important over time as the number of cups increases. The base of the cup (y-intercept) only counts once, just like an initial value.
  • Does it matter what units you use (centimeters or inches)?
  • How can you move students from tabular methods to writing an equation? Discuss the rate of change in the table and initial value of table to help students form an equation.
  • How can you move students from proportional reasoning to writing an equation? Discuss a proportion as a unit rate. A unit rate is the same as slope. Then, discuss with students how we need to consider an initial value of the cup (the base) as well.

Materials

Styrofoam cups (each group gets a stack of 10)

Ruler

Calculator

Graph paper (if requested)

Sequence of lesson

  1. The question

--How many Styrofoam cups would you have to stack to reach the top of your math teacher's head?

--Have every student write down a guess on a sticky note and put it on the board. Now it’s a competition and they really want to know the answer.

  1. Ask them what they need from you

This is a chance to highlight the common core standard of choosing mathematical tools appropriately.

--Some students will ask for hundreds of cups. Offer them ten.

--They'll want a ruler. Offer that, too.

--Optional (if requested): calculator and graph paper

  1. Let it go

The rest of the lesson largely runs itself. Just walk around, ask good questions, and correct faulty assumptions/misconceptions.

Example of a common misconception: “each cup is 8 cm tall, so 10 cups stacked would be 80 cm tall”

– This would be a direct variation model without an initial value. Students need to make the connection that we are dealing with both a rate of change and an initial value, since we have two parts of the cup to consider (the base of the cup and the lip)

  1. Have students share strategies

--Have students share in order of sophistication of strategy used: guess and check, table, graph, and then finally end with equation(s).

--If groups did centimeters and others did inches, have them compare that too.

--This is really meaningful context for linear equations. The lip-height of the cup represents the slope and the base-height of the cup represents the y-intercept. The end goal of the lesson is for students to make connections from their method (be it table, proportion, etc) to representing this algebraically. Students should be able to see where slope fits in with their method and hopefully move towards writing a linear function.

  1. Actually stack them

--This is the best part. They have to know who was the best guesser initially.

--The winning person receives fabulous cash and prizes (or a piece of candy).

Extension (follow up for the next day or two)

  1. Ask them the same question with a different cup. A red Solo cup, a plastic cup, a thin lip cup, and tall base cup, etc. Have them think about if those cups would require more or less stacked to reach the teacher’s height than their original Styrofoam cup.

--This is a great way for students to make connections and to compare properties of two different functions. It highlights the major difference between rate of change and initial value. Students can compare cups with a big rate of change and small initial value versus cups with a small rate of change and large initial value. Depending on the number of cups, students can make conjectures on which cup would be the tallest stacked.

  1. Toss up this graphic.

Have them measure the lip and base of each.

Good questions:

  1. Which will be taller after three cups?
  2. Which will be taller after one hundred cups?
  3. How many cups does it take stack A to rise above stack B? \

--Especially with this last question, you can segue into linear systems of equations, as well. Students may graph both equations to see the intersection point of both cups or they may use a table to find where the height would be equal. Future lessons may include a form of linear substitution with setting both equations in slope-intercept form equal to each other to find the intersection point.