Thick Stacks

Graphing Linear Relations Partial Variation Rates of Change Slope

Finding the Equation of a Line Given Two Points

$Thick Stacks 3 Act Math Task - Finding the Equation of a Line Given Two Points$

This 3 Act Math Task is yet another sequel to the Stacking Paper and Stacking Paper – Sequel 3 Act Math Tasks where we look at linear relations with respect to a real world situation involving packages of paper stacked into a tower.

While the original task begins with a direct variation requiring only basic proportional reasoning skills, the previous sequel had students looking at a partial variation linear relation with an initial value being the height of a table and were required to use the thickness of one package of paper (slope) and the height of the stack on a table (a point) to determine the height of the table. In this task, students are given two stacks of paper (thus two points) and must use them to find the height of a different table.

To access the other tasks in this series, see below:

Stacking Paper – Direct Variation (y-intercept equal to 0)
Stacking Paper Sequel – Partial Variation (y-intercept not equal to 0)
Thick Stacks – Finding the Equation of a Line Given Two Points

Act 1: Introducing The Problem

Stacking Paper on a Table

In the first act, students will view the image of 12 packages of paper and 5 packages of paper stacked on a shorter table than that used in the Stacking Paper Sequel problem:

As usual, give students some time to chat with an elbow partner to decide what question they believe will be asked in this problem. You can try to lead the discussion towards a specific question, or work with what they come up with.

Act 2: Reveal Some Information

Height of 12 Stacks and 5 Stacks on a Table

Give students the following image to ponder with their peers:

Since it is likely that they have completed the Stacking Paper or Sequel tasks, it might be useful to show them that the paper packages stacked here have a different thickness in comparison to the previous two tasks:

The question I want my students to answer here is:

How tall is the table?

At this point, I hope students can quickly identify that the height of the table is the y-intercept of this linear relation and thus can use their prior knowledge to help them get started using the inquiry/discovery process.

Stacking Paper Sequel Inquiry / Discovery

The prior knowledge my students had before completing this task:

an understanding of linear relations,
an understanding of slope as a rate of change,
an understanding of the initial value / y-intercept of a line,
an understanding of how to use their knowledge via logic, table of values and a graph to solve problems, and
an understanding of how to use slope and a point to find the y-intercept in order to create a linear equation.

Here are some approaches my students have used when solving this problem:

Logic / Arithmetic:

Students in my class used this method almost exclusively and all students were successful finding the solution.

$Thick Stacks 3 Act Math Task - Logic Arithmetic Solution$

Table of Values:

A couple of students set up what appeared to be a table of values to help them find the slope of the line by using the differences in the table.

$Thick Stacks 3 Act Math Task - Table of Values Solution$

After finding slope, they could then use the method of finding the y-intercept when given slope and a point as we learned in the Stacking Paper Sequel task to find the height of the table:

$Thick Stacks 3 Act Math Task - Using Slope and a Point to Find Equation$

Algebraic Method Using The Slope Formula:

Although we had previously introduced the formula for slope of a line, we have focused primarily on the use of rise over run as well as finding slope using the differences in a table and thus no students utilized the formula for slope:

$Thick Stacks 3 Act Math Task - Algebraic Slope Formula Solution$

Act 3: Reveal the Solution

The Height of the Table Is…

Sequel / Extenions:

Finding the Equation of the Line

Since the majority of our grade 9 academic and applied curriculum involves linear relations and equations of lines, it seems logical here for students to take their knowledge of the height of the table and find the equation representing this linear pattern.

A possible question could be:

How many stacks of paper you would need to reach the top of the school, flagpole or CN Tower?

Obviously, the more stacks of paper needed to find the answer, the more convenient the equation will seem!

Check out this video summary that goes through the problem:

Comparing To The Textbook:

More and more I am really disappointed with how lame the questions in our textbooks and resource guides are. For years, I have been assigning questions that have no context or purpose. Sure, you could argue that there is no need for us to determine the height of a table, but it sure beats this:

$Thick Stacks 3 Act Math Task - Textbook Problem Given 2 Points$

This Task Is Now in a Multi-Touch Book!

Grab It!

Learn more about what’s inside this multi-touch book for iBooks here. Or, consider using this task in an interactive way via Google Sites.

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About Kyle Pearce

I’m Kyle Pearce and I am a former high school math teacher. I’m now the K-12 Mathematics Consultant with the Greater Essex County District School Board, where I uncover creative ways to spark curiosity and fuel sense making in mathematics. Read more.