## Real World Math in Proportional Reasoning and Linear Relations

Here’s a new Real World / 3 Act Math Task that is related to proportional reasoning and a direct variation linear relation with a partial variation sequel to boot!

Please note that this is **part one** of three tasks in a series:

- 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: What’s the Question?

Students watch the first video and begin discussing what the question will be:

How many packages of paper will it take for the stack to reach the ceiling?

Students can view the first video here:

Students then begin making their best guesses by using a “high/low” strategy to set boundaries for their best guess:

## Act 2: Reveal Some Required Information

*UPDATED DEC 23, 2018*

I’ve updated this task to provide afriendly numbersversion as well as theactual valuesversion so the task can be modified to suit in a wider variety of classrooms.

### Friendly Numbers Versions:

I’ve recently created two different “friendly numbers” versions. The first is very accessible by making the height of 25 cm and height of the room 275 cm for working on mental math and playing with the idea of proportional relationships early on.

In this second friendly number version, we state that 5 packages of paper have a height of 30 cm and the height of the room is 332 cm. This is great for trying to promote solving this problem using quotative division with a double number line, ratio table, or other possible mathematical models.

### Original Version With Decimals

Students can view the following video to gain some required information to help them solve the Stacking Paper 3 Act Math Task:

Alternatively, feel free to provide students with the following images:

## Act 3: Check Out The Solution!

Students can then view the solution here:

Note that the video above has been updated to remove the measurements from the video so you can use it for either the original task or the “friendly numbers” version.

Or, you can simply share this photo.

## Discussion Prompts:

Assuming students use the given information and the assumption that the relationship is linear, they will come up with an answer of around 55 packages of paper. This is a great opportunity for classroom discussion around what could have gone wrong in this situation:

- Why was this not a perfectly linear pattern?
- What do you suppose happened to cause the pattern to vary slightly from the linear pattern?
- What sources of human error could have contributed to this?

Since we are stacking quite a bit of paper, students may realize that the weight will make the thickness or height of each stack near the bottom to decrease.

Something else to consider is the fact that the ceiling is made of ceiling tiles. It is possible that we could have popped out a tile in order to fit the last package.

## Sequel: Making Changes to Slope and Initial Value

In the original question, we are dealing with a proportion or linear relation of the direct variation type. Have your students consider what would happen in the following situations:

What would happen to the equation if we stacked the paper in the same room, but on this table instead of on the floor?

Explain how you got your new equation and then use it to determine how many packages of paper you’d need to reach the ceiling in this new situation.

What would happen to the equation if we stacked thicker packages of paper on the table in the same room?

Explain how you got your new equation and then use it to determine how many packages of paper you’d need to reach the ceiling in this new situation.

## Resources: Math Task Templates

Check out some resources to go along with this task.

### Stacking Paper Tasks Multi-Touch Book

Learn more about what’s in the multi-touch book here. Or, consider using the tasks in an interactive series of Google Sites pages.

### Latest Math Task Template

I designed this most recent version specifically for MFM1P Grade 9 Applied math, but it would also work well with MPM1D Grade 9 Academic and possibly grade 7/8 math:

### Previous Version

This previous version was created specifically with the Learning Goals from the MPM1D Grade 9 Academic Math course in mind:

## Stacking Paper Re-Makes

I love when people share remakes of a 3 act math task and encourage you to try it out for yourself! Once you start, you can’t stop!

Here’s one by Tim Boudreau from the Peel District School Board. Excited to see the rest of this roll out:

## Download All Videos & Resources

You can download all of the resources here.

How did it go in your class? Would love to hear about it!

## New to Using 3 Act Math Tasks?

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!

## Share With Your Learning Community:

## 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.

Access Other Real World Math Tasks

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