Students will identify the proportional relationship between the height and number of packs of paper stacked vertically on the floor to determine how many packs of paper it will take to reach the ceiling.

In this task, students will determine how many packs of printer paper it will take to reach the ceiling in a photocopy room. Using the height of 5 packages of paper as well as the height of the room, students will leverage their understanding of ratios, rates and proportionality to solve for the unknown quantity: the number of packages of paper. 

Some of the big ideas that may emerge in today’s task include: 

• There are two types of ratios; composed unit and multiplicative comparison;
• Thinking of a ratio as a multiplicative comparison involves describing
  a quantity as “a number of times greater than another” or describing a quantity as “a fractional part of another” and often (not always) involves a ratio comparing quantities with the same units;
• Ratios can be scaled in tandem to reveal an infinite number of equivalent ratios;
• Although all ratios can be scaled in tandem, ratios thought of as a composed unit lend themselves more naturally to scaling;
• Quotative division is applied when trying to determine a scale factor between two quantities/measures of the same unit (i.e. packs of paper and packs of paper, height and height);
• Scaling ratios in tandem is thought of as a type of ratio reasoning;
• A variety of mathematical models can be used as a tool for solving problems involving a proportional relationship and as a way to represent or prove your thinking. Some of these mathematical models include double bar/double number line models, ratio tables, two variable graphs, and equations; and,
• An equation of two equivalent ratios is known as a proportion;

Show students this video:

Then, ask students:

What do you notice?

What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group.

Some of the noticing and wondering that came up in a class recently included:

• I notice a stack of paper packages on the ground.
• I notice 5 reams of paper.
• There is a clock on the wall.
• The clock says 11:45. Probably the morning, since it is bright outside.
• It looks like the 5 packs of paper is the same width as 1 block in the wall.
• I wonder how many pieces of paper are in each package?
• How many different rectangles are in the image?
• How many packs of paper would it take to reach the ceiling?
• And many more.

At this point, you can answer any notices and wonders that you can cross off the list right away.

Share the following Estimation Prompt with students:

How many packs of paper does it take to reach the ceiling?
Make an estimate.

We can now ask students to make an estimate as to how many packages of paper it would take to stack to the ceiling. Encourage students to share their predictions and justification with neighbours before welcoming them to the group as a whole.

Monitor student thinking by circulating around the room and listening to the mathematical discourse. Encourage students to use precise mathematical language and listen for the use of additive thinking or multiplicative thinking.

Allow students to share their estimates with neighbours first, then with the class. Write down their estimates on the chalkboard/whiteboard/chart paper so students feel their voices are being heard and so they feel they have a stake in solving this problem. At this point, encourage students to model their thinking so that anyone could be convinced of their estimate.

Since you have already taken some time to set the context for this problem and student curiosity is already sparked, we have them in a perfect spot to help push their thinking further and fuel sense making.

You can verbally share the following information with students while also sharing the :

If 5 packages of paper have a height of 25 cm, how many packs of paper will it take to reach the ceiling height of 275 cm?
Convince your neighbour.

Once again, we are going to promote that students leverage the use of strategies and mathematical models without the use of a calculator to create a convincing argument.

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Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

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Show students the following reveal video animation:

Alternatively, you could simply show the final screenshot of the video below:

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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Consolidation Prompt #3:

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Consolidation Prompt #4:

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Consolidation Prompt #5:

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We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

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