Investigate problems involving proportional reasoning including percentages with the possibility of extending to creating and solving equations.

In this task, students will investigate a problem involving a student council vote between two candidates where they will be required to reason proportionally using percentages. This task will allow students to develop a deeper understanding of big ideas related to percentages including the following:

• A part of a whole represented as a percentage is a part-whole ratio whose whole is 100;
• A percentage is one of infinitely many equivalent part-whole ratios in a ratio relationship;
• Quotative division can be used with percentages to scale in tandem and reveal an infinite number of equivalent ratios in a ratio relationship; and,
• Partitive division can be used to divide a percentage by 100 percent to reveal a rate for the proportional relationship.

Show students the video below showing someone writing “Sonya” on a piece of paper and placing it into a cylindrical container.

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

Write down all of the noticing and wondering from your students. For example:

• Black marker
• Who is Sonya?
• Sonya is only name in bucket.
• I saw a ring on his finger.
• Why all uppercase letters except for the ‘o’?
• Why only 1 piece of paper?
• Why? What does this have to do with math?
• Is this a probability game?
• Is this a vote?
• Why so many papers if you only needed 1?
• Is there other stuff in the bucket?
• And many others…

Take time to acknowledge the noticing and wondering your students have engaged in and try to answer any that you can address right away.

Then show video 2 where you will see a second name, Eli, written on a paper prior to seeing those two names repeatedly written and placed in the container:

At this point, it will become clear to everyone that this is a vote and the question we will land on is:

How many people voted?

Give students an opportunity to make an estimate by first thinking independently and then having them share their estimate with a neighbour.

Ask students to estimate a number of square tiles that they think is too low and a number of square tiles that they think is too high before making their “best estimate”.

After sharing out estimates and recording them on a number line, ask students what information they could use to make their original estimates more precise. You might even consider sharing a few additional images like this zoomed in version of the image to see the pile of ballots that represents the total number of votes that will be cast in this student council election.

Another optional image you might consider sharing to give students a better spatial perspective:

Students can now see the votes being sorted, but not counted. The pile of Sonya votes (left) next to Eli votes (right) will make it apparent that Sonya has won the vote, but by how much?

Let’s give students another opportunity to update their estimates.

After they brainstorm and share out, show the video below:

Follow up with:

Update your estimate.

How might we convince someone that the quantity you come up with is reasonable without the use of a calculator?

Students have now been given enough information to now make calculations using mental math to improve their estimates based on spatial reasoning and/or prior knowledge.

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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.

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After students are finished sharing what they noticed from their exploration, share what really happened by showing the following video and/or image:

Answer: 120 votes.

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

Consolidation Prompt:

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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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