Students will explore the number of passengers per car, and the total number of cars on the ferris wheel through partitive and quotative division.

In this task, students will determine the total number of passenger cars, and the numbers of riders per passenger car on the featured SkyWheel ferris wheel in Niagara Falls, Ontario. This task serves to highlight the difference between the two types of division; partitive and quotative.

Some of the big ideas that will likely emerge in this task include:

• Multiplication and division are related.
• There are two types of division.
• Partitive division reveals a rate.
• Quotative (or measured) division reveals the numbers of parts when the rate is known.
• The dividend from any division sentence can be decomposed into smaller parts to allow for friendlier division by the divisor. This strategy is known as partial quotients. (i.e.: 85 ÷ 5 = 45 ÷ 5 + 40 ÷ 5 = 9 + 8 = 17)

Show students the following 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.

Replaying the video and/or leaving a screenshot from the video up can be helpful here.

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

Finally, allow students to share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other means to ensure students know that their voice is acknowledged and appreciated.

Some of the noticing and wondering that may come up includes:

• I notice a ferris wheel
• I wonder if this is at a fair
• I wonder how tall it is
• I wonder how many people can ride at the same time
• I wonder how many people can ride together
• I wonder how much it costs to ride

At this point, you can answer any wonders that you can cross off the list right away. For example:

• This ferris wheel is in Niagara Falls, Ontario, Canada;
• It costs $12.99 for adults to ride, and $6.99 for children; and,
• It is 53 metres tall.

After we have heard students and demonstrated that we value their voice, we can land on the first question we will challenge them with:

How many passengers can ride this ferris wheel at one time?

Make an estimate.

We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to determine a number of passengers that would be reasonable before determining a more precise answer.

Consider asking students to think about a number of passengers that would be "too low" and a number that would be "too high" before asking for their best estimate in order to help them come up with a more reasonable estimate. Encourage students to share their estimates, however avoid sharing their justification just yet. We do not want to rob other students of their thinking.

Reveal the total number of passenger cars by showing this short video clip:

Alternatively, you could show this screenshot from the final frame of the video:

Now that students know the total number of passenger cars on the ferris wheel, you can prompt them to update their estimates by stating something like:

Now that you know there are 42 passenger cars, how does this impact your original estimate? Consider reflecting on your original estimate and determine whether you think it is still reasonable. You can also consider updating your estimate, if you’d like.

Give students an opportunity to revise their estimates.

Facilitator Note:

Be sure to reiterate that we are not leveraging the help of a calculator here as we are always trying to give students an opportunity to build their fluency and flexibility with operations by leveraging strategies and mathematical models - in this case, for multiplication.

Listen and observe as students work to make updates to their original estimates. Are students aware that multiplication or repeated addition will help them come up with a more precise estimate by selecting a reasonable rate for the number of people per passenger car?

What strategies and mathematical models are students leveraging? Are they:

• Skip counting by their estimated rate of people per passenger car?
• Utilizing partial products through a drawn area model?
• Using multiplicative thinking as they make jumps on a double number line?
• Using an algorithm?

Making note of where students are developmentally as they work through this multiplication problem and ensure that you prompt them to explain why they believe their strategy is going to work to come up with a more precise total number of passengers that can ride on the ferris wheel at one time.

Acknowledge students’ estimates and consider doing a mini-consolidation of multiplication strategies and models students used as tools for thinking or tools to represent their thinking.

By doing this, you will not only show students that you value their voice and their thinking, but also have an opportunity to highlight multiplicative strategies and models that may be helpful as we prepare to dive into the actual intentionality of this lesson.

Share that 252 passengers can ride at one time assuming all 42 passenger cars are full.

Celebrate student estimates that were very close to the actual number of riders using a routine of your choice as we head into the sense making portion of this lesson.

Prompt students by stating:

The total capacity of the ferris wheel is 252 passengers. If there are 42 passenger cars, how many passengers can ride in each car?

Use a mathematical model of your choice to defend the maximum number of passengers per car that can ride on the SkyWheel ferris wheel.

Be sure to remind students that they are not to use a calculator to determine the number of passengers per car as using that tool will rob them of this mathematical experience.

Also be sure to ask that students don’t shout out answers - especially since students who were close with their estimate may now have a number close to the number of passengers per car. Rather, we want students to ensure that they have a mathematical model that will be useful for convincing others in the class.

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After consolidating the learning and allowing students to share out, show this reveal video:

Alternatively or in addition to the reveal video, you might consider sharing this reveal image:

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

Consolidation Prompt:

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Reflect:

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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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