What Do You Notice? What Do You Wonder?

Today’s problem based lesson will begin with a “Would You Rather” prompt by showing this image:

Give students time to discuss with elbow partners and/or their table groups before sharing out to the whole group.

Note that at this point, students may have different opinions and perspectives as to which option they would rather choose.

For example, some students might say:

• I’d rather have option #2 because it is less packaging and less complicated.
• Option #1 because my family likes different toppings, so we usually go with two smaller pizzas to make sure there is no “mixing” of toppings.
• Option #2 because it looks like more pizza and I love pizza!
• Option #1 because I love garlic crust and it looks like more crust.
• I think both options have the same amount of pizza, so I’m easy either way.
• And many more!

Show students you value their voice by acknowledging their thoughts and perspectives.

Estimation: Prompt

Next, begin sharing a story about what matters to you and/or your family/friends when ordering pizza. Something like this could help:

Tomas had mentioned that his family really enjoys the crust of the pizza and in my household, we are also fans of garlic even though it can sometimes leave our breath less than fresh!

So, I’m wondering if Tomas and his family are ordering pizza and they wanted the most crust…

Which option do you think will give them the most crust?

Give students some time to use their spatial reasoning to come up with an estimate that feels good to them. Encourage them to share their initial thoughts and wonders with their table groups before allowing students to share their thinking with the class as a whole.

Although the crust wraps the circumference of each pizza, you might encourage students to make other comparisons by posing purposeful questions like:

• What measurement matters if we are trying to determine which option has the most crust? (i.e.: the distance around the pizza also known as the circumference)
• Are there any other measurements that would be helpful for us to consider or compare as we come up with our initial estimates?
• I wonder how much larger the diameter of the large pizza is compared to a medium pizza?

Rather than placing too much emphasis on the estimate itself, we want to encourage students to reason through this scenario. Through this discussion, students may begin to develop more conviction as to which they believe will result in the most crust.

Facilitator Note:

Listen to students as they discuss with their table groups. Are any students making a connection to our previous work related to the multiplicative relationship between the diameter and circumference of a circle we now know as Pi, π? This is an excellent formative assessment opportunity to inform you of where students are related to this very important concept. Often times, educators believe that connections from previous lessons are solidified when in actuality, they are not. It is very possible that students require more explicit exploration and consolidation of the relationship between the diameter and the circumference than you may have anticipated.

After students have been given adequate time to turn and talk to share with their peers, encourage students to begin sharing out their own thinking or what they are hearing from others to the math community as a whole.

Update Your Estimate

Share the following image revealing the diameter of each medium pizza is 12 units long and the diameter of a large pizza is 18 units long.

Note that we are leveraging non-standard units to make this exploration less about a specific standard unit of measure and more about the relationship between the diameter and the circumference.

Facilitator Note:

Again, this is an excellent opportunity to gain insight as to how students are progressing with their understanding of Pi and circumference of a circle. Use this as an opportunity to document your observations, conversations and student product to help you plan forward as well as to help you when it is time to begin evaluating/reporting.

Give students the necessary time to engage in this exploration noting that if students have not yet developed a deep understanding of the multiplicative relationship between the diameter and circumference of a circle that this activity may take longer than you anticipate. Do not rush students as a lack of understanding here will cause significant challenges later in the lesson as we attempt shifting to finding the area of a circle.

If students are not engaging more deeply with this prompt to come up with a more precise claim, consider asking students not only which has the most crust, but also by how much:

After you’ve updated your estimate, about how much more crust will your selected option give you than the other?

Allow students to share their thinking including their strategies and mathematical models with the group to update their estimate with a more precise claim.

Reveal

Share the following video revealing the circumference of each pizza and which option has the greatest circumference:

Consider leaving the final frame of the video up on the screen for students to discuss and debrief.

Answer:

Option #1 has the greatest combined circumference of approximately 76 units in length (approximately 38 units of circumference per medium pizza) compared to 57 units of circumference length for the large pizza.