Crafting A Productive Struggle: Extension Prompt

Share the EXTEND: PROMPT question:

How many glasses of hot chocolate can be made with 55 scoops of mix?

Intentionality…

This task can be used to explore one of the two types of division; in this case, quotative division.

Some of the big ideas that can be deepened from the first portion of this lesson and newly emerged through this task include:

• Ratio reasoning: the idea that a ratio can be thought of as a composed unit and can be scaled in tandem;
• Quotative division can be used to reveal a scale factor (i.e.: how many groups of one quantity are in another of the same unit). 

Student Approach #1: Multiplication With Known Facts, Then Repeated Addition

I know that 1 glass has 3 scoops, so 10 glasses must have 30 scoops.

That wasn’t enough scoops, so I kept counting by 1 glass and 3 scoops each time until I got to 54 scoops and 18 glasses.

I knew that more glasses would be too many scoops, so the answer is 18 glasses with 1 scoop left over.

Strategy:

Student used known facts to scale by a factor of 10 to 30 scoops and 10 glasses. Then, the student used repeated addition until arriving at 54 scoops and 18 glasses with 1 scoop remaining.

Student Approach #2: Doubling, Then Partitioning

I already had my ratio table drawn for the first question and knew that there were 24 scoops in 8 glasses.

So, I doubled both the number of scoops and number of glasses to get 48 scoops and 16 glasses.

Since I was close, I knew that I could add 2 glasses and 6 scoops to get me to 54 scoops and 16 glasses.

I had 1 scoop leftover, but my teacher asked me how much of a glass that scoop represented. So I drew this picture over here (points to bar model in bottom right).

Big Ideas:

Units can be composed and decomposed. For example, this student composed a new unit by combining 2 glasses for a single unit of 6 scoops. Also, this student decomposed 1 glass into equal parts by partitioning 1 whole glass into 3 thirds of a glass.

Strategy:

Student used a doubling strategy to scale by a factor of 2. Then, made a new unit of 2 glasses and 6 scoops to add. Finally, the student partitioned one glass into 3 parts to determine the fractional number of glasses for 1 scoop.

Student Approach #3: Flexible Division Using Repeated Subtraction

I knew I had to figure out how many groups of 3 scoops were in 55 scoops, so I kept trying to subtract as many groups of 3 as I could at a time.

First, I subtracted 10 glasses of 3 scoops (or 30 scoops) and had 25 scoops remaining.

I knew 5 groups of 3 glasses is 15 scoops, so that left me with 10 scoops remaining.

Then, I removed 2 more glasses of 3 scoops leaving me with 4 scoops left.

So I can remove 1 more glass of 3 scoops, leaving 1 scoop leftover.

I drew a double number line to prove that it works.

Big Ideas:

Although this student may not realize it, they are emerging the big idea that quotative division can reveal the multiplicative comparison (or scale factor) between two quantities of the same unit.

Strategy:

The student approached quotative division through repeated subtraction. Instead of subtracting 3 scoops at a time, the student used known facts to subtract combined groups of 3 scoops (unitizing) to be more efficient.

Consolidation

Consolidate learning using student generated solution strategies based on what you selected and how you sequenced while monitoring during the Sense Making portion of the lesson.

In particular, highlighting how despite the fact that some students may choose to scale the composed unit of 1 glass to 3 scoops in various methods (i.e.: using repeated addition, scaling in tandem, etc.) all the way to 18 and ⅓ glasses and 55 scoops, the number of glasses can also be revealed by dividing the number of total scoops (55 scoops) by the rate (3 scoops per glass) to reveal a scale factor of 18 and ⅓.

This is an example of quotative division (i.e.: knowing the rate or “quota” and determining the number of iterations or “parts” of that rate that compose the total number of scoops).

Reveal

After using student generated solutions to make connections, you can reveal this image.

ANSWER: 18 ⅓ glasses

Reflect

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

Consolidation Prompt #1:

Your relative claims that with 16 scoops of hot chocolate mix, we can make 5 and ⅓ glasses of hot chocolate.

Using division and a model to represent your thinking, convince us whether your relatives claim is true or not.

Consolidation Prompt #2:

One student claims that they could determine the number of scoops in 8 glasses by doubling the number of glasses from 1 glass to 2 glasses, then from 2 glasses to 4 glasses, and finally doubling from 4 glasses to 8 glasses.

What must this student do to the number of scoops in order to see this strategy through?

Share another clever strategy you might use to determine the number of scoops for 10 glasses.

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.