Students will:

• determine the volume of a case filled with boxes of Girl Guide cookies.

In this task, students will explore the volume of a rectangular prism by determining the number of boxes of Girl Guide cookies contained within the larger case. This task will allow students to explore a concrete volume model and develop a deeper understanding of big ideas, including the following:

• Volume and capacity can be measured using non-standard and standard units.
• The measurements of length, width, and height are used to determine volume.
• If there are twice as many groups, the total will be twice as much.
• The relationship between multiplication and division

Show students the video below displaying the case of Girl Guide cookies.

After showing the Spark video, you may choose to leave the still image on the screen:

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 student noticing and wondering. For example:

• A box.
• Girl Guide cookies.
• How many cookies are there?
• How many boxes are in there?
• How much does it weight?
• How much does each box cost?
• 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, land on the following question:

How many boxes of Girl Guide cookies are there in a case?

Follow up that question with:

How might we convince someone that the quantity you come up with is correct?

Let them loose to make some estimates based on what they saw.

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

After sharing out their estimates, have students improve the precision of their estimates by restating the prompt:

Update your estimate of number of boxes of Girl Guide cookies in a case.

How might we convince someone that the quantity you come up with is correct?

Share the following images and they must convince their mathematical community of peers as well as the teacher.

You might consider sharing just the image with all four (4) images in one:

Monitor student thinking by circulating around the room and listening to the mathematical discourse.

Select and sequence some of the student solution strategies and ask a student from the selected groups to share with the class from:

• most accessible to least accessible solution strategies and representations;
• most common/frequent to least common/frequent strategies and representations; or,
• choose another approach to selecting and sequencing student work.

The tools and representations you might see students using to convince their peers and/or the teacher include:

• Connecting cubes to build models of the boxes of Girl Guide Cookies.
• Three-dimensional drawings.
• Polydron Frameworks to build models of the boxes.
• As well as others…

Have students share their strategies and reasoning for determining the number of boxes inside the case. Ask them to convince you and their peers that their answer is correct by sharing mathematical models.

Discuss their strategies and elicit student thinking during your consolidation to build off of their current prior knowledge and understanding rather than “fixing” or “funnelling” student thinking to a strategy and/or model that does not connect to their strategy and/or approach.

I started by building one box of Girl Guide cookies out of linking cubes.

I made the box 11 blocks by 17 blocks by 5 blocks. I decided not to worry about the ½ cm from the 17 ½ cm for now. I didn’t have enough cubes to make more boxes, so I decided to take my one box and start tracing it onto a piece of chart paper.

I traced the first box onto the chart paper, and realized I was making the bottom of the case. So I started tracing more boxes right next to the first one.

Since the first box was 5 blocks wide, I started counting by 5s. I knew the bottom of the case couldn’t be more than 19 blocks by 32 blocks, so once I got to 30 blocks long (or 30 cm), I knew I couldn’t fit any more boxes.

The area of my 6 boxes was 17 cm by 30 cm, which I realized would fit into the bottom of the case.

So I thought there were 6 boxes in the case. But then my partner said that our boxes are only 11 cubes high, and that the case is 23 ½ blocks (or centimeters) high. So we decided we could have a second row of boxes on top of the 6 that we traced, so that would be 12 boxes.

I think that the boxes go all the way across the case. The case is 19 cm wide, and each box is 17 ½ cm long. They would fit across the case, and I think there is enough height for two rows of boxes, one stacked on top of the other.

So I decided to only focus on the front face of the box. I drew the dimensions of the front face. 32 cm by 23 ½ cm. I knew I had to split that face in half, because I think there are two rows of boxes. I know that the length of each box is 11cm. When I cut the height in half, I have 11.75cm of room, so the boxes will fit.

The length of the entire case is 32cm, but each box has a width of 5cm. So I had to figure out how many copies of the 5cm box I can fit within that space, and the answer is 6 boxes on each row, with 2cm of extra space. That means there must be twelve boxes in the case.

I know I need to find out how many of the Girl Guide cookie boxes go into the case. So I decided to find the volume of the case by using the formula, (length x width x height). Then I did the same with the box of cookies. The total volume of the case is 14,288 centimeters cubed, and the volume of one box is 962.5 centimeters cubed.

So I decided to start with ten boxes. If I had ten boxes, that would be 9,625 centimetres cubed from the case, so I knew there was room for more. I tried 5 boxes, and that was 4, 812.5 centimeters’ cubes. When I added those two amounts together it was too much. I decided to try 2 boxes, and realized I could add two more. 10 boxes plus another 4 boxes of cookies gave me a volume of 13,474 centimeters cubed. I was going to add one more box, but that would be too much.

Big Ideas:

When the student asked how many copies of this box of cookies are in that case, that is an example of quotative division. This student accessed that division through multiplication by scaling the number of boxes with the volume of each box.

Facilitator’s Notes:

You might consider asking this student to draw how they see the 14 boxes organized within the case. They are encouraged to keep the dimensions of both the case and boxes of cookies in mind.

After consolidating learning using student generated solution strategies and by extending their thinking intentionally, we can share what really happened by showing the video below.

Alternatively or after showing the video, you can show this still image.

Answer: 12 boxes of cookies.

Revisit the student answers. Ask students why their answers may or may not have been exact.

If you wish to pursue an extension of this activity, you could ask students extension problems such as the following:

Extend #1: Spark

Re-watch the REVEAL video or display the still image showing the 12 boxes of Girl Guide cookies on the counter.

We’ll land on the EXTEND #1: PROMPT question:

How many different cases could you design to hold these 12 boxes of cookies?

Ask students to determine all of the possible cases they could design to hold these 12 boxes of cookies and how the boxes would be organized (12 by 1, 2 by 6…).  Students are encouraged to draw each case. Students should also include the dimensions for each case.

Consider the following extension prompt:

Which design will take the least amount of cardboard to make?

Big Ideas:

This task can be used to explore factors of 12 and reinforce the concept of volume. The extension prompt can also be used to explore the following big idea:

• The closer one gets to a cube when changing the dimensions (length, width, and height), the less surface area is exposed.

Student Strategy #1:

I used linking cubes to build 12 boxes of cookies. I arranged them as many different ways as I could to figure out all of my case designs. I labelled the cases with the dimensions using what I knew about the length, width and height of each box of cookies.

Student Strategy #2:

I decided to draw the cases with the boxes inside to determine all of the ways that I could design the case. And then I labelled the cases with their dimensions.

Student Strategy #3:

I know there are three ways to make twelve. 1 by 12, 3 by 4 and 2 by 6. But then I realized because the case has a handle, each one of those can mean two things. Are the boxes in one row of 12, or 12 rows of one?

Where is the top? 

So I decided to draw them and I shaded in the top of the box to show the difference between 12 x 1 and 1 x 12. But when I was doing that, I realized that depending on how I lay the boxes down in the case, that would actually change the dimensions of the case.

Are the boxes laying down flat or standing up? 

So I drew more cases to prove that the way the boxes are placed in the case would actually change the dimensions of the case. I know that there are more ways, but I ran out of time to draw them all.

Extend #1: Reveal

Play the EXTEND #1: REVEAL video below so students can see some of the different ways to organize the boxes.

ANSWER: There are 6 different ways to organize the boxes:

• 12 rows of 1 box
• 1 row of 12 boxes
• 2 rows of 6 boxes
• 6 rows of 2 boxes
• 3 rows of 4 boxes
• 4 rows of 3 boxes

NOTE:

Consider that the orientation of the boxes within the case can be changed. For example, in the 12 rows of 1 box, are the boxes laying flat, standing up on the long end or the short end?

Changing the orientation of the boxes within the case will change the dimensions and result in more case design options.

Extensions question:

The case with 4 rows of 3 or 3 rows of 4 that looks as close to a cube as possible will have the smallest surface area, therefore require the least amount of cardboard.

Extend #2: Spark

Show the EXTEND #2: SPARK image below.

How Many Cookies in a Box? In a Case?

It would be easy to extend this problem to double-digit multiplication by asking students to predict how many cookies would come in an entire case of Girl Guide Cookies. This would have students making a prediction about how many cookies come in a single box (is it 10? 12? 24? who knows!) and then, they’d use that prediction to multiply by the 12 boxes that come in a case.

This can be a great way to build in opportunities to deepen an understanding of multiplication and automaticity around multiplication facts.

Extend #2: Reveal

Play the EXTEND #2: REVEAL video below so students can see the total number of cookies in a box.

ANSWER: 24 cookies

Extend #3: 

Show the EXTEND #3 prompt below:

How much does a case cost?

Facilitator Notes:

Consider creating a story that a Girl Guide Member actually lost an entire case. How much would she have to repay?