Workshop 2 Facilitator’s Guide– page 1

Background Information for Doing Mathematics, the Main Activity of the Workshop This is a classic geometric problem that can help students develop and consolidate their understandings of basic concepts of two and three-dimensional geometry, and of how geometry connects to basic arithmetic. The problem statement (Participant Handout 2-16): You and your teammates represent the Best Solutions Consulting Company. Out of This World Candies has engaged you to solve the problem described in the following memo: To: Best solutions: To: Best Solutions Consulting Company From: Out of This World Candies Re: Problem to be solved Our company, Out of This World Candies, plans to sell our Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts. * Each wrapped Starburst has a square shape that measures 2 cm on a side and 1 cm high. In your report we want you to tell us:

1. The dimensions in centimeters of all the possible boxes we can use to package exactly 24 Starbursts.

2. The dimensions of the least expensive box for us to produce. 3. An explanation of your answers to parts 1 and 2. 4. A suggestion to us about which one, of all the boxes, you think

would be our best choice. We want to know why you think a particular box is the best choice over all the others.

Workshop 2 Facilitator’s Guide– page 2

Analysis of the “Best Box” Problem: The problem given is not difficult—but because it is complex and because participants need to figure out for themselves what mathematics to use and how to approach the problem, it provides a high-level of cognitive demand, even for participants who are mathematics teachers. The instructional approach suggested here for workshop participants models the approach we recommend for use with English language learners. The work is divided into three steps so that participants can share their mathematical thinking and their plans at several points. Each step allows opportunities for participants to work individually, in pairs and as a large group. They also allow you to assess the progress of the participants and offer any scaffolding needed. We also provide you with some mathematical information and questions that can help you guide the participants if they need suggestions. After reading through the problem and the facilitation suggestions, you should adapt this approach to the group you are working with. Your two most important objectives here are to engage the participants in a lively mathematical challenge; to support them through protocols and questions in mathematical conversations and thoughtful problem solving. • The basic mathematics of this problem is not difficult. Participants should

remember the basic formulas for calculating the volume of a rectangular prism (V = L X W X H) and the area of a rectangle (A = L x W). They should also be able to determine whether a number is or is not a factor of another number.

• Finding possible boxes involves being able to stack Starbursts in rectangular

arrays and visualize and draw different ways that 24 starbursts can be stacked to fit into a box.

• The least expensive box is the one that uses the smallest amount of material.

The surface area of a box is one way to measure the amount of material—the amount of material needed to wrap around the Starbursts.

• The box with the smallest surface area is the one with the most compact

dimensions. The smallest box has dimensions 4 cm x 4 cm x 6 cm and a surface area of 132 cm2. This corresponds to four layers of 6 Starbursts, each layer being a 2 x 3 array of Starbursts. In comparison, the most spread out box with dimensions 1 cm x 2 cm x 48 cm, has the largest surface area, 292 cm2. This corresponds to a single row of 24 Starbursts.

• Calculating surface area is simply a matter of adding up the areas of the six

rectangular faces making up a particular box.

Workshop 2 Facilitator’s Guide– page 3

• Probably the most difficult part of the problem is being able to develop a systematic way of organizing data about different boxes in order to determine whether they have found all the possible boxes (there are ten different possible boxes). It involves keeping systematic records of each new arrangement, and determining whether it fits into a different-sized than any previous arrangement.

One way to find all the boxes is to systematically make stacks of Starbursts arranged in rectangular prisms until no more can be made. Start with all the stacks that are 1 cm high. Make all the possible prisms. Then make a stack 2 cm high and do the same. Continue with 3 cm and 4 cm. Participants who continue to stack Starbursts 6 cm high and 8 cm high will discover that the dimensions of the prisms they find are the same as ones they found using stacks with lower heights. Participants will probably not realize this until they build all the possible stacks, the highest one being 24 cm high. (Dimensions 2 x 2 x 24.) This has the same dimensions as a stack 2 cm high and 12 Starbursts long:

• An excellent way to find out whether all possible boxes have been found is to use the fact that all the dimensions (lengths, widths, heights) must be integers. The volume of a single Starburst (1 cm x 2 cm x 2 cm) is 4 cm3, the volume of 24 Starbursts is 96 cm3. Therefore only integers that are factors of 96 can form boxes that hold 24 Starbursts.

One way to prove this is to choose a length that is not a factor of 96, for example, 10. Since 96/10 = 9.6, at least one of the other dimensions must be a fraction since 10 x W x H must equal 96. But we know that all the lengths must be integers, so 10 cannot be the length of a box. The same will be true of any number that is not a factor of 96.

The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96. So the boxes that can hold 24 Starbursts are limited to all the possibilities for which 3 of these numbers multiply to 96. For example, 3 x 4 x 8 (this could be a 2 x 4 array of Starbursts stacked 3 layers high). Not all factor combinations will result in a box that can hold Starbursts. For example a box with dimensions 1 x 1 x 96, is too thin to hold any Starbursts; a box with dimensions 1 x 3 x 32 can only hold 16 Starbursts, with 8 Starbursts left over that can’t be fit into a space that measures 1 x 1 x 32.

Workshop 2 Facilitator’s Guide– page 4

The following table shows all possible boxes and their surface areas. Dimensions of Box in cm. (the product, h x w x l) must equal the volume, 96 cm3)

Surface Area of Box in cm2 SA = 2 x (h x w + l x w + l x h)

1 x 2 x 48 2 x (2+ 96 + 48) = 292 cm2 1 x 4 x 24 2 x (4 + 96 + 24) = 248 cm2 1 x 6 x 16 2 x ( 6 + 96 + 16) = 240 cm2 1 x 8 x 12 2 x ( 8 + 96 + 12) = 232 cm2 2 x 2 x 24 2 x (4 + 48 + 48) = 200 cm2 2 x 4 x 12 2 x (8 + 48 + 24) = 160 cm2 2 x 6 x 8 2 x (12 + 48 + 16) = 152 cm2

3 x 2 x 16 2 x (6 + 32 + 48) = 172 cm2 3 x 4 x 8 2 x (12 + 32 + 24) = 136 cm2 4 x 4 x 6 2 x (16 + 24 + 24) = 128 cm2

From the table we can see that the box with the smallest surface area is the one whose dimensions are closest to a cube, 4 x 4 x 6. This is analogous to the situation in two dimensions: the shape with the smallest perimeter for a given area is a square. It is impossible to construct a cubic box for Starbursts with a volume exactly 96 because the cube root of 96 is not a whole number. Note: there are boxes possible with Starbursts stacked to heights greater than 4, but all such boxes have dimensions that are already on the list. For example, there are three possible boxes with a height of 6 cm: 6 x 4 x 4, 6 x 2 x 8 and 6 x 1 x 16, all of which are already represented on the list. Put another way, for any heights larger than 4, either the height is not a factor of 96 (5, 7, 10, etc.) or one of the other factors must be 4 or less, in which case the box will duplicate one of the ones already listed. For example, if you started with height = 6, 96 ÷ 6 = 16, so that the other two dimensions have to be factors of 16: 1 x 16, 2 x 8, or 4 x 4. Starting with 8, 96 ÷ 8 = 12, so the other dimensions must be factors of 12: 1 x 12, 2 x 6 or 3 x 4. And so on.

Workshop 2 Facilitator’s Guide– page 5

NOTES FOR STEPS OF THE PROBLEM Step 1 – Unpacking the Problem (Slides 34-39) Participants decide what they already know that will help them solve the problem; What they need to find out; and what the constraints (givens) of the problem are. Individuals share their information with the whole group. This information is posted on three large pieces of chart paper so that everyone has the same information about the problem. Step 2 – Partial solutions (Slides 40-43) Participants divide into pairs and each pair finds one way to put 24 Starbursts in a box. Groups share their results by drawing a sketch of one box on 8-1/2 by 11 paper, and posting their sketch and its dimensions. Participants think about how many more boxes might be possible. Step 3 – Completing the Solution (Slides 44-52). Participants work as teams to develop their plans for finding all the boxes. When the groups compare results they have to decide as a group whether there are any additional possibilities. Then they work out a plan for calculating the surface area of every possible box in order to find the least expensive box. Reflecting on the Problem (Slides 53-58). Participants answer a series of questions, reflecting on the process of problem solving, on the mathematics they used to solve the problem, and on how they would adapt the problem for English language learners. Background Notes for step 1: Here are some sample answers for Handout 2-2. Participants do not need to list everything in the table below in order to start solving the problem. The most important items – for being able to move ahead and begin solving the problem are the items in bold face. All the other items are significant, but they can be discovered while solving the problem. • If participants do not use the term “rectangular prism” you can ask: Does anyone

know the mathematical name for a standard every day kind of box? • If participants do not mention the units of volume and area, you can ask: What

are the units that we use to measure volume for this problem? And, What are the units that we will use to measure area for this problem?

Workshop 2 Facilitator’s Guide– page 6

What specific information is GIVEN in the problem?

What PRIOR KNOWLEDGE

can we use to solve the problem?

What do we need to FIND OUT that will

help solve the problem?

• Each Starburst has dimensions 1 cm x 2 cm x 2 cm

• A box must hold exactly 24 Starbursts. It can’t be larger or smaller

• The shape of a Starburst is a rectangular prism

• A box that can hold Starbursts is a rectangular prism.

• How to draw a sketch of a rectangular prism

• Volume measures the amount of material that can fill something a three-dimensional shape. Volume is measured in cubic units

• The volume of a box in cubic centimeters (cm3) is the length x width x height

• Area measures the amount of material that can cover a flat surface. It is measured in square units.

• The area of a rectangular surface in square centimeters (cm2) is the length x width

• There are several ways that 24 Starbursts can be arranged in layers to fit in a box.

• The surface area of a box will determine the amount material needed.

• The least expensive box will be the one that uses the smallest amount of material.

• The volume of a Starburst

• The volume of 24 Starbursts

• The dimensions of all the possible boxes that can hold exactly 24 Starbursts.

• A strategy or system for making sure we have found all possible boxes.

• The surface area of every box to find the one that uses the least material.

Workshop 2 Facilitator’s Guide– page 7

Background Notes for step 2 In order to finding the dimensions of a box that can hold 24 Starbursts participants must be able to visualize and/or draw an arrangement of Starbursts that form a rectangular prism, for example, spreading them out flat in a rectangle 6 Starbursts long, 4 Starbursts wide and 1 Starburst high. Such a rectangle will have L= 12 cm, W = 8 cm, H = 1 cm; a volume of 12 x 8 x 1 = 96 cm3, and a surface area of 2 x (12 x 8 + 12 x 1 + 1 x 8) = 2 x 116 = 232 cm2. Participants will need to draw a representation of the box, something like this: Once each group has come up with one possible box, there should be several possibilities displayed publicly for all to see. This is a critical moment in the problem solving process. Your goal is to have the group articulate the idea that there must be some more possible boxes, because there are several different ways to arrange the Starbursts so that they form a box shape. The Facilitator’s Notes for Slide 17, provide specific directions and talking points for helping this happen. Background Notes for Step 3. Step 3 has two parts, making sure that the group has found all possible boxes, and can justify that, and helping them realize that the cheapest box is the one with the smallest surface area. One important idea that can needs to be brought out during this step is the need for a system—both a way to keep track of all the boxes in such a way that the dimensions are easy to compare, and a way to divide the work of finding boxes among the different teams or pairs. The second idea that is helpful to bring out at this stage is the idea that all the boxes that can hold 24 Starbursts have the same volume, 96 cm3, and all the possible dimensions for boxes (that is length, width or height) must be factors of 96. Finally, you may need to help bring out the idea that the least expensive box is the one with the smallest surface area. Dividing the work of calculating surface area among all the groups is also helpful here.

L= 12 cm H = 1 cm

W = 8 cm