Cubic conundrums

In this session, students explore combinations with two or more colours in a probability situation. They link the combinations to powers of two and possible outcomes.

1.

Use Cubic conundrums 1 (.pptx) up to the end of slide 2 to introduce the "Townhouse problem." Get a couple of students to make two possible townhouses using cubes of two colours. Two important points are:

• The ground sets a reference point in this problem in terms of deciding whether or not two towers are the same. If the earth is considered as a reference, these two townhouses are not the same:

• Since colours in a set of blocks are limited, use light and dark colours to create the contrast needed, e.g. black and white, blue and yellow, etc. This will allow more students to use the same set.

2.

Allow the students plenty of time to investigate the four-story townhouse problem in small co-operative groups. They will need access to connecting cubes, and square grid paper will support them in recording their solutions. Look for the following:

• Are the students systematic in their approach, or do they just create solutions without checking for uniqueness? If they are not systematic, ask the question:
  • How will you know when you have found them all?
• Do they simplify the task by working through sets of colours? (i.e., divide the search into 4 darks, 3 darks and 1 light, 2 darks and 2 lights, 1 dark and 3 lights, and 4 lights).
• Do they look for symmetry to find new combinations? For example, these new townhouses can be found by reflection (or rotation).

• Do they notice structure in the solutions? For example, there are only four possible positions for one dark cube, so there must be four possible towers with 3 lights and 1 dark. That must also be true of the inverse image of 3 darks and 1 light.

3.

After a period of investigation, bring the class together to discuss the students solutions. Focus on the points above. Listen to the students’ methods first, then use slides 3–6 of Cubic conundrums 1 (.pptx) to discuss other possible strategies.

4.

A "simpler problem" strategy reveals there are two possibilities for the one-story townhouse, four possibilities for the two-story townhouse, and eight possibilities for the three-story townhouse.

• How many possibilities might we expect for the four-story townhouse?

5.

Slide 7 shows the data organised in a table. Do the students recognise that powers of two are involved? For example, 8 = 2 x 2 x 2 = 2³.

6.

Slide 8 shows that the doubling of possibilities occurs as a new storey of two possible colours is added to the bottom. Each existing possibility has two possibilities for the next storey.

• Can students draw the tree diagram for the four-storey houses so they find all 16 possibilities?
• Can they match the arms of the tree diagram to a model made from cubes?

Other questions for investigation include:

• How many five-storey townhouses would be possible with two colours of cladding?
• How many three-storey townhouses would be possible with three colours of cladding?
• How many four-storey townhouses would be possible with three colours of cladding?
• Is there a rule to determine the number of possible townhouses with s storeys and c colours?

7.

Suppose the people who buy the townhouses select their colour scheme by reaching into a bin of cubes, with equal numbers of dark and light. The first colour out is the ground floor, then it is replaced in the bin. The second colour out is the second floor, then it is replaced in the bin, and so on.

• What are the chances that the buyer will get a townhouse of only one colour? (two out of 16 or one-eighth)
• What are the chances that the buyer will get an alternate pattern? i.e., dark-light-dark-light or light-dark-light-dark (also two out of 16 or one-eighth).
• What are the chances of the buyer getting two storeys of each colour? (six out of 16, or three eighths)

Students might make up other possible colour options and work out the probability of that option occurring. They might act out the drawing cubes scenario to see what colour schemes they actually get with 16 attempts.

• Does every possibility come out?
• Why is this unlikely?

High achievers might investigate the connection between the possible townhouses and Pascal’s triangle.

In this session, students use connecting cubes to create growing patterns and attempt to find general rules for those patterns. They also graph the growing patterns using spreadsheets and look for patterns in the ordered pairs.

1.

Introduce the cube model using slide 1 of Cubic conundrums 2 (.pptx). Ask the students to work out the number of cubes required. Highlight the importance of structuring the pattern in parts. Invite suggestions about how to count the cubes. Record the suggestions as equations. Slides 2–6 offer some ways to structure the pattern.

Slide 2: The person sees 4 x 4 + 4 = 20

Slide 3: The person sees 2 x 6 + 2 x 4 = 20

Slide 4: The person sees 4 x 5 = 20

Slide 5: The person sees 4 x 6 – 4 = 20 (Subtract four to allow for overlaps)

Slide 6: The person sees 6 x 6 – 4 x 4 = 20 or 6² – 4² = 20

Slide 7: The slide shows three members of the growing pattern. Ask the students to make models of the pattern members using cubes.

• What changes and what stays constant (the same) as the pattern progresses?
  Students should notice that the side length is growing by one cube each successive figure to the right. That defines one variable, side length (in number of cubes). Students should make other observations, like: 
  • The hollow square in the centre is also getting bigger. Each side length of the square grows by one cube each time.
  • The number of cubes needed to make each model is four more than for the model before it. Why does that happen? (One cube is added to each side of the model).

2.

Pose this problem to the students:

• Imagine someone giving you 96 cubes to build a model in this pattern. How long would the side length of your model be in cubes?

3.

Let the students solve the problem in whatever way they want. Look for the following:

• Some students will choose to build the figure. Ask them to think about what they must consider as they make their model larger. They should note that all four sides must be the same length. Can they predict ahead to anticipate the side length of the complete 96-cube model?
• Some students might use tables to organise their data. If their attendance is according to the ‘going up by four’ recursive rule, encourage them to think ahead without repeatedly adding. Ideally, you would like them to find a relationship between side length (s) and total number of cubes (c).
• Some students will bring through the structure they used to count the number of cubes in the figure with a side length of six. They will imagine what side length might get them close to 96 cubes. For example, a model with 20 cubes on each side has 4 x 19 = 76 cubes in total.

4.

After a suitable time of exploration, gather the class to discuss their ideas. Try to connect different strategies with questions like:

• Why would the sequence going up in fours give you a rule that uses "multiply by four"?
• Would the problem be easier if you had 400 cubes instead? Why? What would you do to solve it?

5.

Cubic conundrums (.xlsx) has two pages: Page one contains the start of a table of values for the pattern, and page three has a lengthy table to n = 25. Start with page one and ask the students ideas about how to extend the table to solve the problem for 96 cubes.

The problem can be solved by auto-filling. Encourage students to suggest formulas that might be used. One possibility is to put the formula =A5+1 into cell A6 and =B5+4 into cell B6. These are recursive formulas that work out members of the sequence from the one before. Copying the formulas into columns A and B will continue the pattern.

6.

Encourage the students to look for a relationship across the table between the values in columns A and B. Since the spatial pattern involves four sides, multiplying the A values by 4 is a good start. The final adjustment is to subtract 4 to allow for overlaps. So the formula = A6 * 4 - 4 in cell B6 will give the total number of cubes if 7 is entered into A6. This formula can be copied down to complete the table.

7.

Page two has a completed table and a graph of the relationship. Ask the students what patterns they notice in the graph and why the patterns occur. They should notice that the points (ordered pairs) lie on a straight line. This occurs because the rate of change, four more cubes for each extra cube of side length, is constant. The rise is four for every run of one. You might ask students to create their own table and graph for the growing pattern using Excel.

8.

Use slides 8–10 of Cubic conundrums 2 (.pptx) to motivate the students to create their own growth patterns from connecting cubes. They need to build three members of the pattern, preferably consecutive members of the pattern. Next, they need to pose a problem for someone else to solve. The problem could be:

• Make the next model in the pattern
• What would be the tenth model in the pattern?
• What model in this pattern would you make with c cubes?
• Create a spreadsheet and graph to predict the number of cubes needed for the 50th member of the pattern.

9.

If possible, use digital cameras to take pictures of the models and use them to publish the students’ problems. If students do publish their problems, they need to create a model answer along with useful hints for other students.

In this session, students explore plan views to represent three-dimensional models made with cubes. They interpret the plan diagrams and create their own diagrams for others to build from.

1.

Begin by building a model with cubes in the shape of an apartment block. Place the model on a desk and invite a student to draw what the model looks like from the front. Use an interactive whiteboard or smart television to show a square grid - this will make drawing easier and more consistent. An A3 sized sheet of Cubic conundrums 1 (.pdf) will also work. Invite other students to draw the "Bird’s eye" and left side (relative to the front) views.

2.

Use Cubic conundrums 3 (.pptx) to introduce the problem of the three views. Draw students’ attention to these features of the plan views:

• What does each square represent? (A single face of a cube.)
• How many cubes across is the building? (Four, as shown by both the front view and satellite image.)
• What is the height of the tallest part of the building? (Four cubes high, as verified by both the front and left plans.)
• How tall is this part of the building (pointing to the front square of the satellite image)? (Two cubes high; it is the only stack of cubes visible on the right of the left image.)

3.

You might want to provide the students with copies of slide 3. Ask them to work in small groups of two or three to build a model that matches the plan views that Simon, the spy, has taken. Look for:

• Are the students systematic, or do they attempt to dismantle their model, and start again at the beginning if it does not work?
• Do they start with one view first, for example, a satellite image, get that view correct, then progressively work on the other two views?
• Do they check all three views to validate that their solution meets all the requirements?
• Do they use recording strategies to record the important information in the problem? For example, using the satellite image and recording the maximum height of each row and column is useful.

4.

After a suitable period of problem-solving, gather the class to share their strategies. Look for some students to create familiar chunks of the building to work with. They might create towers of heights 1, 2, 3, and 4 to match the heights they see in the front and left views, then move those towers around to match the satellite image.

5.

Provide the students with copies of Cubic conundrums 1 (.pdf). Ask them to create three views of the building they create. Restrict the students to 12 cubes, so the problems become manageable. After they have drawn their views, students can exchange their three view plans with another student and ask that student to make their building. Digital cameras can also be used to create model answers for the target solid.

6.

Another good activity is to leave the buildings intact. Each student then receives a set of plans and must tour the room until they find a model that matches their plans.

In this session, students learn to represent models made with connecting cubes using isometric dots on paper (orthogonal diagrams or projections). Students will need isometric dot paper (Cubic conundrums 2 (.pdf)) and connecting cubes. An internet search for "isometric drawing tool" reveals online tools that can be used to experiment with digital isometric drawings.

1.

Begin with an isometric drawing beginner class using Cubic conundrums 4 (.pptx). Using mouse clicks on slide 1, students can progressively draw some simple cube models. Show them slide 2 and invite them to make the model first, then recreate the drawing of it. Students could also draw the building model they created in session 3.

2.

After students have attempted drawing, discuss some important features of isometric drawing:

• Why is the page described as ‘isometric’? ("Iso" means the same, and "metric" means measure. So the dots are all the same distance apart.)
• What shape are the square faces of a cube when drawn on isometric paper? (Non-square rhombi – four sides of equal length.)
• How do you decide which faces to shade? (Imagine a light source and shaded the opposite faces of that source.)
• Can you always see all the cubes? (No, often parts of a cube or a whole cube are hidden behind others.)

3.

Slide 3 of Cubic conundrums 4 (.pptx) introduces the pieces that make up The Steinhaus Cube puzzle, invented in 1950 by a mathematician, Hugo Steinhaus. Ask the students to use the photograph to make all six pieces and draw them on isometric paper. Challenge students to solve the puzzle of putting the pieces together to form a 3 x 3 x 3 cube. Students might video their solution for others. Clips of people solving the puzzle are easily found by internet searches.

4.

As a challenge, students might like to make up their own cube puzzle, draw the pieces, and create an ‘exploded’ solution on isometric paper showing how the pieces connect. If they are unsure what an exploded diagram looks like, ask them to search online for examples. Exploded diagrams are important ways to provide assembly instructions for different products, from flatpack furniture to complex machines. Other isometric drawing challenges can be found in the following Figure It Out activities:

• A different view
• Crazy cubes

In this session, students learn to find the volume of a cuboid (rectangular prism) by visualising the arrangement of cubes that will fill it. They also connect the multiplication of length, width, and depth to the arrangement of cubes.

1.

Slides 1-5 of Cubic conundrums 5 (.pptx) introduce the idea of finding the volume of a cardboard packet by filling it with cubes. The issue of part units is also raised as two edges of the packet measure 7 cm, or the length of 3½ cubes.

Provide the students with an assortment of cuboid shaped boxes of varying sizes, for example, toothpaste, biscuits, cereal, shoes, etc.

Tell the students that you want them to find the volume of each box in connecting cubes.

Let the students work in groups of two or three. Expect them to record measures for the volume of each box and how they found that measure.

For example,

• In a WeightWatchers muesli bar box, 7 cubes high by 2 cubes across = 14 cubes, 8 layers is 8 x 14 = 112 cubes.

2.

As the students work, look for:

• Do the students recognise that the cubes can be arranged in a 3-dimensional array?
• Do the students recognise that a "layer" can be created and mapped into the box as many times as needed to fill it?
• Do the students apply multiplication to find the volume, or do they rely on counting or repeated addition?
• Do they record the measures using both numbers and units? For example, 4 x 5 x 3 = 60 cubes.
• How do the students deal with partial units? Do they work with fractions of a cube to get more precision?

3.

After a suitable period of exploration, bring the class together to discuss the key points above.

Slides 6–8 of Cubic conundrums 5 (.pptx) show how to work out the volume of a box from edge lengths. It also illustrates that two factors must be multiplied first. The first two factors represent a layer that is iterated (copied) through the box.

For each slide, represent the volume calculation with a multiplication equation, i.e., 4 x 3 x 5 = 60 (slide 6), 5 x 3 x 4 = 60 (slide 7), and 3 x 4 x 5 = 60 (slide 8).

Ask the students what they notice. The associative property of multiplication means that the pairing of factors does not affect the product.

4.

You might also look at cubic centimetres (cm³) the standard unit of volume. That is the size of a small place value block. Most likely, the students will have measured their boxes using cubes that are 2 cm x 2 cm x 2 cm.

• How many cm³ fit into one of the connecting cubes? (2 x 2 x 2 = 8).

So if the number of cubes is multiplied by 8, then that gives the volume in cubic centimetres.

5.

If you return to Slides 1–5, the volume of the packet can be worked out more accurately by 20 x 7 x 7 = 980 cm³. This is close to 1 litre, which is 1000 cm³.

6.

Provide additional challenges, like:

• How many different cuboids can be made with the cubes that have a volume of sixty 2 x 2 x 2 = 8cm³ cubes?

Sixty cubes is a good target volume since 60 has many factors. Slides 6–8 give one possible answer of 3 x 4 x 5 = 60 cubes. Uniqueness is an issue. For example, in this investigation, 4 x 3 x 5 should be considered the same cuboid as 3 x 4 x 5.

Do the students approach the task systematically?

7.

Ask:

• How will you know when you have found all the possibilities?

One approach is to close off the first factor, like this:

1 x Possibilities: 1 x 1 x 60, 1 x 2 x 30, 1 x 3 x 20, 1 x 4 x 15, 1 x 5 x 12, 1 x 6 x 10 (All done. How do you know?)

2 x Possibilities: 2 x 2 x 15, 2 x 3 x 10, 2 x 5 x 6 (Why do we not include any ones as factors?)

3 x Possibilities: 3 x 4 x 5 (All done. How do you know?)

• Are there other possibilities? (No. Why not?)

8.

Other target volumes can lead to other discoveries.

For example, a prime target volume such as 29 has only one solution: 1 x 1 x 29. Why?

A target volume that is a power of two will only result in edge lengths that are powers of two, for example, 64 cubes: 1 x 2 x 32, 2 x 2 x 16, 1 x 8 x 8, and so on.