# Shape & Space A Years 2 - 11

A Mathematics Centre Resource
You are encouraged to contribute to these notes. Comments from teachers or students, photos and student work samples will enrich the professional experience for all of us.

Introduction

### Materials

• Tricubes
• Cuisenaire Rods work well for Building Views
• See Equipment List

Picture Puzzles have multiple levels of success. They do not have to be 'finished'. They can be revisited and continued.

### Introduction

There are several ways of representing 3D objects in two dimensions. This unit explores key ones in the context of solving 3D puzzles - real 3D puzzles, not just imitations of them in print or on screen. Students not only learn the skills associated with creating plan, elevation and isometric views, but learn that whichever representation is chosen it must leave out a piece of information that is available in three dimensions. They also learn that this issue can be overcome by using more than one representation for the same object.

Displaying house plans and other 2D/3D drawings around the room will make the unit more purposeful. House plans are particularly useful for showing that several different views of the same object are needed to create a more complete mental image of it.

You might also investigate how 3D printing works. This process is the 'big brother' of the activities on the menu, and in fact, the big brother of the program in which the slides were created.

### Summary

Building Views creates table top 'city blocks' of 'sky scrapers'. You can use linking cubes, other cubes which don't link or Cuisenaire Rods. Initially, students are challenged to create 3D towers from top view images given the extra clue of the tower height. Later they work backwards from side views to work out the heights of the towers on the block. In some cases there may be more than one answer. The final challenge involves minimising the number of cubes needed to create two given side views, then recording the top view in the same way as the Picture Puzzle began.

 Encourage students to get out of their seats, crouch down to look at the required view and cover one eye. Soon students will be able to draw the side views 'without even looking'. When students have finished this Picture Puzzle, they will need to paste recording sheets in their journal.

 Dotted lines show where the buildings separate. This view is a single storey building on the corner with a 5-storey building around the corner to the left and a 4-storey building around the corner to the right.

 It isn't necessary to shade at all to show a correct understanding of this isometric view, however, if students want to shade encourage hatching and cross-hatching (for example) so that minimum time is spent to achieve the effect.

 Colour isn't necessary. Students need only show the towers as rectangles and use a number at the top of each to show height.

 Four views are shown. This is the front view. Students are asked to record the top view. They could add front, left, back and right.

 From the four views students are likely to build as shown. But what goes in the middle. It could be either a 1 or 2 storey building and the four given views would still work.

 But what happens if a zero building is allowed - an open courtyard between the towers. That gives a third solution of zero in the middle; and if you allow that option, then this is also a solution. Any others?

 Cuisenaire Rods can't be used for this challenge. Students must use separate cubes. The solution is not at all obvious. It is based on the same principle as the 0 in back street above. Cubes can be moved within rows and columns and still maintain the given views.

 Once the students have the idea, this one is a little tougher. It's a good idea to try these two challenges yourself to get the feel of how cubes can be moved.

 To learn more about this Picture Puzzle click the photo to visit the Task Cameo. Also Maths300 has a brilliant piece of software called Building Views which offers many challenges similar to this puzzle.

### Summary

Using only six cubes and restricting the way they join by using a square of 4 as the starting point, students make a significant number of objects. The objects are used to introduce plan, elevation and isometric views and, later, to show the need for adding extra information to the plan view to show those cubes that are on top or underneath. There are also spatial jigsaw challenges using all the pieces where the 6 cubes can be seen in plan view.

 Students are asked to draw the plan views partly so they record their work, but also because you may not have enough linking cubes for them to make a collection of all the objects. They need all the pieces to be able to do a challenge later in this Picture Puzzle. If you don't have enough cubes, then drawing the plan views using a scale of 2cm = 1 side of a cube provides cut-outs that are good size for the challenge.

 The object has been rotated 90 degrees from that shown on the previous slide.

 The solutions to these slides are recorded in the task cameo link below. They include an invitation to the first student to find the solution without a repeated shape to send it in.

### Summary

The Soma Cube is a classic 3D spatial puzzle invented by Piet Hein. The pieces are a subset of all the ways to join 3 and 4 cubes. The only ones left out are the straight sticks (3 & 4 cubes) and the square (4 cubes). The seven objects in the subset need 27 unit cubes to make them and that happens to be the number of cubes needed to make a 3x3 cube. The puzzle begins with a search for all the ways to join 2, 3 and 4 cubes. The next challenge is make the Soma Cube and the final challenges are based on a more difficult related cube called a Steinhaus Cube. Recording using isometric and plan views is expected.

 All the ways of joining 2, 3, & 4 cubes are recorded in the task cameo for Soma Cube 1 below.

 There are over 200 ways to make the Soma Cube so we given any answers here (yet). It is a good idea to use drawings, or perhaps photography, to make a gallery of the ones discovered in your class. It will generate much discussion about what constitutes 'same' in this context.

 In essence this challenge is working backwards from someone else's drawing to make the Soma Cube. There are just enough clues in the drawing to be able to do it.

 Note that it would not have been possible to do this drawing without the physical object. The drawing alone in the previous slide does not give enough information about what is unseen at the 'back'.

 We would like to see your students' solutions to these puzzles. The plan view with a digital photograph should be enough to give others clues to each construction.

 This cube can be much harder to make because there are only two solutions. However, the strategy often found by students for the Soma Cube, namely start with the more complex pieces and leave the simpler ones until last, still applies here.

The question remains Which two others?. Counting cubes shows the steps are made from 14 cubes. The L uses 4 of them, so there are ten left for the other two objects. They must both be made from 5 cubes. There are only three of those to choose from so with a bit of experimenting the steps can be made and when they are the remaining pieces fit reasonably easily.

 If students are still frustrated in making the Steinhaus Cube, this slide gives another clue. Now we know where the L-shape goes in the steps. Many students initially take the obvious approach of assuming the 3-tower in the steps is made by the L. Now it is a question of which 5-cube objects make up the rest of the steps. This diagram also shows that it can't be the one shown in black. However even knowing this doesn't necessary mean that the two necessary 5-cube objects fall easily into place in our hands to make the steps.

 Again, without the solid object, this challenge would be virtually impossible.

### Summary

Students are given plan views of objects made from 4 tricubes. They are expected to make the object, then draw it in isometric view showing how the pieces fit together. The extra challenge is for partners to create their own object from four tricubes. Then they draw both its plan view and isometric view and swap the plan views. Now they make their partner's object, draw its isometric view and compare it with the previously prepared view.

The solutions to these challenges are recorded in the task cameo link below.