Cantitowers

This activity gives students opportunities to visualise and build 3-dimensional models based on 2-dimensional diagrams of side views and then to create and draw some of their own. Some will need very little help with this; others will find it surprisingly difficult. You should be prepared to guide the latter group through the decoding process. These notes suggest how you might do this.

Activity

Introduce question 1 by looking together at the diagrams of the first model. Help the students decode the information by asking questions such as:

• What clues do these four diagrams give us about Ainslie’s first model? (Possible ideas are: there are 3 layers; there are 2 blocks on the first and second layers, but only 1 block on the top layer; it’s a long, thin model; 1 side goes up in steps …)
• How many blocks are in the model altogether? (5. In this activity, they’re told the number, but this won’t always be the case.)
• Why does it look like there are only 3 blocks in the two side views? (2 blocks are hidden behind the 3 in front.)

Draw a diagram like this on the whiteboard and say “This is a picture of an 8-stud block, but only 2 studs are shown. Why?” (Because we’re looking at the end of the block.)

To help your students recognise the significance of the studs shown in the 2-D diagrams, draw diagrams on the whiteboard of these two views of a 2-block model and ask:

• What is the difference between these two views? (The left view shows the joining studs on the bottom block.)
• How does this difference affect the 3-D model?
• Using just 2 blocks, can you show me what the model might look like?

Make sure the students recognise that the bottom block in view A must have at least 2 of its studs protruding in front of the top block but that we can’t tell from this view exactly how many. Here are the possibilities for 2 joined blocks:

All the studs on the bottom block in model B are hidden from view. This means that they are either tucked up inside the top block or shielded by it. If we had only the one view, the diagram could represent 1 block sitting squarely on another, or a pair of receding steps. We can’t tell from this view if there is a step, and if there is one, how big it is. Here are the possibilities for 2 joined blocks:

To explore how blocks that are the same distance or different distances from the viewer can appear the same, have the students make the two 3-block models below and then show you the sides of their models that are represented by the 2-D diagram on the right. (Draw it on the whiteboard.) Discuss how the 2-D view doesn’t show whether a block on the bottom layer is further forward or back from its neighbour and warn them to be aware of this when they are making the models in the activity. It is only by checking a model against all four views (front, back, and sides) that we can be sure that it is correct.

Now have the students make Ainslie’s first model and compare it with the given views and a classmate’s model. If they have difficulties, suggest that they start with the top or bottom layer and work systematically, one layer at a time, turning their model and checking all four views each time they make a change.

These activities will show which students are persevering in the face of difficulties. Show that you value this quality and encourage them to keep trying, having a go at each of the models in turn and checking them carefully against all four views before comparing them with a classmate’s. If their model is different from their classmate’s model, get them to look again at the four views and find out what is wrong.

When talking about the models, use these terms and encourage your students to do the same:

• Face: A flat surface
• Edge: Where 2 faces meet (as long as they are not parallel)
• Vertex: A corner, where edges meet (plural: vertices)

Question 2 calls for students to draw the front, back, and side views of three of their own models on square dot paper. This is related to but different from the skill of interpreting drawings done by others, and some may find the task difficult. If this is so, ask them to make and draw the 2-block models illustrated here before attempting their own more complex 5-block models.

If students have difficulties with question 3, remind them that there are 5 blocks in each model and ask them:

• How many blocks are in the top layer of model a?
• How do you know? (There must only be 1 block, because two views show 4 studs [the long sides] and the other two show 2 studs [the short sides].)
• How many blocks are in the bottom layer of model a?
• How do you know? (It must be made up of more than 1 block because the front and back views show 6 studs.)

Encourage lots of trial and checking, emphasising the importance of perseverance and being systematic.

After the activity, ask questions such as these to encourage reflective thinking:

• If you were teaching another person how to understand these 2-D diagrams, what would it be important to tell them?
• Which parts of this activity did you find difficult?
• What did you do to solve the problems that you met?

1.

Practical activity.

2.

Practical activity. Answers will vary.

3.

Practical activity.