This is a level 3 link geometry activity from the Figure It Out series. It relates to Stage 6 of the Number Framework.
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interpret three dimensional drawings
use a table to find a sequential pattern
cubes
These three activities have been designed to help students learn to visualise 3-dimensional objects from their 2-dimensional representations and to make deductions about the hidden parts of the objects based on what they can see. Cantitowers (page 6) is a suitable activity to follow this one.
Before your students begin, you may need to clarify these points:
If students are having difficulty, scaffold the problem by asking:
The key to this problem is recognising that the picture shows three views of the same cube. This means that the red and yellow faces in the illustration at left are the same red and yellow faces in the illustration at right. If students rotate view 3 in their minds one quarter turn, they will give it the same orientation as view 1 and should be able to see that the orange face is on the bottom, opposite the blue.
As a second step, students should imagine view 2 turned so that the blue face is on the top, as for view 1:
Many students find it difficult to visualise and rotate shapes like this in their minds. This doesn’t mean that they shouldn’t try. Let them then test their predictions by recreating the illustrations with cubes and coloured or labelled dots.
Introduce and teach these mathematical words:
If students are having difficulty with question 1, ask them:
Have the students create a 3 x 3 x 3 cube using blocks or multilink cubes.
They can use this to check their predictions.
In question 2, help your students to make links between the 3 x 3 x 3 cube and the larger ones by asking them to explain what is special about the location of each of the 4 kinds of small cube:
Get the students to draw one face of the 4 x 4 x 4 and 5 x 5 x 5 cubes and to colour-code the different types of painted faces, for example:
Ask the students what patterns they see developing. Key patterns are these:
Encourage the students to complete their tables with this level of detail so that the patterns become clear:
Challenge your students to use the patterns they have discovered to predict the number of cubes of each type that there would be in a 6 x 6 x 6 cube or a 10 x 10 x 10 cube. (The numbers get quite large and difficult!)
Swiss mathematician, Euler (1707–1783), found a rule to describe the number of edges a 3-dimensional shape has:
f + v – 2 = e (number of faces + number of vertices – 2 = number of edges.)
Test Euler’s rule on a cube and some other 3-dimensional shapes.
After the students have completed the activities, ask these questions to promote reflective thinking:
Activity One
One of the blocks must have the numbers 0, 1, and 2 on it, and the other must have 1, 2, and 3 on it. Each of the other numbers (4, 5, 6, 7, 8, and 9) can be painted
on any of the remaining faces of either block. The diagram shows possible nets for two cardboard calendar dice. There are many ways of arranging the numbers
so that they work.
Activity Two
Orange
Activity Three
1. For a 3 x 3 x 3 cube:
2. For a 4 x 4 x 4 cube:
Compare this illustration with the one above. Can you
see patterns developing?
For a 5 x 5 x 5 cube:
Printed from https://nzmaths.co.nz/resource/out-sight at 4:38am on the 8th May 2024