Pinning it down
This is a level 2-3 activity from the Figure It Out series. A PDF of the student activity is included.
About this resource
Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.
This resource provides the teachers' notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.
Pinning it down
Required materials
- Figure It Out, Problem Solving, Levels 2–3, "Pinning it down", page 17
See Materials that come with this resource to download:
- Pinning it down activity (.pdf)
Activity
The key to solving each of these problems is to identify how many line segments meet at each point.
For example, in shape a:
Points i and iii have two line segments joining them. Points ii and iv have three line segments joining them. They are called odd points. An odd point can have one, three, five, seven, etc. line segments meeting at it.
A figure can only be drawn if there are only two odd points (nodes), like ii and iv in shape a. If the figure has two odd points, students must start drawing the shape at one odd point and end at the other.
Some solutions for the other shapes are:
As an extension, get students to draw networks like these that cannot be drawn without lifting the pencil.
A table (matrix) is a useful way to organise the results of the clues:
The blue and yellow cups hold an old number of counters. This means that the green and red cups must hold an even number of counters:
|
1 |
2 |
3 |
4 |
|---|---|---|---|---|
Green |
X |
|
X |
|
Yellow |
|
X |
|
X |
Blue |
|
X |
|
X |
Red |
X |
|
X |
|
The green cup has twice as many counters as the red cup. There is only one way this can happen, so the green cup must have four counters, and the red cup must have four counters:
|
1 |
2 |
3 |
4 |
|---|---|---|---|---|
Green |
X |
|
X |
✓ |
Yellow |
|
X |
|
X |
Blue |
|
X |
|
X |
Red |
X |
✓ |
X |
|
The blue cup has three fewer counters than the green cup. This must mean that the blue cup holds one counter and the yellow cup has three counters:
|
1 |
2 |
3 |
4 |
|---|---|---|---|---|
Green |
X |
X |
X |
✓ |
Yellow |
X |
X |
✓ |
X |
Blue |
✓ |
X |
X |
X |
Red |
X |
✓ |
X |
X |
There is no particular best order in which to consider the clues in this problem. Some students may solve the problem by considering the clues without using tables. For example, the last clue defines how many counters are in the blue and green cups.
Students need to attend to the conditions of the problem, especially the fact that they do not need to create new triangle shapes that are the same size. They will need to look for an element in the pattern that, when moved, produces more triangles:
Since the last digit is 7, this eliminates many possibilities. An organised list of possible numbers is:
| 4 2 5 7 | 2 4 5 7 | 5 2 4 7 |
| 4 5 2 7 | 2 5 4 7 | 5 4 2 7 |
Since the machine will eat her card after three wrong tries, Trudy’s mum has three out of six (1/2 or 50%) chances of getting her PIN right. Students could investigate her chances if she didn’t know that seven was the final digit. In that case, there would be 24 possibilities, so she would have three out of 24 chances (1/8).
1.
You can draw all the shapes.
Two solutions for a are:
Solutions and hints for the other shapes are outlined in the notes for this page.
2.
blue: 1
green: 4
yellow: 3
red: 2
3.
4.
a. Answers will vary. There are 6 combinations ending in 7.
b. She has a 50% chance of getting her money.
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