Layer upon layer
These are level 3 number, algebra, and geometry problems from the Figure It Out series. This is focused on solving problems involving sequential patterns and enlargements, and simple multiplicative strategies to solve problems. 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.
Specific learning outcomes:
- Solve problems involving sequential patterns (Problems 1 and 4).
- Use simple multiplicative strategies to solve problems (Problem 2).
- Solve problems involving enlargements (Problem 3).
Layer upon layer
Achievement objectives
GM3-6: Describe the transformations (reflection, rotation, translation, or enlargement) that have mapped one object onto another.
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
Required materials
- Figure It Out, Level 3, Problem Solving, "Layer upon layer", page 6
See Materials that come with this resource to download:
- Layer upon layer activity (.pdf)
Activity
Many students will need to build the model with cubes. This is good visualisation in itself because it involves interpreting a diagram of a solid object. Some students may realise that they can calculate the number of cubes in each layer of the building.
The next layer will be a 5 x 5 square, so it will be made of 25 cubes, giving a total of 55 cubes. Students may be interested in the pattern created by the square numbers, which are central to this problem:
Each difference is the number of extra blocks needed to make each successive square:
The first block, which becomes a corner square, ensures that each difference is an odd number.
Each side of the pentagon is 2.5 centimetres long, so the total perimeter is 5 x 2.5 = 12.5 centimetres. Twice around the pentagon will be 25 centimetres.
Multiplication and division can be used to solve the problem efficiently instead of repeatedly adding 25. Two possible solution strategies are:
1.
4 x 25 = 100 cm (eight times around)
(6 x 4) x 25 = 600 cm (48 times around)
600 + 25 = 625 (50 times around)
Since 50 round trips end exactly on 625 centimetres, Wiremu Wèta’s journey begins and ends at A.
2.
625 ÷ 25 = 25 double trips, which is 50 trips in total.
There is no remainder, so Wiremu Wèta begins and ends on A.
Students may enjoy variations on this type of problem. For example:
Nursery sticks can be used to model the figure and its enlargement.
Ten sticks were needed to make the original figure, so twice as many sticks (20) are needed to make the enlarged figure. The diagram below shows that the area of the shape increases four times.
Students may wish to investigate whether this also occurs with any closed figure when the side lengths are doubled. For example:
Some students may look for a pattern from left to right if they do not read the question carefully. The different pattern this shows is a useful discussion point.
Following the pattern from right to left shows:
Following the pattern from left to right shows:
Students may need more experience using the difference technique to analyse number sequences.
For example:
1.
a. 30
b. 55
2.
A
3.
a. 20
b. The perimeter will be twice as long. The area will be 4 times as big.
4.
14
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