Talk about - Four
These are level 4 measurement and algebra problems from the Figure It Out series. They are focused on finding volume of a cuboid and solving problems involving sequential patterns. 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:
- Find volume of a cuboid (Problems 2 and 3).
- Solve problems involving sequential patterns (Problem 4).
Talk about - Four
Achievement objectives
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
Required materials
- Figure It Out, Level 3, Problem Solving, "Talk About: Four", page 24
See Materials that come with this resource to download:
- Talk a about four activity (.pdf)
Activity
Through this problem, students explore the idea of reciprocals. One is its own reciprocal because dividing by one has the same effect as multiplying by one.
Students will need to discuss what multiplying by a decimal less than one means. Multiplying by 0.5 is the same as finding a half of something. So 0.5 x 2 = 1 (because 1 is a half of 2), 0.5 x 10 = 5, and 0.5 x 18 = 9. Finding a half is the same as dividing by two.
Multiplying by 0.1 is the same as finding a tenth of something. So 0.1 x 3 = 0.3 (because one-tenth of three is three-tenths) and 0.1 x 15 = 1.5 (because one-tenth of 15 is fifteen-tenths, which is 1.5). Finding 0.1 of something is the same as dividing by 10.
Dividing by four is like finding a quarter of something. Since 0.25 = 1/4, multiplying by 0.25 has the same effect as dividing by four. For example:
- 8 x 0.25 = 2 8 ÷ 4 = 2
- 24 x 0.25 = 6 24 ÷ 4 = 6
Finding the volumes of the cuboids shown is more difficult than previous problems involving whole numbers of cubes. Students will need to generalise the method for calculating the volume of a cuboid (length x width x height).
Many of them will need to go back to cube models to realise this:
The boxes shown on page 24 of the students’ booklet have no lines to mark cubic centimetres. Putting these in may help some students visualise the dimensions of each cuboid.
Counting the cubes and parts of cubes verifies that multiplying the sides gives the volume:
- 2.5 x 2.5 x 1 = 6.25 cm3.
Similarly, the volume of the other cuboid can be found. (See Problem One on page 18 of the students’ booklet.)
- Volume = 1.5 x 2 x 3 = 9 cm3
Encourage students to visualise the model without using cubes to build it. Students will have a variety of methods for working out the number of cubes in the model. For example:
Alternatively, the model can be divided other ways:
This problem involves continuing a sequential pattern. Students can model it with hexagonal and triangular pattern blocks. As students build the pattern, they will see that for each new hexagon added, four triangles are added.
This can also be shown in a table:
Hexagons |
1 |
2 |
3 |
4 |
5 |
... |
Triangles |
6 |
10 |
14 |
18 |
22 |
... |
The table could be extended to 20 hexagons by adding four triangles each time.
Students who see repeated addition of four as multiplication by four will find more efficient ways to solve the problem, such as:
- 6 + (19 x 4) = 82
1.
a. Dividing by 10 is the same as multiplying by 0.1.
b. Multiplying by 0.25 is the same as dividing by 4.
2.
The volume of box a is 2.5 x 2.5 x 1 = 6.25 cm3 and the volume of box b is 1.5 x 2 x 3 = 9 cm3.
To find the volume, multiply the height by the width by the depth.
3.
56 cubes
4.
a.
b. 82
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