These exercises, activities and games are designed for students to use independently or in small groups to practise number properties. Some involve investigation (see Related Resources) and may become longer and more involved with subsequent recording/reporting.
- Draw 3D shapes on isometric paper.
- Explore the relationship between edges, faces and vertices of relevant cube structures.
- Explore the relationship between the dimensions of a cube and its volume.
- Interpret drawings of 3D cubes.
- Calculate volume of 3D cube structures.
Multiplication and Division, AM (Stage 7)
- Practice exercises with answers (PDF or Word)
- Isometric paper
- Multi-link cubes
Prior Knowledge.
- Understand what a ‘cube’ is.
- Have a sound knowledge of place value.
- Have a sound knowledge of their basic facts.
Background:
It is important for students to have had early prior knowledge of geometrical concepts so that their understanding is at a level which enables them to be ready for the type of exercises they will encounter in this series of activities. They need to have a spatial awareness of geometric shapes in context. Students will initially have had some difficulty drawing 3D shapes on 2D paper, so it is useful to give them practice opportunities. Algebra patterning is involved in several of the activities, so it is important for students to be able to generalise algebraically and have had exposure to the types of problems that require the development of algebraic thinking skills (a variety of practical activities.)
Comments on these exercises
Exercise 1: Challenges using cubes
Asks students to draw the front, top, and side view of 3D cube structures. Students then count the edges, faces, and vertices to find a relationship. In this exercise the students may require the support of materials in the form of multi-link cubes.
Exercise 2: Four-Cube Houses
Asks students to use isometric paper to find all the different shapes that can be constructed with the same volume. At first glance this problem appears to be quite simple. The task involves finding all the possible four-cube houses you can possibly make. The provisions that the blocks must touch face to face and that the resulting houses cannot have cantilevered rooms will need to be clearly understood by the students prior to them starting the task. The other notion, which is that combinations which are simple rotations around a vertical axis are not counted as they are not considered unique, is something the students are expected to discover. The students are ‘challenged’ to find this answer in Question 3 of the exercise.
The cost structure further complicates the problem and students may find it quite challenging. At this stage it is important to reinforce the notion that it is ‘ok’ to use materials to facilitate understanding.
The challenge of using five cubes may be adapted to become a group activity or project.
Exercise 3: Bigger cubes…
Asks students to discover the relationship between the dimensions of a cube and its volume. For this exercise some students need to have access to multi-link cubes. Each cube will represent one room. Students need to be encouraged to investigate the ‘rule’ for finding the volume. For some students this will only be able to be written in word form. For others the notion of ‘cubed’ will be a natural progression from the knowledge that Volume = width × length × height.
The remaining exercises in this series involve the use of problem solving strategies. These are problems that can be solved by most students, providing they understand the following strategies and skills: make a simpler problem, make a table, make a systematic search, look for a pattern, guess and check, work backwards, draw a diagram, and write an equation.
Students will need time to explore these problems. Any generalisations that are made may be written in word form.
Exercise 4: Painted Cubes
Asks students to investigate shapes that are made up of cubes and explore their properties. This particular exercise scaffolds the strategy of ‘starting with a smaller problem’. This may lead to the student ‘imaging’ some of the tabulated results. (Different numeracy strategy stages will lead to a variety of methods. However, all students should be able solve this problem).
The last part of this task is an extension of the task and involves a further challenge.
Encourage students to see the links between these numbers and the numbers of vertices, edges and faces.
Exercise 5: Castles
Asks students to investigate "castle" shapes that are made up of cubes and explore ways of calculating their volumes. This exercise has less scaffolding and requires the student to apply their knowledge of problem solving strategies in a different context. The last task is an open ended investigation. Students should be encouraged to present their work using words, diagrams, tables, and graphs and the more capable student should be encouraged to use ‘symbols’ to represent a mathematical pattern.
Exercise 6: Staircase
Asks students to investigate "staircase" shapes made up of cubes and explore ways to calculate the volumes. This task requires the use of similar strategies to Exercise 5. It will require the students to have a spatial understanding of what doubling does in terms of the three dimensions we associate with volume. They should have materials available to use if needed.
Exercise 7: Skeleton Tower
Asks students to investigate "skeletal towers" made up of cubes and explore ways to calculate the volumes. This is a more challenging problem and the last task in this exercise requires the students to identify a quadratic pattern. For this reason it should only be given to the ‘more capable’ student. In this task as in Exercise 5 students are also asked to explore and explain different strategies for solving this problem. This task requires the student to apply all previously learned problem solving strategies.