This is a level 2-3 activity from the Figure It Out series.
A PDF of the student activity is included.
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predict a further member in a sequential pattern (Problem 2)
use multiplication facts to solve problems (Problem 3)
use multiplication to find the number of cubes in a cuboid (Problem 4)
FIO, Problem Solving, Levels 2-3, Shaping Up, page 16
Problem One
A combination of strategies could be used to find all the possible shapes. One way is to begin with a simpler problem, that is, to find all the shapes that are possible by joining three triangles.
Only one shape is possible: 
This simpler situation can then be analysed to see where the next triangle could be added to form a four-triangle shape:
Each of the four-triangle shapes can be analysed further to find out which five-triangle shapes are possible:
Students may need access to triangular blocks and isometric dot paper to help them find their solutions for this problem.
Problem Two
This is a typical problem that involves predicting further members of a sequential pattern. It links with the outcomes of the algebra strand. Various strategies are useful, including:
- Building the pattern and systematically counting the matches:
- Making a table and extending the values:
- Writing an equation: 5 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 33
Encourage students to reflect on which methods would be most powerful if the problem were extended. For example, “How many matchsticks would we need to build 15 houses?” In such cases, the table and equation strategies tend to be more efficient than the build-and-count method.
Problem Three
Students will need to apply their knowledge of place value to work out the best placement of digits. In finding the greatest product, it makes sense to have the digits with the largest value in the tens place of the top factor or as the lower factor.
Students could investigate whether this pattern holds for different sets of digits, for example, 2, 5, 9.
Problem Four
Get students to build a model of the building with multilink cubes so that they can demonstrate their methods of counting the number of cubes.
Possible methods include:
Get the students to draw up their own “how many cubes?” buildings for others to solve.
Answers to Activity
1. a.
b. Some possible answers:
2. 33
3.
4. Answers depend on the student’s description of methods. Two methods are by layers or by blocks.