Square Skills

Activity

In this activity, students use multiplication to solve area and perimeter problems. The activity usefully complements Rubber Band Rectangles (pages 14–15) by revising the main ideas. It could also be used as an assessment task.
A co-operative problem-solving approach could be used to work on this activity, with students working in small groups of 3–4 and then reporting back and comparing ideas and solutions with other groups.
Key vocabulary to discuss before the activity:
• Perimeter: the distance around the outside of a shape (imagine an ant walking around it).
• Area: the surface that a shape covers, measured in squares (imagine a square blanket covering the floor).
Some students may need to solve question 1 by manipulating materials such as multilink cubes or squares of card to represent Grandma’s quilting squares.
A key idea underpinning this problem is that of factors. (See Multiples and Factors, pages 16–17.) The students need to use factors if they are to solve these problems using number properties rather than by manipulating materials. Finding all the different rectangles that have an area of 24 squares involves finding all the pairs of factors of 24: namely, 1 and 24, 2 and 12, 3 and 8, and 4 and 6.
After the students have completed the problem, discuss factors and define them together and then ask:
• What are the factors of 24? How do they relate to the rectangular quilts with an area of 24 squares that you drew?
• How could using factors help us to find all the possible different rectangles that have an area of 30 squares? (We could list all the factors and then pair them up with another factor that will multiply together to make 30: namely, 1 and 30, 2 and 15, 3 and 10, and 5 and 6.)
For question 2, encourage the students to find a way of working out the perimeter that uses multiplication rather than simply adding the four sides. Expect responses such as: “I look at how many squares there are in one row and how many rows there are. I add those two numbers together, and then I double the total.”
Return to the idea of using factors to solve question 3a: What rectangular shapes could Grandma make using 36 squares?
When they are doing question 3, your students may not realise that a square is also a rectangle, and for this reason they won’t include a 6 by 6 square in their list. Have them try and define the qualities of a rectangle, and then discuss with them whether a square also has these qualities. It does, so a square is a special sort of rectangle: one with all its sides the same length, just as an equilateral triangle is a special sort of triangle.
To promote generalisations, ask:
• What do you notice about rectangular shapes that need more binding (have a longer perimeter) compared to rectangular shapes that use less binding? (Rectangles that are long and thin have longer perimeters, while rectangles that have the same area but look more square have a shorter perimeter.)
• If Grandma was running short of binding, what would you advise her to do when arranging her patchwork squares? (Try to make a square rectangle, or as close to one as possible, because a square has the smallest perimeter of any rectangle.)

Links

Numeracy Project materials (see https://nzmaths.co.nz/numeracy-projects)
• Book 8: Teaching Number Sense and Algebraic Thinking
Squaring (relating the squaring of numbers to square shapes), page 28
Square Roots (using square roots to find the length of the side given the area of a square), page 29
Prime Numbers (representing prime and other numbers by arranging tiles into rectangles), page 32
• Book 9: Teaching Number through Measurement, Geometry, Algebra and Statistics
Investigating Area (comparing and measuring areas), page 11
Figure It Out
• Basic Facts, Level 3
Factor Puzzles (using basic facts to identify factors), page 11
Stars and Students (game using multiplication facts to identify factors), page 12
• Basic Facts, Levels 3–4
Matrix (finding factors), page 10
A Matter of Factor (factor game), page 12
How Many Factors? (factor investigation), page 15
Primates (finding prime factors), page 22
www.nzmaths.co.nz
A Prime Search (making arrays to explore prime numbers),
https://nzmaths.co.nz/resource/prime-search

 Answers

1. Grandma could make a 1 by 24, a 2 by 12, a 3 by 8, or a 4 by 6 rectangular shape. (Note that a rectangle that is 4 wide and 6 long is the same as one that is 6 wide and 4 long.)
2. a. Note that all the squares have 10 cm sides.
So: 500 cm for 1 by 24 (because the perimeter is 10 + 240 + 10 + 240 = 500 cm);
280 cm for 2 by 12 (the perimeter is 20 + 120 + 20 + 120 = 280 cm); 220 cm for 3 by 8 (the perimeter is 30 + 80 + 30 + 80 = 220 cm); 200 cm for 4 by 6 (the perimeter is 40 + 60 + 40 + 60 = 200 cm)
b. The areas are all 2 400 cm², but the perimeters are all different.
3. a. 1 by 36; 2 by 18; 3 by 12; 4 by 9; 6 by 6
b. i. The 6 by 6 rectangle uses the least binding. (The perimeter of a 6 by 6 rectangle is 6 + 6 + 6 + 6 = 24 x 10 cm = 240 cm. The perimeters of the other rectangles are all more than 240 cm.)
ii. The 1 by 36 rectangle. (1 + 36 + 1 + 36 = 74 x 10 cm = 740 cm)