Mathematical diagrams can support students to represent a situation and assist in finding a solution to a problem.
Teachers can scaffold students' use of a diagram to solve a problem, by asking, "Can we draw a diagram to help solve this problem?"
Understanding the strategy
To implement this strategy, teacher can:
- provide a problem for students to solve (working in pairs or groups)
- observe whether students use a diagram to help them solve the problem
- ask various students to draw their diagram on the board and compare
- lead a class discussion about the benefits of each diagram.
Examples of using a diagram to reason a solution
There are twenty people at a party and they each shake hands once. How many handshakes will there be?
This diagram was drawn by a Year 8 student solving the handshake problem above.
The teacher asked the student to draw the diagram on the board to explain how she used it solve the problem. The student's reasoning has been transcribed.
Reasoning a solution
The circles are people and the lines are handshakes.
Pointing to the diagram on the left:
"When there are four people, there will be 6 handshakes."
Pointing to the diagram on the right:
"If there was a fifth person, there would be an extra 4 handshakes, so 10 in total."
Each time you add a person, the number of handshakes you add is that person's number minus 1. So, the fifth person adds 4 handshakes, the sixth person adds 5 handshakes.
"When I wrote this down, I saw a pattern. You have to add all the numbers up to the number of people minus 1."
"For 4 people, the number of handshakes were 3 plus 2 plus 1 which equals 6; for 5 people, it's 4 + 3 + 2 + 1 = 10."
"For 20 people, it's 19 + 18 + 17 down to + 1."
This strategy supports the
Mathematics proficiencies Reasoning ("adapt the known to the unknown") and Problem solving ("use mathematics to represent unfamiliar or meaningful situations") (VCAA, n.d.)