# Justification of a solution

In Mathematics, teachers should encourage students to focus on more than just the right answer; students need to understand the process and underlying concepts to derive the right answer (Johnson & Watson, 2011). In other words, students need to find and justify their solutions.

To justify a solution, students will need to be able to use appropriate mathematical language to give reasons for the particular approach used to solve a problem. Any time that a student produces a 'solution' in an attempt to solve a problem, that 'solution' needs to be justified. That is, the student needs to explain how they know that their 'solution' is correct.

Justification of a solution can also arise in the context of a class discussion of mathematics, where students will need to explain their solutions orally.

## Understanding this strategy

To support students to justify their solutions, the teacher can:

• have a class discussion about what it means to justify a solution
• The teacher might ask some students to outline how they could justify a particular solution from a previous lesson.

It can be helpful to discuss key terminology related to the mathematical topic being studied to scaffold students with their discussion. These key terms could be sought from the class and written on the board.

• provide a problem to students and have them solve it, recording their justifications

• ask students to work in pairs to justify their solutions

• ask pairs to share and provide constructive feedback about each others' justifications.

## Example using justification

The example below shows how this strategy can be applied to a Year 10 class on linear equations.

### Scenario: charging for doing a task

Compare the following two calculations for charging for a service, where C stands for the cost (in \$) of completing the task and t stands for the time taken (in hours) to complete the task:

Determine when the first equation is cheaper than the second

C = 25t + 200
C = 30t +150

### Developing a solution

Students will work on the solution of the problem. Either graphically, or by solving simultaneous equations, the time for which costs are equal is 10 hours.

Note: The time for which the first rate is less than the second is any time greater than 10 hours.

### Class discussion

Have a class discussion about what it means to justify a solution. Ask some students to outline how they could justify their particular solution.

Key terminology to discuss could be fixed cost, variable cost, hourly rate, etc.

### Justifying a solution

Ask students to work in pairs to justify their solutions.

Students swap solutions with another pair and add suggestions to improve the solutions.

A justification of the solution would involve the following:

• a check of the solution to the simultaneous equations, probably by substitution
• an explanation using graphs or numerical examples of why the solution is t >10 hours

The above example links to VCMNA335 and is also part of the Mathematics proficiency Reasoning, where students "justify strategies and conclusions reached" (VCAA, n.d.)