Modelling Word Problems with Equations

What changes?

Solving an equation is only part of a word problem. First you have to decide what the variable means.

Instead of starting with:

start with a definition:

Let s be the number of students who contribute.

Now each part of the equation has a job:

The variable definition is small, but it prevents a lot of mistakes. If s means students, then the final answer should be a number of students.

Build a model before solving

A tape diagram can show the structure of the equation:

One fixed block is $50. The repeated blocks are $2.50 for each student. Together they must reach $300.

Solve:

Now answer in context:

100 students must contribute.

Check:

The answer works because the left side reaches the target total.

A rental example

A canoe rental costs $30 plus $12 per hour. The total bill is $78. How many hours was the canoe rented?

Define the variable:

Let x be the number of rental hours.

Equation:

Use the balance solver to isolate x.

The answer is x = 4, so the canoe was rented for 4 hours.

Percent models still need variables

Alberta GST is 5%. If the total after GST is $84, the subtotal before GST can be represented by p.

Define:

Let p be the subtotal before GST, in dollars.

Since the customer pays 100% + 5% = 105% of the subtotal:

Check in context:

The subtotal was $80.

Misconception probe

Worksheet

For each problem, define the variable, write an equation, solve, and check whether the answer matches the context.

Going further

Some word problems compare two changing quantities, such as two phone plans or two fundraisers. Those create equations with variables on both sides. The modelling steps stay the same: define the variable, build the equation, solve it, and check whether the answer makes sense in the real context.