# Writing solutions to worded mathematical problems

In mathematics, students need to develop the ability to communicate their solutions and provide appropriate reasoning to support their working. Scaffolding use of appropriate terminology and notation will support students to be able to communicate their mathematical thinking independently, as will explicitly teaching how to write different elements of written solutions and extended worked problems.

## Understanding this strategy

Written solutions to worded problems in mathematics will contain aspects of the following features:

### Formulate the problem mathematically

• Summary of important information given in a problem
• Definition of any variables
• Statement of relevant formulae, equations or functions that will be used in the solution of the problem
• List any assumptions
• Explain the reason for choosing a particular problem-solving strategy

#### Sample strategies

Sample strategies teachers can use to support this component of writing and mathematical thinking:

### Reasoning and calculations

• Reasoning for steps of working, linked to key information given in the problem
• Results of solving equations and performing calculations
• Supporting graphs, tables or diagrams
• Interpretation of graphs, tables or diagrams

#### Sample strategies

Sample strategies teachers can use to support this component of writing and mathematical thinking:

• Statement of a final answer, linked to the context of the problem
• Check to see that answer is reasonable

#### Sample strategies

Sample strategies teachers can use to support this component of writing and mathematical thinking:

Suggested techniques for helping students to produce written solutions to worded problems are also outlined below. These will be helpful in helping students to build their capacity to communicate written solutions in mathematics.

## Example using written solutions

The example below involves ratios and provides opportunities for students to use a dual scale number line to represent a proportional reasoning situation. Students need to unpack the worded problem and decide on a strategy to solve the problem. The solution obtained then needs to be interpreted in the context of the given problem.

### Example: School fete muffins

For the school fete, four Year 9 students are going to make muffins. There has been a donation of 10 kg of flour by a local baker and the school is supplying the other ingredients. The ingredients for one batch, which makes 12 muffins, is given below:

• 240g of flour
• 60 grams sugar
• 1 egg
• ¼  cup oil
• 1 cup milk

The students want to make as many muffins as possible using the 10kg of flour. Work out how much of each ingredient is needed and the number of muffins that can be made. If the cost of the ingredients is estimated to be \$2.50 per batch, excluding flour, how much will they make if they sell the muffins for \$2 each.

### Formulate the problem mathematically

The teacher can model unpacking the problem and recording important information on the board. The teacher could read the first sentence and the list of ingredients, noting that there is a need to record what we can tell from this information. Students could then record their ideas.

Class discussion could result in a summary of this information on the board, namely:

• Each batch needs 240 grams flour
• 10 kg flour in total.
• Each kilogram is 1000 grams, so we need to work out how many lots of 240 grams are in 1000 grams.

Students can then be asked to summarise the rest of the information in the problem, recording what they know. Ask students what assumptions they need to make (i.e., that each recipe will produce 12 muffins).

### Reasoning and calculations

Teachers can use questioning and discussion techniques to ascertain the proportional relationship when scaling up recipes and ask students to discuss mathematical approaches which might be helpful here. Students then need to solve the problem and record their reasoning to communicate how they have solved the problem.

Teachers can discuss the logical sequence for solving the problem. Questioning and discussion will lead students to record the order that they might need to do mathematical calculations and their strategies for solving.

#### Sample student planning

• Work out number of batches
• Work out other ingredients
• Calculate the cost for the number of batches
• Work out the number of muffins
• Calculate the cost
• Calculate the profit

Students first need to work with the quantities of flour to determine the number of batches to be made.

Following this they can determine the amount of each other ingredient used, recording their reasoning and providing the amount of each ingredient with correct units. Teachers could encourage students to tabulate their results.

#### Sample student planning

• 10kg of flour is 10,000g
• 240 g in one batch, so 10,000÷240 = 41.66
• Since we only have 10kg, we can make 41 batches
• Ingredients: For 41 batches, multiply by 41

Ingredient

One batch​
41 batches

Flour

240g
​9840g or 9.84kg

​Sugar

60g
2460g or 2.46kg

​Eggs

1 egg
41 eggs

Oil

1/4 cup
10.25 cups

Milk

1 cup
41 cups

This multi-step problem wi
ll then require students to determine how many muffins can be made, by multiplying the number of batches by 12. Reasoning should be supplied to support the calculation for the total number of muffins. If students are using technology here, they might not have intermediate working out recorded, so it will be important to discuss the need for reasoning.

### Solution

The final step is to calculate the cost of producing all the muffins (by multiplying the number of batches by \$2.50 and then calculate the expected profit).

41 batches of muffins and 12 muffins in each gives a total of 492 muffins

To make 41 batches costs \$2.50 x 41 = \$102.50

If 492 muffins are sold for \$2 each, this gives \$984

Students should provide a statement of the answer at the end of the report. Encourage re-reading of the problem to check that all information has been used and that the problem has been answered.