AC9M2N06: Mathematical modelling for practical problems

Students use mathematics to make sense of practical situations. The important skill is not simply completing a calculation: students decide what the problem is asking, choose a representation and operation, solve it, then explain what the answer means in the original situation.

• Identify the quantities that are known and the quantity that is unknown.
• Represent the situation using materials, drawings, bar models, arrays, number lines, tables or number sentences.
• Choose addition, subtraction, multiplication or division because it matches the structure of the problem.
• Use an efficient Year 2 calculation strategy.
• Check whether the answer is reasonable and whether a remainder needs to be interpreted.
• State the answer with the correct unit and in terms of the real situation.

1. Understand: What is happening? What do we know? What do we need to find?
2. Represent: Draw, model or organise the information so the relationships are visible.
3. Choose: Select an operation and calculation strategy that fits the situation.
4. Solve: Carry out the calculation and keep track of the units.
5. Check: Does the answer make sense? Is it too large, too small or impossible?
6. Communicate: Write a sentence that answers the actual question.

Students should become comfortable moving backwards if needed. If a calculation gives an unreasonable answer, return to the representation or operation choice and revise the model.

Words such as “more”, “left” or “each” can appear in different kinds of problems, so operation choice should come from the mathematical relationship.

• Addition: combine two or more parts to find a total.
• Subtraction: find a missing part, an amount left or a difference.
• Multiplication: combine repeated equal groups.
• Division: share a total equally or find how many equal groups can be made.

Ask students to explain: “I chose ___ because ___.”

A student buys a book for $12 and a game for $8. To find the total cost, combine the two parts: 12 + 8 = $20.

If the student pays with $25, the missing part is the change: 25 − 20 = $5.

A suitable model is a bar showing the $20 cost and the $5 change making the $25 payment. The final answer must include dollars.

Six students each contribute $5 to a class project. The situation has 6 equal groups of $5, so multiplication or repeated addition is suitable:

5 + 5 + 5 + 5 + 5 + 5 = $30, so 6 groups of $5 make $30.

Students can represent the problem using 6 equal bars, an array of 6 groups, skip counting by 5s or a number sentence.

Practical division problems often require students to interpret what happens when the total cannot be divided exactly.

Example: 22 students are placed into teams of 5. Four full teams use 20 students and 2 students remain. The calculation alone is not the final answer; students must explain what should happen with the 2 remaining students in the context.

Possible real-world decisions might include changing the team sizes or forming an additional smaller group. The mathematical model supports the decision but does not automatically choose it.

A calendar can act as the mathematical model when a problem asks how many days remain until an event or until the end of the year.

Students can count the remaining days in the current month, add the days in complete future months, then add any final partial month. The representation should make clear which days are included and which date is the endpoint.

A bare number is usually incomplete in a modelling task. Students should name what the number represents.

• “The class collected $30 altogether.”
• “There are 4 full teams and 2 students left.”
• “Each person receives $9.”

Checking the unit is a simple way to catch many modelling errors.

The curriculum includes modelling equal grouping and sharing through First Nations Australian children’s instructive games, including Yangamini from the Tiwi Island Peoples. When using this elaboration, use curriculum-authorised or community-appropriate source material and avoid inventing cultural details. Keep the mathematical focus on the equal-group relationships, the number sentence and the interpretation of the solution within the source context.

Use a reliable, community-approved source and identify the specific Nation, People or community where the source does. Avoid presenting diverse First Nations knowledges and practices as one generic tradition. Follow local cultural and intellectual property protocols before inviting community participation or reproducing cultural material.

Key idea: Modelling turns a real situation into a drawing, number sentence, table or materials model, then checks whether the answer makes sense in context.

Remember

• Represent situations
• Choose calculation strategy
• Use money contexts
• Interpret the answer
• Communicate solutions

Quick questions

• Model a $6 + $4 purchase.
• What operation fits 4 bags of 5?
• Write a context sentence for 18 ÷ 3 = 6.
• Check if 100 pencils is reasonable for 3 boxes of 4.

If an answer is uncertain, return to the model and explain what each part represents before calculating or drawing a conclusion.

• Choosing an operation from one keyword: ask students to describe the relationship between the quantities first.
• Calculating before representing: a quick diagram or bar model often reveals what the problem is actually asking.
• Ignoring the unit: a correct numerical result can still be an incomplete answer.
• Forcing a remainder to disappear: practical division sometimes needs a decision about leftover people or objects.
• Not checking reasonableness: compare the answer with the original quantities before accepting it.

• Support: provide a six-step modelling checklist and partially completed bar models or Think Boards.
• Core: give mixed everyday problems where students must choose among all four operations and justify the choice.
• Extension: ask students to change one condition in a problem, predict how the model changes and solve the new version.

Content description: use mathematical modelling to solve practical problems involving additive and multiplicative situations, including money transactions; represent situations and choose calculation strategies; interpret and communicate solutions in terms of the situation.

• E1: interpret everyday additive or multiplicative situations, choose an operation and justify why it matches the relationship.
• E2: model and solve simple whole-dollar money problems using addition, subtraction, multiplication or division.
• E3: model practical grouping and sharing problems, including deciding what to do when there is a remainder.
• E4: model “How many days are there left in this year?” using a calendar.
• E5: use First Nations Australian children’s instructive games as a teaching context for equal grouping and sharing, representing relationships with number sentences and communicating solutions in context.

1. A toy costs $14 and a puzzle costs $9. What is the total cost? Show a representation.
2. You have $30 and spend $18. Explain why subtraction matches the situation.
3. Five students each contribute $4. How much is collected altogether?
4. 24 counters are shared equally among 6 people. What does the answer represent?
5. 23 students need teams of 5. How many full teams can be made and what remains?
6. Write one sentence that checks whether one of your answers is reasonable.

Teach the modelling cycle explicitly, then mix additive, multiplicative, money, grouping and calendar problems. Require students to show a representation and a final context sentence so the task remains about modelling rather than calculation alone.

Curriculum equivalents for Mathematical modelling to solve practical problems involving additive and multiplicative...

Mapped skill: use mathematical modelling to solve practical problems involving additive and multiplicative situations, including money transactions; represent situations and choose calculation strategies; interpret and communicate solutions in terms of the situation

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.

• Australian Curriculum Version 9.0
• Victorian Curriculum F–10
• NSW Curriculum

• AC9M2N04: Addition and subtraction strategies
• AC9M2N05: Multiplication and division with equal groups and arrays
• AC9M2A01: Additive patterns