Use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts
use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate the problems using number…
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Learning goalsSay it simply
A mathematical model simplifies a real situation while preserving the relationships needed to answer the question. Strong solutions explain assumptions, operations, units and the meaning of the result.
Learning intention: I can represent a practical problem, choose and explain my calculations, and give a checked answer that makes sense in its situation.
Identify known quantities, the unknown and relevant constraints.
Draw or choose a model and write matching number sentences.
Use an efficient mental, written or appropriate digital strategy.
Check the answer and communicate what it means, including units.
Clean visual examplesOne-page board
Clean visual examples
AC9M4N08 · Models and worked thinking
Example 1
Model a fundraising order with cost and revenue
48 × ticket48 × $12
1 × hall hire1 × $180
1 × materials1 × $95
Revenue 48 × $12 = $576; costs $180 + $95 = $275
surplus $301
The model separates revenue and costs, then subtracts to answer the practical question. Every number is connected to a context and unit.
Formulate: There are 48 paid tickets at $12 each, so income is 48 × $12. Hall hire and materials are one-time costs.
Solve: Income is $576 and costs total $275. Subtract: $576 − $275 = $301.
Interpret and check: With all tickets sold and only these stated costs, $301 remains for the fundraiser. Check $275 + $301 = $576.
Example 2
Compare two possible plans
PlanCapacityCostCost per available placeA40$320$8B60$420$7Need52 studentschoose B
A lower total cost is not always the best model if it fails the constraints. Interpret solutions against capacity, budget and other conditions.
Worked decision: If 52 students need places and these are the two available plans, A is too small. B has 60 places and fits all 52. The $7 entry is cost per available place ($420 ÷ 60), not cost per attending student. The actual group pays $420.
Check the assumptions: If adults need places too, include them before choosing a plan. For 52 students and 5 adults, 57 places are needed, so B still fits.
Example 3
Missing whole or missing part?
To find all books, join the parts: 36 + 24 = 60. If the whole 60 and fiction part 36 are known instead, write 36 + ? = 60 or 60 − 36 = 24. Explain that every number counts books.
Teacher asks: Is the unknown the whole or a part? Look for: the learner labels the whole and both parts before calculating. If the learner adds 60 and 36, rebuild the model with counters and identify the missing part.
Example 4
1. Constraint detective
A trip has 52 students and 5 adults. A 60-seat bus costs $420; a 50-seat bus costs $380. The budget is $450. Learners identify 57 passengers, the seat requirement and the budget before calculating.
Answer: The 60-seat bus meets both conditions and leaves $30. The cheaper bus cannot carry all 57. If a new cost is introduced, revise the model rather than keeping the old conclusion.
Example 5
2. Model comparison
Give pairs 84 counters to represent badges. Of these, 36 are blue and the others are green. Learners draw a bar, make a two-row table and write 36 + ? = 84.
Expected evidence: 48 green badges; all three representations refer to the same whole and parts. The bar shows the relationship, the table organises quantities and the sentence specifies the calculation. Check 36 + 48 = 84.
Example 6
3. Digital verification
A supplier offers these fixed kit bundles. The class needs at least 20 kits and has $150. First write the relationship quantity × price per kit = bundle cost. Then use an available spreadsheet to calculate the row totals.
Set up: Use columns for quantity, price each and total. For data in A2 and B2, enter =A2*B2 in C2; copy the relationship to the other rows. Check one total mentally: 20 × $7.10 = $142.
Conclusion: The 20-kit option meets the quantity requirement and budget, leaving $8. Ten kits are insufficient; the 30-kit option exceeds the budget even though its unit price is lower. A spreadsheet is useful for comparing many such options after the relationship has been chosen.
Curriculum examplesCopied content
The official description and elaborations define this lesson. The worked expansions below show how each requirement is taught.
Content description: use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate the problems using number sentences and choose efficient calculation strategies, using digital tools where appropriate; interpret and communicate solutions in terms of the situation
E1: modelling and solving a range of practical additive problems using materials, part-part-whole diagrams and/or a bar model, and writing addition and/or subtraction number sentences, based on whether a part or the whole is missing; explaining how each number in their number sentence is connected to the situation
E2: modelling practical problems with division, interpreting and representing the situation using a diagram or array to represent what is unknown (the number of groups, or the number per group); writing a division number sentence to represent the situation and choosing an efficient calculation strategy
E3: modelling practical problems involving money, such as a budget for a large event, as requiring either addition, subtraction, multiplication or division and justifying the choice of operation in relation to the situation
E4: modelling and solving multiplication problems involving money, such as buying 5 toy scooters for $96 each, using efficient mental strategies and written jottings to keep track if needed; for example, rounding $96 up to $100 and subtracting 5 ×$4 = $20, so 5 ×$96 is the same as 5×$100 less $20, giving the answer $500 – $20 = $480
E5: modelling situations by formulating comparison problems using number sentences, comparison models and arrays; for example, “Ariana read 16 books for the readathon; Maryam read 4 times as many books. How many books did Maryam read?” using the expression 4 × 16 and using place value partitioning, basic facts and an array, thinking 4 × 10 = 40 and 4 × 6 = 24, so 4 × 16 can be written as 40 + 24 = 64
E1 expansion: represent the unknown
Modelled example: A garden has 94 plants; 57 are flowering plants. The remaining part is 94 − 57 = 37 non-flowering plants. A labelled bar shows 57 and 37 joining to make 94.
Student does: Change the problem so the whole is unknown: 57 flowering plants and 37 non-flowering plants. Write 57 + 37 = 94. Tell a partner what each value measures.
Checkpoint: Why do 94 − 57 and 57 + 37 describe the same collection? Both connect the same two parts and whole. Reteach by joining and separating the labelled parts if the child treats each number as a different collection.
E2 expansion: two meanings of division
Group size unknown: Share 24 counters equally among 4 trays. The four rows can model the trays: 24 ÷ 4 = 6 counters per tray.
Group count unknown: Put 6 counters in each tray. The same array shows 24 ÷ 6 = 4 trays. State what the divisor and answer measure each time.
Efficient strategy: Use the fact 4 × 6 = 24 to check both divisions. Learners make both arrangements with counters and label the answer units; a bare 4 or 6 is insufficient.
E3 expansion: formulate an event budget
An event has $240 available. Four equipment sets cost $32 each and printing costs $28 once. First formulate 4 × $32 + $28 = total cost. Partitioning gives $128 + $28 = $156. Subtract from the budget: $240 − $156 = $84 left.
If six classes share the $156 bill equally, each pays $156 ÷ 6 = $26. Multiplication repeats an equal price, addition joins costs, subtraction finds the remaining budget and division shares a cost equally.
Teacher asks: Does “left” mean a number of items or an amount of money here? Student explains: $84 remains after the stated expenses; no extra costs have been assumed.
E4 expansion: adjust every equal price
Six stools at $97 each cost 6 × $97. Compensation gives 6 × $100 − 6 × $3 = $600 − $18 = $582. Partitioning gives 6 × $90 + 6 × $7 = $540 + $42 = $582. Both strategies preserve all six prices.
Checkpoint: Why is subtracting only $3 wrong? All six prices were increased, so six adjustments are needed. Learners choose either strategy and justify their choice using the numbers.
E5 expansion: compare equal multiples
Ava has 12 cards and Ben has four times as many. Ben has 4 × 12 = 4 × 10 + 4 × 2 = 40 + 8 = 48 cards. In the comparison model, one Ava-sized unit repeats four times.
Four more cards would mean 12 + 4 = 16, a different comparison. Ask learners to write a fresh times-as-many story, draw its equal groups and explain each number in their sentence.
Questions and answersWith answers
Core idea: A model must preserve the practical quantities and answer the actual question. Formulate first, calculate efficiently, then interpret and check.
Important questions and answers
Model income from 48 tickets at $12 each. 48 × $12 = $576. The 48 counts tickets, $12 is price per ticket and $576 is total income.
What does 24 ÷ 6 mean for counters? If each group holds 6 counters, it gives 4 groups. If there are 6 equal groups, it gives 4 counters per group. The story determines the answer unit.
Which constraint can rule out a cheaper transport plan? A bus must have enough passenger places. Include all stated children and adults before comparing feasible costs.
When is a table or spreadsheet useful? When comparing several quantities, prices or plans. Label the columns, derive the calculation relationship and check sample results; a tool does not supply missing information.
What is a complete fundraiser conclusion? With $576 income and only $275 of stated costs, $301 remains. This names the amount, its meaning and the assumptions.
Practice and reviewReady for practice
All numbers used automatically — Include only quantities relevant to the question and model.
Constraint ignored — A mathematically cheaper option may not meet capacity, time or safety needs.
Answer lacks units or context — State what the result means in the situation.
Digital output replaces formulation — Set up the mathematical relationship before using the tool.
Materials: Paper, pencils, counters and access to a calculator or spreadsheet for the digital comparison activity.
Support: Use a labelled bar or counters with a missing part. Ask what each number measures before choosing an operation. Keep prices whole-dollar amounts initially.
Core: Model additive and multiplicative situations, including both division meanings and connected financial calculations. Require number sentences and contextual conclusions.
Extend within this code: Compare several feasible plans, explain a compensation strategy and check a digital formula. Formal algebraic manipulation, percentages and interest calculations are not the target.
There are 128 library books, including 79 storybooks. Draw a model and find the others.
Review hint: Label the whole and known part. Answer: 128 − 79 = 49 other books; the model must show 79 and 49 joining to 128.
Three classes share 84 markers equally. Write the sentence and answer with units.
Review hint: The number of groups is given. Answer: 84 ÷ 3 = 28 markers per class; check 3 × 28 = 84.
Five trays cost $29 each, with a single $12 delivery fee. Is a $160 budget enough?
Review hint: Apply the repeated price first; add delivery once. Answer: 5 × $29 + $12 = $145 + $12 = $157, leaving $3. Require a model/sentence, working and a budget conclusion.
Read the Topic Guide, then project one Classroom View teaching section at a time. Complete the written worksheet tasks, use Practice for feedback and take the Test when ready. Written model or explanation tasks require an adult to inspect the actual work against the task-specific checks.
Curriculum alignmentStart here
AC9M4N08
use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate the problems using number sentences and choose efficient calculation strategies, using digital tools where appropriate; interpret and communicate solutions in terms of the situation
72 beads are packed with 8 per bag. What is unknown? The number of bags: 72 ÷ 8 = 9 bags.
What is 4 times as many as 15? 4 × 15 = 60, shown by four equal 15-unit groups.
Sketch a cost model for 3 tickets at $8 plus a one-time $5 fee. Three $8 parts and one $5 part total $29. Inspect the labelled model and the sentence 3 × $8 + $5 = $29.
Mastery evidence: The learner formulates an appropriate number sentence, explains the model and strategy, and gives an answer with meaning and units. If only a final number is supplied, ask for the missing evidence.
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