Financial Mathematics A: Earning, Saving and Spending Money Stage 5 · Year 10 · Unit of Work

Assumed prior learning, misconceptions and difficulties

Assumed prior learning

The NSW Education Standards Authority (NESA) lists this background knowledge for Financial mathematics A (NESA, 2022a, Financial mathematics A, Teaching advice section). The right-hand column shows the earlier outcome that covers each skill.

Background knowledge (NESA) Earlier outcome that covers it
Converting between percentages, fractions and decimals MA4-FRC-C-01: represents and operates with fractions, decimals and percentages to solve problems
Calculating fractions, decimals and percentages of quantities MA4-FRC-C-01 (as above)
Conversion of time periods such as years to months No single outcome (see the note below)
Substitution into formulas MA4-ALG-C-01: generalises number properties to operate with algebraic expressions including expansion and factorisation
Solving equations MA4-EQU-C-01: solves linear equations of up to 2 steps and quadratic equations of the form ax² = c
Reading and interpreting information from tables and graphs MA3-DATA-02: interprets data displays, including timelines and line graphs. MA4-DAT-C-01: classifies and displays data using a variety of graphical representations

The outcome codes and wording are from the same syllabus (NESA, 2022a). We added a code only where the outcome clearly covers the skill. The Stage 3 time outcome is about 12- and 24-hour time, not converting years to months, so that row has no code.

The Lesson 1 diagnostic quiz checks percentages, decimals, time conversions, substitution into a rule and reading a table before they are needed (see Lesson 1). Solving equations is not checked there, because Lessons 1–3 do not use it. Lesson 5 checks it at the start, before students find a missing rate or time.

Technology. We assume students can open a link and complete an online form (Lesson 1 diagnostic quiz), and can enter values and simple formulas in a spreadsheet (Lessons 6, 8 and 9). If they cannot, the teacher gives a short demonstration and a paper version with the same columns.

Earning money (Lessons 1–3)

Simple interest (Lessons 4–6)

Spending money (Lessons 7–9)

Anticipated misconceptions and difficulties

These are planning predictions, to be checked against what students actually write.

Earning money (Lessons 1–3)

Misconception Why students think this How it is identified Teaching response
5% is written as 0.5 They move the decimal point once instead of dividing by 100. Lesson 1 diagnostic Q1 Key question "Why 0.05 and not 0.5?" in the commission example
Commission is worked out on the retainer "Commission" is a new word, and the retainer is the first number they see. Lesson 1 You do Q5, exit ticket (b) Key question "Is the 5% worked out on the $400 or the $6,000?"
Time and a half means adding $1.50 "And a half" sounds like adding something small. Lesson 2 worked example, exit ticket (b) Reference box links each phrase to a multiplier. Key question "$37.50 or $26.50?"
Hours are added instead of multiplied Two numbers on a line get combined the quickest way. Lesson 2 Spot the Mistake Students explain Jayden's error in a "because" sentence
All overtime hours are paid at the same rate Most examples use one rate for the whole shift. Lesson 2 overtime example Real pay guide headings: first 2 hours, then after 2 hours
Monthly pay = 4 × weekly pay Calendars are often drawn as 4 rows of weeks. Lesson 3 key question, exit ticket (c) 52 ÷ 12 ≈ 4.33 weeks in a month
Net pay is more than gross pay, or tax is added "Gross" and "net" give no clue to their meaning in everyday English. Lesson 3 You do Gross–net diagram. Prompt "Should net pay be more or less than gross pay?"
The tax-free threshold means the whole pay is tax-free The phrase is long and abstract. Lesson 3 key question at $350 $18,200 ÷ 52 = $350, then the table at $350: $0 with the threshold, $62 without
Every worker gets annual leave and leave loading Students' own jobs are usually casual. Lesson 3 key question about Ella Fair Work Ombudsman wording: casual employees do not get paid annual leave

Simple interest (Lessons 4–6)

Treat these as planning predictions. Newman's error analysis distinguishes reading, comprehension, transformation, processing and encoding; use brief questions to locate the difficulty before choosing a response (NSW Department of Education, n.d., introductory section).

Misconception Why students think this How it is identified Teaching response
5% = 0.5 Students may remember a percentage as a whole number or move the decimal one place. Lesson 4 starter Reconnect 5/100 to 0.05
18 months = 1.8 years Decimal notation can be confused with a year-and-month label. Lesson 4 practice Use 18 ÷ 12 = 1.5 years
Interest = total amount Both are amounts of money, and “how much” is ambiguous without reading the noun. Lesson 4 exit ticket Label A = P + I
Annual interest increases each year Everyday “growth” language can suggest that all money in the account earns interest. Lesson 4 table Return to the original principal and fixed rate
r = 0.015 means 0.015% Decimal rate and percentage notation may be confused. Lesson 5 exit ticket Multiply the decimal by 100 to obtain 1.5%
An unknown is found with the wrong operation Students may recall the direct formula without interpreting the substituted equation. Lesson 5 working Substitute known values, solve the resulting equation and verify
The amount graph starts at zero Students may recall I = 0 at time zero and transfer this to A. Lesson 5 graph Identify A = P when n = 0
The gradient is the total rise or a percentage Students may subtract two amounts but omit division by the interval. A coefficient and percentage rate are both described as rates, but have different units. Lesson 6 matching Calculate change in amount ÷ change in time, in dollars per year

Spending money (Lessons 7–9)

These are anticipated explanations, not observed diagnoses. Use the listed checks to distinguish mathematical misunderstandings from language interpretation or arithmetic slips; adjust teaching based on the evidence.

Potential misconception Why students may think this How to identify it Teaching response
Uses 20 rather than 0.20 in a percentage calculation. Percent notation is treated as a whole number rather than a rate out of 100; the decimal conversion has not become secure. Lesson 7 retrieval: 20% of $900. Use 10% as one tenth, build 20%, then reconnect to multiplication by 0.20.
Omits the deposit or treats it as a monthly payment. Students apply a familiar repeated-payment calculation but fail to coordinate the distinct initial payment; deposit language may be unfamiliar. Lesson 7 hinge and labelled working. Separate deposit, repayment total and total paid; show the one-off deposit on a timeline.
Counts years instead of monthly payments. They use the duration number directly, without translating its unit into the payment frequency. Checks on payment count before multiplication. Write duration × payments per unit of time first; one year monthly means 12 payments.
Assumes a smaller monthly repayment is cheaper. The smaller, prominent monthly figure is more immediately visible than the cumulative effect of a longer term. Lesson 7 comparison: $1,500 versus $1,680. Compare complete totals and explain the different durations; distinguish total cost from payment size.
Adds the payment today twice. They interpret today as separate from the advertised count, missing the inclusive meaning of the wording. Lesson 8 retrieval: four payments including today. Mark today as payment 1 of 4, rather than an extra payment.
Assumes interest-free means no additional charges. They generalise absence of interest to absence of every charge, or attend only to the headline. Lesson 8 hinge, advertisement discussion and exit. Read every stated condition; compute principal payments and fees separately.
Confuses a per-payment charge with a total charge. They overlook whether a stated charge applies once or at every scheduled date. Hand calculation and spreadsheet audit in Lesson 8. Label the charge basis; multiply only recurring charges by their stated count.
Treats a late fee as replacing the missed payment. They treat the penalty as an alternative payment rather than a charge added to the original obligation. Compare on-time and one-late-payment scenarios. Retain all required payments and add only the explicitly stated extra fee.
Applies a loan fee percentage to the wrong amount. They use the nearest number or transfer a reducing-balance assumption, without checking the contractual base specified in the task. Lesson 9 model and hinge. Circle “original principal”; state the percentage base before calculation.
Calls total repaid the borrowing cost. They treat all money returned as expense and fail to distinguish returning the principal from paying extra. Lesson 9 retrieval and independent assessment. Use borrowed / returned / extra rows; borrowing cost = total repaid − principal.
Adds fees twice when already included in repayments. They combine two representations of the same fees: the total-cost calculation and instalments that already include those fees. Check consistency between total and payment schedule. Calculate principal plus fees once, then distribute that total into repayments.
Equates meeting payment limits with affordability. They focus on the two supplied ceilings and assume the omitted financial information has no effect. Buyer-profile explanations in all three lessons. Require separate claims about cost, schedule fit and missing income/expense information.
Trusts spreadsheet output without checking the formula. A calculator-like view of the spreadsheet encourages trust in the displayed number without checking cell references or the model. Compare the formula with hand calculations. Audit cell references and charge basis; test a known case before varying an input.

References