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School fair planning investigation - Mathematics Year 5

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Introduction

Can a small stall cover its costs? In this investigation you will build a six-line budget, compare buying choices, draw a dimensioned layout and defend two decisions. Along the way you will retrieve number, fractions, decimals, measures, geometry and data skills from Year 5.

Your brief: Plan a fictional school-fair stall selling badges, bookmarks, cards and pencils. All prices, quantities and dimensions are invented teaching data. Expected sales are assumptions, not orders or observed results. This is a mathematics project, not authority to buy goods or run a real event.

Starting point: Recall multiplication and division, pounds and pence, decimal place value, rectangle area and perimeter, and reading tables. Start on paper or with accessible digital tables; use a calculator after an estimate and written method. Keep a record of what you changed and why. The final product matters more than racing through the quiz.

Build a six-line budget with honest assumptions

A quantity tells how many units are bought or sold. A unit cost is the price for one unit. Purchase cost is quantity multiplied by unit cost. Revenue is money received from sales. Subtracting only the stock purchase cost gives a gross difference; it does not account for every other expense. A budget is a plan with stated assumptions, not a report of things that have happened.

Work entirely in pence first. For 36 badges costing 45p each, split the quantity into 30 and 6: 30 × 45 = 1,350p and 6 × 45 = 270p. Add to get 1,620p = £16.20. An independent check splits the price: 36 × 40 + 36 × 5 = 1,440 + 180 = 1,620p. The estimate 40 × 50p = £20 tells you that £162 is implausible, but an estimate alone does not prove £16.20 exactly.

$36 \times 45 p = 1620 p = 16.20 pounds$

If all 36 badges sell for 70p each, revenue is 36 × 70p = 2,520p = £25.20. The badges alone give £25.20 − £16.20 = £9.00 before other expenses. Alternatively, the difference per sold badge is 70p − 45p = 25p; 36 × 25p = 900p. This second method agrees only because every purchased badge is assumed sold in this particular calculation.

Six-line model budget — all money columns are in pounds

Item	Purchased units	Cost per unit	Purchase cost	Assumed sales	Price per sold unit	Forecast revenue
Badges	36	0.45	16.20	36	0.70	25.20
Bookmarks	40	0.30	12.00	40	0.60	24.00
Cards	24	0.25	6.00	24	0.50	12.00
Pencils	30	0.20	6.00	30	0.40	12.00
Wrapping supply	1	3.00	3.00	0	Not sold	0.00
Stall fee	1	4.00	4.00	0	Not sold	0.00

Audit the columns. Stock purchase cost is £16.20 + £12.00 + £6.00 + £6.00 = £40.20. Adding the two other listed costs gives £47.20 total planned cost. The four revenue entries total £73.20. The forecast surplus after all six listed costs is £26.00. Check by adding the costs back: £26.00 + £47.20 = £73.20.

$73.20 - (40.20 + 3.00 + 4.00) = 26.00$

Do not label £73.20 as profit. Do not call £33.00 the final surplus: that is revenue minus stock cost, before the £7.00 of other listed expenses. Even £26.00 is only the surplus under this model; unlisted expenses and different actual sales could change it. No real commercial forecast is being made.

Guided check. Someone writes £16.02 for 36 × 45p because the product ends in the digits 02. Find the first error: 36 × 45 actually equals 1,620 pence, and 100p = £1. Converting by dividing 1,620 by 100 gives £16.20. Write both units, then compare with the second multiplication method above.

Try it. A customer in the model buys three badges at 70p each and pays £5. Calculate the cost and change before checking. Cost = 210p = £2.10; change = 500p − 210p = 290p = £2.90. Check £2.10 + £2.90 = £5.00. No payment or personal information is needed.

Watch actively: This original Corbettmaths Primary money lesson gives worked examples of monetary calculations. Pause at one example, identify the units and choose a method, then compare the presenter's reasoning. The video's shopping examples are separate from this course's fictional stall. You may study one example at a time; no transcript is required to complete the course.

Compare stock choices and draw a useful layout

A lower unit price does not automatically mean the smallest payment for the quantity you need. Compare quantities, whole packs, invoice totals and unused stock.

Buying-choice model. Supplier A sells loose badges at 45p each; 36 cost £16.20. Supplier B sells packs of ten for £4.00 a pack; packs cannot be split. Three packs contain only 30, too few for the 36-badge brief. Four packs contain 40 and cost £16.00. If you still sell only 36 badges at 70p, revenue is £25.20 and revenue minus this purchase invoice is £9.20. There are four unsold badges, whose future value has not been counted. Replacing only the first cost line in the original budget reduces total cost by £0.20, to £47.00, and raises the planned surplus to £26.20 under otherwise identical assumptions. Delivery, quality and returns are unspecified; ask about them rather than inventing them.

Check a changed forecast. Return to the original six-line budget, not Supplier B. Suppose only 18 of the 36 badges sell while every bookmark, card and pencil still sells as planned. Badge revenue becomes 18 × 70p = £12.60. Purchases have already happened: the original badge cost remains £16.20, not half of it. Total revenue is now £12.60 + £24.00 + £12.00 + £12.00 = £60.60. The surplus after the original £47.20 listed costs is £13.40, with 18 badges unsold. Check: revenue fell £12.60, and £26.00 − £12.60 = £13.40. Stock left over is a count, not automatically extra cash.

$60.60 - 47.20 = 13.40$

Layout model. A rectangular stall footprint is assumed to measure 2.4m by 1.5m. These are supplied dimensions, not measurements you have made. Convert to 240cm by 150cm to check both units consistently. Area measures surface coverage; perimeter measures the boundary.

$A = 2.4 \times 1.5 = 3.6 m^{2}$ $P = 2 (2.4 + 1.5) = 7.8 m$

Check in centimetres: 240 × 150 = 36,000cm². Since 1m² = 10,000cm², that is 3.6m². Perimeter is 2 × 390 = 780cm = 7.8m. A learner who writes 7.8m² has calculated the boundary but attached an area unit; both the method and the unit should match the question.

Draw and label. On a plan let 1cm represent 50cm in real life. The rectangle on paper should be 4.8cm by 3cm: multiply each drawn length by 50 to recover the supplied dimensions. Label the real lengths as 2.4m and 1.5m and write the scale in words. This scale drawing is enrichment that applies Year 5 measure and shape knowledge; the drawing's own area, 14.4cm², is not the stall's real area. If precise drawing is inaccessible, use a dimensioned digital rectangle and explain the same conversions.

$4.8 \times 50 = 240, 3 \times 50 = 150$

Mark a front edge, a display area and a place for an adult to work. State which sizes you assumed. The rectangle alone does not establish safe crowd space, access, fire precautions or permission to hold a real event. An actual school would approve its own layout and arrangements. Here you are defending a mathematical plan, not certifying a venue.

Two decisions with evidence. A defensible buying decision could choose four packs because they meet the quantity and cost 20p less, while disclosing unused stock and missing delivery information. A defensible sales decision could keep a reserve in the forecast because the half-badge-sales case reduces the surplus to £13.40. “I like this option” is a preference; a justification needs a calculation, an assumption and a limitation.

Retrieve across the year and present a checked proposal

An integrated problem asks you to choose methods rather than follow a keyword. First sort information into given, assumed, calculated and observed. Here the prices and dimensions are given fictional data, the sales counts are assumptions and the totals are calculated. No actual sale, measurement or survey has been observed.

Cumulative retrieval. Attempt these eight short tasks without the answer column first. Then mark the first incorrect step, correct it and choose one earlier course for practice. They sample topics; success here alone is not evidence of complete Year 5 mastery.

Topic	Task	Worked check
Place value	What is the value of the digit 7 in 507,042? Round the number to the nearest thousand.	The digit represents 7,000. The last three digits are 042, less than 500, so round to 507,000.
Written operations	Find 43,607 + 8,958 and check the sum.	52,565; check 52,565 − 8,958 = 43,607. An estimate 44,000 + 9,000 = 53,000 is close but not exact proof.
Multiplication and division	Find 36 × 45 and 168 ÷ 7.	1,620 and 24. Check the product by two splits and the quotient with 24 × 7 = 168.
Fractions	Add three quarters and one eighth; find three fifths of 40.	Six eighths plus one eighth = seven eighths. One fifth of 40 is 8, and three fifths is 24.
Decimals and percentage links	Find 0.45 + 0.70 and 25% of 36.	1.15, with tenths and hundredths aligned. A quarter of 36 is 9, so 25% is 9.
Measures	Convert 2.4m to centimetres and £16.20 to pence.	240cm and 1,620p; both conversions multiply by 100, but the units mean different things.
Geometry	Find the perimeter of a 6cm by 4cm rectangle and the angle needed to make a straight angle with 68°.	20cm; 2 × 6 + 2 × 4. The missing angle is 112° because 68° + 112° = 180°.
Data	Forecast sales on three fictional days are 8, 12 and 5 items. Find the total and the difference between greatest and least.	Total 25 items; difference 7 items. There are three entries, not 25 days, and a forecast is not an observation.

Your final proposal. Produce the six-line budget with quantity, unit cost, purchase cost, expected sold quantity, selling price and forecast revenue. Show column totals and a reverse check. Include the dimensioned layout and explain its scale. Add two decisions, each with numerical evidence and one limitation. Finally name one demonstrated strength and a specific next task, such as “practise converting pence to pounds with zeros”.

Compare methods. For 36 badges at 45p you can multiply in pence then convert, or split £0.45 into £0.40 and £0.05 before multiplying. Both can be exact. For whole-pack purchasing, a simple price-per-unit comparison alone misses the minimum number of packs. For a rectangular area, adding sides solves a different problem. Choose methods from the quantities and relationships.

Peer audit without personal data. A partner checks one product using a different split, totals the costs independently, tests the half-sales scenario and traces the layout conversions. Record a disagreement and its resolution. If working alone, hide your first result and use a second method. An unchecked calculator display or agreement between two copied answers is not an independent check.

Transfer challenge. Replace 36 loose badges with 50 badges at 42p each, sold at 65p each. Keep the rest of the six-line model unchanged and assume every new badge sells. Badge cost is £21.00 and revenue £32.50; revised total cost is £52.00, revenue £80.50 and surplus £28.50. Check the change from the original: extra revenue £7.30 minus extra cost £4.80 gives an extra £2.50, so £26.00 becomes £28.50. State every changed assumption; do not mix the two forecasts.

Interactive Tasks

Quiz: Test Your Knowledge

What is the purchase cost of 36 badges at 45p each? (£16.20) (!£16.02) (!£162.00) (!£1.62)

What does the original forecast revenue of £73.20 represent? (Money received if all forecast sales occur) (!Money left after every possible expense) (!The amount paid for wrapping only) (!An observed total from a real event)

What is the surplus after the six original listed costs? (£26.00) (!£33.00) (!£47.20) (!£73.20)

How many packs of ten meet a requirement for at least 36 badges? (Four packs) (!Three packs) (!Two packs) (!One pack)

If only 18 original badges sell, what happens to their already paid purchase cost? (It remains £16.20) (!It falls to £8.10) (!It becomes £12.60) (!It becomes zero)

What is the footprint area of the supplied 2.4m by 1.5m rectangle? (3.6 square metres) (!7.8 square metres) (!3.9 square metres) (!36 square metres)

Which statement correctly separates an assumption from a calculation? (Selling all 36 badges is assumed; £25.20 revenue is calculated from it) (!The £25.20 forecast proves that 36 people bought badges) (!The supplied layout proves that an event is safe) (!Every forecast is an observed measurement)

What change is due from £5 for three badges at 70p each? (£2.90) (!£3.10) (!£4.30) (!£2.10)

What is three quarters plus one eighth? (Seven eighths) (!Four twelfths) (!Three eighths) (!One whole)

What is the strongest mathematical justification? (A checked calculation with stated assumptions and a limitation) (!A calculator answer without units) (!A copied total that matches a friend) (!A preference with no evidence)

Memory Game

Quantity	Number of units
Unit cost	Price paid for one unit
Revenue	Money received from sales
Forecast	Plan based on assumptions
Area	Surface coverage in square units
Perimeter	Boundary length in length units

Drag and Drop

£47.20	Original total planned cost
£73.20	Original total forecast revenue
£26.00	Original surplus after six listed costs
3.6 square metres	Supplied rectangular footprint area
7.8 metres	Supplied rectangular footprint perimeter

Crossword Puzzle

BUDGET	A plan showing costs and expected income
REVENUE	Money received from sales
QUANTITY	How many units there are
SURPLUS	Income remaining above the listed costs
PERIMETER	The length around a shape
ASSUMPTION	Something taken as given for a model without observing it

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

One invoice in pence: Calculate 36 × 45p by splitting 36 into 30 and 6. Convert to pounds and check with a price split.
Change without a till: Find the change from £5 for three badges at 70p. Show a subtraction and an addition check.
Classify the information: Give one given dimension, one assumed sales count and one calculated amount from the model. Explain why none records your own observed sale.
Measure words: For a 6cm by 4cm rectangle, find perimeter and area. Explain why the units differ.

Standard

Audit six budget lines: Rebuild the original six-line budget and calculate £47.20 cost, £73.20 forecast revenue and £26.00 surplus independently. Annotate one reverse check.
Whole packs or loose stock: Compare Supplier A and Supplier B for a requirement of 36 badges. Include whole-pack quantity, invoice and unsold count before deciding.
Dimensioned stall plan: Represent the supplied 2.4m by 1.5m rectangle with 1cm standing for 50cm. Label real dimensions, show conversions and identify one untested layout assumption.
Eight-topic retrieval review: Attempt all eight retrieval rows before consulting the worked checks. Submit corrected reasoning, a strength and a precise next practice target.

Advanced

Sales uncertainty: Keep original purchases but sell only 18 badges and all the other stock. Recalculate revenue and surplus; explain why halving purchase cost is wrong.
Transfer to fifty badges: Replace only the badge line by 50 at 42p bought and 65p sold. Recalculate both totals and compare the new surplus with the original using a change calculation.
Decision with limitations: Write two decisions, one about purchasing and one about sales or layout. Each needs a calculation, an assumption and a limitation; distinguish the plan from a real-event approval.
Construct another scenario: Choose an integer number of original badges sold from zero to 36 while the other stock sells as planned. Recompute the surplus and independently check the change from the all-sales forecast. State what remains unsold.

Learning Check

Original budget submission: Submit all six original rows and the cost, revenue and surplus totals, with an independent reverse check.
Pack comparison: Compare the two badge suppliers for at least 36 badges and explain the limits of the result.
Half-sales case: With the original budget, sell only 18 badges and all other stock. Calculate the revised revenue and surplus.
Layout and unit check: Submit the scale plan, real area and perimeter, with a conversion check.
New badge line: Use 50 badges at 42p bought and 65p sold; keep every other original row. Calculate the revised totals.
Retrieval and reflection: Complete the eight-topic retrieval set and identify a specific strength and next step.
Two justified decisions: Provide two decisions supported by calculation, assumptions and limitations.

Worked checks and assessment guidance

The purchase costs are £16.20, £12.00, £6.00, £6.00, £3.00 and £4.00, totalling £47.20. Forecast revenues are £25.20, £24.00, £12.00, £12.00, £0.00 and £0.00, totalling £73.20. £73.20 − £47.20 = £26.00; check £47.20 + £26.00 = £73.20. A complete table, labelled units and explicit all-sales assumption are required.
A: 36 × 45p = £16.20 for exactly 36. B: four whole packs give 40 for £16.00; three packs are insufficient. B costs £0.20 less but leaves four if 36 sell. Badge revenue stays £25.20; the B invoice difference is £9.20 and the six-line surplus would be £26.20. Unknown delivery or product differences are not assumed away.
18 × 70p = £12.60. Revenue is £12.60 + £24.00 + £12.00 + £12.00 = £60.60. Original costs remain £47.20, giving £13.40 surplus. The forecast falls by the £12.60 lost revenue; £26.00 − £12.60 = £13.40. Eighteen badges remain unsold.
At 1cm for 50cm, draw 4.8cm by 3cm and label real sides 2.4m and 1.5m. Area is 3.6m²; perimeter 7.8m. In centimetres, 240 × 150 = 36,000cm² and 2 × 390 = 780cm, giving the same real results. Inspect the submitted drawing or accessible dimensioned equivalent; written answers alone do not show construction. No venue-safety approval is inferred.
Badge cost £21.00 replaces £16.20, so total cost rises £4.80 to £52.00. Badge revenue £32.50 replaces £25.20, so total revenue rises £7.30 to £80.50. Surplus £28.50 agrees with £26.00 + £7.30 − £4.80. The forecast assumes all 50 sell.
Checks: 7,000 and 507,000; 52,565 with inverse subtraction; 1,620 and 24; seven eighths and 24; 1.15 and 9; 240cm and 1,620p; 20cm and 112°; 25 items and 7 items. Award reasoning as well as results. A useful reflection cites an actual checked response and a next practice task; this small sample does not certify the whole year.
Examples: four badge packs meet the requirement and cost £0.20 less, but include four unused items and unspecified delivery. The reduced-sales scenario still leaves £13.40 after listed costs, but other sales are assumed unchanged and further expenses are excluded. Alternative decisions are acceptable when the arithmetic and assumptions support them. Check the learner’s actual proposal; no automatic endorsement of a real event is given.

Learning evidence

Knowledge: Distinguish quantity, unit cost, purchase cost, revenue, listed-cost surplus, area, perimeter and assumptions.

Skills: Choose and check written operations, handle pounds/pence consistently, evaluate whole packs, compare forecasts and convert layout measures.

Products: A complete six-line budget, a labelled dimensioned plan, a reduced-sales scenario, two justified decisions and an eight-topic retrieval reflection.

Assessment: Use the explicit worked checks and inspect the actual budget and plan. Quiz points support retrieval; they do not prove a construction, observed sale, personal financial competence or complete Year 5 attainment. Accessible representations may demonstrate the same mathematical reasoning.

Transfer: Disclose new assumptions and recalculate affected rows, rather than copying a previous surplus. Keep factual observations separate from fictional plans. An adult-approved real project needs separate practical arrangements; none is claimed here.

OER and Wikipedia

This original integrated investigation applies Year 5 number, measurement, geometry and statistics from Department for Education: mathematics programmes of study. The school-fair setting, supplied prices, scale drawing and linked retrieval sequence are editorial enrichment, not a separately mandated Year 5 unit or an approved commercial plan.

Commons images

Casio electronic calculator.jpg, Leeturtle, CC BY-SA 4.0. Original image unchanged; its display is not the budget result and no manufacturer endorsement is implied.
Illustration for the area of a rectangle.svg, Stephan Kulla, CC0. Original diagram unchanged; the stall dimensions are supplied separately.

Original video

Money - Primary, Corbettmaths, official publication page. Embedded through the provider, not copied or reissued as an openly licensed film. Video shopping examples and all original course data are separate.

No actual pupil, transaction or event data are used. The optional encyclopedia is supplementary reference, not current financial advice.