What percentage of this square is shaded?

50%. Fifty of its hundred equal cells are shaded.

A length exceeds this whole strip. Is its percentage above or below 100%?

Above 100%. The reference strip already accounts for 100%.

Which count belongs below the two heads when writing their fraction?

Five. The denominator counts every coin, including both heads and tails.

Which gives more roti: half the whole roti or half the half-roti?

Half the whole roti. The same fractional share of a larger starting portion gives more.

How many rupees does the 50% fall remove here?

3 rupees. The fall uses the raised price of 6 rupees.

What must match before counting pieces tells us their area shares?

Their areas. Unequal pieces contribute unequal areas, even though each counts as one piece.

What does the 2 rupees represent?

The tax alone. The subtotal must still be added to find the bill.

Can the knight land two squares across and two squares sideways from its start?

No. One direction needs two squares, but the perpendicular direction needs only one.

Teach Fractions in Disguise in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper, scrap paper, a pencil, ruler, five coins, two equally sized rotis and a plate.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Discounts reduce prices by a share of the marked price.
Tax adds a share of the subtotal to a bill.
Equal percentage changes can involve different amounts because their starting amounts differ.

What to teach

How to teach it

Shares and wholes

Half-shaded hundred-square with an enlarged cell divided into ten strips and equivalent share labels.

1. Per Cent, Per Decem and Per Mille

Count the same share in tenths, hundredths or thousandths.

Show: Find the shaded columns, cells and enlarged cell in the diagram.
Ask: How many shaded strips result when every cell becomes ten equal strips?
Say: This half is 5 per decem, 50 per cent, or 500 per mille. Each cell becomes ten smaller parts.
Used in: Battery percentages show the remaining share of a full charge.

If your child says…“Five shaded columns means 5%.”

Ask: When you count cells instead of columns, how many shaded parts are there?
What’s true: Count shaded parts and total parts using the same-sized pieces.

Unfolded paper strip above three partly labelled length, fraction and percentage scales.

2. Bar Models

Equal sections connect lengths, fractions and percentages.

Show: Fold a 20 cm paper strip into four equal lengths. Unfold it. Below it, start separate centimetre, fraction and percentage scales. Mark endpoints and the first crease: 5 cm, one-quarter, 25%.
Ask: What three labels belong at the third crease?
Say: The third crease marks 15 cm, three-quarters and 75%.
Used in: Measured strips help divide shelf space into equal sections.

If your child says…“A percentage cannot go beyond 100%.”

Ask: If this whole strip is 100%, what happens when the length extends beyond it?
What’s true: A length greater than the reference strip has a percentage greater than 100%.

Five coins in a row, two heads-up and three tails-up.

3. Ratios and Percentages

Add all ratio parts before finding each percentage share.

Show: Arrange five coins: two heads-up, three tails-up. Count them all.
Ask: If one tail becomes a head, will the percentage showing heads rise or fall?
Say: It rises. Currently, heads make 40% and tails 60%, together 100%.
Used in: A completed share of a savings target shows how much remains to save.

If your child says…“Two heads and three tails means heads are two-thirds of the coins.”

Ask: Are there three coins altogether, or five?
What’s true: The whole includes both heads and tails.

A whole roti beside half an equally sized roti on a steel plate.

4. Proportions and Absolute Amounts

The amount depends on the percentage and its starting whole.

Show: Plate a whole roti beside half an equally sized roti. After predicting, halve each portion.
Ask: Will half of each portion give equally large pieces?
Say: The larger portion gives more.
Used in: Shopkeepers compare profit with their costs to judge what they earned.

If your child says…“Both are 50%, so both pieces must be the same size.”

Ask: Did the two starting portions contain the same amount of roti?
What’s true: Equal percentages can give unequal amounts when the wholes differ.

Changes, areas and bills

$A_1=4+\frac{50}{100}\times4$

$A_1=6$

$A_2=6-\frac{50}{100}\times6$

$A_2=6-3$

5. Successive Percentage Changes

Each change uses the amount reached after the previous change.

Show: Write these lines. Say, 'A-one names the raised price; A-two names the final price, in rupees.'
Ask: What is the final price?
Say: It is 3 rupees: the fall halves the raised price.
Used in: Successive shop discounts determine what a buyer finally pays.

If your child says…“A 50% rise and a 50% fall cancel.”

Ask: Does the fall remove half of 4 rupees or half of 6 rupees?
What’s true: The fall takes half of a larger amount than the rise used.

Three equal paper rectangles form an L, with the fourth beside the gap.

6. Percentage Area

Compare a region's area with the entire area.

Show: Fold a rectangular sheet into four equal rectangles. Tear along the creases. Reassemble the sheet. Move one piece aside.
Ask: If we put the fourth piece back, what percentage of the original area will they cover?
Say: Together, all four pieces cover 100% of the original sheet. These pieces show shares through area.
Used in: Area percentages help plan how much garden space each crop receives.

If your child says…“Three pieces always mean 75%.”

Ask: What makes each piece here the same share of the sheet?
What’s true: Counting pieces gives area shares only when those pieces have equal areas.

$\text{Tax}=\frac{10}{100}\times20$

$\text{Tax}=2$

$\text{Bill}=20+2$

7. Tax Added to a Bill

Calculate tax from the subtotal, then add it.

Show: Write these lines. Say, 'Ten per cent means one-tenth; amounts are rupees.'
Ask: What total must be paid?
Say: Pay 22 rupees, including tax.
Used in: Checking tax helps explain the bill's total.

If your child says…“The tax calculation gives the whole bill.”

Ask: Where does the original 20 rupees appear in the amount paid?
What’s true: The percentage calculation gives tax alone; the bill also includes the subtotal.

Chessboard movement

Homemade chessboard with a folded paper playing piece near its centre.

8. Peaceful Knights

A knight moves two squares, then one at right angles.

Show: Draw an eight-by-eight chessboard and fold a scrap into a playing piece. Place it near the centre. Demonstrate two squares across and one perpendicular, landing only at the destination. Return it to its start.
Ask: Which other squares can this knight reach in one move?
Say: The knight's landing square is two squares across, then one square at right angles.
Used in: Knowing a knight's possible destinations helps a chess player protect pieces.

If your child says…“Where can it land?”

Ask: Which destination completes both the two-square part and the one-square part?
What’s true: The destination must complete both parts of the move.

A real problem

A trial showed that a polio vaccine worked.

The nationwide Salk vaccine trial was approved in 1954.

Some participants received the vaccine and others received a placebo.

The evaluators did not know which participants received which.

The trial established that Salk's vaccine was effective.

Smithsonian University of Toronto

An unbranded capped syringe on a plain clinic tray.

Concept Lab · Common misconception

The tempting mistake in Fractions in Disguise

Open standalone misconception page →

Does a percent already give a count?

Wrong

A 25% discount always means 25 rupees off.

Why it feels right. I read 25% as a fixed amount.

Right

A percentage is a share of a specified whole.

Worked example

25% of ₹20050 rupees

25% of ₹400100 rupees

One quarter is the same fraction of both prices. The higher price has the bigger rupee discount.

One quarter of ₹200 is ₹50.
One quarter of ₹400 is ₹100.
Both discounts are 25%.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

What percentage of this square is shaded?

50%. Fifty of its hundred equal cells are shaded.
A length exceeds this whole strip. Is its percentage above or below 100%?

Above 100%. The reference strip already accounts for 100%.
Which count belongs below the two heads when writing their fraction?

Five. The denominator counts every coin, including both heads and tails.
Which gives more roti: half the whole roti or half the half-roti?

Half the whole roti. The same fractional share of a larger starting portion gives more.
How many rupees does the 50% fall remove here?

3 rupees. The fall uses the raised price of 6 rupees.
What must match before counting pieces tells us their area shares?

Their areas. Unequal pieces contribute unequal areas, even though each counts as one piece.
What does the 2 rupees represent?

The tax alone. The subtotal must still be added to find the bill.
Can the knight land two squares across and two squares sideways from its start?

No. One direction needs two squares, but the perpendicular direction needs only one.

15 more questions with answers, 15 with pictures

Teach Fractions in Disguise in eight steps.

Why it matters