What number multiplied by zero gives four?

None. Every number multiplied by zero gives zero.

What sign does negative three times negative four have?

Positive. Two negative factors give a positive product.

How can you write seven as a fraction with denominator one?

frac71 Dividing an integer by one preserves its value.

Which is larger: one-half or two-quarters of the same sheet?

Neither; they are equal. The extra fold changes the divisions, while the shaded amount stays unchanged.

What is the absolute value of positive three?

Three. Absolute value measures distance, so a positive number stays positive.

How many rational numbers lie between 1 and frac54?

Infinitely many. Taking another halfway number can continue without end.

Is the square root of 81 rational or irrational?

Rational. Its value is 9, which can be written over one.

For 0.45454… with 45 repeating, which smallest power of ten above one aligns the tails?

100. The two-digit block needs a two-place shift so the tails match.

Teach The World of Numbers in eight steps.

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

Gather paper and a pencil.

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

Why it matters

Signed numbers help account books distinguish money held from money owed.
Equivalent fractions keep ingredient proportions unchanged when recipes are resized.
Square roots give diagonal lengths when square corners must fit accurately.

What to teach

How to teach it

Zero, signs and fractions

$4\div0=x$

$0\times x=4$

$0\times x=0$

1. Zero

Zero counts none; division by zero has no defined answer.

Show: Say: “Call a proposed answer x. Division must undo multiplication.” Then write the line.
Ask: Can four divided by zero pass this multiplication check?
Say: It would require zero to equal four, so no number passes.
Used in: Stock records use zero to show that nothing remains available.

If your child says…“Dividing by zero gives zero.”

Ask: What happens when you multiply your proposed answer by zero?
What’s true: Division by zero has no unique answer that reverses multiplication.

$(-1)\times(1+(-1))=0$

$-1+(-1)\times(-1)=0$

$(-1)\times(-1)=0+1$

2. Brahmagupta’s Laws

Signed arithmetic keeps track of fortunes and debts.

Show: Say: “One plus negative one is zero. Multiplying this sum by negative one gives zero. Multiply each part by negative one.” Then write the line.
Ask: What must negative one times negative one equal?
Say: The product is positive one because it cancels negative one.
Used in: Account books use signed calculations to update balances after borrowing or repayment.

If your child says…“Two negative factors give a negative answer.”

Ask: What must cancel negative one to leave zero?
What’s true: Multiplying two negative numbers gives a positive result.

Number line showing a negative fraction, zero, a positive fraction and an integer.

3. Rational Numbers

Numbers expressible as an integer divided by a nonzero integer.

Show: Look at the diagram’s marked numbers, from left to right. Read each as one integer divided by another. Compare the numbers below the fraction bars.
Ask: Can zero and two belong to the same number family as these fractions?
Say: Yes: each marked number can be written using integer top and bottom numbers. The bottom number must never be zero.
Used in: Fractional food portions make equal sharing possible when whole pieces would be too large.

If your child says…“Seven is not rational because it isn't written as a fraction.”

Ask: What happens to seven when you divide it by one?
What’s true: A number can be rational without already having a fraction bar.

A sheet folded into quarters, with pencil shading on top.

4. Equivalent Fractions and Lowest Terms

Equivalent fractions match. Simplified numerators and denominators share no factor above one.

Show: Fold paper in half, open it, and shade one half. Fold it into quarters and leave it folded.
Ask: When we open it, will the shaded amount have changed?
Say: Open it: two shaded quarters still make one half.
Used in: Equivalent fractions let recipes describe the same ingredient amount using different measuring divisions.

If your child says…“Different top and bottom numbers mean different amounts.”

Ask: Did making more folds add any shading?
What’s true: Multiplying or dividing both parts by the same nonzero number preserves the fraction’s value.

Distances and numbers between

Equal distance brackets connect negative three and positive three to zero.

5. Absolute Value

Absolute value measures how far a number lies from zero.

Show: Look at the diagram. Find negative three, zero and three, then compare the distance brackets.
Ask: How far from zero will each outer point be?
Say: Both are three units from zero, so both have absolute value three. Zero itself has absolute value zero.
Used in: Temperature differences measure the size of a change, whether warming or cooling.

If your child says…“Absolute value always flips the sign.”

Ask: How far from zero is positive three?
What’s true: Positive numbers keep their value because their distances from zero are already nonnegative.

$m=\frac{1+\frac32}{2}$

$m=\frac{\frac22+\frac32}{2}$

$m=\frac52\times\frac12$

6. The Density of Rational Numbers

Another rational number always fits between two distinct rational numbers.

Show: Say: “Call the halfway number m. Add one and three-halves, then halve the total.” Then write the line.
Ask: What fraction will the next line give for m?
Say: Halving the sum, five-halves, gives five-quarters.
Used in: Fractional measurements describe lengths between ruler marks, allowing finer measurement when suitable tools are available.

If your child says…“Eventually, there won't be any fractions left between them.”

Ask: What would stop us adding the new endpoints and halving again?
What’s true: Every pair of distinct rational numbers has another rational number halfway between.

Irrational numbers and decimals

Right triangle with two unit sides and a hypotenuse labelled square root of two.

7. Irrational Numbers

Their decimals never end or settle into a repeating block.

Show: Look at the diagram’s unit sides. Pythagoras gives the sloping length: the square root of two.
Ask: Can a fraction of integers give this sloping length exactly?
Say: No. Squaring a lowest-terms fraction would force its numerator, then its denominator, to be even. Both share two, contradicting lowest terms.
Used in: Square roots give exact square diagonals, which builders approximate when measuring.

If your child says…“Every square root is irrational.”

Ask: Which whole number multiplied by itself gives 81?
What’s true: A root sign alone cannot decide whether a number is rational.

$x=0.\overline3$

$10x=3.\overline3$

$10x-x=3$

$9x=3$

8. Rational Decimals: Terminating and Repeating

Rational decimals stop or repeat; matching repeating tails can cancel.

Show: Say: “Call zero point three repeating x. Ten times x is three point three repeating. Subtracting cancels the matching tails.” Then write the line. Say: “Nine times x is three.”
Ask: What fraction will the next line give for x?
Say: Dividing by nine gives one-third.
Used in: Repeating decimals show when an equal money share needs rounding.

If your child says…“The number of digits shown tells me how far to shift.”

Ask: Would writing more threes change the repeating block?
What’s true: The repeating block’s length determines the shift needed to align the endless tails.

A real problem

A mathematics book tackled currency conversion.

Fibonacci’s Liber Abaci first appeared in 1202.

It addressed merchants’ need to convert between currencies used around the Mediterranean.

It also explained how to change between different representations of fractions.

University of St Andrews Wikipedia

Coins from different countries beside an open handwritten account book.

Concept Lab · Common misconception

The tempting mistake in The World of Numbers

Open standalone misconception page →

Can a calculator's finite display prove a number is rational?

Wrong

√2 is rational because a calculator displays 1.414213562, a terminating decimal.

Why it feels right. I treat the display as the exact value.

Right

A calculator can show a rounded decimal for an irrational number.

Worked example

Rounded to three decimals

1.414→Square→1.999396

Exact number

√2→Square→2

Squaring a shortened display gives a value close to 2. The exact square root is defined by squaring to exactly 2.

The displayed digits describe an approximation.
Squaring 1.414 gives 1.999396, close to 2.
The exact √2 is the number whose square is exactly 2, and it is irrational.

Ask your child eight quick questions.

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

What number multiplied by zero gives four?

None. Every number multiplied by zero gives zero.
What sign does negative three times negative four have?

Positive. Two negative factors give a positive product.
How can you write seven as a fraction with denominator one?

$\frac71$ Dividing an integer by one preserves its value.
Which is larger: one-half or two-quarters of the same sheet?

Neither; they are equal. The extra fold changes the divisions, while the shaded amount stays unchanged.
What is the absolute value of positive three?

Three. Absolute value measures distance, so a positive number stays positive.
How many rational numbers lie between 1 and $\frac54$ ?

Infinitely many. Taking another halfway number can continue without end.
Is the square root of 81 rational or irrational?

Rational. Its value is 9, which can be written over one.
For 0.45454… with 45 repeating, which smallest power of ten above one aligns the tails?

100. The two-digit block needs a two-place shift so the tails match.

15 more questions with answers

Teach The World of Numbers in eight steps.

Why it matters