Teacher guide: Number Foundations

The idea in plain words

The integers are the whole numbers, zero and the negative whole numbers: …, −3, −2, −1, 0, 1, 2, 3, … On a number line, negatives sit to the left of zero.

Two ideas carry the unit:

• Further left is less. −7 < −2 because −7 is further left, even though 7 is more than 2. Zero is greater than every negative.
• Adding and subtracting are moves. Start at the first number. Adding moves the way the second number points: right for a positive, left for a negative. Subtracting moves the opposite way to adding. So 5 − (−3): adding −3 would go left 3, so subtracting it goes right 3, to 8.

Why it matters

Temperatures, heights above and below sea level, money owed and changes in a score all use negatives. In algebra, every table of values with negative inputs, and every graph in four quadrants, needs these two ideas.

What students get wrong

• Comparing by size without the sign. −8 > −2 "because 8 > 2". The board writes the comparison in words: "−8 is further left, so it is less".
• Taking the number's sign as the move. 5 − (−3) = 2, moving left because of the minus sign on 3. The board draws the adding move faintly above a subtraction, so students see the jump turn round.

The words

integer: a whole number, zero, or a negative whole number. negative: below zero. less than ($<$), **greater than** ($>$): the wide end of the sign faces the greater number. further from zero: the size without the sign (later, the absolute value).

The idea in plain words

A square number is a number times itself, the area of a square with that side: $6^2 = 6 \times 6 = 36$. The square root goes back to the side: $\sqrt{36} = 6$. A number that is not a perfect square still has a root between two whole numbers: $36 < 43 < 49$, so $\sqrt{43}$ is between 6 and 7 (about 6.6).

Exponent notation is short for repeated multiplication: $2 \times 2 \times 2 \times 2 \times 2 = 2^5$. The base (2) is multiplied; the exponent (5) counts how many times.

Every whole number above 1 is a product of primes in exactly one way. A factor tree finds it: split into factor pairs until every branch is a prime, then write it with exponents, $60 = 2^2 \times 3 \times 5$. Two factorisations side by side give the HCF (the primes they share) and the LCM (all the primes, shared ones once): for 12 and 18, the HCF is 6 and the LCM is 36.

Why it matters

Squares and roots come back in area and Pythagoras. Prime factorisations make fractions quick: simplifying divides by the HCF, and adding needs a common denominator, the LCM.

What students get wrong

• Squaring as doubling, rooting as halving. $5^2 = 10$, $\sqrt{36} = 18$. Draw the square; square the answer to check.
• The exponent as a multiplier. $2^5 = 10$. Write it the long way first.
• Stopping a tree early. $36 = 4 \times 9$ given as the prime factorisation. 4 and 9 are not prime.
• The product as the LCM. LCM of 6 and 8 given as 48. It counts the shared 2 twice: 24.

The words

square number (perfect square), square root ($\sqrt{\;}$), base and exponent (also index or power), prime (exactly two factors; 1 is not prime), prime factorisation, factor tree, highest common factor (HCF), lowest common multiple (LCM).

The idea in plain words

Fractions are equivalent when they name the same amount: multiply or divide the top and the bottom by the same number ($\tfrac{2}{3} = \tfrac{8}{12}$). Dividing by the highest common factor gives the simplest form ($\tfrac{12}{18} = \tfrac{2}{3}$). A percentage is out of 100 and a decimal counts tenths and hundredths, so one amount has three names: $\tfrac{3}{4} = 0.75 = 75\%$. Divide the top by the bottom for a decimal; × 100 for a percentage.

Every fraction, decimal and mixed numeral has one place on the number line. To find it, split each unit into equal parts (quarters for $\tfrac{3}{4}$, tenths for $0.6$) and count from zero: right for a positive, left for a negative.

A negative mixed numeral or decimal is a whole number and a part, both below zero: $-1.5$ is one unit left of zero and then half a unit further left, so it sits between $-2$ and $-1$. Different names can share a place: $-\tfrac{2}{4} = -\tfrac{1}{2} = -0.5$.

Calculating comes next. To add or subtract, cut both fractions into the same parts first (a common denominator, the LCM of the bottoms): $\tfrac{1}{3} + \tfrac{1}{4} = \tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12}$. To multiply, multiply the tops and the bottoms ($\tfrac{3}{4}$ of $\tfrac{2}{3}$ is $\tfrac{6}{12}$); to divide, count how many fit ($3 \div \tfrac{1}{4} = 12$). A percentage of an amount is a multiplication too: 15% of 80 is $0.15 \times 80 = 12$, and 15% off leaves 85%, so $\times 0.85$. The unit ends by choosing a quick strategy: change the order, change the form ($0.75 = \tfrac{3}{4}$), or split a number.

Why it matters

Scales on thermometers, rulers and graphs are number lines in parts. Ordering mixed fractions and decimals, and later adding and subtracting them, all rest on knowing where each one sits.

What students get wrong

• Adding the same number to the top and the bottom. $\tfrac{2}{3} = \tfrac{3}{4}$. Compare the fraction bars: the amount changes.
• Reading the digits as the percentage. $0.5 = 5\%$. 0.5 is 50 hundredths: 50%.
• Counting the part back towards zero. −1.5 placed at −0.5, between −1 and 0. Ask whether −1.5 °C is colder or warmer than −1 °C.
• Adding the tops and the bottoms. $\tfrac{1}{3} + \tfrac{1}{4} = \tfrac{2}{7}$, which is less than $\tfrac{1}{3}$ alone. Cut both into twelfths.
• Multiplying always makes bigger. $\tfrac{3}{4} \times 20$ judged more than 20. A multiplier less than 1 takes part: 15.
• Multiplying by the part taken off. 20% off $50 worked out as $50 × 0.2 = $10, the discount. What is left is 80%: × 0.8.
• Comparing decimals by digits. −0.3 judged greater than −0.25 because 3 < 25. Write both with the same number of places.

The words

fraction, decimal, mixed numeral ($1\tfrac{3}{4}$): ways of writing numbers between the whole numbers. equivalent: two names for the same number, at the same place on the line.

The idea in plain words

Each place value column is ten times the one on its right, so each is a power of 10: ones $10^0 = 1$, tens $10^1$, hundreds $10^2$, thousands $10^3$. The exponent counts from 0 at the ones column. Expanded notation writes a number as digit times power of 10, added up: $4052 = 4 \times 10^3 + 5 \times 10^1 + 2 \times 10^0$.

To round a decimal to a number of decimal places, find the two numbers with that many places it sits between and take the nearer. Look at the one digit straight after the place: 5 or more rounds up, 4 or less leaves it; halfway rounds up. How much to round depends on the job: to the cent for money, up when you need enough (tins of paint), to what the ruler can show for a measurement.

An estimate rounds each number to its first digit, keeping its size: $38 \times 52 \approx 40 \times 50 = 2000$. Compare it with the exact answer to check it is reasonable.

Why it matters

Place value as powers of 10 leads to scientific notation and to decimals (the columns continue as $10^{-1}$, $10^{-2}$). Rounding and estimating are the checks that catch a decimal point in the wrong place.

What students get wrong

• Counting the exponent from 1. The tens as $10^2$. Count from 0 at the ones.
• Cutting digits off. 3.746 to 2 places as 3.74. Look at the next digit.
• Rounding twice. 2.349 to 1 place as 2.35 then 2.4. Only the digit straight after the place counts.
• Losing the size when estimating. 19.6 as 2, not 20.

The words

power of 10, exponent, place value, expanded notation, decimal place, round, estimate, reasonable.

The idea in plain words

A ratio compares quantities in a stated order: 2 red to 3 blue is $2 : 3$. Part to part (red to blue) is $2 : 3$; part to whole (red to all) is $2 : 5$, the same as the fraction $\tfrac{2}{5}$.

Two ratios are equivalent when both terms are multiplied (or divided) by the same number: $2 : 3 = 4 : 6 = 6 : 9$. Simplest form divides both terms by their highest common factor: $12 : 18 = 2 : 3$.

To share a total in a ratio, add the parts ($2 + 3 = 5$), divide the total by the parts to get one part ($$60 \div 5 = $12$), and multiply for each share ($$24$ and $$36$). The shares add back to the total.

Why it matters

Ratios are everywhere: mixes, maps and scale models, recipes. They lead to rates, proportion and similar shapes.

What students get wrong

• Part to whole as part to part. 4 apples and 3 pears is $4 : 7$ for apples to all the fruit, not $4 : 3$.
• Adding to scale. $2 : 3$ as $4 : 5$. Both terms are multiplied, not added to.
• Dividing by one term. Sharing $60 in 2 : 3 as 60 ÷ 2 and 60 ÷ 3. Add the parts first.

The words

ratio, term, part to part, part to whole, equivalent ratios, simplest form, share, scale.

The idea in plain words

A rate compares two quantities measured in different units: 60 km per hour, $4.50 per kilogram, 80 beats per minute. A ratio compares quantities of the same kind (2 cups to 3 cups). The unit rate is the amount for one of the second quantity: divide the first by the second, and keep both units.

A rate works both ways: total = rate × amount, and amount = total ÷ rate. For speed, distance = speed × time and time = distance ÷ speed. Writing the units in each step (km ÷ km/h = h) shows whether to multiply or divide. Speeds convert with 1 km/h = 1000 m ÷ 3600 s, so divide km/h by 3.6 to get m/s.

Why it matters

Prices, speeds, fuel use, exchange rates and heart rates are all rates. Comparing unit rates is how to find the best buy.

What students get wrong

• Dividing the wrong way round ($0.25 per kg for 3 kg at $12).
• Multiplying distance by speed to find a time.
• Multiplying by 3.6 to get m/s. m/s is the smaller number.
• Calling a rate a ratio. Different units make it a rate.

The words

rate, unit rate, per, km/h, m/s, best buy, exchange rate.

The idea in plain words

Modelling turns a real problem into maths, solves it, and turns the answer back into the situation. Three steps: formulate (what is known, what is unknown, what is the relationship, and which representation: fraction, decimal, percentage or ratio?), solve (estimate first, then calculate efficiently, with a calculator where it helps), and interpret and justify (the answer in the situation's units, sensibly rounded, and why that representation).

For money, a percentage is always of the original amount: profit is a percentage of the cost. Percentages of different amounts do not add: 20% off then 10% off $200 is $200 × 0.8 × 0.9 = $144, not 30% off. For measurement, a mix is shared in parts, and a scale such as $1 : 50$ multiplies from the drawing to the real thing and divides the other way.

Why it matters

Modelling is where Number meets the real world: shopping, savings, building, maps. The skill that matters is the first step, deciding what the problem is, before any calculation.

What students get wrong

• Calculating before formulating. Ask for known, unknown and relationship first.
• The wrong base. Profit out of the selling price, not the cost.
• Adding percentages of different amounts.
• Bare numbers. 33.6 what? Write $33.60.

Year 8: percentage change and money models

A percentage change is a multiplier: up p% is × (1 + p/100), down p% is × (1 − p/100), and one change after another multiplies the multipliers (up 20% then down 20% is × 0.96). Income tax is a model in brackets: each part of the income is taxed at its own rate, so the marginal rate (on the last dollar) is higher than the effective rate (tax ÷ income). The common slips: taking a second percentage of the original amount, and taxing the whole income at the top rate.

Year 8: ratio and rate models

Model a problem by saying what is needed (with units), choosing a ratio (same kind of quantity) or a rate (different units), solving, and interpreting the answer in the situation (round paint tins up). Every model makes assumptions (steady speed, no fees, no starting charge); when it disagrees with real life, name the assumption and refine the model, for example fare = $4.20 + $2.10 × km. The common slips: rounding the wrong way for the situation, and blaming correct arithmetic for a model's mismatch.

The words

model, formulate, interpret, justify, representation, profit, markup, discount, scale.