Number Foundations: recap

On a number line, the number further left is less. That is true below zero too.

$-7$ is further left than $-2$, so $-7 < -2$ ("$-7$ is less than $-2$"), even though 7 is more than 2. Think of temperatures: $-7$ °C is colder than $-2$ °C.

To put integers in order, place each on the line and read from left to right. Zero is greater than every negative number.

To add or subtract integers, start at the first number and move along the number line.

• Adding a positive moves right: $-3 + 5 = 2$.
• Adding a negative moves left: $4 + (-6) = -2$.
• Subtracting moves the opposite way to adding. Adding $-3$ would move left 3, so subtracting $-3$ moves right 3: $5 - (-3) = 8$.

So subtracting a negative gives the same answer as adding the positive.

$a \times b$ is $a$ jumps of $b$ from zero. $3 \times (-2)$ is 3 jumps of $-2$: left to $-6$.

A negative number of jumps goes the opposite way: $-3 \times (-2) = 6$. The pattern agrees: $3 \times (-2) = -6$, $2 \times (-2) = -4$, $1 \times (-2) = -2$, $0 \times (-2) = 0$, so $-1 \times (-2) = 2$.

The rule: same signs give a positive, different signs give a negative. Dividing follows the same rule, because it undoes multiplying: $-12 \div 3 = -4$, since $3 \times (-4) = -12$.

A square number is a number times itself: $6^2 = 6 \times 6 = 36$. It is the area of a square with side 6. The perfect squares to 144 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.

The square root undoes squaring: $\sqrt{36} = 6$ because $6 \times 6 = 36$. It is the side of a square with area 36. (Squared is not doubled: $5^2 = 25$, not 10.)

For a number that is not a perfect square, find the perfect squares either side: $36 < 43 < 49$. Their roots give the whole numbers either side: $6 < \sqrt{43} < 7$.

To get closer, square decimals: $6.5^2 = 42.25$ and $6.6^2 = 43.56$. The closer one wins: $\sqrt{43} \approx 6.6$. (A square root is not a half: $\sqrt{50}$ is about 7, not 25.)

Exponent notation is short for repeated multiplication: $2 \times 2 \times 2 \times 2 = 2^4$. The base (2) is multiplied; the exponent (4) counts how many times. So $2^4 = 16$, not $2 \times 4 = 8$.

Squares are powers with exponent 2 ($5^2 = 25$) and cubes exponent 3 ($2^3 = 8$). Powers of 10 count zeros: $10^3 = 1000$.

A prime has exactly two factors: 1 and itself (1 is not prime). A factor tree splits a number into a factor pair, then splits every branch that is not prime, until all the branches are primes.

The prime factorisation collects them with exponents: $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$. Every tree for 36 ends with these primes. Check by expanding: $4 \times 9 = 36$.

Write both numbers as products of primes: $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$.

The HCF is the product of the primes they share (each to its lower power): $2 \times 3 = 6$. The LCM is the product of all the primes, each shared one counted once (each to its higher power): $2^2 \times 3^2 = 36$. Multiplying the two numbers ($12 \times 18 = 216$) counts the shared primes twice, so it is a common multiple but not usually the lowest.

Fractions are equivalent when they name the same amount. Multiply or divide the top and the bottom (the numerator and the denominator) by the same number: $\tfrac{2}{3} = \tfrac{2 \times 4}{3 \times 4} = \tfrac{8}{12}$. (Adding the same number to both changes the amount.)

A fraction is in simplest form when the top and the bottom share no factor but 1. Divide both by their highest common factor: $\tfrac{12}{18} = \tfrac{2}{3}$ (÷ 6).

Percent means out of 100, so a percentage is a fraction over 100: $75% = \tfrac{75}{100} = \tfrac{3}{4}$. A decimal counts tenths and hundredths: $0.75$ is 75 hundredths.

• Fraction to decimal: divide the top by the bottom ($\tfrac{3}{8} = 0.375$).
• Decimal to percentage: × 100 ($0.5 = 50%$, not 5%).
• Percentage to fraction: over 100, then simplify ($16% = \tfrac{4}{25}$).

Some decimals never end: $\tfrac{1}{3} = 0.333\ldots$

To place a fraction or decimal, split each unit into equal parts and count from zero. Quarters: split each unit into 4. Tenths: into 10.

A negative number counts to the left of zero. $-\tfrac{3}{4}$ is three quarters left of 0. A negative mixed numeral or decimal is further left still: $-1.5$ is one unit left, then half a unit further left, halfway between $-2$ and $-1$.

Once they are on the line, read from left to right to put them in order.

Fractions can be added only when the parts are the same size. Cut both into the same parts with equivalent fractions, using a common denominator (the LCM of the bottoms is the lowest):

$\tfrac{1}{3} + \tfrac{1}{4} = \tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12}$. Then add the tops and keep the bottom. (Adding across gives $\tfrac{2}{7}$, which is less than $\tfrac{1}{3}$ alone.)

For mixed numerals, change them to improper fractions first. For decimals, line up the points so tenths go with tenths.

To multiply fractions, multiply the tops and multiply the bottoms, then simplify: $\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}$. It is $\tfrac{3}{4}$ of $\tfrac{2}{3}$: in a square, 3 of 4 rows of 2 of 3 columns is 6 of 12 parts.

Multiplying by a fraction less than 1 makes a number smaller ($\tfrac{3}{4} \times 20 = 15$). Dividing asks how many fit: $3 \div \tfrac{1}{4} = 12$, the same as $3 \times 4$. To divide by a fraction, multiply by it flipped (top and bottom swapped: $\tfrac{1}{6}$ becomes $\tfrac{6}{1} = 6$).

A percentage is hundredths: $15% = 0.15$. So 15% of $80 is 0.15 × 80 = $12 (or 10% is $8 and 5% is $4).

For a decrease, take it off or multiply by what is left: $80 \times 0.85 = 68$. For an increase, add it on or multiply by $1.15$. To find what percentage 18 is of 24, divide and $\times 100$: $18 \div 24 = 0.75 = 75%$.

A quick strategy usually comes from one of three moves:

• Change the order: multiplying can be done in any order, $4 \times 2.7 \times 25 = 100 \times 2.7 = 270$.
• Change the form: $0.75 = \tfrac{3}{4}$, so $0.75 \times 36 = 27$; $25% = \tfrac{1}{4}$.
• Split a number: $12 \times 2.5 = 12 \times 2 + 12 \times 0.5 = 30$.

Then check with an estimate: is the answer about the right size?

The sign rules for integers work for every fraction and decimal: same signs give a positive, different signs a negative (for × and ÷). Work out the sign first, then the size: $-\tfrac{3}{4} \times \tfrac{2}{5} = -\tfrac{6}{20} = -\tfrac{3}{10}$.

Make the size easy: multiply in any order ($25 \times (-3.7) \times 4 = 100 \times (-3.7) = -370$), or split a number ($6 \times (-2.5) = 6 \times (-2) + 6 \times (-0.5) = -15$).

Every 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 the zeros in the value, and it counts from 0 at the ones column.

A number in expanded notation is the sum of each digit times its power of 10: $4052 = 4 \times 10^3 + 5 \times 10^1 + 2 \times 10^0$. A column holding 0 adds nothing, so it is left out.

To round to a number of decimal places, find the two numbers with that many places that your number sits between, and take the nearer. Look at the one digit straight after the place: 5 or more rounds up, 4 or less leaves the digit as it is. Halfway rounds up. So 3.746 to 2 places is 3.75.

How much to round depends on the job: money to the cent, a ruler measurement to what the ruler can show, tins of paint up so there is enough. More digits than the instrument measures only gives a false sense of accuracy.

An estimate is a quick, close answer from rounding. Round each number to its first digit and keep its size: $38 \times 52 \approx 40 \times 50 = 2000$. If the exact answer is near the estimate it is reasonable; if it is out by a factor of ten, check the decimal point.

Rounding can go up or down, so an estimate can be above or below the exact answer. What matters is that it is close and quick.

A ratio compares quantities in a stated order. With 2 red and 3 blue counters, red to blue is $2 : 3$ (part to part). The whole is $2 + 3 = 5$ counters, so red to all is $2 : 5$ (part to whole), which is the fraction $\tfrac{2}{5}$.

The order matters: $2 : 3$ is not $3 : 2$. The first thing named is the first number.

Two ratios are equivalent when both terms of one are the other's both terms multiplied (or divided) by the same number: $2 : 3 = 4 : 6 = 6 : 9$. Picture it as copies of the same pattern of blocks.

To simplify, divide both terms by their highest common factor: $12 : 18 = 2 : 3$ (÷ 6). Adding the same number to both terms changes the ratio.

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

To work backwards, use a known share to find one part: if 3 parts are $24, one part is $8.

A rate compares two quantities with different units (km per hour, $ per kg); a ratio compares the same kind of quantity.

The unit rate is the amount for one of the second quantity: divide the first by the second, and keep both units. To convert km/h to m/s, divide by 3.6.

Use a rate both ways: total = rate × amount and amount = total ÷ rate. For speed, distance = speed × time and time = distance ÷ speed.

Write the units at each step (km ÷ km/h = h); they tell you whether to multiply or divide. Then check the size makes sense.

Modelling with money: formulate (known, unknown, which representation: fraction, decimal, percentage or ratio), solve (estimate, then calculate efficiently), interpret and justify (the answer in dollars, rounded to the cent, and why that representation).

A percentage is always of something. Profit is a percentage of the cost. Percentages of different amounts do not add: 20% off $200 then 10% off the new price is $200 × 0.8 × 0.9 = $144, not 30% off.

Modelling with ratios: formulate (what is known, what is unknown, what relationship?), solve, then interpret (the answer in the situation's units, and does it make sense?).

For a mix, add the parts and find one part: a dry mix 1 : 3 in 20 L has 4 parts of 5 L, so 5 L of cement and 15 L of sand. For a scale such as $1 : 50$, each 1 cm on the drawing is 50 cm in real life.

An increase of $p%$ multiplies by $1 + \frac{p}{100}$; a decrease multiplies by $1 - \frac{p}{100}$.

For one change after another, multiply the multipliers: each percentage is of the current amount. Up 20% then down 20% is $1.2 \times 0.8 = 0.96$: 4% less.

To model a money problem: decide what to find, choose the calculation, work it out, and check it makes sense.

Income tax is worked out in brackets: each part of the income is taxed at its own rate. The marginal rate is the rate on the last dollar; the effective rate is total tax ÷ income, which is lower (for incomes above $18 200).

To model a problem: say what you need (with units), choose a ratio (same kind of quantity) or a rate (different units), solve, then interpret the answer in the situation.

Paint comes in whole tins and buses in whole buses: round up when you must have enough.

Every model makes assumptions (steady speed, no fees, no starting charge). When the model and real life differ, check the arithmetic, then name the assumption that did not hold.

Refine the model by adding what was left out, such as a starting charge: fare = $4.20 + $2.10 × km. A simple model is good enough when its error does not change the decision.