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GCSE Foundation Non-Calculator Essential Skills

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Number
Order negative numbers
\[ -7, -3, 0, 4 \]
The further below zero, the smaller the number.
Add and subtract negatives
\[ (-3) – 5 = \square \]
Use a number line: subtracting a negative means add.
Multiply and divide negatives
\[ (-6) \times (-4) \]
Same signs give a positive, different signs a negative.
Evaluate powers
\[ 4^3 = \square \]
Multiply the number by itself the right number of times.
Square and cube roots
\[ \sqrt[3]{64} = \square \]
Which number times itself gives this?
Order of operations
\[ 3 + 4 \times 2 \]
Brackets, indices, then × and ÷, then + and −.
Spot a prime, factor or multiple
\[ \text{Prime: } 51, 53, 57? \]
Test each number carefully: odd does not mean prime.
Product of prime factors
\[ 60 = 2^2 \times 3 \times 5 \]
Keep dividing by primes, then write with powers.
Highest common factor
\[ \text{HCF of } 18, 24 \]
The biggest number that divides into both.
Lowest common multiple
\[ \text{LCM of } 6, 8 \]
The smallest number in both times tables.
Decimals and estimation
Multiply and divide by 10, 100, 1000
\[ 3.6 \times 100 \]
Move the digits, do not just add zeros.
Order decimals
\[ 0.4, 0.35, 0.405 \]
Fill with zeros so every number has the same length.
Multiply decimals
\[ 0.3 \times 0.4 \]
Multiply whole numbers, then put the decimal places back.
Divide by a decimal
\[ 6 \div 0.2 \]
Make the divisor a whole number first.
Round to a given place value
\[ 2.96 \to 1 \text{ d.p.} \]
Look at the next digit: 5 or more rounds up.
Round to 1 significant figure
\[ 0.0482 \to 1 \text{ s.f.} \]
Start from the first digit that is not zero.
Estimate a calculation
\[ 48.7 \times 3.1 \approx \square \]
Round every number to 1 s.f., then work it out.
Fractions, percentages and ratio
Decimal to fraction
\[ 0.35 = \frac{7}{20} \]
Use place value to write the decimal over 10, 100 or 1000.
Fraction to decimal
\[ \frac{3}{8} = 0.375 \]
Make the bottom 10, 100 or 1000.
Percentage to decimal
\[ 7\% = 0.07 \]
Divide by 100.
Fraction to percentage
\[ \frac{3}{20} = 15\% \]
Make the bottom 100.
Fraction of a shape
\[ \text{Fraction shaded} \]
Shaded parts out of ALL the equal parts.
Fraction of an amount
\[ \frac{3}{5} \text{ of } 40 \]
Divide by the bottom, multiply by the top.
Percentage of an amount
\[ 35\% \text{ of } 60 \]
Build it from 10%, 5% and 1%.
Add fractions
\[ \frac{2}{3} + \frac{1}{4} \]
Find a common denominator, then add the tops.
Multiply fractions
\[ \frac{2}{3} \times \frac{4}{5} \]
Multiply the tops and multiply the bottoms.
Divide fractions
\[ \frac{2}{3} \div \frac{4}{5} \]
Keep, change, flip the second fraction.
Simplify a ratio
\[ 12 : 18 = 2 : 3 \]
Divide every part by the highest common factor.
Ratio to a fraction
\[ 3 : 5 \to \frac{3}{8} \]
Compare one part with the total of all the parts.
Share in a ratio
\[ \text{£}40 \text{ in } 3 : 5 \]
Add the parts, find one share, then multiply.
Expressions
Simplify products
\[ 3a \times 4b = 12ab \]
Multiply the numbers, then write the letters together.
Collect like terms
\[ 5x + 3y – 2x \]
Only terms with exactly the same letters combine.
Index laws
\[ a^5 \times a^3 = a^8 \]
Multiply: add the powers. Divide: subtract them.
Simplify algebraic fractions
\[ \frac{12x}{4} = 3x \]
Divide the numbers and cancel the letters.
Expand a single bracket
\[ 3(2x – 5) \]
Multiply every term inside by the term outside.
Expand double brackets
\[ (x + 3)(x + 2) \]
Four multiplications, then collect like terms.
Factorise: number factor
\[ 6x + 9 = 3(2x + 3) \]
Take the highest common factor outside the bracket.
Factorise: letter factor
\[ x^2 + 5x = x(x + 5) \]
Take the common letter (and number) outside.
Substitute into an expression
\[ 3a^2,\ a = -2 \]
Swap each letter for its value, then use BIDMAS.
Equations, inequalities and graphs
One-step equations
\[ \frac{x}{4} = 3 \]
Do the inverse operation to both sides.
Two-step equations
\[ 3x + 5 = 20 \]
Undo the + or − first, then the × or ÷.
Equations with brackets
\[ 2(x + 3) = 14 \]
Expand the bracket first, then solve.
Unknown in a formula
\[ v = u + at, \text{ find } a \]
Substitute what you know, then solve.
nth term
\[ 5, 8, 11, \ldots \to 3n + 2 \]
The difference goes in front of n.
Integers in an inequality
\[ -2 < n \le 3 \]
Check whether each end is included.
Read coordinates
\[ (3, -2) \]
Across first (x), then up or down (y).
Gradient and intercept
\[ y = 3x – 4 \]
In y = mx + c, m is the gradient and c the intercept.
Geometry and measures
Name 2D and 3D shapes
\[ \text{Which is the hexagon?} \]
Count sides and check the properties.
Parts of a circle
\[ \text{Which is the chord?} \]
Radius, diameter, chord, tangent, sector, segment.
Angles on a line and at a point
\[ x + 65^\circ = 180^\circ \]
A straight line is 180°, a full turn is 360°.
Angles in a triangle
\[ x + 50^\circ + 70^\circ = 180^\circ \]
The three angles add up to 180°.
Angles in parallel lines
\[ \text{Alternate: equal} \]
F: corresponding, Z: alternate, C: co-interior.
Area or perimeter of a rectangle
\[ \text{Area} = 8 \times 5 \]
Area multiplies; perimeter adds all four sides.
Area of a triangle
\[ \frac{1}{2} \times 10 \times 4 \]
Half of base times perpendicular height.
Volume of a cuboid
\[ 5 \times 4 \times 3 \]
Length times width times height.
Metric unit conversions
\[ 3.5\text{m} = \square\text{cm} \]
Know the conversion fact, then × or ÷.
Statistics and probability
Range
\[ \text{Range of } 7, -2, 5 \]
Largest value take away smallest value.
Mean
\[ \text{Mean of } 4, 0, 7, 5 \]
Add them up, then divide by how many.
Median
\[ \text{Median of } 6, 2, 9 \]
Put them in order, then find the middle.
Mode
\[ \text{Mode of } 4, 7, 4, 2 \]
The value that appears most often.
Probability from a bag
\[ P(\text{blue}) = \frac{5}{12} \]
Number of ways it can happen over the total.
Probability it does not happen
\[ P(\text{not}) = 1 – 0.3 \]
The two probabilities add up to 1.
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