GCSE Mathematics Paper 1 (Non-Calculator) Foundation Tier Summer 2022

💡 Why we do this: We need to convert a larger unit (centimetres) to a smaller unit (millimetres).

We know the conversion fact: \( 1 \text{ cm} = 10 \text{ mm} \).

💡 Strategy: To go from cm to mm, we multiply by 10.

💡 Why we do this: When we add identical algebraic terms together, it is the same as multiplication.

Think of it like adding objects: 1 apple + 1 apple + 1 apple + 1 apple = 4 apples.

💡 Why we do this: A reflection creates a “mirror image”. Every point on the original shape must be the same distance from the mirror line, but on the opposite side.

💡 Strategy: Count the squares from the mirror line for each corner (vertex) of the triangle.

• Top Corner: It is on the mirror line (distance 0). It stays in the same place.
• Bottom Corner: It is on the mirror line (distance 0). It stays in the same place.
• Tip (Right): It is 3 squares to the right of the mirror line.
• New Tip (Left): Count 3 squares to the left of the mirror line.

💡 Why we do this: Each digit in a number has a value based on its position.

Let’s look at the columns:

• 1: Ten Thousands
• 6: Thousands
• 0: Hundreds
• 0: Tens
• 7: Units

The 6 is in the thousands column.

💡 Strategy: It is easier to compare numbers when they are all in the same format. Let’s convert everything to decimals.

Now compare the decimals from smallest to largest:

\( 0.45 < 0.50 < 0.55 \)

Replace them with the original numbers:

\( 45\% < \frac{1}{2} < 0.55 \)

What does the key tell us?

One sun symbol () represents 2 hours of sunshine.

Looking at the row for Saturday, there are 4 full sun symbols.

Why? To find out how many items he bought, we first need to know how much money he actually spent.

He gave £20 and got £6 back. The difference is what the shop kept.

How? Divide the total amount spent by the cost of one candle.

We need to draw two bars:

• May: Height 35 cm
• June: Height 20 cm

What is the scale?

Look at the gap between the labelled numbers (e.g., 15 and 20).

The gap is 5 units.

There is one large grid square representing this gap of 5.

Therefore, half a square represents half of 5, which is 2.5.

Rupa thinks half a square is 0.5, but it is actually 2.5.

So the correct reading would be \( 15 + 2.5 = 17.5 \).

What is the rule?

Each pattern adds 2 squares: one to the right arm and one to the top arm.

• Pattern 1: 1 square
• Pattern 2: 3 squares (1 corner + 1 right + 1 up)
• Pattern 3: 5 squares (1 corner + 2 right + 2 up)
• Pattern 4: 7 squares (1 corner + 3 right + 3 up)
• Pattern 5: 9 squares (1 corner + 4 right + 4 up)

The sequence increases by 2 each time.

Pattern 4 has 7 squares.

Pattern 5 has \( 7 + 2 = 9 \) squares.

Pattern 6 has \( 9 + 2 = 11 \) squares.

What do we know?

• Lowest temperature = \( -15^\circ\text{C} \)
• Highest temperature = Lowest \( + 42^\circ\text{C} \)

We need to add 42 to -15.

Why? First, we need to find the difference between the two meter readings to know how many units (kWh) were used.

We subtract the October reading from the November reading.

Units used = 460 kWh

✓ (M1)

How? Multiply the units used by the cost per unit (16p).

We need to calculate \( 460 \times 16 \).

Total cost = 7360 pence

✓ (M1)

Why? To add fractions, the denominators (bottom numbers) must be the same.

We have 12 and 6. The lowest common multiple is 12.

How? Multiply the numerators together and the denominators together.

Both numbers end in 5 or 0, so we can divide top and bottom by 5.

How? Probability is Number of wanted outcomes divided by Total number of outcomes.

What do we know? The sum of all probabilities must equal 1.

Since there are ONLY green and blue counters:

What do we do? Replace \( x \) with 4 in the equation.

Remember that \( 6x \) means \( 6 \times x \).

How to estimate: Round each number to 1 significant figure.

• \( 92 \) rounds to \( 90 \).
• \( 1.63 \) rounds to \( 2 \) (or \( 1.6 \) or \( 1.5 \) is also acceptable, but \( 2 \) is simplest).

Compare the new calculation to the original:

Original: \( 2.96 \times 3.2 = 9.472 \)

New: \( 29.6 \times 32 \)

• \( 29.6 \) is \( 2.96 \times 10 \)
• \( 32 \) is \( 3.2 \times 10 \)

So the total calculation is \( 10 \times 10 = 100 \) times bigger.

Formula: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)

Convert \( \frac{5}{4} \) hours into minutes.

\( \frac{5}{4} = 1 \frac{1}{4} \) hours.

We know \( \frac{1}{4} \) of an hour is 15 minutes.

So, the journey takes 1 hour and 15 minutes.

✓ (P1, P1)

Add the journey time to the start time (07:30).

Reasoning: If you drive faster (greater speed), the journey takes less time.

Step 1: Start Information

• Total people = 72 (Start node)
• Adults = 32. So Children = \( 72 – 32 = 40 \).

Step 2: Adult Branches

• Adults passed = 20.
• Adults failed = \( 32 – 20 = 12 \).

Step 3: Child Branches

• Children failed = 6.
• Children passed = \( 40 – 6 = 34 \).

We want the probability of picking an adult who failed.

Look at the tree diagram: there are 12 adults who failed.

Total number of people = 72.

Why? We know the recipe for 10 scones, but need it for 25.

How many times bigger is 25 than 10?

Original sugar = 40 g.

Multiply by 2.5.

Strategy: 20% is just double 10%.

To find 10%, divide by 10.

We need to increase 240, so we add the 20% value.

Let’s look at the heights of the shaded regions (assuming height of rectangle = 1).

• Rectangle A: Shaded from bottom up to \( \frac{5}{8} \).
  So the top line is at height \( \frac{5}{8} \).
• Rectangle C: Shaded from top down.
  The unshaded part at the bottom is \( 1 – \frac{9}{11} = \frac{2}{11} \).
  So the bottom line of the shading is at height \( \frac{2}{11} \).
• Rectangle B: The shading is trapped between these two lines.

The shaded fraction of B is the difference between the top limit (\( \frac{5}{8} \)) and the bottom limit (\( \frac{2}{11} \)).

Find a common denominator (88).

First, list all numbers in order from smallest to largest.

10s: 17, 19

20s: 25, 25, 26, 27, 27, 27, 28, 29

30s: 33, 37, 37

40s: 45, 47

The “Stem” is the tens digit. The “Leaf” is the units digit.

Leaves must be in order and evenly spaced.

We need to count the squares to find the dimensions.

• Plan (Circle): The diameter is 6 squares. So the radius (\(r\)) is \( 3\text{ cm} \).
• Front Elevation (Rectangle): The height is 5 squares. So the height (\(h\)) is \( 5\text{ cm} \).

The volume of a cylinder is given by:

We treat the inequality sign just like an equals sign.

Add 27 to both sides to remove the -27.

Divide both sides by 7.

Break the number down into pairs of factors until you only have prime numbers.

Collect all the circled prime numbers.

We have 2, 2, and 31.

Ratio Method: Total shares = \( 3 + 7 = 10 \).

1 share = \( 160 \div 10 = 16 \).

Number of cars = 3 shares.

Find \( \frac{1}{8} \) of 48.

Find 25% of 48.

25% is the same as \( \frac{1}{4} \).

The rest use petrol.

Total cars – Electric – Diesel = Petrol.

The power is \( -3 \), so we move the decimal point 3 places to the left.

\( 1.63 \to 0.163 \to 0.0163 \to 0.00163 \)

Standard form is \( A \times 10^n \) where \( 1 \leq A < 10 \).

For 438 000, we move the decimal 5 places to get 4.38.

Multiply the numbers and add the powers.

• Numbers: \( 4 \times 6 = 24 \)
• Powers: \( 10^3 \times 10^{-5} = 10^{3 + (-5)} = 10^{-2} \)

\( 24 \) is not between 1 and 10.

\( 24 = 2.4 \times 10^1 \)

So, \( 2.4 \times 10^1 \times 10^{-2} = 2.4 \times 10^{-1} \)

Formula: \( \text{Sum of angles} = (n-2) \times 180 \)

For a hexagon (\(n=6\)):

One angle = \( 720 \div 6 = 120^\circ \)

For a pentagon (\(n=5\)):

One angle = \( 540 \div 5 = 108^\circ \)

The angles around a point add up to \( 360^\circ \).

At the vertex where the shapes meet:

Calculate \( y \) for the missing \( x \) values.

• If \( x = -1 \): \( (-1)^2 – 3(-1) + 1 = 1 + 3 + 1 = 5 \)
• If \( x = 2 \): \( (2)^2 – 3(2) + 1 = 4 – 6 + 1 = -1 \)
• If \( x = 3 \): \( (3)^2 – 3(3) + 1 = 9 – 9 + 1 = 1 \)
• If \( x = 4 \): \( (4)^2 – 3(4) + 1 = 16 – 12 + 1 = 5 \)

The solutions are where the graph crosses the x-axis (\( y = 0 \)).

Looking at the graph, the curve crosses between 0 and 1, and between 2 and 3.

Estimates: \( x \approx 0.4 \) and \( x \approx 2.6 \)

Volume of a cube = \( \text{side}^3 \)

• Volume A = \( 3^3 = 3 \times 3 \times 3 = 27 \text{ cm}^3 \)
• Volume B = \( 4^3 = 4 \times 4 \times 4 = 64 \text{ cm}^3 \)

✓ (P1)

Density = \( \frac{\text{Mass}}{\text{Volume}} \)

• Density A = \( \frac{81}{27} = 3 \text{ g/cm}^3 \)
• Density B = \( \frac{128}{64} = 2 \text{ g/cm}^3 \)

✓ (P1)

This is a standard value you should memorize.

\( \sin 30^\circ = 0.5 \) or \( \frac{1}{2} \)