GCSE Nov 2019 Edexcel Foundation Paper 2 (Calculator)

Why we do this: Negative numbers are smaller than positive numbers. The larger the number after the negative sign, the smaller the value (further left on the number line).

The negative numbers are: \( -4, -7, -2 \)

The positive numbers are: \( 8, 1 \)

Why we do this: We need to look at the ‘thousands’ digit and the digit immediately to its right (the hundreds) to decide whether to round up or stay the same.

Number: \( 8375 \)

Thousands digit: \( 8 \)

Hundreds digit (decider): \( 3 \)

Rule: If the decider is 5 or more, round up. If it is less than 5, keep the rounding digit the same.

Since \( 3 \) is less than \( 5 \), we round down (keep the 8).

Why we do this: To convert a decimal to a percentage, we multiply by 100.

How: Use the square root button on your calculator.

Calculator steps:

1. Press √
2. Type 17.64
3. Press =

Check: Calculate \( 4.2 \times 4.2 = 17.64 \).

Why we do this: \( 6^5 \) means multiplying 6 by itself 5 times: \( 6 \times 6 \times 6 \times 6 \times 6 \).

Calculator steps:

1. Type 6
2. Press the power button (usually x^y or ^ or x^□)
3. Type 5
4. Press =

Why we do this: We need to know the full capacity of the cinema first.

Why we do this: We know the total money made and the cost per ticket. Dividing gives the number of people.

Why we do this: Subtract the sold tickets from the total capacity.

Why we do this: Find out how many Harry keeps.

Why we do this: The question asks for the fraction “of the 20 sweets”. This means 20 is the denominator.

Why we do this: Follow the steps from left to right with the input 6.

Why we do this: We know the input (17) and the output (10). We need to see what happens in between.

First box: Add 13. \( 17 + 13 = 30 \).

Second box: We need to get from 30 to 10.

Method: Put the numbers in order from smallest to largest and find the middle value.

Original list: \( 6, 4, 8, 9, 4, 3 \)

Ordered list: \( 3, 4, 4, 6, 8, 9 \)

Method: Count the odd numbers in the list: \( 3, 4, 4, 6, 8, 9 \)

Odd numbers: 3 and 9.

Total numbers: 6.

Fact 1 (Mode is 3): 3 must be the most common number. We already have one 3. To make it the mode, we need at least one more 3.

Let’s assume one hidden card is 3.

Fact 2 (Mean is 5): The mean of 5 numbers is 5. This allows us to find the total sum.

Why we do this: Divide the total cost by the cost per minute to find the duration.

How: There are 60 minutes in an hour. Divide 350 by 60.

Start time: 10:45

Add 5 hours: 15:45

Add 50 minutes: 15:45 + 50 mins

Method: Find ‘3’ on the Stones (horizontal) axis. Go up to the line, then go across to the Kilograms (vertical) axis.

Problem: The graph only goes up to 70 kg, but we need 80 kg.

Method: We can find the value for 40 kg (or 8 kg) and multiply by 2 (or 10), or use the rate.

Why we do this: It is easier to compare and add fractions when they share a common denominator.

Method: Add the two numbers and divide by 2.

What this means: Every side length of the new shape must be 3 times as long as the original shape.

• Vertical side: Original = 2 units. New = \( 2 \times 3 = 6 \) units.
• Bottom horizontal side: Original = 2 units. New = \( 2 \times 3 = 6 \) units.
• Top slanted side: Original goes “3 right, 1 up”. New goes “9 right, 3 up”.

Offer 1: 2 bags of 20 litres = \( 40 \) litres total.

Cost = £3.50.

Offer 2: 3 bags of 40 litres = \( 120 \) litres total.

Cost = £9.00.

To compare fairly, we can find the cost of 40 litres in this deal, or the cost per litre.

Why we do this: The question asks for the answer in cm, and it’s easier to work with whole numbers.

1 metre = 100 centimetres.

Method: A scale of 1 : 24 means the real object is 24 times larger than the model. To find the model size, we divide the real size by 24.

Why we do this: Simple interest is calculated on the original amount (principal) only.

Rate = 1.8% = 0.018

Method: Multiply the annual interest by the number of years.

Why: We are given \( AB = BD \). This means triangle \( ABD \) is an isosceles triangle.

Reason: Base angles of an isosceles triangle are equal.

So, \( \angle BDA = \angle DAB = 64^\circ \).

Reason: Angles in a triangle add up to \( 180^\circ \).

Reason: Angles on a straight line add up to \( 180^\circ \).

\( ABC \) is a straight line, so \( \angle ABD \) and \( \angle DBC \) are supplementary.

Why: We are given \( BC = BD \). This means triangle \( BCD \) is also an isosceles triangle.

Reason: Base angles of an isosceles triangle are equal.

Let the base angles be \( x \). So \( \angle BCD = x \) and \( \angle BDC = x \).

The sum of angles in triangle \( BCD \) is \( 180^\circ \).

Ben: \( n \)

Chloe: “Twice as old as Ben” means \( 2 \times n = 2n \).

Dan: “Five years younger than Ben” means \( n – 5 \).

Why: The question asks for the total age \( T \).

Definition: An identity is true for ALL values of the variable.

• \( 3h + 2 = 14 \): Equation (only true for \( h=4 \)).
• \( 3a + 4b – 2c \): Expression (no equals sign).
• \( A = \pi r^2 \): Formula (relates different variables).
• \( 5m – 3m = 2m \): Identity (always true, because 5 apples minus 3 apples is 2 apples).
• \( x + 7 \leq 12 \): Inequality.

Why we do this: One batch makes 16 biscuits. We need to see which ingredient runs out first (limiting factor).

The lowest number of batches is 2.5 (determined by the flour). She cannot make more than 2.5 batches.

Method: Substitute \( f = 110 \) into the formula.

Why: The question asks to compare the difference to “5% of the real height”.

Real height = 442 m.

Method: Check each midpoint against the frequency table.

• 10-20 (Midpoint 15): Frequency is 6. Graph shows plotted at 8. (Error 1)
• 20-30 (Midpoint 25): Frequency is 21. Plotted correctly.

Rule: Frequency polygons are lines connecting the midpoints at the correct height. They should NOT be closed loops (connected back to the x-axis) unless specifically requested.

The graph connects the first and last points back to the axis, forming a closed shape. (Error 2)

Why: “Correct to the nearest millimetre” means the rounding unit is 1 mm.

To find the bounds, we halve the rounding unit: \( 1 \div 2 = 0.5 \).

The error interval includes the lower bound but goes up to (not including) the upper bound.

Why: Find how many each person has at the start.

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

Fact: When Tom buys stamps from Adam, the total number of stamps (240) stays the same.

New ratio \( 3:5 \). Total parts = \( 3 + 5 = 8 \).

Why: The sample size is 50. The population size is 700.

Scale factor = \( 700 \div 50 = 14 \).

Reasoning: Scaling up a sample only works if the sample accurately reflects the whole group.

Features: A cubic graph of this form goes from bottom-left to top-right and has an inflection point at the origin (0,0). It looks like an ‘S’ shape.

Graph F matches this shape.

Features: This is a reciprocal graph (hyperbola). As \( x \) gets close to 0, \( y \) shoots to infinity. The graph has two separate parts (branches) in the 1st and 3rd quadrants (top-right and bottom-left).

Graph D matches this shape.

Formula: \( 2n^2 – 1 \). Substitute \( n = 1, 2, 3… \)

Formula: \( 40 – n^2 \). Substitute \( n = 1, 2, 3… \)

Note: Since we are subtracting \( n^2 \), the terms will decrease and eventually become negative.

Method: Divide the numbers first, then divide the powers of 10.

Use your calculator for the whole calculation or split it.

Rule: Standard form must be \( A \times 10^n \) where \( 1 \le A < 10 \).

\( 0.456 \) is not between 1 and 10.

Formula: \( d = \frac{720}{n} \). Rearrange to find \( n \): \( n = \frac{720}{d} \).

Ali’s time \( d = 40 \).

Hayley’s time \( d = 30 \).

Dimensions: \( L=18, W=8, H=6 \).

Formula: \( 2(LW + LH + WH) \)

Fact: A cube has 6 identical square faces.

Let side length = \( x \). Surface Area = \( 6x^2 \).

We know SA = 600.

Formula: \( \mathbf{a} – 2\mathbf{b} \)

Instruction: Start anywhere on the grid. Move 1 unit LEFT (because x is -1) and 4 units UP (because y is 4). Draw an arrow.