KS2 Mathematics Paper 2: reasoning 2023

What does “5 minutes past 11” mean?

The time is 11:05.

The hour hand (short hand) should be pointing at 11 (or just slightly past it).

The minute hand (long hand) counts the minutes. Each number on the clock represents 5 minutes. So, “5 minutes past” means the long hand points to 1 (because \(1 \times 5 = 5\) minutes).

Let’s check the clocks from left to right:

• Clock 1: Hour hand is near 11. Minute hand is at 1. This shows 11:05.
• Clock 2: Hour hand is near 1. Minute hand is at 11. This shows 5 minutes to 1 (or 12:55).
• Clock 3: Hour hand is near 11. Minute hand is at 5. This shows 11:25.
• Clock 4: Hour hand is near 10. Minute hand is at 1. This shows 10:05.

Why do we check the negative signs?

The “lowest” temperature means the coldest. On a thermometer, negative numbers are colder than positive numbers. The larger the number after the minus sign, the colder it is.

Let’s look at the negative numbers first: \(-4\) and \(-10\).

\(-10\) is colder (lower) than \(-4\).

So, the order starts: -10°C, -4°C.

Now look at the positive numbers: \(6, 1, 3\).

Ordering these from smallest to largest: 1°C, 3°C, 6°C.

Why do we start with x?

Coordinates are written as \((x, y)\). We always go “along the corridor” (horizontal x-axis) before going “up the stairs” (vertical y-axis).

Look at point C. Follow the grid line down to the bottom axis (the x-axis).

It aligns with the number 6.

Now, go back to point C and look across to the side axis (the y-axis).

It aligns with the number 2.

The key shows that one full circle = 2 children.

This implies that half a circle = 1 child.

Tiger: There are 6 full circles.

\[ 6 \times 2 = 12 \text{ children} \]

Elephant: There are 2 full circles and 1 half circle.

\[ (2 \times 2) + 1 = 4 + 1 = 5 \text{ children} \]

“How many more” means we need to subtract.

\[ 12 – 5 = 7 \]

There are 5 cars. Each car has 4 wheels.

\[ 5 \times 4 = 20 \text{ wheels} \]

There are 3 motorbikes. Each motorbike has 2 wheels.

\[ 3 \times 2 = 6 \text{ wheels} \]

Total wheels = Car wheels + Motorbike wheels

\[ 20 + 6 = 26 \text{ wheels} \]

Stefan counted 28 wheels, but there are only 26. This is why he cannot be correct.

Why subtract?

To find change, we subtract the cost of the item (£1.39) from the amount paid (£5.00).

Calculation: \[ 5.00 – 1.39 \]

Look at the gap between the numbers:

• 75 to 50 is subtract 25.
• 50 to 25 is subtract 25.

The rule is: Subtract 25 each time.

The number is 24,372.

The hundreds digit is 3 (representing 300).

We need to decide whether to stay at 300 or round up to 400.

Look at the digit to the right of the hundreds (the tens column). It is 7.

Rule: If the digit is 5 or more, round up.

Since 7 is more than 5, we round the 300 up to 400.

• Diagram 1 (Top): The line goes all the way across the circle through the centre. This is the diameter.
• Diagram 2 (Middle): The line goes from the centre to the edge. This is the radius.
• Diagram 3 (Bottom): The dashed line is the edge of the circle itself. This is the circumference.

Why use the inverse?

Ken started with a number, divided by 3, and got 72.

To go backward to the start, we must do the opposite of dividing by 3.

The opposite of division is multiplication. So, we multiply by 3.

Calculation: \[ 72 \times 3 \]

Why? “More” means add. We focus on the thousands digit.

Number: 19,039

If we add 1,000 to 19,000, we get 20,000.

\[ 19,039 + 1,000 = 20,039 \]

Why? “Less” means subtract. We focus on the hundreds digit.

Number: 19,039

There is a 0 in the hundreds column. We need to exchange/borrow from the thousands.

\[ 19,039 – 100 \]

Think of it as 190 hundreds – 1 hundred = 189 hundreds.

So, 18,939.

What is a line of symmetry?

It is a line that acts like a mirror. If you fold the shape along the line, both halves match exactly.

Look at the shape vertically (up and down) and horizontally (left and right).

1. Vertical Line: Goes straight down the middle, through the top point and the bottom point.

2. Horizontal Line: Goes straight across the middle.

Imagine the whole number is a chocolate bar divided into 5 equal pieces.

We are told that 1 piece (one fifth) is equal to 22.

If one part is 22, and there are 5 parts in total, we must multiply by 5.

\[ 22 \times 5 \]

How to measure:

1. Place the centre point of the protractor on the vertex (corner) of the angle.
2. Align the zero line of the protractor with the horizontal bottom line.
3. Read the scale that starts at 0 on the right-hand side (the inner scale on most protractors).
4. Follow the scale around to where the second line points.

The angle is wider than a right angle ($90^\circ$) but a straight line ($180^\circ$).

This is an obtuse angle.

The measurement is approximately 130 degrees.

To order fractions, it helps to give them the same bottom number (denominator).

The denominators are 3, 6, 4, 2.

The number 12 is in all of their times tables.

• \( \frac{1}{6} = \frac{2}{12} \)
• \( \frac{1}{4} = \frac{3}{12} \)
• \( \frac{1}{3} = \frac{4}{12} \)
• \( \frac{1}{2} = \frac{6}{12} \)

The number line is divided into 12 small sections.

1. Count 2 marks along: Place \( \frac{1}{6} \).

2. Count 3 marks along: Place \( \frac{1}{4} \).

3. Count 4 marks along: Place \( \frac{1}{3} \).

4. Count 6 marks along (halfway): Place \( \frac{1}{2} \).

We need to find the duration from 6:30 am to 9:00 am.

• 6:30 am to 7:30 am = 1 hour
• 7:30 am to 8:30 am = 1 hour
• 8:30 am to 9:00 am = 30 minutes (half an hour)

Total time = 2 and a half hours (2.5 hours).

The rate is 2 mm per hour.

\[ \text{Rainfall} = \text{Hours} \times \text{Rate} \]

\[ 2.5 \times 2 \]

Double 2.5 is 5.

4 boxes with 50 roses in each.

\[ 4 \times 50 = 200 \text{ roses} \]

We need to divide the 200 roses into groups of 6.

\[ 200 \div 6 \]

We can make 33 full bunches.

There are 2 roses left over, but they are not enough to make another bunch of 6.

We have \( \frac{1}{20} \) and \( \frac{3}{5} \).

To add them, we need a common denominator. We can change fifths into twentieths by multiplying the top and bottom by 4.

\[ \frac{3 \times 4}{5 \times 4} = \frac{12}{20} \]

\[ \text{Total (A+B)} = \frac{1}{20} + \frac{12}{20} = \frac{13}{20} \]

The whole is \( \frac{20}{20} \) (1 whole).

\[ \frac{20}{20} – \frac{13}{20} = \frac{7}{20} \]

If \( 15,000 \div \text{something} = 75 \), then \( 15,000 \div 75 = \text{something} \).

We need to calculate how many 75s go into 15,000.

\( 200 \times 75 = 15,000 \). Correct.

Look at the second row of working: 1 6 1 7 5 0.

This row comes from multiplying the top number (\( \dots 235 \)) by the missing digit in the tens column (represented as \( \dots 0 \)).

Let’s ignore the zero at the end: \( 16175 \).

We know the top number ends in 5.

What digit \( \times 5 \) ends in 5? It could be 1, 3, 5, 7, 9.

Let’s look at the whole number. \( \dots 235 \times ? = 16175 \).

Let’s test 5.

\( 235 \times 5 = 1175 \) (Too small, missing the thousands digit).

Let’s look at the full top number. If we assume the missing top digit is roughly 3 (making it 3235), then \( 3000 \times 5 = 15000 \). This is close to 16175.

So the missing multiplier is likely 5.

Now look at the first row of working: 9 7 0 5.

This is \( \text{Top Number} \times 3 \).

So, \( \square 235 \times 3 = 9705 \).

Let’s divide 9705 by 3 to find the top number.

\( 9000 \div 3 = 3000 \).

\( 705 \div 3 = 235 \).

So the top number is 3235.

The missing digit is 3.

We have 8 feet.

1 foot = 30 cm.

\[ 8 \times 30 = 240 \text{ cm} \]

We have 11 inches.

1 inch = 2.5 cm.

We need to calculate \( 11 \times 2.5 \).

A regular hexagon can be divided into 6 equal equilateral triangles meeting at the centre.

Let’s say the area of one of these equilateral triangles is 1 unit.

So, the total area of the hexagon is 6 units.

The large shaded triangle is exactly one of these 6 triangles.

Area = 1 unit.

The question says the large triangle is double the area of the small triangle.

This means the small triangle is half the area of the large one.

Area of small triangle = 0.5 units (or \( \frac{1}{2} \) unit).

Total Shaded Area = \( 1 + 0.5 = 1.5 \) units.

Fraction = \( \frac{\text{Shaded Area}}{\text{Total Area}} \)

Fraction = \( \frac{1.5}{6} \)

To get rid of the decimal, multiply top and bottom by 2.

Fraction = \( \frac{3}{12} \)

Simplify the fraction by dividing by 3.

Fraction = \( \frac{1}{4} \)

The large box has 20% more than 650g.

First, find 10% of 650g.

\( 650 \div 10 = 65 \text{ g} \).

So, 20% is double that: \( 65 \times 2 = 130 \text{ g} \).

Total mass = \( 650 + 130 = 780 \text{ g} \).

One portion is 40g.

We need to calculate \( 780 \div 40 \).

This is the same as \( 78 \div 4 \).

The result is 19.5 portions.

We need “full” portions, so we ignore the 0.5.

Answer is 19.

The whole pie chart represents 100%.

We know:

• Not important: 12%
• Quite important: 41%

Total known = \( 12 + 41 = 53\% \).

Missing ‘Very important’ = \( 100 – 53 = 47\% \).

We need to find 47% of 1,200 pupils.

47% is \( 0.47 \).

Calculation: \( 0.47 \times 1200 \).

This is the same as \( 47 \times 12 \).

We need to find how many 25s fit into 500.

\[ 500 \div 25 \]

There are four 25s in 100.

So in 500, there are \( 5 \times 4 = 20 \).

Answer: 20 packs.

Total mass = 2.4 kg.

1 kg = 1,000 g.

\[ 2.4 \times 1000 = 2400 \text{ g} \]

There are 500 sheets in the large pack.

Total mass is 2400 g.

Mass of one sheet = \( 2400 \div 500 \).

Cancel zeros: \( 24 \div 5 \).

Formula: \( \text{mass} = 2 \times (\text{age} + 5) \).

Age = 4.

First, do the brackets: \( 4 + 5 = 9 \).

Then multiply by 2: \( 2 \times 9 = 18 \).

Answer: 18 kg.

We know the mass is 16 kg.

Formula: \( 16 = 2 \times (\text{age} + 5) \).

We need to find the age.

1. Undo the multiply by 2: Divide 16 by 2.

\[ 16 \div 2 = 8 \]

So, \( \text{age} + 5 = 8 \).

2. Undo the add 5: Subtract 5 from 8.

\[ 8 – 5 = 3 \]

Answer: 3 years.