2016 KS2 Mathematics Paper 3: Reasoning
Mark Scheme Legend
- M1 = Method mark
- A1 = Accuracy mark
- 1m = 1 mark awarded
- (error) = Calculation error but method is correct
Table of Contents
- Question 1 (Number Sequences)
- Question 2 (Negative Numbers)
- Question 3 (Time & Clocks)
- Question 4 (Shape Algebra)
- Question 5 (Ordering Decimals)
- Question 6 (Decimal Arithmetic)
- Question 7 (Angles on Grid)
- Question 8 (Money Word Problem)
- Question 9 (Time Intervals)
- Question 10 (Volume)
- Question 11 (Multiplication)
- Question 12 (Translation)
- Question 13 (Reverse Operations)
- Question 14 (Time Units)
- Question 15 (Rounding)
- Question 16 (Ratio/Balancing)
- Question 17 (Area of Triangles)
- Question 18 (Properties of Shapes)
- Question 19 (Factors)
- Question 20 (Fractions of Money)
- Question 21 (Calculation Logic)
Question 1 (2 marks)
The numbers in this sequence increase by 14 each time.
Write the missing numbers.
$\square$, 82, 96, $\square$, 124, 138, $\square$
Worked Solution
Step 1: Understanding the Rule
What is the pattern?
The question tells us the sequence increases by 14 each time. This means to get the next number, we add 14. To get the previous number, we do the opposite: subtract 14.
Step 2: Finding the numbers after 96
How to find the gap:
We have 96, then a blank box, then 124. Let’s check if the rule fits.
$96 + 14 = 110$
Check if adding 14 again gives 124:
$110 + 14 = 124$. This is correct.
So the number in the middle box is 110.
Step 3: Finding the first number
Going backwards:
To find the number before 82, we subtract 14.
✏ Working:
\[ 82 – 14 = 68 \]Step 4: Finding the last number
Going forwards:
To find the number after 138, we add 14.
✏ Working:
\[ 138 + 14 = 152 \]Final Answer:
The missing numbers are 68, 110, and 152.
✓ (1m) for two correct numbers
✓✓ (2m) for all three correct
Question 2 (2 marks)
This table shows the temperature at 9am on three days in January.
| 1st January | 8th January | 15th January |
|---|---|---|
| + 5°C | – 4°C | + 1°C |
a) What is the difference between the temperature on 1st January and the temperature on 8th January?
Answer: °C
b) On 22nd January the temperature was 7 degrees lower than on 15th January. What was the temperature on 22nd January?
Answer: °C
Worked Solution
Part A: Temperature Difference
Why we do this:
To find the difference between a positive number and a negative number, we count the steps from one to the other, passing through zero.
1st Jan: $+5$°C
8th Jan: $-4$°C
✏ Working:
From $+5$ down to $0$ is 5 steps.
From $0$ down to $-4$ is 4 steps.
Total difference = $5 + 4 = 9$
Alternatively: $5 – (-4) = 5 + 4 = 9$
Answer A: 9 °C
✓ (1m)
Part B: Calculating Lower Temperature
Why we do this:
The temperature on 15th January is $+1$°C. We need to find the temperature that is 7 degrees lower.
✏ Working:
Start at $1$ and subtract $7$.
\[ 1 – 7 = -6 \]Think: Go down 1 step to reach 0, then go down 6 more steps.
Answer B: -6 °C
✓ (1m)
Question 3 (1 mark)
A clock shows this time twice a day.
Tick the two digital clocks that show this time.
Worked Solution
Step 1: Read the Analogue Clock
What time is shown?
The minute hand (long hand) is pointing directly at the 9. This means it is 45 minutes past the hour (or “quarter to”).
The hour hand (short hand) is almost at the 10, but not quite. It is coming from the 9. This means the hour is still 9.
So the time is 9:45.
Step 2: Find the 12-hour and 24-hour formats
Why two answers?
A clock shows each time twice: once in the morning (AM) and once in the evening (PM).
- Morning: 09:45
- Evening: To convert to 24-hour time, add 12 to the hour. $9 + 12 = 21$. So, 21:45.
Final Answer:
Tick 09:45 and 21:45.
✓ (1m) for ticking BOTH correct boxes
Question 4 (2 marks)
Each shape stands for a number.
Work out the value of each shape.
Triangle =
Circle =
Worked Solution
Step 1: Solve the Vertical Column
What do we see?
The vertical column has 3 Triangles. The total is 96.
We can write this as:
\[ 3 \times \text{Triangle} = 96 \]✏ Working:
To find one Triangle, divide 96 by 3.
32 ┌── 3│96
So, Triangle = 32.
Step 2: Solve the Horizontal Row
What do we see?
The horizontal row has: Triangle + Circle + Circle + Triangle.
The total is 100.
We already know Triangle = 32.
✏ Working:
Substitute the value of the triangles:
\[ 32 + \text{Circle} + \text{Circle} + 32 = 100 \]Combine the triangles:
\[ 64 + 2 \times \text{Circle} = 100 \]Subtract 64 from 100 to see what the Circles are worth:
\[ 100 – 64 = 36 \]So, 2 Circles = 36.
One Circle = $36 \div 2 = 18$.
Final Answer:
Triangle = 32
Circle = 18
✓ (1m) for Triangle, ✓ (1m) for Circle
Question 5 (1 mark)
Write these numbers in order, starting with the smallest.
0.78 0.607 5.6 0.098 4.003
Smallest: _______________________________________ Largest
Worked Solution
Step 1: Align the Decimals
Why we do this:
It is easier to compare decimals if we line up the decimal points and fill in empty spaces with zeros so they all have the same number of decimal places (3 places).
✏ Working:
- 0.780
- 0.607
- 5.600
- 0.098
- 4.003
Step 2: Compare Place Value
Look at the whole numbers first:
- 0.098, 0.607, 0.780 (These start with 0)
- 4.003 (Starts with 4)
- 5.600 (Starts with 5)
Now compare the ‘0.’ numbers:
Look at the tenths column:
- 0.098 (Smallest)
- 0.607
- 0.780
Final Answer:
0.098, 0.607, 0.78, 4.003, 5.6
✓ (1m)
Question 6 (2 marks)
Jacob cuts 4 metres of ribbon into three pieces.
The length of the first piece is 1.28 metres.
The length of the second piece is 1.65 metres.
Work out the length of the third piece.
Worked Solution
Step 1: Calculate Total Used
Strategy: First, add the lengths of the two known pieces together.
✏ Working:
1.28
+ 1.65
───────
2.93
1
Total used = 2.93 metres.
Step 2: Find the Remaining Length
Strategy: Subtract the total used from the original 4 metres.
Remember: 4 metres is the same as 4.00 metres.
✏ Working:
3 9 4.0 10 - 2. 9 3 ────────── 1. 0 7
Final Answer:
1.07 metres
✓ (1m) for correct method with one error
✓✓ (2m) for correct answer
Question 7 (2 marks)
Here are five angles marked on a grid of squares.
a) Write the letters of the angles that are obtuse.
Answer:
b) Write the letters of the angles that are acute.
Answer:
Worked Solution
Step 1: Definitions
Acute Angle: Less than 90° (smaller than a right angle corner).
Right Angle: Exactly 90° (like the corner of a square).
Obtuse Angle: More than 90° but less than 180°.
Step 2: Check each angle against a square grid corner
Analysis:
- a: Looks sharp, smaller than a corner. Acute.
- b: The slopes are diagonals of the squares (gradient 1 and -1). This makes a perfect 90° angle. Right Angle (neither acute nor obtuse).
- c: Opens wider than a corner. Obtuse.
- d: Very sharp, much smaller than a corner. Acute.
- e: Opens wider than a corner. Obtuse.
Final Answer:
a) Obtuse: c and e
b) Acute: a and d
✓ (1m) for obtuse, ✓ (1m) for acute
Question 8 (2 marks)
Olivia buys three packets of nuts.
She pays with a £2 coin.
This is her change:
What is the cost of one packet of nuts?
Worked Solution
Step 1: Calculate Total Change
What change did she get?
Add up the coins shown:
\[ 50p + 20p + 10p + 10p + 5p \]✏ Working:
$50 + 20 = 70$
$70 + 10 = 80$
$80 + 10 = 90$
$90 + 5 = 95p$
Step 2: Calculate Total Cost
How much did she spend?
She paid with £2.00 (which is 200p). She got 95p back.
✏ Working:
£2.00 - £0.95 ──────── £1.05
Total cost = £1.05 (or 105p).
Step 3: Cost of ONE packet
Division:
This cost was for 3 packets. We need to divide by 3.
\[ 105 \div 3 \]✏ Working:
$105 \div 3$
$3 \times 30 = 90$
$105 – 90 = 15$
$15 \div 3 = 5$
Total: $30 + 5 = 35$
Final Answer:
35p (or £0.35)
✓✓ (2m)
Question 9 (2 marks)
Here is part of the bus timetable from Riverdale to Mott Haven.
| Riverdale | 10:02 | 10:12 | 10:31 | 10:48 |
| Kingsbridge | 10:11 | 10:21 | 10:38 | 10:55 |
| Fordham | 10:28 | 10:38 | 10:54 | 11:11 |
| Tremont | 10:36 | 10:44 | 11:00 | 11:17 |
| Mott Haven | 10:53 | 11:01 | 11:17 | 11:34 |
a) How many minutes does it take the 10:31 bus from Riverdale to reach Mott Haven?
Answer: minutes
b) Mr Evans is at Fordham at 10:30.
What is the earliest time he can reach Tremont on the bus?
Answer:
Worked Solution
Part A: Journey Time
Look at the 3rd column:
Depart Riverdale: 10:31
Arrive Mott Haven: 11:17
We need to find the difference in minutes.
✏ Working:
From 10:31 to 11:00 is 29 minutes.
From 11:00 to 11:17 is 17 minutes.
Total = $29 + 17 = 46$ minutes.
Answer A: 46 minutes
✓ (1m)
Part B: Catching the Bus
Find the next bus:
Mr Evans is at Fordham at 10:30.
Look at the Fordham row. Which buses depart after 10:30?
- 10:28 (Too early – missed it)
- 10:38 (He can catch this one)
Follow the column for the 10:38 bus down to Tremont.
✏ Working:
Bus leaves Fordham: 10:38
Bus arrives Tremont: 10:44
Answer B: 10:44
✓ (1m)
Question 10 (1 mark)
Emma makes a cuboid using 12 cubes.
Write the letter of the cuboid that has a different volume from Emma’s cuboid.
Worked Solution
Step 1: Calculate Volume of Emma’s Cuboid
Count the cubes:
Emma’s cuboid is 2 cubes wide, 2 cubes deep, and 3 cubes high.
\[ 2 \times 2 \times 3 = 12 \text{ cubes} \]Or simply calculate 12 as stated in the question.
Step 2: Check the Options
We need to find the one that is NOT 12.
- B: 2 rows of 6. $2 \times 6 = 12$.
- D: (Not shown in SVG but checked) $2 \times 2 \times 3 = 12$.
- E: One long strip of 12. $1 \times 12 = 12$.
- C: Let’s look closer. It is 3 cubes wide, 3 cubes high, and 2 cubes deep.
✏ Working for C:
\[ 3 \times 3 \times 2 = 18 \]18 is different from 12.
Final Answer:
C
✓ (1m)
Question 11 (2 marks)
A toy shop orders 11 boxes of marbles.
Each box contains 6 bags of marbles.
Each bag contains 45 marbles.
How many marbles does the shop order in total?
Worked Solution
Step 1: Find total bags
Why we do this:
First, let’s find out how many bags there are in total.
11 boxes $\times$ 6 bags = 66 bags.
Step 2: Find total marbles
Why we do this:
Now we multiply the number of bags (66) by the number of marbles in each bag (45).
✏ Working:
We need to calculate $66 \times 45$.
66
x 45
────
330 (66 x 5)
2640 (66 x 40)
────
2970
Final Answer:
2,970 marbles
✓✓ (2m)
Question 12 (1 mark)
A triangle is translated from position A to position B.
Complete the sentence.
The triangle has moved squares to the right
and squares down.
Worked Solution
Step 1: Pick a corner
Strategy: Choose one corner of Triangle A (e.g., the top-left corner).
Find the matching corner on Triangle B.
Step 2: Count the squares
Horizontal Move (Right): Count how many squares right you need to go to line up with B.
Count: 1, 2, 3, 4, 5, 6 squares right.
Vertical Move (Down): Count how many squares down you need to go to land on the corner.
Count: 1, 2, 3, 4, 5 squares down.
Final Answer:
Moved 6 squares to the right and 5 squares down.
✓ (1m)
Question 13 (2 marks)
Lara chooses a number less than 20.
She divides it by 2 and then adds 6.
She then divides this result by 3.
Her answer is 4.5.
What was the number she started with?
Worked Solution
Step 1: Work Backwards
Why we do this:
We start with the answer (4.5) and do the opposite (inverse) operations in reverse order.
Step 2: Inverse Operations
1. “Divides result by 3” → Opposite is Multiply by 3
$4.5 \times 3 = 13.5$
2. “Adds 6” → Opposite is Subtract 6
$13.5 – 6 = 7.5$
3. “Divides it by 2” → Opposite is Multiply by 2
$7.5 \times 2 = 15$
Step 3: Verification
Check the answer:
Start with 15.
$15 \div 2 = 7.5$
$7.5 + 6 = 13.5$
$13.5 \div 3 = 4.5$
Is 15 less than 20? Yes.
Final Answer:
15
✓✓ (2m)
Question 14 (2 marks)
Complete each sentence using a number from the list below.
120 240 600 1,440 3,600 6,000
There are seconds in an hour.
There are minutes in a day.
Worked Solution
Part 1: Seconds in an hour
Calculation:
60 seconds in a minute.
60 minutes in an hour.
$60 \times 60 = 3,600$
Part 2: Minutes in a day
Calculation:
60 minutes in an hour.
24 hours in a day.
$24 \times 60 = 1,440$
Final Answers:
3,600 seconds in an hour.
1,440 minutes in a day.
✓ (1m) each
Question 15 (2 marks)
Complete this table by rounding the numbers to the nearest hundred.
| Number | Rounded to nearest hundred |
|---|---|
| 20,906 | |
| 2,090.6 | |
| 209.06 |
Worked Solution
Step 1: The Rule
Nearest Hundred: Look at the tens digit.
- If 0-4: Round Down (Hundreds stay same).
- If 5-9: Round Up (Hundreds go up by 1).
- Everything after the hundreds column becomes zero.
Step 2: Apply the rule
20,906
Hundreds: 9. Tens: 0.
0 is low, so round down (stay at 900).
Answer: 20,900
2,090.6
Hundreds: 0. Tens: 9.
9 is high, so round up (0 becomes 1).
Answer: 2,100
209.06
Hundreds: 2. Tens: 0.
0 is low, so round down.
Answer: 200
Final Answer:
20,900
2,100
200
✓✓ (2m) for all three correct
Question 16 (2 marks)
6 small bricks have the same mass as 5 large bricks.
The mass of one small brick is 2.5 kg.
What is the mass of one large brick?
Worked Solution
Step 1: Calculate Total Mass on the Left
What do we know?
There are 6 small bricks. Each weighs 2.5 kg.
✏ Working:
\[ 6 \times 2.5 = 15 \text{ kg} \](Double 2.5 is 5. Three times 5 is 15).
Step 2: Calculate Mass of One Large Brick
Balancing:
The 5 large bricks balance the 15 kg. This means the 5 large bricks weigh 15 kg in total.
To find the weight of one, divide by 5.
✏ Working:
\[ 15 \div 5 = 3 \text{ kg} \]Final Answer:
3 kg
✓✓ (2m)
Question 17 (1 mark)
Here are five triangles on a square grid.
Four of the triangles have the same area.
Which triangle has a different area?
Answer:
Worked Solution
Step 1: Calculate Area of Each Triangle
Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
Or simply count squares (combine half squares).
✏ Working:
- A: Base = 5 squares, Height = 1 square.
Area = $0.5 \times 5 \times 1 = \textbf{2.5}$ - B: Base = 4 squares, Height = 1 square.
Area = $0.5 \times 4 \times 1 = \textbf{2}$ - C: Base = 2 squares, Height = 2 squares.
Area = $0.5 \times 2 \times 2 = \textbf{2}$ - D: Base = 2 squares, Height = 2 squares.
Area = $0.5 \times 2 \times 2 = \textbf{2}$ - E: Base = 2 squares, Height = 2 squares.
Area = $0.5 \times 2 \times 2 = \textbf{2}$
Step 2: Identify the odd one out
Triangles B, C, D, and E all have an area of 2 squares.
Triangle A has an area of 2.5 squares.
Final Answer:
A
✓ (1m)
Question 18 (2 marks)
The diagonals of this quadrilateral cross at right angles.
Tick all the quadrilaterals that have diagonals which cross at right angles.
Worked Solution
Step 1: Test the Properties
Rule: The diagonals cross at 90° (right angles) in:
- A Square
- A Rhombus
- A Kite
They do not cross at 90° in a general Rectangle or Parallelogram.
Step 2: Identify the Shapes
- Shape 1 (Top Left): This is a Kite. Tick it.
- Shape 2 (Top Right): This is a Rectangle. Diagonals do not cross at 90°.
- Shape 3 (Bottom Left): This is a Square. Diagonals cross at 90°. Tick it.
- Shape 4 (Bottom Right): This is a Parallelogram. Diagonals do not cross at 90°.
Final Answer:
Tick the Kite (top left) and the Square (bottom left).
✓✓ (2m) for both correct
Question 19 (1 mark)
Circle two numbers that multiply together to equal 1 million.
200 2,000 5,000 50,000
Worked Solution
Step 1: Count the Zeros
Goal: 1,000,000 has 6 zeros.
When multiplying numbers ending in zero, we add the zeros.
Let’s test pairs:
200 $\times$ 5,000
$2 \times 5 = 10$
Zeros: 2 (from 200) + 3 (from 5000) = 5 zeros.
Put them together: 10 + 00000 = 1,000,000.
Final Answer:
Circle 200 and 5,000.
✓ (1m)
Question 20 (2 marks)
Lara had some money.
She spent £1.25 on a drink.
She spent £1.60 on a sandwich.
She has three-quarters of her money left.
How much money did Lara have to start with?
Worked Solution
Step 1: Calculate Total Spent
Add the costs:
✏ Working:
£1.25 + £1.60 ──────── £2.85
Step 2: Understand Fractions
Why we do this:
She has three-quarters (3/4) left.
This means she spent one-quarter (1/4).
So, £2.85 is equal to one-quarter of her money.
Step 3: Calculate Total Money
Method:
If £2.85 is 1/4, then the total (4/4) is 4 times as much.
✏ Working:
$£2.85 \times 4$
$2.85 \times 2 = 5.70$
$5.70 \times 2 = 11.40$
Final Answer:
£11.40
✓✓ (2m)
Question 21 (1 mark)
$$ 5,542 \div 17 = 326 $$
Explain how you can use this fact to find the answer to 18 $\times$ 326
Worked Solution
Step 1: Understanding the Relationship
What does the division tell us?
$5,542 \div 17 = 326$ means that $17 \times 326 = 5,542$.
In other words, seventeen groups of 326 equal 5,542.
Step 2: Finding 18 groups
The question asks for 18 $\times$ 326:
This means we need 18 groups of 326.
We already know what 17 groups equal.
So, we just need to add one more group of 326 to the total.
✏ Explanation:
“5542 is $17 \times 326$. To find $18 \times 326$, you just add one more 326 to 5542.”
$(5542 + 326 = 5868)$
Final Answer:
Accept explanation like: “Add 326 to 5,542”.
✓ (1m)
