KS2 Maths Paper 3: Reasoning (SATs Practice)
Key Stage 2 maths has three papers: Paper 1 is arithmetic only, and Papers 2 and 3 are both mathematical reasoning papers of the same length and mark total, drawing on the same broad areas of the Year 6 curriculum, just with a different set of questions. There is no official split of topics between Paper 2 and Paper 3, so both mix number, measurement, geometry and statistics questions inside short problems, including angles, the properties of 2D shapes, position on a coordinate grid, translation and reflection, and reading information from pie charts.
Before you start
Make sure you're comfortable with these topics first:
Method
- For angle questions, learn the four core facts: angles at a point add up to 360 degrees, angles on a straight line add up to 180 degrees, angles in a triangle add up to 180 degrees, and angles in a quadrilateral add up to 360 degrees.
- Find a missing angle by adding up the angles you already know and subtracting the total from the fact that applies (360, 180, or the appropriate shape total), rather than trying to measure it by eye.
- For 2D shape questions, check the properties precisely: the number of sides, whether the sides are equal, and whether the angles are right angles, rather than naming a shape from its general appearance.
- For coordinates, remember the pair is written (across, up): the first number moves along the x-axis, and the second moves up the y-axis. On a full grid, a negative number means left or down from the origin.
- For a translation, move every point of the shape the same distance in the same direction and check the shape has not changed size or turned. For a reflection, count the squares from each point to the mirror line and place the reflected point the same distance on the other side.
- For a pie chart, use the given total and the size of each slice, as a fraction or a stated value, to work out the amount each slice represents, rather than trying to measure the angle of the slice.
- For any reasoning question, write down the fact or rule you are using before you calculate, since a method mark is often available for showing the correct rule even if the final number is wrong.
- Check the answer is sensible for a shape: an angle in a triangle cannot be 200 degrees, and a missing side cannot make the shape's total bigger than a stated perimeter.
Worked example
A triangle has one angle of 90 degrees and another angle of 35 degrees. Work out the third angle.
- Recall the rule: angles in a triangle add up to 180 degrees.
- Add the two known angles together: 90 + 35 = 125 degrees.
- Subtract that total from 180 to find the missing angle: 180 - 125 = 55 degrees.
- Write the answer with its unit: the third angle is 55 degrees.
- Check by adding all three: 90 + 35 + 55 = 180, which matches the rule, so the answer is correct.
- Note that because one angle is 90 degrees, this is a right-angled triangle, which is a useful check that the answer looks reasonable, since 55 degrees is an acute angle, as expected for the other two angles in a right-angled triangle.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Three angles around a point are 110 degrees, 95 degrees and x degrees. Work out x.Show answer
Answer: 155 degrees, because angles around a point add up to 360 degrees, and 360 - 110 - 95 = 155.
Q2A quadrilateral has angles of 100, 85 and 60 degrees. Work out the fourth angle.Show answer
Answer: 115 degrees, because angles in a quadrilateral add up to 360 degrees, and 360 - 245 = 115.
Q3Name a 2D shape with exactly one pair of parallel sides.Show answer
Answer: A trapezium.
Q4A point is at (2, 5). It is translated 3 right and 4 down. What are its new coordinates?Show answer
Answer: (5, 1), because moving 3 right adds 3 to the x-coordinate (2 + 3 = 5) and moving 4 down subtracts 4 from the y-coordinate (5 - 4 = 1).
Q5A point at (-2, 3) is reflected in the y-axis. What are its new coordinates?Show answer
Answer: (2, 3), because reflecting in the y-axis keeps the y-coordinate the same and changes the sign of the x-coordinate.
Q6A pie chart shows how 40 pupils travel to school. The walk slice is exactly a quarter of the circle. How many pupils walk to school?Show answer
Answer: 10 pupils, because a quarter of the circle represents a quarter of the 40 pupils, and 40 divided by 4 = 10.
Q7What is the missing angle on a straight line if the two other angles at that point are 72 degrees and 55 degrees?Show answer
Answer: 53 degrees, because angles on a straight line add up to 180 degrees, and 180 - 72 - 55 = 53.
Q8Explain why a square is always a rectangle, but a rectangle is not always a square.Show answer
Answer: A square has four right angles and four equal sides, which already satisfies the definition of a rectangle (four right angles with opposite sides equal), so every square is a rectangle. A rectangle only needs opposite sides equal, so its two different side lengths do not have to match, meaning it is not always a square.
Exam-style questions
Written in the style of a KS1 and KS2 SATs exam paper, with a full mark scheme.
A quadrilateral has angles of 90 degrees, 90 degrees, 60 degrees and x degrees. (a) Work out the value of x, showing your method. (b) Explain how you know this quadrilateral is not a rectangle.
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Point P is at (-3, -1). It is reflected in the x-axis to point Q, and then Q is translated 5 right and 2 up to point R. (a) What are the coordinates of Q? (b) What are the coordinates of R?
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A pie chart shows how 60 pupils in Year 6 chose their favourite school lunch. Pasta is half the circle, jacket potato is a quarter of the circle, and the rest chose salad. (a) How many pupils chose pasta? (b) How many pupils chose salad?
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Free printable worksheet
Want more practice on paper? Download the ks2 maths paper 3: reasoning (sats practice) worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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