Shape, angles, coordinates and statistics
Geometry and statistics questions make up a good share of the SATs reasoning papers. Most rely on a small set of facts used precisely: the angle totals, the order of coordinates, what each sector of a pie chart represents, and how the mean is calculated.
2-D shapes and their properties
Year 6 pupils compare and classify shapes by their properties: number of sides and angles, parallel and perpendicular sides, lines of symmetry, and whether they are regular. They need to know the properties of the quadrilaterals in particular: square, rectangle, rhombus, parallelogram, trapezium and kite.
Two misconceptions cause most errors. Children think a square isn't a rectangle (it is: it has four right angles), and that a shape only counts if it's drawn in its “usual” orientation. A square turned on its corner is still a square, not a diamond.
Missing angles
Missing-angle problems use known facts to calculate unknown angles. These are the facts children need by heart:
| Fact | Total |
|---|---|
| Angles on a straight line | 180° |
| Angles around a point | 360° |
| Angles in a triangle | 180° |
| Angles in a quadrilateral | 360° |
| Vertically opposite angles | equal |
- Triangle with 65° and 48°: 180 − 65 − 48 = 67°Identify the fact, then calculate.
- Measuring the angle with a protractorDiagrams marked “not to scale” must be calculated.
Multi-step problems usually need an angle that isn't asked for. Encourage children to write every angle they find on the diagram as they go.
Coordinates
Coordinates give a point's position on a grid as (x, y): across first, then up or down. Year 6 uses all four quadrants, so either coordinate can be negative. “Along the corridor, then up the stairs” is still the best reminder of the order. Problems often ask for a missing vertex of a shape, which means using the shape's properties as well as the grid.
Translation and reflection
A translation slides a shape without turning it, described by how far it moves left or right and up or down. A reflection flips a shape in a mirror line, with each point the same distance from the line on the other side. The reliable method for both is to move one vertex at a time, then join them up.
- Translate (2, 3) by 4 right and 5 down → (6, −2)Add to x, subtract from y.
- Counting the gap between shapesCount how far one vertex moves, not the space between the shapes.
Charts and graphs
Year 6 pupils interpret and draw pie charts and line graphs. In a pie chart, each sector is a fraction of the whole, and the angles add up to 360°. A quarter of the circle (90°) represents a quarter of the total, whatever the total is. With line graphs, the skill is reading the scale on each axis carefully and reading between gridlines.
The mean
The mean is a type of average: add all the values and divide by how many there are. The mean of 4, 7 and 10 is 21 ÷ 3 = 7. Year 6 pupils also work backwards: if the mean of five numbers is 8, their total must be 40. That's the step that unlocks most SATs mean problems.
- Mean of 6, 0, 9, 5: (6 + 0 + 9 + 5) ÷ 4 = 5Count the zero as a value.
- (6 + 9 + 5) ÷ 3 = 6.67Leaving out the zero.
In the SATs
All of these topics appear on the reasoning papers. Typical questions include classifying shapes, calculating missing angles in triangles and on straight lines, plotting and reading coordinates in four quadrants, translating or reflecting a shape on a grid, interpreting pie charts and line graphs, and calculating the mean or a missing value from a mean.
Common mistakes
- Using the wrong angle total, or measuring instead of calculating.
- Reversing coordinates: plotting (3, 5) as 5 across, 3 up.
- Counting from the wrong side of zero with negative coordinates.
- Treating pie chart sectors as amounts rather than fractions of the total.
- Misreading a graph's scale, or dividing by the wrong number for the mean.
Teaching ideas
Shape sorting
Venn and Carroll diagrams with criteria such as “has parallel sides” and “has a right angle” force precise thinking about properties. Include rotated shapes on purpose.
Angle facts on the wall
Keep the five angle facts displayed during geometry lessons. For each missing-angle problem, pupils write which fact they used beside their answer.
Coordinate battleships
Play battleships on a four-quadrant grid. Children quickly learn the order of coordinates when getting it wrong means missing.
Class survey
Collect real class data (favourite sport, time spent reading) and present it as a pie chart and a table. Calculate the mean where it makes sense, and discuss when it doesn't.
Try it: five quick questions
SATs-style questions to use as a starter or a quick check. Tap to see each answer.
Two angles on a straight line: one is 127°. What is the other?
Show answer
53°
A quadrilateral has angles of 90°, 85° and 110°. Find the fourth.
Show answer
75°
Reflect the point (−3, 4) in the y-axis.
Show answer
(3, 4)
In a pie chart of 60 children, a sector is 90°. How many children does it represent?
Show answer
15
The mean of four numbers is 12. Three of them are 9, 14 and 10. What is the fourth?
Show answer
15: the total must be 4 × 12 = 48, and 48 − 33 = 15.
Free worksheets for this topic
Printable A4 PDFs with answers. Free to use in class or at home.
