Combining shapes
Two separate pieces, and you pick the shape they make when they're joined edge to edge. Here's how to find it every time, and the traps the wrong answers set.
Chapter 1
What are combining shapes questions?
You’re shown two flat pieces, each black or white. You pick the shape they make when they’re joined edge to edge, like two jigsaw pieces pushed together.
The pieces. Each one keeps its shape, its size and its colour.
The join. One edge on each piece, the same length. That’s where they meet.
Some exam papers letter the edges and use more than two pieces. Chapter 4 shows you how to do those.
Which shape do these two make when joined?
An NVR question from Rush, level 1.
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Edge to edge
A whole edge lies along a whole edge, corner on corner.
The triangle’s bottom is as long as the square’s top, so it sits right on it: a house. No gap, no overlap, nothing hanging past the end.
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The piece Turned Flipped Turned or flipped?
Turning keeps a piece the same. Flipping makes a different one.
This triangle has its square corner at one end. Spin it round and it’s still the same piece. Flip it over, as in a mirror, and the square corner swaps ends. A flipped piece is always wrong.
Chapter 2
Solve it in 3 steps.
Here’s the question, one step at a time.
The question
Which shape do these two make when joined?
An NVR question from Rush, level 1.
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Step 1
Find the matching edges.
Look for one edge on each piece that’s the same length. The semicircle’s flat edge is exactly as long as a side of the square, so that’s the join. The flat edge faces up, so the square sits on top. Picture the joined shape before you look at the options.
The same length, so the square sits on the flat edge.
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Step 2
Check the ends of the join.
The square’s bottom corners must land exactly on the ends of the flat edge. In options 1, 3 and 5 the top piece is slid along, so it hangs past one end and the curve sticks out at the other. Cross them out.
1 2 3 4 5 Slid along: 1, 3 and 5.
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Step 3
Check each piece.
Look at each piece’s shape, size and colour. Options 2 and 4 are left, both a white square on a black semicircle. A square is as tall as the join is long, but option 4’s is only half that: it’s been squashed. So it’s option 2.
2 4 The dashed line shows how tall 4’s square should be.
Three mistakes to avoid
| The mistake | Do this instead |
|---|---|
| Judging by the outline. Options 1 and 5 have a white four-sided shape as tall as the square, but its sides slope. | Check every corner. A square has four square corners. |
| Sorting by colour. Only option 3 has a piece the wrong colour. In Rush at most one option does, and sometimes both pieces are white. | Colour rules out one option at most. Check shape, size and the join too. |
| Missing a squashed piece. Option 4 is joined neatly, in the right colours. Only its square is half as tall as it should be. | Measure each piece against the join: a square is as tall as it, a semicircle half as deep. |
So it’s option 2. The white square stands on the black semicircle’s flat edge, the right size, its corners on the ends of the join.
Remember: Same length. Same ends. Same pieces.
Chapter 3
When it gets harder.
Lookalike pieces come up at every level of Rush. Its hardest level turns the whole shape too. Here’s how to beat both.
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Joined Turned One flipped The whole shape turned
Turned options? Turn each one back before you check it.
At Rush’s hardest level the joined shape is usually given a quarter or half turn. Both pieces turn together. Trap: in the last one the parallelogram is flipped on its own, so it leans the wrong way.
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Lookalike pieces
Check every corner, not just the outline.
Rush swaps a piece for a lookalike on the same edge, the same height, like each pair here. A loaf shape is round on top like a semicircle, but its sides are straight. Trap: at a glance the joined shape still looks right.
Count the corners to check
Picture the joined outline and count its corners. A triangle on the end of a rectangle makes a pencil with 5. Slide the triangle up and you count 7: extra corners mean a piece hangs past the end.
Chapter 4: In the exam
Pieces with lettered edges.
Some practice papers for the 11+ and other entrance exams letter the edges, and pieces may need turning first. Rush doesn’t ask this one, but the same checks work.
Join the edges with the same letters. Which shape do the pieces make?
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Step 1
Match the letters.
The square has a on top and b on its right. So the triangle goes on top and the rectangle on the right. Option 5 has them the other way round: out.
1 2 3 4 5 -
Step 2
Turn each piece. Never flip it.
The triangle’s a is its left side, with the square corner at the bottom left. To face down onto the square, it turns a quarter anticlockwise, and the square corner swings to the right. Option 2 has it on the left: that’s a flip.
1 2 3 4 5 -
Step 3
Check each piece’s size and edge.
Edge b is the rectangle’s long side, as long as the square’s. Option 1 joins its short end instead, and option 3’s triangle is far too tall. That leaves option 4.
1 2 3 4 5
So it’s option 4. The triangle sits on a, turned so its square corner is on the right, and the rectangle stands against b.
Remember: Letter to letter. Turn, never flip.
Other ways the exam asks it
- “Which pieces make this shape?” Start with the shape’s most unusual edge, a curve or a slant. Find the piece that has it, then check the other pieces fill the gap exactly.
- The joined shape turned. Turn each option back first, as in chapter 3.
- Stripes or spots on a piece. They turn with it. Stripes that ran along a piece still run along it after a turn.
- A bit missing or added. Every piece is used once, and nothing else appears. Count the pieces in each option.
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Find the matching edges.
- Check the ends of the join.
- Check each piece.
Remember: Same length. Same ends. Same pieces.
For parents.
What parents ask us about combining shapes.
Questions parents ask
What are combining shapes questions in non-verbal reasoning?
Your child is shown separate pieces and picks the shape they make when joined. Rush uses two pieces that join along their one matching edge. Papers also use pieces with lettered edges, where edge a joins edge a.
How do you know which edges join?
Match the lengths. Compare each straight edge of one piece with the edges of the other, and the pair of the same length is the join. When a paper labels the edges with letters, the letters tell you, and all you need to check is turns and sizes.
Does the joined shape have to be the same way up as the pieces?
Not always. At Rush's level 3 it is usually shown turned, and some papers turn it too. Turn the whole option back in your head before you check the pieces.
Rush: 11+ Practice & KS2 SATsPractise combining shapes in Rush.
- Join Shapes as 100-second games, easy at first and harder as your child improves
- Every miss kept with the right answer and why
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