Year 6 Mathematical reasoning practice for selective school tests

What the section covers

Multi-step problems

Several ordinary steps chained together, where one wrong turn early is fatal.

Rates and proportion

Speed, price per unit, recipes scaled up and down, best-value comparisons.

Space and shape

Area, volume, nets, symmetry and rotations described rather than drawn.

Reasoning with data

Drawing conclusions from a table or graph, and spotting what it does not show.

Sample questions

8 real questions from our Year 6 papers, covering the different ways this section asks things. Answer the ones you can click; the rest show how they work. Nothing to sign up for.

1. Question 1

   Lina maps an island showing areas that have only forest, only lakes, or both. These three categories cover the entire island.

   What area of the island has both forest and lakes? Total area of the island: 200 km² Total area where there is forest (with or without lakes): 130 km² Total area where there are lakes (with or without forest): 90 km²

   
   10 km²15 km²20 km²25 km²30 km²

2. Question 2

   What is the length of one plank?

   Three wooden planks are all the same length. They’re placed in a staircase pattern:

   
   70 cm80 cm90 cm65 cm100 cm

3. Question 3

   You may shade more squares to make a pattern with exactly four lines of symmetry (both axes and both diagonals). What is the smallest number of extra squares you must shade?

   On a 5×5 grid, three small squares are already shaded:

   
   5791113

4. Question 4

   Anna has a square card 24 cm × 24 cm. What is the greatest possible number of these trapeziums she can cut from the card?

   
   2025304080

5. Question 5

   The shape below is made from rectangles (diagram not to scale).

   What is the perimeter of the shape?

   
   78 cm90 cm84 cm76 cm68 cm

6. Question 6

   A large solid cube is built from 125 small cubes (so it’s 5 × 5 × 5). John removes some small cubes. He now see this view when he looks at any of the six faces:

   How many small cubes remain?

   
   3640444864

7. Question 7

   The higher number wins the game. They play 8 games.

   How many games does Alex win? During the 8 games: Alex spins two 1s, two 2s, and four 3s. Blake spins four 1s and four 2s (no 3s). There are no draws.

   
   34567

8. Question 8

   A square piece of paper is folded so that both edges meet in the middle, and then folded in half again, as shown above.

   Which diagram matches the final sheet?