Counting and chance: KS2 and 11 Plus maths practice
Two dice, ways with coins, spinners, counting shapes in a figure, painted cubes and snaking number boards.
OftenIn many papers, especially GL AssessmentAbout 60 seconds a question in the test
The big idea
Counting questions are about being organised. Count one kind at a time, in an order, so nothing is missed and nothing is counted twice.
Two dice give thirty six different throws, because a red two with a blue five is not the same throw as a red five with a blue two. Every throw is equally likely.
A spinner with equal parts gives the chance of a colour as the number of its parts out of all the parts. In a painted cube, the corners have three painted faces, the edges two, the middle of each face one, and the inside none.
Tips that save marks
Make a six by six table for two dice.
When counting squares in a grid, count the small ones, then the next size, then the next.
On a snaking board, find which way the row runs before counting along it.
A worked example
A red dice and a blue dice are thrown. In how many ways can the scores add up to four?
Red one and blue three, red two and blue two, red three and blue one.
Red one blue three is a different throw from red three blue one.
Answer: three ways.
Now try five
Try each one before you look at the answer: answering from memory is what makes it stick. Tap an option to check it. Most children get three or four right at first; five out of five means move on to the next topic.
Question 1Aim for about 60 seconds
This shows the start of a board with 10 squares in each row. The numbers snake: the first row goes left to right, the next right to left, and so on. Which number is directly above 57?
40
39
38
37
36
35
34
33
32
31
21
22
23
24
25
26
27
28
29
30
20
19
18
17
16
15
14
13
12
11
1
2
3
4
5
6
7
8
9
10
Show the worked answer
Answer: E, 64
How to work it out
57 is 4 squares from the left in row 6.
Going up 1 row, the same column holds 64, because each row runs the opposite way to the one below.
Why the other options are wrong
A (63): This is one square along the other way; count the columns again.
B (65): This is one square along from the right answer; count the columns again.
C (44): This is the square directly below, not above.
D (67): This adds a whole row of 10, as if every row ran the same way; the rows snake.
Question 2Aim for about 60 seconds
How many different sets of 5 coins make exactly 37p? (The same coins in a different order count as the same set.)
Show the worked answer
Answer: B, 3
How to work it out
List them in order, largest coin first: 20p, 10p, 5p, 1p and 1p; 20p, 5p, 5p, 5p and 2p; 10p, 10p, 10p, 5p and 2p.
That is 3 sets.
Why the other options are wrong
A (5): This is the number of coins in each set, not the number of sets.
C (100): This counts the same coins in a different order as a new set.
D (2): This misses a set: list them starting with the biggest possible first coin, then the next biggest.
E (4): This counts a set twice or counts one that does not make 37p.
Question 3Aim for about 60 seconds
The spinner is spun once. What is the probability that it lands on red? Give the answer as a fraction in its lowest terms.
Show the worked answer
Answer: C, 3/8
How to work it out
red is 3 of the 8 equal parts, so the probability is 3/8.
Why the other options are wrong
A (5/8): This is the chance of NOT landing on red.
B (1/8): This is the chance of one part; red has 3 parts.
D (3/5): This compares red with the other parts; the chance is red out of all 8 parts.
E (1/4): There are 4 colours, but they do not have the same number of parts.
Question 4Aim for about 60 seconds
This large cube is made of small cubes, 3 along each edge. The outside of the large cube is painted, then it is taken apart. How many small cubes have exactly 2 painted faces?
Show the worked answer
Answer: A, 12
How to work it out
Cubes with exactly 2 painted faces are the edges, not at a corner: 1 on each of the 12 edges, so 12.
Why the other options are wrong
B (27): This is every small cube.
C (6): This is the number with exactly one painted face.
D (8): This is the number with exactly 3 painted faces.
E (1): This is the number with no painted faces.
Question 5Aim for about 60 seconds
This shows the start of a board with 8 squares in each row. The numbers snake: the first row goes left to right, the next right to left, and so on. Which number is directly above 35?
32
31
30
29
28
27
26
25
17
18
19
20
21
22
23
24
16
15
14
13
12
11
10
9
1
2
3
4
5
6
7
8
Show the worked answer
Answer: D, 46
How to work it out
35 is 3 squares from the left in row 5.
Going up 1 row, the same column holds 46, because each row runs the opposite way to the one below.
Why the other options are wrong
A (30): This is the square directly below, not above.
B (43): This adds a whole row of 8, as if every row ran the same way; the rows snake.
C (45): This is one square along the other way; count the columns again.
E (47): This is one square along from the right answer; count the columns again.
Want a plan built around the topics your child finds hardest?
Practise straight away at your child’s school year in maths and English, with verbal and non-verbal reasoning for the 11 Plus, and every answer explained. A free check, whenever you like, shows what to practise next and builds a weekly plan, for their year or for your target schools. Parents follow progress free.