Sorting keys and logic: KS2 and 11 Plus maths practice
Decision tree keys, Carroll diagrams, and true, false or cannot tell.
OftenIn many papers, especially GL AssessmentAbout 60 seconds a question in the test
The big idea
A sorting key is a tree of yes or no questions. Each item starts at the top, answers each question it meets, and follows the Yes or No line until it lands in a box.
A Carroll diagram sorts with two questions at once, one across the top and one down the side, so every number has exactly one box.
Logic questions give a few facts or a table and then some statements. A statement is true if the facts make it so every time, false if they never allow it, and cannot tell if it could go either way.
Tips that save marks
Sorting keys: follow one item at a time with your finger, and say Yes or No out loud at every question.
A missing question must give Yes for everything on its Yes side and No for everything on its No side.
Cannot tell is not the same as false: false means the facts rule it out.
A worked example
I am thinking of a whole number from 1 to 30. It is a multiple of 7. It is even. Is “the number is 14” true, false or cannot tell?
Multiples of 7 up to 30: 7, 14, 21, 28.
The even ones are 14 and 28.
It could be 14 or 28, so we cannot tell.
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
Which box of the Carroll diagram should 40 go in?
Row
a multiple of 10
not a multiple of 10
a multiple of 6
A: 30, 60, 90
B: 6, 84, 96
not a multiple of 6
C: 10, 50, 70
D: 16, 17, 61
Show the worked answer
Answer: B, C
How to work it out
40 is a multiple of 10, so it goes in the "a multiple of 10" column.
40 is not a multiple of 6, so it goes in the "not a multiple of 6" row.
They meet at box C.
Why the other options are wrong
A (D): Box D is for numbers that are not a multiple of 6 and not a multiple of 10; check the column again: 40 is a multiple of 10.
C (B): Box B has both the row and the column wrong for 40.
D (none of the boxes): Every whole number answers each question yes or no, so it always belongs in exactly one box.
E (A): Box A is for numbers that are a multiple of 6 and a multiple of 10; check the row again: 40 is not a multiple of 6.
Question 2Aim for about 60 seconds
Use only the table. Which statement must be true?
The table shows the children who visited the school library at lunchtime one week.
Day
Children
Monday
20
Tuesday
16
Wednesday
7
Thursday
22
Friday
9
Show the worked answer
Answer: A, More children visited the library on Friday than on Wednesday.
How to work it out
Friday has 9 and Wednesday has 7, so it is true.
Why the other options are wrong
B (More children visited the library on Monday than on Thursday.): Monday has 20 and Thursday has 22, so it is false.
C (More girls than boys visited the library.): The table does not split the children into girls and boys, so we cannot tell.
D (The most children visited the library on Tuesday.): Tuesday has 16, but Thursday has 22, so it is false.
E (The most children visited the library on Thursday because it was raining.): The table shows how many, not why. Thursday did have the most, but we cannot tell the reason.
Question 3Aim for about 60 seconds
Use only the facts. Which statement can you NOT tell is true or false?
Five children, Dev, Isla, Finn, Hari and Grace, ran a race and no two finished together. Isla finished before Hari. Finn did not win. Dev finished before Isla. Grace finished fourth.
Show the worked answer
Answer: C, Isla finished before Finn.
How to work it out
The facts allow three orders: Dev, Isla, Finn, Grace, Hari; Dev, Isla, Hari, Grace, Finn; Dev, Finn, Isla, Grace, Hari.
Isla finished before Finn in some of the orders the facts allow but not in others, so we cannot tell.
Why the other options are wrong
A (Grace finished last.): Grace finished last in none of the orders the facts allow, so it is false.
B (Dev finished fourth.): Dev finished fourth in none of the orders the facts allow, so it is false.
D (Dev finished before Finn.): Dev finished before Finn in every order the facts allow, so it is true.
E (Dev finished second.): Dev finished second in none of the orders the facts allow, so it is false.
Question 4Aim for about 60 seconds
Use only the table. Which statement must be true?
The table shows the cups of lemonade sold at a school stall one week.
Day
Cups sold
Monday
30
Tuesday
7
Wednesday
27
Thursday
6
Friday
10
Show the worked answer
Answer: D, Altogether 80 cups were sold from Monday to Friday.
How to work it out
30 + 7 + 27 + 6 + 10 = 80, so it is true.
Why the other options are wrong
A (Each cup cost 50p.): The table shows how many cups, not what they cost, so we cannot tell.
B (The most cups were sold on Friday.): Friday has 10, but Monday has 30, so it is false.
C (More cups were sold on Wednesday than on Monday.): Wednesday has 27 and Monday has 30, so it is false.
E (The most cups were sold on Monday because it was hot.): The table shows how many, not why. Monday did have the most, but we cannot tell the reason.
Question 5Aim for about 60 seconds
One number has been put in the wrong box of this Carroll diagram. Which number is it?
Row
a multiple of 10
not a multiple of 10
a multiple of 3
30, 60, 90
84, 87
not a multiple of 3
20, 80, 100
8, 9, 71, 74
Show the worked answer
Answer: B, 9
How to work it out
9 is not a multiple of 10 and it is a multiple of 3, so 9 belongs where "a multiple of 3" meets "not a multiple of 10", not where it has been put.
Why the other options are wrong
A (20): 20 is not a multiple of 3 and a multiple of 10, which is the box it is in.
C (90): 90 is a multiple of 3 and a multiple of 10, which is the box it is in.
D (71): 71 is not a multiple of 3 and not a multiple of 10, which is the box it is in.
E (84): 84 is a multiple of 3 and not a multiple of 10, which is the box it is in.
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