Which numbers? (1)
I am thinking of three sets of numbers less than 101. They are the
red set, the green set and the blue set. Can you find all the
numbers in the sets from these clues?
Age
7 to 11
Challenge level
Problem
I am thinking of three sets of numbers less than $101$. They are the red set, the green set and the blue set.
Can you find all the numbers in the sets from these clues?
These numbers are some of the red set: $19, 37, 55, 73, 82$, but there are others too.
These numbers are some of the green set: $5, 30, 35, 50, 70,100$, but there are others too.
These numbers are some of the blue set: $18, 27, 54, 72, 81, 99$, but there are others too.
These numbers are in one of the sets: $9, 10, 15, 37, 36, 85$, but there are others too.
These numbers are in two of the sets: $45, 55, 90$.
These numbers are not in any of the sets: $4, 12, 26, 42, 56, 77, 97$, but there are others too.
There are nine numbers altogether in the red set, eleven numbers in the blue set and twenty numbers in the green set.
There are three numbers that are in two of the sets.
There are no numbers that are in all three sets.
There are $63$ numbers altogether that are not in any of the sets.
You can download a sheet of all this information that can be cut up into cards.
Can you find the rest of the numbers in the three sets?
Can you give a name to the sets you have found?
Getting Started
You could start by putting the information you do know onto a hundred square.
Youcould record them on a hundred square like this:
Image
You can find a special hundred square here.
Student Solutions
We had lots of correct solutions to this problem and the majority identified the Green and Blue sets as multiples of $5$ and $9$ respectively. A few question marks arose when looking at the Red set.
Max of St Mary's Catholic High School said:
$\eqalign{
Red &= 10,19,28,37,46,55,64,73,82\cr
Green &= 5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100\cr
Blue &= 9,18,27,36,45,54,63,72,81,90,99}$
So the red set is $9$ times table but adding $1$ on every time.
The green set is the $5$ times table. There are $20$ numbers in this set less than $101$.
The blue set is the $9$ times table. There are $11$ numbers in this set less than $101$.
Max correctly identifes the Green and Blue sets. He also noticed that there was a common difference of $9$ in the red set. However we would need to justify why $91$ and $100$ are not in this set.
Mr Harrison's maths group of Northdown Primary school had a different suggestion for the Red set:
We got a hundred square grid and crossed out all the numbers they gave us at the beginning. We tried to find all the patterns, for example the red set were in a diagonal line.
We looked for gaps in the patterns. We found:
Red set: (digits sum $=10$)
$\{19,28,37,46,55,64,73,82,91\}$
Green set: (Multiples of $5$)
$\{5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100\}$
Blue set: (Multiples of $9$)
$\{9,18,27,36,45,54,63,72,81,90,99\}$
The red set has $9$ numbers and there are no other possible numbers less than $101$ which satisfy the condition Mr Harrison's maths group imposed.
There was a big divide between those of you who agreed with Max and those who agreed with Mr Harrison's maths group.
Megan of Crookhill Primary agreed with Mr Harrison's maths group:
First I plotted the hints on a grid and looked at how many there were in each colour group. Then I looked at the pattern on the board to gain more information. To find out the Green set we knew the numbers that they gave us were already in the $5$ times table. On the grid there were two lines of $10$ numbers which made $20$ in that group so we saw that it was Green set.
Then we solved the Red set. The first thing we spotted was the pattern on the page so then we looked really carefully at the numbers then we solved it. The digits all add up to ten; that was the hardest to spot I think The easist to spot was the Blue set. I enjoy doing my $9$ times table that's probably why so I didn't need to work anything out.
Well done Megan, and I enjoy my $9$ times table as well.
Teachers' Resources
Why do this problem?
This problem requires learners to see the connections between numbers in a set and so find the rest of the set. They will need to make and test hypotheses, and justify their reasoning.Possible approach
You could start by using this interactivity during a 'warm-up' activity. Choose a particular property and drag one number with that property onto the left side of the grid. Invite the group to work out what the property is that you have chosen by calling out other numbers, which you then place on the appropriate side of the grid. Can they decide
upon the 'rule' in as few guesses as possible?
The class could then work in pairs on the problem itself so they can talk through their ideas with a partner. They will need this sheet of information about the numbers, which can be cut up into cards to make it easy to use. Observing how children record as they work
on this challenge will be very informative for you. This special hundred square could help some learners to record if they are struggling to find their own way. The different groups of numbers, the red set, the green set and the blue set could be recorded like
this:
Image
Using a hundred square to record (whether it is the special one or a 'standard' one) will reveal patterns and therefore may help children work out their properties.
Discussion at the end of the lesson could include not only the sets of numbers that have been found, but also the ways that the children approached the problem. What did they do first? What were their first ideas? How did they decide whether these initial hypotheses could be right? How did they record their thinking? Did they work in a systematic way? How did they know that their solution
was correct?
Key questions
What is the same about these numbers? What do these numbers have in common?
What do you know about this number? Is that true of any of the others in the set?
How will you keep track of your thinking?
Can you see a pattern on the hundred square?
Can you see any gaps in that pattern?
Why do those numbers make that pattern?