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Level Five > Number and Algebra

My Son is Naughty

Achievement Objectives:

Achievement Objective: NA5-2: Use prime numbers, common factors and multiples, and powers (including square roots).
AO elaboration and other teaching resources

Specific Learning Outcomes: 

Find factors of numbers

Work systematically

Use logic to explain away certain possible number combinations.

Description of mathematics: 

We have listed this problem at Level 5, not because the number knowledge required is so difficult but that the logic involved is a little subtle.

Here we have a problem that looks impossible to solve at first glance. Ray was certainly right to say that the ‘36 and 13’ part is insufficient information to solve the problem. But surely knowing the additional information that Jack’s smallest son is very naughty can’t help at all? Where is the extra mathematical information there that enables Ray to solve the problem? It’s worth writing down exactly what key pieces of information you actually have.

The trick is in first finding out as much as you can from the ‘36 and 13’ piece of information. Once you have done that what more do you need to know?

This is an example of a problem where seemingly irrelevant information enables the solution to be found. It is a rare kind of problem at school level. But we often get problems in life where apparently useless information turns out to be a key factor in its solution. If you are not convinced by that you should read more detective novels!

Required Resource Materials: 
Copymaster of the problem (English)
Copymaster of the problem (Māori)
Activity: 

The Problem

Jack and Ray were at the rugby. Ray’s team was winning so Jack decided to give Ray a problem to deflate his ego a bit. So Jack said “Did you know that today is my three sons’ birthday?” “How old are they?” Ray asked taking the bite. “I’ll give you a hint. The product of their ages is 36 and the sum of their ages is 13,” Jack replied. “That’s no help,” said Ray. So Jack weakened and gave him another clue. “O.K. My youngest son is very naughty.” “Nothing to it,” exclaimed Ray and he told Jack the correct ages of his sons.

How did Ray figure out the correct answer and what are Jack’s sons’ ages?

Teaching Sequence

  1. Talk about the problem with the class. See if they have any ideas. Before they go to work in their groups you might ask questions like:
    What is the problem asking you to find out?
    What are the important ideas in this problem?
    Can you summarize the problem in your own words?
    Have you done a problem like this before?
    What strategies might you be able to use?
    Do you think it is useful to make a table? If so, how?
  2. Move around the groups to see how they are going and to give them some help. You might ask them questions such as:
    What do you understand from the statement “All my three children are celebrating their birthday today”?
    Do you think the last hint is very important? In what way?
    How many possible answers are there that satisfy the first two hints?
  3. If they have trouble with the problem you might all have to get together to brainstorm. Use the questions above and lead them to see that a table might help.
  4. If a few of the class have managed to solve the problem you might suggest that they go round to some of the other groups and give them some hints on the solution. Warn them not to tell the other groups exactly how to do it but only to give them hints!
  5. When a group has solved the problem ask them:
    Have you considered all the cases?
    Have you checked your solutions?
    Does it look reasonable?
    Are their any other solutions?
    Are there any shortcuts other than making a table?
  6. Give the students time to write up a solution to the problem in their own words before trying the extension.
Extension to the problem
Make up a similar problem to this one. Do this first by seeing if you can find numbers other than 36 and 13 that will work the same way. Then reword the problem using these new numbers.
Solution
There are three key pieces of information here. These are:
  • the product of the ages is 36;
  • the sum of the ages is 13;
  • the youngest of Jack’s sons is very naughty.
Let’s work with the three pieces of information separately.
Suppose that the children are A, B and C. What can we tell about them from the fact that the product of their ages is 36? What three numbers multiplied together give you 36? Or another way, how can you decompose 36 into three factors?
Perhaps the best way to do this is to work systematically as we have done in the table below. Start with 36, 1, 1 and work downwards in the sense that the highest factor gets smaller.
But the second key idea is that the sum of the ages of the children is 13. How can we use this fact? In terms of the factors of 36, this just means that the sum of the factors is 13. In the table we have listed the sums of all of the factors.

 

A’s age
 
B’s age
C’s age
Sum of their ages
36
1
1
38
18
2
1
21
12
3
1
16
9
4
1
14
9
2
2
13
6
6
1
13
6
3
2
11
4
3
3
10

 

From the table we can see that there are two lots of ages (factors of 36) that add up to 13. These are 9, 2, 2 and 6, 6, 1.
It’s not surprising that there are two answers. If there was only one, Ray would have been able to solve the problem without any further clues. But what possible help can it be to know that Jack’s youngest son is very naughty?
The point here is that Jack has a youngest son! He doesn’t have two sons that are the same age. So 9, 2, 2 can’t be the answer. The answer has to be 6, 6, 1, where there is a youngest son whose age is 1.
Ray correctly identified the ages of Jack’s sons as 6 years, 6 years and 1 year.
AttachmentSize
MySon.pdf42.76 KB
MySonMaori.pdf49.3 KB