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My son is naughty

Student Activity: 

Jack says, “Did you know that today is my three sons’ birthday?”

“How old are they?” asks Ollie.

Jack gives him a hint. "The product of their ages is 36 and the sum of their ages is 13.”

“That’s no help,” says Ollie.

Jack gives him another clue. “O.K. My youngest son is very naughty.”

“Nothing to it,” exclaims Ollie, and he tells Jack the correct ages of his sons.

How does Ollie figure out the correct answer and what are Jack’s sons’ ages?

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: 

The subtle logic of this problem places it at this level. 
Students are given three pieces of information which must be synthesised in order to find a solution. They must be attuned to the semantics of one piece of information, and find the right combination of factors for a given product, while looking for addends of a given sum. 

Activity: 

The Problem

Jack says, “Did you know that today is my three sons’ birthday?”
“How old are they?” asks Ollie.  
Jack gives him a hint. "The product of their ages is 36 and the sum of their ages is 13.”
“That’s no help,” says Ollie.
Jack gives him another clue. “O.K. My youngest son is very naughty.”
“Nothing to it,” exclaims Ollie, and he tells Jack the correct ages of his sons.

How does Ollie 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. Ask:
    Can you summarize the problem in your own words? 
    What are the important ideas in this problem?
    What strategies might you be able to use?
    Do you think it is useful to make a table? If so, how?
  2. As they work in their groups you might ask: 
    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. As groups solve the problem, encourage them to record their solutions as they work. Also ask:
    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?
  4. Have students share their solutions 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.
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.

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MySonIsNaughty.pdf134.56 KB
HeMohioAHeraKiTeKoreroMaori.pdf191.66 KB