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Problem
Here’s a subtraction problem. The numbers a and b stand for digits. If the two subtraction sums give the same answer, what digits do a and b stand for?
| 500 | 5ab |
| -ab5 | -500 |
What is this problem about?
Basically this question is about a sound knowledge of place value. It is essential for children to understand the grouping and place value basis of our number system. They must be able to:
- Understand that the same digit is used to represent different amounts
- Interpret the value of each digit according to its position
- Express amounts by using digits
Achievement Objectives
Number (Level 3)
- explain the meaning of digits in any whole number
Mathematical Processes
- devise and use problem solving strategies (guess and check, think logically)
Resources
digit labels available (BSM equipment)
Blackline master of the problem (English)
Blackline master of the problem (Maaori)
Specific learning outcomes
The children will be able to:
- explain face, place and total value of numbers
- solve 3-digit subtraction problems
Teaching Sequence
- Introduce the problem as a puzzle. Ensure that the children understand that the answer to both problems is the same. You may wish to get a child to guess a number for a and then substitute it in the problems as an example.
- Give the children time to work on the problem individually before sharing their work with a partner. Encourage the children to discuss the reasons for their guesses. When the children are working on extensions prompt them to look for patterns.
- As the answer may be "guessed" by some children quickly ask them to write their answer and method. If they have guessed the answer encourage them to think of alternative ways to come to the same answer.
Extension to the problem
This problem can be played with in many ways. First of all spend some time changing the number 5. Replace it in all the subtraction sums by 2 or 8 or ? What happens to a and b? Is there a pattern?
Then look at
| 5000 | 5abc |
| -abc5 | -5000 |
What are a, b and c?
Then look at
| 50000 | -abcd5 |
| 5abcd | -5000 |
Solution
Certainly the problem can be solved using guess and check and some other
approaches. However, we think that the nicest way to do it is to just observe that in the
left subtraction, we get a 5 in the units column. In the right subtraction we get a b in
the units column. Hence b has to be 5 because the two answers are the same.
Now tackle the tens column. Since b = 5, the number that is to be found in the units column of the answer of the left subtraction is 4. So going to the right subtraction we see that a = 4. Just check that there is nothing wrong with a = 4 and b = 5.
Changing 5 to 2 will give a = 1 and b = 8; changing 5 to 8 will give a = 7 and b = 2. Are you getting to see a pattern yet? (5 x 9 = 45; 2 x 9 = 18; …)
Incidentally this is a problem where a lot of knowledge is a dangerous thing. Many secondary students will attempt this using algebra. While algebra works it isn’t the slickest method in the initial stages.
Extension: But now the 5000 problem is very interesting. It doesn’t work! There are no values of a, b, c that will make the two subtractions equal. It’s not often you will have given your children something that won’t work. We tend not to do that in maths classes.
The surprising thing is that the 50000 problem works again! And the pattern here looks a lot like the 500 problem!
This problem might make a good investigation. There’s a lot in it and children generally seem to find it fun to do.