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The 500 problem

Student Activity: 

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?

Achievement Objectives:

Achievement Objective: NA3-4: Know how many tenths, tens, hundreds, and thousands are in whole numbers.
AO elaboration and other teaching resources

Specific Learning Outcomes: 
Explain face, place and total value of numbers
Solve 3-digit subtraction problems
Devise and use problem solving strategies (guess and check, think logically)
Description of mathematics: 

This problem demands a sound knowledge of place value. It is essential for students 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
Required Resource Materials: 
Activity: 

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?

Teaching Sequence

  1. Introduce the problem as a puzzle. Ensure that the students understand that the answer to both problems is the same. You may ask a student to guess a number for a and then substitute it in the problems as an example.
  2. Give the students time to work on the problem individually before sharing their work with a partner. Encourage the students to discuss the reasons for their guesses. When the students are working on extensions prompt them to look for patterns.
  3. As the answer may be "guessed" by some students quickly ask them to write their answer and method. If they have guessed the answer encourage them to find alternative solutions.

Extension to the problem

This problem can be adjusted in many ways.

For example:

Change 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?

Pose these problems:

50005abc
-abc5-5000

Or

50000-5abcd
-5abcd-5000
 
What are a, b and c?

Solution

Several strategies can be used to solve this problem, including guess and check. One way is to just observe that in the left subtraction, there must be 5 ones. In the right subtraction there are b ones. Therefore,  b has to be 5 because the two answers are the same.

Consider the tens. Since b = 5, the number that is to be found in the units column of the answer of the left subtraction is 4. In 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. Perhaps you see that pattern:  (5 x 9 = 45; 2 x 9 = 18; …)

Solution to the Extension:

5000 problem is very interesting. It doesn’t work! There are no values of a, b, c that will make the two subtractions equal. Posing your students a problem that doesn't work is a worthwhile and legitimate challenge.

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 students find it fun to do.

AttachmentSize
500Problem.pdf97.97 KB
HeTangohanga.pdf152.67 KB