Ten Tiles I
Identify numbers divisible by 10;
Count large but relatively straightforward sets.
Devise and use problem solving strategies to explore situations mathematically (be systematic, think, use a simpler case, draw a digram).
This problem is the first of a series of three Ten Tiles I, Ten Tiles II and Ten Tiles III. It might be a good idea to start with At The Movies, (Level 3) though.
This problem is the first of a series of problems involving numbers with a given factor. Students first of all have to be able to identify the property that a number must have, to have this number as a factor. In this first problem, this is relatively easy. A number is divisible by 10 if its last digit is 0.
The next step is to be able to count all the numbers with this property in an efficient manner. As the answer to this question is over 360,000 the counting has to be done by some efficient method. It may help to look back at At The Movies to see how this might be done. On the other hand, using a simpler case might be a good strategy to use here.
Counting is an important part of mathematics and many methods of counting sets of objects have been devised. They are usually to be found in a course on Discrete Mathematics or Combinatorics. Such courses are usually only found at university level.
This problem can certainly be done using a tree diagram but the diagram would be enormous and be too unwieldy to be practical. However, a tree diagram could be used to do a simpler case and hence motivate a pattern that might be followed to obtain the answer.
Problem
Jim has ten tiles with a different digit on each of them. He plays around and discovers that he can make quite a lot of ten-digit numbers that are divisible by ten by using the tiles. In fact how many can he make?
Teaching sequence
- Use a classroom discussion to revise the idea of factors and how you can identify numbers that have given factors.
What are the factors of 24? 36?
What does it mean to say that 35 is divisible by 7?
How do you know if a number is divisible by 2? 10? 5? - Pose the problem.
Can you give me a number that is divisble by 10?
Can you give me a ten-digit number that is divisible by 10?
How about a ten-digit number that contains each of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? - Let the students tackle the problem in their groups. For groups that are having trouble getting started ask
How many numbers can you make using just one 0, 1 and 2, that are divisible by 10?
How about if we use 0, 1, 2, 3 just once each? How many then?
Can you see a pattern here? - After a suitable time, bring the students back together to discuss how they solved the problem. The quicker students might also have had time to tackle the Extension problem.
- Get the students to write up their answers.
Extension
Josey looks at Jim’s tiles. She sees that she can work out how many two-digit numbers she can make using these tiles, that are divisible by two. How many can she make with Jim’s tiles?
Solution
Ten-digit numbers divisible by ten: Such a number has to have a zero in the last place. Then there are 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362880 ten-digit numbers that are divisible by ten. This is because there are 9 possible digits than can be used first. Once that is used there are 8 digits left. For any of the first 9 numbers there are 8 choices for the next number. That gives 9 x 8 = 72 possibilities so far. The pattern continues. (See also At The Movies.)
Solution to the extension:
For any number to be divisible by two the last digit has to be even. Here we have the choice of 0, 2, 4, 6 and 8. This gives us 5 choices. But we now have difficulties. Things are different if 0 is the last digit. For instance, if 5 is the last digit then we have to make sure that 0 is not the first digit. So we have to split the problem into two parts. On the other hand, we don’t have this worry if 0 is the last digit.
If 0 is the last digit, then there are 9 choices for the tens digit.
If 2, 4, 6 or 8 is the last digit then there are 8 choices for the tens digit (10 minus the 0 and minus whatever digit was used in the units position). This gives 4 x 8 = 32 numbers.
Altogether there are 41 even two-digit numbers.
(This can be done by first allowing 0 to be the first digit and then counting how many times 0 is the first digit. We can then subtract the second number – 4 - from the first - 45 – to give the answer of 41. This problem can also be solved using a tree diagram.)
(This problem can also be solved by writing out all possibilities and then counting the result. If a student does it this way that is fine. You might suggest a quicker way though.)
| Attachment | Size |
|---|---|
| TenTiles1.pdf | 38.1 KB |
| TenTiles1Maori.pdf | 46.64 KB |
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