This unit looks at simple rules that will decide when any number has the factors 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12. It also looks at how we can quickly tell what remainder a number has when it is divided by 3 or 9. We give the proofs of these rules and test these ideas using word problems.
From time to time it is useful to know how to test a number to see if it has a given small factor. In this unit we look at such tests for 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12.
You will notice that 7 is omitted from these tests. There are several tests for 7. However, these tests are not simple and have therefore been omitted from the unit.
Websites such as www.primepuzzles.net/puzzles/puzz_101.htm identify many useful divisibility tests. A search for ‘divisibility tests’ or ‘divisibility rules’ leads to several other sites which cover similar ground.
The ability to access calculators easily (e.g. on a cellphone) means that these tests are not as useful as they once were. However, the exploration of the factors of a given number still provides important opportunities to practise arithmetic, and still has important applications to cryptography.
Importantly, these tests also show the power of algebra. At this level, students might have experienced algebra in the context of solving equations. In contrast, this unit uses algebra to prove things (specifically to prove that the tests really work).
For example, let’s consider the ‘3 rule’. We could prove this for 2-digit numbers by showing that it works for all ninety 2-digit numbers. However, the proof that works for a 2-digit number can be turned into algebra. In turn, we can see that it works for all 2-digit numbers with one calculation. This is the real power of algebra! One calculation with letters a and b, can replace ninety calculations with specific numbers.
We can also do the same thing for 3-digit numbers. Nine-hundred specific calculations can be replaced by one algebraic manipulation. This process can be repeated for 4-digit numbers, 5-digit numbers, and so on. However, by using more sophisticated algebra (mainly the idea of summations with subscripts), the ‘3-rule’ can be proved in a few lines for a number with ANY number of digits! So a few lines of algebra can do what might otherwise require an infinite number of calculations.
We have given this general proof in the Teachers’ Notes to session 2. We don’t expect all students to understand it. However, your more knowledgeable learners might be interested in exploring the concept.
Why do all these rules work? It’s simply because we use the decimal system. The Romans couldn’t use these rules because they didn’t have the number system to base them on, nor could they hope for any rules at all that would simplify divisibility. Students might be engaged in investigating the fundamental role that the decimal system plays in this exploration of patterns.
The philosophy used throughout the unit is that of ‘guided discovery’. Within this, students are given the opportunity to find things out for themselves.
The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Ways to support students include:
This unit is focussed on using divisibility rules and as such is not set in a real world context. You may wish to explore real world applications of divisibility rules in teaching sessions following the unit, for example, by framing divisiion word problems in the context of your school or local community.
Te reo Māori kupu such as ture whakawehe (divisibility rule), whakawehe (divide, division), tauwehe (factor), taurea (multiple), toenga (remainder), and ture (formula, rule) could be introduced in this unit and used throughout other mathematical learning.
In this session we review the rules that can be used to decide when a number is divisible by 2 or 5 or 10 and look at divisibility by 3.
Therefore, when we say that a number n is divisible by 2 if and only if the last digit of n is 0, 2, 4, 6, or 8, we mean two things. First, that n is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. Second, if n is divisible by 2, then the last digit of n is 0, 2, 4, 6, or 8.
Logically then, being divisible by 2 and ending in 0, 2, 4, 6, or 8, are equivalent.
In this session we look at proving the rules for divisibility by 2, 5 and 10. We also establish and prove the rule for divisibility by 3.
Once again we get N is a multiple of 3 plus the sum of the digits of N and the argument from the first paragraph follows.
Note that 9…9 and 3…3 means that you have to use i nines and i threes.
This subscript notation is hard to follow at first but can be quite useful. We suggest that you show it only to very knowledgeable students. You will need to explain to them that the sigma is the Greek letter S and that S stands for Sum. So the notation is about summing things. It works like this:
What happens is that you start with putting i equal to 1; then you add what happens when i = 2; then add what happens when i = 3; then add what happens when i = 4; then add what happens when i = 5. Because the number at the top of the sigma is 5, we stop with i = 5.
Bring the class together to consider what they have done. Let them present their ideas in front of the class. Get a good proof constructed on the board. Display this rule on the classroom wall.
NB: The proof we have given first in Teachers’ Notes is reliant on a particular number of digits in the number N. The more knowledgeable students might like to think about how they could prove the rule for ANY number of digits – see Teachers’ Notes.
In this session we give the students the opportunity to discover rules for divisibility by 4, 6, 7, 8, 9.
This session we investigate 11 and 12. If there is time, you might also like to look at how to test for remainders after dividing by 3 and 9.
In this final session we revise the rules introduced throughout unit.
The Number Property game: This is a game that can be played with not just numbers but with almost anything. Let’s talk about the version that we want to play in this unit and then suggest other ways that it could be used.
Here we want to get the students thinking about the divisibility properties that we have developed in this unit. The game is played by you (or one of the students) thinking of a particular divisor. Tell students that you are going to give a sequence of numbers that all have a common property. When anyone thinks they know what the property is they should put up their hand and then give another number with that property. They are not allowed to say what the property is. If the number has the right property then you will say ‘yes’; if it doesn’t you will say ‘no’. Then you will say another number with the same property and then expect someone else to say a number with that property. Keep going until most of the class has contributed a number.
This game can be played with any patterns at all. These may not necessarily be number patterns. For instance, you might use European countries or famous artists or writers. It’s a useful way to focus students’ attention on an idea that you may want to revise or develop.
Dear families and whānau,
Recently we have been investigating divisibility rules and finding whether a number is a multiple of 2, 3, 4, 5, 6, 8, 9, 10, 11, or 12. Ask your child to share their learning with you.
Printed from https://nzmaths.co.nz/resource/guzzinta at 10:35am on the 30th April 2024