In this unit we use rectangular models or arrays to explore numbers from one to fifty. We practice expressing numbers as the product of two smaller numbers and in doing so identify all the factors of numbers systematically. We are also introduced to prime numbers.
- Model the numbers from 1 to 50 as rectangular arrays.
- Identify the factors of the numbers 1 to 50.
- Identify whether whole numbers from 1 to 50 are prime or composite (or a special case, i.e. 1).
This unit looks at the number concepts of factors and prime numbers. These fundamental ideas have a surprisingly wide range of applications. Searching for certain types of prime number has become a test for the speed of new computers and methods of protection of computers for unwanted access. Prime numbers are an integral part of modern coding theory. This allows the easy encryption of words and numbers but means that decoding is quite difficult. Codes are based around the fact that the prime factors of large numbers are hard to find. Such codes are used by banks and the military because they are very difficult to break, even in the age of computers.
Finding factors of a given number can always be done by a systematic search. Starting at 1, each consecutive number is tested to see if it is a factor of the number in question. The search for factors from above and below continues until they converge on the square root of the number. For example, to find the factors of 18, first check 1, then 2, then 3, then 4, and so on until 5, since five is just greater than the square root of 18 (√18 = 4.24, 2dp.). That way all the factors of 18, or any other number for that matter can be found. Systematic searches are important throughout mathematics, especially to verify that all possible answers have been found.
The convergence of factors from above and below to the square root works in this way. We need not test any numbers above 4.24. This is because any number above 4 will be paired with a factor less than 4 and all of these smaller potential factors were tested. For 18 this means that we get 1 and 18 by testing for 1; 2 and 9 by testing for 2; 3 and 6 by testing for 3. Four is not a factor of 18 so the job is done.
- Cubes (multilink or place value)
- A3 sheets of paper (50)
Getting Started
Today we work as a class to investigate the rectangular arrays for some given numbers. We record the appropriate equation with each array.
- Give each student 12 cubes and ask them to form a rectangle.
What size is the rectangle you have made? (Discuss the description of rectangles using rows and columns.)
Have we found all the rectangles? How do you know? (Expect the students to check each of the numbers to 12 although some may realise that you only need to check as far as 6.) Highlight that the factor pairs repeat after 3 (Square root of 12 = 3.46). - As a class make recordings of each of the rectangles using squared paper.
- Attach these rectangles to an A3 page headed with a 12. Record the expression with each rectangle. Organise the rectangles from 1x12 to 12x1. (This will allow for more easy comparison with the factors of other numbers.)
Are each of the rectangles unique? (unique means unlike any others)
If we remove any rectangles that are copies, how many unique rectangles are there? (1 x 12, 2 x 6, 3 x 4).
What name do we give to numbers that multiply to give a number, like 12? (Factors of 12)
You might record the findings using formal notation:
Factors of 12 = {1, 2, 3, 4, 6, 12} - Give each pair of students a number (4-14) and ask them to form as many unique rectangles as they can for their number. As they form the rectangles, first with cubes and then on squared paper, ask questions that focus on the factors of the number.
How many rectangles have you found for your number?
How do you know you have found them all?
Why do some numbers have more rectangles than others?
Are there any numbers that form only one rectangle?
Is it possible to predict these ‘one rectangle’ numbers? - Ask each pair to attach their rectangles to the ‘page’ for their number.
- As a class share the number pages.
Which number has the most rectangles? (12) Why?
Which numbers have only one rectangle? (The primes 5, 7, 11, and 13)
Which numbers have only 2 rectangles? (4, 6, 8, 9, 10, 14)
Can a number have 3 rectangles? (Yes, 12 does)
Note that students may not accept squares as a special class of rectangles, so they may not include rectangles such as 3 x 3. It is important to discuss this point, or the results are invalid. - Formally list sets of factors, e.g. Factors of 8 = {1, 2, 4, 8} and Factors of 11 = {1,11}.
What happens when one of the rectangles is a square?
These numbers are known as square numbers and they have an odd number of factors, e.g. Factors of 9 = {1, 3, 9}.
Numbers that have more than two factors are called composite numbers. What numbers in the range 4-14 are composite? (4, 6, 8, 9, 10, 12, 14)
Numbers that have only two factors are called prime numbers. What numbers in the range 4-14 are prime? (5, 7, 11, 13)
Can a number be neither (not) composite or prime? (Usually this is an ‘either or’ choice except for the special case of one)
How many rectangles can be made with one square? (only a 1 x 1 rectangle is possible, so 1 is a special case, neither composite or prime but it is a square number.)
Exploring
Over the next 2-3 days ask your students to create rectangle charts for each of the numbers from 1 to 50. Use the charts to develop their understanding of factors, multiples and primes.
- Look at the charts from the previous day.
I could say that the factors of 12 are 1, 2, 3, 4, 6 and 12. What does the word factor mean?
You might look up the internet to create a formal definition of the word factor.
I could say that 12 is a multiple of 1, 2, 3, 4, 6, and 12. What does the word multiple mean? Create a formal definition of multiple. - Choose other numbers in the range 4-14 and invite your students to make statements that use the words factor/s and multiple/s.
- Put the numbers from 15 to 50 in a "hat". With a partner the students pick a number from the hat and then work together to construct the rectangular arrays for the number. They record the rectangles on squared paper and then attach these to an A3 piece of paper. As the students work ask questions that focus on their identification of the factors of a number.
How many rectangles have you found for your number?
How do you know you have found them all?
What are the factors of your number?
Is your number prime or composite? How do you know? - When the students have completed a number, they select another from the "hat". (If you run out of numbers continue with the numbers 51-100.)
- At the end of each session look at the developing display of rectangles charts. Invite pairs of students to share their findings with the class.
- You might use a hundred chart to colour code the composite and prime numbers. This may lead to a discussion about which kind of number is most common. Students may notice that primes become less frequent as the range is extended.
- Possibly note the fact that primes tend to be located next to multiples of six. For example, both 5 and 7 are primes. However, being one more or less than a multiple of six does not guarantee that a number is prime. For example, 48 is a multiple of six. 47 is prime but 49 is a square composite.
Reflecting
Today we look at our completed display of rectangle charts and create a newsletter for our families telling them about our findings.
- Display the factor charts, in order, for the class to examine.
- Encourage the students to look at the charts and write statements (in pairs) about their observations. The following questions may be used as prompts for the students.
Which number has the most factors?
How many prime numbers are there less than 50?
Is there a way to predict if a number, like 51 or 57, is prime?
What number do you think is the most interesting? Why?
Which decade has the most prime numbers? Why do you think it is the tens decade? - Share statements.
- Use these statements to form the basis for the newsletter home. In addition to the class statements you may like to include the following brainteaser.
Census Problem
A census taker approaches a house and asks the woman who answers the door.
"How many children do you have, and what are their ages?"
Woman: I have three children. The product of their ages is 36, the sum of their ages is equal to the address of the house next door."
The census taker walks next door, comes back and says to the woman.
"I need more information."
Woman: "I have to go. My oldest child is sleeping upstairs."
Census taker: "Thank you, I have everything I need."
Question: What are the ages of the each of the three children?
This is a very famous problem and the first reaction of people is often that insufficient information is given. However, it can be solved.
First make a systematic is of the sets of three factors that have a product of 36, since the ages of the three children have a product of 36. Begin with one as a factor and increase the size of factors from there:
1 x 1 x 36 (unlikely a child is 36 years old)
1 x 2 x 18, 1 x 3 x 12, 1 x 4 x 9, 1 x 6 x 6
2 x 2 x 9, 2 x 3 x 6,
3 x 3 x 4
Since the only combination possible with four is covered that is the end of the list. Next find the sums by adding the factors.
1 + 1 + 36 = 38,
1 + 2 + 18 = 21, 1 + 3 + 12 = 16, 1 + 4 + 9 = 14, 1 + 6 + 6 = 13,
2 + 2 + 9 = 13, 2 + 3 + 6 = 11,
3 + 3 + 4 = 10.
The only way the census worker needs more information is the possibility that two or more sets of factors might have the same sum. In this case 1, 6, and 6 have the same sum and 2, 2, and 9.
Knowing that there is an oldest child means that 2, 2, and 9 are the ages of the children.
Dear Families and Whānau
This week at school we have been investigating prime numbers. Ask your child to tell you what they have found out.
We are also working on a brainteaser. See if your family can work it out together: if you need, the answer is given below the problem.
Census Problem
A census taker approaches a house and asks the woman who answers the door.
"How many children do you have, and what are their ages?"
Woman: I have three children. The product of their ages is 36, the sum of their ages is equal to the address of the house next door."
The census taker walks next door, comes back and says to the woman.
"I need more information."
Woman: "I have to go. My oldest child is sleeping upstairs."
Census taker: "Thank you, I have everything I need."
Question: What are the ages of the each of the three children?
Solution to brainteaser
For a start we have to find all of the sets of three numbers whose product is 36. These can be found systematically. We do this below but we also find the sum of the factors as this is part of the problem.
Three factors | Sum of factors |
1 x 1 x 36 | 38 |
1 x 2 x 18 | 21 |
1 x 3 x 12 | 16 |
1 x 4 x 9 | 14 |
1 x 6 x 6 | 13 |
2 x 2 x 9 | 13 |
2 x 3 x 6 | 11 |
3 x 3 x 4 | 10 |
From the table the census taker would have known the ages of the children if the number of next door was anything but 13. But he still needed some more information so the number had to be 13.
When the woman said that she had an eldest child then the ages had to be 2, 2 and 9 (rather than 1, 6 and 6). So that’s how the census taker worked out the ages of the children.