A Prime Search
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 the factors of a number. We are also introduced to prime numbers.
Visit ESOL Online for a version of this unit designed to support students for whom English is an additional language.
- model the numbers from 1 to 50 as rectangular arrays
- identify the factors of the numbers 1 to 50
- identify the prime numbers from 1 to 50
This unit looks at the basic number concepts of factors and prime numbers. These relatively straightforward 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. 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 based around the fact that large numbers are hard to factorise are used by banks and the military. Such codes are very difficult to break.
Finding factors of a given number can always be done by a systematic search. Starting at 1, we simply test each consecutive number to see if it is a factor of the number we are looking at. So to find the factors of 18, say, we first check 1, then 2, then 3, then 4, and so on until we get to 18. That way we can find all the factors of 18, or any other number for that matter. Systematic searches are useful throughout mathematics and are useful things for students to know about.
But we can do better than the search above. We don’t have to test all of the numbers from 1 up to the number in question. Look at 18 for example. We needn’t test any numbers above 9. This is because any number above 9 will be paired with a factor less than 9 and we have already tested all of these. So we only have to search half of the numbers less than the given number. 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; 6 and 3 by testing for 6; 9 and 2 by testing for 9.
However, can you see that we can actually do better than this? When we were trying to find the factors of 18 we could have stopped at 3! In fact, in general, we only have to check up to the square root of a number. Take 18 as an example again. The square root of 18 is just over 4, so we only have to check up to 4. Once we get past 4 we can be sure that we’ll only meet the factors that we have already found.
So factors and prime numbers are pretty important and can be fun to play with. What more can you ask?
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.) - 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 equation with each rectangle. Organise the rectangles from 1x12 to 12x1. (This will allow for more easy comparison with the factors of other number s.)
- Give each pair of students a number (1-11) and ask them to form the rectangles 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? - Ask each pair to attach their rectangles to the "page" for their number.
- As a class share the number pages.
Which number has the least rectangles? (1) Why?
Which number has the most rectangles? (12) Why?
Which numbers have only 2 rectangles?
Can a number have 3 rectangles?
Exploring
Over the next 2-3 days we create rectangle charts for each of the numbers from 1 to 50. We use the charts to develop our understanding of factors, multiples and primes.
- Begin by looking at the display of charts from yesterday.
- Write on the board: "The factors of 6 are 1, 2, 3 and 6"
Ask the students to look at the chart for 6 and see if they can work out what a factor is. - Continue to discuss the numbers from 1 to 12.
What are the factors of 8? (1, 2, 4, 8)
What are the factors of 5? (1, 5)
What other numbers have only two factors? (2, 3, 7, 11)
Which number has the most factors? (12)
Which number has the fewest factors? (1) - Write on the board: "Prime numbers have exactly two factors."
- Ask the students to list the prime numbers from 1 to 12. (One is not a prime as it has just one factor.)
- Put the numbers from 13 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 a prime? Why or why not? - 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.
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?
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?
Dear Families
This week at school we have been investigating prime numbers. Ask your child to tell you what they have find out.
We are also working on a brainteaser. See if your family can work it out together:
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.
Similar Resources
Multiples and Factors
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Matrix
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Houses
This unit seeks to connect learning outcomes across all five content strands, number, geometry, statistics, algebra, and measurement. The context of houses is used to develop concepts such as drawing and modelling 3-dimensional objects, using co-ordinate systems to locate position, finding all the possibilities of events, and identifying paths through simple networks. The unit provides an excellent vehicle for students to use a broad range of problem solving strategies.
Highest Common Factors
Identify highest common factors and least common multiples.
