004i - Factors and Prime Numbers

Lesson 4 — Factors and Prime Numbers

Strand: Number | Descriptor: AC9M7N02 | Duration: 45 minutes

Learning Intentions

• To understand what makes a number prime or composite.
• To find all factors of a natural number systematically.

Success Criteria

I can:

1. List all factors of a number using factor pairs, working systematically.
2. Define a prime number and a composite number precisely.
3. Explain why is neither prime nor composite.
4. Use the Sieve of Eratosthenes to identify all primes below .

Warmup

(6 minutes — rectangle building, concrete)

Give each pair a set of counters (or square tiles).

1. Arrange all counters into a rectangle. How many different rectangles can you make? (Count and as the same.)
2. Record the dimensions of each.
3. Now try with counters. What happens?
4. Try with . And with .

Discussion: Numbers that make only one rectangle (a single row) are special. What do and have in common that and do not?

Activities

Activity 1 — Explicit Instruction: Factors and Factor Pairs (10 min)

Definition. A factor of a number divides it exactly, leaving no remainder.

Model the systematic factor-pair method for . Start at and work upwards, stopping when the pair “meets in the middle”:

Try	Divides exactly?	Factor pair
	yes
	yes
	yes
	yes
	no	—
	yes	← stop

Factors of : .

Key question: Why do we stop at ? (Because ; beyond this, every pair simply repeats what we have already found in reverse. The stopping point is .)

We do: Find all factors of and .

You do: All factors of , , , .

Activity 2 — Defining Prime and Composite (8 min)

Definitions.

• A prime number has exactly two distinct factors: and itself.
• A composite number has more than two factors.
• has only one factor, so it is neither prime nor composite.

Sort a set of cards into three columns — Prime / Composite / Neither: .

Answers: Prime — . Composite — . Neither — .

Discussion question: Is really prime? It is even — does that disqualify it? (No. has exactly two factors, and . It is the only even prime, precisely because every other even number has as an extra factor.)

Activity 3 — The Sieve of Eratosthenes (12 min)

Individual work on a printed – grid.

Method, modelled step by step:

1. Cross out — it is not prime.
2. Circle ; cross out every other multiple of .
3. Circle ; cross out every remaining multiple of .
4. Circle ; cross out every remaining multiple of .
5. Circle ; cross out every remaining multiple of .
6. Every number still uncircled and uncrossed is prime — circle them all.

Socratic scaffolding — why stop at 7?

Prompt	Purpose
What are we trying to decide?	Whether we need to sieve by , , and so on.
What would crossing out multiples of actually remove?	, , … — but are these still on the grid?
Have any of them already gone?	Yes — went with the s, with the s, with the s, with the s.
What is the smallest multiple of not already crossed out?	.
Is on our grid?	No — it exceeds .
So what is the rule?	Sieve only by primes up to , i.e. .
Looking back — generalise.	To sieve up to , use primes up to .

The 25 primes below 100:

Noticing prompts: Which rows or columns of your grid are almost empty? Why are there no primes in the column ending in apart from itself? Are the primes evenly spread, or do they thin out?

Checks for Understanding

(5 minutes — exit ticket)

1. List all factors of .
2. Is prime? Justify your answer.
3. A number has exactly two factors. What can you say about it?
4. Explain why is not counted as a prime number.
5. Which is the only even prime number, and why?

Answers: 1. ; 2. No — , so it has four factors; 3. It is prime; 4. It has only one factor (itself), not two; 5. — every other even number is divisible by as well as by and itself, giving at least three factors.

Common Misconceptions

Misconception	How to pre-empt it
” is prime.”	Return repeatedly to the exactly two distinct factors wording. Have students count the factors of aloud.
” can’t be prime because it’s even.”	Address directly in Activity 2. Evenness is irrelevant; the factor count is what matters.
”All odd numbers are prime.”	Immediate counterexamples: , , , . Make students find one themselves.
Confusing factors with multiples.	Anchor language: factors are smaller (or equal) and divide into it; multiples are larger (or equal) and it divides into them. Use the sentence ” is a factor of ; is a multiple of .”
Missing factors through unsystematic listing.	Enforce the factor-pair table starting at and counting up. No random guessing.
Assuming or are prime (they “look prime”).	Explicitly teach divisibility checks up to : , .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is the smallest natural number with exactly factors?

Answer

A number has exactly factors only if it is for a prime (since is prime, the factor count forces ). The smallest is , with factors .

E2 (AMC Junior style). The sum of two prime numbers is . What is the product of those two primes?

Answer

is odd, so one prime must be even — hence one is . The other is . But is composite, so no such pair exists. A good lesson in checking rather than assuming. (Teacher variant with a solution: sum ; then is prime, giving product .)

E3 (Challenge). How many numbers between and have exactly three factors?

Answer

A number has exactly three factors precisely when it is the square of a prime. Below : , , , . That is 4 numbers.

E4 (Reasoning). Twin primes are pairs of primes differing by , such as and . Find all twin prime pairs below . Then explain why can never count.

Answer

— eight pairs. fails because is composite.

Homework

1. List all factors of: (a) (b) (c) (d) .
2. State whether each is prime or composite, giving a factor if composite: (a) (b) (c) (d) (e) (f) .
3. Write down all the prime numbers between and .
4. Find two prime numbers that add to . Is there more than one answer?
5. Reasoning. Jared says “every number that ends in is prime.” Give two counterexamples and explain what is wrong with his rule.
6. Challenge. I am a two-digit number. I am prime, my digits add to , and my tens digit is larger than my ones digit. What am I?

Answers: Q3 — . Q4 — e.g. , , , ; yes, several. Q5 — and . Q6 — .