Factors, Multiples, Primes and Special Numbers: a week of ready-to-teach maths

Five days of lessons for Year 6 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus number and property cards, a sorting mat, a mini-check and every answer.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Cut out the number and property cards once; one set and one sorting mat per pair serves Days 2 to 5.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: whole numbers have a hidden structure. A factor pair is the two sides of a rectangle of squares. A prime number makes only one rectangle, a single row, because its only factors are 1 and itself; a composite number makes more. Every composite number breaks down into primes in exactly one way. Square numbers fill a square and triangular numbers stack into a triangle, and knowing these shapes makes some calculations quick.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Words to watch

The number 1 is neither prime nor composite: it has only one factor. A square number is a whole number times itself: 25 is 5 × 5, written 5² and read “five squared”. In this pack HCF means the highest common factor and LCM the lowest common multiple; say the full words first.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 counts the rectangles a number can make, which sorts numbers by how many factors they have; Day 2 names the primes and composites and sieves out every prime below 100; Day 3 breaks composites down into primes; Day 4 uses shared factors and multiples to solve problems and simplify fractions; and Day 5 meets the square and triangular numbers. It builds on the Year 5 work of writing numbers as products of factors, and it prepares Year 7, where numbers are written as products of powers of primes and square numbers meet square roots.

Materials for the week (one trip)

• From the classroom: scissors, pencils, coloured pencils, this pack printed.
• For Day 1: square tiles, counters or cubes, about 40 per pair (squared paper works too).
• For Day 5: counters for building squares and triangles (optional; the worksheet has dot pictures).
• Cut out once, use all week: the number cards and property cards, with one sorting mat per pair. No maths equipment to buy.

Words for the word wall

factor, factor pair, multiple, prime, composite, prime factorisation, factor tree, common factor, highest common factor, common multiple, lowest common multiple, square number, squared, triangular number

Dear families

This week in maths, Year 6 looks inside whole numbers. We find factors as the sides of rectangles, sort numbers into prime (exactly two factors) and composite (more than two), break numbers down into primes with factor trees, and meet square numbers (1, 4, 9, 16, ...) and triangular numbers (1, 3, 6, 10, ...).

Try this at home

• Pick a number of things, such as 24 biscuits. In how many ways can they be laid out in equal rows?
• Ask whether a house number or a page number is prime. How do you know?
• Stack coins in a triangle: 1, then 2 under it, then 3. How many coins are in a triangle of 6 rows?
• Use factors to calculate quickly: 25 × 12 is 25 × 4 × 3, which is 100 × 3.

My numbers this week

Fill one row a day. Find a number at home and test it.

Printed from the free seegongsik Factors, Multiples, Primes and Special Numbers teaching pack · seegongsik.com/au/y6/number/AC9M6N02/pack

Count the rectangles

A factor pair is the two sides of a rectangle of squares. Children build every rectangle for a number, list its factors in order and count them, and notice the numbers with only one rectangle and the numbers with a square among their rectangles.

We are learning to

• find every factor pair of a number by building or drawing rectangles,
• list all the factors of a number in order, pairing them from the outside in,
• notice which numbers have only two factors and which have an odd number of factors.

Success criteria

• I can find all the factors of a number up to 50 without missing any.
• I can explain why a number with a square rectangle has an odd number of factors.

You need

Square tiles, counters or cubes (about 40 per pair) or squared paper, and the worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with 16 on the board: one of its rectangles, 4 × 4, is a square.

Watch for these ideas

• Missing the outside pair: finding 2 × 6 and 3 × 4 but not 1 × 12. Start every list with 1 and the number.
• Counting 3 × 4 and 4 × 3 as two pairs. It is one pair; the rectangle is only turned.
• Thinking bigger numbers have more factors. 29 has two factors; 24 has eight.

Answers

• Part A: 1 × 18, 2 × 9, 3 × 6; the factors of 18 are 1, 2, 3, 6, 9, 18.
• Part B: 20: 1, 2, 4, 5, 10, 20 (6); 24: 1, 2, 3, 4, 6, 8, 12, 24 (8); 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9); 13: 1, 13 (2); 49: 1, 7, 49 (3); 30: 1, 2, 3, 5, 6, 10, 15, 30 (8).
• Part C: 13 has only two factors, so only one rectangle. 36 and 49 have an odd number of factors: one of their rectangles is a square, and its side pairs with itself.

Count the rectangles

A. Draw every rectangle of 18 squares

The factor pairs of 18: ______________________________________

B. Pull each number apart

C. What do you notice?

Which numbers in Part B have exactly two factors? ______________________

Which have an odd number of factors? What is special about one of their rectangles?

______________________________________________________________

Prime or composite

A prime number has exactly two factors, 1 and itself; a composite number has more than two; 1 has only one and is neither. Children sieve a hundred chart, crossing out multiples, to find the primes below 100.

We are learning to

• say whether a number is prime or composite, and why,
• use multiples to sieve the composite numbers out of a hundred chart,
• explain why 1 is neither prime nor composite, and why 2 is the only even prime.

Success criteria

• I can sort numbers up to 100 into prime and composite.
• I can explain my sort with a factor pair, or by saying there is none.

You need

Coloured pencils, the worksheet, one per child, and the cut-out cards with a sorting mat, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the multiples of 3 in Part A; start Session B with 5 and 7.

Watch for these ideas

• Calling 1 prime. A prime has exactly two factors; 1 has only one.
• Thinking every odd number is prime. 9, 15, 21, 51 and 91 are odd and composite.
• Thinking no prime is even. 2 is prime: its only factors are 1 and 2.

Answers

• Part A: the circled numbers are the 25 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Every composite below 100 has a factor 2, 3, 5 or 7, since 11 × 11 = 121.
• Part B: 25 primes; between 50 and 70: 53, 59, 61, 67.
• Part C: 51 composite: 3 × 17; 53 prime: only 1 and itself; 57 composite: 3 × 19; 91 composite: 7 × 13; 97 prime: only 1 and itself; 1 neither: it has only one factor.

Sieve for primes

A. Sieve the hundred chart

1. Cross out 1. It is neither prime nor composite.
2. Circle 2, then cross out the rest of its multiples: 4, 6, 8 and on to 100.
3. Circle 3 and cross out the rest of its multiples (some are crossed out already). Do the same for 5 and for 7.
4. Circle every number that is left. These are the primes.

B. Count your primes

How many primes are there below 100? ______    Which primes are between 50 and 70? ________________

C. Prime, composite or neither? Show how you know.

Down to the primes

A composite number splits into two factors, and those split again, until every tip of the tree is prime. Whatever the first split, the same primes come out: the prime factorisation, a number’s fingerprint.

We are learning to

• split a composite number again and again until every tip is prime,
• write a number as a product of primes, smallest first,
• check that two different trees for one number give the same primes.

Success criteria

• I can draw a factor tree for a number up to 100.
• I can write 60 as 2 × 2 × 3 × 5.

You need

The worksheet, one per child, the primes list from Day 2, and the cut-out cards with a sorting mat, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with a tree for 48 on the board.

Watch for these ideas

• Stopping at a composite tip, such as 24 = 4 × 6. Keep splitting until every tip is prime.
• Splitting off 1, as in 1 × 24. It changes nothing, so it never helps the tree.
• Worrying that a different first split gives a different answer. Every tree ends at the same primes.

Answers

• Part A: both trees end at 2 × 2 × 3 × 3.
• Part B: 40 = 2 × 2 × 2 × 5; 45 = 3 × 3 × 5; 60 = 2 × 2 × 3 × 5; 84 = 2 × 2 × 3 × 7.
• Part C: 42, 100, 27, 32. A 2 at a tip means 2 is a factor, so the top number is even.
• Two clues: 6, 10, 15 and 21 in the middle, each two different primes multiplied (6 = 2 × 3, 10 = 2 × 5, 15 = 3 × 5, 21 = 3 × 7). Nothing has exactly 4 factors without being composite: four factors is more than two.

Factor trees

A. Two trees for 36: finish both and compare the tips

B. Grow a tree for each number

C. Which number has this fingerprint?

A tree has a 2 at one tip. Can the number at the top be odd? Why? ______________________

Shared factors, meeting multiples

The highest common factor is the largest factor two numbers share; the lowest common multiple is the smallest multiple they share. Children find both and choose the one a problem needs.

We are learning to

• find the common factors and the highest common factor of two numbers,
• find the lowest common multiple of two numbers,
• choose the highest common factor or the lowest common multiple to solve a problem.

Success criteria

• I can find the highest common factor and the lowest common multiple of two numbers up to 50.
• I can say which one a problem needs, and why.

You need

The worksheet, one per child, and the cut-out cards with a sorting mat, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part B; start Session B with the sausage sizzle.

Watch for these ideas

• Mixing up the two. The highest common factor is never bigger than the smaller number; the lowest common multiple is never smaller than the larger one.
• Multiplying to get the lowest common multiple. 4 × 6 = 24 is a common multiple, but 12 comes first.
• Simplifying a fraction in small steps and stopping early. Dividing by the highest common factor finishes in one step.

Answers

• Two clues: 6, 12, 15 and 30 in the middle; each one divides 60 and is in the count of 3s.
• Part A: 20 and 30: 1, 2, 5, 10; HCF 10. 18 and 27: 1, 3, 9; HCF 9. 24 and 40: 1, 2, 4, 8; HCF 8.
• Part B: 6 and 9: LCM 18; 4 and 10: LCM 20; 5 and 8: LCM 40.
• Part C: (1) 24 of each, 4 packs of sausages and 3 of rolls (LCM). (2) 12 packs, each with 2 pencils and 3 rubbers (HCF). (3) After 30 seconds (LCM). (4) 3/4, dividing both by 6 (HCF).

Shared factors, meeting multiples

A. Common factors

B. Count on until the counts meet

C. Which one do I need?

1. Sausages come in packs of 6 and bread rolls in packs of 8. What is the smallest number of each you can buy so that every sausage has a roll? How many packs of each?

________________________________________________________

2. A teacher shares 24 pencils and 36 rubbers into identical packs, with none left over. What is the greatest number of packs? What is in each one?

________________________________________________________

3. Two lights flash together. One flashes every 6 seconds, the other every 10 seconds. After how many seconds do they next flash together?

________________________________________________________

4. Simplify 18/24 in one step. Which number did you divide by? ____________

Square and triangular numbers

A square number is a whole number times itself and fills a square; a triangular number adds the next counting number each time and stacks into a triangle. Children build both, extend the patterns, find where they meet and use them to calculate.

We are learning to

• build and list the first ten square numbers and triangular numbers,
• explain how each pattern grows: odd numbers for squares, counting numbers for triangles,
• use the properties to solve problems and simplify calculations.

Success criteria

• I can say whether a number up to 100 is square, triangular, both or neither.
• I can explain why two triangular numbers side by side make a square number.

You need

Counters for building (optional; the worksheet has dot pictures), the worksheet, one per child, and the cut-out cards with a sorting mat, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B by building the two triangles side by side.

Watch for these ideas

• Doubling instead of squaring: 6 squared is 6 × 6 = 36, not 6 + 6 = 12.
• Adding the same amount each time for triangles. Each new row is one longer: 1, 3, 6, 10, 15.
• Thinking every number fits a shape. Most numbers are neither square nor triangular.

Answers

• Part A: squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100; triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.
• Part B: the odd numbers add to 1, 4, 9, 16, 25 (the first ten odd numbers make 100); the neighbours add to 4, 9, 16, 25, 36, all square.
• Part C: (1) 25 × 4 × 4 = 100 × 4 = 400. (2) 64 = 8 × 8 is square, but not triangular: 55 and 66 are the triangular numbers either side. (3) 1 and 36. (4) 7 rows.
• Two clues: 1 and 36 in the middle, the only cards that are both.

Square and triangular numbers

A. Fill the table

B. Two patterns

What do you notice about all the answers? ______________________ The first ten odd numbers add to ______.

C. Use the shapes

1. 25 × 16 = 25 × 4 × 4 = ______

2. Is 64 a square number? Is it a triangular number? How do you know? ______________________

3. Which numbers below 50 are both square and triangular? ______________

4. 28 children stand in a triangle: 1 in the front row, 2 in the next, and so on. How many rows? ______

Number cards and property cards

Number cards

Cut out the sixteen cards once, with the property cards below. Pairs sort them in a round of Two clues on Days 2 to 5.

Property cards

Cut out the eight cards. Each one names a property a number can have.

Teacher note: the plans put one pair of property cards on the mat each day. Day 2, “prime” and “even”: 2 in the middle. Day 3, “composite” and “has exactly 4 factors”: 6, 10, 15 and 21 in the middle. Day 4, “a factor of 60” and “a multiple of 3”: 6, 12, 15 and 30 in the middle. Day 5, “a square number” and “a triangular number”: 1 and 36 in the middle.

Two clues

A sorting game for two. Print one mat for each pair.

1. Your teacher names today’s two property cards. Lay one on each label of the mat.
2. Take turns to place a number card: in one circle, in the middle where they overlap, or outside both.
3. Your partner checks each card and you give a reason: “16 is a square number, 4 × 4.”
4. When all sixteen are placed, check the middle together. Finished early? Swap one property card and sort again.

Teacher note: a card that fits both properties goes in the middle; a card that fits neither stays in the corner marked neither.

What we know: Factors, Multiples, Primes and Special Numbers

Work on your own. Show your thinking if you can.

1. List all the factors of 28. ______________________________
2. Is 39 prime or composite? How do you know? ______________________________
3. Draw a factor tree for 72, then write 72 as a product of primes.
   72 = ______________________
4. What is the highest common factor of 24 and 32? ______
5. What is the lowest common multiple of 6 and 15? ______
6. Circle the square numbers: 18, 25, 32, 49, 64.
7. The triangular numbers go 21, 28, ______, ______.
8. Use factors to work out 25 × 24 the easy way. ______________________________

Answers and marking guide

Answers

1. 1, 2, 4, 7, 14, 28.
2. composite: 39 = 3 × 13, so it has more than two factors.
3. 72 = 2 × 2 × 2 × 3 × 3 (any tree ends at these primes).
4. 8.
5. 30.
6. 25, 49, 64.
7. 36, 45 (add 8, then 9).
8. 25 × 4 × 6 = 100 × 6 = 600.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, with a factor pair, a tree or a dot picture.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: rectangles and factors, prime or composite, factor trees, shared factors and multiples, and the special numbers.