Maths Year 5 Autumn Multiplication and Division

Each unit has everything you need to teach a set of related skills and concepts. 'Teaching for Understanding' provides whole-class teaching and fully differentiated adult-led group activities. ‘Problem-solving and Reasoning’ develops these skills, and includes questions to enable you to assess mastery. Practice sheets ensure procedural fluency. Extra support activities enable targeted work with children who are well below ARE.

‘UNIT PLAN’ gives you a text version of all parts of the unit to use in your school planning documentation. ‘DOWNLOAD ALL FILES’ gives you that unit plan plus all of associated documents.

Unit 1 Multiples, factors and word problems (suggested as 3 days)

Objectives

Multiples, factors and word problems
Unit 1: ID# 5347

National Curriculum
Mult/Div (i) (v)
Hamilton Objectives
12. Know and recite all times tables including division facts; identify multiples and factors, including common factors of two numbers.
15. Perform divisions mentally in range of tables.
21. Solve problems involving multiplication and division, using knowledge of factors, multiples.

Teaching and Group Activities for Understanding

Day 1 Teaching
Find common multiples. Use the ITP Number Grid to highlight multiples of 3 and 2, and then show that common multiples are all multiples of 6. Repeat for other common multiples.
Group Activities
-- Use a Venn diagram to sort multiples, placing common multiples in the intersection.
-- Find pairs of numbers which have common multiples, using a Venn diagram to help.

Day 2 Teaching
Write 24. What numbers go into 24 exactly? Take feedback. Remind children that numbers which divide into 24 exactly are factors of 24. Together make an ordered list: 1, 24; 2, 12; 3, 8; 4, 6. Point out systematic approach. Repeat for 27 and 36.
Group Activities
-- Find the number less than 50 with the greatest number of factors, find numbers which have an even number of factors and numbers which have an odd number of factors.

Day 3 Teaching
Display word problems one at a time: Agree calculation for each. Solve. Discuss if answer should be rounded up or down. Challenge children to think of a problem where they would round up, and one to round down.
Group Activities
Use the ‘Long As You Like’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Use sketches/cubes to represent division problems to help determine if answers need to be rounded up or down.

You Will Need

  • ITP: Number Grid (see resources)
  • Post-it notes
  • 1-10 number cards
  • 12 x 12 multiplication square (see resources)
  • Division word problems (see resources)
  • Cubes or counters

Mental/Oral Starters

Day 1
Times tables (pre-requisite skills)

Suggested for Day 2
Double and halve numbers to 100 (simmering skills)

Suggested for Day 3
Bar charts (simmering skills)

Procedural Fluency

Day 1
Children play common multiples game.

Day 2
Find factors of 2-digit numbers.

Day 3
Solve division word problems with rounding.

Mastery: Reasoning and Problem-Solving

  • Are there more or fewer common multiples of 2 and 3 under 30 than of 3 and 5? Explain your answer?
  • Write common multiples of 2, 3 and 4 up to 40.
    Write common multiples of 3, 4 and 5 up to 40.
    Which set has more numbers?
  • True or false?
    5 is a factor or 20 and a factor of 40.
    3 is a factor of 7 numbers less than 20.
    15 is a factor of 100.
  • Sunil says that whenever a problem involves working out the number of cars or coaches needed we have to round up. Is he correct?
  • Invent a problem where the remainder is the answer.


In-depth Investigation: Long as You Like
Children create chains of numbers by finding factors and multiples. Their aim is to create the longest chain possible.

Extra Support

Array or Disarray
Making different arrays using a given number as an introduction to factors

Unit 2 Primes, divisibility, mental strategies (suggested as 3 days)

Objectives

Prime numbers
Divisibility rules
Mental division

Unit 2: ID# 5381

National Curriculum
Mult/Div (ii) (iii) (v) (vix)

Hamilton Objectives
12. Know and recite all times tables including division facts; identify multiples and factors, including common factors of two numbers.
13. Identify prime numbers up to 100 and know primes up to 19; understand the vocabulary of prime and composite numbers; identify prime factors.
15. Perform divisions mentally in range of tables; use remainders, fractions, decimal equivalences.
21. Solve problems involving multiplication and division, using knowledge of factors, multiples, squares and cubes.

Teaching and Group Activities for Understanding

Day 1 Teaching
Write: 325, 420, 500, 360, 111, 674, 261 and 472. Discuss how we know if numbers are divisible by 2, 5 and 10. What about 3? Model how to test multiples of 3 and 9. Discuss multiples of 6 and 4.
Group Activities
-- Create a rule for divisibility by 6.
-- Use rules of divisibility for 3, 4 and 5 to write numbers which are divisible by two of the numbers, only one or neither.

Day 2 Teaching
Write the following numbers on the board: 13, 15, 18, 21, 23, 24, 30. Play a game scoring points for numbers which divide into these. Establish that 23 gets almost no points. Discuss prime and composite nos.
Group Activities
Use the ‘Prime Differences’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Work in pairs to try and find all the prime numbers to at least 50.

Day 3 Teaching
Write 68 ÷ 8. Ask children to use tables to find the answer mentally, i.e. 8 r4. Discuss how to write this as 4/8 or ½. Do divisions mentally within tables using fractions and decimal equivalences for remainders.
Group Activities
-- Choose a number on the grid to divide by the number on a dice. Score one point for each of the four different ways they can say the answer (with remainder, as mixed number, as decimal and as money).
-- Come up with divisions with given answers.

You Will Need

  • ITP: Number grid (see resources)
  • Calculators
  • 1–6 dice
  • 1-100 grids
  • blank dice

Mental/Oral Starters

Day 1
Times tables (pre-requisite skills)

Suggested for Day 2
Double numbers to 100 (simmering skills)

Day 3
Multiplication and division facts (pre-requisite skills)

Procedural Fluency

Day 1
Write 3-digit numbers in Carroll diagrams with divisible/not divisible headings.

Day 2
Find prime numbers.

Day 3
Work out mental division writing answers with remainders, as mixed numbers, as decimals and as money.

Mastery: Reasoning and Problem-Solving

  • Write a number above 100 which you can demonstrate is divisible by 3, 4, 5 and 9 but NOT by 8 or 10.
  • Prime numbers are often either side of a multiple of 6. Write the two prime numbers of which this is not true.
  • Explain why 1 is not a prime number.
  • Write 30 as the sum of prime numbers in two different ways.
  • Complete these sentences, writing either a proper fraction or a decimal <1.
    ☐ = 0.25
    1/3 > ☐
    6/8 = ☐
    ☐ < 2/4
  • Write two divisions which have an answer which ends in .75 or 3/4.

In-depth Investigation: Prime Differences
Children find out if prime numbers become less frequent as we consider sets of larger numbers.

Extra Support

This unit has no separate Extra Support activities.

Unit 3 Grid method and short multiplication (suggested as 4 days)

Objectives

Grid method and short multiplication
Unit 3: ID# 5387

National Curriculum
Mult/Div (iv) (vii)

Hamilton Objectives
16. Multiply 2, 3, 4-digit numbers by numbers ≤26 using long or short multiplication or grid method.
21. Solve problems involving multiplication and division, using knowledge of factors, multiples.

Teaching and Group Activities for Understanding

Day 1 Teaching
Write 234 × 6. Remind children of grid method. Children use grid method to do 234 × 8. Rpt for 276 × 6 and 358 × 7. Show how to use rounding to estimate answers. Work them all out.
Group Activities
-- Whole class investigation: work in groups to use the digits 4, 5, 6 and 7 to find the 3-digit x 1-digit multiplication with smallest and biggest answers (grid method).

Day 2 Teaching
Today we learn a new method called ‘short multiplication’ – it takes up less space than grid method. Write 3 × 326 and ask children to work this out using grid. Model short multiplication. Repeat for 5 x 261.
Group Activities
-- Support children using the new method of short multiplication.
-- Identify ranges of multiple of 10 for answers to grid multiplication, and say which multiple of 100 will be nearer.

Day 3 Teaching
How many meals do you eat in a year?! Agree that in most years there are 365 days and we mostly eat 3 meals a day. Together use short multiplication to find the answer. Easy group check using grid.
Group Activities
Use the ‘Excellent Eights’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Begin to use short x to multiply 4-digit numbers by 1-digit numbers.

Day 4 Teaching
Ask diff children to do each stage. Write 6 × £5.24. Estimate an answer. Use grid method to record steps. Stress writing £ and p by each product to be clear.
Group activities
-- Support children using the grid method or new method of short multiplication for multiplying amounts of money.

You Will Need

  • Mini-whiteboards and pens
  • 0 to 9 digit cards
  • Multiplying 3-digit by single-digit numbers (see resources)
  • Whole class practice (see resources)
  • Find the cost (whole class practice), (see practice sheets)
  • ITP: Number Dials

Mental/Oral Starters

Day 1
6, 60, 7 and 70 times tables (pre-requisite skills)

Day 2
Times tables (pre-requisite skills)

Suggested for Day 3
Factors and multiples (simmering skills)

Suggested for Day 4
Find the time later using 24-hour clock (simmering skills)

or
Multiply by multiples of 10 (e.g. 24 x 30) (simmering skills)

Procedural Fluency

Day 1
Multiply 3-digit numbers by 1-digit numbers.

Day 2
Multiply 3-digit numbers by 1-digit numbers.

Day 3
Multiply 3-digit numbers by 1-digit numbers, estimating first.

Day 4
Find the cost of numbers of items using appropriate multiplication method.

Mastery: Reasoning and Problem-Solving

  • Find the product of 274 and 6 using two different methods. Do your answers agree?
  • Explain how you can work out 235 x 4 mentally. Now check your answer using short multiplication.
  • Fill in the missing digits:
    8 4 ☐
    x ☐
    ☐ ☐ 0 5
  • Write a multiplication of one 3-digit number by 6 where the last two digits of the product are consecutive (e.g. 5, 6).


In-depth Investigation: Excellent Eights
Children explore patterns when multiplying numbers with consecutive digits by 8.
The PowerPoint can be used to guide your class through this investigation.

Extra Support

Grid Luck
Using the grid method to multiply 2-digit numbers by 1-digit numbers

Unit 4 Division of big numbers vertical layout (suggested as 3 days)

Objectives

Division of big numbers using vertical layout
Unit 4: ID# 5393

National Curriculum
Mult/Div (vi) (vii)

Hamilton Objectives
18. Divide 2, 3, 4-digit numbers by 1-digit numbers above tables range; choose and use efficient methods; interpret remainders appropriately according to context.

Teaching and Group Activities for Understanding

Day 1 Teaching
Write 196 ÷ 6. Ask children to estimate an answer. Then demonstrate finding out using an empty number line and jumping 30 x 6 (180) then 2 x 6 (12). Agree a remainder of 4. Model how to lay this out vertically.
☐ × 6 = 196
30 × 6 = 180
                16
    2 × 6 = 12
                 4
196 ÷ 6 = 32 r 4
Group Activities
Use the in-depth problem-solving investigation ‘Revision Rules’ from NRICH as today’s group activity.
Or, use these activities:
-- Use the vertical layout of chunking. Make an estimate between two multiples of 10 first, and then subtract the largest multiple of 10 of the divisor first.
-- Use both empty number line and vertical layouts, then choose a preference.

Day 2 Teaching
Display a word problem. Solve by dividing using a vertical layout. Discuss whether to round up or down. Repeat for the next problem.
Group Activities
-- Make sketches to help solve word problems.
-- Solve word problems, the check answer to division with multiplication.
-- Solve a multi-step division problem.

Day 3 Teaching
Children first estimate answers to different divisions. Then write 148 ÷ 3, 148 ÷ 4, 148 ÷ 5, 246 ÷ 5, 246 ÷ 6, 246 ÷ 8. Which of these divisions would you work out mentally?
Group Activities
-- Find answers to divisions, writing the answers as mixed numbers.
-- Spot divisions with the same answer, and write others with the same answer.
Identify which divisions can be done mentally, and why.

You Will Need

  • Mini-whiteboards and pens
  • Division sheet (see resources)
  • Rounding up and down sheet (see resources)

Mental/Oral Starters

Day 1
Multiplication and division facts (pre-requisite skills)

Suggested for Day 2
Place 5-digit numbers on a human number line (simmering skills)

Suggested for Day 3
Convert between units of time (simmering skills)

Procedural Fluency

Day 1
Practise division.

Day 2
Solve division word problems; round up or down.

Day 3
Practise using the vertical layout to divide numbers, spotting the occasional division that can be worked out mentally.

Mastery: Reasoning and Problem-Solving

  • Explain why each of these three divisions is easy to do mentally.
    250 ÷ 5 =
    180 ÷ 6 =
    210 ÷ 3 =
    So now explain how you would do these three.
    255 ÷ 5 =
    186 ÷ 6 =
    213 ÷ 3 =
  • Divide 456 by 8 using just 2 hops along a number line.
    Show the same division using vertical layout.
  • Use vertical layout to find the answer to 234 divided by 6 and say why it is not an easy one.
  • Write a division of a 3-digit number by a 1-digit number where the remainder is 5.


In-depth Investigation: Revision Rules
Make and prove conjectures when dividing larger numbers by a single digit. Revision Rules from nrich.maths.org.

Extra Support

Chunking Challenge
Using chunking to divide, answers between 10 and 20, no remainders

Leftovers
Using chunking to divide, answers between 10 and 20, with remainders