Maths Year 6 Spring 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. These bulk downloads are available to friends and School Subscribers. These bulk downloads are added value for Hamilton Friends and School Subscribers.

Unit 1 Scale factor problems concerning area (suggested as 2 days)

Objectives

Solve problems using a scale factor in areas of shapes
Unit 1: ID# 6481

National Curriculum
Mult/div (iv) (viii)
Rat/Pr (iii)

Hamilton Objectives
34. Solve problems involving similar shapes where the scale factor is known or can be found.
13. Scale up or down by a factor of 2, 4, 5 or 10; solve scaling problems and those involving rates.
35. Solve problems involving simple ratios, using tables facts and knowledge of fractions and multiples.

Teaching and Group Activities for Understanding

Day 1 Teaching
Children sketch a right-angle triangle where the sides along the right angle are 4cm and 7cm long. They measure the longest side, compare, and find the area. Draw a similar triangle with each side next to the right angle twice the length. Compare the longest sides. Define similar triangles and model the scaling effect. Remind children how to find the area of a parallelogram (see below plan), multiplying the base by the height.
Group Activities
-- Scale up rectangles, triangles and parallelograms by a factor of 2 and 3 and see what happens to the area.

Day 2 Teaching
Define similar shapes as identical in shape, but not in size. Show ‘Similar shapes’ (see resources). Ask children to find 2 rectangles which look similar. Measure the sides of each. Ask children to work out the scale factor (1½). Repeat for triangles (scale factor is 3).
Group Activities
Use the ‘Geometry genius’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Identify pairs of similar rectangles and triangles. Use the scale factor to work out side lengths.
-- Draw similar shapes, using a scale factor of 2 or 3 or 1.5. Identify pairs of similar shapes drawn by others and find the scale factor?

You Will Need

  • Mini-whiteboards and pens
  • Rulers
  • ‘Scaling up’ activity sheet (see resources)
  • cm2 paper
  • Similar shapes (see resources)
  • Similar shapes - triangles and rectangles sheets (see resources)
  • ITP: Function Blocks

Procedural Fluency

Day 1
Calculate the dimensions of toys, given scale factors.

Day 2
Identify pairs of similar rectangles and triangles. Use a scale factor to calculate the lengths of sides.

Mastery: Reasoning and Problem-Solving

  • True or false?
    If one triangle is scaled up to have sides 3 times as long as another, the area is also 3 times as large.
    If two rectangles are similar and the scale factor is 4, then the area of the larger rectangle is 16 times that of the smaller rectangle.
  • Calculate the area of the triangle whose sides are half the length of this one (see download). Compare the two areas. What do you notice?
    Explain why the area of the smaller has this relation to the area of the larger.

In-depth Investigation: Geometry Genius
Children use what they know about how to find the areas of triangles and parallelograms to find the areas of rhombi, kites and trapezia.

Extra Support

Function Blocks
Tables knowledge is an important pre-requisite for scaling. Use Function Blocks ITP to generate multiplication by 3. Can children work out the function? Repeat with other single-step x and ÷ functions.

Unit 2 Solve rate and scaling problems (suggested as 3 days)

Objectives

Use mental strategies, factors and multiples to solve problems of rate and scaling
Unit 2: ID# 6491

National Curriculum
Mult/div (iv) (viii)
Rat/Pr (iii)

Hamilton Objectives
9. Know all multiplication and division facts up to 12 × 12; identify common factors, common multiples.
10. Multiply/divide whole numbers mentally, using facts to 12 × 12 and place value; use facts to work with larger numbers.
13. Scale up or down by a factor of 2, 4, 5 or 10; solve scaling problems and those involving rates.
14. Perform divisions mentally within the range of tables facts; divide multiples of 10 and 100 (4500 ÷ 9) and use mental strategies such as halving (450 ÷20).

Teaching and Group Activities for Understanding

Day 1 Teaching
A blue whale is typically 24–30m in length, 4 times as long as the class. It has the fastest growth rate: calves grow 90kg/day, 2.5cm in length per day. So how much in a week? Explain that these are rates: It eats 3 tonnes per day. It grows 90Kg per day. The rate is weight (kg/tonnes) per day (time).
Group Activities
-- Use internet research to find real life rates (e.g. speed of a cheetah, the amount a young lion eats per day ….) and then invent similar problems.
-- Work out how much a blue whale calf will grow in given times and compare to rate of growth for a human child.

Day 2 Teaching
Recap mental strategies to multiply by 5, by 20, by 6, by 8. Show a table of dimensions of scale models of dinosaurs. Each model is 1/20 of the actual height and length. Ask how we work out full size, drawing out multiplying by 20.
Group Activities
-- Work out the scaled-up dimensions of children and classroom furniture: scale factors 5, 6 and 8.

Day 3 Teaching
Recap mental strategies to divide by 5, by 20, by 6, by 8. Say that a toy shop is selling dinosaurs where each measurement is 1/200 of the size of the real dinosaurs. Ask children in pairs to find 1/200 of the actual measurements.
Group Activities
Use the ‘Get to the root’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Scale down dimensions of dinosaurs by factors of 10 or 100 to sketch on paper. Look at relative sizes.

You Will Need

  • Tape measure or trundle wheel to measure the length of the classroom
  • ‘Solve problems involving rate’ sheet (see resources)
  • Access to the internet
  • ‘Scaling up’ sheet (see resources)
  • Mini-whiteboards and pens
  • ‘Dimensions for dinosaur toys’ sheet (see resources)
  • A3 paper
  • ‘Dinosaur measurements’ sheet (see resources)

Mental/Oral Maths Starters

Day 1
Double and halve 3-digit numbers (pre-requisite skills)

Suggested for Day 2
Multiplication and division facts (simmering skills)

Suggested for Day 3
Order of operations and brackets (simmering skills)

Procedural Fluency

Day 1
Answer problems about growth rates for blue whale calves and humans.

Day 2
Work out actual dinosaur sizes, given model dimensions.

Day 3
Work out the dimensions of dinosaur toys, given full size measurements.

Mastery: Reasoning and Problem-Solving

  • Charlie’s uncle drinks chocolate milk at a rate of 3 pints a day. How much milk does he drink in a week? In a year?
  • Find these products using mental strategies:
    (i) 234 × 5
    (ii) 450 × 20
    (iii) 1270 ÷ 8
    (iv) 253 × 6
    (v) 732 ÷ 5
    Explain which you found the most tricky and why.
  • A doll's house is made so that all the furniture is 1/12 size of actual real furniture. Write the dimensions of these items in the doll's house:
    -- Bed (actual size 2.4m by 1.2m)
    -- Sofa (actual size 1.8m by 120cm)
    -- Wardrobe (actual size 2.4m by 1.8m by 60cm)


In-depth Investigation: Get to the Root
Children use their fluency in mental multiplication to explore the patterns of digital roots in multiplication.

Extra Support

Factors and Multiples Game
Use this game where players take it in turns to cross out numbers, at each stage choosing a number that is a factor or multiple of the number just crossed out by the other player. Factors and Multiples Game from nrich.maths.org.

Unit 3 Long division; different remainder forms (suggested as 2 days)

Objectives

Use long division giving remainders as fractions and decimals
Unit 3: ID# 6521

National Curriculum
Mult/Div (ii)

Hamilton Objectives
15. Interpret remainders as whole number remainders, fractions, including decimal fractions where equivalents are known or by rounding up or down
17. Divide numbers with up to 4 digits by 2-digit numbers using a formal written method of long division and giving an appropriate answer.
19. Use estimation to check answers and determine an appropriate degree of accuracy.

Teaching and Group Activities for Understanding

Day 1 Teaching
Write 632 ÷ 14 and discuss how we can first write the multiples of 14 and then use these to solve this problem using long division. Give the answer as 45 r 2. Repeat this process with 532 ÷ 17.
Group Activities
-- Work in a step-by-step way to use efficient chunking to perform long division.
-- Practise long divisions and then create similar divisions using the same number and different divisors.

Day 2 Teaching
Use long division to work out 355 ÷ 15. Agree the answer 23 r 10. We can also divide 10 by 15 to give 10/15. Simplify this to give 232/3. Repeat this process for other divisions, dividing the remainder to give a fraction. Use equivalents to write fraction as a decimal.
Group Activities
Use the ‘Exact Answers’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Divide by 16 using the step-by-step chunking process; write remainders as fractions and as equivalent decimals for halves/quarters.
-- Explore patterns of answers ending .25 or .75 when dividing by 16 and attempt to find dividends with answers ending in .5.

You Will Need

  • Mini-whiteboards and pens
  • Flipchart and pens

Mental/Oral Maths Starters

Day 1
17 times table (pre-requisite skills)

Day 2
Equivalent fractions and decimals (pre-requisite skills)

Procedural Fluency

Day 1
Write multiples and use these to help divide 3-digit numbers by 2-digit numbers (answers with remainders).

Day 2
Write multiples and use these to help divide 3-digit numbers by 2-digit numbers, writing remainders as fractions and decimals.

Mastery: Reasoning and Problem-Solving

  • Which numbers between 1 and 10 does 4566 divide by to leave a remainder? Demonstrate that you have found them all.
  • Write the 22 times table up to 10 x 22. Use this to help divide 5546 by 22.
  • Divide 2752 by 5 and give the answer first as a mixed number and then as a number with a decimal part.

In-depth Investigation: Exact Answers
Children use their knowledge of long division to explore patterns in remainders.

Extra Support

Amazing Multiples
Using chunking to divide (answers less than 100)

Unit 4 Use short/long multiplication in problems (suggested as 3 days)

Objectives

Use short and long multiplication to solve problems
Unit 4: ID# 6529

National Curriculum
Mult/Div (i)

Hamilton Objectives
11. Multiply 2-, 3- and 4-digit numbers by numbers up to 12 using short multiplication or another appropriate written method.
12. Multiply numbers with up to 4 digits by 2-digit numbers using formal long multiplication.
19. Use estimation to check answers and determine an appropriate degree of accuracy; round answers to multiplications and divisions to a specified degree of accuracy.

Teaching and Group Activities for Understanding

Day 1 Teaching
Write 2341, 5372, 4278, 6143. Divide the class into 4 groups. A child from each group takes a 3–9 digit card. Each group decides which number to multiply by the card number, aiming for an answer close to 20,000. Children use short multiplication or the grid method. Closest to 20,000 wins.
Group Activities
-- Use the digits 1, 2, 3, 4 and 5 to create 4-digit by 1-digit multiplications with answers as near to 15,000 as possible.
-- Use the digits 5, 6, 7, 8 and 9 to create 4-digit by 1-digit multiplications with answers as close to 60,000 as possible.

Day 2 Teaching
Write 23 × 367. Revise using long multiplication: multiply by 20 by doing x10 and double, then multiply by three; add the two products. Write 34 × 367: multiply by 30 by doing x10 (put a zero in 1s column) then multiply by 3; multiply by 4 for the second row. Add the rows to get the total.
Group Activities
Use the ‘Riveting reversals’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Measure heart rate then use either the grid method or long multiplication to multiply by 60 and 24 to find approximately how many times their heart beats in a day.
-- Use understanding of multiplying 3-digit numbers by 2-digit numbers to multiply 4-digit numbers by 2-digit numbers.

Day 3 Teaching
A teacher travels 16 miles to and from school for 195 days a year. How far in a year? Draw out 16 × 200 is 3200. Explain how to use long multiplication to work out 16 × 195. Ask the rest of the class to check using the grid method. Repeat for 23 miles × 195 days.
Group Activities
-- Use the context of 24 hours in a day to multiply by 24 to find the number of hours in a year or other 3-digit numbers of days.
-- Use long multiplication to calculate how many days, then hours, children have been alive.

You Will Need

  • 1–9 digit cards
  • Mini-whiteboards and pens
  • Stop watches
  • Flipchart and pens
  • Calendars

Mental/Oral Maths Starters

Day 1
Multiply 1000s by 10s (pre-requisite skills)

Day 2
Mental multiplication by 20, 4, 5 (pre-requisite skills)

Suggested for Day 3
Order of operations (simmering skills)

Procedural Fluency

Day 1
--Use short multiplication to multiply amounts of money, aiming to get answers close to £200.

Day 2
-- Choose grid multiplication or long multiplication to multiply 3-digit numbers by 2-digit numbers then answer word problems.

Day 3
--Use the grid method to multiply 3-digit numbers by 2-digit numbers.
--Use long multiplication to multiply 3- and 4-digit numbers by 2-digit numbers.

Mastery: Reasoning and Problem-Solving

  • Complete each multiplication using a different method.
    (i) 4530 × 23
    (ii) 399 × 25
    (iii) 476 × 6
  • Multiply 531 by 32 using long multiplication or the grid method. Now double it five times to check if this gives the same answer.
  • True or false?
    28 × 4345 is the same as 7 × 4345 plus double 8690.
    1448 × 24 is the same as 36200 – 1448.
    36 × 478 gives the same product as 9 × 478 doubled twice.

In-depth Investigation: Riveting Reversals
Multiply 3-digit numbers with consecutive digits by a 2-digit number; reverse the 3-digit number and repeat. Find the difference between the two answers.

Extra Support

Monster Multiplications
Using known times tables and place value to multiply, e.g. 6 × 4, 6 × 40, 6 × 400

Aim for 2000
Using the grid method to multiply 3-digit numbers by 1-digit numbers

Unit 5 Use short/long division in problems (suggested as 3 days)

Objectives

Use short and long division to solve problems
Unit 5: ID# 6547

National Curriculum
Mult/Div (ii) (iii)

Hamilton Objectives
16. Divide numbers with up to 4-digits by a number up to 12 using short division and giving an appropriate answer.
17. Divide numbers with up to 4 digits by 2-digit numbers using a formal written method of long division and giving an appropriate answer.
15. Interpret remainders as whole number remainders, fractions, including decimal fractions where equivalents are known or by rounding up or down.
19. Use estimation to check answers and determine an appropriate degree of accuracy; round answers to multiplications and divisions to a specified degree of accuracy.

Teaching and Group Activities for Understanding

Day 1 Teaching
Use the ‘bus shelter’ layout to show 4281 ÷ 3. Cover digits 2, 8 and 1. Perform the division, uncovering one number at a time. Say that we can divide the remainder 2 by 3 to give an answer of 8452/3. Write 3341 ÷ 4 and use short division to solve.
Group Activities
-- Use short division in the context of word problems.
-- Solve problems using short division by looking at the patterns of remainders expressed as fractions.

Day 2 Teaching
A comic has 32 pages. The shop can print 875 pages. Estimate the number of comics that can be printed. Pairs list the 32 times table to 10 × 32. Model using long division to find number of comics and the number of spare pages. Repeat for 16 pages.
Group Activities
-- Use the 16 times table to divide 3-digit numbers by 16, using efficient chunking.
-- Use the 24 times table to divide 3-digit numbers by 24, using efficient chunking.

Day 3 Teaching
Write 904 ÷ 22. Children list multiples of 22 up 220. They refer to the list to estimate an answer. Model division on the board, 41 r 2. Say we divide the remainder by 22 to give a fraction, 2/22 and then simplify to 1/11. Repeat for another division.
Group Activities
Use the ‘Why is it so?’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
Choose 3-digit numbers to divide by 18.
Estimate answers, then solve problems involving long division by 32. Use efficient chunking as the least error-prone method.

You Will Need

  • ‘Short division’ sheet 1 (see resources)
  • ‘Short division’ sheet 2 (see resources)
  • Flipchart and pens
  • Mini-whiteboards/large sheets of paper and pens
  • 1–9 digit cards
  • ITP: Tell the Time

Mental/Oral Maths Starters

Day 1
Mental Division by 20, 4, 5 (pre-requisite skills)

Suggested for Day 2
Multiply and divide numbers with up to 2 decimal places (simmering skills)

Suggested for Day 3
24-hour clock (simmering skills)

Procedural Fluency

Day 1
Practise using short division then answer word problems requiring decisions about remainders.

Day 2
Use multiples of 17 to help divide 3-digit numbers by 17.
Use long division of 3-digit numbers by 2-digit numbers to find answers between 30 and 40, then 50 and 60.

Day 3
Write multiples of given 2-digit numbers and use to divide 3- and 4-digit numbers.

Mastery: Reasoning and Problem-Solving

  • If Sally multiplies a number by 12 she gets 9,432. What was her starting number?
  • Tom multiplies his number by 9 and gets 7074. What was his starting number?
  • Which of the following divisions give an answer which ends 1/4 or .25?
    3750 ÷ 24
    2223 ÷ 18
    7300 ÷ 16
  • What do you notice about the two divisors in the two divisions which gave an answer ending ¼?

In-depth Investigation: Why is it so?
Children identify a pattern in the division of a total of six numbers created using the same 3 digits. They then use algebra to explain why it is so.

Extra Support

Mixed up answers
Use the grid method to multiply 3-digit numbers by 1-digit numbers. Begin to use the grid method to multiply 4-digit numbers by 1-digit numbers.

Any left?
Use chunking to divide (answers less than 100).