Maths Year 6 Autumn Decimals and Fractions

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 Add or subtract decimals (suggested as 3 days)

Objectives

Add/subtract multiples of 0.01; add decimals
Unit 1: Code# 6163

National Curriculum
F/De (x)
AS (ix)

Hamilton Objectives
30. Add several decimal numbers using mental or written addition.

Teaching and Group Activities for Understanding

Day 1 Teaching
Use counting stick to count in 0.01s from 2.45 to 2.55 and back. Label 2.5 with a Post-it. Write 2.52 – 0.03. Count back from 2.52 to 2.49. Sketch a jotting to show how we ‘bridge’ 2.5. Repeat for 2.47 + 0.07. Mark the centre as 2, then count in 0.01s from 1.95 to 2.05 and back. Use the counting stick and a jotting to work out 2.02 – 0.05.
Group Activities
-- Game to add and subtract multiples of 0.01 crossing through multiples of 1 and 0.1 using a number line.
-- Repeatedly add 0.06, then subtract 0.07. Challenge to repeatedly add multiples of 0.01 to end on a given number.

Day 2 Teaching
Show 3 DVDs labelled £10.49, £14.79, £12.25. Estimate total cost by rounding. Model compact addition to find total cost. Remind children to leave a line below numbers being added so we can write extra digits. Finish and compare with estimate. Repeat for 3 other prices.
Group Activities
-- Add three amounts of money, aligning digits correctly. Estimate totals first.
-- Add two amounts of money, aligning digits correctly. Estimate totals first.
Write three amounts with a total within a given range.

Day 3 Teaching
Use the distance that snails crawl to generate decimal additions. Children round each distance to nearest metre to estimate the total. Model column addition, showing how to align decimal points.
Group Activities
Use the ‘Adapt Four of the Best’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Add and round three distances in metres (2dp) crawled by snails.
-- Shuffle given digit cards to make additions of distances in metres with 2dp, round and find the total.

You Will Need

  • Counting stick
  • Post-it notes
  • Decimal number lines (see resources)
  • 1 to 6 dice
  • Three DVDs with price labels £10.49, £14.79 and £12.25
  • Whole class practice sheet (see resources)
  • Flipchart and pens
  • Digit cards
  • Mini-whiteboards and pens

Mental/Oral Starters

Day 1
Count on in steps of 0.01 and 0.1 (pre-requisite skills)

Suggested for Day 2
Double numbers with 1 decimal place (simmering skills)

or
Halve numbers with 1 decimal place (simmering skills)

Day 3
Round numbers with 2dp to the nearest 1 and 0.1 (pre-requisite skills)

Procedural Fluency

Day 1
Add and subtract multiples of 0.01, crossing through multiples of 1 and 0.1 including word problems and using a number line.

Day 2
Practise using column addition to add two then three amounts of money.

Day 3
Add distances in metres (2dp).

Mastery: Reasoning and Problem-Solving

  • Jo counts from a number to 4.5 in eleven steps of 0.03. What was her starting number?
  • Write the next 4 numbers in each sequence:
    2.68, 2.69, __, __, __, __
    6.43, 6.42, __, __, __, __
    1.98, 1.99, __, __, __, __
  • Write the missing numbers:
    3.24 + 0.04 = ☐
    ☐ + 0.07 = 3.5
    4.56 + ☐ = 4.76
  • Jim runs 13.85 kilometres and Ann runs 12.78 kilometres to meet him. How far have they run in total?


In-depth Investigation: Adapt Four of the Best
Adapt the numbers in the top row and left column to use two-place decimals, e.g. 1.27, 5.09 etc.

Extra Support

All the 3s
Understanding place value in numbers with two identical places; Adding and subtracting multiples of 0.1 and 0.01 (not crossing 1s or 0.1s)

Take Away Night
Using expanded, then compact addition to add amouts of money

Unit 2 Subtract 1- and 2-place decimals (suggested as 2 days)

Objectives

Subtract numbers with 1 and 2 decimal places
Unit 2: ID# 6173

National Curriculum
F/De (vii)
AS (viii)

Hamilton Objectives
31. Subtract decimal numbers using mental strategies or written counting up.
29. Find the complement to 1, or to next whole number, for a number <10 with up to 3 decimal places.


Teaching and Group Activities for Understanding

Day 1 Teaching
Write 10 – 6.47. Draft an empty number line (ENL) jotting to show a hop from 6.47 to 6.5 then to 7 or one hop straight from 6.47 to 7. Then show a hop from 7 to 10. Children show how to do 9 – 8.63. Then 2.85m + ? = 3.18m and 1.83m + ? = 2.2m.
Group Activities
-- Discuss when to use column subtraction and when Frog (counting up) is preferable when subtracting numbers with one or two decimal places.
-- Spot that given subtractions have the same answer, discuss why and write other pairs of numbers with the same difference.
--Focus on adding hops correctly when using Frog (counting up).

Day 2 Teaching
Write 4.2 – 2.57. Draft an empty number line to show a hop from 2.57 to 3 then to 4.2. Children show how to do 6.5 – 4.78. Then 6.5 – 4.38 and 4.27 – 3.8. Discuss how it is important to add the correct digits together.
Group Activities
Use the ‘Dicey Difference’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Investigate subtracting numbers with consecutive digits, 9.8 – 7.65, 8.7 – 6.54, 7.6 – 5.43 …

You Will Need

  • Mini-whiteboards and pens
  • Flipchart and pens
  • Bamboos growing rate sheet (see resources)

Mental/Oral Starters

Day 1
Pairs of numbers with 1 decimal place and a total of 10 (pre-requisite skills)

or
Say how much is needed to the next metre (pre-requisite skills)

Day 2
Complements to the next whole (pre-requisite skills)

Procedural Fluency

Day 1
Work out how much bamboo grows each day.
Calculate differences in athletics results.

Day 2
Choose how to subtract numbers with one or two decimals places.

Mastery: Reasoning and Problem-Solving

  • Write the missing number in the bar diagram:
4.06
2.68?
  • Write two 2-place decimal numbers which add to 6. Write two different 2-place decimal numbers which add to 4.5.
  • Frog subtracts a number from 3.4. He makes 2 jumps: the first is 0.63 and the second is 1. Write the subtraction he is solving.
  • Work out 7 – 2.89 mentally without writing anything. Check your answer using Frog.

In-depth Investigation: Dicey Differences
Children use two dice with decimal numbers to find largest and smallest possible differences, and use mathematical reasoning to calculate the probability of getting close to a specified target.

Extra Support

Jump to Success
Counting up (using Frog) from numbers with two decimal places to the next whole number

Unit 3 Understand decimals with three places (suggested as 3 days)

Objectives

Understand decimals with three places
Unit 3: ID# 6197

National Curriculum
F/De (vii) (x)

Hamilton Objectives
28. Identify the place value of each digit in a number with up to 3 decimal places; multiply/divide numbers by 10, 100, 1000 giving answers with up to 3-decimal places.

Teaching and Group Activities for Understanding

Day 1 Teaching
Display a place value chart with 10s, 1s, 0.1s, 0.01s and 0.001s. Point at 0.001s; these are 1/1000s. Ring a number on each row. Children write total. Repeat. Use a PV grid 100s, 10s, 1s, 0.1s, 0.01s, 0.001s. Model × and ÷ by 10 and 100.
Group Activities
-- Write place value additions, multiply and divide by 10 and 100 (to include numbers/answers with up to 3 decimal places). (Includes pre-teaching for Day 2.)
-- Play I’m thinking of a number with place value additions/subtractions (numbers with 3 decimal places).

Day 2 Teaching
Give 4 children cards, e.g. 1, 2, 3 and 4. They stand either side of a large decimal point on the flipchart to show 1.234. Ask them to multiply the number by 10 and move accordingly. Repeat to ×/ ÷ by 10, 100 and 1000. Repeat for 4567.
Group Activities
Use the ‘Crack the Code’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
- -Choose a function machine (× 10, ÷ 10, × 100, ÷ 100, × 1000 or ÷ 1000), write the output, partner guesses the input.
-- Select a card with multiplication or division to show on place value grid (× 10, ÷ 10, × 100, ÷ 100, × 1000 or ÷ 1000). Partner guesses the calculation on the card.

Day 3 Teaching
Show a 0–0.1 landmarked line. Use this to place numbers with 3 decimal places on lines; round to the nearest 0.01, 0.1 or 1; Compare 2 numbers using the line.
Group Activities
Order numbers with 3 decimal places, check by placing on landmarked lines.
-- Write numbers in given ranges, order, place on lines and round to the nearest whole and tenth.

You Will Need

  • Place value chart (see resources)
  • Place value grid (see resources)
  • Large number cards (0–9)
  • Additional activity sheets (see resources)
  • 0 to 9 digit cards (with extra 0s)
  • Landmarked lines
  • A 3 to 4 & a 7 to 8 number line

Mental/Oral Starters

Suggested for Day 1
Count in steps of 0.01 and 0.1 through multiples of 0.1 and 1 (simmering skills)

Day 2
Place numbers with 2 decimal places on a line (pre-requisite skills)

Day 3
Round numbers with 2 decimal places to the nearest 1 and 0.1 (pre-requisite skills)

Procedural Fluency

Day 1
Place value additions.

Day 2
Work out the outputs, for × 10, ÷ 10, × 100, ÷ 100, × 1000 and ÷ 1000 function machines.

Day 3
Place numbers with 3 decimal places on lines, then solve ‘Who am I?’ puzzles.
Place numbers with 3 decimal places on lines, then round to the nearest hundredth, tenth and whole.

Mastery: Reasoning and Problem-Solving

  • How many times must I add 0.001 to 2.251 to reach 2.27?
    How many times must I add 0.001 to 9.99 to reach 10?
  • Draw a line from 9.99 to 10. Mark 9.995 and two other numbers, one more than 9.995 and one less.
  • Write the missing numbers in these sentences.
    4.351 × 10 = ☐
    ☐ × 100 = 39.2
    1.871 × ☐ = 1871
  • Write <, = or > between each pair of numbers.
    3.405 3.45
    8.06 8.061
    9.201 9.219
    0.002 2/1000


In-depth Investigation: Crack the Code
Children use logical thinking and number bonds to solve a mathematical puzzle involving multiplying by 10 and 100 and adding decimal numbers.

Extra Support

Dastardly Decimals
Understanding place value in numbers with two decimal places

Unit 4 Add/subtract multiples of 0.1, 0.01, 0.001 (suggested as 2 days)

Objectives

Add/subtract multiples of 0.1, 0.01 and 0.001
Unit 4: ID# 6203

National Curriculum
F/De (vii) (x)

Hamilton Objectives
28. Identify the place value of each digit in a number with up to 3 decimal places; multiply/divide numbers by 10, 100, 1000 giving answers with up to 3-decimal places.

Teaching and Group Activities for Understanding

Day 1 Teaching
Show a partially completed 0.001 to 0.1 grid. Count in 0.001s to 0.01. What’s next? Complete 2nd row. What do we add to move down a square? Fill in one column. Children discuss what number is in bottom right-hand corner. Count to check.
Group Activities
-- Use counting in steps of 0.001 and 0.01 to complete rows and columns in a partially complete 0.001 to 0.1 number grid.
-- Use counting in steps of 0.001 and 0.01 to write numbers in various places in a partially complete 0.001 to 0.1 number grid.

Day 2 Teaching
Start with 34.567. Add or subtract multiples of 10, 1, 0.1, 0.01, 0.001, making sure we do not count through a multiple of 0.1 or 1, e.g. not 34.67 + 0.05 or 34.23 – 0.4. Are children changing the correct digit each time? Re-write 34.567. Together, add 0.005 counting on in 0.001s, taking care when crossing 34.57.
Group Activities
Use the in-depth problem-solving investigation ‘Sometimes We Lose Things’ from NRICH as today’s group activity.
Or, use these activities:
-- Add and subtract multiples of 0.1, 0.01 or 0.001, focusing on passing through multiples of 0.01.
-- Use a counting stick to support counting on and back steps of 0.001.
-- Repeatedly add 0.007, 0.07 and 0.7 to pass targets, predicting how many of each need to be added.

You Will Need

  • Partially completed 0.001 to 0.1 grid (see resources)
  • Mini-whiteboards & pens
  • Add & subtract multiples of 0.1, 0.01 or 0.001 sheet (see resources)
  • Flipchart and pens
  • Counting stick
  • Post-it notes

Mental/Oral Starters

Day 1
Place value in numbers with 3 decimal places (pre-requisite skills)

Suggested for Day 2
Convert from m to cm, cm to mm and vice versa (simmering skills)

Procedural Fluency

Day 1
Fill in sections from a 0.001 to 0.01 number grid.
Use a 0.001 to 0.01 grid to add and subtract 0.001 and 0.01.

Day 2
Add and subtract multiples of 0.1, 0.01 or 0.001.

Mastery: Reasoning and Problem-Solving

  • Write the missing numbers to complete this section of the 0.001 to 0.1 grid
0.0470.0480.049
0.068
0.079
0.087
  • What must be added to 0.089 to make 0.09?
    To make 0.9?
  • True or false?
    3.452 – 0.003 = 3.449
    0.002 + 1.009 = 0.11
    2.985 + 0.005 = 2.99
    0.019 > 0.091


In-depth Investigation: Sometimes We Lose Things
What would happen if we lost all the nines in our number system? Children's understanding of the place value system is put to the test. Sometimes We Lose Things from nrich.maths.org.

Extra Support

All the 3s
Understanding place value in numbers with two identical places; Adding and subtracting multiples of 0.1 and 0.01 (not crossing 1s or 0.1s)

Unit 5 Decimals, fractions: compare, order (suggested as 4 days)

Objectives

Equivalent decimals/ fractions: compare/order; fractions of amounts
Unit 5: ID# 6211

National Curriculum
F/De (i) (ii) (vi)

Hamilton Objectives
21. Use common multiples to generate equivalent fractions, e.g. 4/8 = 1⁄2; reduce fractions to their simplest form using common factors.
22. Use knowledge of equivalence to compare/order fractions
23. Identify simple fraction/decimal/percentage equivalents: E.g. 1⁄4 = 0.25 = 25%, 1/3 = 0.33 = 33%
24. Associate a fraction with division; calculate decimal fraction equivalents, e.g. 4/5 is 0.8, 1/8 is 0.125

Teaching and Group Activities for Understanding

Day 1 Teaching
Use an interactive 1–100 square. Shade 1/100 and the 1/10. Use shading to show that 1/100 = 10/100, 1/10 = 10/100, etc. Then show that 5/10 = 50/100 = 1/2, 25/100 = 1/4, 1/5 = 2/10 = 20/100 etc. 1/8 is half 1/4 or 0.125. Find 2/8, 3/8, 4/8, 5/8, 6/8, 7/8 as decimals.
Group Activities
-- Mark halves, quarters, fifths, tenths and eighths, and the equivalent decimals on a 0 to 1 line marked in 0.1s.

Day 2 Teaching
Use a fraction wall to show equivalent fractions. Draw out that if we multiply the numerator and denominator by the same number we derive an equivalent fraction. Demonstrate how we use this to simplify fractions.
Group Activities
Use the in-depth problem-solving investigation ‘Rod Fractions’ from NRICH as today’s group activity.
Or, use these activities:
-- Use 1–12 (to 1–24) cards to make fractions, and simplify to earn points.
-- Write fractions equivalent to 1/4, then use equivalence to identify fractions less than ¼, and write fractions greater than 1/4.
Day 3 Teaching
Write 2/3 and 3/4. Which is the bigger fraction? But can children prove it? Show how we can change both to twelfths to compare. Use common multiples to compare 1/2 and 3/5, then 1/6 and 2/9.
Group Activities
-- Use a fraction wall to help write fractions as the same type of fractions in order to compare them.
-- Use cards 1 to 6 to make pairs of fractions, both less than 1. Use equivalence to compare them.
-- Equivalent fractions challenge.

Day 4 Teaching
There are 135 animals in a rescue centre. 1/5 are cats, 3/5 are dogs and the rest are ponies. How can we work out the number of cats? Suggest children use short division to find 1/5 of 135 on their whiteboards. Record: 27 cats. Find the number of dogs and ponies.
Group Activities
-- Use the bar model to help solve word problems which involve finding non-unit fractions of amounts.
-- Play fractions of amounts bingo.

You Will Need

  • Mini-whiteboards and pens,
  • Blank 100 square (see resources)
  • ITP: Fractions
  • 0 to 1 line marked in steps of 0.1 (see resources)
  • Fraction wall (see resources)
  • 1 to 12 cards
  • Flipchart and pens
  • 1 to 6 cards
  • ‘Bingo – Fractions of Amounts’ from topmarks.co.uk

Mental/Oral Starters

Day 1
Count in 1/8s along a number line (pre-requisite skills)

Suggested for Day 2
Divisibility by 2, 3 and 5 (simmering skills)

Day 3
Factors and multiples (pre-requisite skills)

Day 4
Fractions of amounts within tables (pre-requisite skills)

Procedural Fluency

Day 1
Use equivalence to identify missing fractions and decimals on a number line.

Day 2
Find equivalent fractions.

Day 3
Write and compare pairs and trios of fractions with the same denominator.
Write and order fractions as fractions with given denominators.

Day 4
Find unit fractions and linked non-unit fractions of amounts.
Find non-unit fractions of amounts and work out fractions of numbers, e.g. ☐ /3 of 81 is 54.

Mastery: Reasoning and Problem-Solving

  • Ben simplified three fractions. They all came out at ¾. Suggest what they could have been.
  • True or false?
    Any fraction which is simplified to give 1/3 has a numerator which is 3 × the denominator.
    36/64 cannot be simplified to 4/9
    1/2 is 500 × 0.001
    1/4 is more than 0.2
  • Write the missing numbers:
    ☐/6 = 4/☐
    6/☐ = ☐/20
    ☐/10 > 1/☐
    ☐/32 > ☐/8
  • Find 3/4 of 202:
202
  • 3/8 of biscuits are wafers, 5/8 are chocolate. How many of each?
344

In-depth Investigation: Rod Fractions
Pick two rods of different colours. How can we work out what fraction the shorter rod is of the longer one? Rod Fractions from nrich.maths.org.

Extra Support

Fraction Families
Finding equivalent fractions within fraction families, e.g. quarters and eighths, thirds and sixths, fifths and tenths, sixths and twelfths

Unit 6 Equivalent fractions: add and subtract (suggested as 3 days)

Objectives

Equivalent fractions: compare, add, subtract
Unit 6: ID# 6221

National Curricuum
F/De (i) (ii) (iii)

Hamilton Objectives
21. Use common multiples to generate equivalent fractions, e.g. 4/8 = 1⁄2; reduce fractions to their simplest form using common factors.
22. Use knowledge of equivalence to compare/order fractions and to add or subtract fractions and mixed numbers.

Teaching and Group Activities for Understanding

Day 1 Teaching
Compare 5/8 and 3/4 by writing both fractions as 1/8s. Then order 5/6, 2/3, 7/12 by writing all three as 1/12s. Discuss how 5/6 is bigger than 3/4 as it is only 1/6 off a whole number, as opposed to 1/4. We prove it by writing them as the same sort of fraction. List common multiples of 4 and 6. We can use this to find a denominator to use. Agree 12 is the smallest common multiple. Ask children to write both fractions as 1/12s then compare them.
Group Activities
-- Look for pairs of fractions they can write as the same sort, confirm with a fraction wall, then compare.
-- Use equivalence to sort the fractions into three groups: < 1/3, = 1/3, > 1/3.
-- Use equivalence to sort the fractions into three groups: less than 1/3, between 1/3 and 2/3, greater than 2/3.

Day 2 Teaching
Add fractions with unrelated denominators by writing 1/2 + 2/3. Point out that these are unrelated fractions. Ask children to discuss what we can do. Take feedback. Draw out changing both to 1/6s. Ask children to do this, then find the answer. Repeat for 2/3 + 2/5 and 1/2 + 2/3.
Group Activities
Use the ‘Domino Fractions’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Look for pairs of fractions with unrelated denominators to add, to give answers of less than 1, and more than 1.
-- Use equivalence to find which pairs of children ate more than a whole chocolate bar and which pairs ate less than a whole bar.

Day 3 Teaching
Write 3/4 – 1/2, 7/10 – 2/5, 3/4 – 3/8 and ask children to use the fact that these fractions are related to subtract them. Then write 3/4 – 2/3 and show that we need to change both to twelfths. Children work out 1/2 – 1/5, 2/3 – 2/5, 4/5 – 3/4.
Group Activities
-- Work out 2/3 – 1/2, 3/4 - 2/3, 4/5 – 3/4, 5/6 – 4/5 … What do children notice about the answers?
-- Work out 1/2 + 1/3, 1/2 + 1/4, 1/2 + 1/5…then 1/2 - 1/3, 1/2 - 1/4, 1/2 - 1/5 … first agreeing in pairs how to write both fractions as the same type.

You Will Need

  • Mini-whiteboards and pens
  • Flipchart and pens
  • Fraction wall (see resources)
  • Fraction cards sheet 1 (see resources)
  • Sheets of paper
  • Fraction cards sheet 2 (see resources)
  • Sheet of fractions (see resources)
  • Chocolate eater cards (see resources)

Mental/Oral Starters

Day 1
Count in 1/8s along a number line (pre-requisite skills)

Day 2
Turn improper fractions into mixed numbers and vice versa (pre-requisite skills)

Suggested for Day 3
Equivalence (simmering skills)

Procedural Fluency

Day 1
Compare pairs then order trios of fractions by writing them as the same sort of fraction.

Day 2
Add fractions with a total of less than 1, then more than 1.

Day 3
Subtract pairs of fractions with related and unrelated denominators.

Mastery: Reasoning and Problem-Solving

  • Write the missing numbers to make each sentence true.
    5/6 + 5/8 = ?
    4/5 – ?/? = 1/4
    1/6 + ?/? = 9/10
  • Write an addition of two different fractions which have a total of 5/8.
  • How many times must I add 1/5 to 3/8 to get a total over 1?
  • Which is larger…
    0.75 – 1/3 or 1/6 + 1/4
  • Mystery fractions.
    Fraction A add fraction B is one whole.
    Fraction A subtract Fraction B is 3/7
  • Write fractions A and B in their simplest forms.


In-depth Investigation: Domino Fractions
Children use dominoes to create fractions. They explore sums of fractions using equivalent fractions and related denominators.

Extra Support

Make the Fractions Friendly
Adding fractions with related denominators

Teeny Subtractions
Subtracting fractions with related denominators