Maths Year 6 Spring 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 Place value in 3-place decimals (suggested as 3 days)

Objectives

Understand place value in numbers with three decimal places
Unit 1: ID# 6619

National Curriculum
Fr (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
Write 4.735 and ask children what the 7, 3 and 5 represent. Agree they are tenths, hundredths, thousandths. Show this on a place value grid. After subtracting 0.005, check that they write 4.73 not 4.730. Emphasize that we only write 0 in thousandths place if there are digits after this. Children add and subtract multiples of 0.1, 0.01 and 0.001.
Group Activities
Use the in-depth problem-solving investigation ‘The Size and Distance of the Planets’ from UC Berkeley as today’s group activity.
Or, use these activities:
-- Add and subtract 0.1s. 0.01s and 0.001s, using a calculator to test predictions.
-- Add and subtract 0.1s. 0.01s and 0.001s from 3.457 to make target numbers.

Day 2 Teaching
Launch the ITP ‘Moving digits’. Use the cards to show 56. Click to multiply by 10 three times, saying multiply by 10, 100, 1000. Discuss how much we have multiplied by and what the effect on the digits is. Discuss the effect of dividing 2340 by 1000. Then multiply and divide other numbers by 10, 100 and 1000, monitoring the effect on the digits.
Group Activities
-- Sequentially multiply and divide a number by powers of 10 to arrive back at the starting number.
-- Divide four numbers by 100 and 1000, then order. Repeat for multiplying by 100 and 1000.

Day 3 Teaching
Show children a counting stick for 2–3. Count from 2 to 3 in steps of 0.1. Children draw a line 2.3–2.4 and mark a number with 2 decimal places. What does this round to? Repeat, for numbers with 3 decimal places.
Group Activities
-- Round numbers with 3 decimal places to the nearest whole kilogram.
-- Generate, then round numbers with 3 decimal places to the nearest whole, tenth and hundredth.

You Will Need

  • ITP: Moving digits
  • Place value addition and subtraction (see resources)
  • Mini-whiteboards and pens
  • Calculators
  • Flipchart and pens
  • 0–9 digit cards
  • Counting stick
  • Dartboard rounding from www.topmarks.co.uk

Mental/Oral Starters

Day 1
Count in steps of 0.01 and 0.1 from numbers with 2 decimal places (pre-requisite skills)

Day 2
Place value in numbers with 3 decimal places (pre-requisite skills)

Day 3
Place numbers with 2 decimal places on a line (pre-requisite skills)

Procedural Fluency

Day 1
Use place value to add and subtract 0.1s, 0.01s and 0.001s.

Day 2
Multiply and divide masses/weights in kilograms by powers of 10.
Identify functions (×/÷ 10, 100 and 100).

Day 3
Mark numbers with 1, 2 and 3 decimal places on lines and round to the nearest whole.
Mark numbers with 1, 2 and 3 decimal places on lines and round to the nearest whole, tenth and hundredth.

Mastery: Reasoning and Problem-Solving

  • Identify the value of the ‘4’ in the following numbers:
    (a) 3.407
    (b) 4.821
    (c) 0.043
    (d) 5.104
    (e) 48,739
  • How many times must Dan multiply 0.048 by 10 to get 48,000?
  • What number is one hundred times smaller than 0.4?
  • Write three numbers, two with 3 digits and one with 4 digits, that round to 4 as the nearest whole number.


In-depth Investigation: The Size and Distance of the Planets
Children will manipulate big numbers to investigate the relative size and distance between planets. The Size and Distance of the Planets from cse.ssl.berkeley.edu.

Extra Support

Deduce the Decimal
Understanding place value in numbers with 3 decimal places

Moving Digits
Multiplying and dividing by 10, 100 and 1000

Unit 2 Add numbers with up to 3 decimal places (suggested as 2 days)

Objectives

Add numbers with up to three decimal places
Unit 2: ID# 6637

National Curriculum
Fr (vii)
Meas (i)

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

Teaching and Group Activities for Understanding

Day 1 Teaching
Children use written addition to work out 4.72 + 3.45. They round each number to the nearest whole to check if the answer seems reasonable. Ensure that children understand that 0.7 + 0.4 = 1.1, not 0.11. Write 45.7 + 3.45 and show how we line up the decimal point to help add place values.
Group Activities
Use the in-depth problem-solving investigation ‘Decimal pyramids’ as today’s group activity.
Or, use these activities:
-- Add pairs of distances (with 2 decimal places and 1 decimal place) to find totals which round to 6m, 7m, 8m, 9m and 10m.
-- Add pairs of weights (with 2 decimal places and 3 decimal places) to find totals under 7kg.

Day 2 Teaching
Use Dartboard rounding and choose kilograms. Discuss which pair might have a total near 20kg. Take suggestions and click to round to the nearest kg to confirm. Work through some additions, talking through adding 0.001s, writing in any ‘carried’ amounts as necessary.
Group Activities
-- Use an online activity to practise adding towards a given total.
-- Use digit cards to create additions with a given target total.

You Will Need

  • Mini-whiteboards and pens
  • Flipchart and pens
  • Dartboard rounding at www.topmarks.co.uk
  • 2–9 digit cards (1 set per pair)

Mental/Oral Starters

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

Suggested for Day 2
Difference between positive and negative numbers (simmering skills)

Procedural Fluency

Day 1
Find pairs of distances with a total of less than 7m [mix of 2 decimal places (dp) and 1dp].
Add pairs of decimals (2 dp + 2 dp, 3dp + 3dp, 3dp + 2 dp)

Day 2
Find pairs of distances, weights and capacities within given ranges.

Mastery: Reasoning and Problem-Solving

  • Add 3.21 and 32.1.
    Add 4.32 and 43.2.
    Add 5.43 and 54.3.
    Before doing the addition, can you predict the answer to 6.54 and 65.4?
  • Write the missing digits in this calculation:
    ☐.6☐8 + 26.☐56 = ☐1.39☐


In-depth Investigation: Decimal Pyramids
Children add numbers with 3 decimal places to give a number with 2 decimal places. They add numbers with 2 decimal places to give a number with 1 decimal place. They then add numbers with 1 decimal place to give a whole number.

Extra Support

Pyramid Pile-up
Adding decimals to complete a pyramid

Unit 3 Multiply/divide 2-place decimal numbers (suggested as 2 days)

Objectives

Multiply and divide numbers with up to two decimal places
Unit 3: ID# 6653

National Curriculum
Fr (viii) (ix)

Hamilton Objectives
32. Multiply numbers such as 4.7 and 0.06 by whole numbers.

Teaching and Group Activities for Understanding

Day 1 Teaching
Use ITP 'Number dials' to show the 6 times table. Then click to select 0.6, hiding numbers in the outside boxes. Reveal the first few. Discuss what children notice. Each answer is 1/10 of answer for the 6 times table. Use this to multiply and divide decimal numbers, e.g. 4.2 ÷ 6, 4 × 0.06, 1.8 ÷ 6, 0.48 ÷ 6 etc.
Group Activities
-- Derive 0.8 times table up to 10 x 0.8; peer-quiz on divisions using these tables. Repeat for the 0.08, 0.7 and 0.07 times tables.
-- Use tables knowledge to write corresponding multiplications and divisions.

Day 2 Teaching
Write 4 × 3.6 on the board and ask children to discuss how they could work this out. Draw out working out 4 × 3 and 4 × 0.6. Write 4 × 3.6 = (4 × 3) + (4 × 0.6) on the board. Repeat for 3 × 0.47.
Group Activities
-- Use this activity as this unit’s in-depth problem-solving investigation:
-- Use the digits 0, 3, 4, 5 and 6 to create and answer as many different multiplications as they can of the form ☐ x ☐.☐ and ☐ x 0.☐☐.

You Will Need

  • ITP: Number dials
  • Flipchart and pens
  • Mini-whiteboards and pens
  • Sets of number cards 0, 3, 4, 5, and 6

Mental/Oral Starters

Day 1
Double and halve numbers with 1 decimal place (pre-requisite skills)

Day 2
Times tables (pre-requisite skills)

Or

Day 2
Division facts (pre-requisite skills)

Procedural Fluency

Day 1
Use place value and tables facts to mentally multiply decimals, e.g. 3 × 0.4 and 3 × 0.04, and divide decimals e.g. 4.5 ÷ 5 and 0.45 ÷ 5.

Day 2
Use partitioning to mentally multiply numbers with 1 decimal place, e.g. 3 × 4.6.
Use partitioning to mentally multiply numbers with 1 decimal place and 2 decimal places, e.g. 3 × 4.6 and 3 × 0.46.

Mastery: Reasoning and Problem-Solving

  • Write the first six facts in the 0.5 times table.
    1 × 0.5 = 0.5
    2 × 0.5 =
    etc.
  • What is 4.5 divided by 0.5?
  • A metal tag is 0.7cm long.
    How many tags can be cut from a strip of metal 6.3cm long?
    How many tags could be cut from a strip of metal 70cm long?
  • Use partitioning to find 28 × 6. Now explain how to multiply 2.8 by 6. Finally, write the answer to 0.28 × 6 without doing any further multiplication.

In-depth Investigation
Use the Group Activity from Day 2.

Extra Support

Adapt Day 2’s Group Activity to practise multiplications of the form: ☐ x 0. ☐, then ☐ x ☐.☐, e.g. 3 x 0.6; 4 x 3.5.

Unit 4 Percentages and fractions of amounts (suggested as 4 days)

Objectives

Recognise and use equivalent fractions; find percentages and fractions of amounts
Unit 4: ID# 6661

National Curriculum
Fr (i) (vi)
Rat/Pr (ii)

Hamilton Objectives
21. Use common multiples to generate equivalent fractions.
22. Use knowledge of equivalence to compare/order fractions.
24. Associate a fraction with division. Calculate decimal fraction equivalents, e.g. 4/5 is 0.8, 1/8 is 0.125.
33. Calculate simple percentages of whole numbers. Solve problems involving use of percentages for comparisons.

Teaching and Group Activities for Understanding

Day 1 Teaching
Which fractions with denominators less than 15 can be written as 1/15s? Take feedback (1/3s and 1/5s). Show how 1/3 is 5/15 and 1/5 is 3/15. Write 2/3 and 3/5 and write these as 1/15s to compare them. Ask children to use equivalence to compare 1/2 and 3/5 then 7/10 and 3/4.
Group Activities
-- Explore fractions with an equivalent number of 1/12s.

Day 2 Teaching
Use a 1–100 bead bar. Say that each bead is one hundredth of the bar. Write 1/100. Say it is also 1% of the bar. Per cent means out of 100. Show 5 beads and write this as 5/100 or 0.05 or 5%. Show that these all mean the same thing. Repeat for other beads then discuss equivalents for 1/3 and 2/3.
Group Activities
Use the in-depth problem-solving investigation ‘Doughnut Percents’ from NRICH as today’s group activity.
Or, use this activity:
-- Use loop cards (dominoes) to explore fraction, decimal and percentage equivalents. Use online activities to consolidate.

Day 3 Teaching
Display a percentages fact web. 50% is equivalent to 1/2 so we can halve £320. Knowing 50%, we can work out 25%. How can we find 10%? It is the same as finding 1/10. Children find 10% and 1% (1/100), then discuss how to find 11%, 20%, 21% and 32%.
Group Activities
-- Explore percentages of an amount of money (£150).
-- Explore percentages of an amount of money (£328).

Day 4 Teaching
Write 1/6 of 150; ask children to discuss how to do it. Draw out dividing by 6. Children do it. Take feedback on methods, e.g. using a written method, or dividing by 3, then halving. What is 5/6 of 150? Draw out multiplying 25 by 5, or subtracting 25 from 150. Find fractions of 150 that give whole-number answers. Find unit and non-unit fractions of 150, e.g. 1/5 then 3/5, 1/15 then 7/15.
Group Activities
-- Investigate unit and non-unit fractions of 240.

Mental/Oral Starters

Day 1
Write improper fractions as mixed numbers (pre-requisite skills)

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

Day 3
Equivalent fractions and percentages (pre-requisite skills)

Day 4
Find unit fractions of amounts within tables (pre-requisite skills)

Procedural Fluency

Day 1

Write fractions with unrelated denominators as 1/10s, 1/6s or 1/12s, then compare.

Day 2
Fill in the missing equivalent fractions, decimals and percentages on a line.

Day 3
Use equivalence with fractions to find percentages of given amounts of money.

Day 4
Use, then draw bar models to find unit and related non-unit fractions of amounts.
Find non-unit fractions of amounts, then answer word problems.

Mastery: Reasoning and Problem-Solving

  • Write three fractions which are equivalent to 3/4.
    Write three fractions which are equivalent to 2/5.
    Now add 3/4 and 2/5.
  • True or false?
    2/5 is the same as 20%.
    0.4 is the same as 4%.
    10% is the same as 0.1.
    30% is the same as 1/3.
  • Find 10% of these prices. Use that to find 20%.
    (a) £14.20
    (b) £1.50
    (c) £99
  • Show that one fifth of 320 is 3 less than one third of 201.

In-depth Investigation: Doughnut Percents
A task involving the equivalence between fractions, percentages and decimals. Members of the group must notice the needs of others and respond. Doughnut Percents from nrich.maths.org.

Extra Support

Special People
Understanding what a percentage is

Fraction Frenzy
Finding fractions of amounts (1/4s, 1/3s, 1/5s, 1/10s)

Unit 5 Multiply and divide fractions (suggested as 3 days)

Objectives

Multiply and divide fractions
Unit 5: ID# 6673

National Curriculum
Fr (iv) (v)

Hamilton Objectives
25. Understand that if two numbers less than 1 are multiplied, the answer is smaller than either of them.
26. Multiply simple pairs of proper fractions, writing the answer in its simplest form.
27. Divide proper fractions by whole numbers, recognising that 3⁄4 ÷ by 2 is equivalent to 3⁄4 × 1⁄2.

Teaching and Group Activities for Understanding

Day 1 Teaching
Sketch a garden divided into quarters on the board. Say that one quarter is for growing fruit. In that part, half of it is for strawberries. Show children how to find 1/2 of 1/4. Demonstrate that this is 1/8. Use other examples to derive a rule for multiplying fractions.
Group Activities
-- Derive a rule for multiplying pairs of fractions, using supporting images.

Day 2 Teaching
Show 3 pizzas. Two children choose a pizza, e.g. the pepperoni, with 3/4 of it left. Draw line to cut remaining pizza in half. What fraction will each child get? Record 3/4 ÷ 2. Repeat for the other pizzas. Repeat with 3 children sharing what is left, then 4.
Group activities
-- Divide fractions by whole numbers using supporting images.

Day 3 Teaching
Write 1/2 × 3/4 and ask the children to remind you how they can do this. We can multiply the numerators then multiply denominators. Demonstrate, finding 3/8 as the answer. Write 2/3 ÷ 3. Draw a cake divided into thirds. Shade 2/3 and then divide each of the shaded thirds into three parts. So, 2/3 ÷ 3 is the same as 1/3 × 2/3. Show that we get 2/9.
Group Activities
Use the in-depth problem-solving investigation ‘Mixed up fractions’ as today’s group activity.
Or, use this activity:
-- Investigate sequences of fraction calculations and a general statement.

You Will Need

  • Mini-whiteboards and pens
  • Multiplying fractions (see resources)
  • Finding 1/4s of 1/5s (see resources)
  • Pictures of pizzas to display (see resources)
  • Flipchart and pens
  • Additional activity sheets (see resources)

Mental/Oral Starters

Day 1
Find 1/4 of numbers (pre-requisite skills)

Suggested for Day 2
Find greatest common factors (simmering skills)

Suggested for Day 3
Find lowest common multiples (simmering skills)

Procedural Fluency

Day 1
Find unit fractions of 1/2, 1/3 and 1/4, using an image. Use an image to begin with to multiply unit, then non-unit, fractions; then use the short strategy to multiply numerators and denominators.

Day 2
Divide fractions by whole numbers using a supporting image.

Day 3
Multiply pairs of fractions and divide fractions by whole numbers.

Mastery: Reasoning and Problem-Solving

  • Find one quarter of
    (a) 1/3
    (b) 2/5
    (c) 3/8
  • A large cake is divided into ten equal pieces. One piece is split into thirds. What fraction is each piece? Another piece is split into quarters. What fraction is each piece?
  • We divide a quarter of a pie into five pieces. What fraction is each piece?
  • Fill in the missing numbers:
    1/4 of ☐ is 1/12
    ☐ of 1/2 is 1/6
    1/5 of 2/3 is ☐

In-depth Investigation: Mixed up fractions
Children add a pair of fractions, multiply the same pair, then find the difference between the two answers, looking for patterns.

Extra Support

Folding Fractions
Finding half of fractions using supporting paper-folding images

Unit 6 Ratios, proportion and percentages (suggested as 3 days)

Objectives

Describe ratios between unequal quantities; link percentages to proportion
Unit 6: ID# 6679

National Curriculum
Rat/Pr (i) (ii)

Hamilton Objectives
23. Identify simple fraction/decimal/percentage equivalents: E.g. 1⁄4 = 0.25 = 25%, 1/3 = 0.33 = 33%.
33. Calculate simple percentages of whole numbers; solve problems involving use of percentages for comparisons.
35. Solve problems involving simple ratios, using tables facts and knowledge of fractions and multiples, e.g. 2 eggs for every 250g of flour.

Teaching and Group Activities for Understanding

Day 1 Teaching
Display a recipe for chocolate chip cookies. Use this to show that if we want to increase the quantity made, we need to increase the quantities in the recipe but keep the ratios the same. Demonstrate how to do this.
Group Activities
-- Use a number of teaspoons of red and yellow paints to mix a larger quantity of the same shade of orange.
-- Use a number of teaspoons of red and yellow paints to mix a larger quantity of the same shade of orange; Paint a colour scale by mixing two paint colours in gradually- changing ratios.

Day 2 Teaching
Show a stick made of unequal numbers of red and blue cubes. Establish the ratio between the two colours and use this to create different length sticks with the colours in the same ratio. Use notation to describe ratio, e.g. 2 red for every 1 blue, 2:1. Ask children to say what fraction of a stick of cubes is a particular colour.
Group Activities
-- Explore ratio by shading 4 by 3 rectangles in two colours. Deduce how many squares are in each colour given ratio clues.
-- Use bar models to help solve ratio word problems.

Day 3 Teaching
Write ‘swimming’ and ‘cycling’ on large cards. Ask 10 children to stand by their favourite. What fraction prefer swimming? What is this as a percentage? Say that we can use fractions or percentages to describe a proportion. Draw a bar model diagram to represent this. Extend to other similar problems, e.g. 70% of 60 children prefer swimming, 75% of 40 children prefer swimming, etc.
Group Activities
Use the in-depth problem-solving investigation ‘Calculating Colours’ from The Royal Institution as today’s group activity.
Or, use these activities:
-- Write equivalent percentages for fractions; then calculate percentages and fractions of amounts.
-- Convert different dance competition scores to percentages in order to compare them.

You Will Need

  • Red and yellow paint, paper, brushes, mixing palettes, teaspoons
  • Red and blue multilink cubes
  • Mini-whiteboards and pens
  • Squared paper and coloured pencils
  • Two large cards
  • Ratio word problems (see resources)
  • Equivalent fractions and percentages (see resources)
  • Dance competition scores (see resources)
  • ITP: Spinners

Mental/Oral Starters

Day 1
Equivalent fractions, decimals and percentages (pre-requisite skills)

Suggested for Day 2
Guardian of the rule (simmering skills)

Suggested for Day 3
Multiply 3 numbers together (simmering skills)

Procedural Fluency

Day 1
Adapt a pizza recipe to make different numbers of pizzas.

Day 2
Colour in squares within rectangles to given ratios.

Day 3
Write equivalent percentages for fractions and work out how many children made each choice. Solve ratio word problems.

Mastery: Reasoning and Problem-Solving

  • Complete the bar models:
    Diagram 1
32 Children
1/4 ? Children3/4 ? Children

Diagram 2

40 Children
40% ? Children60% ? Children
  • Orange paint is mixed using this ratio of red and yellow paints: red : yellow
    2 : 7
    Sam uses 4 litres of red. Assuming he uses the correct amount of yellow, how many litres of orange paint will he make?
  • If 6 children in a class do not like sport, and there are 30 children in the class, what proportion do like sport? Give your answer as a fraction and as a percentage.


In-depth Investigation: Calculating Colours
Children will gain practical experience of combining coloured solutions in varying proportions then describe solutions in terms of fractional (and percentage) composition. Calculating Colours from The Royal Institution.

Extra Support

The Key to Percentages
Finding simple percentages of amounts (1%, 10%, 25% and 50%)

Car Wash
Exploring dividing money in 3:1 and 3:2 ratios