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# Maths Year 5 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. These bulk downloads are available to friends and School Subscribers. These bulk downloads are added value for Hamilton Friends and School Subscribers.

## Unit 1 Place value in decimals; rounding (suggested as 3 days)

### Objectives

Place value in decimals; multiply/divide by 10 and 100; rounding
Unit 1: ID# 5413

National Curriculum
Dec/Fr (vii) (viii) (ix)

Hamilton Objectives
19. Understand the effect of multiplying or dividing by 10, 100, 1000, including 1- and 2-place decimal answers.
29. Write decimal numbers as tenths and hundredths, e.g. 0.71 as 71/100.
30. Locate 2-place decimal numbers on a line; round them to the nearest tenth or whole number.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Revise place value, quizzing children about the digits in 4.03. Find 4.03 + 0.7. Rehearse writing 2-place decimal numbers. Use place value additions to emphasise the value of each digit, e.g. start with 4.56, add one hundredth, then add two tenths, subtract one whole, subtract four hundredths, subtract four tenths. Ask children to show you the new number each time.
Group Activities
Use the in-depth problem-solving investigation ‘Spiralling Decimals’ from NRICH as today’s group activity.
Or, use these activities:
-- Use place value to add and subtract numbers with two decimal places, one calculation at a time or in longer chains.

Day 2 Teaching
Use the place value grid on the ITP Moving digits to show the effect of multiplying and dividing by 10, 100 and 1000. The digits move to the left when multiplying and to the right when dividing. Emphasise that the decimal point does not move.
Group Activities
-- Explore decimal place value and number order when multiplying and dividing by 10, 100 and 1000.

Day 3 Teaching
Show children a counting stick. Say that one end represents 2; the other 3. Use this to demonstrate that we can round 1-place decimals to the nearest whole number, e.g. we round 2.4 to 2 and 2.7 to 3. Write some 2-place decimals between 2.3 and 2.4 and round these to the nearest tenth, then to the nearest whole number. Probe understanding of rounding: Write a number less than 6 with two decimal places that rounds up to 6, etc.
Group Activities
-- Explore rounding 2-place decimals marked on number lines, or while playing Bingo.

### You Will Need

• Place value grid 10–0.01 (see resources)
• Number cards 0–9
• Flipchart
• Whiteboards and pens
• A4 paper for each child
• ITP: Moving digits
• 3-digit number cards (see resources)
• Place value stars image (see resources)
• Place value dice
• × and ÷ starting numbers (see resources)
• Counting stick

### Mental/Oral Maths Starters

Day 1
1-place decimals (pre-requisite skills)

Suggested for Day 2
Adding to the next whole number from a 1-place decimal number (simmering skills)

Suggested for Day 3
Difference between negative numbers (simmering skills)

### Procedural Fluency

Day 1
Place value additions and subtraction; add and subtract 0.1 and 0.01 and multiples of 0.1 and 0.01.

Day 2
Multiply and divide by 10 and 100.
Multiply and divide by 10, 100 and 1000.

Day 3
Round decimals to the nearest whole number.
Round decimals to the nearest tenth and whole number.

### Mastery: Reasoning and Problem-Solving

• Divide 47,310 by 10 repeatedly until you get a number that is less than 100. Write that number.
• Write the next two numbers in each sequence.
0.41 4.1 ____ ____
2.05 20.5 ____ ____
43020 4302 ____ ____
• True or false?
4030 ÷ 100 = 43
1.09 × 100 = 190
0.09 × 10 = 0.9
7000 ÷ 1000 = 0.7
• Rounding to nearest tenth
Tick (a) or (b)
(a) 20.07 → 20
(b) 20.07 → 20.1
Rounding to nearest whole number
(a) 20.09 → 21
(b) 20.09 → 20

In-depth investigation: Spiralling Decimals
Take turns to place a decimal number on a spiral number line. Spiralling Decimals from nrich.maths.org.

### Extra Support

Left or Right?
Multiplying numbers with one decimal place by 10 and dividing 2-digit whole numbers by 10

Left, Left or Right, Right?
Understanding place value in numbers with two decimal places;
Beginning to multiply numbers with two decimal places by 100 and divide 3-digit numbers by 100

## Unit 2 Column addition; 2-place decimals (suggested as 2 days)

### Objectives

Unit 2: ID# 5419

National Curriculum
Dec/Fr (ix) (x)

Hamilton Objectives
29. Write decimal numbers as tenths and hundredths.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Write 4.56 + 2.37 on the board. Ask children to round each number to the nearest whole and add to estimate the total. Demonstrate, using expanded and compact addition, carefully talking through adding 0.07 and 0.06. Ask children to use expanded addition or compact addition to work out other calculations.
Group Activities
Use the ‘Four of the best’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Use expanded and compact layouts to add pairs of numbers with one, then two decimal places.
-- Identify missing digits in additions of 3-digit numbers with two decimal places.

Day 2 Teaching
Show the table of shot put results (see resources). Explain that the total of the shot put throws was used to find the winner. Children estimate who won, and then add two and three decimal numbers to find out.
Group Activities
-- Explore and investigate column addition of 2-place decimals.

### You Will Need

• Whiteboards and pens
• ‘Adding decimals’ sheet (see resources)
• Sticky notes
• ‘Shot put results’ (see resources)
• Number cards 1–9

### Mental/Oral Maths Starters

Day 1
Place numbers with two decimal places on a line (pre-requisite skills)

Suggested for Day 2
24-hour clock (simmering skills)

### Procedural Fluency

Day 1
Written addition of numbers with one then two decimal places.

Day 2
Add two lengths with two decimal places.
Add three lengths with two decimal places.

### Mastery: Reasoning and Problem-Solving

• Sometimes/ Always/ Never… ‘If you add two 2-place decimal numbers, the answer also has two decimal places.’
• Write the missing digits:
4.9☐ + ☐.78 = 8.☐6
• Ali’s homework might need correcting…
Correct any wrong answers and explain what he did wrong.
a) 2.75 + 5.95 = 7.60
b) 3.4.2 + 5.57 = 8.99
c) 4.83 + 1.93 = 5.76
d) 2.47 + 68.5 = 93.2

In-depth Investigation: Four of the best
Children use an incomplete magic square to explore patterns in the addition of four decimal numbers.

### Extra Support

Decimals on a 0–3 Number Line
Mark decimals and fractions on a 0-3 number line and practise counting forwards and backwards in tenths.

## Unit 3 Subtract decimal numbers, e.g. money (suggested as 3 days)

### Objectives

Subtract decimal numbers including money
Unit 3: ID# 5431

National Curriculum
Dec/Fr (vii) (x)

Hamilton Objectives
32. Subtract 1- and 2-place decimal numbers by counting up: 6.2 – 3.5, 13.1 – 9.45.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Write 98 – 21, 46 – 19, 58 – 32, 74 – 6 and 63 – 57 on a flipchart. How do we solve 98 – 21? Subtract 20, then 1. So how do we solve 9.8 – 2.1? Through discussion, establish that we first − 2, then − 0.1. Model how we can use closely-related counting back strategies to solve the other decimal calculations: 4.6 – 1.9, 5.8 – 3.2, 7.4 – 0.6. Note that Frog is sometimes best, e.g. 6.3 – 5.7.
Group Activities
-- Choose and use an appropriate and efficient method for subtraction.

Day 2 Teaching
Display priced items on the board (see resources). Discuss how much change we would have from £100 to buy a chosen item. Model counting up, using Frog, being sure to distinguish a hop which is pence (to the next pound) and a hop which is pounds. Choose another item and repeat. Calculate a total of two items, then the change from £100.
Group Activities
-- Find money totals and change from £100 (or £50).

Day 3 Teaching
Draw a diagram on the board showing differences between amounts. Use Frog to find which pairs of amounts have which differences. Ensure, when counting up, that children are clear that Frog hops to the next pound first, and this hop is in pence.
Group Activities
Use the ‘Pence and pounds reversed’ in-depth problem-solving investigation below as today’s group activity.
Or, use this activity:
-- Explore money differences using Frog.

### You Will Need

• Eight blank cards
• Two large pieces of paper
• 0–9 dice
• ‘Room makeover’ (see resources)
• Number cards 4–9
• Loose pieces of paper

### Mental/Oral Maths Starters

Day 1
Add to the next whole number from a 1-place decimal number (pre-requisite skills)

Suggested for Day 2
Mental division, answer as fractions (simmering skills)

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

### Procedural Fluency

Day 1
Choose whether to count back or count up (Frog) to work out answers to subtractions of numbers with one decimal place.

Day 2
Find the change from £50 and £100, then find totals and the change from £100.

Day 3
Estimate, then find the difference between amounts of money.

### Mastery: Reasoning and Problem-Solving

• Kyla received £7.29 change from a £20 note. How much did she spend?
• Write the missing number in each of the bar models:
Diagram 1

 6.4 2.7 ?

Diagram 2

 9.2 ? 3.5
• Has the shopkeeper given the correct change?
1. Item bought: £13.45
Paid with: £50
Change given: £36.55
2. Item bought: £64.79
Paid with: £100
Change given: £45.31
• Write the missing digits:
£75 – £24.☐6 = £☐.14
• Write two amounts with a difference of £34.76.
The larger amount has 23p.

In-depth investigation: Pence and Pounds Reversed
Children find patterns in the differences when pounds and pence are reversed.

### Extra Support

Change Challenge
Using counting up (Frog) to calculate change from £50

## Unit 4 x/÷ by 10, 100, 1000; rounding decimals (suggested as 2 days)

### Objectives

Multiply/divide by 10, 100, 1000; rounding decimal numbers
Unit 4: ID# 5437

National Curriculum
Dec/Fr (vii) (viii) (ix)

Hamilton Objectives
19. Understand the effect of multiplying/dividing by 10, 100, 1000, including 1- and 2-place decimal answers.
29. Write decimal numbers as tenths, hundredths, thousandths, e.g. 0.71 as 71/100, 0.327 as 327/1000; relate thousandths to tenths and hundredths.
30. Locate 2-place decimal numbers on a line; round them to the nearest tenth or whole number.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Use an IWB calculator to show the effect of multiplying and dividing by 10 and 100. Use place-value cards (arrow cards) or numbers on the ITP Moving digits to show what happens to each digit when you multiply and divide by 10, 100 or 1000.
Group Activities
-- Investigate routes through a grid (× and ÷ by 10, 100 or 1000) with a range of outcomes.

Day 2 Teaching
Display a 3–5 line marked in tenths (see resources). Mark 3.45. Agree how to round this to the nearest tenth. Then show how to round it to nearest whole number. Ask other children to mark numbers with two decimal places on the line. Children round them to the nearest whole, then tenth.
Group Activities
-- Play games to explore positioning and rounding of 2-place decimals.

### You Will Need

• ITP: Moving digits
• Whiteboard and pens
• IWB calculator
• Multiplying and dividing by 10 and 100 grid (see resources)
• Place value grid (see resources)
• Number line 3–5 marked in tenths (see resources)
• Number line 0–1 marked in tenths (see resources)
• Small pieces of paper and envelopes
• 0–9 dice

### Mental/Oral Maths Starters

Day 1
Place value in numbers with two decimal places (pre-requisite skills)
Suggested for Day 2
Place 6-digit numbers on a number line (simmering skills)
Or
Decimal place value addition and subtraction (simmering skills)

### Procedural Fluency

Day 1
Multiply and divide by 10, 100 and 1000. Answers have up to two decimal places and up to five digits.

Day 2
Place numbers with two decimal places on a 0–1 line.
Round them to the nearest tenth and whole.

### Mastery: Reasoning and Problem-Solving

• Fill the empty boxes:
0.15 = 1.5 ☐ 10
5209 = ☐ × 100
☐ ÷ 100 = 4.7
10.08 = ☐ ÷ 1000
• Think of a number that gains a digit when you divide it by 100.
• True or False?
When you divide a 4-digit number by 100, you end up with a 2-digit number.
• What number am I?
A
(i) I round to 5.6 as the nearest tenth.
(ii) I round to 6 as the nearest whole number.
(iii) My digital root is 6.
B
(i) I round to 3.5 as the nearest tenth.
(ii) I round to 3 as the nearest whole number.
(iii) My digits are consecutive.
• Sam wrote:
3.49 rounds to 4 as the nearest whole number because 3.49 rounds to 3.5 as the nearest tenth, and 3.5 rounds up to 4 as the nearest whole number.
Explain why his reasoning is incorrect.

This unit has no separate in-depth investigation.

### Extra Support

Mark and Round
Placing numbers with one decimal place on a number line; Rounding numbers with one decimal place to the nearest whole number.

## Unit 5 Subtraction with decimals, e.g. money (suggested as 3 days)

### Objectives

Subtraction with decimals including money
Unit 5: ID# 5443

National Curriculum
Dec/Fr (vii) (x)

Hamilton Objectives
32. Subtract 1- and 2-place decimal numbers by counting up: 6.2 – 3.5, 13.1 – 9.45.
33. Solve problems involving decimals using known equivalences to help.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Use javelin throws as a context to show how we can count up using Frog from 9.56m to 11.32m. Find other differences in throws measured in metres using Frog to count up. Ensure children are familiar with recording on an empty number line.
Group Activities
-- Explore differences between height and stride length in younger children.
-- Investigate pairs of lengths with given differences.

Day 2 Teaching
Display a table of distances, in which the lengths can include one or two decimal places, e.g. 5.3m and 4.56m. Use Frog (counting up) to subtract pairs of numbers with different numbers of decimal places. Ensure children are comfortable with the empty number line used in this way.
Group Activities
Use the ‘Decimal differences’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Calculate height differences between 2m and 4m.
-- Investigate missing number subtractions.

Day 3 Teaching
Show children a table of prices from two bike shops. Use this to construct problems involving finding differences between amounts. Use counting up (Frog) to solve subtraction word problems.
Group Activities
-- Solve subtraction word problems, choosing an efficient strategy.
-- Rank subtractions in order of difficulty; choose a strategy to solve them.

### You Will Need

• Tape measures
• Paper clips
• Whiteboards and pens
• Flipchart
• Number cards 1–9
• ‘Word problems’ sheet (see resources)

### Mental/Oral Maths Starters

Day 1
Pairs with a total of 1 metre (pre-requisite skills)

Day 2
Subtracting numbers with one decimal place (pre-requisite skills)

Suggested for Day 3

### Procedural Fluency

Day 1
Use Frog to find the difference in tree heights.

Day 2
Find daily heights grown by a bamboo plant.
Find differences between numbers with different numbers of decimal places.

Day 3
Work in pairs to discuss, then solve, single-step and two-step word problems. These all involve subtraction, and some include another operation.

### Mastery: Reasoning and Problem-Solving

• Write the missing length in each bar model:
Diagram 1
 14.8m 7.89m

Diagram 2

 5.25m 3.58m

Diagram 3

 2.3m 1.09m
• Sunil and Zoe were meeting. They cycled 25km between them.
Zoe cycled 11.47km. How far did Sunil cycle?
• Find the differences between:
4.5 and 6.54
3.4 and 5.43
5.6 and 7.65
What’s the same? What’s different?
• If the change from £100 was as follows, how much was the total cost of the two pairs of trainers?
Pair A: change = £34.61
Pair B: change = £28.75

In-depth Investigation: Decimal Differences
Children subtract pairs of numbers with consecutive digits and different numbers of decimal places, and look for patterns in their answers.

### Extra Support

Frog's Teeny Hops
Finding the difference between decimal numbers on either side of a whole number, e.g. 1.8 and 2.3.

## Unit 6 Unit and non-unit fraction problems (suggested as 4 days)

### Objectives

Fraction problems, including unit and non-unit fractions of amounts
Unit 6: ID# 5449

National Curriculum
Dec/Fr (i) (ii)
Mult/Div (xi)

Hamilton Objectives
21. Solve problems involving multiplication and division, using knowledge of factors and multiples, as appropriate.
23. Identify, name and write equivalent fractions; reduce fractions to simplest form.
24. Compare and order fractions where the denominators are multiples of the same number.
33. Solve problems involving fractions.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Display a fraction wall image. Divide the class into three groups. One group writes fractions equivalent to 1/4; one writes fractions equivalent to 1/3; and one writes fractions equivalent to 1/5. Use these to compare different fractions with related denominators, e.g. 2/3 and 7/9, 7/12 and 3/4, 2/5 and 3/10.
Group Activities
Use the ‘Big triangle of fractions’ in-depth problem-solving investigation below as today’s group activity.
Or, use this activity:
-- Order pairs or trios of fractions with related denominators, using equivalence.

Day 2 Teaching
Can we split 148 children equally into three teams? Draw a bar model to represent the problem. Use a mental division strategy with jottings (vertical chunking) to find the unit fraction of this amount: 1/3 of 148. What about five teams? Six teams? Seven teams? Eight teams? Nine teams? Four teams?
Group Activities
-- Investigate divisibility, finding unit fractions of amounts.

Day 3 Teaching
Write 5/6 of 132 on the board. Talk to your partner about how we can work this out. Sketch a bar model to show 5/6 of 132. Take feedback and, together, make a class poster to agree the steps in finding non-unit fractions of amounts.
Group Activities
-- Use unit fractions as a building block to calculate non-unit fractions of amounts.
-- Create non-unit fractions problems for others to solve.

Day 4 Teaching
Display three word problems (see teaching). Ask children to read each in turn and discuss what calculation is needed to solve it. Show how sketching a bar model can sometimes help us to sort out what needs to be done. Solve together.
Group Activities
-- Create and/ or solve word problems involving finding fractions, and using multiplication and division to solve.

### You Will Need

• Fraction wall (see resources)
• Fractions cards (1) (see resources)
• Fractions cards (2) (see resources)
• Large sheets of paper
• Find fractions of amounts (see resources)
• Solving word problems (see resources)
• Flipchart and pens
• Whiteboards and pens
• A4 paper

### Mental/Oral Maths Starters

Day 1
Find unit fractions of amounts - within tables (pre-requisite skills)
'
Fractions of amounts bingo' from http://topmarks.co.uk

Day 2
Division facts (pre-requisite skills)

Day 3
Find unit fractions of amounts (pre-requisite skills)

Suggested for Day 4
Mental division with remainders (simmering skills)

### Procedural Fluency

Day 1
Use equivalence to compare and order pairs or trios of fractions.

Day 2
Find unit fractions of amounts (3-digit numbers).

Day 3
Use bar models to help find unit fractions and then non-unit fractions of amounts.

Day 4
Find fractions, multiply and divide to solve word problems.

### Mastery: Reasoning and Problem-Solving

• Write three fractions equivalent to 3/5.
Look at the pattern in the denominators.
Then write three fractions equivalent to 2/3 and do the same. What can you predict about the pattern in the denominators of fractions equivalent to 5/6?
• Write the missing numbers to make each sentence true.
?/6 > 7/12
?/6 = 5/?
4/? < 5/?
• Draw a bar model to represent each problem. Then find the answer.
(i) 1/3 of 153
(ii) 1/6 of 612
(iii) 7/12 of 144
• Find 3/5 of each of:
(a) 105 (b) 205 (c) 305
Use the pattern to predict the answer to 3/5 of 405.

In-depth investigation: Big triangle of fractions
Children add fractions with related denominators and find equivalent fractions to identify patterns.

### Extra Support

Fraction Facts
Finding simple unit and related non-unit fractions of amounts (1/3s, 1/4s and 1/5s).

## Unit 7 Multiply fractions; decimal equivalences (suggested as 3 days)

### Objectives

Decimal/fraction equivalences; multiplying fractions by whole numbers
Unit 7: ID# 5471

National Curriculum
Dec/Fr (ii) (iii) (v) (vi) (xii)

Hamilton Objectives
21. Solve problems involving multiplication and division, using knowledge of factors and multiples, as appropriate.
27. Multiply proper fractions and mixed numbers by whole numbers supported by materials and diagrams.
28. Identify simple fraction and decimal equivalents.
33. Solve problems involving fractions.

### Teaching and Group Activities for Understanding

Day 1 Teaching
Display a blank 100-square (see resources). Use this to show that one little square is 0.01, which is ≡ 1/100. Shade one row to show that to 0.1 ≡ 10/100 ≡ 1/10. Use shaded parts of the 100-square to show decimal equivalents for halves, quarters and fifths.
Group Activities
-- Create a 0–1 number line annotated with halves, quarters, fifths and tenths.
-- Mark a 0–1 number line with halves, quarters, fifths and tenths; use it to help order fractions and decimals.

Day 2 Teaching
Show a 0–3 line marked in quarters (see resources). Write 6 × 1/4. Draw six hops of 1/4. Six lots of one quarter is six quarters! How else can we write this? Agree that we can write this as 1²/4, or 1¹/2.
Repeat for 9 × 1/4, 6 × 1/8 and 10 × 1/8.
Group Activities
-- Explore multiplication of unit fractions by whole numbers.
-- Explore multiplication of non-unit fractions by whole numbers.

Day 3 Teaching
Sketch two identical pizzas. Two children are eating pizza. They are copycats, if one eats 3/8 of a pizza, so does the other! Divide each pizza into eighths, and cross out 3/8 of each pizza. What is double 3/8? 6/8! Simplify this. Repeat the process to show 2 × 5/8. Derive the ‘rule’ for multiplying fractions by a whole number. 2 × 2/5 = 4/10. What has the person done wrong?
Group Activities
Use the ‘Fraction frenzy’ in-depth problem-solving investigation below as today’s group activity.
Or, use this activity:
-- Investigate patterns when multiplying non-unit fractions by whole numbers.

### You Will Need

• Blank 100-square (see resources)
• ITP: Fractions
• >40cm strips of paper
• Ordering fractions (see resources)
• 0–3 line marked in quarters (see resources)
• 0–2 line marked in eighths (see resources)
• Sticky notes
• 1–9 dice
• Additional activity sheets (see resources)

### Mental/Oral Maths Starters

Suggested for Day 1

Suggested for Day 2
Find the lowest common multiple (simmering skills)

Suggested for Day 3
Units of time (simmering skills)

### Procedural Fluency

Day 1
Identify equivalent fractions and decimals using a shaded 100-square.

Day 2
Use number lines to help multiply 1/3, 1/4 and 1/5 by whole numbers, writing answers as mixed numbers if greater than 1.
Multiply unit fractions by whole numbers, simplifying answers where possible. Predict where answers will be greater than 1.

Day 3
Multiply non-unit fractions by 2, 3, 4 and 5.

### Mastery: Reasoning and Problem-Solving

• Write the missing decimal numbers to make each sentence true. Some can have more than one answer!
☐ = 4¹/2
☐ = 5³/4
13/100 > ☐
7/10 < ☐
☐ = 2/5
• Draw a 0–5 number line. Mark quarters: 1/4, 1/2, 3/4, 1¹/4, etc.
Draw the hops to show seven lots of 1/4.
Draw the hops to show five lots of 3/4.
• Five children each eat 4/5 of their lunches. How many lunches did they eat altogether?
• True or false?
6 × 3/4 = 4¹/2
12 × 1/5 = 25
9 × 2/3 = 6
7 × 1/4 < 2

In-depth investigation: Fraction frenzy
Children multiply proper fractions by whole numbers in a multiplication grid and look for patterns.

### Extra Support

Decimal Pictures
Finding equivalent fractions and decimals (halves, quarters and tenths).

Tiny Steps
Counting in steps of 1/2, 1/3, and 1/4.