Counting by Fractions: a week of ready-to-teach maths
Five days of lessons for Year 4 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus cut-out fraction number lines and mixed number cards, a mini-check and every answer.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Cut out the two card sheets once; the number lines and cards are reused all week.
- Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.
No maths background needed
This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week is short: a fraction is a number with a place on the number line, found by counting equal steps from 0. Keep counting past one whole and the count reaches mixed numerals: five quarter-steps land on 5/4, which is the same point as 1 1/4.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.
How we write and say them
A mixed numeral is a whole number and a fraction side by side: 2 1/4, read “two and one quarter”. A fraction bigger than one, such as 9/4, is read “nine quarters”. Both name the same point on the line, and this pack asks for both.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Children learn and do | On screen |
|---|---|---|---|
| 1 | Step along in fractions | Count on and back in halves, thirds and quarters along a number line, past one whole and up to two | Counting by fractions |
| 2 | Find the fraction on the line | Read the fraction at a marked point by counting steps from 0, and mark fractions on lines | Locating a fraction |
| 3 | Wholes and a bit more | Read and draw mixed numerals such as 2 1/4, and count how many parts they hold altogether | Mixed numbers |
| 4 | Group the pieces into wholes | Turn fractions bigger than one into mixed numerals by grouping pieces into wholes, and back again | Improper to mixed |
| 5 | Mixed numbers on the line | Place mixed numerals between two whole numbers, name each point two ways, and order them | Placing mixed numbers; Counts, mixed and improper |
How the week builds
Day 1 counts in fraction steps; Day 2 reads and marks those steps on a line; Day 3 names amounts past one as mixed numerals; Day 4 moves between mixed numerals and fractions bigger than one; and Day 5 puts mixed numerals on the line and orders them. It builds on Equivalent Fractions, and adding and subtracting fractions in later years rest on it.
Materials for the week (one trip)
- From the classroom: scissors, pencils, coloured pencils, this pack printed.
- For Day 1: a long strip of masking tape on the floor marked 0, 1 and 2, for a walking number line (optional).
- Cut out once, use all week: the three fraction number lines and the mixed number cards in this pack. No maths equipment to buy.
Words for the word wall
fraction, number line, step, count on, count back, half, third, quarter, whole, mixed numeral, improper fraction, between, order
Dear families
This week in maths, Year 4 counts by fractions: in halves, thirds and quarters, along a number line and past one whole. We meet mixed numerals such as 2 1/4 (two and a quarter) and learn that 9/4 names the same amount.
Try this at home
- Count in halves while you climb stairs: a half, one, one and a half, two.
- Cut sandwiches or fruit into quarters. How many quarters make two whole ones? What about nine quarters?
- Spot mixed numbers in recipes, such as 1 1/2 cups, or in times such as 2 1/2 hours.
- Ask which is more, 1 3/4 or 2 1/4, and where each one sits between the whole numbers.
My fraction counts this week
Fill one row a day. Choose a step size and count on five steps from where you start.
| Day | Step size | I started at | My five counts |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Counting by Fractions teaching pack · seegongsik.com/au/y4/number/AC9M4N04/pack
Step along in fractions
Children count on and back in halves, thirds and quarters along a number line. Each step is the same size, so four quarter-steps reach one whole, and the count keeps going past it: five quarters is one and one quarter.
We are learning to
- count on and back by halves, thirds and quarters from any starting point,
- say when a fraction count reaches a whole number (4/4 = 1, 3/3 = 1),
- keep counting past one whole using mixed numerals (1 1/4, 1 1/2).
Success criteria
- I can count by a fraction along a number line, on and back.
- I can say where the count crosses a whole number.
You need
The three fraction number lines (cut-out sheet 1), one set per pair. A floor line marked 0, 1 and 2 (optional). The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Walk the line A child walks the floor line from 0 in quarter steps while the class counts: one quarter, two quarters, three quarters, four quarters. Stop at 1. Ask: “Four quarters brought us to 1. What will the next step be called?” |
| 30 min | Count on the line Show the board (see the box). Pairs put a finger on the quarters line and count to 2, writing each count under its mark (Part A), then do Part B. Ask: “On the thirds line, which count lands on 1? Why three steps, not four?” |
| 10 min | Count back Count back together from 2 in quarters, then children finish Parts C to E. Ask: “Counting back from 1 in quarters, what comes just before 1?” |
Two half-sessions instead? Split after Part A; start Session B by counting in thirds on the floor line.
Watch for these ideas
- Counting 1/4, 2/4, 3/4, 4/4, 5/5. The step size stays a quarter: the next count is 5/4.
- Stopping at one whole, as if fractions end at 1. The line, and the count, keep going.
- Counting the marks, not the steps, so 0 is counted as the first quarter.
Answers
- Part A: 1/4, 2/4 (1/2), 3/4, 1, 1 1/4, 1 2/4 (1 1/2), 1 3/4, 2.
- Part B: 1, 1 1/3, 1 2/3. Part C: 1 1/2, 2, 2 1/2, 3 1/2, 4. Part D: 1 2/4 (or 1 1/2), 1 1/4, 3/4. Part E: 4; 8; 3; 4.
Count by fractions
A. Count by quarters from 0 to 2. Write each count under its mark.
B. Count on by thirds
0, 1/3, 2/3, ______, ______, ______, 2
C. Count on by halves
0, 1/2, 1, ______, ______, ______, 3, ______, ______
D. Count back by quarters
2, 1 3/4, ______, ______, 1, ______, 2/4
E. Steps to a whole
From 0 to 1 is ______ quarter steps. From 0 to 2 is ______ quarter steps.
From 0 to 1 is ______ third steps. From 0 to 2 is ______ half steps.
Find the fraction on the line
Children read the fraction at a marked point by counting the equal steps from 0: the size of a step gives the bottom number, the number of steps gives the top. They mark fractions on lines, including fractions past one whole such as 5/4.
We are learning to
- name the fraction at a marked point by counting steps from 0,
- mark a fraction on a number line by its step size and number of steps,
- read points past 1 both ways: 7/4 and 1 3/4.
Success criteria
- I can read the fraction marked on a number line.
- I can mark a fraction on a number line, even one bigger than one.
You need
The three fraction number lines (cut-out sheet 1) and a paper clip or counter to mark a point, per pair. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Step size first Draw a line from 0 to 1 with four equal steps and mark the third mark. Ask what the step size is, then how many steps. Ask: “How many steps make one whole here? So what is each step called?” |
| 30 min | Read and mark Show the board (see the box). Partners take turns: one clips a mark on a number line, the other names it. Then children do Parts A and B. Ask: “The arrow is past 1 on the quarters line. Name it two ways.” |
| 10 min | Which is further? Children do Part C by finding both fractions on the lines. Ask: “1/2 and 2/4 have different numbers. Why do they land on the same point?” |
Two half-sessions instead? Split after Part A; start Session B by marking 5/4 on the floor line.
Watch for these ideas
- Writing the step count on the bottom: three quarter-steps as 4/3. The steps taken go on top.
- Using the number of marks as the bottom number (5 marks from 0 to 1, so fifths). Count the gaps, not the marks.
- Thinking a fraction cannot be more than 1. 5/4 is past the 1, one step on.
Answers
- Part A: 3/4; 2/3; 3/2 = 1 1/2; 7/4 = 1 3/4.
- Part B: 1/4 is one quarter-step from 0 and 5/4 is one step past 1; 1/3 is one third-step and 4/3 is one step past 1. Part C: 3/4; the same point; 5/4.
Find the fraction on the line
A. What fraction is marked by the arrow?
B. Mark each fraction with an arrow
Mark 1/4 and 5/4.
Mark 1/3 and 4/3.
C. Which is further from 0? Circle it, or write “same”.
3/4 or 2/31/2 or 2/45/4 or 1
Wholes and a bit more
Children meet mixed numerals: a whole number and a fraction side by side. 2 1/4 means two whole ones and one quarter more. They read and draw mixed numerals with bars, and count how many parts they hold altogether: 2 1/4 is nine quarters.
We are learning to
- read and write a mixed numeral for wholes and a part (two and one quarter is 2 1/4),
- draw a mixed numeral with whole bars and a part bar,
- count the parts in a mixed numeral: 2 1/4 is 9 quarters.
Success criteria
- I can write the mixed numeral a picture shows, and draw one.
- I can say how many quarters, thirds or halves a mixed numeral holds.
You need
Paper strips the same length, about four per child. The mixed number cards (cut-out sheet 2). The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Two and a bit Hold up two whole strips and a third strip folded in quarters with one quarter shaded. Write 2 1/4 and read it: two and one quarter. Ask: “Where is the 2 in my strips? Where is the 1/4?” |
| 30 min | Read and draw Show the board (see the box). Children do Part A, then draw the mixed numerals in Part B with whole bars and a part bar. Ask: “1 2/3: how many whole bars do you draw, and how is the last bar cut?” |
| 10 min | Count the parts Cut each whole strip of 2 1/4 into quarters and count all the quarters together: nine. Children do Part C. Ask: “Each whole is 4 quarters. How many quarters in 3 wholes and 1 more?” |
Two half-sessions instead? Split after Part A; start Session B by drawing 1 3/4.
Watch for these ideas
- Reading 2 1/4 as 21 quarters or as 2 times 1/4. It is 2 and 1/4 more.
- Cutting the whole bars as well when drawing. Only the last bar is cut into parts.
- Counting the parts in the last bar only: 2 1/4 as 1 quarter. Every whole holds 4 quarters too.
Answers
- Part A: 2 1/4; 1 2/3; 3 1/2. Part B: 1 whole bar and one bar cut into quarters with 3 shaded; 2 whole bars and one bar cut into thirds with 1 shaded.
- Part C: 1 1/2 = 3 halves; 2 3/4 = 11 quarters; 1 2/3 = 5 thirds; 3 1/4 = 13 quarters.
Wholes and a bit more
Each bar is one whole. A mixed numeral is the number of whole bars and the fraction of the last bar.
A. Write the mixed numeral
B. Draw it with whole bars and a part bar
| 1 3/4 | |
| 2 1/3 |
C. How many parts altogether?
| Mixed numeral | Parts altogether |
|---|---|
| 1 1/2 | ______ halves |
| 2 3/4 | ______ quarters |
| 1 2/3 | ______ thirds |
| 3 1/4 | ______ quarters |
Group the pieces into wholes
Children turn a fraction bigger than one into a mixed numeral by grouping its pieces into wholes: 7/4 is seven quarter pieces, and four of them make one whole, leaving three quarters, so 7/4 = 1 3/4. Then they turn mixed numerals back into fractions by counting every piece.
We are learning to
- explain that a fraction is bigger than one when its top number is bigger than its bottom,
- write a fraction bigger than one as a mixed numeral by grouping pieces into wholes,
- write a mixed numeral as a fraction by counting all its pieces.
Success criteria
- I can change a fraction like 7/4 into a mixed numeral and explain how.
- I can change a mixed numeral like 2 1/3 back into a fraction.
You need
The fraction number lines (cut-out sheet 1) and the mixed number cards (cut-out sheet 2), one set per pair. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Seven quarters Cut seven quarter pieces from paper strips. Children push four together to make one whole and count what is left. Ask: “Seven quarters made one whole and some more. How much more?” |
| 30 min | Group and convert Show the board (see the box). Children do Part A by looping groups of pieces, then Part B by asking how many groups of the bottom number fit in the top. Ask: “11/4: how many fours fit in 11? What is left over, and what is it called?” |
| 10 min | And back again Pairs match each mixed number card with its fraction card, then do Parts C and D. Ask: “3 1/4: each whole is 4 quarters. How many quarters in 3 wholes, and one more?” |
Two half-sessions instead? Split after Part A; start Session B with 11/4.
Watch for these ideas
- Swapping the numbers: 5/4 as 4 1/5. Count the groups of 4 in 5: one group, one quarter left.
- Writing the left-over pieces over the top number: 7/4 as 1 3/7. The pieces are still quarters.
- Turning 3 1/4 into 4/4 by adding 3 and 1. Each whole is 4 quarters: 12 + 1 = 13 quarters.
Answers
- Part A: 7/4 = 1 3/4; 5/3 = 1 2/3; 5/2 = 2 1/2. Part B: 9/2 = 4 1/2; 8/3 = 2 2/3; 11/4 = 2 3/4; 10/4 = 2 2/4 (or 2 1/2).
- Part C: 3 1/4 = 13/4; 2 1/3 = 7/3; 5 1/2 = 11/2. Part D: 7/4 (it is 1 3/4, and 1 3/4 is more than 1 1/2).
Group the pieces into wholes
A. Loop the pieces that make a whole. Write the mixed numeral.
| 7/4 | = ________ | |
| 5/3 | = ________ | |
| 5/2 | = ________ |
B. Write each fraction as a mixed numeral
9/2 = ________8/3 = ________11/4 = ________10/4 = ________
C. Write each mixed numeral as a fraction
3 1/4 = ______/42 1/3 = ______/35 1/2 = ______/2
D. Which is more, 7/4 or 1 1/2? Explain how you know.
______________________________________________________________
Mixed numbers on the line
Children place mixed numerals on a number line: 1 3/4 sits between 1 and 2, three quarters of the way along. They name each point two ways (1 3/4 and 7/4) and use the line to put mixed numerals and fractions in order.
We are learning to
- place a mixed numeral on a number line between the two whole numbers it sits between,
- name the same point as a mixed numeral and as a fraction bigger than one,
- order mixed numerals and fractions by their place on the line.
Success criteria
- I can mark a mixed numeral on a number line.
- I can order mixed numerals and fractions, smallest first.
You need
The fraction number lines (cut-out sheet 1) and the mixed number cards (cut-out sheet 2), one set per group of four. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Between which wholes? Show the first board picture (see the box). Children say the two whole numbers before you press, then where between them the point lands. Ask: “1 3/4 is past 1. Is it nearer to 1 or to 2?” |
| 30 min | Place and name Groups lay the mixed number cards in order along the floor line or the quarters line, pairing each card with the card that names the same point. Then children do Parts A to C. Ask: “7/4 and 1 3/4 sit on the same point. Which name is easier to place, and why?” |
| 10 min | The table and the quiz Show the second board picture, then children do Part D. Finish with the unit’s self-check quiz as a class game. Ask: “Which is more, 5/4 or 1 1/2? Show it on the line.” |
Two half-sessions instead? Split after the card line; start Session B with Part A.
Watch for these ideas
- Placing 2 1/4 at the first quarter mark after 0. Find the 2 first, then step a quarter on.
- Thinking 7/4 and 1 3/4 are different points because the numbers look different.
- Ordering by the top number: 5/4 before 7/4 is right, but 1 1/2 is not before 5/4.
Answers
- Part A: 2 1/2 is 10 quarter-steps from 0; 3/4 is 3 quarter-steps from 0; 1 1/4 is 5 quarter-steps from 0; 2 3/4 is 11 quarter-steps from 0. Part B: 2 1/3 is between 2 and 3; 7/4 is between 1 and 2; 11/4 is between 2 and 3; 1 2/3 is between 1 and 2.
- Part C: 1 1/2 = 3/2; 5/4 = 1 1/4; 2 1/3 = 7/3; 11/4 = 2 3/4. Part D: 5/4, 1 1/2, 7/4, 2.
Mixed numbers on the line
A. Mark each one with an arrow and its name
2 1/2, 3/4, 1 1/4, 2 3/4
B. Which two whole numbers is it between?
2 1/3 is between ______ and ______
7/4 is between ______ and ______
11/4 is between ______ and ______
1 2/3 is between ______ and ______
C. Same point, two names
| Mixed numeral | Fraction |
|---|---|
| 1 1/2 | |
| 5/4 | |
| 2 1/3 | |
| 11/4 |
D. Put them in order, smallest first
7/4, 1 1/2, 2, 5/4
__________, __________, __________, __________
Three fraction number lines
Cut out each line along its dashed border. Each runs from 0 to 3 in equal steps. Write the counts under the marks as you learn them (Day 1), and use a paper clip to mark a point (Days 2 and 5). One set per pair.
Teacher note: only the whole numbers are printed, so children do the counting. The lines are the same length, so 1 1/2 on the halves line sits straight above 6/4 on the quarters line.
Mixed number cards
Cut out the 16 cards: eight pairs that name the same point on the line, one as a mixed numeral and one as a fraction bigger than one. Use them to match pairs (Day 4) and to order along a number line (Day 5).
Cards
Teacher note: the pairs are 1 1/2 and 3/2, 1 1/4 and 5/4, 1 3/4 and 7/4, 2 1/4 and 9/4, 2 1/2 and 5/2, 2 3/4 and 11/4, 1 1/3 and 4/3, and 2 2/3 and 8/3.
What we know: Counting by Fractions
Work on your own. Show your thinking if you can.
- Count on by quarters: 3/4, 1, 1 1/4, ______, ______, 2
- Count back by thirds: 2, 1 2/3, ______, ______, 2/3
- What number is marked by the arrow? ________
- Mark 2/3 with an arrow:
- Write 2 1/4 as a fraction: ______ Write 7/2 as a mixed numeral: ______
- How many thirds make 3 1/3? ______
- 1 3/4 sits between ______ and ______. Is it nearer to 1 or to 2? ______
- Put in order, smallest first: 5/4, 3/4, 1 1/2, 2 ________________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers
- 1 2/4 (or 1 1/2), 1 3/4.
- 1 1/3, 1.
- 5/4, which is 1 1/4 (5 quarter-steps from 0).
- The arrow on the second of the three third-steps from 0.
- 9/4; 3 1/2.
- 10 thirds (3 wholes are 9 thirds, and 1 more).
- 1 and 2; nearer to 2 (three quarters of the way along).
- 3/4, 5/4, 1 1/2, 2.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Counting by fractions (Q1, Q2) | counts on by a fraction up to one whole | counts on and back past whole numbers, naming mixed numerals | counts from any start, in any of the step sizes |
| Locating on a line (Q3, Q4, Q7) | reads a marked fraction below one | reads and marks fractions and mixed numerals, including past one | explains how the step size and the step count give the fraction |
| Mixed and improper (Q5, Q6, Q8) | converts with pictures | converts both ways and orders mixed numerals and fractions | explains a conversion with groups of the bottom number |
Eight questions, three ideas. A child at standard answers most questions and can show any answer on a number line by counting steps from 0.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Counts by a fraction | Reads a marked line | Reads mixed numbers | Converts improper | Places mixed numbers |
|---|---|---|---|---|---|
The five columns are the five days: counting, reading the line, mixed numerals, converting and placing.