Mixed numbers on a number line
Counting past one whole, improper fractions and mixed numbers as one point, converting between them, and reading a measuring scale
About four lessons of 45 to 60 minutes
The jug is full past the 2, but not quite at the 3
You are helping bake, and the recipe needs some water. You fill the measuring jug and it stops above the 2 cup line but below the 3, sitting right on a little mark in between. How much water is that? Not 2 cups, not 3 cups, but two and three quarter cups. That in-between amount is a mixed number.
In Grade 3 you learned to place fractions between 0 and 1 on a number line. Fractions do not stop at 1, though: you can keep counting past a whole, just like the jug fills past the 2 cup line. Today you will count in fractions beyond one whole, meet improper fractions and mixed numbers as two names for the very same point, and read any in-between amount off a scale.
- A jug filled to two and three quarterspast the 2 cup line, that is 2 3/4 cups, or 11/4
- A pencil measuring one and a half unitsbetween 1 and 2 on the ruler, that is 1 1/2
- One whole pizza and 2 more fifths of anotherthat is 1 2/5, the same as 7/5
- One and a quarter laps of the trackone full lap and a quarter more, 1 1/4 laps
What students will be able to do
Students will count in unit fractions beyond one whole, understand an improper fraction and a mixed number as two names for the same point on a number line, convert between the two forms, and read and place mixed numbers on a marked scale.
- I can keep counting in unit fractions past one whole.
- I can place a fraction greater than one on a number line.
- I can name a point as both an improper fraction and a mixed number.
- I can change an improper fraction to a mixed number and back.
- I can read a mixed number off a measuring scale.
Standards this unit teaches
- 4.NF.B.3Common Core (US)A fraction greater than one as a sum of unit fractions
Understand a fraction a/b with a greater than 1 as a sum of fractions 1/b, so that a fraction such as 7/5 is seven fifths joined together, and decompose fractions into sums of unit fractions in more than one way.
- 4.NF.B.3cCommon Core (US)Mixed numbers with like denominators
Replace a mixed number with an equivalent fraction, and a fraction greater than one with an equivalent mixed number, and add and subtract mixed numbers with like denominators. This lettered sub-standard of 4.NF.B.3 has no separate page in the library, so no direct link is given.
- AC9M4N03Australian Curriculum v9 (ACARA)Fractions and mixed numerals on a number line (Year 4)
Count by fractions, including mixed numerals, and locate and represent them as points on a number line, extending counting beyond one whole.
- AC9M5N03Australian Curriculum v9 (ACARA)Compare and order fractions on a line (Year 5)
Use equivalence to compare, order and represent common fractions, including mixed numerals, on the same number line and justify the order. The ordering work at the end of this unit reaches toward this Year 5 descriptor.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 3 fractions teaching unitnaming fractions and placing them between 0 and 1 on a number line
- How to teach visual fractionsreading a fraction from equal parts of a whole
- How to teach the number lineequal gaps, and counting the jumps rather than the marks
- Grade 4 equivalent fractions teaching unitthe equivalence ideas that sit alongside this unit
Words to teach and display
- Unit fraction
- a fraction with 1 on top, one single equal part, such as 1/5
- Proper fraction
- a fraction less than one whole, where the top is smaller than the bottom
- Improper fraction
- a fraction equal to or greater than one, where the top is the same as or bigger than the bottom, such as 7/5
- Mixed number
- a whole number and a fraction together, such as 1 2/5
- Numerator and denominator
- the top number counts the parts, the bottom names how many parts make one whole
- Number line
- a line with numbers at equal spaces, where every fraction has one home
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Counting does not stop at one whole
ConcreteTake two identical paper strips, each one whole, and fold each into 5 equal parts, so every part is one fifth. Count the fifths across the first strip: 1/5, 2/5, 3/5, 4/5, and 5/5, which is one whole strip. Do not stop. Pick up the second strip and keep counting: 6/5, 7/5. You have counted seven fifths, which is more than one whole.
The key idea: the denominator does not change as you count past one whole. Each step is still one more fifth, so after 5/5 comes 6/5, then 7/5. Seven fifths is one full strip (5/5) and 2 more fifths, so it is one whole and 2/5.
- After 5/5, what is the next fifth you count?
- Does the bottom number change when you count past one whole?
2. Two names for the same point
PictorialPut those fifths on a number line from 0 to 2, with each whole split into 5 equal gaps. Count the fifths along the line: after 5 gaps you reach 1, and 2 gaps further along you reach the point for 7/5. That same point is one whole and 2 fifths past it, which we write as the mixed number 1 2/5.
So 7/5 and 1 2/5 are not two different amounts, they are two names for the exact same point on the line. The improper fraction 7/5 counts every fifth from 0, while the mixed number 1 2/5 counts the whole first and then the extra fifths.
- How many fifth-gaps from 0 reach the point for 7/5?
- Why do 7/5 and 1 2/5 sit on the same point?
3. Changing between improper fractions and mixed numbers
AbstractYou do not have to draw a line every time. To change an improper fraction to a mixed number, ask how many whole ones fit. For 11/4, how many groups of 4 quarters (one whole each) are in 11 quarters? 11 divided by 4 is 2 with 3 left over, so that is 2 wholes and 3/4, the mixed number 2 3/4.
To go back the other way, count the total parts. For 2 3/4, each whole is 4 quarters, so 2 wholes is 8 quarters, plus the 3 quarters makes 11 quarters, which is 11/4. Whichever way you go, the amount does not change, only the way it is written.
Change 11/4 to a mixed number, then change it back.
- How many whole ones are in 11 quarters? Divide 11 by 4: it is 2 remainder 3.
- The 2 is the number of wholes, and the remainder 3 is the fifths left over, so 11/4 = 2 3/4.
- Change back: 2 wholes is 2 times 4 is 8 quarters, plus 3 quarters is 11 quarters, so 2 3/4 = 11/4.
Answer: 11/4 = 2 3/4, and 2 3/4 = 11/4.
- In 11 divided by 4 is 2 remainder 3, which number is the whole and which is the leftover parts?
- Change 3 1/2 to an improper fraction.
4. Reading a mixed number off a scale
PictorialBack to the measuring jug from the hook. The jug is marked in quarter cups, so between each whole cup line there are 4 equal gaps. The water stops on the third mark past the 2 cup line. That is 2 whole cups and 3 quarters more, which is 2 3/4 cups.
A scale is just a number line stood on its end, so reading it uses the same skill: find the last whole number the level has passed, then count the extra equal gaps above it. Two whole cups, three quarter-gaps more, gives 2 3/4.
- How many equal gaps are there between each whole cup line on this jug?
- If the water stopped one gap higher, what mixed number would it show?
5. Placing and ordering mixed numbers
AbstractBecause every mixed number has one home on the line, you can compare them by position: the one further to the right is larger. Place 1 1/2, 1 3/4 and 2 1/4 on a line marked in quarters from 0 to 3. First 1 1/2 (halfway between 1 and 2), then 1 3/4 (one gap further), then 2 1/4 (past the 2).
Reading left to right gives the order from smallest to largest: 1 1/2, then 1 3/4, then 2 1/4. Note that 1 1/2 is the same point as 1 2/4, which helps line the halves up with the quarters.
- Which is further right on the line, 1 3/4 or 2 1/4?
- What other name does the point 1 1/2 have in quarters?
Common misconceptions and how to address them
MisconceptionA fraction can never be bigger than one whole.
Why it happens: Every fraction in Grade 3 sat between 0 and 1, so students assume all fractions do.
How to address it: Count fifths past 5/5 on the line: 6/5 and 7/5 are real points beyond 1. An improper fraction and a mixed number both name amounts greater than one whole.
MisconceptionThe mixed number 2 3/4 means 2 times 3/4.
Why it happens: Two numbers written side by side usually mean multiply, as in algebra later.
How to address it: A mixed number means the whole and the fraction added: 2 3/4 is 2 wholes plus 3/4. On the line it is 2, then 3 quarter-gaps more, not a multiplication.
MisconceptionWhen counting fifths past one whole, after 5/5 comes 1/6.
Why it happens: Students think reaching a whole resets the count and bumps the denominator up by one.
How to address it: The denominator names the size of each part and does not change: the parts are still fifths. After 5/5 comes 6/5, then 7/5. Only the numerator keeps counting up.
Misconception7/5 and 1 2/5 are different numbers.
Why it happens: They look completely different on the page, so students treat them as separate amounts.
How to address it: Mark both on the same line: they land on the identical point. One counts every fifth from 0, the other counts the whole first, but the amount is the same.
MisconceptionWhen changing 11/4, the leftover from the division is the number of wholes.
Why it happens: Students grab the remainder and put it in the whole-number place by mistake.
How to address it: 11 divided by 4 is 2 remainder 3. The quotient 2 is how many wholes, and the remainder 3 is the leftover quarters, so it is 2 3/4, not 3 2/4. The whole number is how many complete groups you made.
MisconceptionTo place 1 2/5 you count 1, then 2, then 5 marks along the line.
Why it happens: Students read the three digits as separate steps instead of a whole plus a fraction of the gaps.
How to address it: Go to the 1, then count 2 of the fifth-gaps in the next whole. The 5 is not a distance, it tells you the whole is split into 5 equal gaps. Count the gaps, not the marks.
Guided practice (with answers)
1. Count on in quarters from 3/4 and name the point after 4/4. What is 5/4 as a mixed number?
Answer: 3/4, 4/4 (which is 1), 5/4. And 5/4 is 1 1/4.
2. Write 9/4 as a mixed number.
Answer: 2 1/4, because 9 divided by 4 is 2 remainder 1.
3. Write 3 1/2 as an improper fraction.
Answer: 7/2, because 3 wholes is 6 halves, plus 1 half is 7 halves.
4. On a line from 0 to 2 marked in fifths, where does 7/5 sit?
Answer: Two fifth-gaps past the 1, at the point 1 2/5.
5. A jug shows 1 3/4 cups. Write that as an improper fraction.
Answer: 7/4, because 1 whole is 4 quarters plus 3 quarters is 7 quarters.
6. Which is larger, 1 3/4 or 1 2/4, and how do you know?
Answer: 1 3/4, because it sits one quarter-gap further to the right on the number line.
Independent practice worksheets
Set the matching ChalkBee fraction worksheets for independent work. The answer keys are computed in code, so they are never wrong. Use the visual-fraction sets to keep the part-whole picture strong while students work with fractions beyond one whole.
Differentiation
- Stay with paper strips and pre-marked number lines so counting past one whole is something students can see and touch.
- Keep the denominator small and friendly (halves, then quarters, then fifths) before mixing them.
- Give a line already split into the right number of gaps so the student only counts and marks the point.
- Convert with the strips first: physically join wholes and extra parts before using the divide-and-remainder shortcut.
- Convert improper fractions where more than two wholes fit, such as 17/4.
- Place a mix of improper fractions and mixed numbers on one line and order them.
- Find a missing mixed number a given number of gaps along a scale with an unusual denominator.
- Bridge to Year 5 (AC9M5N03) by ordering fractions with related denominators on the same line and justifying the order.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling converting each way and placing a mixed number on a line.
1. Write 8/5 as a mixed number.
Answer: 1 3/5, because 8 divided by 5 is 1 remainder 3.
2. Write 2 1/3 as an improper fraction.
Answer: 7/3, because 2 wholes is 6 thirds plus 1 third is 7 thirds.
3. Between which two whole numbers does 2 1/2 sit on a number line, and where exactly?
Answer: Between 2 and 3, exactly halfway between them.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 counting past one whole and two names for a point (sections 1 to 2), Lesson 2 converting between forms (section 3), Lesson 3 reading a scale (section 4), Lesson 4 ordering plus the exit ticket (section 5 and assessment).
- This unit assumes the Grade 3 fractions unit: naming fractions and placing them between 0 and 1 on a number line. Revisit it first if placing a fraction on a line is shaky.
- Language to keep saying: the denominator does not change, count the gaps not the marks, two names for the same point, the whole number is how many complete wholes. These pre-empt most of the misconceptions.
- The number-line diagrams label only the whole numbers below the line and put the fraction name above the marked point, so a class that has only just met decimals is not distracted by the tick values behind the fifths and quarters.
- Curriculum note and a US and AU alignment: ACARA states counting by fractions including mixed numerals and locating them on a number line explicitly at Year 4 (AC9M4N03), a clean match. The US spreads this across placing fractions on a line in Grade 3 (3.NF.A.2) and understanding fractions greater than one and mixed numbers in Grade 4 (4.NF.B.3 and 4.NF.B.3c), with explicit ordering reaching into ACARA Year 5 (AC9M5N03).
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.