Back to Lesson

Teacher Guide: Introduction to Mixed Numbers

Learn what mixed numbers are, how to read and write them, how to draw them and where to find them on the number line.

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Fraction Basics. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define a mixed number as a whole number plus a proper fraction
  • Read and write mixed numbers correctly in words and symbols
  • Represent mixed numbers with fraction bars and circles
  • Locate mixed numbers on a number line between two whole numbers
  • Recognize mixed numbers in recipes, lengths and time
Prerequisites
  • • Understanding numerators and denominators (lesson What Are Fractions)
  • • Knowing proper and improper fractions (lesson Improper Fractions)
  • • Placing simple fractions between 0 and 1 on a number line
Discussion Starters
  • 1. Where have you heard someone say "and a half" or "and a quarter"? What did it mean?
  • 2. Why do you think the fraction part of a mixed number has to be less than 1?
  • 3. Is 4124\frac{1}{2} closer to 4 or to 5? How do you know?
  • 4. Would you rather say "seven quarters of an hour" or "one and three quarter hours"? Why?
Common Misconceptions

The whole number and the fraction are multiplied (312=3×123\frac{1}{2} = 3 \times \frac{1}{2})

Remediation: Always read mixed numbers aloud with "and" and write the sum next to them at first: 312=3+123\frac{1}{2} = 3 + \frac{1}{2}. Show 3 full bars and a half bar: clearly more than 3, not less than 2.

The denominator counts all the parts in the picture

Remediation: Ask: how many equal parts are in ONE whole? Color each whole a different color so students see that every whole has the same number of parts.

A mixed number is placed on the number line using only its fraction part

Remediation: Have students first circle the whole number on the number line, then count the fraction parts from there. A floor number line where they physically walk helps.

Differentiation Ideas

For Struggling Students:

  • • Use real objects first: whole apples and apple slices, full and half filled cups
  • • Work only with halves and quarters before other denominators
  • • Read every mixed number aloud with the word "and" and write the plus sign

For On-Level Students:

  • • Write mixed numbers from pictures with denominators up to 10
  • • Place several mixed numbers on one number line from 0 to 5
  • • Change mixed numbers of hours and meters to minutes and centimeters

For Advanced Students:

  • • Find the improper fraction for a mixed number by counting all the parts in a drawing
  • • Explain why 1541\frac{5}{4} is not a correct mixed number and fix it
  • • Create their own recipe or timetable that uses at least four mixed numbers
Standards Alignment
  • 3.NF.A.2 (CCSS.MATH.CONTENT.3.NF.A.2)

    Understand a fraction as a number on the number line; represent fractions on a number line diagram

  • 4.NF.B.3.b (CCSS.MATH.CONTENT.4.NF.B.3.B)

    Decompose a fraction into a sum of fractions with the same denominator, for example 2 1/8 = 1 + 1 + 1/8

Lesson Resources
  • visualFraction Bars and Circles

    Students shade full wholes and a part of one more whole to build mixed numbers

  • activityNumber Line Walk

    Tape a number line from 0 to 3 on the floor and have students stand on mixed numbers called out by the teacher

  • worksheetMixed Numbers Around Us

    Find and write mixed numbers in recipes, timetables and measurements

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A mixed number is a number made of a whole number and a proper fraction written side by side.
234=2+342\frac{3}{4} = 2 + \frac{3}{4}
  • The whole number part (22) tells how many complete wholes you have
  • The fraction part (34\frac{3}{4}) tells what part of one more whole you have
  • The fraction part is always proper: its numerator is smaller than its denominator
We read 2342\frac{3}{4} as two and three quarters (or two and three fourths). The word and stands for the plus sign that is hidden between the two parts.
A mixed number is always greater than 1, because it has at least one whole. In the lesson on improper fractions you saw that the same amount can also be written as one fraction: 234=1142\frac{3}{4} = \frac{11}{4}. The next lessons show how to switch between the two forms.

Worked Examples

What does 3123\frac{1}{2} mean, and how do you read it?
1

Find the whole number part

The big number in front is 33 → 3 complete wholes
2

Find the fraction part

The fraction next to it is 12\frac{1}{2}, a proper fraction because 1<21 < 2 → Half of one more whole
3

Put the two parts together

312=3+123\frac{1}{2} = 3 + \frac{1}{2} → Read: three and one half

Common Mistakes

Thinking 2132\frac{1}{3} means 2×132 \times \frac{1}{3}

Why it's wrong: In algebra, writing things side by side often means multiplying. In a mixed number it does not: the hidden sign is a plus.

Correct: 213=2+132\frac{1}{3} = 2 + \frac{1}{3}. Read the "and" in "two and one third" as plus.

Writing a fraction part that is not proper, like 1541\frac{5}{4}

Why it's wrong: The fraction part must be less than one whole. 54\frac{5}{4} already contains a whole (44\frac{4}{4}), so it belongs in the whole number part.

Correct: Move the extra whole over: 154=2141\frac{5}{4} = 2\frac{1}{4}. In a mixed number the numerator is always smaller than the denominator.

Placing 2142\frac{1}{4} near 14\frac{1}{4} on the number line

Why it's wrong: Looking only at the fraction part forgets the 2 wholes that come first.

Correct: First jump to the whole number (2), then move the fraction part (14\frac{1}{4}) further to the right.

Using the wrong denominator when reading a picture of several shapes

Why it's wrong: With 2 full circles and 3 of 4 parts of a third circle, some students count all the parts and write 1112\frac{11}{12}.

Correct: The denominator is the number of parts in ONE whole (4). The answer is 2342\frac{3}{4}.

Why It Matters

Mixed numbers are the way people naturally talk about amounts that are more than a whole:
  • Cooking: A recipe asks for 1121\frac{1}{2} cups of milk or 2142\frac{1}{4} teaspoons of salt
  • Time: A film lasts 1341\frac{3}{4} hours, a road trip takes 2122\frac{1}{2} hours
  • Length: A shelf is 1141\frac{1}{4} meters long, a room is 3123\frac{1}{2} meters wide
  • Sports: A runner completes 4124\frac{1}{2} laps of the track
Saying "two and a half hours" is much easier to picture than "five halves of an hour". Mixed numbers show at a glance how many wholes you have and how much is left over.

Real World Applications

Cooking and Baking

Recipes use mixed numbers for amounts bigger than one cup, spoon or kilogram.

Example:

A pancake recipe needs 1121\frac{1}{2} cups of milk: fill the cup once completely and then once halfway.

1Try It Yourself

A bread recipe needs 2122\frac{1}{2} cups of flour, but you only have a half cup measure.

How many half cups of flour do you need?

Step 1: Write the mathematical expression

Two halves in each whole cup, plus the extra half:

Measuring Lengths

Builders, tailors and crafters measure lengths that are a few whole meters plus a part of a meter.

Example:

A curtain that is 1341\frac{3}{4} meters long is 1 meter plus three quarters of another meter, so 175 centimeters.

2Try It Yourself

A ribbon is 2342\frac{3}{4} meters long.

How many centimeters long is the ribbon?

Step 1: Write the mathematical expression

Whole meters in centimeters, plus three quarters of 100 centimeters:

Time and Schedules

Timetables and everyday speech use mixed numbers of hours: an hour and a half, two and a quarter hours.

Example:

A train trip of 1121\frac{1}{2} hours takes 60 + 30 = 90 minutes.

3Try It Yourself

Football training lasts 1141\frac{1}{4} hours.

How many minutes does the training last?

Step 1: Write the mathematical expression

One hour in minutes, plus a quarter of 60 minutes:

Key Takeaways

  • 1A mixed number is a whole number and a proper fraction written together, like 2342\frac{3}{4}
  • 2It means the whole number plus the fraction: 234=2+342\frac{3}{4} = 2 + \frac{3}{4}, read "two and three quarters"
  • 3The fraction part is always proper: its numerator is smaller than its denominator
  • 4To draw a mixed number, shade the full wholes, then shade the fraction part of one more whole
  • 5On the number line, a mixed number sits between its whole number and the next one: 2142\frac{1}{4} is between 2 and 3
  • 6The same amount can also be written as an improper fraction, like 234=1142\frac{3}{4} = \frac{11}{4}

Frequently Asked Questions

Is 2142\frac{1}{4} really the same as 2+142 + \frac{1}{4}?

Yes. A mixed number is a short way of writing a sum of a whole number and a fraction. The plus sign is just left out.

Can the fraction part be 44\frac{4}{4} or bigger?

No. 44\frac{4}{4} is one whole, so it should be added to the whole number part. 3443\frac{4}{4} is simply 44, and 3543\frac{5}{4} is 4144\frac{1}{4}.

Why use mixed numbers when improper fractions exist?

Mixed numbers are easier to picture in daily life: 2122\frac{1}{2} hours tells you right away it is a bit more than 2 hours. Improper fractions like 52\frac{5}{2} are often easier for calculations. Both name the same amount.

How can I tell which of two mixed numbers is bigger?

Compare the whole number parts first, and only if they are equal compare the fraction parts. The lesson Comparing Mixed Numbers explains this in detail.

Glossary

Mixed number
A number made of a whole number and a proper fraction, like 3123\frac{1}{2}
Whole number part
The number in front of the fraction; it tells how many complete wholes there are
Fraction part
The proper fraction in a mixed number; it shows the part of one more whole
Proper fraction
A fraction whose numerator is smaller than its denominator, like 34\frac{3}{4}; it is less than 1
Improper fraction
A fraction whose numerator is equal to or bigger than its denominator, like 74\frac{7}{4}; it is 1 or more

Formula Card

Meaning of a mixed number

abc=a+bca\frac{b}{c} = a + \frac{b}{c}

A mixed number is the whole number $a$ plus the fraction $\frac{b}{c}$

Rule for the fraction part

b<cb < c

The numerator of the fraction part is always smaller than the denominator

Position on the number line

a<abc<a+1a < a\frac{b}{c} < a + 1

A mixed number lies between its whole number part and the next whole number

More in This Topic

What Are Fractions

Elementary20 min

Equivalent Fractions

Elementary25 min

Simplifying Fractions

Elementary25 min

Improper Fractions

Elementary20 min