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International Baccalaureate · PYP 4

Fractions

⌚ ~186 min ▢ 21 blocks ✓ 95 exercises ➣ Prerequisites : Division, Fractions
I Fractions Greater than 1

Consider the model below, which illustrates one whole unit and one half: Figure. To express this length as a single fraction, we follow two steps:
  1. Identify the denominator: The units are divided into halves, so the denominator is 2.
  2. Count the parts: In total, there are three shaded parts of this size. This count becomes the numerator.
Therefore, this length is represented by the fraction \(\dfrac{3}{2}\).

Definition — Fraction
A fraction has two numbers separated by a fraction bar: the numerator (on top) and the denominator (on the bottom).
Figure
The numerator and the denominator are whole numbers, and the denominator is never zero: a unit cannot be divided into zero equal parts.
Skills to practice
  • Finding Fractions Greater than 1 in Diagrams
II On the Number Line, Beyond 1

Fractions do not only represent parts of a shape; they can also represent points on a number line. The segment from 0 to 1 is one unit.
If we divide the unit into 2 equal parts, the point in the middle represents the fraction \(\dfrac{1}{2}\).
Figure

Method — Representing a Fraction on the Number Line
To represent the fraction \(\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}\) on a number line:
  1. Draw a straight line and mark the points \(0\) and \(1\).
    Figure
  2. Divide the segment from \(0\) to \(1\) into \(\textcolor{colorprop}{3}\) equal parts (the denominator).
    Figure
  3. Count \(\textcolor{colordef}{2}\) parts from \(0\) (the numerator) and mark the point at \(\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}\).
    Figure
Method — Placing a Fraction Greater than 1 on the Number Line
A fraction greater than \(1\) is placed like any fraction: the denominator says into how many equal parts each unit is divided, the numerator counts these parts from \(0\). To place \(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\):
  1. Mark \(0\), \(1\), \(2\) at equal distances, and divide each unit into \(\textcolor{colorprop}{3}\) equal parts (the denominator).
    Figure
  2. Count \(\textcolor{colordef}{5}\) parts from \(0\) (the numerator). After \(3\) parts you reach \(1\), because \(\dfrac{3}{3} = 1\); \(2\) more parts give the point.
    Figure
To place a whole number plus a fraction, such as \(1 + \dfrac{2}{3}\), go directly to the whole number \(1\), then count \(2\) parts of one third after it.
Figure
The two forms locate the same point: \(\dfrac{5}{3} = 1 + \dfrac{2}{3}\).
Example
Place on a number line:
  1. the fraction \(\dfrac{9}{4}\);
  2. the number \(3 + \dfrac{1}{2}\).

  1. The denominator is \(4\): each unit is divided into \(4\) equal parts. Count \(9\) parts from \(0\): \(4\) parts reach \(1\), \(4\) more parts reach \(2\), and \(1\) more part gives the point.
    Figure
    The point is one quarter after \(2\): $$\begin{aligned}[t] \dfrac{9}{4} &= \dfrac{8}{4} + \dfrac{1}{4} && \text{(8 quarters make 2 units)}\\ &= 2 + \dfrac{1}{4} \end{aligned}$$
  2. Here the fraction added to \(3\) is \(\dfrac{1}{2}\): each unit is divided into \(2\) equal parts. Go to the whole number \(3\), then count \(1\) half after it.
    Figure
    Counting halves from \(0\) gives the same point: $$\begin{aligned}[t] 3 + \dfrac{1}{2} &= \dfrac{6}{2} + \dfrac{1}{2} && \text{(3 units make 6 halves)}\\ &= \dfrac{7}{2} \end{aligned}$$

Skills to practice
  • Finding Fractions Greater than 1 on the Number Line
  • Placing a Fraction Greater than 1 or a Whole Number Plus a Fraction
III Equivalent Fractions

A cake is cut into 3 equal parts. One part, representing \(\dfrac{1}{3}\) of the cake, is set aside.
Figure
Then each of the three parts is cut in half. Now the cake is divided into 6 equal parts. The same amount of cake as before is now 2 of these parts.
Figure
Even though the number of pieces changed, did the shaded amount of cake change?

No—the amount of cake did not change. The models show that the shaded portion is the same in both cases.
Figure\(\quad=\quad\)Figure
Therefore, the fractions \(\dfrac{1}{3}\) and \(\dfrac{2}{6}\) represent the same value.


Definition — Equivalent Fractions
Two fractions are equivalent if they represent the same amount. You get an equivalent fraction by cutting every part into the same number of smaller parts: the numerator and the denominator are multiplied by that number. Grouping the parts back does the opposite: both are divided by the same number, which must divide the numerator and the denominator exactly.
Figure Figure Figure
Figure Figure Figure
Skills to practice
  • Identifying Equivalent Fractions (with Models)
  • Identifying Equivalent Fractions (Symbolic)
  • Writing Equivalent Fractions
  • Finding the Missing Numerator
  • Finding the Missing Denominator
IV Addition and Subtraction of Fractions with Common Denominators

Hugo's cake is cut into \(4\) equal parts (quarters). Hugo eats \(\dfrac{2}{4}\) of the cake \(=\) Figure, then \(\dfrac{1}{4}\) more \(=\) Figure.
What fraction of the cake has he eaten?

The parts are all the same size (quarters), so the eaten parts can simply be counted together: \(2+1=3\) quarters.
Figure
Hugo has eaten \(\dfrac{3}{4}\) of the cake.


Definition — Addition of Fractions with Common Denominators
To add fractions with the same denominator, add the numerators and keep the denominator the same:
Figure
$$\begin{aligned}\dfrac{2}{4} + \dfrac{1}{4} &= \dfrac{2 + 1}{4}\\ &= \dfrac{3}{4}\end{aligned}$$

Another day, \(\dfrac{3}{4}\) of Hugo's cake is left \(=\) Figure. Hugo eats \(\dfrac{1}{4}\) more.
What fraction of the cake is left now?

The parts are the same size (quarters), so subtract the counts: \(3-1=2\) quarters are left.
Figure
\(\dfrac{2}{4}\) of the cake is left.


Definition — Subtraction of Fractions with Common Denominators
To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same:
Figure
$$\begin{aligned}\dfrac{3}{4} - \dfrac{1}{4} &= \dfrac{3 - 1}{4}\\ &= \dfrac{2}{4}\end{aligned}$$ Here the first fraction is the larger one, so these parts can be taken away.
The Whole Cake
Hugo cuts a cake into \(3\) equal parts — and eats them all: Figure. He has eaten \(3\) parts out of \(3\): the whole cake. $$\dfrac{3}{3} = 1$$ Before the first bite, it was the other way round — \(0\) parts out of \(3\), nothing of the cake: Figure. $$\dfrac{0}{3} = 0$$
Report an error
Proposition — When the Fraction is Worth 1 or 0
When the two numbers are equal, every part is taken: the fraction is the whole unit. It is worth \(1\).
Figure
$$\dfrac{3}{3} = 1$$ When the top number is \(0\), no part is taken: the fraction is worth \(0\).
Figure
$$\dfrac{0}{3} = 0$$
Skills to practice
  • Adding Fractions with Common Denominators (with Models)
  • Adding Fractions with Common Denominators (Symbolic)
  • Subtracting Fractions with Common Denominators (with Models)
  • Subtracting Fractions with Common Denominators (Symbolic)
  • Adding and Subtracting: Showing the Steps
V Adding and Subtracting with Related Denominators
Method — Procedure for Adding or Subtracting Fractions
To add or subtract fractions with unlike denominators, follow this three-step procedure:
  1. Find a Common Denominator: Identify a common multiple of the denominators.
  2. Create Equivalent Fractions: Convert each fraction to an equivalent fraction with the common denominator.
  3. Add or Subtract the Numerators: With the denominators now the same, perform the operation on the numerators and keep the common denominator. For a subtraction, the first fraction must not be smaller than the second.
The Sum of Two Positive Numbers
Louis writes \(\dfrac{1}{2} + \dfrac{1}{4} = \dfrac{2}{6}\): he has added the numerators together and the denominators together. On two identical bars, \(\dfrac{2}{6}\) is shorter than \(\dfrac{1}{2}\) alone.
Figure
Yet the sum of two positive numbers is greater than each of them: put end to end, the lengths \(\dfrac{1}{2}\) and \(\dfrac{1}{4}\) make more than \(\dfrac{1}{2}\). Louis's answer is wrong. Only parts of the same size can be counted together: halves and quarters are not the same size, so the half is first cut into two quarters. $$\begin{aligned}\dfrac{1}{2} + \dfrac{1}{4} &= \dfrac{2}{4} + \dfrac{1}{4} && \text{(one half is two quarters)}\\ &= \dfrac{3}{4}\end{aligned}$$
Report an error
When One Denominator Is a Multiple of the Other
When one of the two denominators is a multiple of the other, the common denominator is simply the bigger one, and only one fraction has to be rewritten.
Report an error
Skills to practice
  • Subtracting with Related Denominators
VI Improper Fractions and Mixed Numbers

Fractions can represent values greater than one. Consider the fraction \(\dfrac{5}{2}\), which represents 5 half-sized parts of a unit.
Figure
While this “improper fraction” is a valid mathematical representation, it is often more intuitive to express such numbers as a combination of whole units and a remaining fractional part.

Definition — Proper and Improper Fractions
Fractions are classified based on the relationship between the numerator and the denominator.
  • A proper fraction is a fraction where the numerator is less than the denominator. Its value is always less than 1. Example: \(\dfrac{2}{3}\).
  • An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Its value is always greater than or equal to 1. Example: \(\dfrac{5}{3}\).
Proposition — Whole Number Plus a Fraction
Every fraction greater than \(1\) can be written as the sum of a whole number and a fraction less than \(1\); conversely, the sum of a non-zero whole number and a non-zero fraction less than \(1\) is a fraction greater than \(1\):
Figure
Such a writing, like \(2 + \dfrac{1}{3}\), is called a mixed number.
Definition — Mixed Number
A mixed number is an alternative way to represent an improper fraction. It consists of an integer part (the number of whole units) and a proper fraction part.
The improper fraction \(\dfrac{5}{2}\) can be visualised as two whole units and one half.
Figure
This is written as the mixed number \(2\dfrac{1}{2}\). By convention, the addition sign is omitted:$$ 2\frac{1}{2} \quad \text{is equivalent to} \quad 2 + \frac{1}{2} $$ Caution: A mixed number like \(2\frac{1}{2}\) always means \(2+\frac{1}{2}\), not \(2\times\frac{1}{2}\). If you mean multiplication, write \(2\times\frac{1}{2}\), \(2\cdot\frac{1}{2}\), or \((2)\!\left(\frac{1}{2}\right)\).
Method — Decomposing a Fraction Greater Than 1
To write \(\dfrac{n}{b}\) as a whole number plus a fraction less than \(1\):
  1. Count the whole units: each whole unit is made of \(b\) parts, so divide with remainder: \(n = q \times b + r\), with a remainder \(r\) smaller than \(b\).
  2. The \(r\) remaining parts form the fraction: \(\dfrac{n}{b} = q + \dfrac{r}{b}\), and \(\dfrac{r}{b} < 1\) because \(r < b\).
Conversely, to recompose a single fraction, each whole unit becomes \(b\) parts. On the number line, the decomposition also places the fraction between two consecutive whole numbers: \(q < \dfrac{n}{b} < q+1\) (when the remainder is not zero — if \(r = 0\), the fraction is exactly the whole number \(q\)).
Example
Write \(\dfrac{7}{3}\) as the sum of a whole number and a fraction less than \(1\).

\(\begin{aligned}[t]\dfrac{7}{3}&=\dfrac{2\times 3+1}{3}&&(\text{division of 7 by 3: } 7=2\times 3+1)\\&=\dfrac{2\times 3}{3}+\dfrac{1}{3}\\&=2+\dfrac{1}{3}\\\end{aligned}\) Figure

Example
Write \(3 + \dfrac{2}{5}\) as a single fraction.

\(\begin{aligned}[t]3+\dfrac{2}{5}&=\dfrac{3\times 5}{5}+\dfrac{2}{5}&&\left(3=\dfrac{15}{5}\right)\\&=\dfrac{15+2}{5}\\&=\dfrac{17}{5}\\\end{aligned}\)

Skills to practice
  • Finding Mixed Numbers from Bar Models
  • Decomposing into a Whole Number Plus a Fraction
  • Recomposing into a Single Fraction
  • Finding Fractions from Mixed Numbers
  • Finding Mixed Numbers from Fractions
VII Bounding a Fraction Between Two Whole Numbers
Method — Placing a Fraction Between Two Whole Numbers
Once a fraction greater than \(1\) is written as a whole number plus a fraction less than \(1\), its place on the number line is settled: the whole number is the one just below, and the next whole number is just above. $$\dfrac{7}{3} = 2 + \dfrac{1}{3} \qquad\text{so}\qquad 2 < \dfrac{7}{3} < 3$$
Figure
The two neighbours are read off the decomposition: never compute anything else. If the fractional part is zero, the fraction is that whole number itself, like \(\dfrac{6}{3} = 2\).
Example
Place \(\dfrac{11}{4}\) between two consecutive whole numbers.

\(\dfrac{11}{4}=2+\dfrac{3}{4}\) (division of 11 by 4: \(11 = 2 \times 4 + 3\)), so: $$2 < \dfrac{11}{4} < 3.$$

Example
Place \(\dfrac{29}{4}\) between two consecutive whole numbers.

$$\begin{aligned}[t] 29 &= 7 \times 4 + 1 && \text{(division with remainder by the denominator)} \\ \dfrac{29}{4} &= 7 + \dfrac{1}{4} && \text{(the 7 whole units, then the remaining part)} \\ 7 &< \dfrac{29}{4} < 8 && \text{(the remainder is not zero)} \end{aligned}$$

Skills to practice
  • Bounding Between Two Consecutive Whole Numbers
VIII Going further
Skills to practice
  • Reasoning and Problem-Solving
  • Quiz