International Baccalaureate · PYP 3

Fractions

⌚ ~178 min ▢ 20 blocks ✓ 85 exercises ➣ Prerequisites : Length, Fractions

I Equivalent Fractions

A cake is cut into 3 equal parts. One part, representing $\dfrac{1}{3}$ of the cake, is set aside.

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.

Even though the number of pieces changed, did the shaded amount of cake change?

Answer

No—the amount of cake did not change. The models show that the shaded portion is the same in both cases.

$\quad=\quad$

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.

Method — Recognising Equivalent Fractions

Two fractions are equivalent when they are the same amount of the same unit: on two identical bars, they cover the same length.

Say it. “If, for the same unit, I make parts twice as small and I take twice as many parts, then I take the same amount.” Quarters cut in two give eighths: $\dfrac{3}{4} = \dfrac{6}{8}$.
See it on two identical bars. The first bar is cut into quarters, the second into eighths: $\dfrac{3}{4}$ and $\dfrac{6}{8}$ cover the same length.
Find a missing numerator. $\dfrac{?}{8} = \dfrac{1}{2}$: eighths are $4$ times smaller than halves, so take $4$ times more parts: $\dfrac{4}{8} = \dfrac{1}{2}$.
Find a missing denominator. $\dfrac{2}{3} = \dfrac{4}{?}$: twice as many parts are taken, so the parts are twice as small: sixths, $\dfrac{2}{3} = \dfrac{4}{6}$.

Example

Among the fractions $\dfrac{1}{3}$, $\dfrac{2}{4}$, $\dfrac{3}{4}$, $\dfrac{2}{6}$ and $\dfrac{3}{6}$, which are equivalent to $\dfrac{1}{2}$? Then find the missing numerator: $\dfrac{?}{10} = \dfrac{2}{5}$.

Answer

$\dfrac{1}{2}$ is one part out of two: half of the whole.

$\dfrac{2}{4}$: parts twice as small, twice as many taken: equivalent to $\dfrac{1}{2}$.
$\dfrac{3}{6}$: parts three times as small, three times as many taken: equivalent to $\dfrac{1}{2}$.
$\dfrac{1}{3}$ is less than a half ($3$ parts, only $1$ taken), $\dfrac{3}{4}$ is more than a half, $\dfrac{2}{6}$ is equivalent to $\dfrac{1}{3}$, not to $\dfrac{1}{2}$.

Tenths are $2$ times smaller than fifths, so take $2$ times more parts: $\dfrac{4}{10} = \dfrac{2}{5}$.

Skills to practice

Shading a Fraction of the Bar
Finding the Missing Numerator
Finding the Missing Denominator

II On the Number Line

Discover

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 $\dfrac{1}{2}$.

Method — Representing a Fraction on the Number Line

To represent the fraction $\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}$ on a number line:

Draw a straight line and mark the points $0$ and $1$.
Divide the segment from $0$ to $1$ into $\textcolor{colorprop}{3}$ equal parts (the denominator).
Count $\textcolor{colordef}{2}$ parts from $0$ (the numerator) and mark the point at $\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}$.

Example — Equivalent Fractions

On a number line, two different fractions can mark the same point. Divide the unit from $0$ to $1$ into $2$ equal parts: $\dfrac{1}{2}$ is the middle point. Divide the same unit into $4$ equal parts: $\dfrac{2}{4}$ falls on that very same point.

So $\dfrac{1}{2}$ and $\dfrac{2}{4}$ are two names for the same number. We say they are equivalent.

The Fraction Is a Length

On the number line, a fraction is not only a point: it is also a length. The segment from $0$ to $\dfrac{3}{4}$ is $\dfrac{3}{4}$ of the unit long — three quarters of the way from $0$ to $1$.

A strip of paper as long as the unit, folded into $4$ equal parts, becomes a ruler graduated in quarters: laid along a segment, it measures the segment in quarters of a unit. With a grid, take a unit strip $10$ squares long: each square is one part, a ruler graduated in tenths.

Report an error

Method — Measuring with a Ruler Graduated in Fractions of a Unit

To measure a length that is not a whole number of units:

Graduate the unit. Fold the unit strip into $4$ equal parts for quarters, or take a unit strip $10$ squares long on a grid for tenths. For a length greater than the unit, lay the strip end to end as many times as needed, then read the remaining fraction.
Lay the ruler with its $0$ at one end of the segment.
Read the graduation at the other end, and give the answer in one of three ways:
- a fraction of the unit: “the segment is $\dfrac{3}{4}$ of a unit long”;
- a bound, when the end falls between two graduations: “the strip is between $\dfrac{7}{10}$ and $\dfrac{8}{10}$ of a unit long”;
- whole units and a fraction: “the segment is $2$ units and $\dfrac{1}{4}$ of a unit long”, also written $2 + \dfrac{1}{4}$.

The strip above ends between $\dfrac{7}{10}$ and $\dfrac{8}{10}$: its length is between $\dfrac{7}{10}$ and $\dfrac{8}{10}$ of a unit.

Method — Drawing a Segment of a Given Length

Choose the ruler whose graduations fit the fraction: quarters for $\dfrac{3}{4}$, tenths for $\dfrac{3}{10}$.
Start at $0$ and count the intervals: $3$ quarters for $\dfrac{3}{4}$, that is $3$ graduations after $0$.
If the fraction is not written in the ruler's graduations, use an equivalent fraction. On the tenths ruler, $\dfrac{1}{2}$ is $\dfrac{5}{10}$: count $5$ tenths.
For a length greater than the unit, such as $1 + \dfrac{3}{10}$, draw the whole unit first, then $3$ more tenths.

Skills to practice

Graduating the Unit Strip
Finding Fractions with the Bar Model
Finding Fractions on the Number Line
Placing Fractions on the Number Line
Placing Equivalent Fractions on the Number Line
Reading a Length on the Ruler
Bounding a Length Between Two Graduations
Drawing a Segment of a Given Length

III Comparing Fractions

Discover

Léa and Tom each have a strip of paper. The two strips are identical, and both are cut into tenths. Léa has coloured $\dfrac{3}{10}$ of hers, Tom $\dfrac{7}{10}$ of his. Léa

Tom

Say who has coloured more.

Answer

Tom has. And nothing had to be counted twice: the two strips are identical, cut the same way, so every part is the same size.
When the parts are the same size, only how many are coloured decides. Tom has $7$ coloured parts, Léa has $3$, and $3$ is smaller than $7$. $$\dfrac{3}{10} \lt \dfrac{7}{10}$$

Method — Comparing Fractions on the Number Line

Two fractions placed on the same line compare like lengths: the one further to the right is the greater. Three cases:

Same denominator. Same graduations: the greater numerator goes further. $\dfrac{5}{12}$ is $5$ twelfths, $\dfrac{7}{12}$ is $7$ twelfths: $\dfrac{5}{12} < \dfrac{7}{12}$.
Same numerator. The same number of steps, but the steps are smaller when the denominator is greater: $5$ twelfths are $5$ small steps, $5$ eighths are $5$ bigger steps. $\dfrac{5}{12} < \dfrac{5}{8}$.
One denominator is a multiple of the other. Write the fraction with the bigger parts in the finer graduations: $\dfrac{5}{6} = \dfrac{10}{12}$ (each sixth is two twelfths). Then compare on the twelfths line: $\dfrac{7}{12} < \dfrac{10}{12}$, so $\dfrac{7}{12} < \dfrac{5}{6}$.

Example

Compare $\dfrac{5}{12}$ and $\dfrac{5}{8}$; then $\dfrac{2}{3}$ and $\dfrac{7}{12}$. Finally, order $\dfrac{1}{2}$, $\dfrac{5}{8}$ and $\dfrac{3}{4}$ from the smallest to the largest using the line graduated in eighths.

Answer

Same numerator: $5$ steps each time, but an eighth is a bigger step than a twelfth, so $\dfrac{5}{12} < \dfrac{5}{8}$.
$12$ is a multiple of $3$: each third is four twelfths, $\dfrac{2}{3} = \dfrac{8}{12}$. On the twelfths line, $\dfrac{7}{12} < \dfrac{8}{12}$, so $\dfrac{7}{12} < \dfrac{2}{3}$.
On the eighths line: $\dfrac{1}{2} = \dfrac{4}{8}$ and $\dfrac{3}{4} = \dfrac{6}{8}$. Reading from left to right: $\dfrac{4}{8} < \dfrac{5}{8} < \dfrac{6}{8}$, that is $\dfrac{1}{2} < \dfrac{5}{8} < \dfrac{3}{4}$.

Skills to practice

Comparing Fractions on the Number Line
Ordering Fractions on the Ruler

IV Adding with the Same Denominator

Discover

Hugo's cake is cut into $4$ equal parts (quarters). Hugo eats $\dfrac{2}{4}$ of the cake $=$

, then $\dfrac{1}{4}$ more $=$

.
What fraction of the cake has he eaten?

Answer

The parts are all the same size (quarters), so the eaten parts can simply be counted together: $2+1=3$ quarters.

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:

$$\begin{aligned}\dfrac{2}{4} + \dfrac{1}{4} &= \dfrac{2 + 1}{4}\\ &= \dfrac{3}{4}\end{aligned}$$

Skills to practice

Adding Fractions with the Same Denominator

V Subtracting with the Same Denominator

Discover

Another day, $\dfrac{3}{4}$ of Hugo's cake is left $=$

. Hugo eats $\dfrac{1}{4}$ more.
What fraction of the cake is left now?

Answer

The parts are the same size (quarters), so subtract the counts: $3-1=2$ quarters are left.

$\dfrac{2}{4}$ 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:

$$\begin{aligned}\dfrac{3}{4} - \dfrac{1}{4} &= \dfrac{3 - 1}{4}\\ &= \dfrac{2}{4}\end{aligned}$$ 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:

. 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:

. $$\dfrac{0}{3} = 0$$

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 $1$.

$$\dfrac{3}{3} = 1$$ When the top number is $0$, no part is taken: the fraction is worth $0$.

$$\dfrac{0}{3} = 0$$

Skills to practice

Subtracting Fractions with the Same Denominator

VI Addition and Subtraction with Unlike Denominators

Discover

To add or subtract fractions by combining their numerators, the fractions must first represent parts of the same size, i.e., be written with a common denominator.
Consider the problem of adding $\dfrac{1}{2}$ and $\dfrac{1}{4}$.

$+$

Because the fractions have unlike denominators (2 and 4), the parts are of different sizes. A direct addition of the numerators is not possible. To solve this, we must first express the fractions with a common denominator.

Method — Procedure for Adding or Subtracting Fractions

To add or subtract fractions with unlike denominators, follow this three-step procedure:

Find a Common Denominator: Identify a common multiple of the denominators.
Create Equivalent Fractions: Convert each fraction to an equivalent fraction with the common denominator.
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.

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

Example

We will apply the procedure to calculate $\dfrac{1}{2} + \dfrac{1}{4}$.

Step 1: Find a common denominator. The denominators are 2 and 4. A common multiple is 4.
Step 2: Create equivalent fractions.
- The fraction $\dfrac{1}{4}$ already has the common denominator.
- Convert $\dfrac{1}{2}$ into an equivalent fraction with a denominator of 4:
Step 3: Add the numerators. Now that both fractions have the same denominator, we can add them: $$ \frac{2}{4} + \frac{1}{4} = \frac{2+1}{4} = \frac{3}{4} $$

The visual representation of this process is:

$+$	$=$	$+$
	$=$
$\dfrac{1}{2}+\dfrac{1}{4}$	$=$	$\dfrac{2}{4}+\dfrac{1}{4}$
	$=$	$\dfrac{3}{4}$

The total fraction is $\dfrac{3}{4}$.

Example

Compute $\dfrac{3}{10} + \dfrac{2}{5}$ and $\dfrac{5}{6} - \dfrac{1}{3}$.

Answer

$10$ is a multiple of $5$: the common denominator is $10$. $$\begin{aligned}\dfrac{3}{10} + \dfrac{2}{5} &= \dfrac{3}{10} + \dfrac{4}{10} && \text{(each fifth is two tenths)}\\ &= \dfrac{7}{10}\end{aligned}$$
$6$ is a multiple of $3$: the common denominator is $6$. $$\begin{aligned}\dfrac{5}{6} - \dfrac{1}{3} &= \dfrac{5}{6} - \dfrac{2}{6} && \text{(each third is two sixths)}\\ &= \dfrac{3}{6}\end{aligned}$$ Three sixths is one half.

Skills to practice

Placing the Sum on the Number Line
Adding When One Denominator Is a Multiple of the Other
Subtracting When One Denominator Is a Multiple of the Other

VII Going further

Skills to practice

Shaded Part and White Part
From the Bar to the Ruler
Quiz

\(+\)	\(=\)	\(+\)
	\(=\)
\(\dfrac{1}{2}+\dfrac{1}{4}\)	\(=\)	\(\dfrac{2}{4}+\dfrac{1}{4}\)
	\(=\)	\(\dfrac{3}{4}\)