021. Representing Fractions

Learning Intentions

• To know what the numerator and denominator of a fraction represent in different situations
• represent fractions on a number line
• represent a fraction of a shape by dividing it into several regions and shading some of the regions

Pre-requisite Summary

• Understand that a whole can be divided into equal parts
• Know that equal parts must be the same size for a fraction to be valid
• Recognise basic fractions such as ${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{4}}$
• Understand that the denominator tells how many equal parts the whole is divided into
• Understand that the numerator tells how many of those equal parts are being counted
• Be able to count partitions and shaded parts accurately
• Be able to locate whole numbers on a number line
• Understand that fractions can describe part of a region, part of a set, or a point on a number line

Worked Examples

Worked Example 1

For each fraction, state what the numerator and denominator represent:
a) ${\displaystyle \frac{3}{5}}$ of a pizza
b) ${\displaystyle \frac{2}{7}}$ of a class set of counters
c) ${\displaystyle \frac{4}{9}}$ of a strip divided into equal parts

Worked Example 2

Represent each fraction on a number line from $0$ to $1$:
a) ${\displaystyle \frac{1}{4}}$
b) ${\displaystyle \frac{3}{4}}$
c) ${\displaystyle \frac{5}{6}}$

Worked Example 3

Represent each fraction of a shape by dividing and shading:
a) ${\displaystyle \frac{1}{2}}$ of a rectangle
b) ${\displaystyle \frac{3}{4}}$ of a square
c) ${\displaystyle \frac{2}{3}}$ of a circle

Worked Example 4

A rectangle is divided into $8$ equal regions and $3$ are shaded.
a) Write the fraction shaded.
b) State what the numerator represents.
c) State what the denominator represents.

Worked Example 5

A number line from $0$ to $1$ is divided into $5$ equal parts.
a) Which point represents ${\displaystyle \frac{2}{5}}$?
b) Which point represents ${\displaystyle \frac{4}{5}}$?
c) Explain how the denominator helps place the points.

Worked Example 6

A hexagon is divided into $6$ equal regions.
a) Shade ${\displaystyle \frac{5}{6}}$ of the shape.
b) Explain how many regions must be shaded.
c) Explain why the regions must be equal.

Problems

Problem 1

For each fraction, state what the numerator and denominator represent:
a) ${\displaystyle \frac{2}{5}}$ of a cake
b) ${\displaystyle \frac{4}{7}}$ of a set of marbles
c) ${\displaystyle \frac{3}{8}}$ of a bar divided into equal parts

Problem 2

Represent each fraction on a number line from $0$ to $1$:
a) ${\displaystyle \frac{1}{3}}$
b) ${\displaystyle \frac{2}{3}}$
c) ${\displaystyle \frac{4}{5}}$

Problem 3

Represent each fraction of a shape by dividing and shading:
a) ${\displaystyle \frac{1}{4}}$ of a rectangle
b) ${\displaystyle \frac{2}{5}}$ of a strip
c) ${\displaystyle \frac{3}{6}}$ of a circle

Problem 4

A rectangle is divided into $10$ equal regions and $6$ are shaded.
a) Write the fraction shaded.
b) State what the numerator represents.
c) State what the denominator represents.

Problem 5

A number line from $0$ to $1$ is divided into $8$ equal parts.
a) Which point represents ${\displaystyle \frac{3}{8}}$?
b) Which point represents ${\displaystyle \frac{7}{8}}$?
c) Explain how the denominator helps place the points.

Problem 6

A pentagon is divided into $5$ equal regions.
a) Shade ${\displaystyle \frac{4}{5}}$ of the shape.
b) Explain how many regions must be shaded.
c) Explain why the regions must be equal.

Exercises

Understanding and Fluency

1. For each fraction, state what the numerator and denominator represent:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{3}{4}}$
   c) ${\displaystyle \frac{5}{8}}$

2. A whole is divided into equal parts. State what each fraction means:
   a) ${\displaystyle \frac{2}{3}}$
   b) ${\displaystyle \frac{4}{5}}$
   c) ${\displaystyle \frac{7}{10}}$

3. Represent each fraction on a number line from $0$ to $1$:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{1}{5}}$
   c) ${\displaystyle \frac{4}{5}}$

4. Represent each fraction on a number line from $0$ to $1$:
   a) ${\displaystyle \frac{2}{4}}$
   b) ${\displaystyle \frac{3}{6}}$
   c) ${\displaystyle \frac{5}{8}}$

5. Divide each shape into equal regions and shade the given fraction:
   a) ${\displaystyle \frac{1}{3}}$ of a rectangle
   b) ${\displaystyle \frac{3}{5}}$ of a strip
   c) ${\displaystyle \frac{2}{4}}$ of a square

6. Divide each shape into equal regions and shade the given fraction:
   a) ${\displaystyle \frac{4}{6}}$ of a circle
   b) ${\displaystyle \frac{5}{8}}$ of a rectangle
   c) ${\displaystyle \frac{7}{10}}$ of a bar

7. A shape is divided into equal parts. Write the fraction shaded:
   a) $2$ out of $6$ parts shaded
   b) $5$ out of $8$ parts shaded
   c) $3$ out of $4$ parts shaded

8. A number line from $0$ to $1$ is divided into equal parts. State the fraction at the marked point:
   a) the $2$nd point out of $5$ equal parts
   b) the $3$rd point out of $4$ equal parts
   c) the $7$th point out of $8$ equal parts

Reasoning

9. Explain why the parts of a shape must be equal when representing a fraction.

10. A student says in ${\displaystyle \frac{3}{7}}$ the $3$ tells how many equal parts the whole is divided into. Explain the mistake.

11. Explain why ${\displaystyle \frac{4}{5}}$ on a number line must be closer to $1$ than ${\displaystyle \frac{1}{5}}$.

12. A student shades $3$ parts of a shape divided into unequal regions and says the shaded fraction is ${\displaystyle \frac{3}{4}}$. Explain why this may be incorrect.

Problem-solving

13. A chocolate block is divided into $8$ equal pieces. Mia eats $3$ pieces. What fraction of the block does she eat?

14. A ribbon from $0$ to $1$ metre is marked into $6$ equal parts. At what fraction is the $4$th mark?

15. A rectangle is divided into $12$ equal regions. Shade a fraction showing that $9$ regions are shaded. What is the fraction?

16. A class uses a strip model divided into $5$ equal parts. If $2$ parts are coloured, what fraction is coloured?

17. On a number line from $0$ to $1$, a point is at the $3$rd division when the line is split into $7$ equal parts. What fraction is represented?

18. A circle is divided into $10$ equal sectors. If ${\displaystyle \frac{7}{10}}$ is shaded, how many sectors are shaded?

Potential Misunderstandings

• Students may confuse the numerator and denominator
• Students may think the denominator counts shaded parts rather than total equal parts
• Students may think the numerator tells the total number of parts
• Students may place fractions randomly on a number line instead of partitioning the interval into equal parts
• Students may forget that the number line from $0$ to $1$ must be divided into the number of equal parts given by the denominator
• Students may divide shapes into unequal regions and still label the result as a fraction
• Students may count boundary lines instead of intervals on a number line
• Students may think fractions only describe shaded regions and not positions on a number line

Next: 022. Equivalent Fractions and Simplifying them