Fractions and equivalence of shapes
🎯 In this topic you will
- Explore fractions including thirds, fifths, and tenths.
- Show that different fractions can have equivalent values.
- Understand the relationship between the whole and its parts.
🧠 Key Words
- denominator
- numerator
- equal
- equivalent
- thirds
- fifths
- tenths
Show Definitions
- denominator: The bottom number in a fraction that shows how many equal parts the whole is divided into.
- numerator: The top number in a fraction that shows how many parts are being used.
- equal: Having the same value or size.
- equivalent: Different fractions that have the same value, even though they look different (for example, $\frac12=\frac24$).
- thirds: Parts made when a whole is divided into three equal pieces.
- fifths: Parts made when a whole is divided into five equal pieces.
- tenths: Parts made when a whole is divided into ten equal pieces.
⏰ Fractions in Everyday Life
F ractions help us to tell the time on an analogue clock and to follow recipes when cooking.
🕒 Understanding Quarter Past
T his clock shows quarter past six.
❓ EXERCISES
1. Colour three parts green. Colour five parts yellow.
a. What fraction of the circle is green?
b. What fraction of the circle is yellow?
c. What fraction of the circle is not coloured?
👀 Show answer
The circle has $10$ equal parts.
a. Green: $3$ parts → $\dfrac{3}{10}$
b. Yellow: $5$ parts → $\dfrac{5}{10} = \dfrac{1}{2}$
c. Not coloured: $2$ parts → $\dfrac{2}{10} = \dfrac{1}{5}$
2. How would you share pizza between you and your friends? How many slices would you get?
a. If you and nine friends are sharing the pizza, how many slices would you each get?
b. If you and four friends are sharing the pizza, how many slices would you each get.
c. What is the fraction of the whole pizza that you would each get if
• you and nine friends are sharing.
• you and four friends are sharing.
👀 Show answer
The pizza is cut into $10$ slices.
a. You + $9$ friends = $10$ people → each gets $1$ slice.
b. You + $4$ friends = $5$ people → each gets $2$ slices.
c. Fractions of the whole pizza:
• With $10$ people: $\dfrac{1}{10}$ each.
• With $5$ people: $\dfrac{2}{10} = \dfrac{1}{5}$ each.
3. Use these fraction strips to find equivalent fractions. Write the correct fraction in the boxes.
a. $\dfrac{1}{2} = \dfrac{\square}{4}$ and $\dfrac{1}{5} = \dfrac{\square}{10}$
b. Draw your own fraction strips for quarters and halves and use them to find an equivalent fraction.
👀 Show answer
a. $\dfrac{1}{2} = \dfrac{2}{4}$
$\dfrac{1}{5} = \dfrac{2}{10}$
b. One example: $\dfrac{1}{2} = \dfrac{2}{4}$
4. Some children made a poster about the fraction one-third.
Ask your teacher for a large piece of paper. Choose a fraction and make your own fraction poster.
Show your fraction in as many different ways as you can.
👀 Show answer
This is an open-ended activity. Any clear poster showing one chosen fraction in several different visual ways is acceptable.
5. If this is $\dfrac{2}{4}$ of a shape, draw what the whole shape could look like.
Remember that your triangles must be the same size as these.
👀 Show answer
Since $\dfrac{2}{4} = \dfrac{1}{2}$, the whole shape is made by adding two more identical triangles. Several correct whole shapes are possible.
🧠 Think like a Mathematician — Designing tiles
a. This tile shows a pattern made with fractions.
What fraction of the tile is shaded?
What fraction of the tile is not shaded?
b. Use this square to create a design with a different fraction shaded. Some of the dotted lines may help.
Continued
Design your own tile to show thirds, fifths or tenths.
What fraction did you use?
Share your design with a partner. Ask them to find the fraction that is shaded and the fraction that has not been shaded.
What fraction did your partner use in their design?
👀 Show answers
a. The rectangle is divided into equal triangular regions. The shaded area covers exactly half of the tile.
Shaded fraction: $\dfrac{1}{2}$
Not shaded fraction: $\dfrac{1}{2}$
b. One possible design is to shade $3$ out of $8$ equal parts, giving $\dfrac{3}{8}$. Many different correct designs are possible.
Design your own tile: Example answers:
- Thirds: shade $1$ out of $3$ → $\dfrac{1}{3}$
- Fifths: shade $2$ out of $5$ → $\dfrac{2}{5}$
- Tenths: shade $4$ out of $10$ → $\dfrac{4}{10}=\dfrac{2}{5}$
Your partner’s fraction will depend on their design, but it must match the number of shaded parts divided by the total number of equal parts.