Stage 3

5.16 Compare and write fraction sentences

Lesson

Practice

Lesson

Ideas

Are you ready?
Compare fraction statements
Compare fraction statements of mixed numbers
Number sentences with fractions

Are you ready?

If we can add and subtract fractions , that will help us in this lesson. Let's try a review problem now.

Examples

Example 1

Find the value of \,\dfrac{3}{5}+\dfrac{3}{4}.

Worked Solution

Create a strategy

We need to find the smallest common multiple of the denominators.

Apply the idea

The smallest common multiple of 5 and 4 is 20 since 5\times 4 =20. So the least common denominator of the two fractions is 20.

We need to multiply both the numerator and denominator of \dfrac{3}{5} by 4 to get a denominator of 20.

\displaystyle \dfrac{3}{5}	\displaystyle =	\displaystyle \dfrac{3\times 4}{5\times 4}	Multiply numerator and denominator by 4
	\displaystyle =	\displaystyle \dfrac{12}{20}

We need to multiply both the numerator and denominator of \dfrac{3}{4} by 5 to get a denominator of 20.

\displaystyle \dfrac{3}{4}	\displaystyle =	\displaystyle \dfrac{3\times 5}{4\times 5}	Multiply numerator and denominator by 5
	\displaystyle =	\displaystyle \dfrac{15}{20}

Now we can use these equivalent fractions in the addition.

\displaystyle \dfrac{3}{5}+\dfrac{3}{4}	\displaystyle =	\displaystyle \dfrac{12}{20}+\dfrac{15}{20}	Use the new fractions
	\displaystyle =	\displaystyle \dfrac{12+15}{20}	Add the numerators
	\displaystyle =	\displaystyle \dfrac{27}{20}

Idea summary

Before we add or subtract fractions, we must first make sure that the fractions have the same denominator.

Compare fraction statements

How to compare the size of statements that involve fractions.

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Examples

Example 2

We want to compare \dfrac{8}{12} to \dfrac{1}{12}+\dfrac{2}{4}.

Convert \dfrac{2}{4} into twelfths.

Worked Solution

Create a strategy

Use area models.

Apply the idea

A square that is divided into 4 parts and 2 parts are shaded.

The shaded area here represents \dfrac{2}{4}.

A square that is divided into 12 parts and 6 parts are only shaded.

Cutting the square into 12 parts, we now have 6 parts out of 12 that are shaded.

So the shaded area also represents \dfrac{6}{12}.

Since the areas are equal the fractions must also be equal:\dfrac{2}{4}=\dfrac{6}{12}

Write the symbol, <, > or =, that makes the statement true:\dfrac{8}{12} ⬚ \dfrac{1}{12}+\dfrac{2}{4}

Worked Solution

Create a strategy

Use area models to represent each fraction, similar to the one we used in part (a).

Apply the idea

A square that is divided into 12 parts and 8 parts are shaded.

The area model on the left represents \dfrac{8}{12}.

A square divided into 12 parts and 1 part is only shaded plus a square divided into 12 parts and 6 parts are shaded.

The image on the left represents the right side of the statement \dfrac{1}{12}+\dfrac{6}{12} or \dfrac{1}{12}+\dfrac{2}{4}.

There are 7 squares shaded in total, so \dfrac{1}{12}+\dfrac{6}{12}=\dfrac{7}{12}.

Since 8 \gt 7, we know that \dfrac{8}{12} \gt \dfrac{7}{12}. Since \dfrac{1}{12}+\dfrac{2}{4} =\dfrac{7}{12} the complete statement is:\dfrac{8}{12}>\dfrac{1}{12}+\dfrac{2}{4}

Idea summary

We can use area models to compare the size of statements that involve fractions.

Compare fraction statements of mixed numbers

This video looks at comparing statements that involve mixed numbers

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Examples

Example 3

We want to compare 8\dfrac{9}{10}-2\dfrac{1}{5} to 6\dfrac{3}{5}.

Convert 2\dfrac{1}{5} and 6\dfrac{3}{5} into tenths.

Worked Solution

Create a strategy

Use area models to convert the fraction parts.

Apply the idea

The area models below represent \dfrac{1}{5} and \dfrac{3}{5}.

Two squares that are both divided into 5 parts. 1 square has 1 part shaded and the other has 3 parts shaded.

Cutting the two squares into 10 parts each, we now have 2 shaded parts out of 10 parts for the first square and 6 shaded parts out of 10 parts for the second square.

Two squares that are both divided into 10 parts. 1 square has 2 parts shaded and the other has 6 parts shaded.

So \dfrac{1}{5}=\dfrac{2}{10} and \dfrac{3}{5}=\dfrac{6}{10}.

So we get these equivalent mixed numbers: 2\dfrac{1}{5}=2\dfrac{2}{10} and 6\dfrac{3}{5}=6\dfrac{6}{10}.

Write the symbol, <, > or =, that makes the statement true.8\dfrac{9}{10}-2\dfrac{1}{5} \,⬚ \, 6\dfrac{3}{5}

Worked Solution

Create a strategy

Use area models to subtract and compare the fraction parts.

Apply the idea

On the left side, the whole numbers subtract to: 8-2=6 which equals the whole number on the right side. So we just need to compare the fraction parts.

2 squares divided into 10 parts each with a minus in between them. Ask your teacher for more information.

The image on the left represents the fractional part of the left side of the statement \dfrac{9}{10}-\dfrac{1}{5} or \dfrac{9}{10}-\dfrac{2}{10}.

9 shaded parts take away 2 shaded parts equals 7 shaded parts. So: \dfrac{9}{10}-\dfrac{2}{10}=\dfrac{7}{10}

A square divided into 10 parts and 6 parts are shaded.

The image on the left represents the fractional part of the right side statement \dfrac{3}{5}=\dfrac{6}{10}.

Since 7\gt 6 we know that \dfrac{7}{10}\gt \dfrac{6}{10}.8\dfrac{9}{10}-2\dfrac{1}{5} > 6\dfrac{3}{5}

Idea summary

We can use area models to compare the fraction parts of mixed number statements.

Number sentences with fractions

This video will show us how to write and use number sentences with fractions.

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Examples

Example 4

Hannah had climbed one seventh of the ladder to the roof when she realised she'd forgotten her phone, so went back down to get it.

Complete the number sentence that describes how far in total Hannah has climbed once she has returned to the bottom of the ladder.

Hannah has climbed ⬚ seventh plus ⬚ seventh of the length of the ladder.

Worked Solution

Create a strategy

Use the fact that Hannah will climb the same distance on the way down as she did on the way up.

Apply the idea

Since the fraction of the length of the ladder that Hannah already climbed is \dfrac{1}{7} or 1 seventh, she will need to climb this distance again on the way down.

Hannah has climbed 1 seventh plus 1 seventh of the length of the ladder.

Idea summary

Evaluate the statement and then compare the values.

To help solve the statement, draw a representation (a number line or rectangle).

5.16 Compare and write fraction sentences

Ideas

Are you ready?

Examples

Example 1

Compare fraction statements

Examples

Example 2

Compare fraction statements of mixed numbers

Examples

Example 3

Number sentences with fractions

Examples

Example 4

Outcomes

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