In this unit we are exploring ways to find equivalent fractions. We use the concept of equivalent fractions to convert fractions to the benchmark fractions of halves, quarters, thirds, fifths and tenths. From these benchmark fractions it is easier to convert fractions to decimals and percentages. We use equivalent fractions to compare fractions.
- explore and know equivalent fractions including halves, thirds, quarters, fifths, tenths and hundredths.
- use equivalent fractions to convert fractions to decimals and percentages.
- use equivalent fractions to order fractions with different denominators.
This unit first develops an understanding of why two or more fractions can be called equivalent fractions. The students will explore ways to visualize and show that two (or more) fractions are equivalent. Equivalent fractions have the same ratio, for example 9/12 is the same ratio of 3/4. This can be shown using sets so that 9 is made up of 3 sets of 3 and 12 is made up of 4 sets of 3.
Students will then explore the use of equivalent fractions. In many problems it is useful to be able to convert a fraction to a decimal or percentage. By converting a fraction to a simpler equivalent fraction it is often easier to then convert the fraction to a decimal or percentage. For example 3/12 is equivalent to 1/4 and this is 0.25 or 25%.
By converting fractions to a simpler equivalent fraction it is easier to compare fractions with different denominators. The conversion can be made so the fractions have the same denominator, for example 16/20 and 15/25 can be converted so 16/20 is 4/5 and 15/20 is 3/5. The comparison between 4/5 and 3/5 is straightforward. The idea of a common denominator is introduced to give students another strategy for comparing fractions. It may also be possible to convert fractions to equivalent benchmark fractions to allow an easier comparison of size. For example 4/16 and 3/9 are equivalent to 1/4 and 1/3.
- Begin a discussion by showing the students two grids of the same size marked in different grid spaces. For example
- Ask the students:
Is the same amount of space shaded in both of the grids? (Yes)
What fraction of the space is shaded in the first grid? (3/4)
What fraction of the space is shaded in the second grid? (12/16)
- Discuss with the students that these two fractions are called equivalent fractions.
- Give the students squares of paper. Ask them to fold the paper into quarters and colour in a quarter and then to fold it into eighths Ask the students:
How many of the eighths are coloured in? (2)
What fraction is equivalent to 1/4? (2/8)
- The students may wish to try more examples by folding paper into thirds then sixths or fifths then tenths.
- Try another example using grids.
- Ask the students What fraction of the space is shaded in the first grid? (2/3)
What fractions of the space is shaded in the second grid? (10/15)
How many of the columns are shaded in the first grid? (2/3)
How many of the columns are shaded in the second grid? (2/3)
- Discuss with students that 10/15 are 2/3 are equivalent.
- Show the students 18 counters 12 of which are red.
Ask the students
How is this expressed as a fraction? (12/18)
- Arrange the counters in 3 columns and ask the students:
How many columns are there? (3)
How many of columns are made up of red counters? (2)
- Give the students a pile of 20 counters where 15 are red. Ask them to arrange them in columns so that the columns are the same size and there are complete columns of red counters like the previous example.
- Discuss with the students how they arranged the counters and how they found the equivalent fraction of 3/4.
- As you do more examples together discuss with the students the idea of making complete columns by splitting both the total number of counters and the number of red counters by the same number. Examples could include 4/16, 5/15, 2/8, 12/16, 8/12, 8/20.
- As you work through finding equivalent fractions students may notice that there is often more than two equivalent fractions for example 4/16 is equivalent to 1/4 and to 2/8.
Over the next week or so students will continue to explore finding equivalent fractions. They will use equivalent fractions to convert fractions to benchmark fractions and then to decimals or percentages. They will use equivalent fractions to compare and order fractions.
- Give the students time to continue exploring ways to find equivalent fractions. The students can use counters, folding paper and pictures to solve problems. Pairs of students could draw equivalent fractions pictures for example one picture has 4 sports cars in car park of 6 cars and the other picture has 6 sport cars in a car park of 9 cars. Students could explore the idea of starting with a unit fraction and writing some of its equivalent fractions. They could look at the skip counting pattern for example 1/4, 2/8, 3/12, 4/16 etc. Students can explore using big numbers for example 150/400 = 15/40 = 3/8. Encourage the students to find the simplest equivalent fraction, one that can not be reduced any smaller. Students could write their own questions and discuss what they needed to consider when choosing the numbers. For example ask a friend to colour in the equivalent of 3/4 of the 12 star shapes. Students could explore the relationship that 3/5 of 15 = 9 and 3/5 & 9/15 are equivalent fractions
- Discuss with the students that sometimes it is helpful to be able to convert a fraction to a decimal or a percentage. Work with the students to convert unit fractions of 1/2, 1/4, 1/3, 1/5 and 1/10 to hundredths. These can then be easily converted to decimals or percentages. For example 1/2 is equivalent to 50/100 which can be expressed as 0.50 or 50%. The students can explore converting halves, thirds, quarters, fifths and tenths to decimals or percentages, for example 3/4 = 75/100 or 75%. It is useful for students to know these conversions. Games like snap and memory can help students to learn these. The students can then convert other fractions to these simpler fractions and then to decimals or percentages. For example 36/45 = 4/5 = 0.8 or 80%.
- Discuss with the students the need to be able to compare fractions. Three different strategies can be discussed:
- For example pose the question:
What is a better saving (i) saving $3 on a $5 item or (ii) saving $6 on a $15 item? Discuss that the question is asking "what is bigger (therefore a better saving) 3/5 or 6/15?" Explore with the students how to convert these fractions to equivalent fractions so the denominators are both the same. 3/5 is equivalent to 9/15, 9/15 (or 3/5) is bigger than 6/15 so the $6 item has the bigger saving. It will be easier if you pose questions where one denominator is a multiple of the other, for example 4 and 12.
- A common denominator can also be found by using a common multiple. For example with the fractions 3/4 and 2/6 the common multiple of 4 and 6 is 12. The fractions are then converted to denominators of 12 to give 9/12 and 4/12 respectively.
- Students can also find simpler equivalent fractions and then compare them. For example 4/16 and 3/9 are equivalent to 1/4 and 1/3.
- For example pose the question:
Students could share ways of representing equivalent fractions with materials or pictures. Students could share questions and discuss how they chose the numbers and the ways the answer could be found. The group could make and play games involving equivalent fractions and converting between fractions and decimals, for example snap, memory or bingo.