Purpose
These exercises and activities are for students to use independently of the teacher to practice and develop their number properties.
Specific Learning Outcomes
add and subtract fractions with the same denominator
convert fractions (tenths) to decimals
find fractions of a quantity
Description of Mathematics
Proportions and Ratios, AA (Stage 6)
Required Resource Materials
Practice exercises with answers (PDF or Word)
Activity
Prior knowledge.
- Explain why 10/10 is another name for one whole
- Identify mixed numbers and explain how they relate to improper fractions
- Find fractions of a quantity
Background
In this activity students make the connection between tenths written as a fraction and written as a decimal. Students add together tenths, and find tenths of a set for example 7/10 of 20.
Comments on the Exercises
Exercise 1: Adding tenths
Asks students to solve word problems that involve adding tenths. For example, 7/10 + 1/10. With the exception of the last 2 problems, in this exercise the answers are less than one, and as the questions require students to convert their answers to decimals, simplification of the fractions is not required, but could be discussed as another way of ‘tidying up’ the numbers.
Exercise 2: Adding tenths
Asks students to solve word problems that involve adding tenths. For example, 7/10 + 5/10. It also includes adding mixed fraction. In this exercise, all of the answers are greater than one. Question 7 and beyond represent a slight change in the pattern. Here each rooster is eating more than one worm, so the answers can be sorted by ‘adding like to like’, that is adding the whole numbers and the fractional parts separately.
Exercise 3: Adding and subtracting tenths. Finding “tenths of …”
Asks students to combine the skills of adding fractions to one whole and finding the fraction of an amount. In the last problem, Tim and Ted are equally sharing nine tenths, meaning they get four and a half tenths each. Discussing how to write such fractions should be covered either before the exercise is sat, or afterwards as a debrief, where the students are asked to explain how they met this challenge. Going back to materials (decimats are a good resource here) to display (4/2)/10 and sorting out why it equals 9/20 is useful.
Exercise 4: Finding tenths of coins/working with money.
Asks students to not only work out fractions of an amount, but also work out how much money that have with this number of coins
Fair Shares
These exercises and activities are for students to use independently of the teacher to practice number properties
Fractions ratios and proportions, stages 3 to 4, counting from one by imaging to advanced counting
Prior knowledge. Students should be able to:
Background
These exercises are for students with little fractional knowledge and focus on the ‘parts of a whole’ construct of fractions.
When marking, it is important to discuss with students that it does not matter which pieces are shaded, as long as the correct number are shaded. Likewise for exercise 3, there are a range of different shapes that could be the whole figure. Note that several shapes have been left unshaded in the answers for exercise 1, so students come to you for this discussion.
These activities can be used to follow the teaching episodes based on, Book 7, pages 2 to 4 and are for those students who are able to use the associated number properties.
Fraction wafers
These exercises and activities are for students to use independently of the teacher to practice number properties.
Proportions and Ratios, EA (Stage 5)
Prior knowledge
Background
There are a several significant ideas that arise in this activity, and need to be addressed in the teaching that precedes these activities.
Cutting into 4, 3 and 5 pieces all require different strategies for success
Working with cutting shapes and dividing written lines should be included with the written problems.
Comments on the Exercises
Exercise 1
Asks students to draw pictures to help solve problems that involve a number of people sharing equally a set of items. For example, 4 people share 3 bags of popcorn between them. How much of a bag will each person get. In this questions there are more people than items so the answers are less than 1.
Exercise 2
Asks students to draw pictures to help solve problems that involve a number of people sharing equally a set of items. But in these questions the answers are greater than 1.
These activities can be used to follow the teaching episodes based on, Book7, Wafers and are for those students who are able to use the associated number properties.
Decimal Fractions of a Set
These exercises and activities are for students to use independently of the teacher to practice and develop their number properties.
add and subtract fractions with the same denominator
convert fractions (tenths) to decimals
find fractions of a quantity
Proportions and Ratios, AA (Stage 6)
Prior knowledge.
Background
In this activity students make the connection between tenths written as a fraction and written as a decimal. Students add together tenths, and find tenths of a set for example 7/10 of 20.
Comments on the Exercises
Exercise 1: Adding tenths
Asks students to solve word problems that involve adding tenths. For example, 7/10 + 1/10. With the exception of the last 2 problems, in this exercise the answers are less than one, and as the questions require students to convert their answers to decimals, simplification of the fractions is not required, but could be discussed as another way of ‘tidying up’ the numbers.
Exercise 2: Adding tenths
Asks students to solve word problems that involve adding tenths. For example, 7/10 + 5/10. It also includes adding mixed fraction. In this exercise, all of the answers are greater than one. Question 7 and beyond represent a slight change in the pattern. Here each rooster is eating more than one worm, so the answers can be sorted by ‘adding like to like’, that is adding the whole numbers and the fractional parts separately.
Exercise 3: Adding and subtracting tenths. Finding “tenths of …”
Asks students to combine the skills of adding fractions to one whole and finding the fraction of an amount. In the last problem, Tim and Ted are equally sharing nine tenths, meaning they get four and a half tenths each. Discussing how to write such fractions should be covered either before the exercise is sat, or afterwards as a debrief, where the students are asked to explain how they met this challenge. Going back to materials (decimats are a good resource here) to display (4/2)/10 and sorting out why it equals 9/20 is useful.
Exercise 4: Finding tenths of coins/working with money.
Asks students to not only work out fractions of an amount, but also work out how much money that have with this number of coins
These activities can be used to follow the teaching episodes based on, Book 7, Hungry Birds and are for those students who are able to use the associated number properties.
Related activity
Flipping Decimal Fractions - Memory game based on “Flipping Fractions” (Figure it Out Number Book 1, Level 2 page 17): PDF (74KB) or Word (67KB)
Domino activity: PDF (91KB) or Word (110KB)
Birthday cakes
These exercises and activities are for students to use independently of the teacher to practice number properties.
Using multiplication facts to find a simple fraction of an amount
Proportions and Ratios, AA (Stage 6)
Practice exercises with answers (PDF or Word)
Homework with answers (PDF or Word)
Prior knowledge.
Background
Remember that students who have not reached stage 6 in the "basic facts" domain have not learned all of their basic multiplication facts, so a multiplicative strategy based purely on the numbers may not work for all students. Keeping to simpler numbers has been attempted in the activities for this reason.
Comments on the Exercises
Birthday Cakes 1
Asks students to use multiplication basic facts to find fractions of a set. For example, how many candles on each piece of cake if there are 6 pieces and 30 candles. Asks students to find unit fractions, for example 1/3 of 24 =.
Birthday Cakes 2
Asks students to practice activities parallel to Birthday Cakes 1.
Fraction Benchmarks
In this unit students develop important reference points or benchmarks for zero, one half and one. They use these benchmarks to help compare the relative sizes of fractions, through estimating, ordering and placing fractions on a number line.
This unit builds on the following key conceptual understandings about fractional parts.
An understanding of fractional parts supports students to develop a sense for the size of fractions. This unit helps students develop an intuitive feel for zero, one-half and one, as useful benchmarks for ordering fractions. Understanding why a fraction is close to zero, one half or one helps students develop a number sense for fractions.
Students can be supported through the learning opportunities in this unit by differentiating the complexity of the tasks, and by adapting the contexts. Ways to support students include:
The contexts for this unit are purely mathematical but can be adapted to suit the interests and cultural backgrounds of your students. Fractions can be applied to a wide variety of problem contexts, including making and sharing food, constructing items, travelling distances, working with money, and sharing earnings. The concept of equal shares and measures is common in collaborative settings. Students might also appreciate challenges introduced through competitive games or through stories.
Session 1
In this session students begin to develop benchmarks for zero, half and one.
1/20, 6/10, 10/8, 11/12, 1/10, 3/8, 2/5, 9/10.
As the difficulty of this task depends on the fractions, begin with fractions that are clearly close to zero, half or one.
Why do you think 6/10 is close to half? How much more than a half is it? Why do you think 1/20 is close to zero? How much more than zero? Why is 11/12 close to one? Is it more or less than one? How much less?
3/10, 5/6, 5/9, 4/9, 18/20, 13/20, 2/8, 9/12, 1/5
Once more encourage the students to explain their decision for each fraction.
Session 2
In this session the students continue to develop their sense of the size of fractions in relation to the benchmarks of zero, half and one by coming up with fractions rather than sorting them.
5/6
How do you know that fraction is close to one?
5/6 7/8
How do you know that 7/8 is closer to one than 5/6?
Students might comment on how much needs to be added to each fraction to make it equal to one. "7/8 is closer to one because eighths are smaller parts than sixths and 7/8 is 1/8 smaller than 1 and 5/6 is 1/6 smaller than one. As 1/8 is smaller than 1/6, 7/8 is closer to one."
Do you notice a pattern as the fraction gets closer to one? (5/6, 7/8, 9/10, 11/12)
Which fraction is closer to one 99/100 or 999/1000? Why?
Which of these fractions is closest to zero, 1/3, 1/4, 1/5, or 1/6? Explain why?
Can a fraction have 5 as a numerator but still be close to zero? Give an example.
Session 3
In this session students estimate the size of fractions.
Ask for volunteers to record their estimate on the board. As they record estimates, ask each student to share their reasoning. Listen, without judgement to the estimate and then discuss why any given estimate might be a good one. Encourage students to share their justifications and ask questions of each other. There is no single correct answer but the estimates need to be reasonable. If the students are having difficulty, encourage them to reflect on the benchmarks developed in the previous sessions. Look for creative methods like estimating that the white area combines to 1/4 so the shaded area must be 3/4
Session 4
In this activity students identify which fraction of a pair is larger. The comparisons rely on an understanding of the top and bottom number (numerator and denominator) in fractions and on the relative sizes of the fractional parts. Equivalent fractions are not directly introduced in this unit but if they are mentioned by students they should be discussed.
2/5 or 2/8
Encourage explanations that show that the students understand that the fractions have the same number of parts but that the parts are different sizes. In this example 2/8 is smaller than 2/5 because eighths are smaller than fifths. Some students may know that 2/8 = 1/4. Use Fraction Strips to check students predictions.
4/5 or 4/6.
Encourage explanations that show that the students understand that both these fractions have the same count (numerator) but fifths are larger than sixths. Therefore 4/5 is greater than 4/6. Write 4/5 > 4/6. Use Fraction Strips to check students' predictions.
In this example the students can draw on their understanding of the benchmark of 1 and notice that 10/9 is larger than 1 and 9/10 is smaller than 1.
7/10 or 6/10
6/8 or 6/12
3/8 or 4/10 (more difficult)
9/8 or 4/3 etc
One fraction is greater than the other.
The two fractions have different numerators and different denominators.
What might the fractions be?
Session 5
In this session the students draw on their conceptual understanding of fraction benchmarks (0, 1/2, 1) and their understanding of the relative sizes of fractional parts to line fractions up on a number line.
Dear family and whānau,
This week we are learning about where fractions live on number lines. We have been especially interested in fractions that are close to zero, close to one half and close to one. Ask your child to put the following fractions on a number line and explain to you how they made up their mind about where to place them.
1/30 2/99 44/85 99/100 25/60 3/25 77/80 26/50