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Level Four > Number and Algebra

Fraction Bar

Purpose: 

This unit uses one of the digital learning objects, the Fraction Bar, to support students as they investigate finding fractions of whole numbers. It is suitable for students working at Stage 7-Advanced Multiplicative or Stage 8-Advanced Proportional of the Number Framework. It includes problems and questions that can be used by the teacher when working with a group of students on the learning object, and ideas for independent student work.

Achievement Objectives:

Achievement Objective: NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
AO elaboration and other teaching resources

Specific Learning Outcomes: 

visualise and solve problems which involve finding a fraction of a whole number where the answer is a whole number

solve problems involving fractions, including fractions greater than 1 where the answer is a whole number

Description of mathematics: 

Relevant Stages of the Number Framework

The strategy section of the New Zealand Number Framework consists of a sequence of global stages that students use to solve mental number problems. On this framework students working at different strategy stages use characteristic ways to solve problems. This unit of work and the associated learning object are useful for students in transition between Stages 7 and 8 of the Number Framework, moving from Advanced Multiplicative to Advanced Proportional. At Stage 7 students use a range of strategies to solve problems that involve multiplying whole numbers, while at Stage 8 they can also apply a range of strategies to solve problems that involve multiplying fractions and decimals.

Activity: 

Working with the object with students (fractions < 1)

  1. Show students the learning object and explain that it provides a model for finding a fraction of a number.
  2. Choose 0-1 to work with fractions less than one.
  3. Discuss the first screen of the Fraction Bar with students. Ensure that they understand that the bar represents the number that they are trying to find the fraction of, and that the line below represents an ‘amount’ of that number. Students at this stage should be familiar with the concept of a unit, and it is important that they realise that for the purposes of this learning object, the number line from 0-1 represents the unit.

  4. Scale image
  5. Explain to students that the learning object helps you to find the required fraction of the number represented by the bar. In the following steps the example 3/4 of 28 has been used.
  6. Explain that for every problem, if they ‘just know’ the answer they can always enter it, but that for this practice problem we are going to work through all the steps.
  7. Ask for a volunteer to show where 3/4 is on the number line.
  8. Regardless of whether they click in the right place or not, click on the button saying ‘Show quarters’ to show where all the quarters are located to demonstrate the purpose of this button.
  9. Ask for a volunteer to show where 1 quarter is on the number line. Click on it and then enter 1/4 in the boxes.

  10. Scale image
  11. Discuss how to work out what 1/4 of 28 is. Students at this stage of the number framework should be able to find unit fractions of numbers where the answer is a whole number. Ensure that all students understand that they can work out a 1/4 of a number by dividing it by 4.
  12. Enter 7 in the box at 1/4 on the bar.
  13. It may be useful to demonstrate that you can also locate and label 1/2 on the number line (you can label it as either 1/2 or 2/4).
  14. Ask students how the information now showing on the screen will help them work out the solution to the problem (3/4 of 28). Ensure that all students understand that if 1/4 of 28 is 7, then 3/4 of 28 is 3 lots of 7, which is 21.

  15. Scale image fraction
  16. Enter 21 in the box at 3/4 on the bar.
  17. Depending on your students’ level of understanding you may want to work through one or two more examples as a group before allowing them to work independently.
  18. Ensure that all students understand how they can use the unit fraction of a number to help them work out a non-unit fraction of that number. For example 3/4 of 28 = 3 x 1/4 of 28.
  19. Pose problems for students to solve with and without the learning object. Encourage them to share their solutions.

Working with the object with students (fractions > 1)

  • Choose 0-2 on the Fraction Bar to work with fractions between 0 and 2. Most of the problems in this version of the learning object will involve fractions between 1 and 2. In the following steps the example 5/3 of 21 has been used.

  • scale image and fractions
  • Discuss what the learning object shows:
    Why is the 21 in the middle instead of at the end of the bar?
    Why is there a 1 and a 2 on the line now?
    Where would 5/3 be on the number line?
  • Regardless of whether students can correctly locate 5/3 click ‘Show thirds’ to show all the thirds, and locate and label 1/3 as a reference point.
  • Discuss the similarities between this problem and the problems they have solved with fractions less than 1.
    Is this problem harder?
    Can we solve it the same way?
    What is another name for 5/3? (1 2/3)
  • Students should be able to transfer their understanding from the easier problems to this problem, and see that 5/3 of 21 is the same as 5 x 1/3 of 21, or 35.
  • Depending on your students’ level of understanding you may want to work through one or two more examples as a group before allowing them to work independently.

Notes regarding other methods for multiplication by fractions

The focus of mathematics, and in particular numeracy, education in New Zealand is increasingly on developing an understanding of number properties and how they can be used. The Fraction Bar provides one model which students can use to visualize how to find fractions of numbers, however, in practice we want them to learn how to solve problems such as 5/3 of 21 without needing to rely on a tool or a diagram. Students in your class should understand that there are a variety of ways that they can find fractions of numbers, and that the Fraction Bar does not represent all of them.

  1. Either on the Fraction Bar or on a whiteboard, present a problem that involves multiplying by a fraction.
  2. Discuss how you could work out the answer. Hopefully as a result of experience with the Fraction Bar students will identify that they can divide the number by the denominator of the fraction to find the unit fraction of the amount and then multiply by the numerator.
  3. On the whiteboard present the example 5/3 of 14 and ask students how they could work out the answer. Hopefully they will see that, while they can divide 14 by 3 to find 1/3 of 14, and then multiply by 5 it will not be as easy as in the previous examples because they will need to work with fractions in the answer.
  4. Discuss whether there might be an easier way to solve this problem.
  5. One alternative method for this problem is to recognise that, rather than seeing 5/3 x 14 as 5 x (1/3 x 14), they can see it as (5 x 14)/3.
  6. The Fraction Bar provides a useful model to illustrate one method of solving problems involving multiplication by a fraction, but students should be aware that some problems can be more easily solved using other methods.

Students working independently with the learning object

Because this learning object generates problems for the user, once they are familiar with how it works you could allow individual students or pairs of students to work with the learning object independently. The learning object tracks the number of problems answered correctly so you could challenge students to answer 5 problems involving fractions less than 1, or 5 problems involving problems between 1 and 2.

Students working independently without the learning object

Independent activities that develop the same concepts as the learning object include:

  • Students can work with strips of paper, experimenting with folding them into fractions. For example if they fold the strip into quarters they can see that 3/4 of the strip is three times 1/4 of the strip. If the length of the strip is 28, then 1/4 of the length is 7 and 3/4 is 3 x 7, or 21. Discuss whether folding paper can be used to represent fractions greater than 1.
  • Encourage students to experiment with recording strategies to show how they know that, for example, 5/4 of 24 is 30.