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Level Four > Number and Algebra

Percentage Bar

Purpose: 

This unit uses one of the digital learning objects, the Percentage Bar, to support students as they investigate finding percentages of whole numbers. It is suitable for students working at Stage 7- Advanced Multiplicative to 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 percentage of a whole number

solve problems which involve finding the discounted price using percentages and finding the full price when a discounted price has been given.

Description of mathematics: 

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 Stage 7 and Stage 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: 

Prior to using The Percentage Bar Learning Objects

The unit Getting Percentible would make a useful unit of work leading into this learning objects activity.

Part of the Percentage Bar learning object activity requires students to estimate on a line from 0 to 100% where a particular percentage will be. The learning object does give some prompts using percentage markers. Before doing the learning object it would be helpful for students to revise the link between fractions and percentages so they can estimate where a percentage will be on the line. The students could practice making estimations on lines on paper.

The learning object asks students to solve two types of percentage problem: finding what a new price will be after a discount has been made, and finding the full price when the discounted price is given. The students could practice the easier task of calculating a percentage of a number before attempting the learning object.

Working with the object with students (finding the discounted price)

  1. Show students the learning object and explain that it provides a model for finding a percentage of a number and then the new discounted price.
  2. Choose Total Price Given.
  3. Discuss the first screen of the Percentage Bar with students. Ensure that they understand that the bar represents the number that they are trying to find the percentage of, and that the line below represents the percent. Students at this stage should be familiar with the concept that 100% represents a whole or in this case, the total price.
    Percentage bar
  4. Explain to students that the learning object helps to find the required percentage of the number represented by the bar. In the following steps the example of a 20% discount of $25 is used.
  5. 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.
  6. Ask for a volunteer to answer what percentage of the full price Tom has to pay by simply taking 20% from 100%. Submit the answer.
  7. Ask for a volunteer to click where the 80% belongs on the percentage line. If the student has not estimated close the 80% line, they will be prompted to divide the line into equal parts using the split percentages button.
    Percentage bar with increments
  8. Ask for a volunteer to enter the amount given in the problem ($25) in the correct box and submit the answer.
  9. To find the amount of money equivalent to the 80% students may wish to fill in more of the percentage lines on the graphic. The percentage markers can be added to the graphic at any time using the Split Percentages button.
    percentage bar conintuation
  10. Ask for a volunteer to work out what 80% of $25 is and submit it in the box.
  11. The correct answer can then be submitted in the question box.
  12. 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.

Working with the object with students (Discounted price given)

  • Choose the Discounted price given.
  • Discuss the first screen of the percentage bar with students. Ensure that they understand that the bar represents the full price of the discounted item and that the line below represents the percent.
  • 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.
  • In the following steps the example used is finding the full price of CD that had a 10% discount and now costs $18.
    percentage bar
  • Ask for a volunteer to answer what percentage of the full price Sally has to pay and submit the answer.
  • Ask for a volunteer to click where the 90% belongs on the percentage line. If the student has not estimated close the 90% line, they will be prompted to divide the line into equal parts using the split percentages button.
  • Discuss the similarities between this problem and the problems they have solved when finding the discounted price.
    What is similar about the problems?
    What is different about the problems?
  • Ask for a volunteer to enter the amount given in the problem ($18) in the correct box below.
  • Discuss with the students how to find 10% when they know 90% is equivalent to $18. The strategy of dividing by 9 is simple and useful.
  • Students should be able to transfer their understanding from the easier problems to this problem. 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 calculating percentages.

The percentage bar provides a way to calculate the percentage. It is possible to work out the percentage of a number by using the benchmark of 10%. The term percent is another name for hundredths. In the case of 10% it is renamed as a 10 hundredths or 1 tenth 0.1. Finding one tenth of a number involves dividing the number by ten. It is easy to work out multiples of 10% by multiplying, for example 30% is 10% x 3. It is also possible to calculate 25% using 10% x 2 = 20% and half of 10% = 5%.

By converting the percentage to a fraction it also possible to calculate the percentage. A fraction of a number can be found by dividing. For example, 25% of 40 is equivalent to finding 1/4 of 40 by dividing 40 by 4.

A slightly more complex example using fractions would involve converting 75% to 3/4. 75% of 60 is equivalent to finding 1/4 of 60 by dividing 60 by 4 and then multiplying it by 3.

1/4 of 60 = 15

3/4 of 60 = 15 x 3 = 45

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 a given number of problems involving either the total price given or the discounted price given or a mixture.

 

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 to fold them into percentages of 10% or into other benchmarks like 25% to make the link between fractions and percentages.
  • Student can experiment with strips of paper to design double number lines to find discounted prices and full prices.
  • Give students problems similar to those in the learning object.
  • Encourage students to try different strategies for finding percentages.
  • The work could be extended to working out what percentage discount has been given when the full and discounted prices have been given.