Getting partial to percentages

Purpose

This unit supports students to recognise percentages as equivalent fractions, and to carry out simple calculations involving finding percentages of amounts.

Specific Learning Outcomes
  • Use the percentage bar model to find the percentage that a part is of a whole.
  • Use the percentage bar model to find a percentage amount of a whole.
  • Simplify parts of a whole to common fractions to find percentages.
  • Use percentages to represent the relationship between two different wholes or parts.
Description of Mathematics

In this unit we build on work by Prediger and Pohler (2015) to develop students’ concept of percentages using the metaphor of a data download. The model is linear which has been effective in the teaching of fractions and decimals.

Percentages are a specific type of equivalent fraction with a denominator of 100. The symbol %, is derived from the vinculum of a fraction /, and the two zeros from 100. So, a percentage literally means “for every hundred.”

In many situations a percentage describes a part-whole relationship. For example, Mani lands 24 out of 30 shots in her netball game. What is her shooting percentage? The calculation of her percentage is 24/30 = ?/100. There is an assumption of homogeneity, that Mani will shoot at the same rate.

Percentages are very useful to establish a common base for comparison. Most situations in which part-whole relationships need to be compared, the whole (or bases) are not the same. For example, Mani takes 30 shots, but Alicia takes 28 shots. Some situations involve comparison of different wholes or different parts. In those situations, percentages greater than 100% are possible. For example, there are 12 girls and 18 boys. The number of boys is 150% of the number of girls.

In other situations, percentages are used as a fractional operator, e.g. 30% of $45. That calculation is equivalent to 30/100 x 45 = ?. Common examples of percentages as operators are discounts in shops, and interest on loans.

Required Resource Materials
  • Dot paper, or geoboards and rubber bands
  • Scissors and glue sticks
  • Copymasters One, Two, Three, Four, Five, and Six
  • PowerPoints OneTwo, Three, and Four
  • Access to computer spreadsheets
Activity

Getting Started

Tell the students that they will be investigating percentages in this unit. Ask them for examples of when percentages are used in everyday life. Slide one of PowerPoint One can be used if needed to inspire more ideas. Common contexts are:

  • Sale in shops, e.g. 25% off
  • Sports, e.g. kicking/shooting at 80%
  • Computer downloads, e.g. 75% of 4mb downloaded
  • Livestock births, e.g. lambing percentage of 180%
  • Inflation or interest, e.g. mortgage at 5% interest
  • Statistics, e.g. 46% of the people choose hokey pokey icecream

When we go to a 25% off sale, how much is taken off the normal price?  (one quarter)
How much of the normal price do we pay? (75% or 3/4)

Write these two equations on the whiteboard; 1/4 = 25/100 and 3/4 = 65/100. Discuss that the fractions are equivalent.

What does it mean when you pay 75% of the normal price, but the item does not normally cost $100? Say it costs $60.00?

Discuss that three quarters of the normal price is paid, whatever the normal price is. Take the 75% of 60 example and write; 75/100 = 3/4 = ?/60.

What number replaces the question mark to show how much you pay for the item?

Students should use their knowledge of equivalent fractions, and recognise that multiplying both the numerator and denominator of three quarters by 15 gives 45/60. So, the item costs $45 at the sale.

Look at slide two of PowerPoint One which shows a file download status bar with percentages. Discuss what the bar shows and that the percentages represent the fraction of the data file that is downloaded. For the next three slides discuss what the percentage message is likely to be. Look for students to identify known fractions and try to represent them as percentages. For example, Slide Three shows 10%. Students might realise that five ‘iterations’ of the yellow bar make one half. Therefore, the bar shows one tenth. For each bar write equivalent fraction equations.

Give the students copies of Copymaster One to work on in pairs or individually. Look for your students to:

  • Use benchmarks like 50%, 25%, and 75% to estimate the percentages;
  • Identify the fractions of the whole bar that are shaded, and use equivalence to find the percentages;
  • Iterate a trusted unit like 10% (one tenth) to approximate the percentage;
  • Possibly use a ruler to measure the length of the whole bar and shaded section to approximate the percentage.

After an appropriate time, gather the students together to share their strategies and answers. Highlight the points above in the discussion. Finally, provide one copy of Copymaster Two to each team of students. Challenge them to match the fraction, decimal, and percentage cards. For example, , 0.2, and 20% go together. Ask each team to make a chart by pasting each grouping together and drawing a percentage download bar to match.

Students must be able to justify the percentage to fraction equivalence. For example:

T: How do you know that one quarter equals 25%?

S1: 25 multiplied by four equals 100 so 25% is one quarter of 100%

S2: 50% equals one half, so 25% equals one quarter since half of 50 is 25.

Exploring

In the next two lessons students explore using the percentage bar as a model for solving a variety of part-whole percentage problems. The language of rate, amount, and base are introduced. Calculation steps are derived from the strategies that students use when solving problems with the percentage bar.

Lesson Two

Begin with the download animation to reinforce the fraction to percentage connections from Lesson One. Run the animation on Slide One of PowerPoint Two with instructions:

Put your hand up when (fraction) of the data is downloaded.
Justify why most of you chose this spot (pointing to download bar).

Vary the fractions to include quarters, thirds, fifths and tenths.

Slides Two through Seven of the PowerPoint Two introduce problems of finding a percentage of an amount, using the percentage bar as a visual model. For each problem let the students attempt to solve the problem and discuss the various ways it might be solved. For example, 80% of 35 could be solved by finding 10% first, then multiplying that amount by eight. It might also be solved by finding 20% first then subtracting that amount from 100%.

Every second slide shows how the percentage bar can be used to track and organise calculations. The slides also show the calculation in equation form. You may need to remind students of the process of finding a fraction of a whole number, using simple examples like  first. It is very important to be clear about the meaning of the terms, base, rate, and amount. For every example, bring students back to the meaning of the words in context.

Base means the unit of measurement or comparison. The percentage is calculated as a fraction of the base. Sometimes the fraction is more than one, so the percentage is greater than 100%.

Rate means a relationship between two measures. The relationship is multiplicative not additive. The word per is used to mean ‘for every’. For example, 60 km/h means 60 kilometres pass for every hour. 75% means 75, of the amount, for every 100 of the base.

Amount means the result of applying the rate to the base. For example, if the base is 40 and the rate is 25% then the amount is 25% of 40 = 10.


Slides eight to eleven present a different scenario, shopping. Invite the students to use a percentage bar diagram to solve each problem. The second slide for each problem shows possible solution strategies on a percentage bar and as an equation.

Give students Copymaster Three to work from, individually or in pairs. The worksheet provides a variety of percentage problems about part whole relationships in different contexts. Look for your students to:

  • Use the percentage bar model effectively to represent the problem and work towards a solution.
  • Recognise what is missing in the problem, the base, the rate or the amount.
  • Write equations to represent the calculations that solve each problem.

Lesson Three

Remind the students of the meaning of the words base, rate and amount. Show slide one of PowerPoint Three

Here is a percentage problem for you to solve.
What is the base? (24 people)
What is the amount? (from 9, 5, 4, and 6)
What is missing for you to find out? (the rate or percentage)

Ask your students to work in pairs to work out the correct percentages. Use Slide two to work through the answers. Students should recognise two main strategies:

  • Convert the part-whole fraction to a simple fraction that they know the percentage of, e.g. 6/24 = 1/4 = 25%.
  • Use trusted percentages, equally partition and combine them to reach the required amount, e.g. 50% of 24 = 12, 8 is two thirds of 12 so they need two thirds of 50%.

Provide students with copies of the percentage wheel (Copymaster Four).

Use the marks on the wheel to create a pie chart of the holiday destinations.

Let your students try to mark our sectors of the circle that match the percentages they have calculated.

What fraction of the circle have you given to Fiji? Explain why you did that.
Samoa was chosen by % of students. What fraction is that?

Slide Six shows a model answer. Discuss strategies for accurately drawing the sectors. You might show how a spreadsheet can be used to draw the pie chart. The movie on Slide Seven shows how to do that and show percentages in each sector.

Distribute the tally sheets from Copymaster Five around the room. Tell your students to visit each chart and add a tally to it, that shows their preference. You may need a short lesson on tally charts. Students must choose one of the options for each table.

When the data is collected, allocate one chart to each group of three or four students. Their task is to calculate the percentage for each category. Encourage the use of the percentage strip as a model. Allow the use of calculators.

Once students have calculated the percentages, they can draw their pie chart using the percentage wheel, then use a computer spreadsheet to create a pie chart with percentages.

Look for students to:

  • Apply appropriate benchmarks to estimate the percentages.
  • Use the terms base, rate and amount correctly for their data.
  • Create percentage bar diagrams or equations to support their calculations.
  • Check that their calculations and pie charts match that generated by the spreadsheet.

Reflecting

Lesson Four extends application of the percentage bar to more complex percentage problems. The previous lessons are about part-whole situations. This lesson introduces the use of percentages to compare two different whole or two different parts.

In this story we have two files to download at the same time. One file is 12 kilobytes in size and the other is 18 kilobytes. Let’s look at what happens if the files download at the same rate, in bytes per second.

Slide One of PowerPoint Four has an animation of the respective downloads. You may want to play it twice or three times.

What did you notice about the percentages?

Students might notice that the percentage for the smaller file is one and one half times the percentage for the larger file. Note that 1 ½ = 3/2. The percentage for the larger file is two thirds the percentage for the smaller file. Note that 2/3 is the reciprocal of 3/2, meaning the numerators and denominators are swapped.

What will happen if the animation bars keep growing until the large file is 100% loaded?
What will the percentage be for the smaller file?

Play the animation on slide two then work through slides three and four to show percentage bar models of the relationships. The two different selections for the base lead to different percentage relationships. With the equations ask:

Is the percentage more or less than 100%? How do you know?

Expect students to recognise that if the base is greater than the amount then the percentage is less than 100%. If the base is less than the amount, the percentage is greater than 100%.

Slides five through eight provide other download scenarios in static form. For each example, ask your students to write the relationships using percentages. Encourage them to use percentage bar models if they need to. Allow the use of calculators. Look for your students to:

  • Use common factors to partition the bars into equal parts, e.g. both 12 and 30 divide into threes;
  • Express the relationships using simple fractions, e.g. 12 to 30 can be expressed as 4 to 10;
  • Use the fraction relationships to find the percentages, e.g. 4/10 = 40% and 10/4 = 2 ½ = 250%.
  • Use division on the calculator, to check and to calculate the answers, e.g. 12 ÷ 30 = 0.4 = 40% and 30 ÷ 12 = 2.5 = 250%.

Copymaster Six provides many contextual examples of whole to whole or part to part comparisons. It begins with simple percentage strip diagrams and progresses to everyday situations. Check to see that your students apply the ideas in the bullet points above.


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