The purpose of this unit is to synthesise students’ fraction and decimal place value knowledge to enable them competently understand percentages.
The ability to understand and work competently with percentages depends on the students having a sound understanding of place value, of our decimal number system and of fractions and their operations. It is important that students are given opportunities to explore, recognise, demonstrate and articulate these connections for themselves, and to be able to work fluently between them.
Relational thinking underpins the students' ability to understand that a percentage is used to express how large or small one quantity is in relation to another quantity (which is greater than zero). A percentage is a part to whole ratio.
Percent, from the Latin per centum literally means out of (per) one hundred (cent). The symbol % is made up of the / per sign and the two zeros (00) from the number 100. Therefore, a percentage can be thought of as a way of expressing a fraction of 100, another way of writing hundredths, and a new way of expressing the concept ‘out of 100’, using the % notation. It is important to use a range of physical representations to ensure that the students can clearly see and make the connections between the fraction: eg. 75/100 (or 3/4), percentage 75% and decimal representation 0.75 of the same amount.
Having a sound decimal place value understanding underpins the students’ ability to work with percentages. With this understanding a student can readily see, for example, that 0.6 is 6 tenths, is also 60 hundredths and is therefore 60%, whilst 0.06 is 6 hundredths and is therefore just 6%. Students will come to recognise the advantages of working with decimals rather than fractions when converting to percentages, particularly when the ratio cannot be represented as a common fraction.
Building a ready knowledge of common equivalent fractions for percentages, (1/4 = 25%, 1/2 = 50%, 3/4 = 75%, 1/10 = 10%, 1/5 = 20%, 1/8 = 12.5%), being able to use these as benchmarks to work out some other percentages (eg. 15% = 10% + half of 10%) and simplifying fractions and expressing these as percentages (eg. 9/36 = 1/4 = 25%) and are key outcomes. Using a formula to calculate a percentage requires an understanding of what is happening: the numerator of a fraction is divided by the denominator giving a decimal fraction which is then multiplied by 100 to express a percentage. For example, 72/120 x 100/1 = 0.6 x 100 = 60%. This can also be thought of as 6/10 of 100, which is of course 60.
As in any mathematics, estimating skills are important. Students should anticipate what answer would be reasonable. For example, if asked to find 9% of 450, a student should understand that they could find 10%, which is 45, and 1%, which is 4.5, and suggest that 9% will therefore be close to 40.
In responding to problems related to percentage increase or decrease the student must understand what the percentage is relative to. For example, an increase of 100% means that the final amount is 200% of the original amount, whilst a decrease of 100% means that the final amount is zero. Care must be taken in calculating percentage change. If, for example, the price of $50 were increased by 10% it would go up by $5 to $55. If it were then reduced by 10% it would go down by 10% of $55, or $5.50, resulting in a price of $49.50. The two changes here are relative to different quantities.
Calculating percentage change can be challenging when problems involve reverse percentages. For example, when finding the original price for an item for which you paid $15 following a discount of 25%.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
These ideas are presented in five sessions, however, as the sessions include complex concepts that are fundamental to success with fractions, they can be extended over a longer period of time.
Whilst games are introduced and used within sessions to consolidate ideas, they can also be added to the class or group independent activities, or be sent home for whānau challenges and enjoyment.
The contexts for this unit can also be adapted to suit the interests, experiences, and cultural makeup of your students. The unit includes contexts such as sports games, kai and voting. You could work with the students and whānau to change these suggested contexts to instead link them to meaningful contexts from their lives. Other possible contexts involving percentages could include a marae visit, fundraising event, a fiefia night, or preparing a hāngī. Students could also collect examples or percentages from school data, CensusAtschool information, advertising material and from the media (newpapers, online). This will help them realise just how frequently percentages are used in daily life.
Te reo Māori vocabulary terms such as ōrau (percent), hautau (fraction), and hautanga ā-ira (decimal) could be introduced in this unit and used throughout other mathematical learning.
Session 1: Introductory session
Activity 1: Demonstrate and represent understanding of decimals with the use of different materials.
Activity 2: Explore the language of percentage and express decimal amounts as percentages
Activity 3: Express decimal amounts as percentages
Activity 4: Review and articulate the links between decimals and percentages
Conclude the session by having a student record the fractional representation of the decimal fractions and percentages recorded in Activity 1, Step 2, and Activity 2, Step 2 on the class chart: 10/100, 1/100, 23/100, 50/100, 99/100 and 1/1000 (or 0.1 of 1/100). Have the students read these to ensure they are saying the fraction correctly (hundredths and thousandths).
Record the students’ summary statements about the connections between decimal fraction, fractions and percentages.
Session 2
Make available to the students the percentage equipment from Activity 1, Step 2 above.
Activity 1: Introduction
Have these percentages recorded on the class chart. 25%, 50%, 75%, 10%, 20%.
Pass the Slavonic abacus around the class/group. As one student models the percentage on the abacus, have another record the fraction of the beads moved, in two ways.
25% | 25/100 | 1/4 |
50% | 50/100 | 1/2 |
75% | 75/100 | 3/4 |
10% | 10/100 | 1/10 |
20% | 20/100 | 1/5 |
Activity 2: Calculate percentage amounts using knowledge of common fractions and their percentage equivalents.
Activity 3: Work out 100% from a given percentage.
Pose this problem:
How much is 100% if Manu eats 15 cherries and this is 5% of the bag full?
Have students discuss the solution in pairs. Share and demonstrate one way of thinking, using the abacus to look at the percentages and calculations then apply these to the problem.
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15 x 2 = 30 = 10% | 30 x 10 = 300 = 100% |
Model twice more with a different number eg. 7 is 5%, 12 is 5%
Before Session 3, compile selected student problems created in Activity 2, Step 4.
Session 3
Activity 1: Application of prior learning to student-generated percentage problems.
Review charted conclusions from Session 2.
Distribute student problems created in Session 2, Activity 2, Step 4.
Have students solve these on their own or in pairs. As a group, discuss solutions and have the student writers verify these.
Activity 2: Use knowledge of equivalent fractions to simplify ratios to common fractions and give a percentage percentage amount.
Review the charted list from Session 2, Activity 1, Step 1.
Ask a student to record beside each their decimal equivalent.
25% | 25/100 | 1/4 | 0.25 |
50% | 50/100 | 1/2 (5/10) | 0.5 |
75% | 75/100 | 3/4 | 0.75 |
10% | 10/100 | 1/10 | 0.1 |
20% | 20/100 | 1/5 (2/10) | 0.2 |
Activity 3: Explore and model 1/8 and 1/3 as decimals and as percentages.
Activity 4: Convert an ‘awkward’ fraction into a decimal and show as a percentage.
Ask: Could this help us if a problem does not involve a common ratio or fraction? Pose this problem:
7 of the 49 people in the room voted against the idea. What percentage is this?
First ask students to estimate a percentage.
7/49 = 1/7 : 1/8 is 12.5% and 1/5 is 20% so it will be in between 12.5% and 20% but closer to12.5%
Look at this on the abacus.
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1/5 | 1/7 | 1/8 |
Have students calculate the percentage, using both long division and a calculator.
(1 ÷ 7 = 0 .1428 0.1428 x 100 = 14.28 %
This is: 1/7 x 100/1 = 14.28%
Refer to the original problem and record the answer:
7 of the 49 people in the room voted against the idea. What percentage is this? 14.28 % or round to 14.3%
Activity 5: Explain percentage calculations
Conclude this session by reviewing learning in today’s session.
(Elicit the observation that we can use common fractions to work out some percentages. If a fraction is tricky, we can work out its decimal value by dividing the numerator by the denominator and change the decimal to a percentage by multiplying by 100).
Session 4
Activity 1: Review approaches to percentage problems
Activity 2: Review approaches to percentage problems
Distribute the sets of cards from Copymaster 2 to each pair of students. Purpose: to practise deciding how best to solve percentage problems.
Have them sort these into the two piles indicated, giving the percentage solutions for the “common fractions they know” group as they do so.
Activity 3: Work out percentages by proportionally adjusting ratios that include factors of 10
Ask a student to model this on the abacus.
Establish 20 is a factor of 100 5 x 20 = 100 | Model 7/20 (screen 80 beads) | Model 7/20 five times |
Write 7/20 x 5/5 = 35/100.
Model this again on the abacus.
Have students articulate that the value hasn’t changed as both the denominator and the numerator are multiplied by 5 (and 5/5 =1).
Return to the problem:
Tahu solved 7 of the 20 problems. What percentage is this? 35%
Activity 4: Work out percentages by proportionally adjusting ratios that include factors of 10
Distribute Copymaster 3 to practise deciding how best to solve percentage problems.
List three ways students have to solve percentage problems:
Students can now revisit Copymaster 2 in Activity 2, Step 1 above, this time sorting the problems into three groups. Have them choose several from each group to solve, then write some of their own problems. This could be completed in pairs (tuakana-teina) or independently.
Activity 5: Work with and explain percentages greater than 100%.
Activity 6: Review approaches to percentage problems
Circulate Copymaster 4 to pairs or small groups of students.
Ask groups to decide whether each statement is right or wrong.
They should pair share and take turns to work through the examples and demonstrate and explain their rationale. Make equipment available.
Conclude the session by recording key ideas developed in the session.
Session 5
Activity 1: Calculate a percentage amount from a total amount.
Activity 2: Estimate solutions to decimal and percentage problems and explain rationale.
Activity 3: Understand and calculate percentage increase and decrease problems.
Have them pair share and explain their thinking.
These concepts can be modeled on the abacus, by the teacher or by students as necessary.
For example:
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Pose additional problems.
The original price for the timber for the fence was $397.
Two years later the price had increased by 200%. How much will the fence timber cost now? Discuss.
What if the $397 price had been reduced by 100%?
| This is what they look like if there is a decrease of 100%. Zero. |
Activity 4: Work out an original price from a given percentage discount and discounted amount.
Dear parents and whānau,
We have been working with percentages in class. Ask your child to tell you about what they have been learning. They can teach you to play the game Estimate Match. A copy of the game is attached.
We trust that you find this both challenging and enjoyable.
Printed from https://nzmaths.co.nz/resource/percentages at 9:35pm on the 28th March 2024