This unit encourages students to explore ways of working out problems that involve finding the average percentage of the contents of differently sized containers. It is suitable for students working at Stage 7-Advanced Multiplicative of the Number Framework.
- Solve problems that involve combining different proportions by:
- finding the average percentage of the contents of differently sized containers using the quantities in each container
- finding the average percentage of the contents of differently sized containers by using the percentage in each container.
From previous work in statistical investigations students should be able to find the average of data sets.
This unit introduces how to find the average percentage of differently sized containers. Two strategies are explored, (i) combining the contents of the containers to find the average percentage and (ii) finding the percentage in each container and then averaging the percentages.
It is necessary to use percentages because the containers are of different sizes.
The word average can be used in two ways. It can be used in the same way as the statistical term, mean. The mean is calculated by adding all the numbers together and dividing by the number of numbers. The word average can also be used in everyday language, to mean something that is not out of the ordinary. In this unit we are using the word average in the same way as mean.
The term weighted averages is used because the size of the containers is taken into consideration in finding the average. For example if two containers are different sizes the bigger container will have a greater influence on the average.
- Clear plastic bags
- Measuring jugs
- Begin a discussion about what an average or mean is and how it is calculated. Pose some examples to work out together. For example, the average number of hours of sleep, television viewing, exercise etc.
- Discuss with the students that the total is divided by the number of items combined (ie the number of students) because everyone contributes equally with one piece of data.
- Start by showing the students two clear plastic bags of mixed counters where one bag has 12 counters 8 of which are blue and one bag with 12 counters 4 of which are blue.
- Calculate the average number of blue counters in the bags. 8 + 4 = 12 12 ÷ 2 = 6. Explain that the number of blue counters can simply be added together because the number of counters in each bag is the same. Show this by combining the bags to have 12 blue counters out of a total of 24 counters. This is the same as half. When they are shared equally back into the bags there are 12 counters in each bag and 6 are blue. This combining and sharing out helps to find the average of 6.
- You may wish to work through more examples using this strategy.
- Return to the first example and explain to the students that there is a second way to work this problem out using percentages.
- In the first bag 75% are blue (8/12) and in the other bag 25% (4/12) are blue. The average percentage can be found with 75% + 25% = 100, 100 ÷ 2 = 50, therefore the average is 50%. 50% of 12 is 6, therefore the average is 6 blue counters in each bag of 12.
- Depending on your students’ level of understanding you may want to work through more examples using these methods before moving onto weighted averages.
- Continue to display the bags from the previous example. (8 blue counters in a bag of 12 and 4 blue counters in a bag of 12). Now show the students two clear plastic bags of counters where one bag has 12 counters 8 of which are blue and one bag with 4 counters all blue.
- Ask the students to consider if the average is still 6 and begin to discuss why they can not simply add 8 and 4 together and divide by two in this instance. Combine the bags together. Ask the students to work out.
How many blue counters are there altogether? (12)
How many counters are there altogether? (16)
How can this be expressed as a fraction or a percentage (12/16 = 3/4 = 75%)
- Ask the students to share the counters back into the bags so that 75% of the counters in each bag is blue. The students will need to work out 75% of 12 and 75% of 4.
- Discuss why the percentage is the same but the number of counters in each bag is different.
- Another example to try using this method is a bag 15 counters 8 of which are blue and a bag of 5 counters 4 of which are blue. The average will be 12/20 or 3/5 or 60%. 60% of 15 counters is 9 and 60% of 5 counters is 3.
- The same strategy can be used with any number of containers. It may be helpful to work through a problem with three bags with the students. A bag of 10 counters 8 that are blue, a bag of 4 counters 3 of which are blue and a bag of 6 counters 4 of which are blue. This gives an average of 15/20 or 75%.
- It is acceptable to work out the percentage without going on to redistribute the counters according to the average percentage.
Over the next few days students can work through problems independently or in pairs using the strategy. Students can solve problems using counters or beans. With different sized bags of mixed lollies students could work out the percentage of particular types of lollies. Using the context of capacity students could measure the amount of water in differently sized measuring jugs or water bottles to find the average percentage. The context of mass could also be used. Students can pose their own problems for others to solve.
Getting started on a new strategy
It is also possible to find the average percentage of the contents of differently sized containers by using the percentage in each container.
- Introduce to the students that sometimes it is easier to find the average percentage using the percentage of each container. Show the students two bags, one bag with 10 counters including 6 that are blue and one bag with 20 counters containing 6 that are blue.
- Ask the students to work out the percentage of each bag:
What percentage of the counters in the bag of 10 are blue? (6/10 = 60%))
What percentage of the counters in the bag of 20 are blue? (6/20 = 3/10 = 30%)
- Discuss with the students that like the previous examples the average can not be found simply by adding the percentages together and dividing by two because the bags are of different sizes.
- Ask the students about the different sized bags:
How many counters are there in each bag? (10 and 20)
How much bigger is the big bag compared to the small bag? (two times bigger)
What bag will have the bigger influence on the average? (big)
- Discuss with the students that the combined percentage can be found by combining 60% from the small bag and 30% from the big bag, but the big bag has twice as much influence because it is twice as big. The calculation then becomes 60% + 30% + 30% = 120%, and the average is therefore 120% ÷ 3 = 40%.
- Students could check this answer using the previous strategy. (12 blue counters out of 20 counters is 12/20 and this is 40%).
- Depending on your students’ level of understanding you may want to start with examples where the students are given the percentages to work from. For example container A is 40% full and container B is 25% full but is twice as big. (Answer = 40% + 25% + 25% = 90%, 90% ÷ 3 = 30%).
- The same method can be used with 3 or more differently sized containers. For example in three bags, bags B and C are twice as big as bag A. Bag A is 80% full, Bag B is 50% full and Bag 3 is 60% full. (Answer is the weighted average percentage of 80 + 50 + 50 + 60 + 60 = 300, 300 ÷ 5 = 60%, therefore the answer is 60%).
Exploring the new strategy
Over the next few days students can work through problems independently or in pairs using the strategy. Students can solve problems using different contexts. Problems that give the students the percentage in each container are easier for students than problems where they first need to calculate the percentage of each container.
Students can be extended to explore posing their own questions. They may need calculators if they end up with working with decimal numbers.
Students who have written their own questions can share these with the group. They can discuss how they chose numbers to use in their problems. Students can discuss when decimal numbers arose. Once the students have explored using both strategies they can solve problems using which ever strategy they think it easier. Students can compare ideas and discuss their choice of strategy.