# Mix Up 2

*Keywords:*

This unit uses one of the digital learning objects, Mix Up 2, to support students as they investigate finding average percentage of the contents of differently sized containers. It is suitable for students working at 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.

AO elaboration and other teaching resources

Visualise and solve problems which involve finding the average percentage of the contents of differently sized containers.

### 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 at Stage 8-Advanced Proportional of the Number Framework. At Stage 8 students are asked to solve problems that involve combining different proportions.

### Prior to using The Mix Up 2 Learning Objects

The unit Weighty Averages encourages students to explore ways of working out problems that involve finding the average percentage of the contents of differently sized containers. That unit was designed as an introduction to this unit.

The unit Mix Up 1supports students as they investigate finding the percentage of contents of equally sized containers.

### Working with the learning object with students (Bag)

Show students the learning object and explain that it provides a model for finding the average of the percentages in the bags or jars.

- Choose bags and then start by selecting two bags.

- Discuss the first screen with students and clarify with them what the problem is asking.
- Discuss with the students ways to find an average when the bags are of different sizes. The combined percentage can be found by combining 76% from the small bag and 60% from the big bag, but the big bag has three times as much influence because it is three times as big. The calculation then becomes 76% + 60% + 60% + 60% = 256%, and the average is therefore 256% ÷ 4 = 64%.
- Alternatively, on this problem the number of objects in each bag could be used to solve the problem. Bag A has 19 blue objects. There are a total of 100 objects, 19 blue in Bag A and 45 in Bag B. This means a total percentage of 19 + 45 = 64. The calculation then becomes 19 + 15 + 15 + 15 = 64 and the average is therefore 16 counters. 16 blue counters in Bag A and 48 blue counters (16 + 16 +16) in Bag B are both equivalent to 64%.
- At any stage students can click on the Mixer button to see a demonstration of how to work out the answer using the first method described in step 4.
- If the second attempt is wrong the students will be prompted to consider that there are 100 objects altogether.
- If the third attempt is still not right the students will be prompted to click on the Mixer button to see a demonstration of how to work it out.

- The mixer button shows the bags being combined and then being shared equally. Discuss this with the students. This could be demonstrated using counters and containers.
- The before and after button shows average is the measure of the middle.

- Choose New and this time choose a different number of bags.
- As the students work through the activity discuss how the problem is similar and how it varies when the number of bags changes.
- 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.

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### Working with the object with students (Jars)

- Choose the Jars and then start by selecting two jars.
- Discuss the first screen of the jars problem with students and clarify with them what the problem is asking.
- Discuss with the students ways to find the answer. Again like the bag problem the combined percentage can be found by combining 56% from the small jar and 40% from the big jar, but the big jar has three times as much influence because it is three times as big. The calculation then becomes 56% + 40% + 40% + 40%= 176%, and the average is therefore 176% ÷ 4 = 44%.
- Ask for a volunteer to discuss an answer and submit.
- If the answer is not right they will be prompted to imagine some of the liquid moved between the jars. Discuss with the students how this could help. In this example, to imagine moving the liquid from one jar to another could involve taking 6% from Jar A to give 50%, the 6% will add only 2% to Jar B because Jar B is three times bigger than Jar A. Jar A now has 50% and Jar B has 42%, again taking 6% from Jar A and adding 2% to Jar B resulting in 44% in each jar.
- If the answer is still not right they will be prompted to use the mixer to see the liquid from the jars being combined and then shared equally. Again the before and after button shows the average is the measure of the middle.
- Choose New and this time choose a different number of jars.
- As the students work through the activity discuss how the problem is similar and how it varies when the number of jars changes.
- Discuss with the students the similarities between the bags and jars problems.
- 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 averages of percentages.

There are a number of ways of finding the average percentage of the contents of differently sized containers.

- The students can find the average of differently sized containers by adding the percentages together remembering that different samples have different weightings in the calculation. For example, if there are 2 samples and one sample is three times as the other then its percentage will contribute three times to the calculation. If Sample A has 76% and Sample B has 60% and is three times is big then the percentage when equally shared is 76% + 60% + 60% + 60% = 256%, 256 ÷ 4 = 64%.
- The students can find the average of differently sized containers with discrete objects by combining the number of objects, redistributing by the size of the samples and then converting the number to a percentage.
- The students can find the average of differently sized containers by adjusting the percentages in proportion to the size of the samples.
- The learning object demonstration empties the containers into a mixer and then shares the contents between the number bags or jars in proportion to their size.

### 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 bags or jars.

### Students working independently without the learning object

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

- Give students the opportunities to manipulate materials to find the average percentage. Give students problems similar to those in the learning object.
- Encourage students to try different strategies for finding the average percentage.