This type of activity is known as Practice. Please read the guidance notes here, where you will find useful information for running these types of activities with your students.
The purpose of this activity is to provide students with practice of calculating the mean from a list of data (with 5, 6 and 7 numbers), but at the same time giving them an opportunity to think harder about what effect specific changes to each set of data will have on the mean. For example, between the first two questions we can draw students’ attention to what effect doubling each piece of data has, followed by adding 1 to each piece of data.
When I have run this activity with groups of students, I have encountered an example of the hypercorrection effect between questions 3 and 4. Some students reflect that 5 has been taken off, and so expect that the mean will also go down by 5. Subsequent discussion often reveal that this expectation is justified because the numbers increased by 1 in the previous example, and the mean went up by 1. When students check by working out the mean, they are often surprised to find that the mean actually only decreases by 1, and we can have a useful discussion as to why this might be the case, with students offering visual representations to convey the notion that the 5 is spread out across each piece of data. Because I feel this concept is a critical one, notice how the next question covers the exact same ground, in order both to consolidate this and give students a boost in confidence.
Another notable moment in this activity seems to come between questions 7 and 8. Many students (and teachers!) are unable to form an expectation as to what happens when the number 18 is added to the set of data. This is no problem at all. I tend to encourage students to try to expect something less specific – like will the mean in crease, decrease or stay the same – and then work out the answer. Having discovered that the mean increases by 1, they can then look back at the question and try to see why this would be the case. Just because students are not always able to form an expectation does not mean they miss out on the opportunity to reflect and harder.
Finally, notice how this activity ventures into the areas of decimals and negative numbers. This is my way of interleaving. This can be left out if your students are not comfortable in these areas. Likewise, there is nothing stopping this sequence of questions moving into areas such as fractions, algebraic expressions, or even surds and indices. This could be something that the students who finish early could be encouraged to explore.
1. Example-Problem Pair
2. Intelligent Practice
3. Answers
4. Downloadable version
5. Alternative versions
- feel free to create and share an alternate version that worked well for your class following the guidance here