Eyeball Estimates

Estimating is an important skill that can be used in many real-life situations, and it’s one that improves rapidly with practice. This activity gives students opportunities to develop their estimating skills in a variety of different scenarios.
It is useful to recognise three kinds of estimation processes (and skills):
1. Measurement. This kind involves a measurement (continuous) variable and requires the use of suitable benchmarks. See Judgment Calls for examples.
2. Computational. This kind involves simplifying numbers in an appropriate way to reduce the complexity of a calculation, often to the point where it can be done mentally.
3. Numerosity. This kind involves estimating the number of items in a given situation (always a discrete or whole number variable). This is done by benchmarking and often goes hand in hand with computational estimation. Examples include the number of people in a crowd, coins in a stack, or fish in an aquarium.
When tackling the challenges in Eyeball Estimates, students should try and distinguish which of the three kinds of estimation are involved.

Activity

Use question 1 as the subject for class discussion or small-group discussion followed by a report back and class discussion. It is important that students understand that estimate and guess are not the same thing (see the introduction to Judgment Calls, pages 23–24) and why they need to know how to estimate.
Estimating is the process of coming up with an approximate answer that is accurate enough for our purpose. Estimates are useful:

• when we don’t need or want an exact answer (for example, how far it is to the dairy)
• when, for some practical reason, it is difficult or impossible to get exact numbers or measurements (for example, the number of people in a crowd)
• when we don’t have a measuring device with us (for example, the dimensions of the bedroom we will be moving into in our new house)
• when we have measured or calculated something and want to make sure that our result is sensible.

After looking at how Lakisoe and Sara estimated the number of people that would fit into their school hall, ask the students to suggest situations where they have used estimation or observed others using estimation.
Ask them why it was more appropriate or useful to estimate in those situations than to make an accurate measurement or do an exact calculation.
Question 2 gives students a number of scenarios and asks them to choose three. This means that they can choose ones that interest them and have an appropriate level of difficulty. They are told to write their methods down; this is part of the learning process, and the written notes are needed for the discussion that takes place in question 3.
In each case, the estimate must be based on some real information with numbers in it; otherwise it is a guess, not an estimate. This information may be a known fact (a benchmark), or it may need to be gathered:

• If a student gets a classmate (who says their height is 156 cm) to stand by a tall tree and uses their eye to judge that the tree is 8 times the person’s height, they can then estimate that the height of the tree is 1.5 x 8 = 12 m. In this case, the real information was a known fact (the height of the classmate).
• If a student times a classmate bouncing a ball and finds that it takes 42 seconds (about of a minute) to bounce it 50 times, they can then estimate that it will take about x 40 = 30 minutes to bounce it 2 000 times (50 x 40 = 2 000). In this case, the real information had to be gathered (the time taken for 50 bounces).

You will need to decide how much help you should give your students before sending them off to make their estimates. One approach is to put them into pairs, give them time to decide which of the estimates they are going to work on, and get them to write down the methods they plan to use. They could then combine with another pair (who may have chosen different challenges) and critique each other’s strategies. Or you could get the whole group together to say which challenges they are taking on and to share plans before putting them into action.
Your students may struggle with some of the calculations if they aren’t allowed to use a calculator, but wherever possible, they should try to cope without. They won’t see the point of simplifying the numbers (for a computational estimate) if they can equally easily work out the exact result. Encourage them to use a pencil and paper to jot down intermediate working stages.
Questions 3 and 4 are for follow-up. Question 3 should highlight the fact that there is normally a variety of different ways of arriving at an estimate and that there is no one “correct” estimate for any given situation.
You will need to make it clear that one estimate may be better than another (because it gives a better approximation of the actual measurement or number) but that both may be completely acceptable estimates. This is not the same as saying that any estimate is OK. (10 is not an acceptable estimate for 27!)
Try these questions to promote reflective thinking:

• Can you explain the difference between estimating and guessing?
• Give an example of where you would estimate something instead of measuring it. Why would estimating be more appropriate or better in this situation than measuring?
• Why might different pairs have come up with different estimates for the same challenge?
• Can you tell which of two estimates is closer to the truth without actually doing the measuring?
• What could help you improve your estimating skills?

Answers to Activity

1. Answers will vary. One possible definition is: “Estimation is when we work out a rough (but good enough) answer.” Some reasons why we estimate:
i. In the circumstances, an estimate may be all we need or want.
ii. For some practical reason, it may be difficult to get exact numbers or  measurements.
iii. We don’t have a calculator with us, and the numbers are too difficult to work with in our head.
iv. We’ve worked out the answer on our calculator and want to check if it is sensible.
2. Answers and methods will vary, even for exactly the same challenge (for example, two students estimating the height of the same tall tree).
3. Practical task. The important thing is that you can explain to your classmates how you arrived at your estimates and why you chose the methods you did.
4. Practical task