How Many Things?

Task 99 ... Years 4 - 8

Summary

The challenge in this task is to find efficient methods for estimating number and apply the mathematician's question Can I check this another way?. The final check is to count all the beans, but this is only as confirmation.

Note: Whether the task contains beans, beads or macaroni depends on availability of supply.

Materials

1 small cup and a bag of beans or similar
1 grid board
Your own scale capable of measuring to 50 grams

Content

estimating number
counting
basic arithmetic skills, particularly times tables and addition
measuring mass

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Estimation methods are frequently used in the world, so it is important that students become confident with accurate methods of estimating. In this case the estimation is about number, but other experiences can be designed around mass, volume, perimeter, area and so on. Students seem to enjoy the sense of not being able to fail because they are only being asked to make a guess and yet, having made the guess, it becomes the stimulus for finding the accurate answer. Students also appreciate that the more they practice, the more accurate they become.

When students have finished the card, ask them:

to try to develop an estimation method different to those given.
to write a journal entry comparing their estimation methods.
to apply what they have learned to a challenge such as calculating the number of people living in their district/suburb/area and comparing their estimate to official figures.

Estimating number is also used to check the reasonableness of answers to calculations. For example the answer to 743 + 159 could be estimated as:

700 + 200 = 900, or
750 + 150 = 900

Use the task to encourage students to invent calculations for each other to estimate and then check with their calculator. How close can they get by estimating? Are they better at estimating answers to addition, subtraction, multiplication or division? How about calculations involving mixed operations, ...fractions, ...decimals?

Whole Class Investigation
Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
School storerooms will often have quantities of small objects which can be packed into cheap containers. Or, as in the photo above, use small pasta which is cheap to purchase in large quantities. You will need containers for groups of four students as a minimum.
In preparing your material consider whether you want each group to have the same size and shape container.
This whole class investigation might generate from a fund raising activity such as guess the number of lollies in the jar. Discuss the methods students use for estimating in these circumstances. From here challenge the students to devise at least three ways to estimate the number in their container. Hold off counting to check until after a class discussion about methods and results.

Do you want to try another method based on what you have heard from your classmates?

Would the method you choose depend on the material you are counting? For example, this jar is full of small, flat round counters and this one is full of 1cm cubes.

Finalise this lesson by opening and counting the material. Students make a journal record of estimates, counted total and any significant 'feeling moments' through the lesson. For example, moments of confidence, concern, relief, interest - anything which acknowledges that more of themselves than just their brain has been involved. Why? To continue developing the understanding that mathematicians are humans with feelings not machines processing algorithms.
Consider making the lesson part of unit on estimation. At this stage, How Many Things? does not have a matching lesson on Maths300. However Lesson 145, Estimating Averages is a direct follow up and Lesson 150, Fermi Problems places estimation of number into real world situations. Estimating the population of the local region, as suggested above, is an example of a possible Fermi Problem.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The How Many Things? task is an integral part of:

MWA Number & Computation Years 3 & 4

MWA Chance & Measurement Years 7 & 8

Follow this link to Task Centre Home page.