Lesson 6

Could Be or Crazy

Est. Class Sessions: 1

Developing the Lesson

Part 1: Estimating and Measuring with Links

Estimate and Measure with a 100-Link Chain.
Draw attention to the 100-link chain you prepared prior to the lesson. See Materials Preparation. Tell the students you think it is 100 links long, but you need them to check it to make sure.

Ask:

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  • How can you check the length of this chain? (Possible response: skip count by tens)

If students do not suggest it, ask them to skip count by tens to verify the length of the chain. Remind students of Math Practices Expectation 6, Use labels. Stress that it is important to include the correct unit when reporting their measurements so that is clear what their numbers mean.

Review with students that a reasonable estimate is a "good guess." Tell them that sometimes they will have to measure something or figure it out exactly and at other times they will need to be able to make a "good guess." They need to practice both things so that they become good estimators as well as good measurers. See Content Note.

Have them look at the 100-link chain and ask:

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  • Would it be a good guess to say that a first-grader is 100 links tall? Could that be possible or is it crazy? How about three links tall—could be or crazy? (Both estimates are crazy.)
  • What do you think would be a "good guess" for a first-grader's height in links?
X

Students sometimes resist making estimates. They are much more comfortable with questions that have a clear right answer. If an estimate they make turns out not to be the exact measurement or quantity, they may conclude that it is the "wrong" answer. It is important to stress early on that there are some situations when estimates are appropriate and "correct," and that having the exact right answer does not matter. This occurs when an exact answer is not possible or when exactitude is not necessary. For example, the statement "Mary lives about ½ mile from school" is an appropriate estimate. It would be onerous to measure the distance exactly in yards, feet, and inches; in addition, such exactitude is not necessary. Be careful to discuss student estimates in terms of reasonableness and strategy, not in terms of whether an estimate turns out to be the "right" answer or not.

Give the students a few minutes to think about the question. Encourage them to discuss their thinking with their classmates. Write the students' estimates on the board, no matter how rough. A wide range of reasonable estimates for a first-grader's height might be anything from 25 links (about 32 inches) to
50 links (64 inches). It is important to engage students in a discussion about their reasoning.

Ask:

X
  • How did the 100-link chain help you estimate a
    first-grader's height?     (Possible response: I looked at the chain and compared it to my own height. The 100-link chain is much taller than I am.)
  • How did you decide to "go higher" or "go lower"?

Ask a student of average height to come forward and have another student measure him or her in links. Emphasize counting the links by tens and then counting on the additional links. Ask another student to verify the measurement.

Have the students compare the actual measurement with the estimates on the board.

Ask:

X
  • What do you think about my first estimates of 100 links and 3 links? Were those good guesses or were they crazy? Why? (Possible response: Crazy guesses. 3 links was much too short and 100 links was much too tall.)
  • What about your estimates? Were they better guesses? (Answers will vary.)

Estimate and Measure with 10-Link Chains. Distribute a 10-link chain to each student. See Materials Preparation. Ask students to look at the teacher's desk. Make clear which part of the desk they are to consider, such as the length of the top of the desk.

Tell students to think about the 10-link chains and the 100-link chain while determining an estimate for the length and ask:

X
  • What is a crazy estimate in links for the desk's length, an estimate that could not possibly be true?

Write the "crazy" estimates on the board and see if there is agreement in the class that these estimates are in fact "crazy." See Figure 2.

Then ask:

X
  • Talk with a partner and decide on a reasonable estimate in links for the desk's length, an estimate that could be possible.

Have each pair write their estimate on a piece of paper. When all have finished, ask them to tell the class their estimate. Write the estimates on the board from lowest to highest as in Figure 2.

Ask:

X
  • How did you use the chains to help make an estimate?

Ask a student to measure the desk with links. Ask:

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  • Where would we place the actual measurement in the list of estimates on the board?
  • Which numbers are smaller and which numbers are larger?

Write the measurement at the appropriate interval among the list of estimates and mark it clearly as the actual measurement.

Evaluate the Reasonableness of an Estimate. Remind students that an estimate is not the same as an actual measurement and that it is not expected that it will be the same as the measurement. However, we do want to have some way to decide whether we think an estimate is a good guess (it could be) or whether it is not close enough.

Ask:

X
  • What do you notice about the estimates on the board?
  • Are the estimates larger than our measurement, smaller, or some of each?
  • Are there a lot of estimates that are close to the actual measurement or only a few?
  • How can we decide which estimates are close enough to our measurement and which ones we think are too far away?
  • Where would you draw the lines between good enough guesses and estimates that are not close enough? Give the reasons why you think so.

Analyze with the class whether the estimates cluster around the actual measurement or if they are more spread out. If there is an identifiable cluster, draw a circle around the cluster and suggest that maybe the estimates inside the circle could be considered close enough and those outside not close enough. If there is no identifiable cluster around the measurement, have the students discuss where to draw the lines between the estimates they think are close enough and those not close enough. Allow for a wide range of estimates to be considered close enough. See Figure 3.

Adventure Book:
Student Activity Book:
Master:
Listing students' estimates
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A cluster of estimates that are "close enough"
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