Lesson 3

Close Enough!

Est. Class Sessions: 3

Developing the Lesson

Part 2. Strategies for Computational Estimation

Find Convenient Numbers. Explain that rounding is one strategy that can be used to find a reasonable estimate when an exact number is not needed. When rounding to the nearest ten or hundred, the numbers that you round to are sometimes called benchmark numbers. Benchmark numbers can also be called convenient numbers. Convenient numbers are numbers that are easy to use when you solve problems using mental math.

Ask:

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  • How does rounding a number to the nearest ten or hundred make it a convenient number to add or subtract? (Numbers that end in 0 are often easier to think about and add or subtract in your head.)

Display the number 123.

Ask:

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  • If you round 123 to the nearest ten what number will you round to? (120)
  • What number do you get if you round 123 to the nearest hundred? (100)

Help students think about other strategies to find convenient numbers.

Ask them to again think about the number 123 and ask:

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  • How could you use 125 as a convenient number for 123? (You could think about quarters. 123 is close to five quarters.)

Use Convenient Numbers to Estimate Sums and Differences. Ask students to work with a partner to read the vignette and complete Question 1 on the Close Enough! pages in the Student Guide. After students have had time to work together, ask them to share their thinking.

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Some students may find it helpful to make notes or draw a number line when finding an estimated answer.

For Question 1, students may round all of the numbers to the nearest hundred, so that the estimate is 200 + 500 + 300 or 1000. This is a reasonable estimate, since rounding 247 down and 457 up will make for a fairly good estimate. It is important for students to look back at their estimates in this way to see if the convenient numbers they used balance out.

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When asked to estimate, students often find the exact answer, then round it to the nearest ten or hundred. In order to encourage students to develop flexible estimation strategies and mental skills, students are asked to estimate answers to addition problems for three-digit numbers before they have developed formal methods for numbers this large. This will challenge students to invent strategies based on their knowledge of 2-digit addition methods and place value. For students who are frustrated finding and using convenient numbers for problems involving three digits, revise the problems so that these students can apply the strategies to two-digit numbers.

Students might prefer front-end estimation in which only the left-most digits are used to estimate. One advantage to this method is that students can visualize the largest base-ten pieces in each number. For example, the numbers in the vignette can be seen as 200 + 400 + 300 = 900 (2 flats + 4 flats + 3 flats = 9 flats). However, if students think again about the numbers in the problem (and their representations with base-ten pieces), they will realize that their estimate is low since they did not take into account any of the tens and ones (skinnies and bits) and they might want to revise their estimate upward.

Have students work with a partner or in small groups to read the estimation strategies used by Carla, Levi, and Sara in the Student Guide and complete Question 2. Ask students to choose one of the three strategies and prepare a brief presentation using base-ten shorthand (or pieces), number lines, pictures, numbers, or words that will clearly explain the strategy they chose. For example, for Carla's strategy, students may show the numbers using base-ten shorthand and count the flats to explain how Carla knew “200 + 300 + 400 is 900.” Then they can combine the remaining pieces and show that they will make more than a flat or “more than 100” to make the answer more than 1000, so Sam's estimate of 1050 students is reasonable. Encourage students to ask questions of the presenters if the presentation is not clear enough for others to understand.

After students have finished their presentations, ask:

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  • How are all of these estimation strategies similar? (Possible response: They all used convenient numbers to find out that Sam's estimate was reasonable.)
  • How are they different? (They all used a different strategy. Carla just added all of the 100s and then estimated that the other numbers would add up to more than 100. Levi rounded the numbers to the nearest ten and then added them. Sara rounded the first two numbers to the nearest 100 and added to get 700. She then added the 300 and then the 22 to see if she agreed with Sam.)

Ask students to share any other estimation strategies that could be used.

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Some students may find the actual sum instead of an estimate. This is fine, particularly when the numbers lend themselves to mental strategies that students can easily use to compute exact answers. However, point out that some problems may be too difficult to solve mentally or students may not always have enough time to find the exact answers.

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Students may use the symbol ≈ which means “is about equal to” when rounding numbers or sums. For example, 241 ≈ 250 or 235 + 255 ≈ 500.

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Explain to the class that you have used the word “estimate” in many ways. Note that the word can be used as both a verb, “Estimate this sum,” and a noun, “Here is the estimate.” We can estimate a number such as 312 to be about 300, and we can also estimate a sum.

Discuss Strategies for Finding Convenient Numbers. Ask students to work in pairs to solve Questions 3–4 on the Close Enough! pages in the Student Guide. As students work, encourage them to round appropriately and choose friendly numbers that will make the computation easy to compute mentally but produce a reasonable estimate.

Use prompts similar to the following:

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  • What numbers did you use in your estimate? What were you thinking when you chose them?
  • Are the numbers you chose easy to add in your head? If not, what other numbers can you choose that will still give you a good estimate? Can you use some other numbers besides tens and hundreds?
  • I see that you rounded all your numbers down. That makes it easy to add, but did you take into account the tens and ones in the numbers? If you look back at the numbers in the problem, do you think there is a way to revise your estimate so that it is more reasonable?

Discuss student solutions for Question 4. To solve this problem students may choose to round both 457 and 322 to the nearest ten. They can then add 460 + 320 = 780, and use this number to decide how many buses. Since each bus holds 50 students, 16 buses are needed to transport students from Clark School and Lincoln School. Students might also round both 457 and 322 to the nearest hundred and then add 500 + 300 = 800 to decide the number of buses (16) that are needed. A third strategy would be to decide how many buses Clark School needs (10) and then how many Lincoln School needs (7) and then add the buses for each for a total of 17 buses needed.

Assign Check-In: Questions 5–7 in the Student Guide for students to complete independently. Remind students to use the strategies they have learned to find convenient numbers that will help them solve each problem.

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Use Check-In: Questions 5–7 in the Student Guide to assess students' ability to estimate sums using mental strategies including rounding using benchmarks and finding convenient numbers [E5].

Student Guide: 133 134
Answers
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