Estimating Sums

Est. Class Sessions: 2

Developing the Lesson

Part 2: Use Estimation to Determine Reasonableness

Estimate Sums. In Unit 7, students learned that estimation is very helpful in determining if an answer is reasonable. Pose the estimation problem in Question 1 on the Setting Goals pages in the Student Activity Book. Ask student pairs to estimate the answer and be ready to explain the strategies they use. Emphasize that they have to find out if the second grade classes are close to 1000 laps and that they do not have to find the exact answer. Have copies of the Open Number Lines Master and base-ten pieces readily available.

Upon completion, have students share their strategies for solving the problem. Ask students to give names to the strategies they used and add the strategies to the chart. See Figure 1 for possible estimation strategies.

It is important to give students the opportunity to invent strategies for estimating answers and to understand that different estimation strategies can result in different correct answers. Some common computational estimation strategies are:

Front-end Estimation: When estimating sums or differences, the most important digits are the ones to the left. Looking at the first one or two place-value columns (e.g., adding the hundreds digits) will usually result in a reasonable answer.
Rounding or Using Friendly Numbers: Rounding two- or three-digit numbers to the nearest ten or hundred helps students use numbers that are easier to add or subtract. Rather than teaching rigid rules for rounding (e.g., numbers halfway between should always be rounded up), allow students to be flexible in their thinking and select numbers that are easier to add or subtract or numbers that will help them compute the answer quickly.
Compatible Numbers: Compatible numbers are two or three numbers that can be grouped together to make it easier to compute the answer. For example, for the problem 82 + 59 + 19, we can group 82 and 19 to equal about 100, and then add 59. The estimate is 159.

Make a list of strategies students use to solve the problems and add them to the Estimation Strategies for Addition chart. In Figure 1, some of the strategies used were:

Using Friendly Numbers (Closest 100)
Using Friendly Numbers (Closest 10)
Thinking about Base-Ten Pieces
Counting On by Hundreds and Tens
Adding Hundreds
Composing and Decomposing Numbers

Use What Works. This lesson focuses on having students invent strategies for estimating sums. Encourage students to use estimation strategies that rely on their understanding of place value. Focus on using mental math to check for reasonableness of a sum by having students use strategies that make sense to them. Although finding the closest ten or hundred for a number is one of the strategies mentioned in this lesson, mastery of this concept is not expected at this level.

Display the second Setting Goals page in the Student Activity Book. Explain that the third-graders thought the second-grade walkathon was such a good idea that they wanted to have their own walkathon. Read the introduction and ask student pairs to use estimation to solve the problem presented on the page. Encourage students to use some of the strategies on the Estimation Strategies for Addition chart and to be ready to explain the estimation strategies they use.

Upon completion, ask students to share their solution strategies.

Ask:

How did you begin to solve the problem?
What type of estimation strategies did you use?
What kind of addition strategies did you use?

Use Sample Dialog 1 to guide the discussion.

Use the Sample Dialog to guide your discussion of Question 2 on the Setting Goals pages in the Student Activity Book.

Teacher: How did you begin to solve the Setting Goals problem?

Rosa: We knew that if the kids had this many laps in 2 days we would have to double them to see where they might be in 4 days.

Luis: But before we doubled, we found the friendly numbers for 158, 121, and 201.

Teacher: How did you find the friendly number for 158?

Kim: 158 is close to 150 and 150 is easy to double.

Diana: We didn't choose 150. We said it was closer to 160.

Teacher: Kim and Diana, how might your estimates be different?

Kim: Our total might be a little bit less than Diana's.

Teacher: Yes, but I think both of your estimates are reasonable. How about 123?

Josh: We found the closest ten on the number line and it was 120.

Teacher: And 201?

Josh: That was easy. We just found the closest 100 for 201 and it was 200.

Teacher: Okay. How did you decide whether or not the third-graders met their goal?

Diana: We doubled the friendly numbers and then added them all up.

Teacher: What kind of addition strategy did you use?

Diana: For each number, we thought of the expanded form and added the hundreds and the tens. Like to double 160 we thought 100 + 60 + 100 + 60 = 200 + 120 = 320. After we doubled the 2-day totals, we got 320 and 240 and 400 laps. Adding the hundreds, 300 + 200 + 400, they had 900. Adding on the tens, 900 + 20 + 40 is about 960 laps in 4 days.

Teacher: Did anyone use a different addition strategy?

Miguel: We didn't double the friendly numbers right away. We added them up and then doubled our answer. 150 + 120 + 200 = 470. We estimated that 470 was close to 500 and 500 + 500 is 1000.

Teacher: Great strategies! Do you think the third-graders will make their goal of 1200 laps in 4 days?

Yolanda: Not if they keep walking at this pace. They will need to walk more laps in the next two days to reach a goal of 1200.

Determine Reasonableness of a Sum. Display the following problem and have student pairs discuss which answer is the most reasonable:

552
+623

Ask:

Do you think the answer is under 1000 or over 1000? (Possible response: I think it's over 1000 because if you add the hundreds it is 1100. That's over 1000.)
Here is another problem with three possible answers. Which answer is the most reasonable?

679
+233

8112
900
1200

See Sample Dialog 2 for possible estimation strategies students may use to determine a reasonable sum.

Use the Sample Dialog to guide your discussion of estimation strategies and checking for reasonable sums.

Teacher: Which answer is the most reasonable for this problem?

Levi: We think 900 is the most reasonable because we started at 679 and counted on 200 and we got 879 for an estimate. Our estimate is closest to 900.

Roberto: We also think 900 is the closest. We found friendly numbers for 679 and 233. The closest ten for 679 is 680 and the closest ten for 233 is 230.We used the number line to add. We added 600 and 200 and that's 800. Then we added the 80 from the 680 and that brings us to 880. Then we hopped 30 more and we landed at 910. That's closest to 900.

Kathy: We said 679 is close to 700 and 233 is close to 200, so we thought 900 was the most reasonable.

Megan: We think our strategy is the easiest. We just looked at the hundreds. We added 600 + 200 = 800. We said that 900 is the closest to 800, so 900 is the most reasonable.

Teacher: Why did you think that 1200 wasn't a good estimate for this problem?

Johnny: It wasn't as bad as 8112, but when we estimated, we knew that 900 was a better estimate.

Teacher: Why?

Johnny: We added the hundreds: 600 + 200 = 800. Then we looked at the tens and 70 + 30 = 100 and 800 + 100 = 900, so we knew the answer was close to 900.

Teacher: Why do you think it's a good idea to estimate your answers?

Julia: When I estimated, I knew right away that 8112 was not a reasonable answer. If I got that answer, I would know that I was way off and I should check my answer to see what I did wrong.

Natasha: It helps us to see if our answers are reasonable and it helps us to find sums quickly.

Teacher: Can you think of an example of when it's important to find a sum quickly?

Jason: When I go to the store and I find two toys I want to buy and they cost $3.99 and $1.99, I can estimate that they cost about $6.00.

Teacher: Great answer! Estimation helps us to arrive quickly at an answer close to the actual sum and to check the reasonableness of an answer.

Have student pairs estimate the sums in Questions 1–6 on the Find the Best Answer pages in the Student Activity Book. Upon completion, have students share their estimation strategies.

Adventure Book: