Estimating Sums and Differences

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In this lesson, we build on some ideas explored in the modeling addition and subtraction lesson. Suppose we want to estimate the sum modeled by the combining of these two collections (the small square represents one unit):

There are several ways to approach this:

• One is to simply combine the largest pieces (hundreds) in the two collections to get an estimate of 500. (When we look at the largest pieces, represented by digits at the left of a number, it is referred to as front-end estimation.)

• Another is to round the two collections to the nearest hundred (300+300) to get a total of 600.

• We can apply front-end estimation to the hundreds and tens (300+200+10+50), to get a total of 560.

• Yet another method is to round both collections to the nearest ten (320+260) for an estimate of 580.

All of these are reasonable estimations for the sum of 316 and 255. In symbolic terms, these estimations would be written as (‰ means "approximately equal to"):

• 316+255‰500
• 316+255‰600
• 316+255‰560
• 316+255‰580

None of these estimates is a 'best' estimate. It depends on the situation. For example, in estimating how much money to take on a trip to cover motel and food expenses, it might might be better to overestimate, as in the second example.

Now, suppose we want to estimate the difference between the following two collections:

Some possibilities are:

• Use hundreds only to get a difference of 300.

• Use hundreds and tens to get a difference of 360.

• Round both to the nearest hundred to get a difference of 400.

• We could also use compatible numbers (575 and 225) to get a difference of 350.

Again, these are all reasonable estimates for finding the difference of 584 and 225, depending on the situation, and in symbolic terms would be written as:

• 584-225‰300
• 584-225‰360
• 584-225‰400
• 584-225‰350

Here are a some situations to think about. How would you perform the estimation in these cases? Front-end? Rounding?

• Becky earned $127 in one week of babysitting and has $248 in the bank. She wants to buy a $400 bicycle.
• Jason's favorite comics cost $2.17 on sale. He wants to buy three of them.
• The school basketball team averaged 76.8 points over a four game series. The school record for a four game stretch is 306 points.
• Shawna has $153 in the bank. She has to pay her mom back for a $49 pair of tennis shoes, and wants to buy her brother a $28 logo t-shirt for his birthday. Her share of horse camp this summer will be $65.

Rounding and mental computation are important estimation skills. We use estimation every day in many contexts, so it's a good idea to practice with your child often in everyday situations at home, in the store, and on the road. Estimation is also a great way to verify that the results of calculator and pencil-and-paper computations in math problem solving are reasonable!

Don't forget - The above methods for estimating both sums and differences could be used with decimal numbers. In that case, the large square is the unit, and the strips and small squares represent tenths and hundredths, respectively. (See the lesson on modeling decimal numbers.)