3.8: Use Estimation
Introduction
Summer Job Benefits
Jose has enjoyed working all summer. He loved helping Mr. Harris and his recycling idea ended up being very profitable.
Jose began the summer with an estimate of how much money he thought he would make.
Jose earned $7.00 per hour and he worked ten 30 hour weeks.
Jose ended up earning $2100.00 for the summer, and he is very pleased with his accomplishment.
Now that the summer is over, Jose wishes to spend part of his money on new clothes for school.
He has selected the following items.
$19.95
$32.95
$46.75
Jose brought $100.00 with him to purchase the items.
If he estimates the total cost, what would it be?
How much change will Jose receive from the $100.00?
Using estimation will help Jose with his purchases.
Let’s look at some situations where estimation makes the most sense, then we will come back to this problem to help Jose with his shopping.
What You Will Learn
In this lesson, you will learn to use the following skills:
- Read and understand given problem situations
- Develop and use the strategy: Use Estimation
- Plan and compare alternative approaches to solving problems
- Solve real-world problems using selected strategies as part of a plan
Teaching Time
I. Read and Understand Given Problem Situations
We can use estimation in several different problem situations. To use estimation, we need to read and understand the problem. There will be clues in the problem to let us know if estimation is a good option for solving that specific problem.
Let’s review what it means to estimate.
Estimating means that we are going to be finding an answer that is an approximate answer.
When estimating, our answer must make sense, but it does not need to be exact.
We can find an answer that is reasonable to provide us information for our problem.
When looking at a problem, we need to read the problem to see if estimating is a good option in the problem.
We can look for key words to help us with this.
Here are some of the key words that we use when estimating:
- Close to
- Approximate
- Estimate
- An answer that makes sense
- About
If you see these words in a word problem, you can use estimating to find the answer.
Let’s look at an example.
Example
Kelly wanted to get an idea how much she was spending at the store. On the way to the checkout she looked at the items in her cart. Here are the prices of the food in her cart: $.50, $2.50, $ 3.45 and $ 6.79. About how much will Kelly spend when she checks out?
Are there any key words in this problem?
Yes, the word ABOUT lets us know that we can estimate to find our answer.
Now that we know that we can estimate, how do we use estimation to solve this problem?
II. Develop and Use the Strategy: Use Estimation
Once you have figured out that you can estimate to solve the problem, you will need to apply the estimation strategy.
We can do this in one of two ways.
- Rounding
- Front – end Estimation.
For the problem that we just looked at, let’s use rounding.
Here is the problem once again.
Example
Kelly wanted to get an idea how much she was spending at the store. On the way to the checkout she looked at the items in her cart. Here are the prices of the food in her cart, $.50, $2.50, $ 3.45 and $ 6.79. About how much will Kelly spend when she checks out?
Next, let’s round each price.
.50 becomes 1
2.50 becomes 3.00
3.45 becomes 3.00
6.79 becomes 7
Now we can add up the rounded answers: 1 + 3 + 3 + 7 = 14
Our answer is $14.00. Kelly will spend approximately $14.00 at the store.
III. Plan and Compare Alternative Approaches to Solving Problems
There are many different ways to approach solving a problem. In the last example, we used rounding and estimation. We know that this is an approach that works when we are looking for an approximate answer.
If we had been working with large numbers in the thousands, we would have been using estimation and front – end estimation.
Sometimes, we will need to draw a picture to solve a problem. That is what will make the most sense.
Let’s look at an example where we would draw a picture to solve an estimation problem.
Example
Carl is working on building a small cd rack out of wood. He can buy material in a 6' \begin{align*}\times\end{align*} 8' rectangular piece of plywood. Carl needs to build two sides from one piece of wood. The sides have the dimensions 2' \begin{align*}\times\end{align*} 4'. If Carl buys one sheet of plywood, will he have enough wood for the two sides of the cd rack?
Hmmmm…. How can we work on this problem?
We don’t need an exact measurement we just need to know the rough dimensions to figure out if the two sides of the cd rack will fit on piece of plywood.
We can use estimation to do this.
First, let’s underline the important information in the problem.
Carl is working on building a small cd rack out of wood. He can buy material in a \begin{align*}\underline{6' \times 8'}\end{align*} rectangular piece of plywood. Carl needs to build two sides from one piece of wood. The sides have the dimensions \begin{align*}\underline{2' \times 4'}\end{align*}. If Carl buys one sheet of plywood, will he have enough wood for the two sides of the cd rack?
Notice here that we show three pictures.
The first one is of the rectangular piece of wood that is 6 \begin{align*}\times\end{align*} 8.
The second two are the two rectangles that will make up the side of the cd rack.
This is a visual way to estimate whether the two pieces will fit on the one piece of plywood.
Visually it looks like it will work. Visual estimation is one strategy.
What about if we want to be sure our estimate is accurate?
We can estimate the dimensions of the two sides of the cd rack combined.
2 \begin{align*}\times\end{align*} 4 + 2 \begin{align*}\times\end{align*} 4 = 4 \begin{align*}\times\end{align*} 4
We need a piece of wood that is 4 \begin{align*}\times\end{align*} 4 to build the sides of the cd rack.
Since our piece is 6 \begin{align*}\times\end{align*} 8 it will work for us.
Our visual estimation is accurate.
Real Life Example Completed
Summer Job Benefits
Now we can help Jose with his shopping. Shopping is a great real life example where estimation is very useful. We can get an idea of how much we are spending as well as about how much change we can receive when estimation.
Let’s take another look at the problem.
Jose has enjoyed working all summer. He loved helping Mr. Harris and his recycling idea ended up being very profitable.
Jose began the summer with an estimate of how much money he thought he would make.
Jose earned $7.00 per hour and he worked ten 30 hour weeks.
Jose ended up earning $2100.00 for the summer, and he is very pleased with his accomplishment.
Now that the summer is over, Jose wishes to spend part of his money on new clothes for school.
He has selected the following items.
$19.95
$32.95
$46.75
Jose brought $100.00 with him to purchase the items.
If he estimates the total cost, what would it be?
How much change will Jose receive from the $100.00?
We could use a couple of different strategies to estimate the total of Jose’s purchases.
We could use rounding or front – end estimation.
Let’s use rounding first.
$19.95 rounds to $20.00
$32.95 rounds to $33.00
$46.75 rounds to $47.00
Our estimate is $100.00.
Hmmm. Ordinarily, rounding would give us an excellent estimate, but in this case our estimate is the amount of money Jose wishes to pay with.
Because of this, let’s try another strategy. Let’s use front – end estimation and see if we can get a more accurate estimate.
\begin{align*}19 + 32 + 46 & = 97 \\ 1 + 1 + 80 & = 2.80\end{align*}
Our estimate is $99.80.
With front – end estimation, we can estimate the Jose will receive .20 change from his $100.00. While he isn’t going to get a lot of change back, he is going to receive some change so he does have enough money to make his purchases.
Technology Integration
Khan Academy Estimation with Decimals
Time to Practice
Directions: Look at each problem and use what you have learned about estimation to solve each problem.
1. Susan is shopping. She has purchased two hats at $5.95 each and two sets of gloves at $2.25 each. If she rounds each purchase price, how much can she estimate spending?
2. If she uses front – end estimation, how does this change her answer?
3. Which method of estimation gives us a more precise estimate of Susan’s spending?
4. If she brings $20.00 with her to the store, about how much change can she expect to receive?
5. If she decided to purchase one more pair of gloves, would she have enough money to make this purchase?
6. Would she receive any change back? If yes, about how much?
7. Mario is working at a fruit stand for the summer. If a customer buys 3 oranges at $.99 a piece and two apples for $.75 a piece, about how much money will the customer spend at the fruit stand? Use rounding to find your answer.
8. What is the estimate if you use front – end estimation?
9. Why do you think you get the same answer with both methods?
10. If the customer gives Mario a $10.00 bill, about how much change should the customer receive back?
11. Christina is keeping track of the number of students that have graduated from her middle school over the past five years. Here are her results.
2004 – 334
2005 – 367
2006 – 429
2007 – 430
2008 – 450
Estimate the number of students who graduated in the past five years.
12. Did you use rounding or front – end estimation?
13. Why couldn’t you use front – end estimation for this problem?
14. Carlos has been collecting change for the past few weeks. He has 5 nickels, 10 dimes, 6 quarters and four dollar bills. Write out each money amount.
15. Use rounding to estimate the sum of Carlos’ money.
16. Use front – end estimation to estimate the sum of Carlos’ money.
17. Which method gives you a more accurate estimate? Why?
18. Tina is working to buy presents for her family for the holidays. She has picked out a cd for her brother for $14.69, a vase for her Mother at $32.25 and a picture frame for her father at $23.12. Use rounding to estimate the sum of Tina’s purchases.
19. Use front – end estimation to find an estimate for the purchases.
20. Which estimate is more accurate?
21. Why?
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