1.2: Whole Number Estimation
Introduction
The Penguin Estimation
After figuring out how many fish to order for the seals, Jonah went to meet his friend Sarah for lunch. Sarah is also a zoo volunteer. She has been working in the penguin arena. There are 57 penguins at the city zoo. One of Sarah’s jobs is to feed the penguins.
“Wow, what a morning. I had to figure out how much seafood to order for the seals. My whole morning has been solving problems. I thought I left math at school,” Jonah says, biting into his peanut butter sandwich.
“What’s so hard about that? I could figure out how much to order for the penguins without even using a piece of paper,” Sarah states.
“What! How can you do that?”
“Estimation. The penguins eat about 18,000 fish per month,” Sarah says, biting into her sandwich.
“18,000 fish!! How do you know that?”
“I told you, estimation. There are 57 penguins who each eat about 8 – 10 fish per day. You don’t need an exact number, just be sure to have enough fish. Once you know that, the rest is easy,” Sarah smiles and takes a sip of her water.
Jonah is completely perplexed.
How did Sarah do that so quickly?
What is estimation all about anyway? Could he have used estimation to solve his own problem?
You will learn all that you need to know to help Jonah to understand how Sarah figured out the penguin food so quickly by reading through this next lesson.
Pay close attention. At the end of this lesson, we’ll revisit this problem and see how she did it.
What You Will Learn
In this lesson, you will learn the following skills:
- Estimating sums and differences of whole numbers using rounding
- Estimating products and quotients of whole numbers using rounding
- Estimating to find approximate answers to real-world problems
- Using estimation to determine whether given answers to real-world problems are reasonable
Teaching Time
In the real world problem in the introduction, you saw how puzzled Jonah was when Sarah was able to use estimation to help her solve the penguin problem.
Estimation definitely seemed to save Sarah some time.
What do we mean by estimation? When can we use it and when shouldn’t we use it?
To estimate means to find an answer that is close to the exact answer.
The key with estimation is that you can only use it in instances where you don’t need an exact answer.
When we estimate, we want to find an answer that makes sense and works with our problem, but is not necessarily exact.
Let’s start by looking at estimating sums and differences.
I. Estimating Sums and Differences
Remember back in the first lesson, we used the word sum and the word difference.
Let’s take a minute to review what those two words mean.
A sum is the answer to an addition problem.
A difference is the answer to a subtraction problem.
To estimate a sum or a difference, we can round the numbers that we are working with to find our estimation.
What does it mean to round a number?
When we round, we change the number to the nearest power of ten (times a whole number), such as ten or hundred or thousand, etc.
Let’s look at an example.
Example
69
Let’s say that we want to round this number to the nearest ten. Well, we can look at whether 69 is closer to 60 or to 70. These are the two numbers in the tens surrounding 69. It is closer to 70, so we would change the number to 70.
Example
53
If we want to round this to the nearest ten, then we can look at the numbers surrounding 53 which are multiples of ten. Is 53 closer to 50 or 60? It is closer to 50, so we would “round down” to 50.
When rounding, we can follow the rounding rules.
If the number being rounded is less than 5, round down.
If the number being rounded is greater than 5, round up.
In the examples, we were rounding to the tens, so we use the number in the ones place to round. Using 69, since 9 is greater than 5, we round up. In the case of 53, 3 is less than 5, so we round down.
Let’s apply this.
Example
128 Round to the nearest ten.
Look at the number. We are rounding to the tens, so we look at the ones place.
8, is greater than 5, so we round up to 130.
What does this have to do with estimating sums and differences?
Well, when we estimate a sum or a difference, if we round first, it is easier to add.
Example
\begin{align*}58 + 22 = \underline{\;\;\;\;\;\;\;}\end{align*}
We want to estimate this answer.
If we round each number first, we can use mental math to find our estimation.
58 rounds to 60
22 rounds to 20
Our estimate is 80.
Here is one with larger numbers.
Example
\begin{align*}387 + 293 =\underline{\;\;\;\;\;\;\;}\end{align*}
We want to estimate our answer by rounding to the nearest hundred.
387 rounds to 400
293 rounds to 300
Our estimate is 700.
This worked for addition. What about subtraction?
We can estimate differences by rounding too.
Example
\begin{align*}56 - 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
We want to estimate this difference by rounding to the nearest ten.
56 rounds to 60
18 rounds to 20
Our estimate is 40.
We can estimate differences with larger numbers too.
Example
\begin{align*}990 - 211 = \underline{\;\;\;\;\;\;\;}\end{align*}
We want to estimate our difference by rounding to the nearest hundred.
990 rounds to 1000
211 rounds to 200
Our estimate is 800.
Here are a few problems for you to try on your own.
- \begin{align*}17 + 27 =\underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}290 + 510 = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}78 - 16 =\underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}592 - 411 = \underline{\;\;\;\;\;\;\;}\end{align*}
Take a few minutes to check your work with a peer.
II. Estimating Products and Quotients of Whole Numbers
We just finished estimating sums and differences. What about products and quotients?
Those are vocabulary words from the first lesson. Let’s review what they mean before we continue.
A product is the answer to a multiplication problem.
A quotient is the answer to a division problem.
How do we estimate a product?
We can estimate the product of a multiplication problem by rounding the factors that we are multiplying.
We use the same rounding rule as with sums and differences.
Example
\begin{align*}12 \times 19 = \underline{\;\;\;\;\;\;\;}\end{align*}
Let’s estimate by rounding each factor to the nearest ten.
\begin{align*}& 12 \ \text{rounds to} \ 10\\ & 19 \ \text{rounds to} \ 20\\ & 10 \times 20 = 200\end{align*}
Our estimate is 200.
This may seem a little harder than adding and subtracting, but you should be able to use mental math to estimate each product.
We can estimate a quotient in the same way.
Example
\begin{align*}32 \div 11 =\underline{\;\;\;\;\;\;\;}\end{align*}
Let’s estimate by rounding each value to the nearest tenth.
\begin{align*}& 32 \ \text{rounds to} \ 30\\ & 11 \ \text{rounds to} \ 10\\ & 30 \div 10 = 3\end{align*}
Our estimate is 3.
Example
\begin{align*}869 \div 321 = \underline{\;\;\;\;\;\;\;}\end{align*}
Let’s estimate by rounding each value to the nearest hundred.
\begin{align*}& 869 \ \text{rounds to} \ 900\\ & 321 \ \text{rounds to} \ 300\\ & 900 \div 300 = 3\end{align*}
Our estimate is 3.
Sometimes, when working with division, we need to find a compatible number, not just a rounded number.
What is a compatible number?
A compatible number is one that is easily divisible.
Let’s look at an example that uses compatible numbers.
Example
\begin{align*}2321 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
This one is tricky. Normally, we would round 2321 to 2300, but 2300 is not easily divisible by 8.
However, 2400 is easily divisible by 8 because 24 divided by 8 is 3.
2400 is a compatible number.
Let’s round and estimate.
\begin{align*}& 2321 \ \text{becomes the compatible number} \ 2400\\ & 8 \ \text{stays the same}\\ & 2400 \div 8 = 300\end{align*}
Our estimate is 300.
Sometimes, it can be a little tricky figuring out whether you should round or use a compatible number. You have to do what you think makes the most sense.
Here are a few problems for you to try on your own.
- \begin{align*}34 \times 18 =\underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}187 \times 11 = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}122 \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}120 \div 11 = \underline{\;\;\;\;\;\;\;}\end{align*}
Take a few minutes and check your work with a peer.
Did you catch the compatible numbers?
Real Life Example Completed
The Penguin Estimation
Now we can apply what we have learned about estimation to our real world problem.
Let’s go back to Sarah and Jonah having lunch.
Here is a review of the conversation that they had.
“Wow, what a morning. I had to figure out how much seafood to order for the seals. My whole morning has been solving problems. I thought I left math at school,” Jonah says, biting into his peanut butter sandwich.
“What’s so hard about that? I could figure out how much to order for the penguins without even using a piece of paper,” Sarah states.
“What! How can you do that?”
“Estimation. The penguins eat about 18,000 fish per month,” Sarah says, biting into her sandwich.
“18,000 fish!! How do you know that?”
“I told you, estimation. There are 57 penguins who each eat about 8 – 10 fish per day. You don’t need an exact number, just be sure to have enough fish. Once you know that, the rest is easy,” Sarah smiles and takes a sip of her water.
Let's pause here for a minute and underline any important information. This has been done for you in the paragraph above.
Sarah claims that she can estimate to figure out how much fish the penguins eat.
Sarah begins by saying that the penguins eat about 18,000 fish per month.
Now that we know all about estimation, let’s look at how she used estimation to come up with this number by learning some more of the story.
Jonah sat puzzled for a long time. Then he finally gave up.
“Okay, I give up. How did you figure it out?” he asked.
“There are 57 penguins in the pen. I began by rounding 57 to 60 because 57 is a tough number to work with,” Sarah said smiling.
“The penguins each eat 8 to 10 fish per day. Well, 10 is a much easier number to work with than 8, so I rounded up to 10.”
“If there are 60 penguins, each eating 10 fish per day-that is 600 fish per day. I estimated that product by multiplying in my head.”
“There are 30 days in a month. So I estimated 600 per day times 30 days. My final answer is 18,000 fish.”
Sarah looked at Jonah, whose mouth was open. Then he smiled.
“That’s great for you,” he said. “But that wouldn’t have worked for my problem. I needed a closer answer. I would have ended up with way too much seafood.”
Is he correct? Let’s take a look.
Here is what the math looked like in Sarah's problem.
\begin{align*}& 57 \ \text{penguins rounded to} \ 60 \ \text{penguins}\\ & 8 - 10 \ \text{fish rounded to} \ 10 \ \text{fish}\\ & 60 \times 10 = 600 \ \text{fish per day}\\ & 30 \ \text{days in one month}\\ & 600 \times 30 \ \text{days} = 18,000 \ \text{fish}\end{align*}
Sarah’s answer makes sense. She did not need an exact answer, so this was the perfect opportunity to use estimation.
What about Jonah? Would estimation have worked for his problem?
Let’s revisit it. Here are the facts.
There are 43 seals at the zoo.
Each seal eats 11 lbs of seafood per day.
How many 25 lb buckets does Jonah need to order?
We can estimate to find our answer.
43 rounds to 50-if we round down some seals won’t eat
11 rounds to 10
\begin{align*}50 \times 10 = 500 \end{align*} pounds per day
\begin{align*}500 \times 7 = 3500 \end{align*} pounds per week
\begin{align*}3500 \div 25\end{align*} lbs per bucket \begin{align*}= 140\end{align*} buckets
Jonah figured out using exact math that he needed to order 133 buckets of seafood.
Estimating, Jonah would have ordered 140 buckets.
\begin{align*}140 - 133 = 7\end{align*} buckets \begin{align*}\times \ 25\end{align*} pounds of fish \begin{align*}= 175\end{align*} extra pounds of fish
That would have been a lot more seafood than he would have needed.
This is an example of the key things to think about when estimating:
- The answer must make sense for the problem.
- It must be reasonable.
- We need an answer that is close to the exact answer.
- If the answer does not make sense, then we have to use exact math.
Vocabulary
Here is the vocabulary in this lesson. Remember, you can find these words in italics throughout the lesson.
- Estimation
- to find an approximate answer to a problem
- Sum
- the answer to an addition problem
- Difference
- the answer to a subtraction problem
- Round
- to change a number to the nearest ten, hundred or thousand etc.
- Product
- the answer to a multiplication problem
- Quotient
- the answer to a division problem
- Factors
- the numbers being multiplied in a problem
- Compatible number
- a number that is easily divisible by the divisor in an estimation problem.
Technology Integration
Khan Academy Rounding to Estimate Sums
James Sousa on Estimating Sums and Differences
James Sousa on Estimating Solutions to Multiplication and Division
http://www.teachertube.com/members/viewVideo.php?video_id=115862&title=Estimating_Whole_Numbers - You will need to register with this website. This website looks at estimating addition, subtraction, multiplication and division using rounding and compatible numbers.
- www.teachertube.com/members/viewVideo.php?video_id=115862&title=Estimating_Whole_Numbers - This website looks at estimating addition, subtraction, multiplication and division using rounding and compatible numbers.
Time to Practice
Estimate the following sums, differences, products, and quotients.
1. \begin{align*}45 + 62 = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}32 + 45 = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}21 + 54 = \underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}103 + 87 = \underline{\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}101 + 92 = \underline{\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}342 + 509 = \underline{\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}502 + 307 = \underline{\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}672 + 430 = \underline{\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}201 + 303 = \underline{\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}678 + 407 = \underline{\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}23 - 9 = \underline{\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}46 - 8 =\underline{\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}58 - 12 = \underline{\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}76 - 9 = \underline{\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}204 - 112 = \underline{\;\;\;\;\;\;\;}\end{align*}
16. \begin{align*}87 - 65 = \underline{\;\;\;\;\;\;\;}\end{align*}
17. \begin{align*}98 - 33 = \underline{\;\;\;\;\;\;\;}\end{align*}
18. \begin{align*}354 - 102 = \underline{\;\;\;\;\;\;\;}\end{align*}
19. \begin{align*}562 - 112 = \underline{\;\;\;\;\;\;\;}\end{align*}
20. \begin{align*}789 - 99 = \underline{\;\;\;\;\;\;\;}\end{align*}
21. \begin{align*}17 \times 12 = \underline{\;\;\;\;\;\;\;}\end{align*}
22. \begin{align*}22 \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
23. \begin{align*}9 \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
24. \begin{align*}7 \times 23 = \underline{\;\;\;\;\;\;\;}\end{align*}
25. \begin{align*}36 \times 40 = \underline{\;\;\;\;\;\;\;}\end{align*}
26. \begin{align*}13 \times 31 = \underline{\;\;\;\;\;\;\;}\end{align*}
27. \begin{align*}9 \times 27 = \underline{\;\;\;\;\;\;\;}\end{align*}
28. \begin{align*}11 \times 32 = \underline{\;\;\;\;\;\;\;}\end{align*}
29. \begin{align*}19 \times 33 = \underline{\;\;\;\;\;\;\;}\end{align*}
30. \begin{align*}22 \times 50 = \underline{\;\;\;\;\;\;\;}\end{align*}
31. \begin{align*}43 \div 6 = \underline{\;\;\;\;\;\;\;}\end{align*}
32. \begin{align*}19 \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
33. \begin{align*}44 \div 5 = \underline{\;\;\;\;\;\;\;}\end{align*}
34. \begin{align*}72 \div 7 = \underline{\;\;\;\;\;\;\;}\end{align*}
35. \begin{align*}17 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
36. \begin{align*}43 \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
37. \begin{align*}62 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
38. \begin{align*}122 \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
39. \begin{align*}345 \div 11 = \underline{\;\;\;\;\;\;\;}\end{align*}
40. \begin{align*}678 \div 22 = \underline{\;\;\;\;\;\;\;}\end{align*}