Imagine this scenario:
You’re sitting in a restaurant finishing dinner with friends, and the waitperson hands
you the check. It’s your job to figure the tip on the $238.00 total. You reach for your tip table and your partner stops you and says, “What are you doing?! Just estimate! Take 10 percent of 230 and then double that.” You sheepishly put your tip table away and realize that mental math and the power of estimation did the job. And it also made you think.
In the real world, we estimate about half the time and need an accurate answer about half the time. Don’t believe me? Try this: Make a list of the things you do in your life (outside of school) for which you need to use arithmetic (adding, subtracting, multiplying, dividing).
Now check out your list. Put an ‘E’ next to the things for which you can make an estimate, and an ‘A’ next to those things for which you need an accurate answer.
Here’s my list:
I’ve done this exercise with teachers hundreds of times and without exception, they are surprised at how much we use estimation. But what happens in math class? We ask our students to find accurate answers most of the time.
Why is Estimation so Important?
Giving our students many opportunities to estimate prepares them for using it in the real world. As students gain skill and experience with estimating in different contexts, they bring more and more number sense to tasks. They learn to use benchmarks: I know how many beans are in the small scoop so I can use that information to estimate how many beans are in the larger scoop. They begin to get a feel for quantities that are reasonable: There are 30 students in our class, and it looks like there are enough chairs in the auditorium for 5 classes. They develop a sense of relative magnitude: If I have 150 cookies, will 8 serving plates be enough to hold them all? (Bresser & Holtzman, 2018). Can you imagine bringing 2 serving plates for that many cookies?!
As students gain experience with estimation, they begin to be able to decide when accuracy is essential and when an estimate will be good enough (or even better). And being good at estimation can really help students when they take multiple choice tests and have to narrow down possible answers. For example, before solving a problem like 4,268 + 3,229, students can do ‘front end’ estimation to get a ballpark answer by just adding 4,000 and 3,000.
Estimation Activities for the Classroom
It’s easy to integrate estimation activities and questions into your school day. One thing you can do is ask students to estimate before figuring the exact answer.
“Is the answer going to be greater than or less than 50 (or 100, or…)? Why?”
“The answer will be about/around….”
“What estimate would be too low? Too high? Close enough?”
These questions and prompts can give you a lot of information about a student’s number sense and their estimation abilities. Are their estimates ‘in the ballpark’ or are they way off?
What’s important is that students have many opportunities to estimate because the skill develops over time and with experience. Following are some of our favorite estimation activities for a range of grade levels.
How many are in the jar?
This estimation activity is appropriate for any grade, depending on how many things you put in the jar. We recommend putting the same size thing in the jar. For example, the jar in the picture above has all the same things in it and they are the same size.
You can start with the jar full or empty. If it’s full, you can ask students to estimate and collect their estimates by writing them down on a class chart. Take some out, and then ask students to re-estimate. This gives them a chance to think about their estimate again, letting the number of things left in or out of the jar guide them. Using benchmarks is an important skill when estimating. For example, a student might reason, “At first, I thought there were 100 things in the jar. We took out 10, and it’s now about half full. Now I think there are about 20 things in the jar.”
Another idea is to show the class 3 jars and ask, “Which one has about 100 (or whatever quantity is appropriate)? Then ask them to explain.
Estimation for Very Young Children
In her work with preschoolers and kindergartners, Brenda Mercado uses estimation as a starting point in her assessments of children’s counting abilities. She begins by showing a child a jar full of rocks.
When she asks a 5-year-old how many rocks he thinks there are in the jar, he first says “Three,” and then changes his mind and guesses “Ten.” Not far off! She has him spill the rocks on a table and count them, looking for evidence of one-to-one correspondence, cardinality (the last rock touched and counted stands for the total amount), knowledge of number names, and an ability to count in sequence.
If you teach preschool, TK, or kindergarten, incorporate estimating during the school day. During snack time for example, show students a handful or cupful of snacks, like fish crackers, and ask them, “How many do you think there are?”
During math time, have children draw a shape on a piece of paper and ask them, “About how many color tiles will cover the shape? Or have them build a train with cubes and ask them, “About how many cubes long is your train?”
How Many Beans in the Jar?
With a group of fourth graders, I held up an empty jar, a scoop, and a bag of dry pinto beans and asked, “About how many scoops of beans do you think will fill the jar?” After students made their initial estimates, I put 10 scoops of beans in the jar and asked the students to re-estimate. Their estimates got much better!
On another day, I showed the class a different empty jar, a scoop, and a bag of dry pinto beans and asked, “How many beans do you think are in the jar?” After we counted how many beans in one scoop, I put 20 scoops of beans in the jar until it was about half full and asked the class to write about their thoughts. Megan explained:
In Megan’s explanation above, she used some computation when making her final estimate.
When engaging students in computational estimation, you can pose a problem, give them some possible ‘about answers’ and then ask them to talk with a partner about which ones are reasonable and which are unreasonable.
For Third Graders… For Fourth and Fifth Graders…
560 + 1,409 365 x 80
The answer is…. The answer is…
About 1,000 About 500
About 2,000 About 2,500
About 3,000 About 30,000
About 4,000 About 350,000
Greater than 1 or Less Than 1?
In Marilyn Burns and Lynn Zolli’s Listening to Learn assessment tool, they interview students to learn about their thinking. One question they ask is about the problem 3/4 + 1/3. They want to know if students think that the answer is greater or less than one whole. Rather than ask for an exact answer, the interviewers want to see how the students estimate and are curious about their reasoning. One student says that “It’s greater because 3/4 is equal to 0.75. And 1/3 is around 0.32 more or less. So, if you add them together it’s more than one.”
Another student says that the answer to 3/4 + 1/3 is greater than one, “Because 3/4 plus 1/4 is equal to one, and 1/3 is greater than 1/4, so 3/4 + 1/3 has to be greater than one.”
Asking students to estimate and explain their reasoning can tell us a lot about what students understand and what strategies they use to solve problems.
Tell Me All You Can
Tell Me All You Can is kind of like a Number Talk. Here’s how it works.
Write an arithmetic problem on the board. For example, 12 x 7.
Ask them what they can say about the answer without figuring the exact answer. You want them to estimate, so providing sentence frames can help.
“The answer is going to be between ____ and ____ because ______.”
“The answer is going to be greater (or less) than ____ because _______.”
“The answer is going to be close to ____ because __________.”
“The answer is going to be about ____ because ___________.”
Students might say things like,
“I think the answer is going to be less than 120 because 12 times 10 is 120.”
“I know that the answer is going to be greater than 60 because I know that 12 x 5 is 60.”
“I know it’s going to be greater than 70 because 10 x7 is 70, and 12 is more than 10.”
Tell Me All You Can works for almost any grade level to give students practice with arithmetic and estimation skills. Students typically do some sort of initial computation to make their estimate, and the activity pushes them to use their place value knowledge to make those estimates as seen in the examples above.
Developing Good Estimators
We want our students to become good estimators because they’ll use this skill in real-life and it helps them develop their number sense. Good estimators are flexible thinkers. They have a sense when an answer is reasonable or not. They have a good feel for the relative magnitude of numbers (how big or small numbers are in comparison to other numbers). And they know when an estimate is too big or too small or just right.
The simple question, “What do you think the answer will be?” can put a student on their journey to becoming a good estimator. What are some good estimation activities that you can share with us?