Teaching Fractions, Week 3: Fraction Benchmarks and Estimation
Using Fraction Benchmarks
So, if we can incorporate the reinforcement of benchmarks, estimating, and the understanding of ‘reasonable’ it will be a huge benefit to our middle school students.
4th Grade: Using Fraction Benchmarks to Compare
Let’s look at the 4th grade concept of using benchmarks to compare fractions. When students are asked to compare fractions like 5/8 and 2/5, for example, the standard would like them to be able to look at 5/8 and think:
- Hmm, 4/8 is half (benchmark) and 5/8 is more than that
- 2.5/5 would be exactly half, and 2/5 is less than that
- So, if 5/8 is more than half and 2/5 is less than half, that means 5/8 must be greater than 2/5
This is great logic. I have found, however, that some students don’t know how to find half of a number and even if they do know, they have trouble with half of an odd number. Have you found this too? That may be something for another post!
Using Fraction Benchmarks in 5th & 6th Grades
When it comes to using fraction benchmarks in 5th and 6th grades and beyond, we might not be using them as much to teach comparing fractions; but we can teach using benchmarks (and finding half) in the context of estimating answers for adding and subtracting fractions (and eventually for multiplying and dividing).
Fraction Benchmarks to Estimate Fraction Addition
1) Before adding, ask students to estimate the answer by changing each fraction to a benchmark of 0, ½, or 1.
2) Estimating 5/8 prompts the question ‘how many 8ths would be in exactly half?’ (bulleted items are a sample discussion between teacher and student)
- Student: 4/8
- Teacher: How do you know?
- Student: because 8 divided by 2 is 4
- Teacher: That’s right, to find half of a number, we divide by 2. So, is 5/8 close to 4/8?
- Student: yes
- Teacher: Is 5/8 closer to 0,½, or 1?
- Student: ½
3) Same process for 2/5: how many 5ths would be in exactly ½?’
- Student: 2.5
- Teacher: How do you know?
- Student: because 5 divided by 2 is 2.5
- Teacher: is 2.5/5 close to 2/5?
- Student: yes
- Teacher: Is 2/5 closer to 0,½, or 1?
- Student: ½
4) Summarize the estimation of the addends to get to the estimation of the sum:
- If 5/8 is close to ½ and 2/5 is close to ½, about how much should the final answer be?
- Student: 1
- Teacher: Why?
- Student: because ½ and ½ = 1
Include Models for Fraction Estimation
If you have fraction strips to show these fractions quickly, that’s great. If not a quick drawing will help. (Personally, I might have some difficulty making these quickly and accurately, so I created a PDF download with the fraction strips to go with the examples in this post.)If students are having difficulty with finding half of 8 and 5, a visual model like this will allow them to line up the
½ mark on the ½ strip with the 8ths and 5ths, and see exactly where the fraction falls. They can see more clearly how close the fractions are to 0, ½, and 1.While estimation may cause the fraction addition or subtraction problems to take a little longer, the extra time and modeling reinforces several concepts: finding half, comparing fractions with benchmarks, and estimating to see if the answer is reasonable.
Include Models for Fraction Estimation: Example 2
In this case, the number line may be even more helpful than in the previous example.
These fractions ARE closer to ½, but not as close as those in the previous example:
- Students may think that 3/10 is closer to 0 than ½ if they can’t see it.
- Students may think 2/3 is closer to 1 than ½ if they can’t see it.
Reasoning for 3/10:
- Half of 10 is 5, so 5/10 is halfway
- 3 is closer to 5 than it is to 0, so ½ would be the better estimate.
Reasoning for 2/3:
- Half of 3 is 1.5
- 2 is closer to 1.5 than it is to 3, so again ½ is the closer benchmark
How will you use Fraction Benchmarks?
If you have any fraction tips, please share in the comments:-)
Do you need a few tools to help your students master the fraction basics? I’ve got a ‘Fraction Toolkit’ for you!You can grab the Fraction Toolkit and learn how to use it in this post!
