**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:-)

You can grab the Fraction Toolkit and learn how to use it in this post!