**Fraction Series, Week 3: Benchmarks and Estimation**

In 4th grade, we have โcomparing fractions with different numerators and denominators by using common denominators or benchmarks.โ

When we move to the 5th grade standards, we see benchmarking is also included in 5.NF.A.2:

- โSolve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.โย

**Using Fraction Benchmarks**

I’m not sure if your experience is the same as mine, but over the years, I’ve observed that this understanding of whether or not answers make sense is definitely an area of weakness for students (not just in the area of fractions).

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.

**Letโs 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 thatย
- So, if 5/8 is more than half and 2/5 is less than half,ย that means 5/8 must be greater than 2/5

โThis is great logic. I have found, however, that some students donโt 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!

When it comes to using 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, ยฝ, or 1.

2) Estimating 5/8 promptsย the question โhow many 8ths would be in exactly half?โ (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: Thatโs 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,ยฝ, or 1?
- Student:ย ยฝ

3) Same process for 2/5: how many 5ths would be in exactly ยฝ?โ

- Student: 2.5
- Teacher: How do you know?
- Student: because 5ย divided by 2 is 2.5
- โTeacher: is 2.5/5 close to 2/5?
- Student: yes
- Teacher: Is 2/5ย closer to 0,ยฝ, or 1?
- Student:ย ยฝโ

4) Summarize estimation of the addends to get to estimation of the sum:

- If 5/8 is close to ยฝ and 2/5 is close to ยฝ, about how much should the final answer be?
- Student: 1
- Teacher: Why?
- Student: because ยฝ and ยฝ = 1โ

**Include Models for Fraction Estimation**

If you have fraction strips to show these fractions quickly, thatโs 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

ยฝ mark on the ยฝ 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,ย ยฝ, 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**

Again providing a visual model is very helpful.

In this case, the number line may be even more helpful than in the previous example.

These fractions ARE closer to ยฝ, but not as close as those in the previous example:

- Students may think that 3/10 is closer to 0 than ยฝ if they can’t see it.
- Students may think 2/3 is closer to 1 than ยฝย 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 ยฝ 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 ยฝ is the closer benchmark

If you have any fraction tips, please share in the comments:-)

Check out the course,ย

**Fractions: From Foundations to Operations.**