Using Benchmark Fractions to Estimate Sums & Differences

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SWBAT estimate the sums and differences of fractions using benchmark fractions to guide the reasonableness of their answer.

Big Idea

Students apply their knowledge of benchmark fractions to estimate sums and differences of fractions. They also learn how to show their thinking using number lines.


5 minutes

Right before today's math class the students had PE. and I'm using this to help me make connections between the real world and math class.  I start with a quick conversation about fractions that they encountered in PE.  The students had many great thoughts about fractions in kick ball.  

• " I ran 3/4 of the bases without getting out"

• "No one kicked 7 out of 11 pitches that I threw"

• "We played kickball for 2/3 of the time we were in gym class"

Next, I draw a blank number line on the board and ask students to label the 3 benchmark fractions.  When students write 0 and 1 on the board, I ask, "Are you sure that 1 is a fraction?Can you give an example?"

When the benchmark fractions are labeled on the number line.  Students are invited to write any random fraction in their notebook.  Then, five students challenge a classmate to place the fraction they wrote on the number line.  Students must explain their reasoning in the placement of the fraction.  I encourage students to use benchmark fractions as a reference. 

This part of the lesson moves very quickly today, because we did the same thing yesterday when benchmark fractions were introduced.  Since benchmark fractions are critical to establishing the number sense of fractions, I spend time revisiting this activity so students can review these strategies.

I extend my number line beyond 1 at this point, to demonstrate that the line continues past a whole.  However, the way the curriculum map for our school is designed, we will work with mixed numbers in a future topic. 

Launch/Guided Practice

15 minutes

During the launch, I want to move the students from placing fractions on a number line using benchmarks as guides to using benchmark fractions as a way to estimate sums and differences.

Estimating the sums and differences of benchmark fractions is easy.  Using number lines to represent this  process is more challenging.  Today, my focus will be on helping students understand this process.

Since we will be working with benchmarks to help estimate addition, subtraction, multiplication and division, I emphasize these benchmark fractions because these the the fractions that they will be able to use most comfortably and confidently when gauging the reasonableness of an answer.  This is the purpose of estimating, so I don't want to over complicate this by asking students to try to use more fractions as benchmark fractions.  Also, our text book refers to 1/2s as the benchmark fractions.  I align my teaching with this to maintain consistency for the students.  You could use thirds and fourths as additional benchmark fractions as well.

To transition from placing single fractions on a number line to modeling the sum of benchmark fractions. I call on two students to share the fractions they created.  

I write 5/7 and 4/6  on the board.  The students estimate these fractions as each close to 1.  So our estimation equation is 1 + 1 = 2.  How can we express this on a number line?

Looking at this equation, what number should my number line end with?  (2).  After marking the end of the number line with a 2, place other benchmarks on the number line. (0, ½, 1, and 1 ½)

I model my thinking, using a think out loud, to show how to represent 1 +1 on a number line.  Starting at 0 and drawing a line that jumps to 1.  Then another line that jumps to 2.  

Together, we complete a few more examples of modeling this thinking.  I consciously only model addition problems.  I know when the students move on to the independent work, there are a few examples with subtraction.  As student will have the support of working in pairs, I use this as an opportunity to challenge them to apply what they know to think about how to represent subtraction. 

Independent Practice

20 minutes

Students work in pairs estimate the sums and differences of fractions with unlike denominators.  They model their thinking on number lines. 

I provide the students with blank number lines.  An important part of the modeling process is determining what numbers to put on the number line.  I like to get the students thinking flexibly about the whole from the beginning.  For this lesson, some number lines need to represent two wholes, and others only need to show 1/2.  

The various approaches students use also contribute to the group share at the end.

While students are working, I circulate to to each group.  I know in advance that students will have a hard time representing 1-1 = 0 because I did not model this type of problem.  I want the students to struggle a bit with this, then learn from a mistake if they make it.  Be aware that many students start at 2, and then show 2 "jumps" of 1 to get 0.  This is because they do the opposite picture of 1 +1 =2.  To help students model subtraction.  I ask them to consider where they should start on the number line.  Then mark this place with a *.  This scaffolds students to see that in 1 - 1, one is the whole (or start) then they jump backwards.  

I provide enough time for all students to model between 3 and 5 examples before moving on to the group share.

Group Share

5 minutes

While students are working in pairs to complete the independent practice, I invite different groups to show their thinking on the board.  These groups lead the group share at the end.  

Each of the examples that I ask students to model show unique situations that they had to really work through to represent.  

1 -1 = 0

1 - 0 = 0

½ + 1 = 1 ½

The students are proud of their accomplishments.

Ticket Out

5 minutes

I call on 5 students to share something they learned today that they did not know before class started.  

One student said that he learned that ½ and 1 ½ are not really the same thing, he always thought they were.  This quick and easy reflection prompt helps students process information from the day when the time is limited.