Using Number Lines to Discover Benchmark Fractions

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SWBAT make generalizations about benchmark fractions using evidence from number lines.

Big Idea

Benchmark Fractions help establish quantities, developing fraction number sense and judgment of the reasonableness of estimates.


7 minutes

The focus of today's lesson is the use of number lines.  Number lines are a valuable tool for students to use when building an understanding about fractions.  To get started, I want to determine where my students are (in terms of using number lines) so I can meet them there and build on.

The knowledge I am looking for here is the understanding of the layout of a number line, the labeling of a number line, and the placement of values on a number line. The skills students need complete this task is  to divide by halves to create halfway points between whole numbers, dividing halves in half to create fourths, etc.

In order to gain this understanding.  I provide the students with an open ended prompt for their math journal.  "Make a number line and label any fractions that you can.  If you want a challenge, extend your number line to the number 5."

Students work for 5 minutes on their own.  

Then, we set up at math museum so the students can walk around the room and make observations.  

These observations are recorded on the board.  

From this activity, I am able to see that students need some direct instruction about making a number line. 


10 minutes

After the warm -up, I quickly survey the students about their feelings toward number lines.  I ask them to put themselves into one of the 3 categories.  1.  Number lines are great! 2.  Number lines are a tool I can use, but don't always choose too.  3.  Number lines are challenging to work with.

The response was pretty balanced with about 1/3 in each of the groups.  I explain to the class, that we are going to work with number lines today, because they are an awesome tool.  But we have to really understand them in order appreciate them.

I use a structured resource to demonstrate the number lines are not unlike fraction tiles.  I draw rectangles over the number line to show that each of the spaces represents a fraction tile.  For example, the first number line has 4 spaces.  I draw 4 rectangles over these spaces to represent 4ths.  

I spend time explaining how a number line is created.  "To make a number line that shows 4ths.  You first have to identify 0 and 1, then break the whole into 4 equal parts. 

I work through the rest of these examples with the whole group.  As we move through these number lines, I gradually stop drawing tiles over the number lines.  I call students up to label the fraction that is represented in each and ask them to explain how they knew. 

Make a Number Line: Guided Practice

15 minutes

Now that students have a better sense of how a number line can be used as a tool.  I help the students work as a class to make their own number lines.  I pass out a sheet of paper, cut in 1/2 (horizontally), to each of the students.  

Using the following steps, we create a "stacked" number line. That shows equivalent fractions. Be sure to model this activity as you go.  Students can get very confused if there is no visual to guide them.  Since the goal is to help students appreciate this tool, it is very important they they are not frustrated by this activity.

1. Draw a line horizontally across the middle of your paper. This will be your number line.

2. Label 0 and 1 on this paper.  0 is at the far left and 1 and at the far right.  

3. Fold your number line in half.  What fraction does this line represent?  (1/2). How do you know?  (Because there are two equal sections on each side)

4. Label 1/2 on your number line.

5. Now fold your number line in 1/2 again.  How many spaces are there between the 0 and 1 mark? (4)  What fractional part is this line broken into now? (4ths).  Call a student to the board to label 1/4 on the model.  Ask students to do the same on theirs.  Draw the number tile representation on the model number line to help students understand that the 1/4 label should be placed on the fold, not in a space. 

6.  Ask students to talk with their group members about where 2/4 should go.  Be aware that some students might skip the 1/2 mark and label 2/4 in the 3/4 place.  Students with less fraction number sense may need to be reminded that 2/4 and 1/2 are equivalent and therefore are represented by the same place on the number line. 

7. Repeat these steps until 8ths are labeled.

8. Some students will need more support than others to get to this point.  While helping those who need it, here are some  ideas that I use for possible extensions:  Give students another strip of paper (already folded into 3rds) ask students to label the fractions.  Then add more fractions to the number line. OR allow students to continue to break the 8ths into smaller parts.

Highlight marks on the number line that are represented by more than one fraction.  Ask students use some vocabulary to explain why there is more than one fraction for each of these lines. (Equivalent fractions).

Benchmark Fractions: Guided Practice

10 minutes

The purpose of constructing a number line, with equivalent fractions represented, is to help the students make generalizations about benchmark fractions.  The following activities will help students answer these three focus questions:

How do you know if a fraction is 1/2 

How do you know if a fraction is 1 whole? 

How do you know if a fraction is close to zero, close to 1/2, or close to 1?

Ask students to list the fractions that are equivalent to 1/2 on their number line.  Then ask students to share other fractions that are also equivalent to 1/2 (but are not on the number line that was made in class).  

Students are able to state fractions that are equivalent to 1/2 with no trouble.  After there are about 5 examples on the board, ask students to make generalizations about these fractions.  The focus question, "How do you know a fraction is 1/2?" can help start this discussion.  The end goal is to get students to recognize that when the numerator is half of the denominator, the fraction always represents one half.

Repeat these same steps with the fractions that make 1 whole. The end goal here is to get students to articulate that fractions that are equal to 1 whole have the same number in the numerator and denominator.


15 minutes

Students have now developed a generalization about fractions that are equivalent to 0, ½, 1. These generalizations help students to develop a strong understanding of the basic benchmark fractions. At this point in the lesson, it is important to introduce the term "benchmark fraction" and explain their significant role in estimating the value of fractions. 

Note: Benchmark fractions are any fraction that can be easily identified  -  and at the elementary level this is commonly understood as quarters and thirds as well as halves and wholes. Some interpret it as all the fractional units represented on a ruler (⅙, ⅛, 1/16). Students may have worked with benchmark fractions in the past.  It is important to recognize this and determine a working "class definition" of benchmark fractions in your room.  Also, if students are able to reason with benchmark fractions aside from halves, it is important to develop this further.

This is a summary of the script I use to explain benchmark fractions to my students.  Benchmark fractions are the 3 fractions that everyone in this school knows about.  Even those kindergarteners.  I said 3 benchmarks fractions.  Is 1 a fraction?  Is 0 a fraction?  (students respond with hesitation, but then prove that 0 is a fraction using their number line as evidence 0/2, 0/4, 0/8) and the same with 1.  Benchmark fractions are fractions that we know well, so it is easy to use them to estimate the value of tricker fractions.

Next, to get the students involved and moving, I draw a number line on the SMART Board and label the 3 basic benchmark fractions.  I call on one student to say any random fraction and I write that on the board.  Then, I call on a different student to come up, and move the fraction to the place on the number line that this fraction belongs.  

Do you think it is between 0 and 1/2?

Is is close to 1/2?

Do you think it is between 1/2 and 1?

Is is more than 1? 

Then, they are asked to explain why they placed the fraction where they did.  This exercise helps students work with the 3 focus questions to estimate fractions using benchmark fractions.

Ticket Out

5 minutes

Today, I asked to students to look at 7 questions in the textbook about estimating fractions using benchmark fractions.  Since time was limited, I had the students meet in pairs to practice without pencils.  The students used the 3 focus questions to determine an estimate for these fractions. 

Tomorrow, we will start with these same 7 questions. 

1. 9/10 + 5/6

2. 11/12 - 5/6

3. 2/3 -1/8

4.  5/9 +5/6

5. 7/10 +1/3

6. 1/4 + 1/4*

7. 19/20 - 9/6

* Depending on how students estimate 1/4, this will provide interesting discussion.  I will connect it with a context to help students determine the most reasonable way to estimate this ( closer to 0 or a 1/2).