Missing Fractions

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Students will be able to add and subtract fractions with unlike denominators by creating equivalent fractions.

Big Idea

You have to see it visually to make it concrete.


I use literature and math all the time – actually every time I can find a story book to go with a math concept I will.  In this lesson I use Elinor Pinczes’s Inchworm and A Half.  It is a lower Lexile level but that is what I want.  I don’t want my students struggling to understand the story and miss the math.  I also know that I am going to challenge my students reading comprehension in other areas. 

I start by reading the story to my students, asking questions about the story and then working on addition with fractions with unlike denominators.  Since this is in story format the students are figuring out word problems or 5.NF.2. This story is great to challenge students to solve for a missing fraction.  I am also looking forward to using this book with our little buddies - taking the measurements right out of the book, creating worms of each length, and having my class help the little buddies measure. 

I also think I will be rereading this story when I introduce making a ruler during the measurement unit. 

Finding the Missing Fraction

30 minutes

I started this lesson by reading Inchworm and a Half by Elinor J. Pinczes, with a focus on having the students predict what fraction the author will be adding (5.NF.1). 

After completing the book, tell your students you have a challenge for them.  They are going to be creating fraction addition problems that are missing a fraction piece. Working with a partner, they are to figure out what is missing, then exchange fraction pieces for equivalent fractions.   Be sure students are writing as well as talking about what they are doing. MP1 make sense and persevere in solving problems, MP2 reason abstractly and quantitatively and MP3 construct arguments and critique reasoning of others. 

Students build a whole fraction, using fraction pieces and then remove one (MP4 model with mathematics).  Once they have removed a fraction piece, they write the number sentence with the missing fraction and then read the fraction sentence to a partner.  The partner writes the fraction sentence down and then recreates the problem using their fraction circles - figuring out what the missing fraction is (MP7 look for and make use of structure).  They then change out the missing fraction with any equivalent fraction pieces, making sure to list all the equivalent fraction pieces (MP1).  The next step is to trade out duplicate fractions for their equivalent to see how few pieces they can use. (5.NF.1)

Start by modeling an incomplete fraction sentence and have students use their fraction bar models to complete the sentence (MP4).

1/4 + 1/4 + 1/8 + 1/8 + 1/16 + 1/16 + 1/16 + N = 1










What could N be?  If a student gives you 1/16, ask if there is anything else it could be. 

At this point of the year, I'm easing into variables. I have stopped using a question mark to represent an unknown in an equation, I now use a letter.  This way when we start to focus on algebra it isn’t a surprise nor will it be perceived as a new, separate, idea.

You may want to do a couple of these to make sure your students understand what they are doing and then send them off with a partner.  Be sure they are explaining why verbally and in writing why their answers make sense.  It is in this discourse where students will show the depth and complexity of their understanding. 

Extensions or continuation:

Have your students write other fraction sentence that are equivalent to yours (with the missing piece or pieces).

Have one student model a fraction sentence and then their partner use one fraction to answer the sentence.  An example would be ¼ + ¼ + 1/8 + 1/8 + 1/16 + 1/16 + 1/16 + N = 1 and the other student would replace all fraction pieces with 1/16 to come up with 1/16 + 1/16 + 1/16 + 1/16  + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 = 11/16

Have one student put out 2 fraction pieces and the other student put out 2 fraction pieces and have them compare with >, <, or =.  Again having them write it down. 

I gather evidence in the student journals (see work sample) to show how the students added or subtracted and if they exchanged for equivalent fractions.  

Student Reflection

5 minutes

Student reflection is an important part of a lesson. It builds critical thinking skills and doubles retention.   This is why I include a student reflection section in every one of my lessons. Three years ago I choose to make it a goal of mine to include student reflection in my lessons.   Still, I sometimes forget, or don't leave enough time for reflection. Now that it is a habit or a pattern to my lessons (teacher MP7), when I don't do student reflection a student will raise their hand and either share what they learned automatically or remind me we need to reflect.  Last year, some of my colleagues made including reflection in all of their lessons and reaped the same benefits I have.  Increased retention on materials, deeper critical thinking skills, and because we include reflection on behavior as well as content, there is an increase in on-task behavior. 

Ask your students questions about what they learned, how they learned it (describe their thinking MP6) and about their behavior. 

This lesson was going so well, the kids engaged and quiet, and we ran out of time to reflect. But, when they came back from specials one student raises their hand, saying, "My partner and I found out that we could always exchange fraction pieces with another one when there are two like fractions."  Another student raises their hand, commenting, "When we exchanged fraction pieces for equivalent fractions the denominators got smaller."  We also went on to discuss behavior, one student commenting, "I appreciated it when my partner explained how to exchange my pieces for equivalent fractions." 

When you have established a routine in your classroom your students help you stick to it!