## INSPIRATION

I’ve been tinkering with area models for awhile, storing countless versions in my teacher.desmos.com account.  A few of my area models support the conceptual thinking of multiplying fractions.  Recently, I was approached by a coaching colleague, Crystal, to teach a lesson using one of the models.  I was thrilled to put a model in the hands of students.

Prior to teaching, I presented the classroom teacher, Mrs. Barnes, with two different fraction area models.  The first is a commonly used model that’s also promoted in her textbook.  The second was a pared down version of the first. Mrs. Barnes selected the version not promoted in her textbook, for two reasons. The pared down version was

1. less cluttered.
2. easier to apply with word problems.

In total, I ended up teaching three lessons in two different 5th grade classes using both whole class and a rotation format.  This post embodies the essence of the collective experiences.  The area models are in the Desmos Activity Builder found here.  I used the first slide only.  For each new problem, students would reset the area models by moving the sliders to 1.  Students revisited the activity builder to complete the word problems.

## The Lesson(s)

• I walked into the office and saw a cake.  About 1/3 of the cake was left.  Draw a square and shade the amount of cake left.
• We circulated the room checking pictures.
• I recreated the picture using Desmos.
• When I reached the front of the room, I continued my story.
• I grabbed the cake and decided to share it with a couple of students.
• Teacher tip:  I asked for students who have a December birthday to raise their hand.  One student, Ellie, raised her hand.  Ellie was now worked into the situation.
• I gave Ellie 1/4 of the remaining cake.  In your drawing, circle the piece of cake given to Ellie.
• We circulated the room checking pictures.
• I called on students to guide my Desmos representation
• They first told me to section the red region into 4 parts.
• Then shade in one of the smaller pieces and circle it.
• I instructed them to label their circled part as Ellie’s piece.

• Here’s my question:  What is the fraction size of Ellie’s piece?
• We circulated the room, listening to student conversations and asking additional questions. I heard 1/4, 1/6 and 1/8
• When I’d hear, 1/12, I’d ask:
• Where did the 12 come from?
• Student responses
• “We need equal parts.”
• “I drew imaginary lines.”
• “If you extend the lines, to create equivalent pieces, you get 12.”
• “The first row has 4 sections.  I counted 4, 8, 12.”
• “The first row has 4.  There are 3 rows, so those rows need to have 4 sections too.  You multiply 4 by 3.”
• Being a middle level math teacher for 20 years, I was unfamiliar with 5th graders. I didn’t know what to expect.  Needless to say, I was blown away by their responses, especially the last two – which were explaining the algorithm.
•  At one table I noticed a student with the answer of 1/6.  When I asked him how he arrived at 1/6, he pointed to the 6 sections.  His table-mate immediately added, “You need equivalent pieces!!”
• Once back at the front, student shared their thinking.
• From our discussion, we seem to agree that Ellie received 1/12 of the whole cake. I’d like you to slide the black dot from OFF to ON.
• Student responses
• “I knew it!”
• “I was right”
• “Yes!”
• Overall excitement
• Their response was completely unexpected and a pleasant surprise.
• Lastly, we connected the algorithm with the visual.
• 1/3:  The original cake was divided into 3 sections, one of which was left.
• 1/4:  The remaining cake was divided into 4 sections, one of which was given to Ellie.
• Multiply numerators:      1 section of  1 section =  Ellie’s portion    1
• Multiply denominators:  4 sections x 3 sections = total sections    12

## The Textbook Version

Earlier, I mentioned presenting two area models to Mrs Barnes.  The post outlines the structure of the pared down area model.  The other model, found in the textbook, uses an overlapping approach (shown below).  One-third of a square is vertically shaded red.  One-fourth of a square is horizontally shaded blue.  When the two squares are combined, the overlapping region represents the area.  Both models area found in the Desmos Activity Builder found here.

Figure 1                                          Figure 2                                  Figure 3