Teresa's Pizza (Fraction Decomposition)

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SWBAT decompose fractions into sums of fractions in more than one way.

Big Idea

Fractions can be broken into sums of fractions.


5 minutes

Rationale for teaching with a task:

After I have worked directly with the students on a skill, I like to use a task.  A task gives the students more practice on the skill while working in groups.  Allowing the students to work in groups gives the students different perspectives from their classmates.  Students can learn from each other.  As the students work on a task, I am the facilitator, walking around monitoring and questioning the students to lead them to the solution.


I let the students know that today we will do a task.  I remind the students of the structure and routine of a task.  First, the students will have private work time to think about and plan how to solve the task.  Next, the students will work in groups to explore the concept of the lesson.  Finally, the students will share/analyze/and discuss the task as a whole class.  Each student should have a copy of the task at their desk, as well as a tool kit that contains fraction strips.  

I find that this type of lesson is crucial to the students becoming critical thinkers.  When they have to discover things for themselves, then the concepts stay with the students.


5 minutes

Teresa's Pizza (Fractions).docx:


Teresa has 3/4 of a pizza.  How can Teresa write 2 addition equations that show the amount of pizza that she will eat?

1.  Write 2 addition equations using fourths that show the amount of pizza that Teresa will eat (4.NF.B.3b).

2.  Draw a model of each addition equation. (MP4)

3.  Write to Explain:  Use words and numbers to explain how you solved this problem.


Give the students about 5 minutes of independent time to read and plan to solve this task (MP1).  The students should have a tool kit with fraction strips at their desk.  The students can use the fraction strips at this time to plan how to solve the task (MP5).  After the 5 minutes of independent planning, the lesson will go to the next phase of group exploration.

Group Exploration/Discovery

20 minutes

During the group exploration/discovery phase, the students will work in groups of 3.  Each group will have a copy of the task.  The students must work together to complete all requirements of the task.  The students will have to reason abstractly and quantitatively by attending to the meaning of 3/4 (MP2).    During this phase, the students will not receive direct instruction.  This is a discovery lesson.  I find that students who discover concepts have a better conceptual understanding of the lesson.  The students will be guided to the conceptual understanding through questioning by me and their classmates.

The students will be required to decompose the fraction 3/4 (4.NF.B.3b) into two different addition equations.  The students must communicate with each other and agree upon the equations.  This will take discussion, critiquing, and justifying of answers to get all 3 students to understand the concept (MP3).  Each group will have a tool kit with fraction strips.  During this part of the lesson, the students should use the fourths strips to decompose 3/4 (MP5).  Once they have come up with their 2 addition equations, they must draw a model to represent each equation (MP4).  The model must be accurate and show that fraction pieces are equal in size (MP6) 

During the phase, I will monitor and assess the students' progression of understanding through questioning.  Possible questions to help lead to the concept are as follows:

1.  What is the denominator in this fraction?  What does it mean?

2.  Will the answer be a whole number of a fraction?

3.  How much pizza did Teresa have?

4.  What operation do they want for the equation?  What clue word tells you this?

5. How does you model connect back to the fraction?

6.  How many fourths does it take to get 3/4? (MP7)

7.  What can you tell about the sizes of the fraction strips?  What does it tell us when we draw our fractions?




15 minutes

During this phase of the lesson, student solution paths will be shared.  While the students were working in groups and I was walking around questioning, I identified solution paths to be shared as a whole class for this phase.  Some of the solution paths are correct, but there is one that I wanted to share that had a misconception that I wanted to address.

From the picture (Misconception - Student work (MP6) you can see that one group added 1/4 + 2/4 = 3/4.  I wanted to address this with the whole class because I wanted to talk about MP6 (attend to precision).  In the rectangular models, the students' models do not represent that fractions are equal sized pieces.  In the equation 1/4 +2/4= 34, the 1/4 fraction piece is a lot larger than the 2/4.  This is an excellent teaching opportunity to talk about equal size pieces in fractions referring to the same whole.

 Possible questions that can help guide the students to the understanding that fraction are equal sized pieces:

1.  Teresa at 3/4 of a cake.  What was the amount of the whole cake? 

2.  How many fourths are there in 3/4?

3.  What can you tell about the size of these fraction pieces?  (Have fourths put up on your Smart board for the students to see)

4.  Should I draw 1/4 larger than 2/4?  Why or why not?


10 minutes

After the share/discuss/analyze phase of the lesson, I close the lesson out by having the students do an exit ticket.  This will enable me to see how well the students understood how to decompose a fraction into a sum of of fractions with the same denominator.  

The students will receive an Exit Ticket Decomposing Fractions.docx to complete their answers.  I will collect these exit tickets to evaluate the students' understanding.  Those students who need remediation will work with me in small group the next day.