Using Shapes to Model Fractions as Multiples of Unit Fractions

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SWBAT model multiplying whole numbers and unit fractions to get a multiple of the unit fraction.

Big Idea

A unit fraction is a fraction with a numerator of 1. Fractions are multiples of unit fractions.

Whole Class Discussion

10 minutes

In today's lesson, the students learn to apply and extend previous understanding of multiplication by multiplying a fraction by a whole number (4.NF.B4a).  The students learn that fraction a/b is a multiple of 1/b.  The students demonstrate Math practice 4 (modeling with mathematics) because the students draw models of their fractions. 

Before we begin this lesson, I review with the class what has already been learned about fractions.  This is done through questioning.  I let the students know that today we will use shapes to model fractions as multiples of unit fractions.  First of all, give me an example of a multiple? Student response(s):  3, 6, 9, 12, 15.  That means that 15 is a multiple of 3 because we can multiply 3 x 5 to get 15.  Multiples are a list of the products of a number.  We can have multiples in fractions.  The lesson is about multiples of unit fractions.  I remind students that a unit fraction is a fraction that has a numerator of 1.  We can break fractions down to the unit fraction.  If I have the fraction 4/8, and I say that fractions are multiples of unit fractions, that means that I can multiply some number by my unit fraction of 1/8 to get 4/8.  This lets me know that 4/8 is a multiple of 1/8.  A student asked an excellent question, "Where did you get the 4?"  I let him know that this is exactly what the lesson is about today.  I ask for students to answer the question.  (This will give me an idea of how much the students already know about this skill.)  One student responds, "I think that 4/8 is split in half.  Another student responds, "When you look at the numerator, I know you can multiply it by 4."  (This is one of my higher level students.)  I ask the class did they agree with this?  The majority did not agree that 1/8 can be multiplied by 4 to get 4/8.  (This let's me know that the students are not familiar with this skill, and this lesson is necessary.)  

As we talk through the problem, I am writing on the board to show the students.  I continue to probe the students to get them to understand the concept.  One student says that we can get 4/8 by adding the unit fractions.  I ask, "What must I add if I am using unit fractions?"  The students tell me to add 1/8 + 1/8 + 1/8 + 1/8= 4/8.  Now that we have the addition sentence, we go back to what the student told us about multiplying by 4.  I remind the students that multiplication is repeated addition.  I tell them to look at our addition sentence and tell me what is repeating.  Students' response:  1/8.  How many times is it repeating?  4.  What is my multiplication sentence?  1/8 x 4= 4/8.  


Skill Building/Exploration

20 minutes

For this activity, I let the students work independently first, then share with a partner after they have had time to work on the concept.  (By doing this, it allows me to see what each student is doing on their own.  Also,  the students have a chance to hear their classmates thinking on the skill.)  


Baskets of shapes are set at each table.  (In a previous lesson, the students learned the fraction represented by each of the shapes.  They will use this information to determine that fraction a/b is a multiple of 1/b.  Each student pulls 3 fractions from a bag.  The students must determine the unit fraction.   After determining the unit fraction, the students must display a model of the fraction using the shapes.  Based upon the model, the students write an addition sentence and multiplication sentence for the model.  After completing these steps for all 3 fractions, the students write to explain, "How multiplication is used to create fractions." 

I give each student an activity sheet and shapes.  The students must write the addition and multiplication sentences for each fraction, then dislay a model of the fraction using the shapes. (I decided at the last minute to have the students also draw a model of the fractions.  This is because the students will not have the shapes at home to help them.)

 As they work, I monitor and assess their progression of understanding through questioning. 

1. What is the unit fraction?

2. How many shapes will it take to create your fraction?

3.  What is the addition sentence?

4.  What is the multiplication sentence?

As I walk around the classroom, I am questioning the students and looking for common misconceptions among the students.  Any misconceptions are addressed at the point, as well as whole class at the end of the activity.

 Any student that finishes the assignment early, can go to the computer to practice fractions  at the following site until we are ready for the whole group sharing:




15 minutes

To close the lesson, we review the answers to the problems.  This gives those students who still do not understand another opportunity to learn it.  I like to use my document camera to show the students' work during this time.  Some students do not understand what is being said, but understand clearly when the work is put up for them to see.

I feel that closing each of my lessons by having students share their work is very important to the success of the lesson.  In the Video - Fractions as Multiples of Unit Fractions, you hear a student explaining her work.  Students need to see good work samples (Student Work - Fractions as Multiples of Unit Fractions.jpg and Student Work), as well as work that may have incorrect information.  More than one student may have had the same misconception.  During the closing of the lesson, all misconceptions that were spotted during the activity will be addressed whole class.  

Evidence being used to determine students' understanding of the concept:  addition and multiplication sentences, written explanations, and student responses.  I collect all papers from the students.  All struggling students identified as I monitored during their independent activity will receive further instruction in small group.

My Observations:

The students did well on this activity.  I feel that once they learned that a fraction can be multiplied by a whole number, they understood the concept.  As I questioned the students, their responses validated for me that they understood that fractions are multiples of unit fractions.