Multiplying a Fraction by a Whole Number

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SWBAT multiply fractions by whole numbers.

Big Idea

Students understand that they can group the multiples of fractions in different ways to multiply by a whole number and get the same answer.

Whole Class Discussion

15 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.B4b).  The students understand that a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.  

Before we begin this lesson, I review with the class what they learned on the previous day.  I remind the students that fractions are multiples of unit fractions.  I ask for a volunteer to tell me the meaning of a unit fraction.  Student response:  The numerator is 1.  In yesterday's example, we used 3/4 as a fraction.  We said that 1/4 is the unit fraction for 3/4.  We learned to write an addition sentence for our fraction.  In this example, the addition sentence was 1/4 + 1/4 + 1/4= 3/4.  We discussed that multiplication is repeated addition, therefore, if a number repeats, we can write a multiplication sentence.  Our multiplication sentence in this example was 1/4 x 3 = 3/4.  I remind the students that we write the whole number as a fraction by placing it over a denominator of 1.  Our division rule tells us that any number divided by 1 is that same number.  

For more practice, we do a problem together.  The students are at their seats with a piece of paper.  I am writing on the Smart board as we review.  I tell them that if we have the fraction 4/10, we can add 1/10 + 1/10 + 1/10 + 1/10= 4/10.  Our multiplication is 4 x 1/10 = 4/10.  Today, we learn that we can find another multiplication sentence that gives us the same fraction.  I give the students the example of making a multiplication sentence using 2/10.  I ask, "What do I multiply this by to get 4/10?"  Student response:  2.  Together, we multiply 2 x 2/10 to get 4/10.  I let the students know that 2 x 2/10 and 4  x 1/10 equal the same fraction.  I ask the students to explain to me why they are the same.  Student responses:  1) The numerators in both fractions are the same, and 2) They have the same denominator.  Look at the relationship of those two number sentences, and then explain what you notice.  (The students had a difficult time explaining what they noticed about the two number sentences. At this point, I'm hoping that the students will see a connection when they do the activity.  I do not want to tell them the answer.  I want them to be able to tell me because it will show that they have gained understanding.)  

Before letting them work in groups, I remind them that we learned that fractions are multiples of unit fractions.  I need them to remember this as they write equivalent equations.  On the Smart board, I draw a model of 4  x 1/10.   To show the students the connection between 4 x 1/10 and 2 x 2/10, I group the model of the 4/10 into groups of 2.  The students can see 2 groups of 2/10 (See Model - Multiplying Fractions.JPG).  

I let the students know that they are going to practice the skill with a partner.

Skill Building/Exploration

20 minutes

For this activity, I let the students work as pairs.  I give each pair a Multiplying a Fraction by a Whole Number.docx activity sheet, along with a Multiplication Chart.pdf (MP5).  The students must match equations to discover that a multiple of a/b as a multiple of 1/b.  After they learn this, they must use their new knowledge to write their own equations. 


a.  2/3 x 2                       b.  1/3 x 5                             c.  4/8 x 2                  d.  3/8 x 3

Part 1: Multiply the problems listed above (a, b, c, and d) to find the products.  Match the correct equations from a, b, c, or d to the equations for problems 1 and 2 below.  Write the letter on the line.  Explain how the equations are equivalent.  

______  1.          1/3 x 4 = 4/3                                      

______ 2.         1/8 x 9= 9/8

Part 2:  Write another equation that equals the fraction.  Draw a model of each equation.  Explain why they are the same.

3.  3/4 x 4= 12/4

4.  2/6 x 4 = 8/6


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

1. What is the unit fraction?

2.  What is another multiple of the unit fraction?

3. What number can you use to multiply by the numerator to get the same product?

4.  Do both equations equal the same fraction?

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:

 My Findings:

As I monitored the pairs, I questioned them to assess their understanding.  The pairs could match the equations easily.  However, when it came to writing their own equation some found it a little difficult.  The goal of this lesson is for the students to understand that they can group the multiples of the fractions in different ways to multiply by a whole number and get the same answer.  (Thus, multiples of a/b are multiples of 1/b.)


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 by closing each of my lessons by having students share their work is very important to the success of the lesson.  Students need to see good work samples (Multiplying a whole number and fraction.jpg), 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.  In the Video - Multiplying a fraction by a whole number, you can hear a student explain his answer.

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.