Adding Mixed Numbers

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SWBAT add mixed numbers with like denominators by changing them to improper fractions. They validate their answers with a visual model.

Big Idea

A mixed number can be solved by changing it to an improper fraction.

Whole Class Discussion

15 minutes

In today's lesson, the students learn to add mixed numbers with like denominators.  This relates to 4.NF.B3c because the students add mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

Before we begin the lesson, I give the students a review by asking a series of questions.  First, I ask, What is a mixed number?   Student response:  A mixed numbers is when you have a whole number and a fraction.  I let the students know that today, we are adding mixed numbers with like denominators.  What are like denominators?  Student response:  When the denominators are the same.  If the denominators are the same number, what must we do to solve the problem? Student response:  Add the numerators.  I explain to the students that we are not working with unlike denominators today, so we do not have to find a common denominator.  

The Adding Mixed Numbers with Like denominators.pptx power point is displayed on the Smart board.  The students are sitting at their desk with paper and pencil.  I let the students know that we are learning two ways to add mixed numbers. I let the students know that they can solve the problem in either way that is easiest for them.  I instruct them to write down the sample problem from the power point, 5 1/5 + 3 2/5.  I want the students to work with me as we discuss the power point.  (I find that this method works well for my students because it keeps them focused on what is being said in class.)  Be sure to line up the fractions under the fractions and the whole numbers under the whole numbers.  We learned in addition that we line numbers up according to place value.  This also applies to fractions because fractions can be written as decimals.  

When you add, always begin with the fraction.  I remind the students that when they learned to add in earlier grades, they learned to start with the place farthest to the right.  Since the denominators are the same, we can begin by adding the numerators.  The students add 1 + 2 to get 3.  On their papers, they write the 3 as the numerator.  What is the denominator in this problem?  5.  I was curious to see if the students remembered why we do not add denominators.  Therefore, I asked, Can someone explain to me why the denominator did not change?  I asked 3 students before I could get one student to tell me that the denominator tells us how many pieces the whole was divided into.  (At this point, I changed my original plans for this lesson.  I originally planned to have the students just solve the problems by adding and using multiplication to help with the division - if they chose to use improper fractions.  However, I decided at this point to have the students draw models because a lot of them were still not connecting the model to the problems.)  I reminded the students that in our previous lesson on changing mixed numbers to improper fractions, we drew the models to show the mixed numbers and improper fractions.  We learned that the denominator told us how to divide the whole.   I explain to them that in the problem that they are currently working on, the denominator stays fifths because the whole is cut into fifths.  

Back to the problem, next we add the whole number.  What is 5 + 3?  8. We have 8 3/5 as our answer.  I instruct the students to draw 5 1/5.  What did we learn about the wholes?  Student response:  They have to be the same size.  As the students draw their models, I draw models on the board as well.  (A teacher modeling for the students can help give those lost students a starting point if they are completely lost.)  Together we shade 5 whole and 1 out of 5 for our 5 1/5.  Next, the students draw a model of 3 2/5, then shade 3 whole and 2 out of 5 to show 3 2/5.  Now, let's see if our model connects to our answer.  We said that the answer to the problem was 8 3/5.  Let's count to see how many "wholes" are shaded.  The students count to get 8.  Next, count the boxes that represent the fraction.  The students count 3/5 shaded.  Therefore, our model and answers connect.

Another Way:

 I remind the students that this is not a new skill.  We have learned this skill in a previous lesson.  The students write the problem down again, 5 1/5 + 3 2/5.  We can change a mixed number to an improper fraction.  To change a mixed number to an improper fraction, you multipy the denominator by the whole number, then add the numerator.  The students and I work together to change 5 1/5 to 26/5.  Next, we change 3 2/5 to 17/5.  I tell the students that now we can add the numerators because we have like denominators.  The numerator is 43.  Our fraction is 43/5.  I ask, "Can we leave our answer as an improper fraction?" No.  "No, we can not.  We must change our improper fraction to a whole or mixed number by using division.  We have learned that multiplication can help us with division.  We should divide 43 divided by 5.  "How many times can 5 go into 43?"  8.  "What multiplication fact can help you with this?"  8 x 5 = 40.  "What is the remainder?"  3.  The quotient is your whole number.  The remainder is your numerator.  Your divisor is your denominator.  We write our mixed number as 8 3/5.  So that the students see the connection between the mixed number and improper fraction, I have the students go back to the model that they drew earlier.  I have them count the total number of pieces shaded.  The students see that there are 43/5 shaded in their model of 5 1/5 + 3 2/5.  

Skill Building/Exploration

20 minutes

For this activity, I let the students work independently.  I give each student an Adding Mixed Numbers with Like Denominators.docx activity sheet.  The students must add mixed numbers with like denominators in one of two ways:  1) by adding the mixed numbers, then simplifying the fraction, or 2) by changing the mixed numbers to improper fractions.  The students must draw a model of their mixed numbers (MP4).  

The students work on real-world scenarios to add mixed numbers with like denominators.  The students are guided to the conceptual understanding through questioning by me.  As I walk around the classroom, I am questioning the students and looking for common misconceptions among the students.  Any misconceptions are addressed at that 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 the skill at the following site until we are ready for the whole group sharing:

 My Findings:

In today's lesson, the students learned to add mixed numbers in two ways.  As I monitored and answered questions from the students, I noticed that some of the students still had a hard time drawing the models.  I encouraged those students to solve the problem first, then draw a model last.  Most of the students could solve the problems, but as stated earlier, the models were giving a few of them a problem.  After I kept driving home the point that the denominator tells you how to divide the whole, it became better for most of the ones having problems.  

I was surprised that several of the students used equivalent fractions to add the mixed numbers.  I expected more of the students to just add the fractions and whole numbers.  Overall, I was pleased with the lesson.  



15 minutes

To close the lesson, we review the answers to the problems as a whole class.  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, as well as work that may have incorrect information.  From the Video - Adding Mixed Numbers, you can hear a student explain how she solved a problem.  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.  


-Drawing models correctly (as stated earlier)