Modeling Adding and Subtracting Mixed Numbers

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The SWBAT use shapes to model adding and subtracting mixed numbers.

Big Idea

Adding and subtracting mixed numbers can be modeled by using shapes.

Whole Class Discussion

15 minutes

In today's lesson, the students learn to model adding and subtracting mixed numbers with like denominators.  This relates to 4.NF.B3c because the students add and subtract 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.

I begin by letting the students know that our lesson for today is modeling adding and subracting mixed numbers with like denominators.  To model adding and subtracting mixed numbers, we will be using shapes.  I ask someone to raise their hand and explain to me the definition of a mixed number.  (The students should remember this because we have already learned to change a mixed number to an improper fraction in a previous lesson.)  Student response:  A mixed number is an extra number with a fraction.  At this point, I believe the student has an idea of what I'm talking about, but he is not saying it correctly.  Therefore, I ask, "Will someone tell me what he is trying to say.  He said that it is an "extra" number.  Is it called an "extra" number?"  Student response:  A mixed number is a whole number with a fraction.   

I let the students know that a mixed number is a "whole number part and a fractional part."  I reminded the students that we have already learned how to add fractions with like denominators.  In our lesson today we will work with mixed numbers with like denominators.  I ask, "What is a "like" denominator?"  Student response:  Fractions with the same denominator.  I tell the students that we are adding and subtracting mixed numbers with like denominators.  We are using models to show our answers.  

I remind the students that when you're talking about fractions, you must be talking about same "whole."  I can't talk about a large cake and a small cake.  If I'm adding fractions, those two cakes have to be the same size.  We also learned the meaning of a denominator.   I ask, "What does a denominator tell you?"  Student response:  How many pieces it (whole) was cut into.  I go on to ask, "What does the numerator tell you?"  Student response:  The amount of pieces you need to shade.  I add to the student's response by letting them know the numerator tells you the amount of pieces you are describing.  For example, the number of pieces eaten from a cake, or the number of pieces left in a cake.  One more thing I discuss with the students before beginning our lesson.  "What did we learn about the size of the pieces?"  Student response:  They have to be the same size.  All of these things are used in this lesson.  


Skill Building/Exploration

20 minutes

For this activity, I let the students work in pairs.  I give each student a Modeling Addition and Subtraction of Mixed Numbers.docx activity sheet, along with shapes (MP5).  The students must model adding and subtracting mixed numbers with like denominators, then write the answers in simplest form.  This applies to MP2 because proficient students make sense of quantities and their relationships in problem situations. Because the students are working in pairs and must agree upon the answer, this will take discussion, critiquing, and justifying of answers by both students (MP3).  

Because this lesson will be complex for some of the students, we work together on the first problem.  I let the students know that the yellow hexagon is our 1 whole.  I tell them that we need to figure out the fraction for the other shapes.  We can use the hexagon (1 whole) and the other shapes (fraction) to make our mixed numbers. Each pair starts with a hexagon.   I instruct the pairs to reach into the basket and take out green triangles.  I have the students see how many green triangles it take to equal the 1 whole hexagon.  The students work together to form a hexagon out of the triangles.  The students figure out that it takes 6 triangles to make 1 hexagon.  So I ask (as I hold up 1 triangle), "What is the fraction for this 1 triangle?"  The students tell me 1/6.  The students write this information on the paper.  Next, the students take out trapezoids.  Again, the students put the shapes together to make a hexagon.  "How many trapezoids does it take to make a hexagon?"  Student response:  2.  "What is the fraction for 1 trapezoid?"  Students' response:  1/2.  Last, the students repeat this process with the blue rhombuses.  The students find that the fraction for the rhombus is 1/3.  

 Now that the students have discovered the fractions for the shapes, I work with them on the first problem.  The students take 2 hexagons and 4 triangles out of the basket.  Using what we have learned, the students must figure out the mixed number.  The students lay the shapes in front of them.  I ask, "What is the whole number?"  2.  "What is the fraction?"  The students called out incorrect information, so I instructed them to look back at their papers to see what fraction the triangles represented.  They tell me that a triangle is 1/6.  So I ask, "How many triangles do you have?"  Students' response:  4.  So, I ask again, "What fraction does the triangles represent?  The students realize that it is 4/6.   The first mixed number for problem one is 2 4/6.  

The students take the shapes out of the basket for the second part of the problem.  They lay 3 hexagons and 1 triangle before them.  We find that the whole number is 3, and the fraction is 1/6.  The second part of the problem is 3 1/6.  The students write 2 4/6 + 3 1/6 on their papers.  Before we add on our paper, I tell the students to combine their shapes for both parts of the problem.  In front of the students, they see 5 hexagons and 5 triangles. "How much is a triangle worth?"  1/6.  We have 5 triangles, "What fraction do we have?"  5/6.   "How much is a hexagon worth?"  1 whole.  We have 5 wholes. The students tell me that the mixed number is 5 and 5/6.  Next, I tell the students that when we add mixed numbers on paper, we begin with the fraction.  Together, we add 4/6 + 1/6 to get 5/6.  Last, we add 2 + 3 to get 5.  The mixed number is 5 5/6.  The students have a visual to validate their answer.

 I tell the students that now they will explore to find the answer to the remaining problems.  I will walk around to monitor and help lead them to their answers. I also remind the students to simplify their fractions to simplest form, as well as change improper fractions to whole numbers or mixed numbers.

From the Video - Modeling Adding and Subtracting Mixed Numbers, you can hear the students discuss the problem and agree upon the answer to the problem.  As the pairs discuss the problem, they must be precise in their communication within their groups using the appropriate math terminology for this skill.  As I walk around, I am listening for the students to use "talk" that will lead to the answer. I am holding the students accountable for their own learning.  


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

1. How much is the shape worth?

2. Can you group the shape to make a whole?

3.  Is your answer in simplest form?

4.  Is your fraction improper?  If so, how do you change it to a whole or mixed number?

As I walk around the classroom, I am questiong 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.




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 (Student Work (Student Work - Modeling Adding & Subtracting Mixed Numbers.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.  


I found that some students had a hard time changing improper fractions to mixed numbers on paper.  If the fraction had the same numerator and denominator, they knew that it equaled 1. If the numerator was larger than the denominator, then it proved a little more challenging.  To help them with the concept, I had them to show me with the shapes.  They could come up with answer with the shapes.  I told the class that on tomorrow we will learn to add mixed numbers using the relationship between adding and subtracting.