Using Decomposition As a Tool to Prove The Why Behind Simplifying

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SWBAT add and subtract fractions using decomposition.

Big Idea

This introductory lesson to decomposition of fractions is a straightforward whole group instruction that teaches students the "why" behind the decomposition of fractions in addition and subtraction of mixed numbers.

Warm Up: Factor Pairs

10 minutes

In order to fulfill the standard of students being able to demonstrate their ability to find all factor pairs from 1-100, I have allowed daily work and drill in this area to help with mastering this standard. Once a week, they are given a test to find factor pairs for random numbers between 1-100 . This test then reveals their fluency in those facts. This test occurs on Wednesdays. I keep track of their progress by using a spreadsheet. To master the goal, my team and I decided that 80% overall was proficient.

In addition to that goal, however, between weekly assessments of factor pairs, all students who didn't achieve 100% on the last assessment practice finding factor pairs by writing a given product down in their math notebook and write out the factor pairs of each. I check their answers and ask them to check the online factor pair calculator to be sure they have found all of them.

Today, I have numbered note cards 20-100. I throw them up in the air and they land around the room. Students choose one they haven't worked on before and begin to factor the number.

They list the number in their notebook like this: They have chosen a card with 24 on it. I instructed them to prove that they considered each factor possibility by listing it like this. And, if they don't know, to show their division work.



2x 12

3 x 8

4 x 6

6 x 4 > This is the commutative pair. I know I am done.

Whole Group Instruction: Core Lesson

15 minutes

Decomposition of Mixed Numbers Addition and Subtraction Teacher Notebook File I developed this SB file to teach this whole group instruction about decomposition of fractions as the next step to mastering the standard. I chose whole group instruction because this concept is really the first time students have been exposed to thinking about taking apart or putting together fractions to help them solve addition and subtraction problems of mixed numbers. This lesson became a very complex discussion. Students were engaged from start to finish because of their desire to understand what decomposition is. I heard students say that they really enjoyed this lesson. 

I have used some of the SB graphics and given them life with speech bubbles. We discussed what the characters on the first page were questioning about prior knowledge. We talked about the value of prior knowledge and that those concepts learned play a key role in the success of today's lesson.

One student wanted to know what I meant by using "number sense." I asked for other student's input to see what some of them were thinking. A student commented that she thought it meant that adding and subtracting fractions needed to " be thinking about if the numbers make sense." I explained that indeed that we need to think about what we are doing and what those numbers mean.  I knew from past teaching experience that they would soon be trying to change fractions of mixed numbers to subtract. I explained that we need to think about the value of each number we add or subtract. I asked them if they thought that they needed to think about the total when using fractions to add or subtract? Did that concept we learned in whole numbers change? This was answered as the whole class said no, it was the same. I questioned this because in order to completely satisfy CCSS, I know I must continually bring back past concepts learned and build upon them with each new lesson or experience with numbers. I strive to continually focus on this instruction as part of connecting math practice standards to the standards of mathematical content. This is a huge difference in the way I used to teach.

Decomposition of Mixed Numbers Addition and Subtraction Classroom Notes shows how each page has been taught and the pages after page three show classroom whole group examples and work. They took notes in their notebooks as they did the sample problems right along with me. On page two, I introduced the concept of we would have to do something with mixed numbers in order to subtract and compared it to borrowing from the next place value;except this time, we are borrowing from the whole. I confused them and so in the next page, I introduce the concept of decomposing the fractions to do the job. As soon as they saw how the wholes were pulled apart, I heard the glorious sounds of light bulbs going on in their heads.

I emphasized that each number had meaning. I said that 8 1/5 meant 8 wholes and 1/5 of another. That assigned value could not change just like 108 is not 180 just because it would make subtraction easier because we might not have to regroup. I am always amazed at what my students talk about in terms of value. 

The next practice example, I used a fraction that simply could be added and decomposed. ( The addition sign is not there.) In the next samples after that, we experimented with decomposing each number before we started to subtract. They could see how numbers were combined to make wholes and create the decomposed answer. We discussed how perhaps this method was a longer method. But, it works! On the last slide, we arrive at the concept that we decompose each number, create an improper fraction and then subtract and that works well too!




10 minutes

We took time to practice a few problems together on the board and draw pictures to review prior knowledge and then apply it to the new mixed number addition. Practice and Struggle demonstrates how there can be stuggle with this concept. While we worked, two other students sat on the floor with whiteboards and worked with us. We finally solved the problem as students supported each other- Finishing the Problem. Students who were ready to work independently logged onto Level F R.11 Add and Subtract Mixed Numbers with Like Denominators I told them to work in partners and solve using decomposition and they could also use the Virtual Manipulatives app  to help them support each other master decomposing, or fractional models. I roved the classroom as students worked. This student explained her work:Explaining Decomposition.

After students had a chance to work and ask questions, I told them I wanted a quick quiz to show their understanding of adding, subtracting and decomposition.

I wrote this problem on the board: 'Add 2 6/8 + 3 7/8. Explain what you think decomposition is and how it helps you solve the problem'. These photos are some samples of the type of answers I got. This shows their accountability for their learning and the healthy learning struggles that we experience to master this standard. I need help!& Decomposing is...& I get it!


Homework and Video Support Through Educreations

10 minutes

After students spent time practicing adding and subtracting fractions with mixed numbers, I assigned the same IXL lesson for homework. Students were assigned 10 problems to do at home proving their answers on loose leaf paper.

To help parents support them at home, and to tutor some absent students, and support students in general, I created this Educreations Video to assist and be "live" notes.