Understanding Composing and Decomposing: Day 1

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SWBAT compose and decompose fractions by using fractional models.

Big Idea

Virtual Manipulatives is a dynamite ap that helps students quickly understand the composition and decomposition of fractions. This first day is an introduction and is just the right lesson to get them started at this Common Core Concept!

The Word Problem of the Day

15 minutes


Our word problem solving skills are weak! "Larry," our word problem bear, enticed a student to choose a word problem. This is a great way to attack some weak areas and for me to see who has progressed in reading and understanding word problems. Students find it really fun to reach in Larry's mouth and pull out a slip of paper with a random word problem on it.

After choosing a word problem, he photographed it, displayed it on the SB and read it aloud. Students copied the problem in their math journal. Some took a photo of the word problem to easily copy it into the journal from their desk. I required them to highlight the key word to develop a better habit of identifying the correct equation operation(s) to use. I roved and checked progress as they solved and stopped to discuss their choices.

 Students had a great time solving this problem and finding that the key word was "decreased." Two students confused increased with decreased and added to solve. One student focused on the words " how much" and assumed it was addition without using the "Turning on the TV" (visualization strategy) to make sure he knew  what the numbers meant. Turning On the TV !  is a strategy that transfers from reading skills we have practiced all year. It works beautifully for word problem solving, but my students have had to be really coached to remember to do this. The immediacy of students wanting to solve right away without thinking about the meaning of the numbers is still rampant! As we transition to Common Core, I am hoping that we see our students thinking more deeply and develop better habits in solving word problems.

I assessed each student and required each to make corrections. I counseled those who struggled to see it was a subtraction problem and reviewed using key words.


Core Lesson: Composing and Decomposing Fractions

20 minutes

After reviewing their pretests for this module, I assessed that 90% of my students know how to add and subtract fractions with like denominators! They understand how to add unit fractions and some have a firm concept of understanding that 1/5 five times is to equal one whole.

Knowing this, it didn't make sense to proceed with basic lessons for unit fractions. The one area that no one understood was decomposing fractions. Thus, today's lesson was born!

I guided this lesson to the whole group using SB file, Decomposing Fractions. We reviewed the first page as I activated their prior knowledge and developed more vocabulary, making sure they had examples of each. They took notes in their journals. We worked through to page 4 and stopped. I wanted them to be able to manipulate some fractions using their iPad and practice  composing fractions using unit fractions.

Common Mistake surfaced that this app made easy for the student to discover she was not thinking correctly. The student added 1/2 and 1/6 and got 2/8. With the app, she manipulated the pieces to check herself, discovering that it was not true. So we labeled it "Common Mistake" and screen saved it. She understood now why it doesn't make sense to add denominators. Now, she has a reminder to not make that mistake again!

I introduced the Virtual Manipulatives app and demonstrated how the app is used on the Smart Board using my iPad and Apple TV. Students logged onto their app and copied what I did on the board. They could screen shot any work they did on the app and save it. This essential feature helps students go back to reflect or discuss their work. (We noticed that when we tried to save the page to 'your photos', it turned out to be just a black photo. So, the screen shot works great.) I continued as we added and composed fractions using this app. I started with composing 8/ 8ths and then slipped the whole bar over all the eighths.

I switched back over to the SB file and opened up page 5. We talked about improper fractions and the concept that the line between the numerator and denominator means "divide." I asked them to create 25 5ths using their virtual manipulatives app. Everyone worked together. One student raised his hand and showed his work. I switched over to Apple TV and he logged onto the screen showing everyone his work. He had put the fifths together in rows like an array and then slide the whole bar over to reveal 5 wholes. This concept was easy for them to see that 25 5ths were equal to 5 wholes. I told them that I knew they knew that 25 ÷ 5 was 5, but wanted them to be able to decompose and prove their work!



Practicing and Finding Two Ways to Decompose Fractions

10 minutes

I turned to the last page on the SB file to practice the decomposing of those fractions on that page.

I taught them to decompose 4 4/8 by using the white board next to the SB so they could look at both. Showing 3 3/4 through manipulatives and Two different ways to decompose shows the process that one student used to decompose 3 3/4. Through this students could see easily how improper fractions are equivalent to a mixed number. The student used wholes or broke it apart with fourths, easily proving where the improper fraction is derived. This is so cool! It brings forward the "why" of improper fractions so beautifully!

Each student worked independently on their Virtual Manipulatives app, pulling apart the improper fraction, or composing an improper fraction. One student started to add the denominators and I could see that the Virtual Manipulatives app proves to the student  why we don't add denominators. She had taken mixed numbers and decomposed them and added the denominators: 1 1/2 was decomposed as 2/2 + 2/2 + 1/2 as 5/6. I easily slid over unit fractions of sixths and lined up 1 whole and 1/2. She easily saw the difference and we talked about how that is the common mistake that students make. I stopped the independent work and reviewed it with the whole class to support anyone who may be making the same mistake.

 We got tired! It was time to stop, and so I closed the lesson with the understanding that we would start again tomorrow and practice some more by using addition and decomposing sums into mixed numbers. This is a great way to practice a very rigorous concept for our fourth graders!