What does 3/3 represent?  Section 5: Student SelfAssessment  Math Graffiti Boards
Subtracting Fractions with Common Denominators
Lesson 6 of 9
Objective: SWBAT to subtract simple fractions with common denominators using a number line. They will express their developing understanding of fractions that are less than, equal to, and greater than one whole.
Opener
"Subtracting fractions has much in common with adding fractions, in the same way that addition and subtraction of whole numbers are related. Addition and subtraction are the inverse of each other and both operations share a set of properties. 4 + 3 = 7 is the inverse of 7  3 = 4. Think in your head and remember a definition for inverse."
I count to 20. I may or may not call on someone.
"Remember, in third grade, we will always be adding and subtracting fractions with common denominators. Look at these equations with fractions that have common denominators, and then write a definition in your math journal/on your whiteboard."
I write up the equations below and give students the following sentence stem:
12/13  5/13 = 
90/100 – 75/100 = 
4/25 – 1/25 = 
When two fractions share a common denominator it means that...
Guided Practice
I use simple number lines to show two subtraction problems, 7  5 = 2 and 7/8  5/8 = 2/8.
I repeat what I said in the addition lesson, and make an effort to keep this language consistent so my English language learners aren't bombarded with language complexity on top of new math concepts.
"The denominator stands for equal parts that make up one whole unit or group. As the number of pieces that make up one whole remain the same, the denominator remains unchanged when we are adding. In addition (and subtraction) we are working with the numerator, the number of those equal pieces that are represented."
Then I have students work through the following examples with me. Something I like to make sure they remember is that every single value doesn't need to be represented by a tick mark.
2/4 + 2/4 = 4/4

1/3 + 1/3 = 2/3

1/8 + 3/8 = 4/8.

I ask guiding questions:
 "What did you notice about the numerator?"
 "What did you notice about the denominator?"
 "Is the sum greater than one whole? How do you know?"
 "Is the sum less than one whole? How do you know?"
 ** "Which is larger? 5/4 or 2/3?" (What I'm looking for here is that 5/4 is larger because it is larger than one whole, whereas 2/3 is less than one whole. The comparison does not need to go beyond a recognition of fractions that are less than, equal to, or greater than one whole."
** This is a much more difficult question and I wouldn't start with this or ask this of a struggling student.
Then I draw number lines. I explicitly make the tick marks and count the spaces. I ask the students to assist me as I label the fractional part represented by each line. Then a I ask students to model the addition problem with me. After we've done 5 to 7 of those, I have them do 5 to 7 more on their own as I watch.
1/2 + 1/2 = 
1/2 + 2/2 = 
1/2 + 3/2 = 
1/6 + 4/6 = 
1/3 + 3/3= 
2/3 + 2/3 = 
4/8 + 6/8 = 
2/4 + 3/4 = 
1/8 + 5/8 = 
When I cross the "one whole" mark I ask questions at least half of the time instead of just stating it. "Is 3/2 larger or smaller than one whole? How do you know?"
Independent Practice
For independent practice, I have students work with a partner to draw out number lines to solve these subtraction equations.
I have them draw out number lines to show these addition equations:
3/4 – 1/4 =

4/4  2/4 =

6/4  4/4 =

3/3  1/3 =

6/3  3/3 =

9/3  3/3 =

2/2  1/2 =

4/2  2/2 =

8/8 2/8 =

8/8  4/8 =

Create your own. 

As I monitor students, I adjust the complexity of the problems based on individual needs. Students who grasp the concept and are able to effectively draw open number lines are give problems with fractions of a greater overall value. (24/4  12/4, for example). I keep students who are really struggling to fractions with a value of one whole or less.
After these two days of practice with the fundamentals of adding and subtraction benchmark fractions I close out with a graffiti board activity. I write a series of preplanned questions on construction paper and place the papers throughout the room. Then I use the oldfashioned counting off technique to quickly and randomly put students into 5  7 different groups. They rotate around the stations and in each place they need to draw a picture and/or write a sentence, as individuals or with their group, that represents their understanding of the question.
Sometimes I put a student error into a question, like this: Is the 3 in 3/4 a third? Why or why not? This gives the students who made this mistake the benefit of peer explanations. Asking what does 3/3 represent? lets me gauge the degree to which they've processed the basic concept that an equal numerator and denominator signify one whole! I will revisit questions such as why is 1/2 a bigger piece on the number line than 1/4? throughout the year because I have found that some students revert to thinking that 1/4 (for example) is larger due to the four even if they have previously expressed an understanding. There is something fundamentally counterintuitive about fractions, as far as children process it, and I try to ask questions to test for understanding of what a denominator is in different ways. This is a similar question: Would you rather have 1/2 or 1/20? Why? Finally, as part of the process of developing their awareness of their own thinking about math, I ask, What do you know about fractions now that you didn't know before winter break?
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