Lesson 22

Adding and Subtracting Positive and Negative Fractions on a Numberline

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SWBAT add and subtract signed fractions using a number line

Big Idea

Do sums of positive and negative fractions behave the same way as integers? Students explore this question using (mostly) common denominator fractions and a number line.


10 minutes

I begin the lesson by asking the essential question:  How does adding and subtracting positive and negative fractions compare with adding and subtracting integers?  I then will provide a quick review of the method that we have been using for adding and subtracting on a number line. (MP5


Many of the problems are written with mixed numbers but I will encourage my students to rewrite them as improper fractions.  This is how we will work with fractions in the procedural lessons to come.  Students will be asked to examine the number line to determine the increments.  By now, they should quickly determine that the increments are fourths.  If this confuses a student, I will ask them to pick 2 consecutive integers.  Then I will draw arches from one hash mark to the next until I have connected the two integers.  This makes it very easy to count 4 spaces between each whole step.  

Next we will work through the addition and subtraction problems on the page.  I want to make sure students are able to solve problems on this page so they can answer the questions in the next section that relate to the essential question of the lesson.    

The pointers that I made have the word add on one side with a right facing vehicle or person.  On the other side is the word subtract with a left facing vehicle or person.  As students are working on this, I will be on the lookout for students who are incorrectly using their pointer.

The majority of the fractions on the page have the denominator of 4, but I have included a denominator of 2 and a whole number.  This will be a good opportunity to remind ourselves about equivalent fractions.  How many fourths make one-half?  How many fourths are in two wholes?

Problem Solving

15 minutes

The problem solving section starts with 6 questions that we have already answered in previous lessons on integers and signed decimals.  It is a good review but also helps see that signed fractions behave in the same manner.  

This section does not necessarily require students to write all of their answers, but I may choose to do that.  It could be just as effective (and save time if needed!) to have a conversation in a series of turn-and-talks for each question.  I can then be the recorder by writing responses on the SmartBoard.  Actually, I'll type them in because my handwriting isn't the best.  

After the questions have been discussed, students are to solve the 10 problems on the page.  These are similar to the problems from the introduction section of the lesson, but now the denominators are sixths.  I have included a couple problems with thirds and halves.  Students then match each problem number to any of the questions for which they provide evidence.  This is to help them learn to provide evidence and examples when constructing a viable argument (MP3).

Independent Practice

20 minutes

The first three problems of independent practice ask student to determine if a sum will be positive or negative without calculating an answer.  I think this helps students start to make sense of quantities (MP2); it is a check to see if a calculation makes sense.  This also applies to problems 4--6.  However, these questions present inequalities that students must identify as true or false.  

Problems 7-14 are purely computation problems.  Students may continue to use the number line but some may have already found a procedure during the last activity.  Question 15 has students find a correct sum.  The correct answer is given in three representations:  a mixed number, an improper fraction, and a decimal.


While students are working, I will be monitoring their work to find some common misconceptions that I may need to have students discuss.  I anticipate most of the problems will come with problems 4-6.  By this point, we will have done a lot of work on the number line, so I will ask students to place their pointers on a number line at a point called -a, or -b, or c - based on the problems.  Then I'll ask, what happens when we subtract a positive or negative value?  If necessary I'll continue with the following questions.  Which way do we move the pointer?  Is this greater than or less than?  This questioning usually helps students. 


The extension consists of one-step equations involving fractions with like denominators.  Students are encouraged to solve these in any manner that makes sense to them.

Exit Ticket

5 minutes

The exit ticket has three problems.  Students are asked to make a sum or difference given certain parameters.  It may be helpful to have the class do a similar problem together before working on the exit ticket.  If there is not time to do this I will give a few hints to the class.   


For problem 1 I'll say:  "You just added and ended up at -2 1/10.  Which way is your pointer facing [answer: right].   Now pick another point and add to it a value that returns you to -2 1/10.  If you use a positive and a negative number, you have your answer. "


I will guide them through a similar set of questions that pertain to question 2.


Problem 3 is another question to assess students' understanding of the additive inverse.  It also assesses their understanding of equivalence, that -3 3/5 is equivalent to -3.6 and -36/10.