Lesson 23

Adding and Subtracting Positive and Negative Fractions Using Counters

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Objective

SWBAT add and subtract signed fractions using counter models

Big Idea

Students model sums and differences of signed common (mostly common) denominator fractions to discover that integer procedures also work with fractions.

Introduction

10 minutes

I will start the lesson by asking the essential question:  How can we use a procedure to add and subtract positive and negative fractions?  The purpose of these activities are to get students to see by using counters (MP5) that signed fractions sums and differences can be found using the same procedures as integer sums and differences.

I will use black counters for positive values and red counters for negative values.  I will hold up a black and red counter and ask the class the value.  This is a reminder to the class of the zero pair.  In preparation for the first 8 problems, I will tell them that the black counter will be assigned the value of 1/4 and the red counter will be assigned the value of -1/4.  Students will then be given 2-3 minutes to solve the first 8 problems.  I will insist that students write all mixed numbers as improper fractions.  I want students to start seeing how the numerator behaves like integers.

The second set of problems in this section involve subtraction.  Now we will assign the value of 1/3 and -1/3 to the black and red counters respectively.  I want students to follow the procedure for subtracting as outlined in the resource because it creates a visual representation of how a difference relates to a sum.  The usual method is to only add as many zero pairs as necessary to remove the subtrahend.  I want the students to add zero pairs into the value of the subtrahend.  

Problem Solving

15 minutes

This section is leading students directly to using the procedure they already know to solve signed fractions problems.  Students are asked to explain the procedure they used to add integers with the same sign, different signs, and for subtracting integers.  After each questions, students are asked to try the procedure with fractions and then to verify using counters.  For each question, I will insist that students use precise language (MP6).  The terms absolute value, greater than, less than, additive inverse, etc.  should be a part of the explanations.

We conclude this part of the activity with a brief discussion to confirm that integer procedures work for signed fractions.

Independent Practice

20 minutes

For the first four problems of this section, I require students to draw the counter model.  I think it is useful to create opportunities for students to draw such models.  They are a great tool to use when trying to solve an integer problem.

For problems 5-8, students may solve using their physical counters, but by 9-14 I make the numbers too large to encourage modeling.  Now I want students to put to use their most efficient method.

As in the previous sections of this lesson, the majority of problems use a common denominator.  I want students to be focused on understanding the operations.

The extension has a problem about stock prices.  Students are asked to write two different expressions that show how the stock price changed.  The first expression is to use only positive numbers and the second expression is to use only addition.  This is so that students can see how sums and differences of positive values can equal an addition expression with positive and negative values.  It will be necessary to ask students if they see this connecition when discussing answers.

Exit Ticket

5 minutes

The exit ticket has 3 questions.  Students are to explain how to find the two different sums and a difference. They are encouraged to use a counter drawing to help explain their answers.  A few students have developed their own less common procedures for solving integer problems.  They will be encouraged to use those methods in their explanation if they so choose.  However, they must make sure to clearly explain the method.  

I will remind them that precise language is necessary in the explanation.  Again the terms absolute value, less than, and greater than are probably going to be part of a good explanation.  That being said, many of my students have trouble clearly expressing themselves in writing.  A student may have a successful exit ticket - meaning they understand a way to add and subtract fractions - even if they don't have the best written explanation.  I'll make suggestions for improvement of explanations, but in the end I am looking for an understanding of this lesson's essential question.