I'll begin with the essential question: How can you add and subtract positive and negative fractions using a procedure? I'll then ask about our conclusions from the previous lesson. In this lesson, we concluded that adding and subtracting signed fractions works in the same manner as sums and differences of decimals and integers. Students take this conclusion to fill in the procedures for fraction sums and differences.
There are then 4 examples. Now I will assess whether or not students can apply the rules above to solve the problems on their own. I will remind students to convert whole numbers and mixed numbers to improper fractions first. Two of the four problems have unlike denominators. I will be looking to see that students can quickly find a common denominator and correctly rewrite the fraction. Students who can find the common denominator sometimes forget to rewrite the numerator, so I'll be on the lookout for that. Even less developed students may still try to add or subtract the unlike denominators. Drawing a simple model may help them see the error here. For example we could a model to represent 1/2 + 1/4 so that students see the sum is not 1/6.
I'll monitor the progress of students on these 4 problems. If the majority of students are easily able to solve these problems, I know that we can skip the Guided Practice section and go straight into independent practice. I'll save the guided practice for the few students who may still be struggling.
This section of the lesson may be omitted if the majority of students are successful on the previous 4 examples. These may be saved for the few students who are still struggling after the 4 examples.
The problems presented consist of the various combinations of positive and negative values that students may encounter when adding or subtracting. Four of the six problems have unlike denominators. I have tried to use problems that can easily be modeled with a drawing in case there is a misconception. So for now there are no problems like -5/13 + 4 7/21.
While working on these problems I will ask students to consider the quantities before solving by asking:
1) Will the sum or difference be greater than or less than the first addend or the minuend? How do you know?
2) Will the sum be positive or negative? How do you know?
These questions come from the essential questions of the unit. Students can use these questions as a way to see if their answers make sense (MP1).
I will also look for students to distinguish between adding fractions with the same sign compared to different signs and whether students are making mistakes with the additive inverse.
As mentioned in a previous lesson, students can find it confusing that they use subtraction to add rational numbers with different signs and that they use addition to solve subtraction problems which then may result in using subtraction again. This is where it helps to recall either the number line model or the counter model to help students make sense of the problem.
The first two problems require students to make a sum of -1 1/2. The first requires a positive and a negative value, the second requires two negative values. If students struggle with these, I may suggest they use a number line model. I'd ask them where did we end up? Answer: -1 1/2. What are some ways we could have arrived here? If the first addend is less than -1 1/2 is the second addend positive or negative? If the first addend is greater than -1 1/2, is the second addend positive or negative? Prove it.
The main thing for students to understand here is that trial and error is a fine strategy to solve these problems; making revisions based on a trial that does not meet certain requirements is what good problem solvers do. (MP1)
The rest of the work is an exercise in fluency of these operations. I have selected fractions with slightly less model friendly values now.
The extension has one-step addition and subtraction equations involving signed fractions. Students will be encouraged to solve these using fact families or inverse operations. Mental math and model drawing are okay as well, but students should be expected to clearly detail their process for solving.
The exit ticket consists of two addition problems and two subtraction problems. Each of these problems involve fractions with unlike denominators. In two of these problems only one fraction will be needed to be rewritten. The problems assess the 4 basic relationships that result when adding or subtracting (using the additive inverse) rational numbers. By these I mean there are two sums where the signs are different, however, in one problem the negative value has the greater absolute value and in the other problem the positive number has the greater absolute value. The other two problems result in a sum of addends that have the same sign.
I consider a student to be successful with this lesson if they score at least 3 out of 4 on the exit ticket. Students who only miss 1 problem usually have made a silly error or have a strong enough understanding that they will quickly learn from their mistake.