Lesson 8

How Addition and Subtraction are Related (Part 2 of 3)

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SWBAT write a difference as a sum using counters as a model

Big Idea

Students will use counters to prove that a difference can be written as a sum.


10 minutes

As an opening, I will want to quickly review what we have already learned about how adding and subtracting positive and negative numbers changes a value.  I may ask:  When adding two numbers do we always end up with a greater value?  When subtracting two numbers do we always end up with a lesser value?  Explain.  These can be brief turn-and-talks followed by a discussion.  I don't want to spend too much time with this but I want students to make the connection between what they already have learned - that you can end up with a greater or lesser value depending on the sign of what you add or subtract - and the day's essential question.  

Next, I will teach students the counter model for subtracting integers (MP5).  The way I teach the  model is slightly different than most texts.  In step 2, I ask students to add zero pairs until the subtrahend can be seen.  Most texts (that I've seen) suggest adding only enough zero pairs as necessary.  I choose my method because I think it helps students to more easily see the connection between addition and subtraction.  For example, in problem 1 there is no real reason to add zero pairs to model 5 - 3, but if we want to see the relationship to addition it is necessary.  

Ideally, each student will have 10 positive counters and 10 negatives counters to use themselves.  After step 3, I will make sure to emphasize that what is left is evaluated as a sum.  We will then write the sum and its value (along with the difference).  Students may get confused that we evaluate the counters as a sum.  I will tell them that the set of integers is always evaluated as a sum, it is how we add or remove counters that determine the operation we are modeling.

During this section, I will dictate when to move on to the next problem.  We will go through each problem one at a time.  I want to make sure every students understands how the model works before moving on to the next section.  




Problem Solving & Questions

15 minutes

Now students have the opportunity to explore the relationship between subtraction and addition with their partner(s).  Students will fill in a table for 8 subtraction problems.  They use the method in the introduction to write a sum, draw a representation of the sum, and evaluate the sum (and its equivalent difference).  Students then answer questions that should lead them to "discovering" that to subtract a number you can "add its opposite".  They will be able to put this into their own words, however.  This activity brings out MP7, where students look for the structure of an addition problem and its equivalent subtraction problem.  I would argue that MP8 is also necessary to come up with a general rule for re-writing a difference as a sum.


We will share out answers to the questions.  I will especially focus on #2.  As students present their answer, I will present an example problem to the class that we will solve using the method presented.  This will help us to critique the reasoning of others (MP3).  A clear solution will also call upon clear language (MP6).    

Independent Practice & Extension

20 minutes

The independent practice gives students a chance to practice rewriting differences as sums.  Students may use counters as needed, but hopefully from the discussion in the previous section students will have all found the more efficient method by now.  In fact, that is the point of problems 9-12, where they only have the differences of two variables.  The counter model would be a bit difficult in this case!  If students struggle with this, I will ask them what we concluded in the previous part of the lesson.

Problems 13-14 ask students to rewrite a difference of 3 values as a sum, and then explain how the commutative property would help to evaluate the problem.  

The majority of students should be able to reach the extension today.  These problems all call upon MP2 as values are presented in a somewhat abstract manner, yet students are asked to determine the truth of a statement or determine their sign based on principles already learned.  A logical explanation (MP3) is required for all of these.


After about 15 minutes of work, I will quickly review solutions (as needed)  and then spend the most amount of time discussing the extension.

Exit Ticket

5 minutes

The exit ticket has 5 problems.  Students are to select any sum or difference that is equivalent to the difference given.  Each answer choice has the sum twice (expect for #3).  The second sum is always a rewrite of the sum using the commutative property.  Before they begin, I may tell my students to look out for the commutative property.  I may also tell them that almost every problem has more than 1 correct answer.  If students are able to find only 1 correct answer for each, for example -6 - (-4) = -6 + 4, I know they understood the lesson.  However, there is still a need to understand the properties since they failed to realize that -6 - (-4)  also equals 4 + (-6).