Lesson 10

Algorithms for Subtracting Integers

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Objective

SWBAT subtract integers using the additive inverse or their own algorithms

Big Idea

Students use the additive inverse to subtract integers and then develop some of their own algorithms.

Introduction

10 minutes

I'll begin the introduction by asking the essential question:  "How can we subtract integers using a procedure?".  I want students to consider a difference like -857 - 457 and ask why it would be helpful to have a procedure.  I will lead students to the answer that a procedure is more efficient than modeling.  

Next, we will reflect on two essential questions from previous lessons.  I will most likely present these questions one at a time using a turn and talk.  I will listen to responses from each group.  As usual, I will be looking for precise language (MP6) in all responses.  If a response is not clear, I may ask another student "How could you add to or refine that statement?".   This reflection will serve two purposes: 1) It will allow students to have a way to make sure theirsubtraction answers make sense; 2) It will lead them to seeing how using addition is a useful way to solve integer subtraction problems.  In fact, we will conclude the reflection by writing "To subtract an integer, add its opposite".  Students will then solve 12 subtraction problems using equivalent addition.  

As students work, I will be looking for a sum being written and a correct solution.  If I see an error, I will ask students to verify an answer using a counter model or number line model.  

Problem Solving

15 minutes

Using the additive inverse for subtraction is useful, it is not the only method.  The problem solving section is to help students develop other ways to subtract two integers with the same sign.  It is an exercise in MP2, MP7, and MP8 among others.  For each set, students reason about the relationship of the quantities in a difference and look at these structures to find an efficient algorithm.    

Students work with their partner(s) to answer questions about 3 different categories of differences.  To help with pacing, we will focus on 1 group at a time followed by a brief discussion.  Each group should take about 5 minutes including the discussion.  Students should be encouraged to find 2 things in common for the first question of each group.  This is to lead students to see the signs of the values and the relationship between the values.

After group A, I hope students can see that the difference of a less positive and a greater positive is the opposite of the difference of the greater positive and the negative positive.  In other words 3-5 is like 5-3 only the opposite.

 

After group B, I hope students can see that subtracting two negatives is similar to subtracting two positives.  So -5 - (-3) = -2 and -3 - (-5) = +2.  The latter example is similar to group A. 

 

Group C, becomes very apparent when using a counter model.  When subtracting a lesser negative by a greater negative, just remove the greater amount of negatives.  -5 - (-3) means remove 3 negatives for a difference of -2.

 

If time permits, I will post two more groups.  Group D would be differences of a different sign where the minuend is positive [ 5 - (-3) ] and group E would be differences of a different sign where the minuend is negative [ -5 - 3].  Students may conclude that you can add the absolute values of the two numbers and keep the sign of the minuend.

What I enjoy about this section is that students see that there are other methods for subtracting integers and that some of these may be easier for them.  These are useful for mental math and making sense of quantities.  

 

Independent Practice & Extension

20 minutes

The purpose of this section of the lesson is for students to individually practice their fluency with subtracting integers.  They may use whatever method makes the most sense to them.  There are only 10 problems, so I will set a timer for about 5 minutes.  We can then quickly review the answers as needed.  If only a few students are struggling with these 10 questions, I may send them off to work with a teacher assistant (if I have access to one!) while the rest of the class moves onto the extension.

There should be plenty of time for students to work on the extension.  Problems 11-13 ask students to find values to equal a difference of either -1 or +1.  Students must make sense of the problem (MP1) and reason abstractly and quantitatively (MP2) to solve these three problems.  These problems will show who really understands how different quantities affect the difference of integers.

The last 4 problems present statements that students must label ALWAYS TRUE, SOMETIMES TRUE, or NEVER TRUE.  An explanation is required to justify their answer (MP3).  This explanation could be framed around any essential question that we have answered so far in the unit along with an example.

Exit Ticket

5 minutes

The exit ticket assesses the main purpose of the lesson - to see if students can efficiently subtract integers.  While students have seen more cognitively challenging questions in the problem solving and extension section of this lesson, the main point of the lesson is to teach students to efficiently subtract integers.  I have presented the various combination of subtracting problems in the first 4 problems.  The last problem is a difference of 3 values.