This section of the lesson has 2 purposes. The first is to make sure students know the essential question of the lesson: How can you determine if the difference of two numbers will be less than or greater than the minuend? The second is to make sure students know how to use the number line model for subtracting integers (MP5). Then, we spend a bit of time practicing subtracting on the number line. Again, this is just to make sure students know the mechanics of subtracting, but the purpose of the lesson is the essential question. MP2 and MP7 are at the heart of this lesson as students consider the relationship between quantities in a difference and notice how the structure of a difference expression can help determine the relative value of the difference itself.
To make this a bit fun, students will be able to use popsicle sticks with pictures of vehicles or people to keep track of their movements down the number line. For subtraction problems, they must use the side that shows the vehicle pointing to the left. This is labeled on the stick.
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Now students fully engage in answering the essential question. Students will work with their partners to complete 11 rows of a table related to a subtraction problem. Here students consider how the subtrahend affects the difference (MP2 and MP7).
As students are working, I will walk around to check in with groups for a few reasons. First of all, I want to make sure students are completing the table correctly. If I notice mistakes, I will ask a student in the group to model how they solved the problem by suggesting one partner read each step while the other partner acts it out on the number line.
I also will want to look for a variety of right and wrong responses to bring to the whole group when we discuss the questions. The questions are designed to lead students to the conclusion that when subtracting by a positive we get a less value and when subtracting by a negative we get a greater value. As we are discussing questions, students will engage in MP3 as they explain and listen to the various repsonses. Also for the sake of clarity, clear language around terms subtrahend, minuend, difference, greater than, and less than will help make sure we are all discussing the same parts of a problem (MP6). We can then refine answers.
By the end of this section, students should be able to answer the essential question and be ready for independent practice.
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Now students start to put their understanding of the essential question to practice.
The first 3 problems are straight forward examples. I purposely use large numbers to make modeling on a numberline less practical.
Questions 4-9 use algebraic inequalities and equations. Here students must pick any value that would make the inequality true. These problems require students to use MP2 and MP7. MP2 is in practice as students view a more abstract representation of quantities (example 7-a<7) and they must consider a quantity or quantities that will make the statement true. MP7 is in practice as students must see the structure of a difference as representing either a single value that is less than, greater than, or equal to some other value.
Problems 10 - 15 continue in the same manner as problem 4-9 yet the representation is even more abstract as the given values are represented as variables on a number line. If students struggle with these problems it may be helpful to ask them: 1) What are the signs of the values in the inequality? 2) How do you know if a sign is positive or negative? ... Once this is established, ask the students to refer to the conclusions from the problem solving section - the answer to the essential question.
Problems 16-18 begin the extension. These problems ask students to apply the answer to the essential question to rational numbers other than integers. Problems 19-21 are simple one-step equations that students are encouraged to solve using mental math. Knowing the answer to today's essential question should make this problem easier.
The last question asks students to critique the reasoning (MP3) of another student.
The first two questions of the exit ticket ask students to find an unknow value that will make a difference greater than or less than the given minuend. These are similar to the first set of problems in the independent practice.
The third problem is similar to problems 10-15 of independent practice.