To begin this lesson I write the following operation on the board large enough for all to see.
–3 – ^{– }8
I ask the class to think about the answer mentally, but by no means shout the answer out. I wait a few seconds and then ask the class to raise their hands to one of the answers to the following question:
How confident are you that your answer is correct?
Students are usually quite sincere in responding, so gauging student understanding with this question gives valid feedback. But even if I get a good show of "confident" hands, I will only know from sufficient student work how much they've mastered the topic.
I ask someone to try and describe their thinking when figuring out the answer. Some students will say that they know, "Minus a minus equals a plus." Others will say, "Subtracting a negative is the same as adding." Their answers give me a good sense of where individual students stand with respect to meeting the learning objectives of 7.NS.A.1.
Next, I project the following two cartoons up on the SMART Board.
For these cartoons, I use the Think, Pair, Share discussion technique. I ask students to decide which student in each cartoon has the best understanding of addition or subtraction. When students share their responses with the class, some students may want to go up to the board to justify their response. Volunteers should clearly demonstrate with an explanation or an example that Patty and Miguel are two students with the best understanding in the cartoons.
Despite having models to explain addition and subtraction of signed numbers, my experience in teaching my students is that many struggle more with understanding subtraction with integers, than multiplication with integers. Despite this recognition on my part, I still find it a difficult topic to teach. I've never been fully satisfied with student outcomes after teaching integers, and, I have tried multiple approaches. Unfortunately, too many students carry weaknesses in this area far past middle school, which is one of the reasons I include this review lesson for 7.NS.A.1 in my Grade 8 class.
I personally think that students need much time and support in their earlier years, starting at the lower elementary level, to conceptualize negative numbers, so that by the time they reach middle school, performing operations with signed numbers is grasped more easily and of course, fully understood.
If you have thoughts or lessons that work for you, I would love to hear from you. I continue to look for and try new ideas.
I now divide the class into small groups, preferably pairs and hand each group a set of Signed Numbers cards to be cut out. (Some years, I cut them out in advance to save time in the class.) I ask each group to read the cards and organize them into 3 groups:
As students work on this task, I walk around looking and listening. For some students these phrases will be familiar enough that they can quickly organize them. I instruct these students to write an expression representing the situation on the front, and, to write the final value on the back of each card. (Eventually, all groups should do the same.) Some ELL students may have trouble with phrases lie "break even" or "withdrawal". I help by giving an example of the situation. I may also give them the Spanish translation, if Spanish is their first language.
Once students are done I make 3 columns on the board and call on students to go up to the board and write the operations in the corresponding columns.
Teaching Note: Students may use the common word, "minus". I tell students to use the term negative because minus can be misleading when working with variables. -x is not necessarily a negative number, so I ask students to read -x, as "the opposite of x", or simply negative x. (For more on this topic, See my Terms for Signed Numbers Reflection.)
Once the operations for all 12 cards are organized into the 3 columns on the board, I ask if someone wrote a different operation for one of the situations. For example, for the first situation students may have written -8 -6 or -8 + (-6).
I expect my students t0 have trouble with situation 5. The expected number sentence is -25 - ^{-}12 = -13, which is similar to the question from the Launch. I try and make students understand that the landlord is subtracting a debt (a negative value) and therefore this is mathematically equivalent to receiving +12. As a result, -25 - (^{-}12) can also be thought of as -25 + (+12). Students may or may not use parentheses to group the -12, which is fine.
Some suggestions about talking and writing “signed numbers”:
I always use, and have students use, the terms positive number and negative number instead of plus or minus.
I try to use the word subtraction instead of minus to refer to the operation of taking away a quantity from another quantity. This is to help students distinguish between the two different meanings of the "minus" sign.
We know that the minus sign (“-”) is used both as a sign of subtraction, the operation, and as a sign indicating a negative number. I like to use, and I motivate students to use in their writing, a superscript negative sign (and positive sign) to distinguish it from the subtraction sign (“-5” or “+5”)
After a fair amount of group work and discussion early in this lesson, it is time for some independent practice. I distribute the Activity 2 Application sheet and I ask students to complete as much of the activity as they can on their own before turning to a partner.
This sheet has six questions that poke into students' understanding of addition and subtraction of positive and negative numbers. Question 5 is the toughest due to variables involved. In Question 6, students may be tempted to use some of the real life situations they've seen in the lesson to complete the task. This is ok, yet I do encourage some originality. I tell students that the thermometer intervals are drawn to scale and that they can substitute values in, if they get stuck.
I walk around assessing students while they work. Again, ELL students may have difficulty with Question 6, so I definitely help out here. I ask that they use our previous situations as templates for their own.
When students are done, I collect Activity 2 to assess at home before I hand out the homework assignment. These 6 questions are worth going over together, so I make sure that I do so at the start of our next class.
Over the next several years students will consistently encounter signed numbers in algebraic expressions. Tonight's Homework (Those Negative Numbers) is a basic practice in fluency with simplifying numerical expressions.