Lesson 8 of 24
Objective: SWBAT solve integer addition puzzles by using patterns that determine whether the sum is positive, negative, or zero.
This lesson is at a critical point in adding and subtracting integers. Students have had some experience with it now in class and in homework. Some of them have noticed some potential "shortcuts" and asked "can't we just subtract the numbers?" (when adding numbers with opposite signs) or "can't we add +3 instead of taking away -3?" It is definitely time to explore all of these questions as they come up and make sure all the students are part of the discussion, but it is important not to mistake a few students being ready for the whole class being ready to start requiring this. Students are used to using short cuts without having any idea why they work, so if they forget them they have no way of figuring it out and run a huge risk of using them incorrectly. This is why I ask them to explain why, use examples, generalize with variables to express the pattern for any situation, and use a number line. Some students really need more time and experience with adding and subtracting integers before the "shortcuts" are understood and internalized.
During our lesson today students will be struggling to find the integer addends of a sum. As they work together they will be sharing their understanding of the number relationships as they relate to negative numbers. The more deeply they understand the relationships between the integers the more readily they will recognize the patterns that lead to "shortcuts".
The warmup consists of 4 problems:
x + (-x) = x + (-x) + (-10) = -6 + 6 + 20 = 3 + (-5) =
Before we go over the warmup I direct attention to a larger expression on the board.
-7 + -6 + -5 + -4 + -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8
I remind them that this comes from earlier work we did with the consectutive sums problem from several earlier lessons. I remind them that they discovered they could make any number by adding consecutive numbers in this way. I ask them how they know this long string of addition will equal 8 without doing the addition? The explanation may take several students because I want them to explain that all the numbers between -7 and 7 form pairs of opposites which cancel each other out or add up to zero and, since the opposite of 8 is not being added that the problem is really just 0+8.
When we go over the first warmup problem I ask them to show or explain how they know it is zero. I expect them to do one of three things:
- give examples by assigning values to x
- explain that any number when added to it's opposite makes zero
- use a number line and show that we move equal distances in opposite directions which brings us back to zero.
Problem two they may just explain that since x + (-x) is zero this problem is just 0+(-10). Similarly, for the third problem -6 + 6 makes this problem 0+20. I want them to show me the whole thing on a number line as well to help see the pattern in the next problem.
After they work out the last problem to equal -2, I ask them how this problem is different from all the rest. (there are no opposites being added). I tell them that actually there are, but the -3 is hiding in the -5. I draw a number line and ask them to model the problem on a number line like we did for the last two. Starting at zero, we go right 3, then left 5. I ask them what part of the problem on the numberline makes zero and they should indicate the 3 jumps right and 3 jumps left. I write 3 + (-3) and ask them how we finish writing the rest of the problem. 3 + (-3) + (-2). I ask then, where the -5 is from our original expression? [(-3)+(-2)]. I go back to the original expression and ask if 3 of the 5 negatives canceled out the 3 positives to zero (yes). "And how many more negatives are left?"(-2).
Today students will be solving my "secret number" triangles Classwork secret numbers. I tell them they don't have to be triangles, they can be any shape made up of straight lines, but these ones are the easiest to start on. They are given a triangle with numbers on each side and told that there is a secret number hidden at each vertex of the triangle. On each side of the triangle is written the sum of the two numbers at each end. There job is to uncover the secret numbers. They may choose to do the work on white boards, but I encourage them to write them down on paper so they can take them home and try to get their families to figure them out.
The first one has three sides labeled 7, 0, and -3. Since we've been working with opposite pairs, giving them a side equal to zero tells them that one end has a positive number and the other has its opposite. As they struggle with this I circulate and encourage them to start trying something. I ask questions like "what do these given numbers tell you about the secret numbers?" "Does that mean that this one has to be positive? negative? Try it and see", etc.
During this time I also have students taking aSimplifying retest after completing and correcting intervention.
As students finish the first problem I go around and give them a new one from my secret numbers list. When several students solve the first problem I ask someone to give me just the secret number from the first vertex (5) and tell the rest of the class to figure out the rest. After the first one the rest fall into place, but they all get a sense of accomplishment after struggling with it. Then I give them the next one: 2, 0, 8. I want to keep one of the sides zero, because it gives students a starting point and many have not solved one on their own. As students move through the problems that I give them I may ask them to start creating their own.
I tell them that if they are going to create their own they should start by choosing numbers for each vertex to find the sums for the sides, then rewrite it without the vertex numbers, making them secret. As they begin to create their own I may start to put their creations on the overhead for the class to try. At the end of class I tell them to try to create one secret number problem to bring back to the class and I remind them that it doesn't have to be a triangle and I draw some crazy shape like the one at the bottom of the secret numbers.