The purpose of this lesson is to give students the opportunity to "play" with integers to produce a sum that is either positive, negative, or zero. Students should become familiar with the relationships required to produce such a sum. I want them to notice that the sum of two integers will be negative if more negatives are being added than positives, or if only negatives are being added and vice versa for a positive sum, and that adding opposite numbers results in zero. This lesson puts students in the "driver's seat" by having them create the problems that will result in a specified sum.
In this warmup secret number patterns students are given three "secret number" triangles telling them that a "secret number" is hidden at each vertex of the triangles. The number on the sides of the triangles are the sums of the two numbers each end. In these triangles students are not given any of the "secret numbers" and are not given the numbers on the sides except whether they are positive, negative, or zero. Students are supposed to find "secret numbers" which will result in sums that match the given criteria. Answers may vary, but so long as the "secret numbers" create a sum that matches the criteria it is correct.
The warmup is replaced with positive negative zero sum set of two "secret triangles" with nothing given but the sign of the "secret number" and students are asked how they know if the sums will be positive, negative, or zero.
One triangle has all negative "secret numbers" and the other has two negative and one positive.
Students discuss in their math family groups what the options are for each side of the triangles. This is a great way to generate argumentation by having them defend their claim with an example or an explanation or by having them challenge the claims of others by requiring them to use evidence. I ask them to look for all the options, so they don't just stop at one possibility. If they say it can be negative, I ask if it could also be possible to get a positive or zero sum.
For the first triangle, with all negative "secret numbers" the sums can only be negative, but for the other triangle, two of the sides could be either positve, negative, or zero, while one can only be negative. I have students discuss and explain the relationship between the numbers that produces each given possibility.
The next patterns in addition gives two of the "secret numbers", 2 at the top and -6 in the lower right (which makes the right side of the triangle a sum of 4) and I change the criteria for the sums on the other two sides and ask them what the third "secret number" (n) could be in each case.
First I ask what value of "n" would make the bottom side of the triangle a sum of zero (+6) and what would that make the left side (+8). Then I ask what value of "n"would make the left side of the triangle a sum of zero (-2) and what would that make the bottom side (-8). This gives them a feel for the type of questions I'm going to ask them. I ask what "n" could be that would give both remaining sides a negative sum. They should come up with several possibilities which we try and I write down (n=-3, -4, -5,etc) I ask them why it can't be -2 or a positive number. I ask them how they could describe ALL the possible values. I am looking for something like "any negative number between -3 and -infinity" or "all the numbers less than -2 or less than or equal to -3". If they say "greater than", I show them on a number line that they are less.
We go through the same process when I ask them what value of "n" could make the bottom side a negative sum and the left side a positive sum, both positive. I ask if there is a way to make the left side negative and the bottom side positive (no). The key is making them explain why and why not. If they are really picking this up I like to extend it a little by showing them the possibilities on a number line and in an inequality (-2 < n < +6).
I let them get started on their homework patterns addition for the remainder of class. I tell that for the first two, which are "secret number" triangles, they only have to come up with one solution, but for extra credit they can find all the possible solutions.