This lesson asks students to compare and contrast specific elements of a subtraction problem and it's equivalent addition problem. The goal is for students to look closely and notice patterns of similarities and differences so they can explain why adding the opposite can be used to solve subtraction problems. So often they learn the "trick" with out making sense of why it works. Without a sense of how the relationships between the numbers and operation apply they can't internalize the "trick". Instead they just use it because the teacher said so, but don't understand or believe that it is really equivalent. It is important for students to be able to explain and justify every step they take mathematically in order to make sense of the math. (mp1 & mp3)
This is also a fairly self directed lesson which makes it easier for a sub. The warm up consists of two open ended questions with multiple solution possibilities. This engages students more and gives them more ownership of the activity when it allows them choice and creativity. It also raises questions within the math family groups and encourages students to check each other's solutions (mp3).
The Warm up gives students two open ended questions about integer addition.
1. Vanessa added three numbers together and got a sum of zero. What might the 3 numbers have been.
The sub is told to have students share their solutions and have the explain or show how it works or have the rest of the class verify it. I also have him/her ask students if it is possible for all three numbers to be negative? positive? (no) and have them explain why not. I expect them to say that in order to cancel out to zero the numbers must combine both positive and negative numbers. They may say that in order to go back to zero on the number line they have to move in the opposite direction which would involve adding numbers with both positive and negatives. These sub notes are included on an Warm up answers.
2. Caleb was adding integers and he knew that the sign on the sum would be negative. What could that tell you about the numbers being added.
Students may think all the numbers being added have to be negative, in which case the sub is directed to ask if they could make a negative sum if one of the numbers were positive? If two of the numbers were positive? (it doesn't say how many numbers are being added).
I really want students to grapple with the relationships between the numbers here, which is why I don't give them specific numbers. These open ended questions force students to test and adjust the numbers and learn the limits which helps them generalize the patterns or rules.
After going over it I ask the sub to have students take a second look at the front page and draw their attention to the pairs of equivalent equations. Ask students to check these to see if they follow the same patterns as were found in the comparisons on the back.
For the remainder of class students may work together on their homework subtraction int patterns. One side asks students to solve addition in one section, subtraction in the next, and then match the equivalent equations. They are also asked to explain why it makes sense that subtracting positives is equivalent to adding negatives and vice versa.
The second page asks students to compare equivalent equations, numberlines, etc. as they did last night. This assignment has students compare each part separately so that they must say there are "no differences" in the solution, the number line, etc. It makes it very clear what is different and emphasizes the "sameness". This is important because it helps them see why they can choose to do the simpler problem.