Magic Witch Hats

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SWBAT add the opposite instead of subtracting integers.

Big Idea

Students will understand that subtracting is like "canceling out".

Intro & Rationale

Most students at this point should be understanding that adding the opposite is equivalent to subtraction and can be used as a tool. However, I expect some students to need another way to look at it. This lesson is supposed to equate subtraction with canceling. When we have established that subtraction is like canceling we can ask questions like "what number would cancel out the -7 we are trying to subtract?" Students can think of it as changing subtraction into a canceling problem rather than an addition problem. For some students the idea of adding and subtracting being equivalent is tough.

Seating Activity

5 minutes

As students enter they get an index card with an addition or subtraction problem on it. They must solve it and find the table group with that solution taped to it. There are four equivalent cards to form groups of 4.

The Seating activity chart shows the solutions at the top and the equivalent expressions in each column that I write on index cards.

Students are reminded to check with other members to make sure they belong together and to help each other find their seats. This is a quick way to get students focused on math and working together. As they work on the warm up section I check that they are all in the right groups.

Warm up

15 minutes

They work on the warm up canceling subtraction as they find their seats, but I cover the last question for them to do after we have gone over and modeled the problems. Before we go over the warm up I have them take a look at an expression written on the board:


They came up with this during our earlier Consecutive sums exploration. They determined that they could make the sum of any number as long as they added a whole string of opposites to equal zero. We review the idea of canceling opposites then we go over the warm up.

The first two problems: -3 + ___ = 0 and 8 + ___ = 0 get students thinking about opposite pairs that cancel out to make zero. I model warm up canceling subtraction modeling the first one with + and - symbols and the second with a number line.

The third and fourth problems are equivalent and I model number 3 with symbols and 4 with a number line.

Then I uncover the question "How is 'canceling' like subtraction?" and have them discuss it in their groups. I tell them to use the symbolic and number line models to help them with their discussion. In general their responses all sound something like "subtraction and canceling are alike because they are both taking away from what is there". They point out how the canceling of the symbols removes positives and negatives from the original numbers and how steps are "taken away" on the number line.

Magic Witch Hats

34 minutes

On the overhead I draw a "magic witch hat", which looks like a black witch hat with a peak on top and bottom (upside down) of the brim to familiarize them with the format of each problem. I tell them there are good witches who are giving and I put a plus sign inside the top hat. Then there are bad witches who are takers and put a minus sign inside the upside down hat. I place two numbers on either side of the brim (i.e. -3 and -5) so the top (good witch) hat says -3 + -5. I tell them the sum goes at the top of the hat.  I place the same numbers under the brim on either side of the bad witch hat so it says -3 - -5 and tell them that the difference goes at the bottom. Then I tell them that there is good in everyone and ask if they can find a way to see the bad (taking) witches as good (giving) witches.

I remind them that subtraction is like canceling, so if we are going to change the bad witch into a good giving (adding) witch we have to add the number that will cancel -5. (+5).

I model a couple more examples, having them choose the numbers. I emphasize that we are changing the subtraction into a canceling problem so we look for the number that cancels the number we are trying to subtract.

The first two or three people who finish correctly become my graders and runners. Graders get answer keys to grade papers as they come in. Runners go pick up papers when students hold them up to show they are done. Graders turn in correct papers and return papers with mistakes for corrections. As students make corrections they can take it back to the graders.

As students finish I give them our equivalent expressions cards classwork matching equivalent expressions that they sorted in an earlier lesson (Equivalent Expressions ). I tell them to try to sort them faster and more acurately than before. If some partners complete it faster I have them do it again on their own. I expect them to make fewer mistakes this time.