The story for this series of lessons is modified from the Interactive Math Program. You might recognize the lesson image from one of the high school lessons. I love this story because students can relate to the temperature model. The idea of adding and removing hot and cold makes sense to them and helps them reason through the math when it feels a bit abstract. I have incorporated written symbols to give students something concrete to model. I also have several practice vignettes that break down the operations.
This warmup consecutive integers on a number line helps to reinforce with students what consecutive numbers look like when we include negatives. Some students that -2 for instance is consecutive with +3, so I have them look to see if they are next to each other on the number line. Their homework consecutive sums negative last night was to start looking for what types of numbers they can and can't make adding consecutive numbers. They have done the consecutive sums problem using only positive whole numbers (Number System Assessment, Garden Design, Power of factors) and found that they were unable to make the number 1 and the powers of 2. This raised several questions in their minds about whether or not they could make some of these numbes if they were allowed to use negative numbers.
I think it is really important to let student questions play a part in guiding instruction because it makes the instruction more meaningful for them and it also encourages more question asking in students which is so important to making sense and perseverence. I expect students may have made a couple of different mistakes in homework and some may have given up. If the numbers they added are not consecutive I tell them to check if they are next to each other on the number line. Some students may have made addition errors with positive and negative numbers and I tell them to work with our Mathmaster Chef today and that might help. I tell the class this is a big problem and that we can make it smaller for ourselves just by working on a few target numbers tonight. I suggest trying to make the numbers -10 through +2 and if they get some others by accident, great!
The warm up starts with a number line similar to yesterday's lesson (Equivalent Expression Assessment) and asks students to label the numbers around 0. The next two problems ask them to fill in the blanks of addition problems with consecutive numbers from the number line, then complete the sum. The second one has zero in the middle, so they need to fill in 1 and -1 on either side. The last problem asks them to write an addition problem of their own using consecutive numbers on the number line for one of their math family members to solve. Most students will use only 3 numbers, but they can use any amount.
When they switch I tell them to double check that the numbers are consecutive before they find the sum. If they are not they have to return it to the originator to correct it. Once they have solved the expression they need to show their partner how they added the numbers on the number line.
Students share some of the sums their partners gave them and, when I write them on the overhead I model using the number line or let them do it. If the sum includes opposites I circle that part and ask what this part equals. For example: -2 + -1 + 0 + 1 + 2. It will be important for them to understand that adding opposites equal zero.
This is the first in a series of lessons that uses the same context. Today we are going to go through the first 11 slides of the powerpoint Hot and Cold cubes. This follows a story about a Mathmaster chef and how he cooks his food using hot and cold cubes in a cauldron. I tell them how the judges on the cooking contests are always complaining about under or over cooked food. I tell them how important it is to get the temperature just right to bring out all the flavors of the food. I tell them our Mathmaster chef adjusts the temperature in his cooking by using cold cubes which are like ice cubes but they never melt and hot cubes which are like charcoal brickettes but they never lose heat. Slide 3 shows a pot with 3 hot and 3 cold cubes, which equals zero degrees. I ask students to think of other ways to make a temperature of zero and ask for one way from each group. I ask them how they know all these methods will give a temperature of zero so that they articulate that as long as there is an equal number of each the temperature will be zero.
The next four slides show what happens when one cold cube or one hot cube is added or removed. At each slide I ask students to predict how they think this will change the temperature and why before it appears on the slide. To show me their prediction I tell them to hold up the number of fingers they think the temperature will increase and hold the same number down if they think it will decrease that number. Predicting is important because it helps them synthesize the information they have and make sense of the problem. (mp1)
After slides 4 & 5 have shown them that adding 1 hot or removing 1 cold will both result in an increase of 1 degree I ask them how they can make the temperature increase 2 degrees. This helps introduce the idea that subtracting negatives is equivalent to adding positives. Some students may just see one way, but someone is likely to see the other as well. I ask each time how they know their suggestion will work. Similarly, after slides 6 & 7 have shown them that removing one hot or adding one cold cube will both result in a decrease of 1 degree I ask them how they can decrease the temperature 2 degrees.
The next four slides reinforce the equivalence of subtracting and adding the opposite. After each slide I ask what the temperature change would be and how they know. After slide 9 I ask for two different ways to increase the temperature 2 degrees and ask them to explain why they are equivalent. Similarly, after slide 11 I ask for two different ways to decrease the temperature 3 degrees and ask them to explain why they are equivalent.
I ask them to explain this equivalence within the context of the model first because it is so much easier to understand and explain than with the abstract. This concept of adding the opposite is so useful, but also so difficult to understand in the abstract and this temperature model is one which all students have experience with (heating and cooling water). More students have had experience with ice cubes and charcoal than with borrowing and lending money. When we get to subtracting amounts that aren't there like -2 - (-4) or -2 - 5 my hope is that students will be more willing to add the opposite after using this model which I think gives them the best understanding of this equivalence. I think most mistakes that students make in this regard is that they are not entirely certain that the two really are equivalent.
This is something I do with students when we have a few minutes at the end of class and I don't want to move forward. I have several different flash cards made of recycled halved manila folders. The ones I might choose now could be review like exponent cards, order of operations, or I might preview missing side lengths of irregular shapes for our geometry unit. Since we just finished the assessment on distributive property (assessment ) and I know that students made some mistakes factoring out the greatest common factor this would be a good time to hold up pairs of numbers for students to figure out the GCF.
I ask all students to stand behind their desks and figure out the answer silently. Students do not raise hands and do not shout out. When I call on a student I want the answer in 1 second or less, which means everyone needs to get their answer ready. This ensures that most students are actually motivated to do the problem even if I don't end up calling on them. Theoretically it gets all students answering the question rather than just the one I call on. When they get the right answer they may sit down and are out. Students that I think need the most practice tend to stay in the game longer, because once they've answered correctly sometimes students stop participating, but surprisingly, not many.