SWBAT use a number line and a temperature model to represent operations with negative numbers.

Students will see the relationship between integers and operations and will use multiple equivalent methods.

In day one with our Mathmaster Chef students learned that he adjusts the cooking temperature by adding or removing hot or cold cubes. Today's lesson introduces the number line as a new mathematical tool to represent operations with integers. The main idea is to reinforce the equivalence of subtracting and adding the opposite. This is a really confusing idea for my students that addition and subtraction can be equivalent. The number line is another way for them to visualize what it happening mathematically.

14 minutes

While students are working on the warm up I circulate to check in homework consecutive sums negative and to see what new numbers they were able to make with adding consecutive numbers. I keep track of which numbers between -10 and +2 they were able to make which is what their target numbers were.

The warm up asks students to use a vertical number line as a thermometer to show the temperature changes made by the chef. First they must realize that their starting point is zero since we start with equal numbers of cold and hot cubes in the pot. I ask students to demonstrate on the projector and ask them where they think they should start and why. If the students don't know where on the number line to start I ask:

- "what is the starting temperature of our pot?"
- "how do we know that?"

**After each subsequent move I ask students to explain:**

- what each action (made by the Mathmaster chef) does to the temperature
- what direction to move on the number line
- how they knew which direction to go.

When the model is complete I ask them what direction we move on the numberline when we add hot cubes (up - hotter). I ask them what other action caused us to move in the same direction as adding hot cubes (removing cold cubes). **I make sure to ask why that makes sense.** (they both increase temperature) I ask the same questions about removing hot cubes and adding cold cubes. Not only does this help them understand the equivalence, but it also gives them a model to help them explain to others the equivalence (mp3) and it gives them a context to help them make sense of harder problems later (mp1).

40 minutes

This lesson follows the powerpoint Hot and Cold cubes slides 12 - 17 and introduces mathematical modeling for the temperature changes and for the hot and cold cubes. My goal is to focus on multiple methods so that students can get a feel for the equivalent methods and also gain some ownership through creating the problems themselves. As they work through each slide students are working together. I expect some dissagreements as students explore these positive and negative changes and we take this opportunity to help them use evidence to support their claim and critique or counter the other claims (mp3).

Slides 12-15 show combinations of adding and removing different numbers of hot and cold cubes. Students are asked to figure out what the temperature change will be. I ask all students to show me how much the temperature will change by holding up that number of fingers, up if it is a temperature increase and down if it is a decrease. I write all the answers on the board, there should really only be 2 or 3 different answers. The first time we have a dissagreement students want me to tell them which one is right, but this would not engage them in argumentation. I ask them if they see an answer they strongly believe in and ask them to take a moment to think of why they are convinced it's right. The individual white boards are really useful for this, especially if they want to show a number line. Then I ask if they see an answer they strongly dissagree with and ask why they are so convinced. I tell them to explain their reasoning to their math family team and listen to the reasoning of others to see if they can convince or be convinced. Then I ask who changed their minds and ask them to explain what they thought at first and what was said that changed their minds. After deciding on the right answer I ask "how can we represent the amount of that change mathematically?" which they show me on their white boards. I might need to clarify "without words, just with symbols"(-5) and "what is another way to make a change of -5?" I follow the same line of questioning with slides 13, 14, & 15. I expect the argumentation and convincing to happen before I ask them to show me this time and I would circulate to make sure and to highlight for the class aspects of good argumentation "Hannah just shared some convincing evidence", "Johnnie is using a number line to help show his partners", etc.

Slide 16 asks how we can represent each of the cubes mathematically with symbols. I ask them what the hot cubes do to the temperature and what symbol might represent that change? (+) and similarly (-) for the cold cubes.

There may not be time for slide 17, which asks how to decrease the temperature 8 degrees by adding and removing some cubes. I would rather spend more time on the argumentation process and the explaining of multiple methods than rush through to get to this last slide today.