The Warm-Up shows a lot of spiraling, both forwards and backwards. At this point, students should be able to tackle problems (1) and (2) and problem (3) finally shows them a piecewise function and asks them to graph it.
For the first problem, I have a number line posted and refer to it when students struggle. The multiple representations help them catch their own mistakes, so before I give them feedback about whether their answers are right or wrong, I ask them if they can use a different representation to check their own answers (MP1).
The second problem will be challenging simply because of the words, though it is similar to the overtime problem from yesterday. Again, ask students to make a data table to get started (MP4).
Hopefully they can figure out the third problem, and I tell them this: “See if you can figure out the third problem without asking me to show you how.” This pushes them to make sense of the problem without being told how to do it. They will usually struggle with the endpoints, so I leave this part open to discussion: “What happens when x = 2? Why?” This function is not continuous, so this will be a challenge. See if they can figure it out themselves, and if they can’t, you still don’t need to show them. They will have more days to work on this before they are expected to master this skill. The more they can internalize the idea that they don’t have to figure things out right away, the more comfortable they will become really thinking about problems without immediately solving them. This is a powerful shift in classroom culture, so I explicitly say it constantly: “You don’t need to figure out this problem today. Just see what you can make sense of so far. You will have many more chances.” I also tell them that I will only give them feedback if they can show me evidence of their collaboration and their thinking. When they ask me, “Is this right?” I ask them, “Why do you think it’s right?” It’s essential to ask this question whether their answer is right or not, to make sure that they don’t look for clues in your voice!
The two closing questions in the Exit Ticket are really important to discuss today to help students understand the big ideas of the lesson. The whole idea of a progressive tax structure is a perfect way to remember the big idea because the whole point is that people pay different tax rates based on their income level. Students should be able to connect the different income levels to the inequalities and the rates paid to the function rules.
The second question is much deeper: how do the two functions relate to each other? The best possible answer is that the rate function tells the slope of the amount function. If students understand this, they are really grappling with calculus. At this point, students may be able to say that the rate function consists of horizontal lines for each income bracket while the amount function consists of lines with positive slopes. Start a class discussion about this to see what students come up with. At the very least, we want them to understand that these two functions are different and that we can generate both of them using the information given about the tax structure.