The focus of the first part of today's class is helping students to understand that inequalities do not work exactly like equations and building their conceptual understanding of why this is so. We begin today's class by taking a look at a basic inequality like 4 < 6. I ask students to add the same number to both sides of the inequality and check to see if it remains true. Next, I ask them to subtract the same number from both sides and so the same thing. Students will almost always exclusive add and subtract the same positive number from both sides here. We then do the same thing with multiplication and division. Again, students usually don't use negative numbers so we're left with four inequalities that are still true. I ask students, "What if we used negative numbers instead?" I ask them to go back and do the four operations again, this time with negative numbers and see if the inequalities remain true. Of course, they find that the multiplication and division will make the inequalities false.
I hand out 4 < 6 and ask them to follow the same steps, but with a different true inequality. They can make up their own or you can suggest one for them to try. I want students to see that this is not a fluke, the same thing happens again.
We discuss what's happening here by using a number line. We look at how the inequality "jumps" on the number line but stays true whether or not we add or subtract a positive or negative number. Then we look at what happens with multiplication and see how when we multiply be a negative number, our greater and less than numbers switch places. Many students may have already learned about "flipping the sign" but they are not likely to understand why. Here, we start to tap into the conceptual reasons for treating inequalities slightly differently than equations.
Next, students begin work on Cafeteria Conundrums. This task asks them to generate inequalities based on a list of variables with meaning and a context. Students often struggle with a number of these questions. I encourage them to really think through what each question is asking and perhaps substitute numbers they make up to help them understand what the question is asking. Students struggle in particular with Question #3 where they have to write two equations first and then substitute them into an inequality in order to use only one variable. Students often struggle to write a compound inequality for Question #6. This can be a good point for whole group discussion in the next section of the lesson.
We don't have a lot of time for today's discussion so we get right to the meat of the questions. I have students share out how they solved Question #3, paying special attention to how they knew how to write equations first and not inequalities. This can lead to a nice discussion about the difference between equations and inequalities and how we know when to write which one. Question #6 is also a good question to discuss as students often have trouble combining both constraints into one inequality. Lastly, we focus on Question #9 and work through it step by step and working on putting the two expressions together.
Cafeteria Conundrums is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.