Today I’m going to go over the worksheet from yesterday while showing the students how to use their graphing calculators to graph inequalities and systems of inequalities.
I’m going to have them do questions 1 – 8 from the textbook’s review section for homework. Never got there. Just passed out papers and let them go.
Today I’m going to call students up to my desk to go over the previous day’s work with them while they work on the second review sheet. I’ll be grading this one for accuracy rather than effort to get the kids to put in a little more effort.
I was at IFL training today, so the students had to do a review worksheet which is below.
Because I won’t be in class today due to an IEP meeting, I made a very simple notes page for the substitute to give to the students that should help them along. Since they’ve spent three days on graphing inequalities, I hope that they can get through this easily and see some similarities to solving systems of equations.
For homework I will have them do the More Practice Your Skills worksheet for 5.7.
On (another) day that I’ll be gone, I’m giving the students a worksheet to graph twelve inequalities from the Algebra with Pizazz booklet. On the board I’ve written the steps:
In two periods I gave the Extra Practice worksheet to do. They turned it in at the end of the period. The work was done pretty well. When helping the students, I told them when graphing inequalities to focus on three things:
For question 4b, the students will need to solve for y first. All in all they did pretty well, although some students are much slower than others at getting them done.
For the afternoon classes, I had them do the same worksheet in groups as a quiz. One class did very well and the other did very poorly. I’m unsure why there was such a difference, but the fact that the worksheet said “sketch the inequality” meant “just draw a line” for some kids.
I put two inequalities on the board and we spent a good deal of time graphing them on pre-scaled graph paper (to save time):
paying special attention to when and why we used dashed vs solid lines and when we shade above or below. After doing those two, the students worked on numbers 4 and 5 in the book.
During my afternoon lessons, instead of “giving” them points to check (like the book does for questions 4 and 5) I let them choose which they would like to do. In my morning class, I had each person pick their own point and find whether the inequality was true or false, but when they came back riddled with errors I gave up on that.
I’m going to go over the homework from the previous day and then give the students a quiz to do in a group. The quiz itself is just the Lesson 5.5 worksheet from the “Practice Your Skills” section of the supplementary materials.
I’m going to show the kids how to solve inequalities, bringing them back to the investigation where the inequality sign changes direction if both sides need to be multiplied or divided by a negative sign. I have a PowerPoint presentation to help move things along.
Then I’m going to have them do questions 1 – 11 in class.
Before I did the investigation I talked about the prefix “in” by using ineffective and inappropriate so that the students could tell me that an inequality means “not equal”. I then put two numbers on the board that are not equal to each other and asked what the students could tell me about the numbers being more specific than “not equal” so that they would use “bigger than” or “smaller than”.
I then wrote an equation such as x = 4 on the board and said, “read this to me”, which the students did quickly. I then wrote x < 4 on the board and said, “read this to me” which was only done by a few kids and much more slowly. I told the students that by the end of the chapter, they should be able to read the inequalities (which are then put on the board) just as fast as they read the equation.
For reasons I have idea why at this point, I thought that the investigation in the book required the students to continually go back to (2,4) to do the next operation. I didn’t realize that each operation should be done in succession. So I wrote my own operations that (as it turns out) do the same thing as the book. I put them on a worksheet for the kids to do while I had kids at the board underneath the number line.
I then had the students graph inequalities on a number line. The first one is t > 5. I told them that we have to graph every number that is greater than 5 so I begin to ask for some. Most give me 6, 7, 8, 9, …. and so I put those points on my number line. Then kids start to shout out 5.5 and 5.1, so I graph those. I zoomed in with the document camera around 5 and then said that I’m going to plot 5.00000001. Then I ask if I can put a dot on the 5 – the kids were good about saying no. But then I asked how it would be possible for someone looking at what I had on the board to tell that there wasn’t a dot on the 5 when I’m so close to it already. That led me into using the open circle to show that “we get very close, but don’t touch” the 5.
For some classes I did the translating of phrases to inequalities, but in no classes did I get to the very last part – solving the inequalities for y. In one class I put very simple inequalities on the board to show what happens when you add, subtract, multiply and divide by positive numbers, and multiply and divide by negative numbers. I also showed them problems like 3x < -12 to show that it isn’t the -4 for the “solution” that determines if the inequality sign changes, but the number that you divide by (the three).
Oh, when going over the translation pieces, I found it easier to make a list of numbers that made the list true before writing the inequality rather than just translating the words. For example, students got stuck on “at most 2″ so I asked them to imagine telling their mothers that they would be out with friends for at most 2 hours. How long could you be gone? Then they could get to x < 2…with the phrase “most” they tend to think of “more than” so they’ll write x > 2.
Day 1:
In order to show the kids why we have elimination as a method, I had the students solve three systems with me using substitution:
The first two were pretty straight-forward, the third is very difficult because you must solve for y (or x) first. By showing how easy it was using the elimination method, they bought into it right away and got mad at me for not showing them earlier.
I did warn them that the elimination method should be used only sometimes…part of the art of being a mathematician is to recognize when substitution should be used and when elimination should be.
The presentation below was used – I got through the whole thing, but it took every minute of the class period. The worksheet could be used instead of the presentation – I had them take notes from the presentation instead.
5.3 Homework to go with the presentation
5.3 – Notes and Homework Together
Day 2:
I went over the previous night’s homework by having the kids come to the board and write their solutions. Then I gave them ten more systems to solve. I had them come up to me after they had the first four done to show me what they have done and to make any corrections before they tried the rest.
Day 3:
We will do the Pennies and Paper clips investigation and then go over the example in the book (two numbers sum is … and difference is ..; find the numbers) and then do some of the problems in the textbook: 1 – 5, 7.
Day 4:
I went over the homework from the book. Despite many not even attempting it, I was pleasantly surprised at the number of students that did well on it. I went over all six problems on the board, and then gave them the Skills Worksheet from the publisher for section 5.3 and had them do the “odd” lettered problems.
Day 5:
I’m at another meeting and they are doing another skill worksheet from the Algebra with Pizazz book.
Example 2 from the book is extremely difficult. I would avoid it and instead, focus more on types of problems like questions 9 and 11 in the text.
Day 1:
After going over the homework from section 5.1 and giving the quiz, there wasn’t much time to introduce the lesson. So in order to prepare the students for the substitution method, I gave them nine equations with the variable x on both sides to show them how they had to be solved.
Day 2:
Today I gave six systems to solve…very easy ones. All equations were in slope-intercept form and each solution had whole number coordinates. My focus was on explaining what the “substitution” part was (I talked about substitution of margarine for butter and about substitutions in sports) and how to show their work properly. The last problem has no solution and I talked in length about why there was no solution.
5.2 Drill and Kill Substitution (Easy)
Day 3:
More problems with substitution. This time one equation was in slope-intercept form and one was in standard form. These are the easiest ones to show what substitution is all about. The final question was an introduction into using substitution when you have two equations in standard form, which is difficult, but I wanted the students to see that they are hard to do so I can have a good lead into using the elimination process. However, students have not had practice with solving equations for either x or y when when the equations are written in standard form….
5.2 Drill and Kill Substitution (Harder)
Day 4:
So on the fourth day, I began the class by solving standard-from equations for x or y and later showing them that’s how some systems can be solved. I’m not going to assess them on solving systems by substitution this way because they aren’t meant for elimination anyway.
Over the last four days I’ve put as much emphasis on symbol manipulation and showing work properly as on the substitution process.
5.2 Drill and Kill Substitution (Hardest)
Day 5:
I’m at a meeting, but the students are doing questions 1 – 7, 11 in class. Nice to finally give problems with some context. But I think their symbol manipulation skills needed a boost so that they aren’t setting problems up correctly just to have them not be able to solve them algebraically.
Day 6:
Going over homework and giving a quiz.
My students aren’t great about getting calculators despite the fact that they can sign them out of the library. So I plan on showing the students how to graph them on the calculator, but I need to show them how to do them by hand as well.
But, the homework in the book is difficult without the students being able to take calculators home. So I just had the students do number 9 at home, which is like the one I do in class (see lesson) Number 8 could be done as well.
Day 1:
I only got through the first problem on the PowerPoint lesson, but it went well. I tried to spend a great deal of time working on the vocabulary. Some of the points:
For homework I gave them question #9.
Day 2:
I finished the PowerPoint lesson and then gave question number 8 for homework. I’ve also started calling the brace for the system a “rubber band” so the kids understand that the equations are really part of one thing.
