Welcome

The weblog here is a synopsis of my 2007-2008 school year – the first year that I taught Algebra 1 in many years. I used Discovering Algebra from Key Curriculum Press.

I was told that chapter 0 is easier after chapter 1 by teachers that had done it previously, so that is the order that I did it in.

I come back to this website to this year to:

reread what I did last year in order to improve on what I did then, and to upload worksheets or quizzes that I use in 2008-2009 when I am teaching algebra for the second time.

Here is my 2007-2008 Calendar (we have 171 student contact days)

          2007 - 2008 School Year   | 2008 - 2009 School Year
==============================================================
Chapter 1 (04 Sep - 26 Sep) 17 days |
Chapter 0 (27 Sep - 12 Oct) 11 days |
Chapter 2 (15 Oct - 14 Nov) 20 days | (02 Sep - 26 Sep) 19 days
Chapter 3 (15 Nov - 07 Dec) 14 days | (29 Sep - 22 Oct) 15 days
Chapter 4 (10 Dec - 18 Jan) 21 days | (23 Oct - 09 Dec) 30 days
Chapter 5 (28 Jan - 07 Mar) 27 days | (10 Dec - 30 Jan) 25 days
Chapter 6 (10 Mar - 25 Mar) 11 days | (02 Feb - 20 Feb) 14 days
Chapter 7 (07 Apr - 23 Apr) 13 days | (23 Feb - 20 Mar) 18 days
Chapter 8 (24 Apr - 14 May) 14 days | (23 Mar - 27 Apr) 20 days
Chapter 9 (15 May - 06 Jun) 16 days | (28 Apr - 04 Jun) 26 days
Finals    (09 Jun - 11 Jun)         | (05 Jun - 09 Jun)

All that is 164 days. We spent 2 days for quarter 2 and quarter 3 finals (one day review – one day testing) and 3 days for finals (one day review, one day my exam, one day the district’s exam)
Believe it or not, that accounts for all 171 days!

Protected: Final Exam (Day 2)

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Protected: Final Exam (Day 1)

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Final Exam – Practice

I didn’t realize until last week that the end of year exam was comprehensive – I assumed that the test would be on the quarter only. Also, it is in two parts and I’m pretty sure that they both can’t be done in one day, so they’ll have to be done on Tuesday and Wednesday.

For practice, I’ll use the twenty-question multiple choice exam that I wrote for quarter four after going over the chapter 9 test.

Quarter 4 Exam (Chapters 7, 8, and 9)

Protected: 9.9 – Chapter 9 Exam

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9.9 – Chapter Review

No classes (team mosaic day) on Wednesday, so review has to be done in one day – today.

Chapter 9 Review Worksheet

9.7 – Quadratic Formula

Due to time constraints, we had to skip completing the square, which leaves us no way to derive the quadratic formula. So today I put y = x² + 6x + 8 on the board and had the students factor it. We then found the roots and confirmed it by graphing it on Sketchpad. Then I changed the problem to y = x² + 6x + 7 and we found that we could not factor this one. I did graph the function and did show the students that there were x-intercepts, just that they were not rational. I told the students (this was after the quadratic formula quiz) that was what the formula was for – to find the roots of a quadratic even when the roots were irrational.

So I used the quadratic formula to solve y = x² + 6x + 8 (already shown at the beginning), y = x² + 6x + 7 (which we found that we couldn’t factor), then y = x² + 6x + 9 (double root) and then y = x² + 6x + 10 (no real solutions).

I then asked the students to tell me what it was about the quadratic formula gave away how many roots the quadratic had – I then wrote that:

• if b² – 4ac > 0, then there are two real roots
• if b² – 4ac = 0, then there is one real root
• if b² – 4ac < 0, then there are no real roots

I assigned questions 1, 2, 5, 6, and 7 for homework.

9.4 – Factored Form (Day 3)

I went over the first three problems on the worksheet below.

#1 is like the ones that we did last week. Given general form, factor to get factored form then find the x-intercepts and the vertex.

#2 has an a-term that is not equal to one

#3 cannot be factored

That took about 20 to 25 minutes. The rest of the period they worked on the worksheet.

9.4 Worksheet

9.4 – Factored Form (Day 2)

I began the day by going over the homework – I explained that although it was just arithmetic, it would help with our algebra later today.

I then put a quadratic on the board in factored form and we discussed what it told us, went to general form and talked about what it told us, and then we went from factored form to vertex form and talked about why we knew it was right using the roots of the factored form to get the coordinates of the vertex. I had the function graphed on Sketchpad on the screen for us to refer to.

I began the new lesson with a “triangle diagram” showing the three forms of quadratics with arrows going from one to the other for the changes from one form to another that we could do. I then drew in an arrow going from standard form to factored form in another color to show that was what we were going to do today.

I wrote a quadratic in standard form on the board and using the area rectangle we figured out what the factored form would be. I then showed them the same problem using the CPM circle diagram from the previous night’s homework. Finally, I wrote underneath the equation what we needed our constants to add and multiply to (assuming a = 1) like this:

And we did lots of them – one at a time – as I walked around the room checking their work. The biggest hangup wasn’t the procedure, but their arithmetic skills. It is very difficult for them especially when there are negative values involved.

In one class, I had enough time at the end to go back and ask what the x-intercepts of each equation was after they were written in factored form and then asked what the x-coordinate of the vertex would be. I didn’t have time to give any homework.

9.4 – Factored Form (Day 1)

In one class I began by going over the homework from the textbook and in the other I went over their quiz. I think going over the homework was more helpful because question #5 led into what were going to do.

I put a equation on the board in factored form and then graphed it with Sketchpad. We discussed what we noticed about the graph and the equation, then confirmed it with other examples – that factored form gives us the x-intercepts (or roots). So I put a sketch of a parabola with two roots on the board and I labeled the roots, the y-intercept, and the vertex.

Using factored form and from what they did for homework, we changed the equation into standard form. I then had them look at the graph on the screen and got them to notice that the constant term was the y-intercept. I mentioned that was because if x = 0, then the x² and x terms would become zeros.

Then using what they learned in 9.2, we went from factored form to vertex form. Then, we transformed vertex form to general form. We also showed that it was the same as we got from our factored form. All the while, I had been putting each form in Sketchpad and graphing them so that the students could see that they were all the same graph. I reiterated that all three forms mean the same thing, but they just give us different information about the graph of the quadratic – the x-intercepts, the y-intercepts, and the vertex.

After that, I put another factored form on the board and had them find the general form as I walked around the room helping them. Then they had to find the vertex form. This was repeated once more – all kept on the board as well, before I gave them their homework, which were the CPM circle – factoring worksheets which I told them would help them go from standard form to factored form which we would do the following day.

9.3 – From Vertex to General Form (Day 2)

I went over the Practice Your Skills worksheet. I only had the students do every other letter: a, c, e, … and told them that I would give them a quiz on the other questions.

For homework I gave them 1 – 7 from the book.

9.3 – From Vertex to General Form (Day 1)

I’m not sure of the wisdom of having the students find which expressions are polynomials or not in this section, which is really just a continuation of quadratics. It probably should be left out for later on in the chapter.

The students really like the area method of expanding binomials. They also like working backward. The trick, of course, is for the students to recognize when they should use them instead of merely writing that (x + 3)² = x + 9².

9.3 Presentation

9.2 – Finding the Roots and the Vertex (Day 2)

I went over the homework from the previous day and then I gave them the Practice Your Skills worksheet for homework.

9.2 – Finding the Roots and the Vertex (Day 1)

I put a graph on the board using Sketchpad with “obvious” x-intercepts of -2 and 6 (using a slider) and we talked about “roots” and “line of symmetry”. By eyeballing the graph, we found that the vertex had to be at x = 2. Moving the sliders around, we found that the x-intercept of the graph would always be exactly halfway between the roots. We talked about how the average of the two roots would give you that value. We also discussed how we could find its y-value by using the equation, which was the first time that we could actually use the standard form in class due to lack of calculators.

Most of the rest of the lesson we just went back over solving quadratics, then I let them begin their work.

I gave questions 1 – 3, 5 – 6, 10, and 13.

9.1 – Solving Quadratic Equations (Day 4)

Quiz Day. I went over yesterday’s worksheet with them, then gave them the quiz.

I assigned questions 2 -4 and 11 – 12 for homework.

9.1 Quiz

9.1 – Solving Quadratic Equations (Day 3b)

Same as the previous day….due to a block schedule these two days, some didn’t see me yesterday so I’ll do yesterday’s work with them today.

9.1 – Solving Quadratic Equations (Day 3a)

I went over the homework from the previous day (Practice Your Skills worksheet) in its entirety since they’ve had a weekend without it. Then they were given the following worksheet that they did with minimal help from me because I had to work on their mosaic project with some students.

9.1 Worksheet

9.1 – Solving Quadratic Equations (Day 2)

Since I spent so much time talking yesterday, I quickly went over the sets of numbers once again, and then let the students get to work on the Practice Your Skills worksheet. I spent the entire day walking around helping them get this done.

9.1 – Solving Quadratic Equations (Day 1)

I began class explaining what a quadratic was. I showed some examples – some in vertex form (which the kids were used to) and some in standard form just to show them that the highest exponent of x was 2. There is a section on polynomials later so I didn’t go into standard form too much.

I then spent awhile on the Venn diagram of real numbers that appears at the bottom of page 499. Strangely(?!), the students seemed to like this exercise.

Next, I explained how quadratic equations could have 0, 1, or 2 solutions. I did so by showing what (x + 1)² = 10 would look like on a graph and then asked how we could change the “10″ so that we had just one or zero roots.

I then spent some time explaining what what square roots were and how they “undo” squaring. Going back to simple equations such as x² = 9, I showed the students that there were two possible solutions, ±3. When solving quadratics in vertex form, I focused their attention on the fact that 6 ±√3 is actually two different numbers.

Once we had x = 6 ±√3 I wrote “(exact answers)” after it and then below that I wrote the approximate solutions and wrote “(approximate answers)” there. I know that students have a difficult time thinking that 6 +√3 is a simplified, exact answer – they’d rather find “one” number despite me trying to get them to think of “6 +√3″ as a single number. I think that will come with time.

By now the class period was near the end, so I gave them two quadratic equations to solve to take home.

Protected: 8.6 – Chapter Test

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