4.6 – Chapter Review

4.1 – A formula for Slope

• Investigate and solve real-world problems that involve the slope of a line
• Learn how to calculate slopes with slope triangles and the slope formula
• Learn about slopes of rising, falling, horizontal, and vertical lines.

4.2 – Writing a Linear Equation to Fit Data

• Draw a line that fits or models a set of points
• Write a intercept equation that fits a set of real-world data

4.3 -Point-Slope Form of a Linear Equation

• Learn the point-slope form of an equation of a line
• Write equations in point-slope form that model real-world data

4.4 – Equivalent Algebraic Equations

• Learn and use the distributive property
• Rewrite equations to determine whether they are equivalent
• Formalize algebraic properties
• Identify properties as they are used in solving equations
• Introduce factoring as the reverse of the distributive property

4.5 – Writing Point-Slope Equations to Fit Data

• Write linear equations in point-slope form that model real-world data.
• Discover strengths of the point-slope form for linear equations
• Learn to deal with variation in linear data

Comments:

• Not sure why the book can’t call the “intercept” form the “slope-intercept” form.
• Not sure if teaching students the point-slope form is a good one. They didn’t get real comfortable with the slope-intercept form. If they stayed with only the slope-intercept form, they could use a point to substitute to find the “a” (y-intercept term) by solving. A positive about point-slope is that they can use the algebraic properties such as the distributive property and combining like terms to find equivalent equations.

Chapter 4 Review Worksheet – Day 1

Chapter 4 Review Worksheet – Day 2

Chapter 4 Review Worksheet – Day 3

4.5 – Writing Point-Slope Equations to Fit Data

With most students still without calculators, I wasn’t too enthused about graphing the data in the textbook. Plus lines-of-fit would be very difficult, so I wrote my own problem to do.

In class I worked through the Extra Practice worksheet with the students from the section. Question #1 is review, and the kids certainly needed it. It was if slope and the point-slope form was something that they haven’t ever done before.

I then told them that we have drawn lines-of-fit in section 4.2, but then we were using slope-intercept form. The only thing different is that were going to use point-slope form now. So working through question #2 was not too difficult.

For homework I gave them the following sheet. I told them that sometimes it isn’t appropriate to start scales at zero so to begin this one at 60 on the x-axis and go up to 85. One question on the worksheet has them try to explain why we can’t get the y-intercept from the graph if the x-axis doesn’t begin with 0.

In all, I think the worksheet hits on the following goals for the chapter:

• Draw a line that fits or models a set of points
• Write a intercept equation that fits a set of real-world data
• Write equations in point-slope form that model real-world data
• Learn and use the distributive property
• Rewrite equations to determine whether they are equivalent
• Write linear equations in point-slope form that model real-world data.
• Discover strengths of the point-slope form for linear equations
• Learn to deal with variation in linear data

4.5 Homework Worksheet

4.4 – Equivalent Algebraic Equations

Day 1:

I put a graph of a line up on the board and wrote the equation of the line under the heading “Point-Slope Form”. Then underneath that I wrote the equation of the line under the heading “Slope-Intercept Form”. I then explained how these two lines were just different names for the same thing, just like 1+8, 11-2, 3², and 9 all look different but have the same value.

I told them that their goal was to be able to prove that the two equations were the same using algebra skills. Then I taught them the distributive property with a couple of examples and how to combine like terms. With these, I did show the students with their help that the two equations on the board were equivalent.

For classwork and homework, I had them do an “Algebra with Pizazz” (DD9 and DD15) worksheet that only uses positive numbers. This helped immensely, although I warned them that the textbook would use negative values as well.

Day 2:

I was gone for a meeting this day. They did two things:

First, the investigation in the book had some equations written in standard form which I wasn’t sure that the students would remember how to solve for y without me there, so I re-wrote those problems, instead using equations with combining like terms.

4.4 – Equivalent Equations Investigation

And then I had them do the bookwork, questions 1 – 5 for homework. The following day they had a quiz:

4.4 – Quiz

4.3 – Point-Slope Form of a Linear Equation

Day 1:

First day back after winter break.

I gave each kid a piece of graph paper and then had them fold it into four parts. I had them graph lines using slope-intercept form as a review. I then wrote on the board, “The two things you need to know to graph a line are” and asked them what I needed to put down. The kids were good about saying that we need a y-intercept and the slope of the line, so I wrote them on the board along with “starting point” and “direction” underneath them.

I then put the ordered pair (5,1) on the board and had them plot the point in a new coordinate plane. Once I checked to see that this point was placed correctly, I put “slope = -2/3″ on the board and had them plot more points and then had them draw a line through them.

After this was done, I asked them once again what was needed to graph a line. All four classes said that we needed two things:

1. y-intercept and
2. the slope of the line.

I then pointed out that we had just graphed a line but we never knew the y-intercept, and because of the point and slope I chose, it was difficult to even tell what the y-intercept was. Then the students were okay with saying that we need two things:

1. a point on the line
2. the slope of the line

to graph the line.

From there, I went over the slope formula once again, paying particular attention to what the subscripts mean in the formula:

slope = (y2 – y1) / (x2 – x1)

And then using our example with only one point, I derived the equation of the line

-2/3 = (y – 1) / (x – 5)

y = 1 + -2/3(x – 5)

It was then time to go, so I didn’t give any work over night.

Day 2:

The second day I wrote on the board:

And then went through the following problems with them underneath the camera. I had them

1. Name the slope and one point on the line that each equation represents.
2. Write an equation in point-slope form given its slope and one point that it passes through.

Algebra 04-3 Notes

I then assigned questions 1 – 4 from the textbook.

4.2 – Writing a Linear Equation to Fit Data

I was in the middle of preparing a presentation on PowerPoint to do the example on page 227 when I was called to go get a kid at daycare, so I had to make a “notes” sheet for the kids to do when I was gone, which is below.
Algebra 04-2 Notes

Upon return, I went over the notes with the students including going over questions 1, 2, and 3 and gave the homework here, which is a little review. In class, they spent the entire period doing question #4 on page 230. I had the students do the work using graph paper rather than the calculator because their scales were so poor. I had the students come up to me and show me their scales before they began to plot the points. If their scales don’t begin at zero, then finding the y-intercepts by using the graph won’t be done correctly.

Algebra 04-2 Homework Day 2

The next day I had the kids work on the Extra Practice for section 4.2 in class while I called the kids to the desk and went over the 4.1 quiz and question #4 with them.

As if that wasn’t enough, we did one last worksheet for review before I gave a quiz right before winter break.

Algebra 04-2 Review

Note: I spent a total of nine days on the last two sections which were just the idea of slope with the slope-intercept form thrown in. My students do not have a grasp on the concept, though. The “idea” of slope comes up again and again throughout the chapter so I will go over it again and again, but I’m not satisfied with the level of understanding that the students have after all this time. I’m unsure how I will teach it next year, but I know that it is something that has to be understood well. I might use the term “slope” in chapter two in addition to “rate of change”.

To make matters worse, I finished 4.2 just as winter break came, so they’ll have two weeks to forget whatever they’ve picked up during the last two.

4.1 – A Formula for Slope

Day 1:

I began class by putting the intercept rule, y = a + bx on the board and asking what the a and b represented. All classes began with “starting point” for a, which I put on the board in small writing. Then I asked them what the “starting point” on the graph of the line would look like, to which they answered “y-intercept”. I wrote that in larger words. For b, they were more vague. I was hoping for “rate of change” but their answers were more along the lines of “how much y goes up by”. I tried (once again!) to get them to say “for an x-change of one”, which was somewhat successful. For a couple of classes I put a table of values up quickly to help them through. I then wrote “slope” in large letters underneath the b in the equation.

Next I talked about what slope is by talking about steepness. And how we can describe exactly what a line looks like if we know where to “start” and what its slope is. With that I helped them through what the slope formula really was and how it related to the tables where they had to find the rate of change from the previous chapter.

I then put two lines on graph paper underneath the camera to find the slope of using slope triangles. I was very careful to tell kids to follow a path on the slope triangle so that their positive and negative signs would be correct. I also always wrote the sign, whether positive or negative, in the numerator and denominator of the slope formula, such as slope = +5/-3. I talked about how it is important to try to leave the answers in simplified fractional form so that they’ll always be able to see what the “change in y” and the “change in x” are.

We then did the Extra Practice worksheet for section 4.1 together. We didn’t do every problem, but most of them were finished.

For homework, I gave them questions 1, 2, and 3.

Day 2:

I am going to go over the homework and then work through the investigation for the section, which I have in a power point document. I’m going to stress that on a line, it doesn’t matter what two points you choose on a line, since the line always goes in the same direction, the slopes will always be the same. I also am going to talk about the equations of horizontal and vertical lines, which aren’t mentioned in the text, but appear in problem number 5 on the homework. I’m assigning questions 5 – 9, and I’m planning on giving question 10 as a mini-quiz thing either at the end of the class or at the beginning of the class tomorrow.

4.1 Investigation and Equations of Horizontal and Vertical Lines

Day 3:

Still going! After helping the students through the text’s questions yesterday in class, I’m convinced that they still need another day, but I don’t want to do a drill-and-kill assignment, so I’ve made a worksheet that hits on different types of problems. I’ve also included a problem that leads them into the next lesson on lines of best fit.

I’m going to give a quiz tomorrow – which is just question #10 from the textbook.

Day 4:

Short day, so I went over the homework, gave the quiz, and then gave them a two-question worksheet to take home.

Algebra 04-1 Homework Review