8.5 – Chapter Review (Day 2)

Chapter Review Worksheet #2

8.5 – Chapter Review (Day 1)

At the beginning of class, I went over each of the types of transformations that we’ve done throughout the chapter. I wrote each one on the board and wrote down what and with what we’d have to replace a variable with to translate, reflect, or vertically stretch a graph.

With the second class, I began by doing question number 1 with them since it was difficult. About a third of the class period finished the work which they had about 30 or 35 minutes to do it in.

The biggest problem that I see is that students will try to write the answer with no work shown despite me showing them all the steps on the board over and over. For instance, I’ll write down something like this:

Then there is the problem with order of operations – I’ve been making sure that when there is more than one transformation that I’m asking the students which (if any) must come first. This is easier to do with Sketchpad open.

Review Sheet #1

8.4 – Stretching and Shrinking Graphs (Day 2)

I’m out of the building today so the substitute will show the answers to the students for the homework from the previous day. Then they will do #1-3, 9, 11, 16, and 17 from the textbook.

8.4 – Stretching and Shrinking Graphs (Day 1)

Because of the lack of calculators here, I have to do the investigation on paper, which isn’t so bad since the students still have trouble identifying coordinates of points. I’m using the notes below as my instruction, which I will do with the students.

For homework, they will do the Practice Your Skills worksheet for section 8.4.

8.4 Notes

8.3 – Reflecting Points and Graphs (Day 4)

Quiz day. I went over the worksheet from the previous day in its entirety and then gave the students a quiz using some questions from the worksheet. The last two questions on each were not from the worksheet – given graphs of functions, students had to write their equations.

8.3 Quiz v1

8.3 Quiz v2

8.3 – Reflecting Points and Graphs (Day 3)

I went over both translations and reflections using the homework (Practice Your Skills worksheet). Then I had them work on the following worksheet for the rest of the period.

8.3 Classwork Worksheet

8.3 – Reflecting Points and Graphs (Day 2)

I went over the previous night’s homework in detail – I also added question 6 to discuss from the book. The most difficult by far was question #7, where they would look at an equation and have to write how the parent function was transformed. Because a “-(y – 5) = |x| was written as y = -|x| + 5, the students had a difficult time realizing that the variable y was the one being transformed.

For the rest of the period the students worked on the 8.3 Practice Your Skills worksheet, which they turned in at the end of the period.

8.3 – Reflecting Points and Graphs (Day 1)

This was a tough lesson, and there is no way that I can expect kids to have this figured out in a day.

I began with a review of the four basic transformations, once again writing them all on the board – mentioning that we’ve done translations and now we’re going to do reflections. To begin, I had a linear/piecewise generic function on the screen using Sketchpad. I had the students tell me what each point’s coordinates were (this was not done well on the previous day’s quiz), and using the text function wrote them on the graph. Then I reflected the function over the x-axis (after asking what quadrant the image would be in) and we wrote down the coordinates of the images and compared them. On the board, I wrote that a reflection over the x-axis made (x,y) → (x,-y).

Next I selected the points of the function in Sketchpad and told the program to print the coordinates for me to save time. I then reflected the graph over the y-axis and wrote the rule down. I also did the same thing for reflection over the line y = x. The students had a difficult time noticing that the original image, which was entirely in the first quadrant would remain in the first quadrant when flipped over the line y = x. But once it was on the screen and I pointed out where each point was going, they seemed to get it. One kid in one class asked what would happen if I reflected the graph over both the x- and y-axis, which I did. I also pointed out, using Sketchpad, that it was the same thing as a 180 degree rotation, which I showed by having Sketchpad to a 175, 176, 177, 178, 179 and 180 degree rotation in succession, which the kids thought was cool.

When I was done, we had a table like this on the board, which I had them write down.

Once that was complete, I turned the screen off and began plotting functions, then once the equation was hidden away, I’d turn the screen back on so that they could try to look at a function and figure out its equation.

For each one (I only could do three in each class because it is very time consuming) I’d have the students write out the descriptions (in words), the coordinates in (x, y) form, and the equation.

I didn’t get to do too many and it was apparent that many kids were not getting it, so I’ll have to do lots more examples tomorrow. But, I did assign homework, #1 – 3, 5, and 7, which I can use for more examples tomorrow.

8.2 – Translating Graphs (Day 2)

Shortened class period and quiz day. The quiz that I’m using is the 8.1, 8.2 quiz from the assessment materials.

After going over the homework I had this table up on the board:

And with them I filled out the table while showing the translations of the absolute value and the parabola on Sketchpad. The function with the square around x isn’t a real function, but I wanted to point out that it could represent any function at all, so I drew a weird function on the board and explained that it would just be translated like the other ones from the rule that we used.

8.2 – Translating Graphs (Day 1)

Long lesson.

I began the lesson by talking about parent functions. So I put y = |x| on the board and put its graph up on the overhead screen using Sketchpad.

I then asked the students how they would think I would have to change y = |x| so the the graph would move up three units. They didn’t have any idea. I asked them, “would the x-part or the y-part change if you were moving a graph up?” They knew that it would be the y. So I then asked, “Would you change the y to a “y + 3” or to a “y – 3″? This is a leading question, of course, but I thought that it would be much quicker than making a t-table and making a rule plus the same question will be asked so many times over the course of the day that they might get used to asking themselves the question.

The students all said that “y + 3″ would be appropriate. So I told the students to read the parent function back to me but to replace the y with a y+3. I then told them that I could not type in a function into the computer unless it was solved for y, so we got y = |x| – 3. I then graphed it and we saw that we didn’t get the new graph (graphed in red) to move up, but down. So we changed the original y + 3 to a y - 3 and re-graphed the problem, and this time it was correct.

From then on I did lots and lots of examples with the absolute value function….trying to get the students to understand that if I wanted to translate one way, then their substitution of the form “x + h” or “y + k” would be counterintuitive – left means +h, right means -h, and so on.

After those, I began doing the problems again with y = x² making sure that the students understand that it doesn’t matter what parent function we are using, the rules are the same for translations.

Once those were done, I did one example with an exponential.

Then I put the problems on the PowerPoint on the board – graphs of absolute value, parabolas, and exponentials that have been translated and we came up with the equations. For these, I would ask the students to tell me what the parent function was. I’d write that on the board. Then I would ask how the function has been translated in words and write that on the board. Next I’d ask for what needs to be substituted such as changing x to a x + 3. Then I would physically erase the x in the equation and replace it with an x + 3 to show that step. Once we had the final equation, I’d put a rectangle around it and say that it was the equation for the graph on the screen.

For homework, I gave the Practice Your Skills worksheet for the section.

8.2 Presentation (PowerPoint)

8.1 – Translating Points

I began the class by talking about geometry and, specifically, the geometry of movement. While doing so, I used Sketchpad to do the following transformations while I defined them on the board:

Types of Transformations

• Dilations (which they had done with the Wumps worksheet previously)
• Rotations
• Reflections
• Translations (which they will do today)

Then I went through the PowerPoint presentation with them writing the rules (using ordered pair notation) as we went along. The lesson was only about 15 minutes, much of it with Sketchpad. Then they worked on their homework which was 3, 6b, 7 – 13 from the textbook.

8.1 Presentation (PowerPoint)

8.0 – Introduction to Transformations (Day 2)

I gave this worksheet so that kids would have practice with finding y-values given x-values and so that students will have a better understanding that you can graph a function just by picking points to get a pattern, which I don’t think that they realize.

I think that the worksheet is too long, and maybe because it was a Friday with a substitute, but very little of it was finished.

8.0 – Worksheet

8.0 – Introduction to Transformations (Day 1)

I was going to be in Duluth so I gave the students CMP materials – graphing the Wumps. We changed the material a bit so that students were graphing translations and dilations while working with graphing points.

We gave the students the graph paper already with scales on it to make it go a little more quickly.

The students seemed to understand what was happening with the dilations. However, with the transformations, most didn’t realize that anything had changed at all. The fact that the images were in different places didn’t seem to register….so we talked about what a translation was (in section 8.1) upon my return.

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7.7 – Chapter Review (Day 3)

I’ve found these to be the biggest issues.

• The “faster and faster” or “slower and slower” for rates of change is confusing for students. When they encounter “faster and faster”, I ask whether you would have to start off going slowly or quickly if you need to go faster and faster, which they answer correctly. Still, they don’t seem to ask themselves the same question when they see the problem.
• The students are still having a difficult time realizing that when they are given f(3) and they find that the answer is 7, then that is the same thing as knowing that the point (3,7) is part of the function.
• Because there are two solutions to both |x| = 4 and x² = 4, the students sometimes think that if f(x) = |x| and g(x) = x² then f(4) =  ±4 and g(2) =  ±16.
• Students have a very difficult time solving equations such as |x+5| = 9 or (x+5)² = 9. To solve them on the board, I cover up the “x+5″ with my hand and ask “what number would have to be behind my hand to make this equation true?”. They usually say “9″, so I have to ask, “or…..” to which they say negative 9. While they are saying that, on the board with my other hand I write
  “________ = 9 or _________ = -9″. Then I fill in the blank with the x +5 and we solve each equation. Since we begin with easy ones such as |x| = 2 and the answer is x = ±2,  sometimes when they solve x+5=9 and get 4 they assume that the solution to x+5=-9 would be x = -4.
• The students will change |-4 + 5| to 4 + 5. I tell the students that the absolute value sign works just like parentheses in that you have to do the operations on the numbers inside first, before finding its absolute value so that |-4 + 5| = |1| = 1.
• Domain seems to make sense to most kids (although many write -2 to 6 rather than -2 ≤ x ≤ 6) but they have a difficult time with the range. I’ve been drawing rectangles around the entire function so that the domain and range can be seen more easily.

7.7 Review Sheet #3

7.7 – Chapter Review (Day 2)

Chapter 7 Review Sheet (Day 2)

7.7 – Chapter Review (Day 1)

7.1 – Secret Codes

• Investigate the concept of function through secret codes

7.2 – Functions and Graphs

• Learn a definition of function
• Learn about properties and geometric representations of functions

7.3 – Graphs of Real-World Situations

• Construct and interpret graphs that describe real-world situations
• Learn the terminology of independent and dependent variables
• Describe, read, and interpret graphs of real-world situations using the terms linear, nonlinear, increasing, decreasing, rate of change, continuous, and discrete.

7.4 – Function Notation

• Learn function notation
• Evaluate functions by substitution, by using the graphs drawn by hand, and on the graphing calculator
• Write and interpret functions that describe real-world data

7.5 – Defining the Absolute-Value Function

• Investigate the concept of absolute value
• Construct and interpret graphs of absolute-value functions
• Learn the piecewise definition of the absolute value function
• Evaluate expressions containing absolute values

7.6 – Squares, Squaring, and Parabolas

• Learn about the squaring and square root functions
• Graph parabolas
• Compare the squaring function with other functions
• Relate the squaring function to finding the area of a square

For review I’ve scheduled three days since I went through the chapter rather quickly.

Chapter 7 Review Sheet (Day 1)

7.6 – Squares, Squaring, and Parabolas

Very similar to yesterday’s lesson. I stressed that the number of solutions to the equations will either be two, one, or zero, just like with the absolute value function. I also told them many times that giving me one of two solutions is not sufficient.

For homework, I only gave 1 – 4.

7.6 Notes

7.5 – Defining the Absolute-Value Function

Homework: 1 – 4, 11, 16

Notes Worksheet

7.4 – Function Notation (Day 2)

Because of state MCA testing and kids coming in late (and some classes were canceled) instead of going on to do 7.5 I gave the students the 7.4 Function Notation Extra Skills worksheet and had them do it as a group. I gave them a answer sheet to write all the answers on and graded each of them.