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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.

7.4 – Function Notation (Day 1)

Rather than working with the graphs in the book, I decided to make my own since I like to tell the students that one reason we use function notation rather than “y=” is because if there is more than one function, you don’t want to have to say “the graph on the top” or “the second list”. So I put a lot of problems using function notation on the board for the kids to go through – r(4), g(10), what t value makes b(t) = 100, etc. I did discuss old vocabulary throughout the discussion such as “is this a function?”, “what is the domain?”, “what is the range?”, “continuous or discrete?”….

(click on the image)

Then I went through the example on page 414 for them. They still have trouble recognizing when they have to work backward to find a value of x given a value of y.

Homework: 1 – 7, 10

A better version on the graph above is here.

7.4 – Graph for Presentation

7.3 – Graphs of Real World Situations (Day 3)

Quiz Day

I went over the homework first – I wrote every answer on the board. I spent time writing the inequalities out so that the kids understand that “2 – 8″ isn’t a good way to write a domain. The students still have a difficult time with questions where they have to name the dependent and independent variables from a story.

They also have problems figuring out when to write a single number for an answer (when is it at its highest level, rising the fastest) compared to using inequalities (when it is decreasing) like in question #8.

I then put a series of hand-drawn graphs under the camera. I asked the following for each of them:

• Is the relation a function?
• Is it increasing or decreasing (or neither or both)
• What is the rate of increase/decrease? Constant?
• What is its domain?
• What is its range?
• Is it continuous or discrete? (Discrete graphs have domains and ranges that are lists, so we went through that together)

The quiz takes only a few minutes, so I spent a good time reviewing before the quiz

7.3 Quiz (two versions)

7.3 – Graphs of Real-World Situations (Day 2)

I went over the two “real world graphs” in the textbook and focused on the definitions from yesterday. I wanted to spend a bit of time writing the inequalities, which I know that the students struggle with. So, using the graph on page 404, I wrote the inequalities on the board like:

Decreasing: 0 < t < 6

Increasing: 6 < t < 11

Decreasing 11 < t < 16

The book states that the inequalities should be ≤, but at the points 0, 6, and 11, the graph is neither increasing nor decreasing so they should be left out.

I assigned problems 2 – 6, 8 for homework. I’ll be giving a quiz on sections 7.1 to 7.3 tomorrow.

7.3 – Graphs of Real-World Situations (Day 1)

I began the lesson by putting the four sets of two vocabulary words on the board like this:

Linear		Continuous
Nonlinear		Discrete
Increasing		Dependent Variable
Decreasing		Independent Variable

and I went through each pair, except the last one in a little detail. I also threw in that all functions don’t have to be increasing or decreasing and asked the kids what else could happen so that I could get them to say something about horizontal lines and slopes of zero.

Then I handed out the worksheet “7.3 Notes (Definitions)” and we went through them one-by-one, which was an exercise that they liked, although it took a little longer than I thought. And that was fine because I was putting their sketches underneath the document camera for others to look at. Since I was constantly telling them to draw functions, I was picking up examples of sketches that weren’t functions to point them out.

After that exercise, they turned their papers over and I went through the PowerPoint notes on dependent and independent variables. For each situation, the students had to determine which was what variable – “Does the temperature of the milk depend on how much time it has been on the counter or does the time spent on the counter depend on the temperature of the milk?” Then together we decided whether the graph would be linear or nonlinear, increasing or decreasing, and continuous or discrete. Then we labeled the axes and sketched the graph of each situation.

I didn’t finish that part, so we will finish the section tomorrow and then assign the homework.

7.3 Independent-Dependent PowerPoint

7.3 Notes (Definitions)

7.2 – Functions and Graphs

I began the class by going over the homework. I left the definitions on the board from the previous day which I kept pointing and referring to.

For the lesson, I had the students look at the table on page 396 and come up with the rule on their own, which I then talked with them about the equation and the graph. For the graph, I was picking points on it and writing them down so that the kids would understand that the y value was one more than the x value. I didn’t spend any time at all on the “diagram”…what’s that?! I did say that instead of x and y they wrote “domain” and “range” respectively.

Then I spent a good deal of time on the investigation. For Table 1 and Table 2 we derived the equations ourselves and I did my best to make them understand which were functions and which were not. I would say, “I’m going to give you an input, and you give me the output”. So I’d yell, “1″ and they would say “3″. If they could say more than one output for my input, then we would say that we don’t have a function. But I did spend lots of time on this, again referring to the definition on the board. I also brought up the word counterexample over and over so that the kids would be able to give me any input that made the relation a non-function.

Then I told them that their goal by the end of the lesson would be to tell whether a relation was a function or not by only looking at the graph of the relation. So I scrolled down the online book to the graphs in the investigation.  Pointing to the counterexamples they came up with above, I showed them while holding a metre-stick vertically that there were two points where a certain input was chosen. And that’s where we came up with the vertical line test.

After that, I put the nine graphs of relations on the screen from my PowerPoint and I called on kids to tell me whether or not they were functions. After that, I assigned the homework. For my last two periods, I had to help them graph 1a…their graphing skills are quite poor. I think that the decimals in the problem really threw them for a loop.

7.2 Vertical Line Test Presentation

Homework:

• 1-4 (need graph paper)
• 6 (vertical line test)
• 7 (different way of finding out it students intuitively know what a function is – counterexample is defined)
• 10, 11 (difficult – need graph paper)

7.1 – Secret Codes

I began with the idea of what a letter shift code. I put a simple Excel spreadsheet up with two rows of letters:

Input

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

Output

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

Then I took the second row of letters and slid them to the right

Input

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

Output

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

and then cut the end of the W-Z off the second row and attached it to the beginning of the row.

Input

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

Output

U

V

W

X

Y

Z

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

Then I began with the investigation. I did the first two steps by having the students think of a three-,four-, or five-letter word and code them after I did one example. They then gave them to a partner and had them decode them. I did not have the students create their own letter-shift grid because I thought that it would be too time consuming. Then I had the students do another word using the grid on page 390, which most students realized that there “was something wrong” with.

From there I wrote the definitions for relation, function, domain, and range on the board. I did relation and function first and then we used the examples on page 391 to explain the terms a bit more. I also wrote the equation y = 2x on the board and explained why that was a function. Once I had finished that, I defined domain and range on the board and we wrote the domains and ranges of the tables in the example using set notation.

For homework, I gave:

• 1 – 6 (codes),
• 13, 14 (domain and range), and
• 17 (review of exponent properties)