I’m surprised there are no transformation programs/applications for the graphing calculator. Did you focus more on the computer or just not find any for the calculator? Does anyone know of any applications on the calculator?

Trig Functions

Have you thought about how you would teach radians vs. degrees? Would you use the technology for them to explore it? It was something that always seemed to cause problems

Also how did everyone learn the definitions of the trig functions? I learned it as SOHCAHTOA = “soak-a-toe-a” (if you say it it makes more sense)

• What does constructing triangles have to do with ratios, proportions, or percents?

Graphing

• If you only teach the students to graph using technology, what happens when the students have to graph by hand?

Inequalities

• Is there a way to do inequalities in Excel?

It’s also vital to find the right number for your data. If you start to look at too much data it can be information overload. And if there is not enough data no real conclusions can be drawn. I feel like this is where some groups had problems. Some had too much information they were trying to deal with. They were not able to come to a conclusion in the given amount of time. On the other hand there were some groups who didn’t have enough data and were making inferences that could not be backed up by their data. Some groups also had trouble finding the line (or curve) of best fit. This just has to do with knowing what technology was available and how to use it.

I really wish we had more data for snowfall in Alabama than just three cities. While we were able to draw conclusions for each of those cities, it would be more interesting to have a wider range to start to see how the snowfall changes as you move across the state. We came up with formulas that could give an approximation for the amount of snowfall in a given month. When discussing as a class, we realized that our domain was limited to 1-12 to represent each month. We had to limit it to months because the data we were given combined each year to give an average for a given month. (So basically, the year doesn’t make any difference.)

The unit goals as stated in this lesson are:

• Ideas of infinity come to light when considering number and geometry, the worlds of the discrete and the continuous.
• Incommensurability is the idea that there is no measurement unit that fits into some two quantities a whole number of times.
• Incommensurability led to the discovery of irrational numbers.
• Irrational numbers have decimal expansions that never end and never repeat.
• Two sets are the same size if their elements can be put into one-to-one correspondence with one another.
• The size of a set is its cardinality.
• There is more than one type of infinity.
• The sets of rational and real numbers are examples of two different sizes of infinity.
• To properly describe the different sizes of infinity, a new definition of number is required.
• Given a set of any size, one can create a larger set by taking the subsets of the original set.’

The video that goes with this lesson had some important points. It first starts out with history of the idea of infinity including stories that describe infinity. While we haven’t given the name infinity to this insanely large number, we have had this idea for a long time. The video talks about how the Greeks didn’t like the idea of infinity because they used geometry and liked having tangible measurements. Infinity isn’t tangible though.

While looking at the diagonal of a square, the Greeks came up with the idea of an irrational number. However, this diagonal is still tangible.  But when trying to find an exact value, they couldn’t. The decimal for the √2 continued on infinitely, just like for Pi. THis meant that numbers that were “unimaginable” were now able to be written.

Some of the great thinkers of the past have had a hard time working with infinity. Aristotle flatly refused to work with the idea of infinity. He said this was becuase there were something that was greater than the heavens. Galileo also thought that infinity should obey different rules for infinity than finite number

It was John Wallace who introduced the “love knot” or “lazy eight” for infinity.

One important idea that lead to the idea of infinity was the question “Can I count the numbers we count with?” The thing about this is that we can keep counting even numbers or odd numbers. These sets can keep growing. This is the idea of infinity. There is no limit to it. It just keeps growing and getting bigger.

I don’t really feel like the video answered the question that was posed. At no point did they talk about how big infinity was or how to explain that to students. The idea is important for students to understand. It is a big issue that many students struggle over. Infinity keeps coming up though and it is vital that students understand.

1st offense: 1 day of ISS and the phone will be confiscated for 5 days

2nd offense: 1 day of OSS and the phone will be confiscated for 10 days

3rd offense: 1 day of OSS and the phone will be confiscated for 15 days

4th offense: 1 day of OSS and the phone will be confiscated for 30 days

I also find the dress code for different schools very interesting. My high school was strict about the things they cared about but compared to my younger cousin’s school, they were pretty lax. So I found out the dress code for the school I’m working at,

“Students not in compliance will be written a detention or disciplinary form and sent to the office at the time to correc the violation. Time out of class to correct dress code violations shall constitute an unexcused tardy or absense for the student for that class, depending on the amount of time taken to correct the violation. The dress code applies any time a student is on camps, a bus and/or while participating in a school-sponsored event. Pupils are expected to be clean and appropriately dressed for school. Dress and appearance bust not cause disruption or present health or safety problems. Shoes must be worn by all pupils. The following, though not all-inclusive, are examples or inappropriate dress for students:

1. Strapless dresses or blouses
2. Picks and combs in hair
3. Hats, beanies, bandannas, doo-rags, headbands, wraps, or other types of headgear
4. Long chains and/or large medallions
5. Sunglasses
6. “Spaghetti” straps
7. Halters
8. Clothing so skin-tight or revealing to provoke, distract or offend others
9. Cut-off jeans or cut-off shorts of any kind
10. Heels unless approved for field trips or presentations
11. Tank tops (anything less than 2 inches on the shoulders)
12. Belts and suspenders unfastened
13. Clothing with decals or slogans that are provocative, suggestive/offensive, contain profanity, and/or refer to immoral/illegal behavior
14. Clothing with holes, rips, or tears above the knee
15. Tops that expose midriff (when arms are raised completely above the head)
16. Any clothing worn that exposes underwear
17. “Sagging” or loose fitting pants worn without a belt
18. Jerseys cut too low in front or beneath the arms
19. Skirts, shorts, culottes, pants, dresses and similar items of clothing that are not of an appropriate or modest length
20. Sleep clothing, pajama pants, dresses and similar items of clothing that are not of appropriate or modest length
21. Facial piercing
22. Tights or leggings used as pants. Skirts worn with tights or leggings must be appropriate length.

I find the word “modest” interesting in the dress code. This is very subjective and can be taken any way. We also didn’t have anything in our dress code about sunglasses facial piercings. In fact one of my best friend had facial piercings. I guess this is the difference of not being in a small town, southern school. At least this school does not have anything about dyed hair. I have heard of some schools where students weren’t allowed to have hair color that isn’t a natural color. So no pink, green, blue, etc. Again, this was not ever in my school. It was a way for students to express themselves.

After section 3: Solving Multi-Step Inequalities, we created a lesson to review what the students had discovered so far. This lesson is called More Practice Graphing One Variable Inequalities. At this point the students have discovered that inequalities are equivalent if they add or subtract numbers to both sides. They also have found that if they multiply each side by a positive number, the inequality is still equivalent to the previous one. However if they multiply by a negative number, the sign in the inequality must be switched to keep the two equivalent. The students have also learned the definitions of a closed and open interval and solution set.

With this lesson the students explored an interactive program on the internet that covers graphing one variable inequality on a number line to establish set notation for the solution set. They also have learned set notation by using open and closed intervals and the solution set. The TEAM-Math and Alabama Course of Study objectives this lesson covers are:

TEAM-Math: A1f. Solve equations and inequalities including:

• Multi-step linear
• Radical
• Absolute value
• Literal
• Linear systems in two variables
• Factorable quadratics
• Using the quadratic formula

g. Write in set notation and graph solutions of an equation or inequality.

A2. Model real world problems by developing and solving equations and inequalities, including:                inverse and direct variation, systems of equations, and simple number theory.

Alabama Course of Study: 7. Solve multi-step equations and inequalities, including:          linear, radical, absolute value, and literal equations.

a. Writing the solution of an equation or inequality in set notation.

b. Graphing the solution of an equation or inequality.

c. Modeling real-world problems by developing and solving equations and inequalities, including those involving direct and inverse variation.

During this lesson the students will go to the website http://mathcasts.org/gg/enliven/ns/numberline/inequality1/inequality1_index.html (handout is attached). Here they will work with open and closed intervals to see how the properties they discovered the days before affect the intervals.  The first three activities allow the students to explore how the graph of an inequality of a single variable looks on a number line.

This is one of the concepts that students need to understand before they learn to graph inequalities with two variables. In this concept students learn whether or not the dot is open or closed at the point the inequality ends at. This will lead to the discussion on how we represent the line of strictly less than vs. less than or equal to. The last activity in this lesson has the students explore how to graph two inequalities on one number line and how they relate to one another. This is an important concept when we cover absolute values because absolute values use the idea of “and” when graphing on a number line.

The students we were working with hadn’t spent any time in the computer labs. Like I said earlier, they don’t use any technology in their classroom. This gave them a new adventure and a different way of looking at topics we had covered in class already. Different students learn in different ways and the more approaches we can provide, the better for everyone. Even if a student understood the topic the first day we covered it, by going over it again, they get to have more practice with the concept.

Another lesson we came up with for Chapter 6 was to address absolute values. When we first saw this section in the chapter with inequalities we were confused as to why it was included. We then remembered that to graph inequalities on a number line includes properties from graphing inequalities (using the word “and”).  The lesson we wrote is called Absolute Value Inequality Number Line with Geogebra.

The TEAM-Math and Alabama Course of Study objectives covered in this lesson are similar to the first lesson.

TEAM-Math: A1.f. Solve equations and inequalities including:

• Multi-step linear
• Radical
• Absolute value
• Literal
• Linear systems in two variables
• Factorable quadratics
• Using the quadratic formula

g. Write in set notation and graph solutions of an equation or inequality

A2.  Model real world problems by developing and solving equations and inequalities, including:               inverse and direct variation, systems of equations, and simple number theory.

ACOS: 7. Solve multi-step equations and inequalities, including: linear, radical, absolute   value, and literal equations.

a. Writing the solution of an equation or inequality in set notation

b. Graphing the solution of an equation or inequality

c. Modeling real-world problems by developing and solving equations and inequalities, including those involving direct and inverse variation

In this lesson the students will explore features of the absolute value function. According to their teacher, the students have worked with absolute values before. I have noticed though, that just because they have worked with absolute values in the past they do not necessarily understand what they are doing.  The students will also explore basic transformations of the absolute value function. For example they will add, subtract, multiply, and divide different numbers to the function to see how the function changes.

The students will first discuss what they think an absolute value means. This concept is very different from most mathematics concepts and comes up often, so it is important that the students fully understand what is going on.  They will then go to the website http://teachers.henrico.k12.va.us/math/HCPSAlgebra1/Geogebra/Abs_Value_Inequality.html. Here the students have a screen with the absolute value a| x+ b | ≤ c.

There are three sliders to change the values of a, b, and c. There is also a slider to change the sign. The students can make the inequality anything they want or need to. As they change these values and signs, the graph below also changes. Sometimes the graph is one dot (as shown above). Then the students can create an inequality that has no solutions.

They can also create an inequality that has all the real numbers except for one number.

The last graph the students can come up with is a graph of all real numbers.

In this lesson they are to change the numbers and signs and figure out how the graph changes because of those alterations. After everyone finishes the investigations, the class comes back together to discuss the observations they made. They also discuss why they think the graph changed like it did. So far the students have been really good at making observations and connections.

This is another activity that gives the students a hands-on approach to learning absolute values and seeing the graphs. The students can easily change whatever needs to be changed. Every student in the class (even the inclusion students) can take part in this lesson. They can all move the mouse to adjust the numbers. And so they can all make observations to what changes.

At the end of the chapter I prepared a lesson on graphing inequalities. I creatively named this lesson Graphing Inequalities with Calculators. Here the students learned how to graph the solution of an inequality on the coordinate plane, determine the boundary, model real-world problems by solving inequalities, solve multi-step inequalities, and learn how to use a graphing calculator to graph inequalities that have two variables. The TEAM-Math and Alabama Course of Study objectives it covers are:

TEAM-Math: A1c. Analyze linear functions from their slopes, equations, and intercepts: Graph two-variable linear equations and inequalities on the Cartesian plane.

A1g. Write in set notation and graph solutions of an equation or inequality

Alabama COS: 2c.  Graphing two-variable linear equations and inequalities on the Cartesian plane

7b. Graphing the solution of an equation or inequality

8. Solve systems of linear equations and inequalities in two variables graphically or algebraically.

The previous day, the students had learned to graph inequalities on the coordinate plane by hand. This lesson lets them explore how to do it on the calculator. I first go through a quick introduction to the graphing calculator. I wasn’t sure how many of the students had used a graphing calculator before so I wanted to make sure they knew all that it could do. The class then goes into their investigation with their small groups. (handout is at the end) They first try to figure out which side of the graph the shading will be on and if the line is dashed or solid. This reinforces what they had done the day before.  The graphing calculator shades whichever side of the graph that is indicated by the inequality. By using the “Inequalz” application the students can insert a <, >, ≤, or ≥. The calculator can figure out which side of the line needs to be shaded.

Here is a picture of the inequality y<x- 3. Students would be able to see that the line is dashed since it is “less than” and not “less than or equal to.” The graph is also shaded underneath because it is less than as opposed to greater than.

On the other hand, here is a graph of the inequality y≥x-3. Here, the graph is a solid line and the shading is above the line.

Many students have difficulty in finding which side should be shaded. This lesson will help them discover and see why that side is shaded. It also allows students who are struggling with graphing equations stay up with what the class is doing. Students have to have a good understanding of how to graph equations to learn how to graph inequalities. However if there is a student with a disability who was not great at the part before on graphing equations, they will be able to see the graphs for the inequalities.

The final lesson is for the next chapter, Chapter 7: Solving Systems of Linear Equations and Inequalities.  This is when students are given a few equations and find where they are equal. If the students are graphing these equations it is where the equations intersect. The lesson I created for this section was Finding Intersections to Solve Systems of Equations. The TEAM-Math and Alabama Course of Study objectives are:

TEAM-Math: A1: f. Solve equations and inequalities including:

• Multi-step linear
• Radical
• Absolute value
• Literal
• Linear systems in two variables
• Factorable quadratics
• Using the quadratic formula

A2.  Model real world problems by developing and solving equations and inequalities including inverse and direct variation, systems of equations,

ACOS: 8. Solve systems of linear equations and inequalities in two variables graphically or algebraically.

a. Modeling real-world problems by developing and solving systems of linear equations and inequalities

The students will determine whether a system of linear equations has zero, one, or infinitely many solutions. This is the number of times that the lines intersect. If there are no solutions, then the lines do not intersect. This means they are parallel lines. All other lines intersect at least once. This is the time that the system of equations has one solution. For it to have infinitely many solutions, the lines have to actually be the same line. The students will also solve systems of equations by graphing.

The students will first enter equations into their calculators to graph. They will graph each line individually and make a hypothesis as to how many times the lines will intersect. Then they will graph the lines together to see if they were correct. Then using the calculate function, the students will find where the lines intersect.  The students will discuss what this point means and test what they come up with.  The point for the students to notice is that when the x value is put into both equations, the y value is the same.

Here are two views of screens of what the students should be finding. The first is when the two equations are y=x-1 and y=2x+1. The intersection point is (-2,-3). When -2 is substituted in for x in either equation, y=-3.

The second graph is of the lines y=-x+5 and y=2x+1. To test the intersection point (1.333333, 3.6666667) the students would substitute the x value into both equations and get the y-value.

By using the calculator to find solutions to systems of equations, students can find more exact answers. If they graphed the equations by hand, they would have to estimate at where the lines intersect. Sometimes the intersection point can look as if it is at one point but that is not exact. In the last example shown, 1  looks very close to 1 and 3  looks very close to 4. It would not be surprising if students, who did not use the calculator to find the exact values, would guess the solution was (1, 4). When these numbers are substituted in though, the equations are no longer true. The other day in class a student and I discussed how in mathematics, we need to be exact and so we need to use decimals. He wanted to round his answer to the closest number.

References:

Glencoe Algebra 1 Textbook, Chapter 6, 7-section 1 (pgs. 18-374).

Prentice Hall Mathematics Algebra 1 Textbook, Chapter 7- Section 1 (pgs. 343-345).

TEAM-Math. Retrieved November 6, 2009, from TEAM-Math Curriculum

Guide Web site: http://team-math.net/curriculum/index.htm

Handouts

INVESTIGATION: Graphing Inequalities on a Number Line

Algebra 1A

GeoGebra

Name:

Getting Started

• Open a new browser in Microsoft Explore.
• Type in the following web address – http://mathcasts.org/gg/enliven/ns/numberline/inequality1/inequality1_index.html
• Look at Activity 0.  (Be sure to read the description before entering the site.)

Activity 0

• Take a few minutes to read the instructions listed on the left of the number line.
• Once you have read the instructions, take a few minutes to explore any inequality that you like.  Try doing one for each of the inequality signs.  What do you notice about the graph as you are changing your inequality?
• Have you tried looking at the line from a different view?  Re-read step two of the instructions and take note of what happens.  Do you think the line has an end?  Why or why not?
• Try at least two of the examples listed under the inequality input box.

1. Which two did you choose?
2. How did the graphs differ?

• How can you summarize graphing an inequality on the number line?

Activity 1

• Go back to the main menu. (The menu button is located above the number line.)
• Now we are going to look at Activity 1.  (Be sure to read the description before entering the site.)
• Take a few minutes to read the instructions listed on the left of the number line.
• Once you have read the instructions, look at the equation that is already on the number line.
• Going back to steps two and three, what inequality symbol would you use to go in the place of the “=” to represent the red ray?  Check your answer.

1. A pop up box will notify you if you have the correct answer.  (Press OK to close the box.)
2. If you used the wrong inequality symbol, go back and try it again.  Be sure to jot down a few notes on why you choose your first choice and then why you choose your second choice.

• Go back to step 1 and click “New Inequality”.

1. Repeat the step above.  What do you notice about the inequality that was different from the last inequality?

• Do about three more “new inequalities”.

1. What do you notice about each?

• How can you summarize determining the correct inequality symbol by looking at the graph on the number line?

Activity 2

• Go back to the main menu.  (The menu button is located above the number line.)
• Now we are going to look at Activity 2.  (Be sure to read the description before entering the site.)
• Take a few minutes to read the instructions listed on the left of the number line.
• Once you have read the instructions, look at the equation that is already on the number line.
• Going back to steps two and three, which endpoint and arrow direction would you use to graph the inequality?  Check your answer.
• Go back to step 1 and click “New Inequality”.

1. Repeat step 5 above.  What do you notice about the inequality that was different from the last inequality?

• Do about three more “new inequalities”.

1. What do you notice about each?

• How can you summarize finding the endpoints on the line, the point type, and arrow direction of a given inequality?

Shooting Inequalities

• In a different browser window in Microsoft Explore, type in the following web address: http://www.ltcconline.net/greenl/java/BasicAlgebra/inequalityGame/inequalities.html
• Read the directions CAREFULLY!
• After you have read the directions, locate the arrows on your keyboard.

1. The LEFT arrow will move the shooter left and the RIGHT arrow will move the shooter to the right.  The UP arrow will shoot down the segments, dots, and arrows.
2. Brief reminder of interval notation: ß Where have they seen this before? Is this prerequisite or something that you are introducing?

• (a, b) indicates that both end points are not included in the solution set.
• [a, b) indicates that only one end point, a,  is included in the solution set.
• (a, b] indicates that only one end point, b,  is included in the solution set.
• [a, b] indicates that both end points are included in the solution set.
• Begin exploring this site.
• Be sure to write down your solutions, as we will share them at the end of class.
• Once you have finished a round, make notes on how you did.
• Press the “Next Problem” button located in the right corner to try another problem.

INVESTIGATION: Absolute Value Inequality Number Line

Algebra 1A

GeoGebra,

Group Members:

Getting Started

• Open a new browser in Microsoft Explore.
• Type in the following link – http://teachers.henrico.k12.va.us/math/HCPSAlgebra1/Geogebra/Abs_Value_Inequality.html.
• Read the instructions in the box located above the graph.
• Take a few minutes to explore the site.

Let’s explore the equation.

• First, set a = to 1, b = 5 and c = to 0 (1|x + 5|  0) Make sure the inequality sign is set for .

1. What do you notice about the graph and the solution set?

• Next, graph the point 1|x – 5|  0.

1. How did the graph change?
2. Did the solution set change?  How?
3. What did you notice about the first graph that was different from the second graph?  What about the solution set?

• What if we keep the equation in number two and use the slider to change the inequality sign to.  What do you notice about the graph and the solution set?
• Now change the inequality sign to .  What happened?
• What do you think will happen if we changed the inequality sign to >?

Changing the coefficient.

• First, think about what you did in B and how changing the “a” will affect the graph?  The solution set?
• Then, graph the following equation, 2|x – 5| 0.

1. What do you notice about the graph and the solution set?

• Try at least two numbers for the “a”.

1. Write down your observations.
2. What do you notice about the graph?  The solution?

• What happens to the graph if you slide “a” back and forth?
• Change “a” back to 2.  Now, try changing the inequality sign.

1. What do you notice about the graph?  The solution?

• Why do you think the graph is as it is for this inequality?

Changing the constant.

• Go back to the equation 1|x – 5|  0.
• Then, change “b” to equal -3.

1. What do you notice about the graph?
2. What do you think will happen if we changed “b” to equal 3?

• Try at least two different numbers for “b”.

1. Write down your observations.
2. What do you notice about the graph?

• What happens to the point on the graph if you slide “b” back and forth?
• Change “b” back to 3.  Now try changing the inequality sign.

1. What do you notice about the graph and the solution set?

• Why do you think the graph is as it is for this inequality?

Changing the solution.

• Looking at a different view, let a = 1, b=-2, and c=3 using the less than or equal to inequality sign.

1. What do you notice about this graph and its solution that is different from the others?

• Try changing the inequality sign.

1. What do you notice about the graph and the solution?
2. Try the other signs and make observations about what is happening?

• Now let’s see what happens if we changed c =3 to c=-3.  Make observations about the changes, if any.

1. If we changed the inequality sign, will the solution still be the same?

• Using the greater than or equal to sign, what happens to the graph when you slide “c” back and forth?

1. Does this happen with the other inequality signs?

• Taking it a step further, what would happen if we changed “a” to 3?
• Slide “a” back and forth and notice what is happening to the graph and the solution.

1. What has changed in the problem that was different from the previous problem involving “a” in C?
2. Why do you think the problem changed?
3. Has your opinion about “a” changed since then?

Putting it all together.

• Predict what the following graphs will look like.  Then verify your answers using GeoGebra.

1. i.      | x | > 2
2. ii.      5| x + 4 | < 3
3. iii.      | x – 2 |  3
4. iv.      2| x + 1|  4
5. v.      Summarize the effect of a, b, and c on the inequalities listed in #1.
6. vi.      Which of the graphs above demonstrate a union?  An intersection?

• Create two of your own absolute value inequalities.

1. vii.      Was anything surprising about the graph of your inequalities?
2. viii.      Write any final observations and remarks that you have about this investigation.

Graphing Inequalities on a Calculator Investigation

Make sure you follow each step carefully. When you start graphing, make sure you draw the first five inequalities yourself on graph paper (by hand) and then on the calculator to see if the graphs are the same.

1. Find the Apps button. Scroll down till you come to “Inequalz.” Hit enter twice. You are now prepared to enter inequalities.
2. Graph y<x-3 by hand on graph paper. Then do it on the calculator. To get the correct sign, click alpha and then the button that is under the sign you want. To get an x, use the button right next to the alpha key that says “X,T,θ,n.” Are the graphs the same? How do you know these are correct?
3. Every time you will need to clear what you put into Y₁. Now graph y≥3x+1 both by hand and then on your calculator. Are they the same?
4. Graph y>1. Check some of your points using trace to make sure they are the same.
5. Graph x≤5-y. How is the graph different based on if there is the equals part?
6. Graph x+3>y
7. Graph y-x≤5. What do we have to do first to graph this?
8. Graph 2y+x>6
9. Graph 3y-6x≥12

10.  Graph x<4

11.  Graph 5y+3x≤14

Finding Intersections to Solve Systems of Equations

Turn on your calculator. Go to “y=.” Enter in the two equations individually. Hit “graph” and observe the line. After graphing the two equations, hypothesize how many intersections they will have and where they will intersect. Then graph the two equations together.

To find the intersection, push “2^nd” + “Calc” (above the trace button). Select intersect (#5). It then asks for the first curve. This means to be on one line at where you think the intersection is. Click “enter”. Do the same for the second line. Then click “enter” for guess. This is asking if you want the calculator to guess where the intersection is.

Write down the intersection.  What do you think this combination of numbers means? Test that theory.

1. Y= -X+5         Y=X-3
2. Y= -X+5         2X+2Y= -8
3. 2X+2Y= -8     Y= -X – 4
4. Y= -X +8        Y=4X-7
5. X+2Y=5          2X+ 4Y =2
6. Monica and Michael Gordon both want to buy a scooter. Monica has already saved $25 and plans to save $5 per week until she can buy the scooter. Michael has $16 and plans to save $8 per week. How many weeks until Monica and Michael have saved the same amount of money? How much money will each person have saved at that time?
7. We have one equation that is X-2Y=10. Find an equation that gives no solution for that system.
8. Find an equation that gives one solution for the above equation.
9. Find an equation that gives infinitely many solutions for the above equation.

10.  Create your own system of equations for your classmates to solve. Make sure you know what the answer will be and can show why.

This school has four computer labs for the teachers to reserve. They have to be reserved in two weeks in advance. Availability is only dependent on who gets the computer lab first. Each computer lab has about 30 computers since that is the most that the classes have. The computers have Excel and GSP which we explored in our classroom. There are also two sets of mobile computer labs. These are sets of 30 laptops that can be reserved for a classroom. These computers only have GSP on them though so they can only be used in a geometry classroom.

Here is an example of the type of calculators our classroom has. As you can see, this is not a graphing calculator. However it is more than just a four-function calculator. It can be used to find trig values, square roots, exponential values, along with the traditional functions of a calculator. These calculators are useful depending on what you want to use them for.

A smartboard is pretty an impressive piece of technology. It can be used to project a video onto the screen for the entire class to see. It can also be used as a black board. The teacher and students can write on it as if it were a regular board. In other classes, we have used it to put up data for everyone to copy down. Teachers are also able to write notes and save them. This way if the teacher has a website or blog they can post the notes they completed that day with the class. Then if any student missed the class or were confused about something, they can go back and look. It is actually kind of sad that my cooperating teacher does not have a smartboard. He puts up notes every day in class and works through the problems for the students. The students would then be able to see what they missed and review what they did in class.

My teacher does not use any technology in his classroom. This is mainly because the technology is not easily accessible. He also does not allow his students to use a calculator. This has to do with the fact that his students do not know how to work with the numbers. They still have problems working with negatives and he obviously wants them to learn how to deal with this part of mathematics before using calculators. For example, the students do not know how to add or subtract negative numbers. Using a calculator will not help this stumbling block though.

We first looked at wolframalpha. Here you can put in the term “plot ____” where you put in a function. This doesn’t require actual graphing on the students’ part but does let them have a quick glance at what the graph will look at.

The next technology we used was a graphing calculator. Here students can put in each equation as long as it is in the form of “y=___.” The students can find an intercept if they are graphing more than one equation. They can change the screen window as they like. Graphing calculators are something that can most likely be acquired for each student and let them have something at their hands. It is a chance to see each equation as it is entered.

Excel was the third form of technology we used for graphing. This is different from the graphing calculator. Here students will have to have entries to graph. Excel takes the entries from each cell and plots them. I feel like it’s sometimes more trouble than it’s worth. It depends on which entries you have in which cells for the graph to come out correctly. I’m not a fan of excel graphs.

The final technology we used was Geometer’s Sketch Pad. I really liked playing with this program. The size of the cells can be changed so we can get a better view of how the graph acts. The program also allows you to change certain parameters. Then through the animation, you can see how those changes change the graph. It’s a lot of fun to play around with and I feel like students would find it really interesting without realizing it’s math that they’re enjoying (which is hard for some students). The only problem with GSP is that you can’t find an exact intersection point between two functions. You can estimate a point, but you’ll never be completely sure it was the exact point of intersection.

One thing I’ve always wondered about statistics is how trustworthy they are. I know there are some times that I’m asked to fill out some form, and since it is taking up my time that I need to be working on something else I do the easy answer and not always my true answer. So does everyone realize that when they’re collecting data? Or when we read data in papers or journals, has it taken those feelings among people into account?

I was talking to my roommate about how she wasn’t being productive because she was wasting time on the computer. I started wondering how much time do college students spend on the computer? And how is that time divided up between the more popular sites, such as Facebook, internet, game sites, and sites to watch t.v. online? The internet has become such a large part of our lives, especially those of us in college, so I’d like to see how much of our lives it takes up.