Investigation #1 - Post by February 6th

For this first investigation, you will be using graphing technology to examine multiple graphs, make conjectures about your observations, and use reasoning to either prove or disprove your conjectures. You can use any of the TI graphing calculators (83, 83+, 84....) or emulator (free download for a PC), Geometer's Sketchpad, Geogebra, or other online graphing software to help you complete your first write up.

Before you decide on which problem to work on, I ask you to examine all of them.

SELECT TWO PROBLEMS TO WRITE UP AND POST TO YOUR WEB PAGE.

1) Investigating linear equations

Graph the following on the same axis:

Part 1.

y = 5x + 4

y = 5x + 3

y = 5x + 2

y = 5x + 1

y = 5x + 0

y = 5x + (-1)

y = 5x + (-2)

What are your observations?

What do you think students would notice?

If you were to present these equations in a lesson, what do you think your objective(s) are? How did you decide on the objective(s)?

Think about the difference between graphing all of them at once verses one at a time.

Use GSP or Geogebra to graph an animation that includes all of the graphs above.

Part 2.

y = 5x + 6

y = 4x + 6

y = 3x + 6

y = 2x + 6

y = x + 6

y = 0x + 6

y = (-1)x + 6

y = (-2)x + 6

What are your observations?

What do you think students would notice?

If you were to present these equations in a lesson, what do you think your objective(s) are? How did you decide on the objective(s)?

Think about the difference between graphing all of them at once verses one at a time.

Use GSP or Geogebra to graph an animation that includes all of the graphs above.

Part 3.

y = (x - 2) + 2

y = (x - 2) + 1

y = (x - 2) + 0

y = (x - 2) + (-1)

y = (x - 2) + (-2)

What are your observations?

What do you think students would notice?

If you were to present these equations in a lesson, what do you think your objective(s) are? How did you decide on the objective(s)?

Think about the difference between graphing all of them at once verses one at a time.

Use GSP or Geogebra to graph an animation that includes all of the graphs above.

Part 4.

y = 3(x - 2)

y = 2(x - 2)

y = 1(x - 2)

y = 0(x - 2)

y = (-1)(x - 2)

y = (-2)(x - 2)

What are your observations?

What do you think students would notice?

If you were to present these equations in a lesson, what do you think your objective(s) are? How did you decide on the objective(s)?

Think about the difference between graphing all of them at once verses one at a time.

Use GSP or Geogebra to graph an animation that includes all of the graphs above.

2) Investigating quadratic equations

Graph the following on the same axis:

Part 1.

y = x² + 3x

y = x² + 2x

y = x² + 1x

y = x² + 0x

y = x² + (-1)x

y = x² + (-2)x

What are your observations?

What do you think students would notice?

If you were to present these equations in a lesson, what do you think your objective(s) are? How did you decide on the objective(s)?

Think about the difference between graphing all of them at once verses one at a time.

Use GSP or Geogebra to graph an animation that includes all of the graphs above.

Part 2.

y = 3x² + 3x

y = 2x² + 2x

y = 1x² + 1x

y = 0x² + 0x

y = (-1)x² + (-1)x

y = (-2)x² + (-2)x

What are your observations?

What do you think students would notice?

If you were to present these equations in a lesson, what do you think your objective(s) are? How did you decide on the objective(s)?

Think about the difference between graphing all of them at once verses one at a time.

Use GSP or Geogebra to graph an animation that includes all of the graphs above.

3) Family of Quadratic Equations

Part 1. Examine the graph of y = ax² + bx + c for different values of a, b and c (where a, b, and c are rational).

Try fixing two of the values and changing the values of the third.

• What do you notice?

• Why does the graph move the way it does?

• Is it possible to change two or third values at once?

Part 2. Examine the equation y = ax² + (a² - a)x + a² , when a is rational.

• What do you notice if you graph a series of these parabolas on the same axis (a = -2, a = -1, a = 0, a = 1, a = 2, a = 3)?

• Is there anything you can say about the vertex of the equation as you change the a values?

4) Working in a Cuisenaire Environment #1

Part 1. The figures below represent a pattern. Use the Cuisenaire environment to recreate the pattern and determine what the next stages would look like.

• How can you describe the pattern?

• Translate the pattern into numbers and generalize.

• How does this task relate to more complex mathematical tasks?

Part 2. The figures below represent a pattern. Use the Cuisenaire environment to recreate the pattern and determine what the next stages would look like.

Stage 1 Stage 2 Stage 3 Stage 4

• How can you describe the pattern?

• Translate the pattern into numbers and generalize.

• How does this task relate to more complex mathematical tasks?

5) Working in a Cuisenaire Environment #2

Part 1. The figures below represent a pattern of squares.

Stage 1 Stage 2 Stage 3 Stage 4

• Recreate the pattern with more squares and describe how it is made.

• Describe some patterns within the pattern.

• Determine if there is other way to make different size squares. Discuss the new pattern if there is one.

Part 2. A fault-line is when you can split a figure into pieces that have congruent areas and are made up of the same components.

Here is an example of a 6 by 3 rectangle that has a vertical fault-line.

• How is a fault-line different then a line of symmetry? Provide examples

• Is it possible to make a figure with a fault-line from six, 2 by 1 rectangles? If so, show as many figures as you can.

• Is it possible to make a rectangle without a fault-line from one white (1 by 1) and four red (2 by 1)?

• Can you make a rectangle without a fault-line from red (2 by 1) and green (3 by 1)?

• Generalize that you are finding out about rectangles without fault-lines. Provide support.

6) Investigating Exponential functions

The general equation for an exponential function is: f(x) = a*b^(c*x+d) + e, where a,b,c,d and e are rational

Investigate how each constant (a, b, c, d, and e) change the behavior and appearance of the function. Hint: Only change one constant at a time by choosing values for the other two. Example: f(x) = a*3^x + 4 and investigate how the graph changes as "a" changes. Make sure to look at many different rational values.

7) Logarithmic functions and properties

The general equation for an logarithmic function is: f(x) = a*log(b*x +c) + d, where a,b, c, and d are rational

Part 1. Research and explain what logarithms are and why they are useful.

Part 2. Investigate how each constant (a, b, c, and d) change the behavior and appearance of the function. Hint: Only change one constant at a time by choosing values for the other three. Make sure to look at many different rational values.

8) Interesting exponential functions

i) What does y = x^x represent and look like? Are there any negative values defined in the domain or codomain?

ii) What does y = x^-x represent and look like? How is this different then the graph in part i?

iii) What does y = (-x)^(-x) represent and look like? How is this graph different then the others?

iv) Explore what y = 0^x represents and looks like. Discuss and explain your observations.