Activity: A Careful Balancing Act!
ACOS/TEAM Math Objective: UNIT: 2 Triangles – Classification, Congruency, Similarity and Trigonometry: Identify special segments of triangles, their intersections and their measurement properties.
Audience: Geometry students
Materials: Computers with GSP, wooden triangles, pole to balance the triangles on.
Launch:
Watch this video: http://www.youtube.com/watch?v=owCjDR0vXEQ
As usual, behind every cool thing there’s some math that helps explain how it’s possible. Any object has what’s called a center of gravity – also known as a center of mass or a center of balance. While I won’t be teaching you how to balance plates, I do want us to explore how to balance our favorite shape, what we’ve been talking about for the past unit: triangles.
Exploration:
Webpage:
Send the students to the webpage, which has instructions for the exploration. Students will work as pairs on GSP to discover the centroid of a triangle, while encountering other centers of the triangle along the way.
Share and Summarize:
- Which “center” of the triangle turned out to be the center of balance of a triangle? (centroid)
- How do you construct a centroid of any triangle? (connect each vertex with the midpoint of the opposite side)
- Do you think you could find the center of balance of a triangle if I asked you to? (The question isn’t rhetorical. Each pair will get a wooden triangle, a yardstick, and some sharpies. They must construct the centroid of the triangle and balance it on a vertical pole.)
- Note that along the way, we found four types of centers of a triangle? What were they again? (incenter, circumcenter, orthocenter, and centroid) Tomorrow we will talk about each of these further, and discover their mathematical properties.
Evaluation:
Each pair must answer questions on the webpage, as well as balance their triangle on the vertical pole. Grading will be based on their answers and their construction of the centroid of the wooden triangle.
Two Graphing Calculator Investigations in Algebra I TAKE 2
Presented by Steven Clontz
CTSE 6040, Write-up #1
September 19, 2008
Problem
From the assignment:
Some educators, parents, and mathematicians are concerned that the access of the graphing calculator can be used as a "crutch" instead of developing mathematical understanding.
My task is to choose a unit from either Algebra I or Algebra II and find a way to enrich it with the use of graphing calculators, without letting students merely use it as a crutch.
The unit I chose is Unit 6 from the 2006 TEAM Math Curriculum Guide: Factoring and Quadratics.
Activity 1: Factoring Trinomials
This activity uses the graphing of equations in order to help students learn how to factor trinomials, and help them to recognize that a factors of a trinomial represent the x-intercepts of its graph.
The activity begins with the teacher starting with a factored trinomial. He or she FOILs it out, to result in the expanded trinomial. To reinforce that the two expressions are equivalent, the teacher then asks the class to graph both equations on their calculators.
The equation of y=(x+1)(x+2)=x^2+3x+2
The teacher then asks the class to recall the factoring of natural numbers. He or she claims that just like 3*7=21, and 21 can be factored out to be 3*7, if a trinomial is formed by multiplying two binomials like (x+1) and (x+2), the resulting trinomial can be factored to get those original binomials.
The teacher then explains how to find the factors of a trinomial, taking all the factorizations of the constant term and seeing which ones add up to get the coefficient of the x term. Work is checked by first FOILing out the resulting factorizations, and then by graphing the factorizations with the original trinomials to make sure they have the same graph, as above.
The teacher then poses the QUESTIONS: why do we care? Why does factoring a trinomial help us sometimes? The teacher revisits the original trinomial y=x^2+3x+2. He or she has the class zoom in on the x-intercepts.
The equation of y=(x+1)(x+2)=x^2+3x+2
The class is guided to the conclusion that the x-intercepts of the graph are exactly the numbers that make either factor equal zero by using some, none, or all of the following QUESTIONS:
- What are the points on the graph where the curve crosses the x-axis?
Answer: (-2,0) and (-1,0)
- So what happens when you plug in -2 or -1 into the equation?
Answer: the equation equals zero, because one of the factors equals zero.
- So what do the x-intercepts correspond to?
Answer: the numbers that make the equation equal zero.
The class can then make rough graphs of parabolas with x-intercepts based on the trinomial’s factorization.
Another application of factorization can be illustrated by asking the class what they think the graph of y=(x^2-3x-4)/(x-4) looks like. The class will probably think it’s rather complicated, but then the teacher has the class graph it on their calculators.
The equation of y=(x^2-3x-4)/(x-4) looks like a line!
Why does the graph look like a line? (THIS QUESTION IS RHETORICAL since the class will probably not have anyone clever enough to see the factorization immediately.) The teacher then has the class factor the numerator, to discover that the equation is equal to y=(x-4)(x+1)/(x-4). The teacher then tells them that the (x-4) terms can be cancelled, warning that the resulting equation is only valid when x-4 doesn’t equal zero. The result is that the graph is equal to y=x+1 when x is not 4.
Activity 2: Perfect Square Trinomials
This activity uses simple graphing of equations in order to help students recognize perfect square trinomials. Before the beginning of the activity, the teacher should recall the factoring of trinomials, and how if a trinomial is factorable, the factors represent the x-intercepts of the graph of the trinomial.
Next, the teacher reviews the concept of perfect square numbers - an integer that can be expressed as the square of another integer. The teacher then asks what the class thinks a perfect square trinomial would be, guiding toward the answer that it is a trinomial that can be expressed as the square of a binomial.
The teacher then gives the class a list of second-degree polynomials and asks them to graph them separately on their calculators. The class should not factor the equations yet.
- y = x^2 + 8x + 12
- y = x^2 - 7x + 30
- y = x^2 + 4x + 10
- y = x^2 - 6x + 9
- y = x^2 - 25
- y = x^2 + x + 1
Upper Left: The equations entered into the calculator
Upper Right: Equation 1 graphed on the calculator
Lower Left: Equation 3 graphed on the calculator
Lower Right: Equation 4 graphed on the calculator
The teacher then asks which of the graphs is the graph of a perfect square trinomial. If no one in the class is clever enough to figure it out, the teacher can urge the students to look at the x-intercepts of the graphs. (They need to be sure the display is square in order to keep the image from being skewed vertically or horizontally.) The students should recognize that the graph of y = x^2 - 6x + 9 only has one x-intercept. In turn, if it is factorable, it can only have one unique factor, and it must be (x-a) where a is the x-intercept of the graph.
(At this point, the teacher could also remind the students that equations 1, 5, and 6 may be factorable, but aren’t necessarily as in the case of equation 6. Equations 2 and 3 are not factorable, because they lack x-intercepts.)
The class can then use their usual techniques to factor y = x^2 -6x + 9 to y = (x-3)^2.
The teacher then presents the class with some perfect square trinomials, and asks them to use their calculators to graph them in order to check that they are perfect squares.
- y = x^2 - 6x + 9
- y = x^2 +4x + 4
- y = x^2 -12x + 36
- y = x^2 - 2x + 1
- y = x^2 + (8/3)x + (16/9)
Left: The graph of equation 2
Right: Zoomed-in graph of equation 5
The class can then set up a table of equations and x-intercepts.
Table of Equations and X-Intercepts
Equation y=
|
X-Intercept of the Graph |
| x^2 - 6x + 9 |
3 |
| x^2 +4x + 4 |
-2 |
| x^2 -12x + 36 |
6 |
| x^2 - 2x + 1 |
1 |
| x^2 + (8/3)x + (16/9) |
-(4/3) |
What the class can then recognize is that the x-intercept is the square root of the constant term in the equation, with the sign being the opposite of the coefficient of the x term. Or if they can’t the teacher asks:
- "What is the x-intercept of the first equation?"
Answer: 3
- "What is the constant term of the first equation?"
Answer: 9
- "How are those numbers related?"
Answer: 9 is 3 squared
- Repeat for enough equations that the class recognizes the relation.
Then the class can notice that the x-intercept is also half of the coefficient of the x term.
- "What is the x-intercept of the first equation again?"
Answer: 3
- "What is the coefficient of the x-term in the first equation?"
Answer: 6
- "How are those numbers related?"
Answer: 3 is half of 6
- Repeat for enough equations that the class recognizes the relation.
The class can then take a guess as to the requirements of a perfect square trinomial, leading towards:
- The constant term is a perfect square
- The coefficient of the x term is double the square root of the constant term
The teacher can end by proving this, by expanding (x+a)^2 to get x^2 + 2ax + a^2.
Implications
This section is not applicable according to the criteria set up by the write-up format. That is, it asks to describe what math might be learned from an investigation and how you might set up an activity based on the investigation, both of which are covered in the above Activities explicitly.
References
All activities in this write-up are original works of the author. No outside works were referenced in the making of this write-up.
Man Caught Fondling Himself Near the Bear’s Grave
Authorities were called yesterday as a cemetary caretaker caught a local man with his pants around his knees near Paul William "Bear" Bryant’s grave. He was shortly detained and accused of public indecency, Tuscaloosa County’s Prosecutor’s Office announced Friday.
The accused man, who only identified himself as Billy-Joe, spent nearly twenty minutes next to the legendary football coach’s grave, wearing nothing except a houndstooth hat, a #4 Crimson Tide football jersey, and a pair of ratty blue jeans pulled down to his ankles.
"It’s kind of tragic, really," the arresting officer said. "The guy didn’t really didn’t want to leave his ‘Papa Bear’, and kept screaming as much as we pulled him away."
Despite the obscene nature of the exposure, Tuscaloosa authorities are not pressing charges.
"I mean, can you blame him?" asked Tuscaloosa’s district attorney. "Coach Bryant won us more national championships than I can count! I mean, literally, what comes after three again?"
This is the fourth such incident reported this football season.
