Day Three

What standards were addressed yesterday?

MELT wiki: melt-institute-resources.wikispaces.com

Apply a dilation by a factor of 2, centered at P, to the picture.

*You can measure, but can you do it without measuring?

Apply a dilation by a factor of 2, centered at P, to the picture.

What about a dilation by a different factor? How could we apply it?

a. In the picture, locate the images A’, B’, and C’ of the points A, B, and C under this dilation.

b. Based on your picture in part a, what do you think happens to the line (let’s call it l when we perform the dilation)?

c. Based on your picture in part a, what appears to be the relationship between the distance A’B’ and the distance AB? How about the distances B’C’ and BC?

d. Can you prove your observations in part c?

Why does measuring not count as a geometric proof?

What does this task allow us to conclude about similar triangles?

Are They Similar?

In the picture on the handout, line segments AD and BC intersect at X. Line segments AB and CD are drawn, forming two triangles.

In each part of the problem, determine whether the given assumptions are enough to prove that the two triangles are similar. If so, give the correspondence of the vertices and prove it by describing the sequence of similarity transformations

Are They Similar?

a. The lengths AX and XD satisfy the equation 2AX=3XD.

b. The lengths AX, BX, CX, and DX satisfy the equation AX/BX = DX/CX.

c. Lines AB and CD are parallel.

d. Angle XAB is congruent to angle XCD.

Break Time

But before we go, what Math III standards were addressed this morning?

What are Trigonometric Functions?

If similar triangles have constant ratios between sides, then there is a relationship between angle measures and side length ratios.

In a right triangle, it only takes one non-right angle to determine the ratios. Why?

In the book Life of Pi, the protagonist is stranded on a lifeboat adrift at sea. He finds a survival guide on the lifeboat that says you can see about 2-3 miles to the horizon. Is that true? If you’re stranded on a lifeboat, how far can you see to the horizon?

*The radius of the earth is 3959 miles

Lunch Time

But before we go, what Math III standards were addressed this morning?

Cognitive Demand

Lower-Level Demand

Memorization•Reproduce learned facts, rules, formulas, or definitions

•No ambiguity or connection to concepts or meaning

Procedures Without Connections•Algorithmic, use of procedures.

•No connection to concepts or meaning

•No explanations

Procedures With Connections•Attention focused on using procedures to develop deeper understanding

•Multiple representations

•Engagement with concepts underlying procedures

Doing Mathematics• Complex

nonalgorithmic thinking

• Exploratory, need analysis and synthesis of relevant knowledge

• Self-monitoring required

Higher-Level Demand

Low-Level Demand

Smith & Stein (1998)

Factors Associated with Maintenance of High-Level Cognitive Demand

• Scaffolding of student thinking and reasoning is provided

• Students are given the means to monitor their own progress

• Teacher of capable students model high-level performance

• Teacher presses for justifications, explanations, and meaning through questioning, comments, and feedback

• Tasks build on students’ prior knowledge• Teacher draws frequent conceptual connections• Sufficient time is allowed for exploration

Factors Associated with Decline of High-Level Cognitive Demand

• Problematic aspects of the task become routinized• The teacher shifts the emphasis from meaning,

concepts, or understanding to the correctness or completeness of the answer

• Not enough time is provided to wrestle with the demanding aspects of the task, or too much time is allowed and students drift into off-task behavior

• Classroom-management problems prevent sustained engagement in high-level cognitive activities

• Task is inappropriate for a given group of students• Students are not held accountable for high-level

products or processes

What is a Radian?

• Draw a circle, and mark its center (by the way, how would you find the center if you didn’t already know it?)

• Bend a pipe cleaner equal to the length of the radius. Be as accurate as possible.

• Mark a starting point on the outside of the circle, and use the pipe cleaner to measure an arc length equal to the radius. The resulting angle has a measure of one radian

What is a Radian?

• Continue laying the string around the circumference of the circle. How many does it take to go around? Be as precise as possible.

• What does this tell us? What can we deduce from it?

Building Graphs of Trig Functions

• If we can define angles larger than 90 degrees (π/2 radians), how can we define trigonometric functions of them?

• Can we graph the function?

Break Time

But before we go, what Math III standards were addressed this morning?

A Ferris Wheel revolves every 10 seconds and it’s center is 40 feet high. The lowest point clears the ground by 15 feet. The wheel revolves every 10 seconds. Can you model the height of a rider over time? Assume the rider begins at the lowest position and moves counterclockwise.

What if it’s a double Ferris wheel, where the middle bar has the dimensions given earlier and the small wheels at either end of the bar have a diameter of 20 feet and revolve every 7 seconds.

Wrap-up

What Math III standards were addressed this afternoon?

Write down:I have learned …

I wonder …I wish …