Day TWo

The set of all tangent lines to a parabola is the envelope of the parabola. Here’s another way to create one:

• On a blank sheet of paper, draw an angle (two rays). Create equidistant marks on each ray – you get to choose how far apart they should be, but it needs to be uniform. Number them from the vertex out.

• Connect the largest on each ray to the smallest on the other, then the second largest to the second smallest, and so on.

Which of your tangent lines is tangent to the vertex (let’s call it the vertex tangent)? How do you know?

• Construct perpendicular lines to the rays at the points where vertex tangent intersects the rays (try doing this with just a compass and straightedge – no ruler markings, protractors, or squares). Where do they intersect (let’s call it F) ?

• Estimate the points of tangency on each line. Choose one (call it P) and create a line through P that is perpendicular to the vertex tangent.

• Also create a line segment from P to F. What do you notice about the lines through P? Try and prove any relationships you might notice.

Break Time

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

We’ll reconvene after break in Room 103A

Lunch Time

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

This morning, we constructed perpendiculars. Start by creating a line segment and construct a perpendicular bisector.

Can we prove that this construction works?

So, we used the fact that the points we created to find the perpendicular bisector were equidistant from the endpoints of our segment.

It also didn’t matter which points we created (we could have set our compass to any length bigger than half the segment).

This makes me wonder – is any point on a perpendicular bisector equidistant from the endpoints of the segment?

Work (on your own) on a proof of this fact. Convince yourself, then justify it formally.

Trade with someone else. Read the proof in front of you and identify any issues with the proof.

Productive Discourse Moves

• Waiting• Revoicing• Asking students to restate someone else’s

reasoning• Creating opportunities to engage with

another’s reasoning• Prompting students for further participation –

inviting participation• Probing a student’s thinking

• In whole group, small group, or partner talk

Break Time

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

Find the roots of the quadratic function:

Q(x)=ax2+bx+c

Apply as many of the methods you’ve created as can reasonably be applied.

What about the roots of the quadratic function:

P(x)=2x2-8x+20

TIME Out

What’s a complex number? Where the heck did that come from?

Closure

• A set is closed under an operation * if a*b is in the set for every a,b in the set

• What operations are these number sets closed under?– Natural Numbers {1,2,3,…}–Whole Numbers {0,1,2,3,…}– Integers {…,-3,-2,-1,0,1,2,3,…}– Rational Numbers {a/b|a,b are integers, b≠0}

– Real Numbers

A complex number a+bi can be represented as a point in the complex plane

5+3i

-3+i

7-2i

-1-6i

Real part

Com

ple

x

part

Find the following and plot the results on a complex plane:

1. (2-3i)+(4+2i)

2. (-1+2i)+(5-4i)

3. (2-3i)-(4+2i)

4. (-1+2i)(4+2i)

5. (2-3i)(5-4i)

6. (2-3i)÷(-1+2i)

Create each of the following, if possible:• A quadratic function that has 2 real roots• A quadratic function that has exactly 1 real

root• A quadratic function that has at least 1 real

root and at least 1 complex root• A quadratic function that has 2 complex

roots• A quadratic function that has more than 2

roots

For each one you create, graph the function and show its roots in the plane.

Create each of the following, if possible:• A 3rd degree polynomial function that has

3 real roots• A 3rd degree polynomial function that has

exactly 2 real roots• A 3rd degree polynomial function that has

at least 1 real root and at least 1 complex root

• A 3rd degree polynomial function that has at least 2 complex roots

• A 3rd degree polynomial function that has more than 3 roots

Consider the polynomial function

P(x)=x4-3x3+ax2-6x+14

If (x-2) is a factor of P(x), what is the value of a?

Let P be a polynomial of degree d>0

a. If P(0)=0, show P is evenly divisible by x

b. If P(1)=0, show P is evenly divisible by (x-1)

c. If r is a real number and P(r)=0, show P is evenly divisible by (x-r)

d. Use part c to show that P can have at most d distinct roots.

Wrap-up

What Math III standards were addressed this afternoon?

Write down:I have learned …

I wonder …I wish …