**Next Quarter**

The class next quarter will be on 10 Monday nights same
time

(5:00 PM -8:15 PM) same place (Munroe School)

starting Monday, March 30 and ending Monday, June 8. Class

will not be held on Memorial Day, Monday, May 25 or Monday

April 6.

**Today**

inequalities in two
variables

‘conjugates’ and rational/ real coefficients

Yao and what is ‘ solving an equation ’

Harel and ‘guess, check, and generalize’

minimax problems

evaluations

questions

exam discussion

**Homework analysis**

1. Solve the fol lowing system inequalities graphically
using your

graphing calculator and turn in a sketch of the graph with the

region satisfying the inequalities shaded.

What sort of object is
the solution?

How can it be described? What are the
intersection

points?

calculator screen size; use z-box

**Difference of Squares - conjugates**

CME 665: 12, 13

Extend to i. Return to 664: 10, 11

**Imaginary roots and quadratics **

Ask someone to explain the correct solution to CME 664

numbers 10-11.

**Yao**

Consider the following argument.

Let H be your height, Y be Yao’s height and A be the average

of the two heights, (H + Y )/2. Then H + Y = 2A so that:

or

or

or, adding A ^{2} to both sides of the equation

Thus, H − A = Y − A or H = Y . You are indeed as tall as

Yao! If you do not think you can ever be as tall as Yao, find

the f law in this argument .

**What is a written solution of an**

equation/inequality

It is a series of deductions about any number(s) that
might

satisfy the

equation

inequality

system of equations

system of inequalities

**Moral**

If the solution is correct, each line implies the next.

In the case of Yao, the conclusion that if two numbers have the

same square they are equal was incorrect.

In other cases, (next quarter), the implication may be correct

but The two statements may not be equivalent .

So the result is a candidate solution.

Any solution will satisfy the last line.

Not every number that satisfies the last line has to be a

solution.

**Harel**

See handout. Taken from Harel’s DNR as a conceptual

framework.

**Maximizing area**

CME 710 problem 8

Look at problem 10 on that page.

Make up a similar but more complicated problem where there

are different constraints on the perimeter (including internal

walls) but the area is maximized.

**Course Summary for Exam **

Each topic may be addressed either in a concrete situation
or

as pure mathematics problem.

Solution of systems of linear equations .

Graphs of linear equations

modeling

What does it mean to have no or infinitely many solutions

Solution of systems of linear inequalities ( in one unknown )

Solution of systems of linear inequalities (in two

unknowns)

absolute value including solutions of inequalities and

equations in one or two variables

Finding exact solutions; finding approximate solutions

graphically

Normal forms for linear and quadratic equations

Properties of quadratic functions and their graphs

Solutions of quadratic equations (factoring, completing

the square, quadratic formula )

representing functions by formulas, graph, table

Transformation of quadratics and the effect on the graph

understanding the logic of equation/inequality solutions

the zero product property and other properties of the real

numbers

minimizing or maximizing quadratic functions

the function notion

Find at least one problem from the homework or from the CME

text that addresses each of these topics and solve it. Some of

the topics are broad enough so that you should look at several

different problems. Look for connections between the various

topics. Try to see how the same concrete situation can be

solved using different topics.