# Tuesday, Mar 21

In class today the lesson focused on the relationship between the Pythagorean Theorem and the equation for a circle. Students relate the distance between $\dpi{300}\inline (a, b)$ and $\dpi{300}\inline (x, y)$ to the pythagorean theorem, and realize that $\dpi{300}\inline d = \sqrt{(x - a)^2 + (y-b)^2}$ and then we note that the equation of a circle whose center is at $\dpi{300}\inline (a, b)$ and which has a radius of $\dpi{300}\inline r$ is

$\dpi{300}\inline r^2 = (x - a)^2 + (y-b)^2$

or

$\dpi{300}\inline r =\pm \sqrt{(x-a)^2 + (y-b)^2}$

Note, I just now realized that I did not include $\dpi{300}\inline \pm$ in the notes from class.

Handed back today: Chapter 8 Assessments on Criteria A and C. Students received two scores on the latest assessment.

Today's assignment: 9.5 on p. 489: 1-12

Upcoming: Assessment on Chapter 9 at end of week. See me if you plan to be absent on Thursday/Friday.

# Tuesday, Mar 21

Hi All –

In class today, I passed back a number of tests which I graded over the weekend and were on the progress reports yesterday. We also studied properties of reflections and perpendicular bisectors. Students took notes on how to find the equation of a perpendicular bisector line between any two points. This relates to reflections (the reflection line is a perpendicular bisector between any point and its reflection) and the perpendicular bisector can also be used to find the center of rotation.

Question: How would you find the equation of the perpendicular bisector line for the points A(4, 2) and B(2, 7)?

Answer: First find the slope between these two points $\dpi{300}\inline \dfrac{7 - 2}{2- 4} = \dfrac{5}{-2} = -\dfrac{5}{2}$, now note that the slope of a line perpendicular must be the negative reciprocal slope, so our perpendicular bisector will have slope $\dpi{300}\inline -\left(-\dfrac{2}{5}\right) = \dfrac{2}{5}$.

Now, we need to find the midpoint of A and B – so we find the average of their x and y- coordinates: $\dpi{300}\inline \left(\dfrac{4+2}{2}, \dfrac{2+7}{2}\right) = (3, 4.5)$

Ok, so we know the slope of the perpendicular bisector and one of the points on the line, we can use point-slope form to write the equation of the line in the form y = m(x – x1) + y1

$\dpi{300}\inline y = \dfrac{2}{5}(x - 3) + 4.5$

If we want this equation in slope-intercept (y = mx + b) form, we just need to simplify the equation above.

$\dpi{300}\inline y = \dfrac{2}{5}(x) + \dfrac{2}{5}(-3) + 4.5$ (distribute the 2/5)

$\dpi{300}\inline y = \dfrac{2}{5}x - \dfrac{6}{5} + \dfrac{9}{2}$ (simplify 2/5 times -3)

$\dpi{300}\inline y = \dfrac{2}{5}x - \dfrac{12}{10} + \dfrac{45}{10}$ (common denominators for like terms)

So, finally, in slope-intercept form, the perpendicular bisector between points A and B is

$\dpi{300}\inline y = \dfrac{2}{5}x + \dfrac{33}{10}$

or

$\dpi{300}\inline y = 0.4x + 3.3$

Today's assignment: is to complete the "Reflections are neat!" packet for tomorrow.

# Wednesday, Mar 8

In class today we learned about the relationship between central angles, inscribed angles and arcs.

Assignment: 6.3 PYS worksheet

# Wednesday, Mar 8

In class we reviewed for the module 5 test tomorrow. Today's assignment is a review worksheet and students had their choice of what they chose to work on.

Topics on the test

• Solving systems of equations by graphing, substitution or elimination
• Solving systems of inequalities
• Solving contextual problems which can be represented by systems of inequalities/equations (including profit problems)

# Tuesday, Mar 7

In class we worked on a few maximum profit problems. Tonight's assignment involves solving a problem about the proper amount I spend sleeping and awake.

Mr. Hershman's Get-Enough-Sleep Dilemma!
====================================

Mr. Hershman is a busy, busy person (at least he thinks he is). He has to divide his time after school between sleeping and taking care of his kids, cooking, cleaning, and tending to the home front, spending time with his wife, keeping up with his friends, taking pictures at the climbing gym, and writing math problems – let’s just call it _____ (choose a description of Mr. Hershman's non-sleeping activities to fill in this and all remaining blanks).

Determining how to balance his time between _____ and sleeping is a difficult task and he needs your help!

To begin with Mr. Hershman has a basic allotment of time each day he spends outside of school which is roughly from 4pm until 8am, so his total _____ and sleeping time falls within this interval. Each hour spent _____ he accomplishes 5 units worth of work and he decides to place a value of 3 work units on each hour spent sleeping. He must not let his total number of work units fall below 30 for any day because then he would feel “behind” and that’s not good.

It’s also important for Mr. Hershman to feel rested, he decides that time spent _____ can be restful in the sense that productivity can create the space for relaxation, so he also decides to count each hour _____ as 1 restfulness and each hour spent sleeping as 3. He’s a busy person though so he can’t accumulate more than 34 restful points in any day without feeling lazy. And, yup, laziness is no good either.

As if being productive and restful weren’t all, it is important that he maintain some ability to converse with other humans. Being awake / doing the things that encompass _____ yield roughly 5 topics of possible conversation per hour. While each hour of blissful sleep Mr. Hershman snoozes causes him to forget 3 topics. Caution is important though, too many topics of conversation can make it difficult to discuss anyone in particular, and so he chooses 40 as his arbitrary limit for the number of topics of conversation that he should accumulate in any single day.

So, in the end, Mr. Hershman wants to balance his total hours between ______ and sleeping. He must pay attention to his productivity, restfulness, and conversation. Ultimately he believes that each hour ______ should be given  a weight of 3 happiness units and each hour spent sleeping should be given a weight of 5 happiness. How many hours should he spend sleeping and how many hours should he spend _____ in order to maximize his happiness???

Instructions: Define two variables, write a set of inequalities to represent the constraints, plot the inequalities carefully on the graph, identify the feasible region and determine the point of maximum happiness. Can you think of a real life dilemma faced by a friend or family member – a situation in which two or more options must be put in balance? Create your own optimization problem.

This problem is tonight's homework.

# Monday, March 6th

In class today students had about 1/2 hour to work silently on the chapter review assignment and then time to compare. I answered questions on a few area problems. One nice question was whether a square peg in a circular hole is a tighter fit than a circular peg in a square hole? The ratio of the area of a square peg to the smallest round hole it can fit through is $\dpi{300}\inline 2 : \pi$ which we can reason must be less than $\dpi{300}\inline 66 \dfrac{2}{3}\%$ while the ratio of the area of the round peg to the smallest square hole's area is $\dpi{300}\inline \pi : 4$ which we can reason is greater than 75%. So, the round peg in a square hole is a tighter fit.

Study Guide

• Be able to explain the area formula for any 3- or 4-sided polygon and also any regular n-gon.
• Explain why the area of a circle is $\dpi{300}\inline \pi r^2$ and the circumference is $\dpi{300}\inline \pi \cdot D$ or $\dpi{300}\inline 2 \pi r$
• Explain why the lateral surface area of any pyramid with a regular polygon base is $\dpi{300}\inline \dfrac{P \cdot l}{2}$
• Explain why the formula for the surface area of a cone is $\dpi{300}\inline \pi r^2 + \pi r l$ or $\dpi{300}\inline \pi r(r + l)$
• Solve problems involving area of geometric shapes, express answers in exact form (in terms of $\dpi{300}\inline \pi$, in simplest radical form, using fractions)
• Understand problems similar to homework problems throughout the unit.

Assignment: Chapter 8 Review on p. 455: 1-47 (odds) + 38. Priority problems (to be done first) were 25, 38, 39, 45.

# Monday, March 6

In class we are about to wrap up our unit on Systems of Equations and Inequalities. Students should be able to solve a system of equations by

• Making a graph
• Using the substitution method
• Using the elimination method

The unit also began by leading students to be able to solve systems of inequalities by graphing the inequalities and shading the graph to find the feasible region. This skill is also useful in solving maximization problems. In our problem set today students had to write a set of inequalities to find the optimal number of two sizes of cookies for a shop to make in order to maximize profits given three different sets of constraints.

Full size of this example here. In order to solve this kind of problem students need to

• Identify the appropriate variables to use in representing the constraints and profit — select variables which represent the quantities you are looking to balance (in this case small/large cookies)
• Represent each constraint using an inequality using the variables you defined above. For example, in this problem, since Jorge has 50 raisins, we notice that each large cookie uses 8 raisins, and each small: 3 raisins. So we can write $\dpi{300}\inline 3x + 8y \leq 50$
• Graph the constraints and identify the feasible region — where all points represent viable values of $\dpi{300}\inline x$ and $\dpi{300}\inline y$. Pairs which satisfy all constraints.
• Locate the vertices of the feasible region and compute the profit at each of these points. In a linear optimization problem, the maximum profit will be found either at one vertex of the feasible region or along an edge of the feasible region.

Assignment: First six problems in the handout from class.

# Wednesday, March 1

In class students asked questions and we solved problems to get ready for the Chapter 6 Test. The chapter Review on p. 351: 1-10 is today's assignment.

Study Guide On the ch. 6 test students will need to be able to:

• Perform matrix arithmetic, including addition, subtraction, multiplication (by a scalar or another matrix) by hand. This includes determining when two matrices cannot be added / multiplied etc. due to their dimensions.
• Represent and solve a system of equations using an augmented matrix and row reduction by hand (quickly solving a 2 variable, 2 equation system, and able to solve a 3-variable, 3 equation system given sufficient time)
• Solve a system of inequalities by graphing the inequalities and clearly finding the feasible region (again by hand)
• Solve a linear programming problem by writing a system of inequalities to represent the constraints, finding the feasible region, and determining a point to maximize or minimize a profit function.
• Recognize situations in which a transition diagram and transition matrix should be used. Use transition matrices and matrix operations to solve real-world problems.