# Mon Feb 27

Wrote yesterday didn't send:( Here's a short update on class from yesterday.

In class I checked HW, reiewed 6.4 and 6.5 and we completed an example linear programming or profit/optimization problem. Students have a Chapter 6 test on Friday. We will spend Wednesday's class working to solve practice problems.

HW:
Due Wednesday:
6.6 on p. 347: 4-8(linear programming/optimization), 11-14 (review)

Upcoming (to be assigned Wednesday) Chapter 6 Review on p. 351: 1-10

# Mon Feb. 27

In Class today we solved area problems involving sectors, segments and anuluses.

The trick in many area problems is to make a plan and think of ways to add and subtract calculable areas to find difficult to find areas. For example, a segment is just a sector minus a triangle.

One of the tasks students are assigned today is to find the measure of an angle given information about the area of a sector:

So, We would compare the area of the sector to the area of the circle

$\dpi{300}\inline \pi r^2 = A$, so since $\dpi{300}\inline r = 24 \text{ cm}$ we have $\dpi{300}\inline \pi \cdot (24)^2 = \pi \cdot 576 = 576 \pi \text{cm}^2$

Now, this means that the shaded area, as a fraction of the total area of the circle is

$\dpi{300}\inline \dfrac{120 \pi \text{ cm}^2}{576 \pi \text{ cm}^2} = \dfrac{5}{24}$

So, since the shaded area is 5/24 of the area of the circle, the central angle must make up 5/24 of 360 degrees, so we find

$\dpi{300}\inline 360^\circ \cdot \dfrac{5}{24} = 75^\circ$

So, the value of $\dpi{300}\inline x$, or the central angle of the sector is $\dpi{300}\inline 75^\circ$

Tonight's assignment is to complete 8.6 on p. 439: 1-12, 17-22

# Monday, Feb 27

In class we corrected errors from Friday's exit ticket by looking over copies of 32 examples of student work. I solved a few example problems for the class and we had time to practice solving some system

Today's assignment is to complete 4 problems from the proficiency leveled worksheet "Substitution #8"

# Thu Feb. 23

In class we covered the definition and method for deriving identity and inverse matrices. We learned to solve systems of equations like

$\dpi{300}\inline 2x + 3y = 8$

$\dpi{300}\inline 4x - y = 10$

Using matrices

$\dpi{300}\inline \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 10 \end{bmatrix}$

We would then solve the system. Multiplying on the inverse on both sides

$\dpi{300}\inline \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}^{-1} \cdot \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}^{-1} \cdot \begin{bmatrix} 8 \\ 10 \end{bmatrix}$

We get

$\dpi{300}\inline \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{19}{7} \\ \frac{6}{7} \end{bmatrix}$

So, since $\dpi{300}\inline \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is the 2-by-2 identity matrix, we have

$\dpi{300}\inline \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{19}{7} \\ \frac{6}{7} \end{bmatrix}$

We also looked at solving systems of inequalities

Assignment:
6.4 on p. 331: 3, 6-10, 15, 17, 18
6.5 on p. 339: 5-9, 11-15

# Thu Feb. 23

In class today students worked on solving problems involving the area of circles and leaving answers in terms of $\dpi{300}\inline \pi$. In 5th period I taught students how to derive the formula for the area of a circle by "slice n dice" and treating a circle like a regular polygon.

Assignment: 8.5 on p. 435: 1-21 + optional mathcounts worksheet

# Thu Feb. 23

Hi All —

I had a curriculum meeting this morning, so Ms. Isikbay taught class. The goal is to work on solving systems of equations by substitution.

The substitution worksheet is today's assignment and there is an optional mathcounts challenge which students can do as well for tomorrow.

# Wed. Feb 22nd

Hi All —

In class today we solved a problem involving the area of triangles, circles and radicals. We took notes and practiced simplifying radicals, and students had time to work on the handout I gave yesterday on 8.4 and 8.5.

The area problem we worked on was to find the area of the shaded region.

As an example of where students are working to get by the end of the chapter, I had them solve this problem with me in notes.

To start breaking down this problem into some skills we can work on in class students need to be using equations in their process, simplifying radicals and drawing on background knowledge of properties and relationships in triangles.

We practiced simplifying radicals like $\dpi{300}\inline \sqrt{20}$

First, we find the prime factorization of 20: $\dpi{300}\inline 2 \cdot 2 \cdot 5$, so we can say

$\dpi{300}\inline \sqrt{20} = \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{5}$

And since $\dpi{300}\inline \sqrt{2}$ is defined as the number which when multiplied by itself gives 2, $\dpi{300}\inline \sqrt{2} \cdot \sqrt{2} = 2$ So,,

$\dpi{300}\inline \sqrt{20} = 2 \cdot \sqrt{5}$

1. What is the simplest form of the square root of 98?

$\dpi{300}\inline 98 = 2 \cdot 49 = 2 \cdot 7 \cdot 7$, so $\dpi{300}\inline \sqrt{98} = 7\sqrt{2}$

2. What is the simplest form of the square root of 600?

$\dpi{300}\inline 600 = 6 \cdot 100$ so $\dpi{300}\inline \sqrt{600} = \sqrt{100} \cdot \sqrt{6} = 10\sqrt{6}$

3. What is the difference between doing the work on your calculator and writing out the steps?

Assignment: 8.4/8.5 handout

# Wed. Feb 22

Hi All —

In class today we worked on a tricky motion problem and started to learn the algebraic methods for solving a system of equations. By the end of the module students should understand how to solve a system of equations by a variety of methods including graphing, substitution, and elimination. I think it is also an opportunity to study situations involving motion and rates.

Today's Motion Problem

Aryahi and Sophia are about to start a race around a 400m track. They plan to run in opposite directions starting from the same spot. Aryahi can run a lap in 1.25 minutes and Sophia can run a lap in .8 minutes. How far from the start will Sophia be when she passes Aryahi for the third time?

I solved this problem with the class and had students discuss the steps at their tables. It would be a great chance for you to ask them to explain it back to you. Some questions you can ask:

• How do we find that Aryahi can run 320 meters per minute and Sophia can run 500 meters per minute? Use the formula d = rt, substitute d = 400m and t = 1.25 min for Aryahi, and d = 400m and t = 0.8 min for Sophia. Solving the equation 400m = (r) * (1.25 min) we get r = (320 m)/(1 min) so 320 meters per minute. Similarly for Sophia, 500 meters/min.
• What is the significance of 820m per minute? Since they are running in opposite directions, 820 m per minute is the speed at which they are covering the distance between them around the 400 meter track.
• \$400 m div 820 m/min approx 0.4878\$ what are the units for the answer here? They are minutes. So 0.4878 min times 60 seconds per minute is 29.268 seconds. It takes the two runners about 29 seconds to cross paths the first time.
• How would you explain the significance of the moments at which they cross paths? When they cross paths for the first time, the total distance run by the two runners must be exactly 400 m in total.
• Can you show me how to finish solving the problem? The correct answer in the end is about 68.3 meters

Today's assignment: complete all problems from 5.7 Task, Set, and Go.

# I’m back! Also, HS Forecasting info and current units

Hi All —

I had my first day back at CPMS after family leave today. It was really nice to see everyone and it is of course tough to leave the cuddly baby and family but also good to be back. My son, Evan Paul, was born 8 lbs 8 oz and 22.5 inches long. He is healthy and really good at sleeping right now, even though his big sister likes to loudly announce his  every move 🙂

If you have an 8th grader then they should be completing their forecasting sheets this week. They will need my signature for whatever math class they hope to take. If you are interested in more information about the HS courses you can find more info about HS courses on my Geometry website at

http://teacher.nickhershman.com/?p=geo

Forecasting forms are due to Advisory teachers by Friday.

I did not assignment any hw today. In AGS 1 we will be focusing on learning about systems of equations in the later sections of Module 5. Geometry students will finish the unit on Area/Surface Area of 2-d and 3-d shapes. Adv. Algebra 2 will be finishing the unit on Matrices and Linear Programming.

# I’m Back! Also, HS Forecasting, and current units

Hi All —

I had my first day back at CPMS after family leave today. It was really nice to see everyone and it is of course tough to leave the cuddly baby and family but also good to be back. My son, Evan Paul, was born 8 lbs 8 oz and 22.5 inches long. He is healthy and really good at sleeping right now, even though his big sister likes to loudly announce his  every move 🙂