Symmetry, Day 1

Today's lesson started with a review of identifying different types of symmetry in images. We looked at 17 different logos and determined if they have line symmetry or rotational symmetry.

For line symmetry, the image needs an imaginary line that presents mirror images. Many figures have vertical line symmetry, but the line can come in at any angle.

For rotational symmetry, the image needs to align with itself when it is spun around by less than 360° (every image has 360° rotational symmetry). To calculate the rotational symmetry, divide 360° by the number of points on the image that it can turn to.

For our graphs, we consider images that have vertical line symmetry at the y-axis to be even. And, images that have 180° rotational symmetry about the origin to be odd.

Homework: Worksheet on Graphs of Even & Odd Functions

Remember, there is a TEST scheduled for Thursday!

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Rationals - Day 1 (Thursday & Friday)

The last function that we will graph and translate this year is the rational function. This function is created when a variable is located in the denominator of a fraction. The parent function is

And its graph is

Notice that this graph is two separate curves. They do not connect in the middle. In fact, the center of the curves, (0, 0) in this graph, has two asymptotes running through it. An asymptote is a line on the graph that the curves do not cross, but instead they always approach. This means that the distance between the curves and the asymptotes is narrowing, but they do not intersect. On the graph of the parent function, the asymptotes are x = 0 (the y-axis) and y = 0 (the x-axis).

The graph of a rational function can be translate similar to our previous transformations. The general form of the rational function is

In this form, the graph translates according to:

• "a" stretches the graph if its absolute value is larger than 1;
• "a" compresses the graph if its absolute value is between 0 and 1;
• "a" reflects the graph upside down if it is a negative value;
• "h" shifts the graph to the left and right (remember, use the opposite of the sign that is "seen" in the function);
• "k" shifts the graph up and down.

Because of the nature of this graph, the characteristics of the graph are quite different from the characteristics of our previous functions.

• This graph has a center at (h, k) that is used to determine the asymptotes. The previous functions each have a vertex at (h, k).
• This graph has 2 asympotes: one vertical and one horizontal.
• This graph has a domain of all real numbers except the x-value of the vertical asymptote.
• This graph has a range of all real numbers except the y-value of the horizontal asymptote.
• This graph has end behaviors approaching the y-value of the horizontal asymptote.
• This graph is either increasing or decreasing, not both. And there are 2 separate intervals for the graph to be increasing (or decreasing).

Example: Graph the following function.

This graph has a center at (5, 3).

This graph has a vertical asymptote at x = 5.

This graph has a horizontal asymptote at y = 3

This graph is reflected upside down.

To find additional points for the graph, substitute in values for x around 5.

Homework: Students were to complete the notes that were handed out on Thursday. 1st period has already had this homework checked. 3rd and 4th periods will have this checked on Monday.

There will be a quiz on Thursday, March 25th on radical and rational functions.

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Radicals - Day 3 (Wednesday)

Sorry for the delay in posts. I was having trouble connecting to the blogging site from work. And while I can email blogs in, I have not figured out how to get the images to appear correctly when I do. Here is Wednesday's lesson from last week:

Today we brought together the previous two lessons of radicals to make a connection between the graphs and solutions of equations. In a radical equation, the two sides of the equation can be graphed as separate functions. Then the solution to the equation is located at the x-coordinate of the intersection of the two graphs.

Example: Solve the following equation by graphing.

Step #1: Break up the equation into two functions, one for each side of the equation:

and

Step #2: Graph the two equations.

Step #3: Find the intersection. For this graph, the functions intersect at (9, 6).



Step #4: Find the solution. The solution is the x-coordinate of the intersection. For this equation, x = 9.

We extended this out to finding what is called the zeroes of the function. The zero of a function is the solution of the function equal to 0. When graphing g(x) = 0, we find that it is the x-axis. So, solutions of the equation equal to 0, and zeroes of a function, and x-intercepts of a graph are all the same thing!



Homework: Worksheet on Solving and Graphing Radical Equations

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Radicals - Day 2

In class, students were reminded to take the online practice EOCTs every week. Many students currently have 2 zeros recorded for the last two weeks.

Today's lesson continued the work with radicals: we solved equations that contain radicals. To do this, follow these steps:

1. Isolate the square root by removing every value that is on the side with the square root. Remember, undo addition/subtraction first. Then undo multiplication/division.
2. Square both sides of the equation to undo the square root.
3. Solve the equation. Our equations today are so basic, this step is not needed. The equation will be solved in step #2.
4. Check the solution.

It is always important to check solutions for equations to verify that no mistakes were made. It is doubly important with radical equations because it is possible to accurately arrive at a "solution" that doesn't actually solve the equation. This occurs when step #1 sets up the square root to be equal to a negative number, which is not possible. However, if step #2 is completed, students will still arrive at a value that presents itself to be the solution. When the extraneous solution is substituted into the original equation, the student should arrive at a false equation (where the two sides are not equivalent).

Homework: Worksheet on Solving Radical Equations (found on the back of the notes)

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Radicals - Day 1

Today's lesson incorporated our graphic transformations (that we studied with the Quadratic and the Absolute Value function) with the square root function. The parent function looks like half of a parabola, turned on its side.

The generic form of the equation to graph a square root function is:

In this form, transformations are determined by:

• if the absolute value of a is greather than 1, then the graph is stretched taller;
  
• if the absolute value of a is between 0 and 1, then the graph is compressed shorter;
  
• if the value of a is less than 0, then the graph is reflected over the x-axis (flipped upside down);
  
• if the value of b is less than 0, then the graph is reflected over the y-axis (flipped over right to left);
  
• The value of h is always the opposite of whatever is "seen" within the square root (just like we did for the last 2 weeks) and this value shifts the graph to the left or right;
  
• The value of k is exactly whatever is "seen" after the square root (just like we did for the vertex form) and this value shift the graph up or down.

Students should always use a table of values to find points for their graphs. Pick x-values that will make the radicand (the inside of the square root) equal 0, equal 1, equal 4, and equal 9. We use these values because they are perfect square, so their square roots won't have us plotting decimal values for coordinates.

Example: Graph the following function:

• a = -2, so this graph will be flipped over and stretched taller by 2


• h = 1, so this graph will be shifted to the right 1 unit


• k = 4, so this graph will be shifted up 4 units


• use a table of to have points to graph: input x = -1, x = 2, x = 5, and x = 10.

Homework: Pages 140 & 141, #2 - 24 even

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Test day

Students are taking their test today on the quadratic concepts that we have been working with for the last 2 weeks.

This weekend is a math "holiday". Sunday's date is 3.14 which is also the common approximation used for pi. To celebrate this, students have been offered an extra credit opportunity to wear a shirt on Monday with some notes about pi on it. They received a handout with the information in class today.

Next week, we will start working with radicals (square roots) and rationals (fraction). Our next test is scheduled for April 1st.

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Review

We started class by going over homework. Then, student receive their Unit 4 Part 2 tests. For the most part, these grades were an improvement over previous tests. Students also receive a grade printout showing their current grade (with this most recent test). The statistics projects are not graded yet so those scores do not appear on the handout. Students should get their parents to sign off on the grade sheet.

Students received a review packet in class today. It is due tomorrow for a homework check and then we will go over it to prepare everyone for Thursday's test.

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Systems of Equations - NonLinear

We used the review of systems of linear equations on Friday to jump to systems of non-linear equations today. Our lesson has systems that are either 1 linear & 1 quadratic equation OR 2 quadratic equations.

Students should be able to find solutions on a graph. To do this, identify where the graphs meet. There could be no solutions (no intersection), 1 solution, 2 solutions, or infinitely many solutions (when the graphs overlap completely).

Students should also be able to find solutions by working with the equations. To do this follow these steps:

1. Confirm that both equations are solved for y. If not, get y by itself.

 

2. Substitute the first expression for y in the second equation. Solve for x. This can be done by either (1) setting the equation equal to 0, factoring, and solving for x from the factors; or (2), only when b = 0, set x² equal to the constant value and then square rooting both sides. Remember, when using method #2, the solution is the positive & negative of the value acquired through square-rooting.

 

3. Substitute the solved value(s) for x into either of the two equations and solve for y.

 

4. To check the solutions, substitute the ordered pairs into both equations to verify the values are correct.

Example:

Homework: Worksheet on Solving Systems of Non-Linear Equations.

There will be a TEST on Thursday covering all quadratic concepts from last week and this week. Students will receive a review worksheet tomorrow in class.

 

 

Systems of Equations - Linear

We discussed the online test that all students should have finished at least once by today. Students are encouraged to retest their *missed problems* to raise their score. I do not suggest that students completely retake a *new* test. They should build on the test that they already answered some of the problems correctly. I will record and only use the highest score that they achieve. At this time, I have not received a test score from every student. They have until midnight to complete a test and not receive a ZERO for this week's assignment.

Today's lesson is review of 8th grade concepts: solving systems of linear equations. Their handout has directions and examples for solving a system of linear equations with three different methods.

Solving a System by Graphing:

• Step 1: Graph each equation on the same coordinate plane.


• Step 2: Identify where the two graphs meet (if anywhere). The point where the graphs meet is a solution of the system. It is possible for the two lines to be parallel and never intersect. Parallel lines have no common points, so the system has no solution.


• Step 3: Check the solution by substituting the values of x and y into both equations. Verify that it solves both equations.


• Example:

Solving a System by Substituting:

• Step 1: Solve one of the equations for either of its variables.


• Step 2: Substitute the expression from step 1 into the other equation and solve for the other variable.


• Step 3: Substitute the solved value from step 2 into either of the original equations and solve for the remaining variable.


• Step 4: Check the solution by substituting the values of x and y into both equations. Verify that it solves both equations.


• Example:

Solving a System by Eliminating:

• Step 1: The goal is to make one of the variables have opposite values for its coefficients in the two equations (like -5x and +5x ... or +4y and -4y). If the equations do not already have this requirement, then select one of the variables to affect. Multiply everything in the equation to get the desired coefficient on the selected variable. You may need to multiply both equations by different values to meet this requirement.


• Step 2: Add the two equations together by collecting like terms. Some students remember this step from last year when they wanted to either get "0x" (zero-x, but called ox) or "+ 0y" (plus zero-y, but called toy). Solve for the remaining variable.


• Step 3: Substitute the solved value from step 2 into either of the original equations and solve for the remainging variable.


• Step 4: Check the solution by substituting the values for x and y into both equations. Verify that it solves both equations.
• Example: Note ... the step numbers in the example below are each one value higher than the way I numbered my steps above.
  
  
  

Homework: Worksheet on Systems of Linear Equations

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Quadratics, Day 3

Our third lesson with quadratics is our final version of graphing quadratics. Today we discussed the vertex form of the quadratic function: y = a (x - h) ² + k. This form makes is the easiest for identifying the vertex of the parabola.

• The vertex is located at (h, k). Students need to remember that they switch the sign of what the "see" inside the parenthesis in the equation for h. The value of k is exactly what they "see" to the right of the parenthesis in the equation.
• Use a table to select values near the vertex. Substitute these values in for x and solve for y. Get at least 5 points on the graph.
• Plot these points and draw a U-shaped curve.

Homework: Worksheet on Graphing Quadratics in Vertex Form

Don't forget to do your vocabulary and your online EOCT practice test. Both are due tomorrow!!!

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Delayed Start today

Due to the 2-hour delay today, Milton High School started with 3rd period classes. This means that I did not see my 1st period class.

My other IAA classes took a notebook quiz and a county wide Checkpoint Test. 1st period already took the notebook quiz last week. And, 1st period just will not take the checkpoint test due to our schedule. The Checkpoint Test is a diagnostic test to assess how much information from the entire year the IAA students know right now. The results of this test will NOT affect a student's grade in anyway. Its purpose is to show me what information I may need to reteach before our final exam.

Yesterday, students were given a handout explaining how to access a website (www.usatestprep.com) that has practice tests correlated to our final exam. They have a weekly assignment to take the "Large Test" for "Mathematics 1" each week by Friday.

There is no new homework today. If students did not finish yesterday's worksheet, then they have tonight to do it. Or, they can use tonight to take care of their weekly EOCT Practice Test assignment.

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Quadratics - Intercept form

Students found out yesterday that graphing quadratics in standard form does not lend itself well for locating the x-intercepts. Sometimes the graph crosses between two "nice" integers. Sometimes the graph crosses the x-axis a significant distance away from the vertex and requires extra work to locate. So, today's lesson is about the intercept form which identifies the x-intercepts very easily.

Intercept form: y = a (x - p) (x - q)

• To find the x-intercepts in this form, set the factors equal to 0 and solve for x: (p, 0) and (q, 0).
• To find the vertex and the axis of symmetry in this form, remember that the graph is symmetrical, meaning that the AOS will have to be in the middle of the intercepts. To find this location, average p and q. This calculates the x-value of the vertex. Substitute this into the equation and solve for y.
• Pick another value for x near the vertex and find it corresponding y value. Plot all points and draw a smooth U-shaped curve.

Homework: Worksheet on Graphing Quadratics in Intercept Form

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New Unit!

We started a new unit today. Unit 5 has us going back to algebra concepts so we will review concepts from the fall and build on them.

Today's concept is graphing quadratic equations given in standard form, y = ax² + bx + c. Much of this lesson is review. However, one part of the lesson is new:

• To find the vertex of a parabola, first find -b/2a. This value equals x for the vertex. Then, plug this value into the equation and solve for y. These values locate the vertex.

Once the vertex has been located, use a table or chart to organize your work picking values for x, substituting them in, and solving for y. Plot these points to find the shape of the parabola.

Along with graphing the quadratic equation, we expect students to be able to answer the following questions:

• Opens: the graph opens up when a > 0 and the graph opens down when a < 0 ;
• Vertex: the point found using the directions above;
• Min/Max: the vertex is a minimum when the graph opens up and the vertex is a maximum when the graph opens down;
• AOS: the axis of symmetry is the line "x = (-b/2a)";
• X-intercepts: the points where the graph crosses the x-axis. This can be none, one, or two points. When the graph crosses between two integers, simply state "between (m, 0) and (n, 0)" where m and n are the integers;
• Domain: the domain is the set of all possible x that can be used in the equation. For the quadratic functions it is {all real numbers};
• Range: the range of a quadratic function is based on the vertex and whether it is a maximum or a minimum. When the vertex is at the top of the graph (maximum), then the range is {y ≥ the y-value in the vertex}. When the vertex is at the bottom of the graph (minimum), then the range is {y ≤ the y-value in the vertex};
• End Behavior: the arrows at the ends of the graph of a quadratic function either both point up (going toward positive infinity or +∞) or both point down (going toward negative infinity or -∞);
• Intervals of Increasing and Decreasing: the intervals are determined by where your pencil draws downward (decreasing) and where it draws upward (increasing) when making the parabola. If the interval is to the left of the AOS, then write {x < the x-value in the vertex}. If the interval is to the right of the AOD, then write {x > the x-value int the vertex}.

Homework: Page 107, #24 - 30 even; Page 109, #16 - 26 even. For all graphs, also include x-intercepts, domain, range, end behavior, and intervals of increasing and decreasing.

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Statistics, Day 5

On Monday, we looked at where the data come from for the statistical calculations. We discussed how surveys occur and how the people are selected from a larger group to participate in the survey. Students need to be able to identify ...

1. The population: The larger group that the survey is trying to share information about.
2. The sample: The smaller group that actually participates in the survey.
3. The sampling method: One of the 5 methods that we discussed in class: random, stratified random, self-selected, systematic, and convenience samples.
4. Whether or not the sample has bias: Does the sample leave out part a group from the population? Is one group over represented or under represented within the sample?
5. Whether or not the question has bias: Does the question project the expected or desired response of the interviewer?

Homework: Page 361: #2 - 10 even; Page 362: #2 - 10 even

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Statistics, Day 4

On Friday, we went further into the concept of mean absolute deviation. We looked at two sets of grades for students that have the same average or mean (75%) and the same range (from 60% to 90%). However, they were very different students. One had grades more balanced out in the span from 60 to 90; while the other had grades that either were high 80s to 90 or were low 60s. I used these sets to introduce the statistic that measures the deviation away from the mean, or a set of data's variability. To calculate the mean absolute deviation:

1. Calculate the mean.
2. Find the distance from every number in the set away from the mean. These are the deviations.
3. Find the average of all deviations by adding them together and dividing by the number of data in the set. This is the mean absolute deviation(MAD).

The higher the value of the MAD, the less consistent the data are around the mean. Or, the more variability the data has away from the mean.

Homework: Page 365; #9 - 14 and 17 c & d

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Statistics, Day 3

We spent much of classtime today going over the Box & Whisker plot and its characteristics. We discussed where the box can appear with respect to the median. The box, which is defined by the lower and upper quartiles, can appear anywhere around the median, even with one end lined up with the median, for instance when the lower quartile and the median turn out to be the same number. But the box cannot occur completely below or completely above the median because the lower quartile must be less than or equal to the median and the upper quartile must be greater than or equal to the median.

We also talked about the significance of the range and IQR. While the mean, median, and mode show the central tendency of the data, the range and IQR show the spread of the data. We talk about the data being dispersed or spread out when the range or IQR are larger values. We consider the data to be condensed, dense, or consistent when the range or IQR are smaller values. For example, consider two students who both have 75 averages. One has grades spanning from 72 to 78, and the other has grades spanning from 60 to 90. While they have the same average, the differing ranges imply that the second student is capable of better grades but is not consistently working at their potential (high Bs). The first student is much more consistent with their grades and is probably a solid C student.

We started discussing the mean absolute deviation (MAD) today. We will continue to discuss it and work through calculations tomorrow.

Homework: Complete the bottom of the front page of the notes. The assignment listed at the top of the notes is due Monday, February 22nd.

Remember, project topics need to be approved by tomorrow.

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Statistics, Day 2

Wednesday, Febuary 17th



We discussed the concepts of the 5-Number Summary, Interquartile Range (IQR) and the Box-and-Whisker Plot. Students have made Box & Whisker plots in previous math courses. To make this graph, students must first find the 5-Number Summay. The 5-Number Summary is made up of the following 5 numbers from the data set: minimum, lower quartile, median, upper quartile, and maximum. The interquartile range is used to show how the middle 50% of the data values are spread or are compacted. To find these values, the students need to follow these steps:

1. Order the data from least to greatest.
2. Identify the median.
3. Identify the median of the lower half of the data values. (Do not include the overall median in the lower half of the data values.) This is the lower quartile, also called the first quartile or simply Q₁.
4. Identify the median of the upper half of the data values. (Do not include the overall median in the upper half of the data values.) This is the upper quartile, also called the third quartile or simply Q₃.
5. Identify the smallest value (minimum) and the greatest value (maximum) of the data.
6. The interquartile range is found by subtracting the third quartile minus the first quartile.

To make a Box and Whisker Plot, follow these steps:

1. Create a number line or x-axis that spans at least from the minimum to the maximum values.
2. Use a short, vertical tic-mark to indicate the numbers in the 5-Number Summary in the space above the number line.
3. Draw a segment connecting the first two tic-marks for a "whisker" connecting the minimum to the first quartile.
4. Draw a segment connecting the last two tic-marks for a "whisker" connecting the third quartile to the maximum.
5. Connect the second and fourth tic-marks across the top and across the bottom to create a "box" around the middle 50% of the data values.

The numbers in the 5-Number Summary show how the data are spread or compacted. Each number occurs 25% of the way through the data values, however some of the distances between the actual data values are less (we talk about the these spans being more dense) or the distances between the actual data values are more (we talk about these spans being spread out).

Example:

Homework: Page 371, #1 - 9. Additionally, create a Box & Whisker Plot for #1 - 4 and #6 - 7.

Students received a project today. They need to select a measurable idea to compare between 2 populations (example, how many hours do boys vs. girls spend on Facebook?) and make a hypothesis about the expected results. They will have to collect at least 16 data values for each population (a minimum of 32 data values total). Then, they have to generate all of our statistics with each set of data: mean, mean absolute deviation (to be introduced tomorrow), and the 5-number summary. They need to make a Box & Whisker plot for both sets of data. They may make a histogram for extra credit. They then need to draw a conclusion based on the statistics. The final product can be a report, a poster, a PowerPoint, etc ... This project is due on Wednesday, February 24th.

There will be a test on Statistics (Unit 4 Part 2) on Friday, February 26th.

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Statistics, Day 1

Tuesday, February 16th

We reviewed the concepts of central tendency: mean, median, and mode. The mean is the average of the values in a data set. The median is the middle number of the values in the data set when it is arranged in numerical order. The mode is the value that occurs the most often. These concepts were taught to the students in previous math courses. However, we now want them to be able to pick one (or more) of those values that best represents the data. Students should consider whether or not outliers move the mean further up or down away from the center. They should also consider if having a dense group of numbers positions the median further away from part of the data values.

We also reviewed the concepts of a frequency table and a histogram. Both of these can have the data values grouped together or listed individually. They should have between 5 and 10 spans or individual data values. A frequence table lists the data values or spans of the data values in one column and then records the number of times those values occur in the data. A histogram shows the data values or spans of the data values on the x-axis and then has bars going up with a height equal to the frequency for that value. A histogram is a bar graph, however it is slightly more specific: it must have a continuous set of numbers on the x-axis. (A bar graph can have words recorded along the horizontal axis.)

Homework: Histogram Worksheet and Page 365, #1 - 8, 15 b & c, 16, and 17 a & b.

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Compound Probability Day 2

We started working our way through a packet today which will progress our understanding of compound probabilities. The problems in this packet will utilize contingency tables and Venn diagrams to help us see the numbers where events overlap and where events are disjoint.

Homework: Complete Example 2, Example 3, Practice 1, and Practice 2.

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New Unit!

We started our lessons for Unit 4 today. This unit included probability and statistics concepts. Today's lesson was on the Fundamental Counting Principle. There are two parts to this principle. First the multiplication counting principle shows that to find the total number of outcomes from a certain sequence of events can be found by multiplying the number of ways each event can occur.

Example: Find the total number of ways to order a dinner when selecting one entree from 10 options, one drink from 5 options, and one dessert from 6 options.

Answer: 10 entrees * 5 drinks * 6 dessert = 300 different dinners

The addition counting principle indicated that the sums from events can be grouped in such a way that the groupings do not share possible outcomes, then the total number of ways each group can happen should then be added together.

Example: Find the total number of ways a 3 symbol code made of digits or letters can be made if at least one symbol must be a letter.

Answer: There are 26 different letters and 10 different digits

1-letter and 2-digit code (A # #): 26 * 10 * 10 = 2600 A## codes.

But, the letter does not have to be in the first spot.

It can occur in any of the 3 spots, so we multiply the total by 3.

26o0 * 3 = 7800 different 1-letter 2-digit codes

2-letters and 1 digit code (A A #): 26 * 26 * 10 = 6760 AA# codes

But, the digit does not have to be in last spot.

It can occur in ay of the 3 spots, so we multiply the total by 3.

6760 * 3 = 20,280 different 2-letter 1-digit codes

3-letters (A A A): 26 * 26* 26 = 17,576 AAA codes

Now, add together the total from each group:

7800 + 20,280 + 17576 = 45,656 different 3-symbol codes

(with at least 1 letter)

There are some typical modifications to the above problem, such as limiting the digits to only evens (5 options), or not allowing a symbol to repeat. For the latter modification, the number of symbols available decreases for one that is already used.

Example: How many 3 letter codes can be made if no letter can be repeated?

Answer: 3-letters (A A A): 26 * 25 * 24 = 15,600 different 3-letter codes

(with no repeated letters)

There will be a quiz on Friday covering the lessons from this week: counting principles, permutations (Tuesday), and combinations (Wednesday).

Homework: Page 340, #2 - 10 even, #11; Page 341, #2 - 8a even (omit 8b)

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Catching up!

Sorry to have missed so many days. It has been a busy January!!

Today's notes finished up our "special segments" in triangles with medians and altitudes. Medians are segments that connect a vertex of a triangle with the midpoint of the opposite side. Every triangle has 3 medians. The medians will always all meet at a single point inside the triangle called the centroid as shown below.

The centroid has two properties:

1. It is known as the "balancing" point of the triangle.


2. It occurs exactly 2-thirds of the distance from the vertex to the midpoint.

An altitude is the perpendicular segment to a side of the triangle and ending at the opposite vertex. This segment is also known as the height of the triangle. Every triangle has 3 altitudes. The altitudes will always meet at a single point called the orthocenter. This point can occur inside, on, or outside the triangle depending on whether it is acute, right, or obtuse, respectively.

Acute Triangle example:

Obtuse Triangle example:

Homework: Page 280, #1 - 6; #10 - 12; and #14 - 26 even

There is a review session tomorrow morning in room 5207, tomorrow afternoon in room 5202, and Friday morning in room 4303. Students will get their quizzes back tomorrow and a review packet for Friday's test.

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Quadrilaterals Day 3

Today's lesson began with a discussion and an activity investigating parallelograms with more properties: Rectangle, Rhombus, and Square. These are parallelograms, so they still have the 5 properties specific to parallelogram and the 2 properties specific to all quadrilaterals. In addition to them, we added more.

For Rectangles:

1. Definition: a parallelogram with 4 congruent angles (all 90º)
2. diagonals are congruent

For Rhombuses:

1. Definition: a parallelogram with 4 congruent sides (think of a diamond shape)
2. diagonals are perpendicular
3. each diagonal bisects a pair of opposite angles

For Squares:

1. Definition: a parallelogram with 4 congruent sides AND 4 congruent angles
2. Because a square is BOTH a rhombus and a rectangle, it has ALL properties previously mentioned!

Homework: Page 319, #2 - 18 even; Page 321, #2 - 18 even, #28

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Quadrilaterals Day 2

After reviewing the 5 properties specific to parallelograms, we turned those properties around. If those properties are true about a quadrilateral, then the quadrilateral MUST be a parallelogram. So, if a figure of a quadrilateral shows that ...

1. both pairs of opposite sides are parallel,
2. both pairs of opposite sides are congruent,
3. both pairs of opposite angles are congruent,
4. one angle is supplementary with both of its consecutive angles,
5. diagonals bisect each other,
6. or, one pair of opposite sides is BOTH parallel and congruent

... then the quadrilateral is a parallelogram.

Again, students are expected to be able to identify which property is needed as well as to create and solve necessary equations, much like Day 1.

Homework: Page 313, #2 - 12; Page 315, #1 - 14, #19

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Quadrilaterals Day 1

On Tuesday, our first day of the new semester, we reviewed the properties of Quadrilaterals:

1. Definition: 4 sided polygon
2. the sum of the 4 interior angles is 360º

After that short review, we investigated the properties specific to parallelograms by measuring the angles, side lenghts, and diagonal lenghts of a few parallelograms. We found and discussed the following properties specific to parallelograms:

1. Definition: a quadrilateral with both pairs of opposite sides parallel
2. both pairs of opposite sides are congruent
3. both pairs of opposite angles are congruent
4. consecutive angles are supplementary
5. diagonals bisect each other

From these properties, the students are expected be able to identify which one is shown in a figure and be able to use the relationships to create and solve equations. When approaching these problems, students should first identify which part of the parallelogram the problem measures. Then, apply the property for those parts. Remember, congruent means the measures/lengths are equal. Supplementary means that the angle measures sum to 180º.

Homework: Page 306, #2 - 26 even; Page 308, #2 - 20 even (omit #10)

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