Geometric Justifications

Today's notes introduce the students to many of the reasons that we will use to justify our steps in a geometric problem. We discussed several properties of equality, the Segment and Angle Addition Postulates, and the Reflextive, Symmetric, and Transitive Properties of Congruence.

Students will use these statements as reasons to justify their steps throughout this unit.

Homework: Page 219, #1 - 8; Page 221, #1 - 8

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Last of the Triangle Inequalities

Tuesday's notes started with a review of the Triangle Exterior Angle Inequality theorem:

From there, we discussed that the measure of an exterior angle must always be greater than the measure of either non-adjacent interior angle. The students made this conclusion because it takes both added together to make an equivalent amount to the exterior angle.

Homework: Worksheet "Triangle Sums". This worksheet is review of concepts taught earlier in this unit.

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More Triangle Inequalities

We continued working with inequality relationships among the sides and among the angles in 1 or 2 triangles. Today, the first theorem that the students explored is that the angle in a triangle that is opened the widest is always opposite the longest side of the triangle (and similiarly for the smallest angle & shortest side, and for the middle angle & middle side). This relationship allows for us to organized the sides and the angles of a triangle from least to greatest even if we only know one set of measurements (all of the sides or at least 2 of the angles). This lesson can be found in the textbook as lesson 5.5

The next theorem that we discussed is the Hinge Theorem (found in lesson 5.6). This theorem is applied to two triangles that have 2 sets of congruent sides. Whichever triangle has the longer of the two remaining sides must have an angle opened wider (the HINGE). We call that angle the "included angle" because it is between the two sets of sides that are congruent. This theorem was demonstrated for the students by using the hinge in the classroom door. As the hinge was opened for a wider angle, then the doorway had more open space across the opening.

Homework: Page 287, #4 - 12 even, #28; Page 289, #2 - 6 even, #18; Page 294, #2 - 10 even.

The next test will be on Thursday, December 10th. It will cover all Geometry topics taught through that date. There will be a review session Wednesday, December 9th, at 7:45 in my classroom (4303).

Remember, there are 6 help sessions offered to our IAA students each week:
Monday, 7:45am with Mrs. Bearden in 4210
Tuesday, 7:45am with Mr. Roth in P123
Tuesday, 3:30pm with Ms. Gonding in 4303
Wednesday, 7:45am with Mrs. Bearden in 4210
Thursday, 7:45 am, with Ms. Dufresne in 5207
Thursday, 3:30 pm, with Mrs. Martina in 5202

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Tuesday

Review!

We reviewed today to get ready for tomorrow's quiz. Students received a half sized handout with some review problems and made their own flipbook to organize the necessary formulas and relationships.

Homework: Study!

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Polygon Angles

Today, we first reviewed the names of polygons with 3, 4, 5, 6, 7, 8, 9, 10, or 12 sides (any other number of sides to a polygon (#) is just called by #-gon). Then we explored how to break up all of the shapes, no matter how many sides they have, to have only triangles inside them. Because we know that the sum of the measures of the three angles inside a triangle is 180°, we can figure out the sum of the measures of the angles inside any polygon based on how many triangles it takes to fill the same shape.

Example: A pentagon can be filled with 3 triangles:

Because each triangle has 180° total for its three angles, then the 3 triangles have 3 * 180° or 520° for the 5 angles inside the pentagon.

Students discovered that there is relationship between the number of triangles it takes and the number of sides: There are always 2 fewer triangles to fill the shape than there are sides. Using this relationship, students can find the sum of the interior angles of any convex polygon by using the formula 180°(n - 2), where n is the number of sides.

Example: Find the sum of the interior angles of a 23-gon.

A 23-gon has 23 sides, so 21 triangles are needed to fill the shape.
180° * 21 = 3780°

Students also learned that the sum of the exterior angles of a convex polygon is always 360° (no matter how many sides it has). The model for this that I gave students was to consider walking around the shaping and turning to follow the next corner. As I made my way around my desk, I had to turn all the way around by the time I got back to the beginning. They know that all the way around a full circle is 360°

Example: There are 4 exterior angles shown in this quadrilateral.

Homework: Page 300, #2 - 24 even; Page 302, #2 - 14 even, #34.

There will be a quiz this week. It may be as early as Thursday. It will cover the lessons on angles (both interior and exterior) given on Friday, today, and tomorrow. There will be a review session Wednesday morning in room 4303 for this quiz.

Remember: Vocabulary definitions are due on Wednesday!
Remember, also: The Logic Project is due on Friday!

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Friday's lesson ...

On Friday, the class reviewed a concept that they learned a few years ago in math: that the sum of the measures of the three angles in a triangle is always 180 degrees. We even tore apart triangles and placed the three angles together and noted how they fit together, side by side, to make a straight line (or a straight angle which measures 180 degrees). If students are given a problem with expressions for the three angles inside a triangle, they should add together the measures for the three angles and set it equal to 180 degrees. Then, they should solve that equation.

The class also learned that an exterior angle (an angle formed by extending a side of a triangle beyond the triangle) has a measure equal to the sum of the two non-adjacent (meaning "not next to") interior angles. If the students are given a problem with an expression for an exterior angle of a triangle, they should add together the measures of the two angles that do not touch the exterior angle and set that equal to the measure of the exterior angle. Then they should solve that equation.

Homework: Triangle Angle Worksheet

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Logic Statements

Yesterday's lesson introduced the manipulations and truth value of logic statements. Much of this lesson is knowing and applying the definitions: Conditional statement, converse, inverse, and contrapositive.

The conditional statement has the "if - then" format. The "condition to be met" is called the hypothesis and follows "if". The "result" is called the conclusion and follows "then"
Example: If all students are present on the day of a quiz, then they will get 2 bonus points. (This is true for my classes.)

The converse switches the hypothesis and the conclusion.
Example: If all students get 2 bonus points, then they were all present on the day of a quiz. (This could be true, but it could be false if they all did a bonus problem correctly and that is why they got the bonus points. This is called a counterexample, it is an example that counters the argument.)

The inverse is formed by negating both the hypothesis and the conclusion of the original conditional statement.
Example: If all students are not present on the day of a quiz, then they will not get 2 bonus points. (Again, this may not be true if they still answer the bonus question for the extra points.)

The contrapositive is formed by switching and negating both the hypothesis and conclusion of the original conditional (this is a combination of the converse and the inverse).
Example: If all students did not get 2 bonus points, then they were not all present on the day of a quiz. (This is true for my classes.)

The second day of this lesson brings in the final logic statement. The biconditional statement is formed with both the conditional statement and the converse are true. It is not written in "if - then" form. Instead, this statement places "if and only if" in the middle of the hypothesis and conclusion (sometimes abbreviated "iff"). Definitions are often written in the biconditional format in our class.
Example: An angle is a right angle if and only if it measures 90 degrees (the definition of right angle).

Homework: Yesterday's - Page 207, #1 - 10 and Page 209, #1 - 8. Today's - Page 207, #11 - 19 and Page 209, #17 - 19.

Students were also given a project today. This project is called "The Logic of Advertising". Individually, or in pairs, the students must find 3 slogans to convert to conditional statements and then write the converse, inverse, and contrapositive. All 12 sentences (4 sentences for each of 3 ads) must be typed and clearly identified on a single sheet of paper. Then, a presentation can be made (power point, poster, video) to share the sentences with the class. This project is due on Friday, November 20th.

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A New Unit!

The IAA classes are now all done with Unit 2. They completed their tests on Friday and Monday, and grades should now appear in ParentConnect (unless your child was absent). Today started the third unit: Geometry! This is a nice break for many students. We will put much of the algebraic skills away for a while and work with rules about the definitions and relationships among 2-dimensional shapes (mostly triangles and quadrilaterals). However, because the final exam in December (a short 5 weeks away!) is comprehensive for the whole semester, students should be sure to the get help that they need to understand the previous 13 weeks of algebra concepts & information. To finish out Unit 2, we had a notebook check today. Students are now encouraged to take all unit 2 material out of their notebooks and set it aside in a folder (to be used later in preparation for the final exam).

Today's introduction to geometry had absolutely NOTHING to do with shapes. The classes were told that math isn't about numbers and shapes. Math is about the logic and rules that show we can work with numbers and shapes and their relationships. So, our handouts were about logic. There are two types of logic: Deductive Reasoning and Inductive Reasoning. Deductive reasoning uses rules, definitions, laws, and properties to make a logical argument (think of the television shows like CSI, NCSI, or NUMB3RS). Inductive Reasoning looks at patterns and draws a conclusion from them (think of shows like Criminal Minds or Lie To Me). We did two riddle-type logic questions and then looked at a larger logic problem that uses a grid to organize the information and answers. Students are to finish the logic problem for homework.

Also, the unit 3 vocabulary was passed out today. Students are to definite the 27 terms by next Wednesday, November 18th.

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Rational Expressions: Adding & Subtracting

Adding and subtracting of rational expressions requires that the fractions have common denominators. If they do, then add or subtract the numberators and keep the same denominator.

Example:

However, if the fractions do not already have a least common denominator, then the LCD must be made. To do this,

1. Factor the denominators.
2. Multiply each denominator by what the other denominator has but it doesn't already have.
3. Remember, whatever you multiply by on the bottom of a fraction, you must also multiply by in the top of a fraction.

Example:

Homework: Page 173, #1 - 10, 13 - 17, 20; Page 174, #1 - 3

It was very nice to see so many students take advantage of today's help sessions. Remember, every Tuesday and Thursday morning & afternoon there is an IAA teacher willing to work with our IAA students!!

The test will be on Friday covering square roots, polynomials, factoring, and rationals. Students will receive a thorough review sheet tomorrow. There is a review session Thursday morning at 7:45 am, in room 4303.

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Rational Expressions: Multiplying & Dividing

After reviewing the steps to find excluded values and to simplify rational expressions (from Friday), we looked at multiplying and dividing rational expressions.

To Multiply Rational Expressions:

1. Write each expression as a fraction.
2. Factor each expression. Remember, GCF first. Then try to factor a quadratic polynomial.
3. Cancel common factors either vertically or diagonally. But NOT horizontally.
4. Multiply the remaining expressions by multiplying "straight across". Numerator times numerator. Denominator times denominator.

Example:


To Divide Rational Expressions:

1. Write each expression as a fraction.
2. Change to a multiplication problem by flipping the second fraction over (reciprocal!). Many students know this step as "KFC" for Keep, Flip, Change.
3. (Now follow multiplication steps.) Factor each expression
4. Cancel common factors vertically or diagonally.
5. Multiply the remaining expressions straight across.

Example:

Homework: Page 167, #2 - 8 even, #10 - 17 all; Page 168, #2, 4, 8, 10, and 14

Extra Help: The IAA teachers are offering extra help for 30 minutes Tuesdays and Thursdays, before and after school. Students can attend any session, no matter who their teacher is.

Tuesday, 7:45 am, with Mr. Roth in P-123
Tuesday, 3:30 pm, with me, in room 4303
Thursday, 7:45 am, with Ms. Dufresne in room 5207
Thursday, 3:30 pm, with Mrs. Martina in room 5202

Also, if students need more practice with a lesson they can try Purple Math.com or the textbook website for additional tutorials & practice.

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Rational Expressions: Excluded Values & Simplifying

No post yesterday. It was just the quiz.

Today, we worked with rational expressions. Rational expressions are expressions that can be written as a fraction. However, even when the fraction is made with polynomials, we have to remember that the denominator cannot equal 0. So, we have to list any exclude values.

Example: Find the excluded values from the following rational expression.

1. Set the denominator equal to zero: 25u² - 36 = 0
2. Solve the equation to find the excluded values. Look back at Tuesday's notes to remember how to solve by factoring.

Here: (5u - 6) (5u + 6) = 0

So, 5u - 6 = 0 or 5u + 6 = 0

5u = 6 5u = -6

u = 1.2 or u = -1.2

Excluded Values: 1.2 and -1.2

We also discussed how to simplify rational expressions. This concept was taught last Friday, so today's lesson reviewed and reinforced that previous understanding.

Homework: Page 163, #2 - 14 even and #19; Page 164, #3 - 8 and #11 - 13

There will be a test next Friday covering ALL of unit 2: Radicals, Polynomials, Factoring, & Rationals.

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Applications & Review

We went over the homework on solving quadratic equations. These problems required students to first factor (same methods that we have been working with for a week now) and then solve easier equations. Then we looked at some application/word problems. We discussed how the students can do guess-and-check or a quadratic equation to solve the problems.

Homework: Review worksheet for tomorrow's quiz.

There is a review session tomorrow morning in room 4303. The students should bring their completed review worksheet.

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Solving by Factoring

Today's lesson revisits the concept of solving quadratic equations. Last time, we used square roots to solving quadratics (September 30th). This time, we are using the Zero Product Property. This property states that the only way to multiply 2 (or more) factors together for a product of 0 is if at least one of the factors is already 0. (Zero times any number results in zero!)

1. Set the equation equal to zero.
2. Completely factor the polynomial expression.
3. Set each factor equal to zero and solve the equations.

Example: Solve r² + 12r - 15 = 30

Step 1 - Set equal to zero, here subtract 30 from both sides: r² + 12r - 45 = 0

Step 2 - Factor the polynomial: (r + 15) (r - 3) = 0

Step 3 - Set each factor equal to zero and solve: r + 15 = 0 or r - 3 = 0. So, r = -15 or r = 3.

Here, we find out that there are 4 vocabulary terms that are very similar in use/definition: solutions, roots, zeros, and x-intercepts. The methods are almost the exact same for finding any one of those terms. However, when we are presented with a function in function notation, step 1 - Set equal to zero simply becomes rewrite f(x) as 0.

Example: Find the zeros of f(x) = 5x² + 15x - 50.

Step 1 - Set equal to zero, here rewrite f(x) as 0: 0 = 5x² + 15x - 50

Step 2 - Factor the polynomial, here a gcf and then a quadratic:

First, 0 = 5 (x² + 3x - 10)

Then, 0 = 5 (x + 5) (x - 2)

Step 3 - Set each facgtor equal to zero and solve: 5 = 0 (not possible!) or x + 5 = 0 or x - 2 = 0. So, x = -5 or x = 2.

Homework: Page 79, #2 - 12 even and #22 - 30 even. Page 84, #10 - 24 even.

Remember: Quiz Thursday. Review session Thursday morning, 7:45am, in room 4303

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Monday's Notes ...

I was out sick. However, thanks to the help of my colleagues, the students didn't miss out on the lesson.

Monday's lesson went over the last two types of factoring that we will use this semester.

Differences of two squares: This type of quadratic polynomial has two perfect square terms that are being subtracted. The factors of this type look like the conjugates that we have seen several times in the last month and a half. To make the middle (linear) term of the quadratic drop out, the factors must have had the exact same terms, but opposite signs.

a² - b² = (a - b) (a + b)

Example: Factor 16x² - 25 = (4x - 5)(4x + 5)

Factor by grouping: This method of factoring is often used when there are 4 (or more) terms in the polynomial.

1. Create groups of the terms in such a way that each grouping has a gcf.
2. From each group, factor out the gcf.
3. The remaining factors in the parenthesis must be the same to continue. Now that the parenthesis have the exact same terms, that is a gcf of both parts of our polynomial expression. Factor the parenthesis out to the front and create another parenthesis with what is left behind.

Students saw a similar problem to Step 3 in notes from last Wednesday: Factoring by Using a Greatest Common Factor, example "l".

Example: Factor 12x³ - 8x² - 18x + 12

Step 1 - Group: (12x³ - 8x²) + (-18x + 12)

Step 2 - Find GCFs in the groupsing: 4x² (3x - 2) + -6 (3x - 2)

Step 3 - Factor out the GCF Parenthesis: (3x - 2) (4x² - 6)

Homework: Finish the Handout

Remember - there will be a quiz on Thursday covering all uses of Factoring! There will also be a binomial expansion problem (from last Monday). We will have a review session Thursday morning at 7:45 am in room 4303 to prepare for this quiz.

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Bringing it together ...

Today's guiding notes worksheet brings together the concepts from last week (dividing polynomials by cancelling common factors) and this week (doing the factoring on our own). In last week's problems, the division problems were already factored. Today's lesson requires us to factor first, and then cancel the common factors.

NOTE: Many students experience difficulty with today's homework because they are not reading the directions. The steps are clearly spelled out and nothing is new on its own.

1. Remember, it may be helpful to rewrite a division problem as a fraction. To do this, the dividend (what is being divided into) goes in the numerator. The divisor (what is doing the dividing) goes in the denominator.
2. Factor the polynomials. First, look for a gcf. If there is one, pull it to the front. Then, if the polynomial is a quadratic, factor it like yesterday's lesson.
3. Cancel any common factors that appear in both the numerator and the denominator.
4. Write the simplified answer.

Example: Simplify the following by first factoring out GCFs.

Example: Simplify the following by first factoring the quadratic polynomials.

Example: Simplify the following by first factoring out GCFs and then factor any remainging quadratic polynomials.Homework: Finish the handout.

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Factoring Quadratic Trinomials

We continued factoring polynomials today, but we kept them to the form ax² + bx + c. These are called quadratic polynomials because of the ax² term.

Example: Factor x² + 5x - 36

1. Consider factors of the constant term (c, or the last term). Here: -36 is -1*36; -2*18; -3*12; -4*9; -6*6; -9*4; -12*3; -18*2; -36*1.


2. Choose the pair of factors whose sum is the coefficient of the linear term (b, or the middle term). Here: -4 + 9 = 5


3. Create your factors (2 sets of parenthesis) by filling in the variable and the factors chosen in step 2. Here: (x - 4) (x + 9)

For our Hopewell Middle School students, we made the link from today's lesson to the Diamond Puzzles that they did several times last year. They were presented with a diamond with two numbers. They were instructed to find two numbers that multiply to make the upper number and add to make the lower number. The following diamond puzzle is similar to the above example.

The two numbers needed to complete the puzzle are -9 and 4, as shown below.

Homework: Page 83: #1 - 9; Page 84: #1 - 6; Page 91: #1, 7, and 13

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Wednesday's Notes ...

Yesterday, we reviewed how to find the greatest common factor (gcf). We added to this concept how to factor it out of the polynomial.

Example: Factor 9a³ + 12a² - 15a

1. Look at each number and identify the largest number that can go into each evenly. Here, 3 goes into 9, 12, and -15 evenly.
2. Look at each variable and identify how many each term has in common with the others. The students were told to find the smallest exponent because that term limits how many the others can have in common. Here, each term has one a-variable. While the first and second terms have more a-variables, the last term only has one so it limits the common factor.
3. Write the GCF and then start parenthesis after that: 3a ( ______ )
4. To find what polynomial goes inside the parenthesis, determine term by term either "what is left when we divide by the gcf" or "what multiplies with the gcf to make the term". Here, 3a * 3a² makes the first term of 9a³. 3a * 4a makes 12a². And, 3a * -5 makes -15a. So, the final answer after factoring out the gcf is 3a (3a² + 4a - 5).
5. We can check every factoring problem by multiplying the factors back together. Their product should be the original polynomial. Remember, add exponents when multiplying bases. Here: 3a * 3a² + 3a * 4a + 3a * -5 = 9a³ + 12a² - 15a.

Homework: Page 79: #13 - 21 and Page 80: #10 - 18

There will be a quiz on Thursday, October 29 on factoring and solving by factoring (for the next week).

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Grades ...

Today in class, after going over Monday's homework, all graded papers were passed back: Quiz #3, Quiz #4 and the EOCT Practice Test. We took a good deal of class time to go over these assessments. The EOCT Practice Test is a good indicator of how well each student understands and retains the concepts from the first 9 weeks of school. This test will not hurt a student's average. I entered these grades in parent connect in a manner that won't affect the grades, but it allows parents and students to see the grades. If a student has a quiz grade lower than this EOCT practice test grade, then it replaced one lower quiz grade.

Students received a grade print out with all of their grades. Most of these progress reports also have notes on them about the EOCT Practice Test and the quiz replacement. These papers need to be signed by the parents and returned to school.

Notice: We were going to have a test on Thursday. That test is cancelled. We will move on to the concept of factoring polynomials tomorrow. More information to come soon!

Homework: Square Root and Polynomial Review Worksheet

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Notes from Monday ...

Yesterday's lesson was on Binomial Theorem, which is a short cut for raising binomials to exponents. We simplify this process as much as we can. One way we simplify this is to use Pascal's Triangle. This "triangle" is an arrangement of numbers that continues forever. Each number is made by adding together the two numbers above it. Here is the triangle through the first 5 rows (the top number is considered to be row 0). The 6th row would be 1, 6, 15, 20, 15, 6, 1.


Example: Simplify (x + y)⁴

This problem would require us to multiply out (x + y)*(x + y)*(x + y)*(x + y). The students saw through a problem in class that this is a lengthy process with many places to have errors. The Binomial Theorem and Pascal's Triangle makes this much simpler.

1. Note the exponent and go to that row of Pascal's Triangle. Here, the exponent is 4 so we will use row 4: 1, 4, 6, 4, 1. These numbers become our coefficients (numbers in front of the variables). We write the numbers down followed by 2 blanks each: 1_ _ 4 _ _ 6 _ _ 4 _ _ 1_ _
2. To start filling in the blanks, use the first term inside the parenthesis (here, x). This term will start with the exponent of the problem (here, 4) and then count down as it fills in the blanks from left to right: 1 x⁴ _ 4 x³ _ 6 x² _ 4 x¹ _ 1 x⁰ _
3. To finish filling in the blanks, use the second term inside the parenthesis (here, y). This term starts with the exponent of 0 and counts up to the exponent of the problem as it fills in the blanks from left to right: 1 x⁴ y⁰ 4 x³ y¹ 6 x² y² 4 x¹ y³ 1 x⁰ y⁴ . Notice that each term has exponents that add together to 4 (the exponent in the problem).
4. Simplify, if possible. Remember, any non-zero value raised to the 0 power is 1, so we remove that from our answer. If there are any numbers raised to powers, this is the step to simplify those powers and multiply times the coefficients. Here: 1 x⁴ 4 x³ y¹ 6 x² y² 4 x¹ y³ 1 y⁴
5. Enter the signs throughout the polynomial answer. If the binomial has addition, then all signs in the answer are plus signs. If the binomial has subtraction, then all signs alternate plus and minus, with the first term being positive. Here: 1 x⁴ + 4 x³ y¹ + 6 x² y² + 4 x¹ y³ + 1 y⁴

Homework: Page 75, #2 - 16 even and #24

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Quiz #4

Today, the IAA classes took a quiz covering polynomials. Students had the option to come in this morning for a review session. Because there is no handout at the review sessions, students need to be sure to have their own paper and copy down the notes/examples that Ms. Como provides as review.

Students will not have math class tomorrow. Instead they will take the PSAT in their homerooms. I will be out Thursday and Friday of this week at a math teacher conference. Thursday, students will have a guide to working with polynomial division. Friday, students will take a practice EOCT. This practice test will be in a multiple choice format. It will be cummulative with problems from the last 9 weeks. This practice test cannot hurt a student's average. However, it may be used to raise a grade.

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Quiz #4 Review

We played a jeopardy game to review polynomials today. Students should study the polynomial vocabulary as well as the steps to add, subtract and multiply polynomials. All of these concepts were in the lessons last week. Tomorrow we will have a quiz.

There will also be a review session tomorrow morning at 7:45 in my classroom (4303).

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Multiplying Polynomials

After going over last night's homework assignment (to finish yesterday's packet), we compared 4 different ways to multiply polynomials: tiles, Punnet Square, FOIL, and distribution. Students are encouraged to use whichever method works the best for them. We discussed how to multiply monomials (one term times another).

Example: 4x²y * -6x³y²

1. Multiply the coeffiecients together: 4 * -6 = -24
2. Multiply the variables together by adding exponents: x² * x³ = x⁵ and y * y² = y³
3. Write the product: -24x⁵y³

We also practiced multiplying polynomials. See yesterday's blog for examples of the tiles and Punnet square methods. FOILing and distributing requires us to multiply each term in the first polynomial times each term in the second polynomial. Finish by collecting like terms. Remember, do not change the exponents when collecting like terms (adding/subtracting). [FOILing stands for First, Outer, Inner, and Last. Those are the pairs of terms based on location to multiply. It ONLY works when multiplying a binomial (2 terms) times another binomial (2 terms) such as (4x + 2) (5x - 8).]

Example: (4x + 3) (5x² - 2x + 10)

= 4x (5x²) + 4x (-2x) + 4x (10) + 3 (5x²) + 3 (-2x) + 3 (10)

= 20x³ - 8x² + 40x + 15x² - 6x + 30

= 20x³ + 7x² + 34x + 30

Homework: Page 67: #2 - 20 even AND Page 71: #2 - 18 even

Remember, there will be a quiz next Tuesday covering this week's lessons.

Enjoy your 3 day weekend from school!!!

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Multiplying with tiles

We revisited the algebra tiles today. This time, the times set up a rectangle of given dimensions and we found the area inside the rectangle by counting up the tiles inside the rectangle. Because area is also found by multiplying length times width, this model finds the product of the two polynomial dimensions.

Example: Find the product of (x - 3) * (x + 2).

1. Set up the dimensions to the side of the work space (as shown above and to the left of the diagram below).


2. Fill in the space with shapes matching the dimensions. Or, continue the lines down and across that are started in the dimensions.


3. Record the values of the shapes in the resulting rectangle as the product. Simplify zero pairs.

Product = x² - 3x + 2x - 6 = x² - x - 6

We also found the product of the two polynomials by using a chart that is similar to the Punnett squares that the students learned in Biology in middle school (and will see again this year).

Example: Find the product of (x + 2) * (x² + 3x + 1)

1. This process again requires the students to set up the polynomials on the top and side of the work space (this time as collected terms rather than individual tiles).


2. Each box in the grid is found by multiplying the term above and to the left.


3. Collect like terms for the answer (usually on diagonals).

We will continue working with this concept tomorrow. There is a quiz scheduled for Tuesday covering adding, subtracting, and multiplying polynomials.

Homework: Finish today's packet

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Adding & Subtracting Polynomials

Our lesson today covered polynomial vocabulary. Then we again found sums and differences with polynomials. This builds on yesterday's lesson with the tiles, but we just used symbols today. Remember, for subtraction problems we change it to addition and change all signs in the second polynomial. This step distributes the minus through the second polynomial. To add the polynomails, collect like terms. Do NOT change the value of the exponent!

Examples:
(4x² + 5x - 8) + (2x² - 3x - 4)
= 6x² + 2x - 12

(9x² - 2x - 5) - (7x² - 6x + 4)
= (9x² - 2x - 5) + (-7x² + 6x - 4)
= 2x² + 4x - 9

Homework: Page 61: #1 - 14 and Page 62: #14 - 19

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Catch up post! (Sorry!)

So, Thursday we used word problems to set up more quadratic equations. We solved these equations using the same steps as we learned on Wednesday. There was no homework.

Friday, we had a quiz covering square roots. There was no homework.

Today (Monday), we used colored tiles called Algebra Tiles to learn the processes of adding and subtracting polynomials.
For adding polynomials:

1. Group/Collect like terms - the terms that have the same variable(s) to the same exponent
2. Add the coefficients, keep the term like what they started as - same variable(s) to the same exponent
3. Write the resulting polynomial

For subtracting polynomials (KFC or "add the opposite"):

1. Keep the first polynomial the same
2. Flip the signs of the second polynomial (positive to negative, and vice versa)
3. Change the problem to addition
4. Follow the steps for addition

Homework: finish today's packet. Students could checkout a packet of Algebra Tiles from me. They may use the tiles on next week's quiz (but they do not have to use them).

Quiz: Next Tuesday, October 13th covering adding, subtracting, and multiplying polynomials.

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Solving Quadratics by Square Rooting

We went over homework and answered any questions from the assignment that were asked. If students are still struggling with simplifying square roots, they should make time to see a teacher outside of class for extra help. Our lesson today was on solving quadratic equations by using square roots to cancel the squared variable.

Homework: page 121 #2 - 26 even (not #20) and page 122 #1 - 6

Reminder: The quiz will be Friday. There is a review session available Friday before school in room 4303.

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Reviewing square roots

The students needed some more practice working with simplifying square roots, so we took today to answer questions, do a practice worksheet, and not move forward with a new concept. This will push our quiz off to FRIDAY. This will also push our morning review session off to FRIDAY. It will still be in room 4303 at 7:45am.

Homework: Finish worksheet "Lesson 3.4 Practice" and Unit 2 Vocabulary

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Dividing with square roots

We spent time today going over the assignment from the weekend. After that, our lesson covered how to work with division of square roots.

1. Simplify any radicals with perfect squares
2. Reduce the fraction, if possible ... check again for perfect squares
3. Rationalize the Denominator - this is a process of eliminating square roots from the denominator of a fraction. This can take two forms. One form has us multiply the top & bottom of the fraction by a radical that will make the denominator into a perfect square. The other form has us multiply the top & bottom of the fraction by the conjugate of the denominator. Conjugates are specially made to eliminate square roots when they are multipied together. Conjugates have the same numbers & radicals, but they have opposite signs in the middle.

Examples of conjugates: (4 + √3) and (4 - √3); also (6 - √10) and (6 + √10)

For more help rationalizing denominators, try http://www.purplemath.com/modules/radicals5.htm

In fact, all of the lessons at Purple Math are well written with several examples.

There will be a quiz on Wednesday. We will hold a review session Wednesday morning at 7;45 in room 4303 for that quiz.

Homework: page 144 #10 - 18 & page 145 #8 - 15

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Square Roots

We reviewed simplifying square roots. We built on their previous knowledge of square roots by showing how to add, subtract, and multiply with radicals. It is important to remember that we must have like terms to add or subtract radicals (exact same square root). When multiplying radicals, we multiply their coefficients (the numbers on the outside) and we multiply the radicands (the numbers on the inside) separately.

Homework:
Due next Thursday - Unit 2 vocabulary definitions
Due Monday - Page 144: #1 - 9, #19 - 24 and page 145: #1 - 7, #16 - 20

There is a quiz scheduled for next Wednesday that will assess the students' understanding of simplifying square roots and using square roots to solve quadratic equations (lessons from today through Tuesday).

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TEST

Yep, today is the test. Before the test, students had to total their earned points from the 9 assignments that we had for this half of Unit 1. Once those points were totaled, the pink homework stamp sheets were turned in to me. They will get a new homework stamp sheet tomorrow for the new assignments for the first half of Unit 2.

A few students needed the full classtime to complete the test. But, most students finished with enough time left in the period to start their homework assignment.

Homework: "Why Did Krok ..." worksheet with problems on simplifying square roots. These problems are review problems. Students must show work for each problem to get full credit tomorrow. Just a reminder: break up the number under the square root sign to find factors that are perfect squares so that they can be square rooted.

Example: √75 = √25 * √3 = 5 √3

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Review

Class began today with a notebook quiz. The students had to have completed notes and assignments from the last 3 weeks to be able to answer the questions asked. After that was finished, we went over the review sheet and answered questions to prepare for tomorrow's test.

Homework: STUDY!!!

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No school!

I hope every one is making the most of their day off from school.

The review session that was originally planned for today will now happen tomorrow morning, 7:45 am, in room 4303. We will push the test off to Thursday.

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Rainy Monday Review

We began class by going over the homework from the weekend. Then we reviewed the differences in recursive and explicit forms of the sequence formulas. We identified formulas as one or the other form, and we found the first 5 terms of the sequence. We also built both of the forms of the sequence formulas for a couple given sequences. Finally, we started on a review worksheet for this week's test. This needs to be finished for homework.

Reminder: tomorrow morning, review session, at 7:45 am, in room 4303

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Sequence formulas

Today, we discussed how to define formulas (or rules) for sequences. This can be done with a recursive formula that defines each new term in the sequence by how to make it from the previous term. For example, the recursive formula for 5, 8, 11, 14, ... is:

• a₁ = 5 ← This is the starting or first term
• a_n = a_n-1 + 3 ← This is the what the sequence does, it adds 3 to the previous term.

This can also be done with an explicit formula which defines each new term in the sequence based on its position. For example, the explicit formula for 5, 8, 11, 14, ... is:

• First, we recognize that there is a rate of change of 3 for the terms in the sequence. This becomes the slope for the formula. We will start with a_n = 3n
• But, when we substitute n = 1 to find the first term, 3 * 1 = 3. This is 2 units short of the first term. We will add 2 to our formula to bring the output values up to the sequence terms.
• a_n = 3n + 2

Homework: Complete the handout with Sequences: Part 5 and Sequences Homework on the front and back. We will spend Monday reviewing all of the sequences concepts. We have a test scheduled for Wednesday. There will be a review session in the morning, Tuesday the 22nd, at 7:45 in room 4303.

Have a great weekend!

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Sequences, day 1

We started learning about sequences today. We identified patterns in sequences of numbers. We also discussed how the term is each number, shape, or item in the pattern, and the index is the placement of the term in the sequence (1st, 2nd, 3rd, etc...).

Homework is ...
page 185: #1 - 3
page 187: #2 - 18 even
page 188: #2 - 16 even

We will have a test on Wednesday, September 23rd covering Transformations, Rates of Change, and Sequences. There will be a review session on Tuesday, September 22nd at 7:45 in room 4303. We will also review in class

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Recovery information

All quizzes and tests have now been returned to the students for them to keep. They already learned of their grades for these tasks when they saw the grade handouts on Monday. However, we discussed the process for RECOVERY today. Recovery is only available for failing test grades. To be eligible, students must complete 90% of the homework assignments from the material on that test. I do not limit the number of times a student may use the recovery process. However, they are limited to only earning up to a 70% from recovery.

For the test on Unit 1 Part 1, the students must have earned 35 out of 39 points on the assingments. To do recovery on this test, they need to schedule time with me either before or after school to rework the problems that they missed. Recovery must be done by next Wednesday, September 23rd.

Recovery is not available for quizzes. However, there will be other opportunities later in the semester to improve quiz grades.

There is no homework for math today!

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More Average Rates of Change

Today we continued working with average rates of change. However, the numbers were presented this time in function notation format rather than word problem. This lesson also appears in the textbook on pages 180 and 181, in lesson 3.12.

Homework is in the textbook. Page 183: #1 - 7. Page 184: #1 - 6

Remember, tomorrow is an Early Release Day. Tests and Quizzes will be returned along with going over the homework.

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"Walking, Falling ..."

Today, I passed out grade sheets. These grade sheets should be shown to your parents, signed, and returned for a stamp. All entered grades for our class can be seen, including the most recent test and quiz. Those graded tasks will be returned on Wednesday. I am just waiting for everyone to have taken them.

We went over "Walking ..." from last Wednesday. This handout introduced the concepts of rate of change and average rate of change. We investigated how Dwain's speed remained constant at 4 feet per second, while Bryan's speed changed each of the first 5 seconds. However, they both ended their race in a tie.

We ended class by starting "... Falling ..." Remember, although there is a constant change to the speed of the falling ball, it is still a changing speed. It has a constant increase or acceleration of 9.4 (this is the acceleration due to gravity on Earth). The formula to calculate the average rate of change is:

change in distance

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change in time

Homework: Finish "... Falling ..."

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Quiz Day!!!

Today, we had a quiz on Parent Functions and Function Transformations. We have been working with these concepts since Tuesday last week.

Remember to bring "Walking ... Part 1" to class with you on Monday. You were assigned to finish the handout on Wednesday. We will go over that and continue discussing Rates of Change next week.

Have a great weekend!

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