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
Tuesday
Monday, November 30, 2009
Wednesday, November 18, 2009
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!
Homework: Study!
Monday, November 16, 2009
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:
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.
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°
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!
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!
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
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
Thursday, November 12, 2009
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.
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.
Tuesday, November 10, 2009
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.
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.
Tuesday, November 3, 2009
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,
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.
Example:
- Factor the denominators.
- Multiply each denominator by what the other denominator has but it doesn't already have.
- Remember, whatever you multiply by on the bottom of a fraction, you must also multiply by in the top of a fraction.
Example:
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.
Monday, November 2, 2009
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:
To Divide Rational Expressions:
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.
To Multiply Rational Expressions:
- Write each expression as a fraction.
- Factor each expression. Remember, GCF first. Then try to factor a quadratic polynomial.
- Cancel common factors either vertically or diagonally. But NOT horizontally.
- Multiply the remaining expressions by multiplying "straight across". Numerator times numerator. Denominator times denominator.
To Divide Rational Expressions:
- Write each expression as a fraction.
- Change to a multiplication problem by flipping the second fraction over (reciprocal!). Many students know this step as "KFC" for Keep, Flip, Change.
- (Now follow multiplication steps.) Factor each expression
- Cancel common factors vertically or diagonally.
- Multiply the remaining expressions straight across.
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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