Suffolk Public Schools Mathematics Honors to AP Preparation Marks Success

Honors Geometry

Pre-requisite: B or better in Algebra I

Expectations: Students will be provided with hands-on experiences to explore such concepts as isometric drawings, tessellations, and three-dimensional figures. The computer will be used to explore transformations of geometric figures with the aid of Geometers Sketch pad. In addition, students will be working on activities designed to increase their level of thinking and problem solving ability and to explore the international applications of mathematics.

Suggested websites: www.regentsprep.org www.pearsonsuccessnet.com

https://odyssey.spsk12.net/clologin.aspx www.khanacademy.org

Summer Assignment:

Assignment Objective: The goal of this assignment is for you to be comfortable with the Algebra 1 topics so that we can focus on the Geometry. Upon return to school, you will be given a quiz on these topics.

Assignment Evaluation: This assignment will be due the first day of class and counted as your first quiz grade worth 35 points. Be sure to give your very best effort, please understand that this is an individual assignment, so it is expected that you do your own work.

Directions: There are a total of 35 questions. You will need five pieces of loose leaf paper. Fold your paper like a “hotdog” once and a “hamburger” twice, resulting in eight rectangles on the front and eight on the back. Number the rectangles, left to right so the odd numbered problems are on the left and even numbered problems are on the right. You must show all calculations in order to earn credit. A sample example is provided for each topic; if you need additional help please refer to the suggested websites listed above.

I. Solving Multi-Step Equations:

Example: solve the equation: -18 – 6k = 6(1 + 3k) (given problem)

-18 - 6k = 6 + 18k (distributive property)

-18 = 6 + 24k (combine like terms)

-24 = 24k (combine like terms)

-1 = k (division property)

II. Solving Multi-Step Inequalities:

Example: solve the inequality: -p + 6p ≤ 4 + 6p (given problem)

5p ≤ 4 + 6p (combine like terms on same side)

-p ≤ 4 (combine like terms)

p ≥ -4 (multiply or divide by negative #, flip symbol)

III. Finding Slopes and Graphing Lines:

Example 1: find the slope of the line going through the points (5, -3) and (8, 5).

Use ; ; if we had to find parallel slope it would be the same: ; if we had to find the perpendicular slope (negative reciprocal of it): .

Example 2: find the slope and y-intercept of the given line and then graph the line using slope and y-intercept.

Given: , first solve the equation for y so that it is in the form of y = mx + b, where m = slope and b = y-intercept. When we solve the above equation for y, we get: ; so the slope is and y-intercept is -3. Now, use the y-intercept and the slope to graph this line.

IV. Solving Systems of Linear Equations:

Example 1: Solve the system using “Substitution Method”: ; with this method, you want to solve for the variable that has the coefficient of 1, so in this example, we would want to solve for the y in the first equation. When we do that, we get: . Now, use this equation to represent the y in the second equation. So, we get:

(distributive property)

(combine like terms)

(division property)

Thus, we substitute x=-1 in the equation y=2x+3 and get y=1. We write our answer in ordered pair form, because this is the point where the two lines intersect. The solution is (-1, 1).

Example 2: Solve the system using “Elimination Method”: ; with this method, you want to make sure that all your variables and constants are line up underneath each other. Your goal is to make one variable coefficient the same with opposite signs. This way you can add the two equations and have only variable to solve for. Then, use substitution to finish the problem. For our example: I want to eliminate the y, so I will have to multiply the top equation by 9 and bottom equation by 4.

(after doing the above)

Now, add the two equations to get: . Solve for x to get x=6. To find y, you must plug 6 in for x in one of the original equations. I will use the top equation to get: , solve for y to get y = -15. Place answer in ordered pair form: (6, -15) is the solution.

V. Factoring:

Example 1: Factoring special cases like difference of squares and perfect square trinomials:

A. Factor: ---- this is difference of squares, where the pattern is when we

have

B. Factor: or we can write as ---- this is a perfect square

trinomial, where the pattern is when we have

Example 2: Factoring Trinomials – there are so many different methods that are used to accomplish this, I

will show you only 1, it is called “Slide and Divide”.

Factor

(Slide the coefficient to the back by multiplying the last term by it, so 2 x 9 = 18)

Now look for factors of -18 that will give us 3 (that is the middle term), we get 6 and -3, because

6 x -3 = -18 and 6 + (-3) = 3

(set up our factors, use the first term as the variable that is given in the problem)

(we divide the factors by the number we slid back, in this case it was 2)

(Reduce, we can reduce 6/2 to 3 and 3/2 cannot be reduced, so leave it)

(Write the factors properly by slipping the 2 in front of n in the second factor and that

is our final answer.)

VI. Solving Quadratic Equations by Factoring and Quadratic Formula:

Example 1: Solve by Factoring: (given problem)

(must write in standard form by setting equation equal to 0)

(factor out the GCF)

(factor the trinomial)

(set each factor = to 0)

Example 2: Solve by using the Quadratic Formula:

(given problem)

(list a, b, and c values from equation)

(plug in a, b, and c values into the formula)

(simplify)

(take the square root of 169)

(set up the two solutions and simplify)

VII. Simplifying Radicals and Dividing Radicals:

Example 1: Simplify. (given problem)

3100∙2∙m2∙m2∙n (rewrite 200 using perfect squares & as multiples of )

3∙10∙m∙m2n (take square root of 100 and the two s)

(simplify)

Example 2: Simplify. (given problem)

45∙55 (cannot have on the bottom, rationalize, which means to multiply the top and bottom by )

(multiply top and bottom)

(simplify all radicals and make sure all fractions are reduced)

Final answer:

I. Solve the following equations:

1) 2(4x – 3) – 8 = 4 + 2x 2) -3(4x + 3) + 4(6x + 1) = 43 3) x – 1 = 5x + 3x – 8

II. Solve the following inequalities:

4) 4(8 – 2x) – 2x ≤ 32 5) 7 + m ≤ 2 or m + 1 > 2 6) -53 < 9v + 1 < -26

III. Find the slope of the given line, and then find the parallel and perpendicular slope.

7) A(7, 9) and B(-2, -1) 8) C(0, -2) and D(4, -2) 9) E(6, -5) and F(6, 10)

Find the slope and y-intercept for each line, and then graph the line, exactly!

10) 11) 12)

m=______, b=______m=______, b=______m=______, b=______

IV. Solve the following systems of equations; make sure to place your solution in an ordered pair form.

13) 14) 15) 16)

V. Factor the following:

17) 18) 19)

20) 21) 22)

23) 24) 25)

VI. Solve the following by factoring:

26) 27) 28)

Solve the following by using the Quadratic Formula:

29) 30) 31)

VII. Simplify and Rationalize completely.

32) 33) 34) 35)