Name Geometry 6 Prerequisites Dear Incoming Geometry Student, Listed below are fifteen skills that we will use throughout Geometry 6. You have likely learned and practiced these skills in previous math classes. Although we will briefly review a skill in class the first time we use it, you are ultimately responsible for the mastery of the skill. The following pages describe the target skills and provide some worked-out examples along with an opportunity for you to practice the skill on your own (answers are on the last three pages of the packet). You should work out the problems by yourself. This packet is not mandatory, but we highly recommend that you work enough problems from each section to reassure yourself that you have mastered the skill. If you need help please visit with me outside of class, or visit the Titan Learning Center (in the GBS library) to meet with a tutor. There are also many online options such as Khan Academy. Numbers Target N: Target N: Target N: Target N4: I can multiply a fraction and a whole number without a calculator. I can simplify the square root of a whole number without a calculator. I can multiply square roots without a calculator. I can add square roots without a calculator. Algebra Target A: Target A: I can multiply polynomials (monomials and binomials) without a calculator. I can solve linear equations without a calculator. Target A: I can solve proportions, including those that lead to x = n (where n is a whole number), without a calculator. Target A4: Target A: Target A6: I can solve a system of linear equations by the elimination method. I can solve a quadratic equation with factoring and without a calculator. I can solve a quadratic equation with the quadratic formula. Geometry Target G: Target G: Target G: Target G4: Target G: I can determine the coordinates of a point in the xy-coordinate plane. Given two ordered pairs in the xy-plane, I can calculate the distance between the points, the slope of the line connecting the points, and the coordinates of the midpoint. Given two side lengths of a right triangle (a triangle with a 90 angle in it), I can use the Pythagorean Theorem to calculate the exact length of the missing third side. I will not attempt to use the Pythagorean Theorem unless I have evidence that it is a right triangle. Given the graph of a line with a definite y-intercept and slope, I can write the equation of the line in slope-intercept form (y = mx + b). If the line is vertical, I can write its equation in the form x = #. I can accurately graph a line given in slope-intercept form without a calculator. Geometry 6 Prerequisites page
Numbers Target N: I can multiply a fraction and a whole number without a calculator. Examples: (6) 7 (4 9) C) (7+64) 8. Multiply the following, without a calculator. (4) 8 4 C) 4 (4) D) (8 9) E) ( 7) F) (8 ) 4 G) 6 (4+ 48) H) 9 (6+ 4) I) 7 (49+6) Target N: I can simplify the square root of a whole number without a calculator. You undoubtedly know that will simplify to, but what will 4 simplify to? In the screen shot shown below you can see an exact answer and an approximate answer. Exact Approximate Geometry 6 Prerequisites page
When the square root of a number does not have a whole number answer, it may still be possible to simplify it. First we need to review the following property of real numbers: If a and b are both non-negative real numbers, then a b = a b. To simplify a square root you first factor the number into primes and then (if possible) look for any perfect square factors that you can take the square root of. The factorization of 4 into primes is 4 =. Using the property above we can write 4 = 4 6 = 4 6. We do this is because we 4 = and we leave 6 know that as it is. Therefore can be simplified to 6. Many numbers e.g.: 6, 7, 0 do not have perfect square factors and cannot be simplified. Examples: simplify the following square roots without a calculator. 4 8 7 C) 90 D) 00. Simplify (if possible) the following square roots without a calculator. 99 8 C) 4 D) 0 E) 0 F) 6 G) 6 H) 0 I) 00 J) K) 80 L) Geometry 6 Prerequisites page
Target N: I can multiply square roots without a calculator. To multiply square roots we need to review the following property of real numbers: If a and b are both non-negative real numbers, then p a q b = p q a b. The "outside" numbers multiply together, and the "inside" numbers multiply together. Since it might be necessary to simplify the square root, you should not actually multiply the a and b, but rather use them as the factor tree to see if there are any perfect squares. Examples: multiply (and simplify) the following square roots without a calculator. 48 7 C) 6 D) 0 6. Multiply (and simplify) the following square roots without a calculator. 8 7 4 C) 0 6 D) 4 6 E) 0 F) G) H) 6 8 I) 8 Geometry 6 Prerequisites page 4
Target N4: I can add square roots without a calculator. Square roots can be added only when the number under the square root is the same for both terms: p a q a is the same as ( p q) a. Note that adding square roots is similar to adding like terms: x + 4x = ( + 4)x = 9x, and 7 +4 7 =(+4) 7 =9 7. Examples: add (and simplify) the following square roots without a calculator. 0 6 6 C) 8 D) 4 7 4. Add (and simplify) the following square roots without a calculator. +7 0 C) + 6 D) + 8 E) 0 + 80 F) 0+ G) + 7 H) 7 + I) 0 + 0 Geometry 6 Prerequisites page
Algebra Target A: I can multiply polynomials (monomials and binomials) without a calculator. Examples: 9x 4x x(4x 7) C) (x + 6)(x ) D) ( x +0). Multiply the polynomials without a calculator. 7x 8x (4x)(x) C) 6x x D) x(6x 4 x) E) (4x 9) F) -6(x ) G) x (4 x ) H) x(x + 9) I) x (x 0) J) (x + )(x + 0) K) (x )(x 7) L) (x + 9)(x 4) M) (x + )(x + ) N) x x + 4 6 O) ( x ) P) (x+ ) Geometry 6 Prerequisites page 6
Target A: I can solve linear equations without a calculator. Examples: 4x + 0 = 7x x +0 = 40 C) 4 (x ) = 9 6. Solve each equation for x without a calculator. 8x + = x 9 (x + ) = x + 8 C) 4 + 6(x + 8) = 6 D) x + 8 + (x ) = 6 E) 4 (x + ) = 7 F) 6 (x + ) = 4 G) x (x + 8) = 9 H) x 7 = I) x + = 0 4 J) x += K) x 9 = 4 L) x = x+8 Geometry 6 Prerequisites page 7
Target A: I can solve proportions, including those that lead to x n without a calculator. = (where n is a whole number) Examples: 7 = x +6 4 x = C) 9 x x = x 7. Solve each proportion for x without a calculator. x = 8 = x 4 C) 4 = x +7 9 D) 4 = x x E) x = 0 x F) 0 x = x Target A4: I can solve a system of linear equations by the elimination method. Examples: x+ y=7 x y= 8 x+ y= x 4 y= C) x+ y= x + = -y Geometry 6 Prerequisites page 8
8. Solve each system of linear equations (you may use a calculator for arithmetic). x y=0 4 x+ y= 6 x+ y= 9 x y=4 C) x+ y = x+ y = 6 D) 4 x+ y= x y=4 E) x+6 y= 4 y + x =7 F) y x= 8 x+ y= - G) 4 x+ y= 9 x y= H) 4 x+6 y=0 6 x+ y= I) x+ y=0 x+7 y= Geometry 6 Prerequisites page 9
Target A: I can solve a quadratic equation with factoring and without a calculator. Examples: x 9=0 x 8 x = 0 C) x 9 x+0=0 9. Solve each quadratic equation with factoring and without a calculator. x =0 x 6 =0 C) x 8 x=0 D) 4 x +0 x=0 E) x =x F) x +7 x+0=0 G) x +4 x+=0 H) x 9 x+4 =0 I) x 8 x =0 J) x + x=8 K) x +6 = x L) x = x 40 Geometry 6 Prerequisites page 0
Target A6: I can solve a quadratic equation with the quadratic formula. If a quadratic equation is in the form ax +bx +c = 0, then there are two solutions to this equation: - b+ b 4ac x= a and - b b 4ac x= a. Examples: solve the quadratic equations with the quadratic formula. x + x = 0 x + =7 x 0. Solve each quadratic equation with the quadratic formula (you may use a calculator for arithmetic). 7 x + x = 0 x x+ = 0 C) 4 x + x+= 0 D) x +76 = 60 x Geometry 6 Prerequisites page
Basic Geometry Target G: I can determine the coordinates of a point in the xy-coordinate plane.. Use the xy-plane to determine the coordinates of each point. A(, ) B(, ) C(, ) D(, ) E(, ) F(, ) G(, ) H(, ) Target G: Given two ordered pairs in the xy-plane, I can calculate the distance between the points, the slope of the line connecting the points, and the coordinates of the midpoint. For two points ( x, y ) and ( x, y ) The distance between the points is x x + y y The slope of the line connecting the points is y x y x The coordinates of the midpoint is x + x y + y, Example: (,7) and x y (,7) y x Distance: Slope: Midpoint: Geometry 6 Prerequisites page
. Determine the length of the segment, the slope of the segment, and the midpoint of the segment. Distance: Slope: Midpoint: C) Distance: Slope: Midpoint: D) Distance: Slope: Midpoint: Distance: Slope: Midpoint: Geometry 6 Prerequisites page
E) Distance: Slope: Midpoint: F) Distance: Slope: Midpoint: Target G: Given two side lengths of a right triangle (a triangle with a 90 angle in it), I can use the Pythagorean Theorem to calculate the exact length of the missing third side. If the right angle is missing, I will not attempt to use the Pythagorean Theorem. In the right triangle shown at right, the Pythagorean Theorem states that (leg #) + leg # = hypotenuse (leg # and leg # are interchangeable and always intersect at the right angle) leg # leg # hypotenuse Example: determine the length of the missing side. cm cm Geometry 6 Prerequisites page 4
" 0. km. Use the Pythagorean Theorem (when appropriate) to determine the exact length of the missing side. C) 6" cm 0 ft 4. ft 4 cm D) E) F) 09 ft cm 9 ft 0 km 8 cm G) H) I) 6" 9." mm 4' 7' mm Geometry 6 Prerequisites page
Target G4: Given the graph of a line with a definite y-intercept and slope, I can write the equation of the line in slope-intercept form (y = mx + b). If the line is vertical, I can write its equation in the form x = #. Example: write the equation of the line shown at right in slope-intercept form 4. Write the equation of each line (use the slope-intercept form when possible). C) D) E) F) Geometry 6 Prerequisites page 6
G) H) I) Target G: I can accurately graph a line given in slope-intercept form. Example: accurately graph y= - x+6.. Graph each line. Include the y-intercept and at least two other integer valued coordinates on the line. y = x y = x + C) y = - x + 4 4 D) y = - x E) y = -x + 6 F) y = - Geometry 6 Prerequisites page 7
Answers. 6 C) 6 D) 4 E) 6 F) 48 G) 8 H) I) 0. C) D) E) F) 4 G) 7 H) 6 I) 0 J) K) 4 L) 4. 0 4 C) D) E) 6 F) 4 G) 4 H) 60 I) 60 4. C) + 6 D) G) 8 E) 6 F) 0+ H) + I) 7 0 =. 0. 6x x 44 C) 8x 4 D) 6x E) x 7 F) -x + 8 G) x x H) 4 x + x I) 9 x x J) x + x+0 K) x 0 x+ L) x + x 6 M) x + x+ N) x +7 x 0 O) x 6 x+9 P) 4 x +0 x+ 6. x = -0 x = 7 C) x = -6 D) x = 6 E) x = - F) x = G) x = 9 H) x = 6 I) x = 4 J) x = 9 K) x = L) x = 7 7. x = x = =. C) x = 0 = 0. D) x = -6 E) x = -0 or x = 0 F) x = - or x = Geometry 6 Prerequisites page 8
8. x = 6 and y = x = 9 and y = - C) x = - and y = 7 D) x = 8 and y = 0 E) x = 9 and y = F) x = - and y = G) x = and y = - H) x = - and y = I) x = 4 and y = - 9. x = - or x = x = -6 or x = 6 C) x = 0 or x = 8 D) x = 0 or x = - E) x = 0 or x = F) x = - or x = - G) x = - or x = - H) x = or x = 7 I) x = - or x = J) x = -7 or x = 4 K) x = 6 L) x = or x = 8 0. x = - or x = 7 x = or x = C) x = - or x = - 4 D) x = 8 or x = 4. (, ) (-, 4) C) (-, 0) D) (, -6) E) (-, -4) F) (0, -) G) (, -) H) (, 0). Distance: 4 Slope: 0 Midpoint: (0, 8) Distance: Slope: undefined Midpoint: (, ) C) Distance: Slope: D) Distance: 4 Slope: 4-8 Midpoint: (8., 8) Midpoint: (8, ) E) Distance: 9 Slope: F) Distance: 4 Slope: 0 Midpoint: (40., -6) 7 - Midpoint: (., -4.). 8 cm 7. ft C) 6 in D) 4. km E) Impossible to complete F) 60 ft G). in H) mm I) 6 ft 4. y = x 6 y = - x C) y = x+8 4 D) y =- x E) y = 7 F) y = -x + 0 G) y = x H) x = - I) y = x+ 4 4 Geometry 6 Prerequisites page 9
. C) D) E) F) Geometry 6 Prerequisites page 0