SUMMER WORK FOR STUDENTS ENTERING HONORS MATH 3. Dear Honors Math 3 student, Welcome to your summer work assignment for Honors Math 3.

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1 SUMMER WORK FOR STUDENTS ENTERING HONORS MATH 3 Dear Honors Math 3 student, Welcome to your summer work assignment for Honors Math 3. Instructions: No calculator (unless specified) should be used on any of these problems. You should be able to complete the entire packet, as all the skills below are required for Honors Mathematics 3. These skills can and will be sprinkled across future tests and/or pop quizzes. If necessary, you may reference other sources (i.e. textbooks, notebooks, KhanAcademy), but all the work must be done on your own, without the use of any device (e.g. graphing calculator, Geogebra, etc.). Simplify all answers completely (i.e. combine like terms, reduce). Show all your work and box (or circle) your answers. Print the following pages and complete the work in the space below each problem. Here are a few resources that might be helpful for you. Khan Academy: Purplemath: Math is fun: Within the instructions for different problems in this assignment, you will see key terms in bold. These terms can be used in the search box when exploring the suggested resources above. During the first week of school, your teacher will check your summer work assignment for effort, completion and correctness. You will then be given the opportunity to discuss some of the problems with your classmates, and you will also receive an answer key to help you double check your work. Finally, you will take a summer work assessment at some point during the first two weeks of school. Throughout this assignment it is very important that you show your work. Please do not simply write the final answers and argue that you did the work in your head. Your teacher wants to understand your thought process. Please document it! You may complete this entire problem set directly on these sheets (you do not need to use separate paper). If you have any questions about this summer work assignment, please feel free to contact the Math Department Chair, Christina Kennedy, via ckennedy@pingree.org Topic 1 : Order of Operations Following the proper order of operations is crucial to correctly solving problems in math. 1. Evaluate the following. a) 5 2 b) (16 ( 2 4 ))

2 c) 4(1 3(3 6 ) + 5) d) ( 23 )( 34 ) e) ( 5x 3) f) 2 + 4(x 3) Topic 2 : Distance, midpoint, slope Three important formulas that relate to lines are the distance, midpoint and slope formulas. When given two points ( x 1, y 1 ) and x, y ) the formulas are: ( 2 2 Distance: (x 2 x ) 2 y ) ( 2 y 1 x 1+ x2 Midpoint: ( 2, y2 y1 Slope: x 2 x 1 y 1+ y 2 2) Note : It doesn t matter which point is ( x 1, y 1 ) and which is x, y ) as long as you are consistent when using each formula. ( 2 2

3 Given the following points answer the questions below. Leave all answers for this section in simplified radical form if applicable. 1. Find the midpoint of AB. A (0, 1) B (5, -1) C (9, 2) D (4, 4) 2. Find the slope of BD. 3. Use the distance formula to find the length of BC. 4. Are AB and CD the same length? Back up your conclusion with mathematical evidence, meaning show your work. 5. Are AD and BC parallel? Back up your conclusion with mathematical evidence. 6. Are AD and DC perpendicular? Back up your conclusion with mathematical evidence. 7. Do AC and BD bisect each other? Back up your conclusion with mathematical evidence.

4 8. Find the midpoint and the distance between the following two points, R and W, given in three dimensions. R(-2, 5, 0) W(1, -3, -4) Topic 3 : Right Triangle Trigonometry When working with right triangles there are three important ratios, called the trigonometric ratios, which are used to find the values of sides and angles. You may use a calculator for this section. Round answers to two decimal places if applicable. 1. Solve each of the following. Round to two decimal places as necessary. a) sin (46 o ) = x 10 b) tan (50 ) o 15 = x c) c os (θ) = Draw and label triangle ABC. Then find all missing angles and sides given that AB = 7, C = 90, A = Draw and label triangle DEF. Then find all missing angles and sides given that DE = 4, F E = 8, and E = Draw a triangle that shows the following identities. Label all sides and angles. 1 a. s in(30 ) = 2 2 b. c os(45 ) = 2

5 Topic 5 : Algebra 1. Solve for the unknown variable a) w 2 + 4w 2 = 3 ( 4 w) 2 b) 5t 3 (t + 4(2 t )) = 10 1 c) 5 (3x 5 ) = x Find the equation of the line through the following two points. Write in point-slope form and slope-intercept form : (-1,4) and (-2,-4)

6 3. Find the equation of the line through the point (2,1) and a) parallel to 4x 2 y = 3 b) perpendicular to 4x 2 y = 3 4. Simplify the following expressions completely. Key words for this might include exponent rules or negative exponents. 0 a) 2(13dm) m 1 b) 7p ( 3 q 2 q 4 2 ) 3 ( p 7 ) 2 c) 4(3x 3 ( ) 2) 1 d) 12x 3 y 4 4x 2 y 4. Completely factor the following expressions. a) u u + 24 b) ( k 1) 2 49

7 Topic 4 : Quadrilateral definitions and properties Below are definitions for key quadrilaterals (four-sided polygons). You will want to know the definitions in the table below and properties when you arrive back to Pingree in September. Good definitions are as short as possible and describe only the features that make the figure what it is. Extra features of a polygon that result from the definition are properties of the figure. Be careful to know the distinction. For example, the definition of an equilateral triangle is just that: a polygon with exactly three sides all of which are equal in measure (congruent). A property of the figure is that all three angles are equal too--but that is NOT part of the definition. (Note: Definitions below are inclusive. This means that a rhombus, for example, is considered to be a special type of kite (Rhombuses are included in the definition of a kite.). A rhombus is also included as a parallelogram--just as a parallelogram is included as a special type of trapezoid--because it does have one pair of sides that are parallel--even if the other sides are parallel also.) Some texts try to write definitions that exclude this sort of overlap, but we do not. Shape name and definition Quadrilateral A polygon with exactly four sides. Parallelogram A quadrilateral with two pairs of parallel sides. Rhombus A quadrilateral with four congruent sides. (Or, equilateral quadrilateral) Rectangle A quadrilateral with all angles congruent. (Four 90 degree (right) angles.) Square A quadrilateral with congruent sides and angles. Kite A quadrilateral with two pairs of consecutive congruent sides. Trapezoid A quadrilateral with one pair of parallel sides. Isosceles Trapezoid A trapezoid with non-parallel sides that are congruent. Properties All angles add up to 360 o! (However, try connecting all four midpoints of any quadrilateral, and see what figure is created...) Diagonals bisect each other. Opposite sides, and opposite angles, are congruent. Diagonals are perpendicular bisectors of each other. Opposite angles are congruent. Opposite sides are parallel. Diagonals are congruent and bisect each other. Opposite sides are congruent and parallel. The diagonals are congruent and perpendicular bisectors of each other. Diagonals are perpendicular. Non-vertex angles are congruent. The diagonal between the vertex angles bisects the other diagonal. The midsegment (the line formed by connecting the midpoints of the non-parallel sides) is parallel to the bases and the average of their lengths. Diagonals are congruent. The midsegment is parallel to the bases, and the average of their lengths.

8 1. Use the definitions and properties above to fill in the following chart. Write Yes in the box if the quadrilateral in the top row always satisfies the property in the 1st column. Write No in the box if the property is not satisfied. Hint : If you have trouble determining if a property is always true sketch a picture! Kite Isosceles Trapezoid Parallelogram Rhombus Rectangle Each pair of opposite sides parallel Opposite sides congruent Opposite Angles congruent Diagonals bisect each other Diagonals are perpendicular Diagonals are congruent Exactly one line of symmetry Exactly two lines of symmetry

9 Topic 5 : Logic Puzzles KenKen is a type of mathematical logic puzzle, similar to Sudoku. The rules are simple and the puzzles can be played at a range of levels of difficulty. Being able to pull together given information to make new conclusions will become very important in Math 3 this fall during our unit on proofs. 1. Go on the website below to learn the how to play KenKen. You should be able to fill in squares without guessing. Start with the easiest level (3x3) and progress to at least a 5x5. Then complete the puzzle below. With a bit of practice, it should take less than 10 minutes. Number each box with a small subscript number in the order in which you were able to fill out the puzzle. Website:

Period: Date Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means

Period: Date Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means : Analytic Proofs of Theorems Previously Proved by Synthetic Means Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of