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In order to make our new students have an experience with mathematics at Sage Hill that is meaningful and productive as possible, we need to ensure that you are placed in the appropriate mathematics class. Your mathematics class will be determined by a placement test, which will be administered on **Saturday, ****May 16, 2020 at 9:00 AM on the Sage Hill School campus**.

Placement tests are necessary for many reasons. We have found that middle schools differ greatly in curriculum and standards. Taking the appropriate mathematics class based on your skills and abilities is the best path to success. As an additional benefit, our carefully designed placement tests will reveal gaps in knowledge and areas of strength. Your mathematics teachers can use results of the placement exam to give them insight into your strengths and weaknesses as you progress through the Sage Hill mathematics curriculum.

It is extremely important that you RSVP for the math placement exam by **Wednesday, May 1**, so that the Mathematics Department can properly prepare for the day. If you do not RSVP, we will assume that you do not intend to take any placement tests and you will be placed in Algebra I as your first math class at Sage Hill School.

Read on for instructions on which test is appropriate for you, and for details on each test.

Please email Drew Ishii at ishiid@sagehillschool.org if you have any questions.

Drew K. Ishii, Ph.D

Mathematics Department Chair

Sage Hill School

(949) 270-2318

Contents

For all of the placement tests, in order to be consistent, **no calculators are allowed**. For a couple of sections, this made crafting the questions a bit tricky, but we have made all of the tests and problems doable without the aid of a calculator. You will have to multiply and such in some places, but none of the calculations are that daunting, or at least they *shouldn’t* be that daunting. Do __not__ bring a calculator with you to the placement test.

The **only** things you need to bring are sharpened pencils (that’s pencils -- plural -- as it is quite common for people to break a lead here or there) as well as a good eraser.

Each of the placement tests are used to place you **out** of a class. So if we are talking about the Algebra I placement exam, we are talking about taking and passing *out* of Algebra I, not *into* Algebra I.

You will need to take and pass the Algebra I placement test.

The Algebra I placement test has a one hour time limit.

You will need to take and pass both the Algebra I placement test **and** the Geometry placement test.

Each test has a one hour time limit, so be prepared to be here for a total of two hours.

You will need to take and pass the Algebra I placement test, the Geometry placement test, **and** the Accelerated Algebra Supplemental test.

The Algebra I and Geometry placement tests are one hour each. The supplemental test is allotted 30 minutes. Therefore you need to set aside two-and-a-half hours for tests that morning.

You will need to take and pass both the Geometry **and** Algebra II placement tests. We will assume you can manage the Algebra I level material and not require you to take that test that day, **however**, if you fail both the Geometry and Algebra II placement tests, then we may need to revisit that and you may need to come back and take the Algebra I test.

Each test has a one hour time limit, so be prepared to be here for a total of two hours.

Placement into the Accelerated class requires a very good score on the Geometry placement exam and a near perfect score on the Algebra II placement exam.

You will need to take and pass both the Algebra II and Precalculus placement exams. We will assume you can manage the Algebra I and Geometry material and not require you to take those tests that day, **however**, if you fail both tests, then we may need to revisit what will need to happen with your placement.

Each test has a one hour time limit, so be prepared to be here for a total of two hours.

If you do pass both tests, the level of calculus you are placed in depends on how well you do on the tests.

If you did not do well on the placement test or after looking at the course roadmap you have decided that you want to get ahead in the curriculum, you will have time to enroll in summer school.

Summer school runs June 24 to July 26, 2019.

The courses offered in summer school are:

- Algebra I
- Geometry
- Algebra II
- Precalculus

The results should be mailed out on Monday following the test. You should receive them shortly thereafter.

Once you know which class you have qualified for, you can look up how your Sage Hill mathematical career will progress on the Mathematics Course Roadmap in your Course Catalog.

The Algebra I placement test is a 60 minute exam. Here are the topics that you will be expected to know for the exam.

- Definitions of the following types of numbers: integers, naturals, wholes, irrationals, reals, and rationals
- Adding, subtracting, multiplying, and dividing fractions with only numbers in the numerators and denominators
- Adding, subtracting, multiplying, and dividing fractions with numbers and/or variables in the numerators and denominators
- Proper order of operations
- Finding the prime factorization of numbers
- Using the distributive property
- Combining like terms
- Adding, subtracting, multiplying, and dividing exponents
- Properly dealing with negative exponents
- Adding, subtracting, multiplying, and dividing radicals
- Simplifying radicals, including rationalizing denominators
- Expanding algebraic expressions (whether through FOIL or similar methods)
- Solving for
*x*in rational linear equations - Solving equations with absolute value
- Solving inequalities
- Given an equation with numerous variables involved, being able to solve for any of the variables
- Factoring all manner of algebraic expressions (up through quadratic expressions, no need for cubic or more)
- Solving systems of equations (should be able to solve through graphing, substitution, or elimination)
- Finding the slope between two points on the Cartesian plane
- Knowing the standard form of a linear equation, and being able to successfully graph an equation in standard form
- Knowing the slope-intercept form of a linear equation, and being able to successfully graph an equation in slope-intercept form
- Knowing the point-slope form of a linear equation, and being able to successfully graph an equation in point-slope form
- Converting between the different forms of a linear equation
- Distinguishing between functions and relations
- Finding the equation of a line passing through two points
- Finding equations of a line parallel and perpendicular to another line
- Finding slopes of horizontal and vertical lines
- Finding equations of horizontal and vertical lines
- Graphing equations and successfully labeling axes, the origin, the intercepts, and the graph
- Given the graph of a linear equation, being able to come up with the equation of the graph
- Graphing absolute value equations
- Graphing inequalities
- Graphing simple linear programming problems
- Taking a simple quadratic equation and being able to find all of the intercepts, the coordinate of the vertex point, and sketching the resulting graph
- Translating words into mathematical equations
- Solving simple word problems

The Geometry placement test is a 60 minute exam. Here are the topics that you will be expected to know for the exam.

- Know the definition, symbol, and/or how to label all of the following -- point, line, line segment, plane, ray, opposite rays, endpoint, length, congruent, similar, midpoint, vertex (of an angle, polygon, etc.)
- Segment bisectors and angle bisectors
- Complementary and supplementary angles
- Vertical angles
- Conditional statements (a.k.a. if/then statements)
- Hypothesis, conclusion, conditional, converse, inverse, contrapositive, and biconditional

- Properties
- Addition, subtraction, multiplication, division, distributive, reflexive, symmetric, transitive, substitution

- Postulates
- Segment Addition, Angle Addition

- Perpendicular lines and planes
- Parallel lines and planes
- Skew lines
- Alternate interior angles, corresponding angles, and same-side interior angles and their properties with parallel lines
- Polygons
- Vocabulary -- triangle, quadrilateral, pentagon, hexagon, octagon, decagon,
*n*-gon, regular, convex, non-convex - Exterior angles
- Remote interior angles
- Should be able to calculate the sum of all interior angles in a polygon, compute the measure of each interior angle in a regular polygon, compute the measure of each exterior angle in a regular polygon

- Vocabulary -- triangle, quadrilateral, pentagon, hexagon, octagon, decagon,
- Classification of triangles:
- By angles (acute, obtuse, right, straight, equiangular)
- By sides (scalene, isosceles, equilateral)

- Given three potential lengths of a triangle can you determine:
- Can it form a triangle at all?
- If so, is it an acute, right, or obtuse triangle?

- Given a triangle, can you use a ruler to draw in the medians, perpendicular bisectors, and altitudes.
- The five ways to prove triangles congruent -- SSS, SAS, ASA, AAS, HL (the first three being universal and the most important, the last two not as universal and thus not as important)
- Corresponding parts of congruent triangles are congruent
- Definitions and properties of all kinds of quadrilaterals -- parallelograms, rectangles, rhombuses, squares, trapezoids, kites
- The five ways to prove a quadrilateral is a parallelogram
- Show opposite sides are parallel
- Show opposite sides are congruent
- Show opposite angles are congruent
- Show diagonals bisect each other
- Show that one pair of sides is both congruent and parallel

- Ratios
- Proportions
- Scale factor
- Segments divided proportionally
- The three ways to prove triangles are similar -- AA~, SSS~, SAS~
- Deductive vs. inductive reasoning
- Proofs (both algebraic and geometric) -- the only expectation is they can successfully complete two-column proofs (no need for paragraph or inductive proofs)
- Arithmetic vs. geometric mean
- Pythagorean Theorem
- Properties of a 45°-45°-90° triangle and a 30°-60°-90° triangle
- Right triangle trigonometry (using sine, cosine, tangent and their inverses)
- Angle of elevation and depression
- Circle vocabulary and properties of all of the following -- center, radius, diameter, chord, secant, tangent (lines, rays, and segments), points of tangency, central angles, minor arc, major arc, semicircle, inscribed angles
- Concentric circles and concentric spheres
- Polygons inscribed in a circle and circles circumscribed about polygons
- Normally a Geometry curriculum includes a full unit on constructions, but since we are not having people bring compasses, there won’t be any constructions on the placement exam
- Regular polygons -- central angles, radii, central angles, apothems
- Must be able to calculate the perimeter and areas of all of the following -- squares, rectangles, triangles, parallelograms, rhombuses, trapezoids, regular polygons, and circles
- Note: There are a lot of formulas to know. You do not get a formula sheet on the exam. You are expected to have them memorized.

- Calculating the arc length in a circle
- Calculating the area of a sector of a circle
- Geometric probability
- 3D shape vocabulary and properties of all of the following -- prisms, bases, lateral faces, lateral edges, base edges, altitudes, right vs. oblique, pyramids, cylinders, cones, spheres
- Must be able to calculate the lateral area, total area, and/or volume of all of the following -- right prisms, cubes, pyramids, cylinders, cones, and spheres
- Note: There are a lot of formulas to know. You do not get a formula sheet on the exam. You are expected to have them memorized

- Similar solids

The supplemental exam is just that -- supplemental. The Algebra I placement exam is really the first part, and this goes above and beyond that exam. You definitely need to pass the Geometry placement, but really that is all. However, in order to make it to Accelerated Algebra II, you need to have **stellar** algebra skills. Therefore the Algebra I and Accelerated Algebra Supplemental exams are of utmost importance and thus need to have extremely good scores. The Algebra I placement exam should be nearly perfect.

There are really two parts to consider with respect to the Accelerated Algebra Supplemental exam.

First is the material. This test definitely covers the hardest stuff from Algebra I, however, you can only go so far with that. There really isn’t that much farther that Algebra I goes, but this test certainly pushes the limits to what might be seen in an Algebra I course.

The second part about this exam is the length versus the time limit. 30 minutes is not a lot of time for this test. You need to be able to work fast. The pace of the Accelerated Algebra II class is __very__ fast, so we need to make sure you can do algebra not only accurately but quickly as well. Therefore the time limit becomes a major factor in this test. In order to get through everything, you need to move fast.

Here are the topics that you will be expected to know for the exam.

- There are a lot more word problems that are much more difficult in nature
- Fractional exponents
- Solving for
*x*in cubic equations - Factoring cubic expressions

The Algebra II placement test is a 60 minute exam. Here are the topics that you will be expected to know for the exam.

- Dealing with exponents, including negative and fractional exponents
- Adding, subtracting, multiplying, and dividing radicals
- Simplifying radicals (not just square roots), including rationalizing denominators
- Expanding algebraic expressions (whether through FOIL or similar methods)
- Solving for
*x*in all kinds of equations (not just linear or quadratic ones) - Solving equations with absolute value
- Solving inequalities
- Given an equation with numerous variables involved, being able to solve for any of the variables
- Factoring all manner of algebraic expressions (up through cubic expressions)
- Solving systems of equations (should be able to solve through graphing, substitution, or elimination)
- Finding the slope between two points on the Cartesian plane
- Knowing the standard form of a linear equation, and being able to successfully graph an equation in standard form
- Knowing the slope-intercept form of a linear equation, and being able to successfully graph an equation in slope-intercept form
- Knowing the point-slope form of a linear equation, and being able to successfully graph an equation in point-slope form
- Converting between the different forms of a linear equation
- Distinguishing between functions and relations
- Finding the equation of a line passing through two points
- Finding equations of a line parallel and perpendicular to another line
- Finding slopes of horizontal and vertical lines
- Finding equations of horizontal and vertical lines
- Graphing equations and successfully labeling axes, the origin, the intercepts, and the graph
- Given the graph of a linear equation, being able to come up with the equation of the graph
- Graphing absolute value equations
- Graphing inequalities
- Taking a simple quadratic equation and being able to find all of the intercepts, the coordinate of the vertex point, and sketching the resulting graph
- Logarithms and natural logarithms -- including all of the properties of logarithms and solving equations with logarithms
- Word problems
- Direct, inverse, and/or joint variation, including problems were you must solve for the constant of variation
- Finding inverses of functions
- Composite functions
- Complex numbers
- Conics (if given an equation, should be able to quickly determine which conic it describes, even if it is not in standard form)
- Parabolas -- putting equations into standard form, graphing, finding the vertex, finding all of the intercept(s)
- Circles -- putting equations into standard form, graphing, finding the center, finding the radius
- Ellipses -- putting equations into standard form, graphing, finding foci, minor axis, major axis
- Hyperbolas -- putting equations into standard form, graphing, finding foci, finding asymptotes

The Precalculus placement test is a 60 minute exam. Here are the topics that you will be expected to know for the exam.

- Interval notation
- Solving systems of equations (should be able to solve through graphing, substitution, or elimination)
- Factoring both algebraic equations (not just quadratic ones)
- Rational root theorem
- Differentiating between functions and relations
- Finding the slope between two points on the Cartesian plane
- Knowing the standard form of a linear equation, and being able to successfully graph an equation in standard form
- Knowing the slope-intercept form of a linear equation, and being able to successfully graph an equation in slope-intercept form
- Knowing the point-slope form of a linear equation, and being able to successfully graph an equation in point-slope form
- Converting between the different forms of a linear equation
- Finding the equation of a line passing through two points
- Finding equations of a line parallel and perpendicular to another line
- Finding slopes of horizontal and vertical lines
- Finding equations of horizontal and vertical lines
- Taking a function or relation and being able to:
- Find the domain
- Find the range
- Find the
*x*- and*y*-intercepts - Finding the zeros
- Finding the intervals where the function is increasing and/or decreasing
- Finding the local minimums and local maximums
- Finding the absolute minimums and absolute maximums
- Sketching the graph
- Sketching the asymptote(s), if there are any
- Finding the inverse
- Analyzing the inverse just like the original function/relation (find domain, find range, etc.)

- Given a function, solving for when the function is positive or negative
- Solving inequalities
- Graphing inequalities
- Composite functions
- Piecewise functions
- Given a trigonometric function, being able to:
- State the period
- State the amplitude
- Sketch the graph over a specified period
- Stating whether it is odd, even, or neither

- Radians
- Calculating exact trigonometric values (this test assumes no calculator, so the test only covers trigonometric values that can be found exactly without a calculator)
- Calculating inverse trigonometric values (sin
^{-1}, cos^{-1}, etc.) -- like the previous bullet point, since this test is meant to be done without a calculator, the test only covers the values that can be found exactly without a calculator) - Verifying trigonometric identities and simplifying trigonometric equations
**Note:**These two topics usually require a formula sheet covering a ton of trigonometric identities. Here is a breakdown of the identities you should have memorized and which you do not need to have memorized, and we will provide it to you if you need it.**Trigonometry Identities You Need to Have Memorized**- Reciprocal Identities (sin
*u*= 1/csc*u*, etc.) - Pythagorean Identities (sin
^{2}*u*+ cos^{2}*u*= 1, etc.) - Quotient Identities (tan
*u*= sin*u*/cos*u*, etc.)

- Reciprocal Identities (sin
**Trigonometry Identities You Do Not Need to Have Memorized**(Anything else. If you need anything else we will provide a formula sheet for you, but here are some specifics.)- Co-function Identities
- Even-Odd Identities
- Sum-Difference Formulas
- Double Angle Formulas
- Power-Reducing/Half Angle Formulas
- Sum-to-Product Formulas
- Product-to-Sum Formulas

- Finding all values of
*x*which satisfy a trigonometric equation in a given interval - Evaluating summations written with capital-sigma notation
- Conics (if given an equation, should be able to quickly determine which conic it describes, even if it is not in standard form)
- Parabolas -- putting equations into standard form, graphing, finding the vertex, finding all of the intercept(s), etc.
- Circles -- putting equations into standard form, graphing, finding the center, finding the radius, etc.
- Ellipses -- putting equations into standard form, graphing, finding foci, minor axis, major axis, etc.
- Hyperbolas -- putting equations into standard form, graphing, finding foci, finding asymptotes, etc.

- Logarithms (converting to exponent form, evaluating, solving equations involving logarithms, converting sums and differences of logarithms, etc.)
- Given a logarithmic or exponential function, being able to:
- Find the intercept(s)
- Find the asymptote
- Sketch the graph

- Complex numbers
- Direct, inverse, and/or joint variation, including problems were you must solve for the constant of variation
- Finding limits, both from an expression and from analyzing a graph
- Successfully filling out the values on a unit circle

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