Westerville City School District COURSE OF STUDY Math Lab ... · congruent. G-CO.10 Prove theorems...

Westerville City School District COURSE OF STUDY

Math Lab for Geometry MA 102

Recommended Grade Level: 10, 11

Course Length: Semester

Credits: .25/semester mathematics elective credit

Course Weighting: 1.0

Course Fee: None

Course Description

Math Lab is designed to help students be successful in their Geometry course. Math lab is a .25 elective credit per semester. In order to receive credit, students must earn a grade of “satisfactory.” Math Lab is designed to give additional opportunities to learn and apply algebraic and geometric concepts. The class will be activity based and students will need to fully participate. Some homework help and review for Geometry assessments will also be incorporated into the course.

Course Rationale Students may enter Geometry having not been successful in previous math courses or not having the prerequisite knowledge and skills necessary to be successful in Geometry. The goal of Math Lab is to support each student to be successful in the Geometry classroom by providing the opportunity to acquire the background understandings necessary as well as to front load the Geometry content. Math Lab will provide opportunities for students to fill gaps and improve on their foundational mathematical knowledge and skills. Some students may be ready to exit this course during or at the end of the year based on the grades in their Geometry class.

Math Lab does not have a scope and sequence in the traditional sense. As each student will have an individual path through the school year, the units below are meant as general guidelines; the work in the classroom must necessarily be responsive to the gaps and needs of the particular students in each individual section.

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Scope and Sequence

Topics of Study Estimated Time (in weeks)

1 Evaluating Expressions and the Order of Operations 1 week 2 Shapes and Transformations 3 weeks 3 Angle Relationships and Measurement 3 weeks 4 Solving and Writing Linear Equations 2 weeks 5 Similarity, Right Triangles, and Trigonometry 4 weeks 6 Solving Linear Systems of Equations 1 week 7 Probability 2 weeks 8 Congruence and Proof 2 weeks 9 Review for Semester Exam 1 week 10 Factoring & Multiplying Polynomials 1 week 11 Quadrilaterals and Coordinate Geometry 4 weeks 12 Polygons and Circles 5 weeks 13 Applications of Probability 1 week 14 Writing Exponential Equations 1 week 15 Three Dimensions 2 weeks 16 Writing Equations of Circles 1 week 17 Laws of Exponents 1 week 18 Review for Semester Exam 1 week

Primary Material Recommendation

Text: ● CPM Texbooks - Course 3, Algebra 1, and Geometry

Other Resources: ● ALEKS for diagnostic and gap filling; assessment tool ● braingenie - online individualized practice; assessment tool ● kahoot - assessment tool ● quizizz - assessment tool ● desmos - online graphing utility

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Topics of Study

Topic #1: Transformations and Congruence

Content Standards

G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Essential Questions

1. Which transformations preserve the integrity of the shape? 2. What is symmetry? 3. What characteristics can a shape have?

Enduring Understandings

Rigid transformations preserve angle measure and distance relationships and are at the foundation of the concept of congruence.

Key Concepts/ Vocabulary

reflection, rotation, translation, rigid transformation, symmetry, right angle, circular angle, acute, obtuse, straight angle, protractor, regular, line of reflection, prime notation, perpendicular, isosceles, line segment, midpoint, polygon, triangle, quadrilaterals, pentagons, hexagon, octagon, decagon, parallel, perpendicular, scalene, venn diagram, image, equilateral

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Content Elaborations

Build on student experience with rigid transformations from earlier grades.

Students will also need fluency in these checkpoints:

Course #1

Checkpoint #6: Locating Points on a Coordinate Graph

Course #3

Checkpoint #8: Transformations

Algebra

Checkpoint #1 & #4: Solving Linear Equations

Checkpoint #5B: Writing the Equation of a Line

Learning Targets I can:

● use my spatial visualization skills to investigate reflection, rotation, and translation.

● use prime notation for corresponding parts of transformations. ● show that objects and their images are equidistant from the line of

reflection, and that the line segment connecting a point with its reflected image is perpendicular to the line of reflection.

● recognize that the slopes of perpendicular lines are opposite reciprocals.

● use symmetry to discover relationships within basic geometric shapes such as rhombus, square, parallelogram, isosceles triangle, right triangle, kite, and dart.

● classify shapes by their attributes using Venn diagrams.

Assessments Performance on Geometry assessments in their regular Geometry course and the online benchmark and progress monitoring tools determined for the course.

Instructional Strategies and

Materials

● CPM Geometry textbook

● CPM Checkpoints from previous courses

● CPM Parent Guides

● ALEKS dynamic computer software for tracking skill mastery

● Study team strategies

● Teacher created additional practice

Considerations for Intervention and

Acceleration

This is an intervention class. The teacher will modify, as needed, on an individual basis.

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Topic #2: Proving Theorems

Content Standards G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Essential Questions

1. What are the conditions necessary for angle relationships to hold? 2. What are the preconditions to use the Pythagorean Theorem?


Students will understand the necessary connections between parallel lines and the angle relationships. Students will be able to use the Pythagorean theorem to solve problems.


theorem, linear pair, vertical angles, alternate interior angles, same-side interior angles, corresponding angles, perpendicular bisector, supplementary angles, complimentary angles, equidistant,, midpoint, mid-segment, angle bisector, altitude, isosceles triangle,bisector, congruence properties, Pythagorean Theorem

Content Elaborations

Fluency in the following checkpoints from Course 1, 2, and 3 needed to be

successful in this course:

Course 3

Checkpoint #5 Solving Equations

Algebra 1

Checkpoint #10B Factoring Polynomials

Learning Targets I can: ● use the angle relationships to solve problems.


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Materials

● Students will be given checkpoint materials from College Preparatory

Mathematics Course 1,2, and 3

● Students will use ALEKS dynamic computer software for tracking

skill mastery.

● CPM checkpoint materials

● CPM parent guide


Acceleration

This is an intervention class. The teacher will modify, as needed, on an individual basis.

Topic #3: Similarity And Trigonometry

Content Standards G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.

1. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformation the meaning of similarity for triangles a s the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.2 Apply concepts of density based on area and volume in

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modeling situations (e.g., persons per square mile, BTUs per cubic foot). G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Essential Questions 1. What are the conditions for similarity? 2. What are trigonometric ratios? Where and when can they be used? 3. How can trigonometric ratios be used to solve problems involving

right triangles?


Students will understand that Trigonometry is a special case of similarity. Students will understand the importance of Pythagorean Theorem and its applicability to many real world contexts.


transformation, dilation, scale factor, similarity, AA criterion, ratio, corresponding angles, corresponding sides, proportionality, SSS, SAS, vertex, Pythagorean Theorem, congruence, hypotenuse, leg, Pythagorean triples, Trigonometric ratios, sine, cosine, tangent, opposite side, adjacent side, right triangle

Content Elaborations Fluency in the following checkpoints from Course 1, 2, and 3 needed to

be successful in this course:

Course 1

Checkpoint #8A Rewriting and Evaluating Variable Expressions

Course 2

Checkpoint #6 Writing and Evaluating Variable Expressions

Checkpoint #8 Solving Multi-Step Equations

Course 3

Checkpoint #2 Unit Rates and Proportions


Checkpoint #7 Solving Equations with Fractions

Algebra 1

Checkpoint #6A Rewriting Equations with More than One Variable


● use trigonometric ratios to solve real life problems.

● use trigonometric ratios to solve problems involving right triangles.

● use similarity to define the relationships between sides and angles

of right triangles.

● model real life situations using Trigonometry.

● use the Law of Cosines and Sines to solve problems.

Assessments Performance on Geometry assessments in their regular Geometry course and the online benchmark and progress monitoring tools determined for

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the course.


Materials



● Checkpoints from previous courses

● Study Team Strategies



Acceleration

Please see attached documents addressing the challenges and response to exception students of all levels.

Topic #4: Circles

Content Standards G.C.1 Prove that all circles are similar. G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ) lies on the circle centered at the origin and containing the point (0, 2). G.MG.1 Use geometric shapes, their measures, and their properties to

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describe objects (e.g., modeling a tree trunk or a human torso as a cylinder.

Essential Questions 1. Are all circles similar? 2. What is a sector and segment of a circle and how does one find it’s

area? 3. What is a radian? 4. What is the difference between arc measure and arc length?


Students can use the properties of circles to solve problems.


diameter, radii, chords, arcs, arc measure , arc length, inscribed angle, inscribed shape, circumscribed shape, sector, circumference, radian, degree, segment of circle, central angle



Course 1

Checkpoint #8A Rewriting and Evaluating Variable Expressions

Course 2

Checkpoint # 8 Solving Multi-Step Equations

Course 3

Checkpoint #4 Area and Perimeter of Circles and Composite Figures

Algebra 1

Checkpoint #11 The Quadratic Web

Completing the Square


● decide if a point lies outside, inside, or on a given circle.

● find an equation of the line tangent to the circle at that point given

the equation of a circle and a point on it.

● complete the square in circle problems

● derive the equation of a circle in a variety of ways



Materials








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Acceleration

Topic #5: Connecting Algebra and Geometry Through Coordinates

Content Standards G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ) lies on the circle centered at the origin and containing the point (0, 2). G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point.) G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula

Essential Questions 1. How does Algebra support Geometry? 2. Can the tools of Algebra the student in their investigation of

Geometry?


Students will come to understand that Algebraic Geometry and Synthetic Geometry are mutually supportive. Algebra and Geometry are working towards the same goals,addressing the regularities found in our world, from different prospectives.


slope, distance formula, vertex, midpoint, equations of parallel, perpendicular, or intersecting lines, slope-intercept form, point-slope form, quadrilaterals: parallelogram, rectangle, rhombus, square, tangent



Course 1

Checkpoint #6

Course 2

Checkpoints #1, 6, & 7A

Course 3

Checkpoints #4, 5, & 6

Algebra 1

Checkpoint #5B & 6B

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● draw conclusions about figures in the coordinate plane using linear

relationships.

● use the Pythagorean theorem, slope, and the properties of

coordinate geometry to model real world situations.

● use special right triangles in a myriad of ways.



Materials







Acceleration


Topic #6: Extending to Three Dimensions

Content Standards G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

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G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Essential Questions 1. What situations in real life can be modeled with geometric concepts?


Many real world problems can be solved using area, surface area, and volume to model the problem situation. Many everyday objects can be formed by rotating geometric shapes.


circle, circumference, perimeter, diameter, area, polygon, radius, base, prism, cylinder, pyramid, cone, sphere, volume, height, volume



Course 2

Checkpoint #1 Area and Perimeter of Polygons

Checkpoint #6 Writing and Evaluating Algebraic Expressions

Checkpoint #7A Simplifying Expressions

Course 3

Checkpoint #4 Area and Perimeter of Circles and Composite Figures


Algebra 1

Checkpoint #4 Solving Linear Equations with Fractional Coefficients

Checkpoint #5B Writing the Equation of a line

Checkpoint #7A Solving Problems by Writing Equations


● use my prior knowledge develop formulas and equations.

● solve many real world problems using area, surface area, and

volume.



Materials






Considerations for Please see attached documents addressing the challenges and response to exception students of all levels.

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Intervention and Acceleration

Topic #7: Probability

Content Standards S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or”, “and”, “not”). S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if your are a smoker with the chance of being a smoker if you have lung cancer. S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A) P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Essential Questions 1. How can probability be used to predict outcomes? 2. What are independent events?

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3. What methods can be used to organize data? 4. What real life situations are influenced by probabilities? 5. What are the rules of probability ? How and when can they be

applied? 6. How does the order of events affect whether or not it is considered

a different outcome? 7. How can we determine whether a decision is fair? 8. How can probablility be utilized to analyze the impact of a possible

decsion?


Probability is a real life tool for predicting outcomes. Organizing data into tables is helpful for analysis of independence and estimating probabilities. Interpreting probability can be useful in real life situations Probability can be used to make fair decisions in real life situations. Probability can be used analyze and assist with decision making..


expected probability, fair decision, probability, events, outcomes, subsets, sample space, unions, intersections, complements, independent, dependent, conditional probability, frequency table, random, data compound events, mutually exclusive, addition rule, multiplication rule, permutation, combination, venn diagram, tree diagram



Course 1

Checkpoint #7A Multiplication of Fractions and Decimals

Checkpoint #8B Division of Fractions and Decimals

Checkpoint #9A Displays of Data

Course 2

Checkpoint #3 Multiplying Fractions and Decimals

Checkpoint #7B Displays of Data


Course 3


Checkpoint #9 Satterplots and Association

Algebra 1

Checkpoint #7A Solving Problems by Writing Equations

Checkpoint #8 Interpreting Associations


● I know that probability can be represented as a fraction or decimal

on the interval .

● decide if two events are independent or dependent.

● organize data into tables is helpful for analysis of independence

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and estimate probabilities.

● interpret probability in real life situations

● explain the rules than govern probability relationships.

● find expected value and use it as a tool to determine a situation is

fair for all participants.



Materials







Acceleration


Acknowledgements

It is through the hard work and dedication of high school Mathematics team that the Westerville City Schools’ High School Geometry Math Lab Course of Study is presented to the Board of Education. Sincere appreciation is extended to the following individuals for their assistance and expertise.

Central HS North HS South Hs District

Jennifer Horn

Brittany Barnhart

Jessica Martin Kasandra Sliney

Richard Gary

Michael Huler

Westerville City School District COURSE OF STUDY Math Lab ... · congruent. G-CO.10 Prove theorems...

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