Unit 1: Lines and Planes Grade: 10
Number
Performance Indicators to be mastered in this unit:
Review Review Review G.G.1 G.G.2 G.G.3 G.G.4 G.G.5 G.G.6 G.G.7 G.G.8 G.G.9
Performance Indicator Undefined terms: point, line, plane, straight line Properties of real numbers Planar geometry term: rays, angles, polygons, etc. Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them Know and apply that through a given point there passes one and only one plane perpendicular to a given line Know and apply that through a given point there passes one and only one line perpendicular to a given plane Know and apply that two lines perpendicular to the same plane are coplanar Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines Know and apply that if two planes are perpendicular to the same line, they are parallel
Major Concepts: To understand relationships between points, lines, and planes. To know how to use the notations for point, line, plane, parallel, and perpendicular. To understand the meaning of coplanar and collinear. To review basic vocabulary of geometry from previous years. Essential Understandings: • How do points, lines, and planes describe or relate to the world we live in? • How do we know if lines or planes are parallel or perpendicular? Essential Questions: • Are the lines and planes in a given situation parallel or perpendicular? • How many planes can be perpendicular to a line through a given point? • How many lines can pass through a point perpendicular to a given plane? • When are two planes perpendicular/parallel? • When can two lines be perpendicular/parallel? Essential S-VE Exit Behaviors: self-discipline Skills needed for mastery on performance indicators (& possible teaching strategies): • Review the vocabulary terms that students recognize from past years. • Study the meaning of parallel and perpendicular lines and planes. • Look at situations involving lines, planes, and points and determine what result could be had (is it parallel or perpendicular?) when we add another line or plane with a certain stipulation into the
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problem. • Look at intersections of lines and planes and make conclusions regarding parallelism or perpendicularity. • **Use pictures and objects to help with the imagery of these problems.** Key Terms: Alternate Interior Angle, Alternate Exterior Angle, Axiom, Collinear, Coplanar, Corresponding angles, Line, Linear Pair, Parallel, Perpendicular, Plane, Point, Postulate, Ray, Same side interior, Segment, Skew Lines, Straight angles, Supplementary angles, Transversal, Undefined terms, Vertical angles
Unit 2: Logic Grade: 10
Number
Performance Indicators to be mastered in this unit:
Performance Indicator
G.G.24 Determine the negation of a statement and establish its truth value G.G.25 Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true G.G.26 Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences G.G.27 Write a proof arguing from a given hypothesis to a given conclusion Major Concepts: To understand and determine truth values of statements. To differentiate between conditional, bi-conditional, conjunction, and disjunction and know the truth tables for each. To find the inverse, converse, and contrapositive of a conditional statement. Essential Understandings: • What type of conditional always has the same truth value as the original? • What are the differences/similarities between conjunction and disjunction? • What are the differences/similarities between conditional and bi-conditional? Essential Questions: • Is the following statement true or false? • What is the inverse/converse/contrapositive of the statement? • Determine the truth value for this compound statement. Essential S-VE Exit Behaviors: self-discipline Skills needed for mastery on performance indicators, (& possible teaching strategies): • Be able to differentiate the difference between a statement and an open sentence. • Be able to determine if a given statement is true, false, or neither. • Be able to recognize and use the symbol for negation. • Be able to understand the meaning of a compound statement. • Be able to recognize and use the symbol for conjunction, disjunction, conditional, and biconditional. • Be able to complete a truth table for conjunction, disjunction, conditional, and bi-conditional. Spencer-Van Etten School District
Revised 5/25/10
• • •
Be able to determine the truth value of compound statements. Be able to determine the hypothesis and conclusion of a conditional statement. Be able to recognize the four different forms of a conditional statement (conditional, inverse, converse, and contrapositive) and change from one form to another. • Be able to define and use the term logically equivalent. • Be able to find a logically equivalent statement of a given conditional statement. • Be able to draw a valid argument (find the truth value) from the given statements and their associated truth values. Key Terms: Conjunction, disjunction, conditional, bi-conditional, inverse, converse, contrapositive, logically equivalent, and, or, negate, statement, truth value, truth table, hypothesis, conclusion, logic, valid argument
Unit 3: Congruent Triangles Grade: 10
Number
Performance Indicators to be mastered in this unit:
Performance Indicator
G.G.28
Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles G.G.29 Identify corresponding parts of congruent triangles G.G.27 Write a proof arguing from a given hypothesis to a given conclusion Major Concepts: • Students will understand the meaning of “congruent triangles” • Students will understand how to determine if triangles are congruent • Students will understand how congruent triangles may be used in the “real world” • Students will understand how to use deductive proof-writing to prove two triangles are congruent • Students will be able to name and label corresponding parts of congruent triangles Essential Understandings: • What are the different ways to determine if two triangles are congruent? • How is the concept of congruent triangles used in the construction trade? Essential Questions: • What parts of these two triangles are congruent? • Which congruence method can be used to prove these two triangles congruent? • What do we know about these two triangles? • How do we know that the sides/angles of two triangles are congruent? • Give three examples of where congruent triangles are used in construction. Essential S-VE Exit Behaviors: self-discipline Spencer-Van Etten School District
Revised 5/25/10
Skills needed for mastery on performance indicators, (& possible teaching strategies): • Develop the basic understanding of congruence and proof by proving simple statements. • Begin incorporating new terms and use algebra with them to ensure students become comfortable with the concepts. • Use new terms for mini proofs (prove line segments congruent or angles, etc.) • Discuss what it means to be congruent triangles. • Be able to find congruent sides and angles of two given triangles. • Explore the ways to prove triangles congruent (geo sketchpad). • Begin an idea of how to prove two triangles congruent (flowchart) • Discuss why we need a reason for every statement that we make and what possible reasons could be used. • Write two-column proofs for congruent triangles. • Introduce CPCTC and use it to prove sides or angles congruent. • Research the construction field and look into why they would need congruent triangles. Key Terms: Alternate Interior Angle, Alternate Exterior Angle, Axiom, Corresponding angles, Postulate, Same side interior, Straight angles, Supplementary angles, Transversal, Vertical angles, Reflexive property, SSS, SAS, AAS, ASA, HL, Partition postulate, Addition/Subtraction property, Symmetric property, Transitive property, Substitution property, Complementary angles, Angle bisector, Median, Midpoint, Linear pair, Segment bisector, Congruent, Perpendicular, Hypotenuse, Leg
Unit 4: Triangles Grade: 10
Number
Performance Indicators to be mastered in this unit:
G.G.21 G.G.30 G.G.31 G.G.32 G.G.33 G.G.34 G.G.35 G.G.42 G.G.43
Performance Indicator Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle Investigate, justify, and apply the isosceles triangle theorem and its converse Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem Investigate, justify, and apply the triangle inequality theorem Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle Investigate, justify, and apply theorems about the centroid of a triangle, dividing each
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median into segments whose lengths are in the ratio 2:1 G.G.27 Write a proof arguing from a given hypothesis to a given conclusion G.G.48 Investigate, justify, and apply the Pythagorean theorem and its converse Major Concepts: • Know and understand all aspects of triangles Essential Understandings: • Sum of interior angles = 180◦ • Isosceles triangle, exterior angle, triangle inequality theorems • Longest side is opposite largest angle • Median, altitude, angle bisector, perpendicular bisectors, midsegments, centroid • Pythagorean theorem Essential Questions: • Find a missing angle given certain circumstances • What side is the longest? • Which is the largest angle? • How long is this mid-segment? • What is the distance between the centroid and a given vortex? • Name the triangle? Essential S-VE Exit Behaviors: life long love of learning Skills needed for mastery on performance indicators (& possible teaching strategies): • Sum of a triangle’s angles is 180 degrees • External angle theorem • Isosceles triangle theorem • Sides of a triangle • Largest angle is across from longest side • Triangle inequality theorem – can the sides make a triangle? (the sum of the two smallest has to be greater than the largest) • Pythagorean Theorem & its converse – is this a right triangle and why? • Review parallel lines cut by a transversal • Special relationships – segments and triangles • Medians, altitudes, angle bisectors, perpendicular bisectors, midsegments in triangles • Midsegments connect midpoints (half distance of parallel side) • All three medians intersect at the centroid and divides median in a 2:1 ratio Key Terms: median, altitude, angle bisector, perpendicular bisector, isosceles triangle, converse, exterior angle, inequalities, vertex angles, base, legs, base angles, mid-segments, centroid, midpoint, hypotenuse, Pythagorean theorem
Spencer-Van Etten School District
Revised 5/25/10
Unit 5: Coordinate Geometry Grade: 10
Number
Performance Indicators to be mastered in this unit:
Performance Indicator
G.G.62 G.G.63 G.G.64
Find the slope of a perpendicular line, given the equation of a line Determine whether two lines are parallel, perpendicular, or neither, given their equations Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line G.G.65 Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line G.G.66 Find the midpoint of a line segment, given its endpoints G.G.67 Find the length of a line segment, given its endpoints G.G.68 Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment Major Concepts: • Parallel lines have equal slopes. • Perpendicular lines have negative reciprocal/inverse reciprocal slopes. • Midpoint, slope, and length of line segments. • Finding equations of lines with a given stipulation (parallel/perpendicular to another line). Essential Understandings: • Students will be able to use the slope formula to determine if two lines are parallel, perpendicular, or neither. • Students will be able to find an equation of a line that is parallel/perpendicular to a given linear equation. • Students will be able to describe and find the length, midpoint, and distance of line segments. Essential Questions: • How do you determine if two lines are parallel? • How do you determine if two lines are perpendicular? • What is the length/midpoint/slope of the line segment connecting these two points? • What is a midpoint? • What is distance/length and how does the Pythagorean Theorem relate? • What does slope really mean? Essential S-VE Exit Behaviors: self-discipline
Spencer-Van Etten School District
Revised 5/25/10
Skills needed for mastery on performance indicators, (& possible teaching strategies): • Students will review the concept of slope and finding slope of an equation and of two given points. • Students will look at horizontal/vertical lines to determine their slopes. • Students will look at slopes of parallel and perpendicular lines to discover the relationship between them. • Students will work with manipulating equations and solving for a variable (this is to help them to find the y-intercept for the next skill). • Students will find the y-intercept given a point and a slope. • Students will find the equation of a line that is parallel or perpendicular to a given line and containing a specific point. • Students will use the midpoint formula to find the midpoint between two points and to find a missing endpoint given one endpoint and the midpoint. • Students will look at the distance formula and see how it relates to Pythagorean Theorem. • Students will put radicals into their simplest form. • Students will find lengths/distances of line segments between two given points. • Students will find an equation for a line perpendicular to and bisecting two given points. Key Terms: Abscissa, Bisector, Coordinates, Distance, Endpoint, Evaluate, Length, Midpoint, Perpendicular, Perpendicular bisector, Pythagorean Theorem, Radical, Radicand, Rational vs. Irrational, Simplify, Simplest Radical Form, Square, Square Root, Coefficient, Equation, Horizontal, Linear, Negative Reciprocal, Inverse Reciprocal, Negative Slope, Parallel, Point-Slope Form, Slope, Positive Slope, Slope-intercept Form, Standard Form, Undefined Slope, Vertical, Zero Slope rise run Δy Δx
Unit 6: Polygons Grade: 10
Number
Performance Indicators to be mastered in this unit:
G.G.36 G.G.37
Performance Indicator Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons
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G.G.38 G.G.39
Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals
G.G.40
Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals
G.G.41
Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas
G.G.69 G.G.27
Write a proof arguing from a given hypothesis to a given conclusion
Major Concepts: • All polygons have exterior angles that sum up to 360 degrees. • Each different number sided polygon has a different number of total degrees inside. • Regular polygons have all sides and angles equal. • Parallelograms include rhombi, squares, and rectangles and all have properties that overlap. • Trapezoids are not parallelograms. • It is able to be proven which kind of shape we have by using slope, distance, and midpoint of sides and diagonals. Essential Understandings: • Students will be able to determine the number of degrees in a given polygon. • Students will be able to determine the number of degrees of a given interior/exterior angle in a regular polygon. • Students will know properties of parallelograms, rhombi, rectangles, squares, trapezoids, and isosceles trapezoids. • Students will be able to use coordinate geometry to prove that given points create a certain type of triangle or quadrilateral. Essential Questions: • How many degrees are in an octagon? (any type of polygon) • What does it mean to be a regular polygon? • Describe the process for finding the number of degrees in an exterior angle of a regular decagon. (any type of polygon) • What do we know about the sides/angles of parallelograms? • What properties do squares and rectangles have in common? • How can we prove that these points create a rhombus and not a square? Essential S-VE Exit Behaviors: self-discipline
Spencer-Van Etten School District
Revised 5/25/10
Skills needed for mastery on performance indicators, (& possible teaching strategies): • Begin with what a polygon is and is not. • Develop the idea of a regular polygon. • Look at degrees of a triangle and how we can break any polygon up into a number of triangles. • Look at many shapes and discover how we can find exterior angles when given interior angles. Find the sum of these exterior angles. • Find the number of degrees of both interior and exterior angles in a given regular polygon. • Look at triangles and the types based on side lengths and angle measures. • Determine what makes a triangle one type as opposed to the others. • Use coordinate geometry to prove that given coordinates create a certain type of triangle. • Look at quadrilaterals and use a visual organizer to help keep the 6 types in order. • What are the similarities/differences between any two given quadrilaterals. • Use coordinate geometry to prove that given coordinates create a certain type of quadrilateral (and maybe doesn’t create and different one!). Key Terms: Polygon, interior, exterior, regular polygon, parallelogram, rhombus, square, rectangle, trapezoid, isosceles trapezoid, diagonals, sides, angles, triangle, scalene triangle, isosceles triangle, equilateral triangle, right triangle, distance/length, midpoint, bisect, slope, inverse reciprocals/negative reciprocals, parallel, perpendicular, congruent
Unit 7: Transformations Grade: 10 Code G.G.54 G.G.55 G.G.57 G.G.60 G.G.56 G.G.61
G.G.58 G.G.59
Performance Indicator Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation. Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections) Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90º and 180º, reflections over the lines x = 0 , y = 0 , and y = x , and dilations centered at the origin Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries) Investigate, justify, and apply the properties that remain invariant under similarities
Major Concepts: Reflections, Rotations, Translations, Dilations, isometry, and compositions Essential Understandings: Shape never changes under a transformation. Some properties may or may not change under transformations.
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Revised 5/25/10
Essential Questions: Which transformation(s) change(s) orientation? Which transformation(s) keep(s) the same sized shape? Where do you see transformations used in real life? How do you …? Essential S-VE Exit Behaviors: life-long learner Skills needed for mastery on performance indicators (& possible teaching strategies): Notation of letters after a transformation is performed. Recognize/Identify the four transformations by graphs or pictures. Know the properties that remain constant and those that change for each of the four transformations. Define isometry (direct/opposite) and orientation. Know and apply the formal notation of transformations (R, r, T, D) to points in the coordinate plane. Rotate, reflect, translate, and dilate figures and points in the coordinate plane. Use the terms image and preimage correctly when talking and writing about transformations. Find the transformation used to get from point A to point A’ and be able to apply the same transformation to a given point B. Graph, find a composition of transformations (including glide reflection). Justify distance using Pythagorean theorem or distance formula. Find the properties that do or do not remain invariant (parallelism, distance, angle measure, betweenness, collinearity, orientation, etc.) Key Terms: isometry, orientation, image, preimage, invariant, reflection, rotation, dilation, translation, glide-reflection, direct and opposite isometry, composition of functions ( ο)
Unit 8: Similar Triangles Grade: 10
Number
Performance Indicators to be mastered in this unit:
G.G.44 G.G.45 G.G.46 G.G.47
Performance Indicator
Establish similarity of triangles, using the following theorems: AA, SAS, and SSS Investigate, justify, and apply theorems about similar triangles Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle Investigate, justify, and apply theorems about mean proportionality: o the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse o the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg
Spencer-Van Etten School District
Revised 5/25/10
G.G.27 Write a proof arguing from a given hypothesis to a given conclusion Major Concepts: • Students will understand the meaning of “similar triangles”. • Students will understand how to determine if triangles are similar. • Students will understand how to use deductive proof-writing to prove two triangles are similar. • Students will be able to name and label corresponding parts of similar triangles. • Students will understand that similar triangles are dilations of each other. • Students will be able to find all missing lengths of two similar triangles, given a proportion is already available or able to be found. • Students will be able to solve problems involving similar triangles. Essential Understandings: • How to set up a proportion • How to solve a proportion • How to prove triangles are similar • Know what makes some triangles similar; altitude of a right triangle drawn to the hypotenuse create 3, line parallel to a side creates 2, connections of midpoint of 2 sides creates 2, medial triangle creates 4 congruent triangles that are similar to original, dilation creates similar triangles • Angles never change in similar triangles; sides are proportional • Perimeters have the same ratio of similitude as the sides, areas are in a ratio squared, volumes are in a ratio cubed Essential Questions: • What are the different ways to determine if two triangles are similar? • What ratio are the sides of these two similar triangles in? • What happens to the angles of similar triangles? • What parts of these two triangles are congruent? • What parts of these two triangles are similar? • Which method can be used to prove these two triangles similar? • What do we know about these two triangles? • What is the length of side x, given two similar triangles with a similarity ratio? • Given the ratio, what is the ratio of the areas, volumes, perimeters? Essential S-VE Exit Behaviors: self-discipline Skills needed for mastery on performance indicators (& possible teaching strategies): • Review ratios and proportions, linear and quadratic, operations with radicals (simplify) • Idea: read the book “Cut Down to Size at High Noon”, give out puzzle pieces and have students make dilations of them to cut out and put together as a class • Review/teach midsegment theorem • Understand what makes triangles similar; angles always stay the same • Understand that dilations make similar figures • Be able to use similarity theorems to prove triangles similar; SAS, AA(AAA), SSS • Solve for a missing side, shadow word problems, other word problems • Know that an altitude drawn to the hypotenuse of a right triangle makes 3 similar triangles • Know that the altitude drawn to the hypotenuse of a right triangle is the mean proportional between the segments of the hypotenuse • Teach medial triangle a a a2 • Know that, given a ratio of , the perimeter of similar triangles is , area is 2 , and volume b b b Spencer-Van Etten School District
Revised 5/25/10
a3 is 3 b Key Terms: Alternate Interior Angle, Alternate Exterior Angle, Axiom, Corresponding angles, Postulate, Same side interior, Straight angles, Supplementary angles, Transversal, Vertical angles, AA, SSS, SAS, Perpendicular, Hypotenuse, Leg
Unit: Circles Grade: 10
Number
Performance Indicators to be mastered in this unit:
G.G.49
G.G.50
Performance Indicator
Investigate, justify, and apply theorems regarding chords of a circle: o perpendicular bisectors of chords o the relative lengths of chords as compared to their distance from the center of the circle Investigate, justify, and apply theorems about tangent lines to a circle: o a perpendicular to the tangent at the point of tangency o two tangents to a circle from the same external point o common tangents of two non-intersecting or tangent circles
G.G.51
Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is: o inside the circle (two chords) o on the circle (tangent and chord or 2 chords) o outside the circle (two tangents, two secants, or tangent and secant)
G.G.52 G.G.53
Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines Investigate, justify, and apply theorems regarding segments intersected by a circle: o along two tangents from the same external point o along two secants from the same external point o along a tangent and a secant from the same external point o along two intersecting chords of a given circle
G.G.27 Write a proof arguing from a given hypothesis to a given conclusion Major Concepts: To understand relationships between segments, arc lengths, and angles when a circle and two lines or line segments intersect Essential Understandings: • Know how to find the measure of an arc formed by two lines intersecting a circle and creating an angle • Know how to find the length of segments that intersect a circle • Know relationship between chord length and central angle measure
Spencer-Van Etten School District
Revised 5/25/10
Use similar and congruent triangles to prove relationships between segments and angles Be able to determine angle measures based on the measure of its intercepted arc(s) Be able to determine the length of segments based on the location of the point of intersection of the two line segments Essential Questions: • How many places does the segment intersect the circle? What type of line segment is this? • Where is the vertex; inside, on, or outside the circle? • What does it mean to bisect? • What type of angle is formed? • What do we know about arcs formed by parallel lines? • **What’s the rule given this situation? Essential S-VE Exit Behaviors: self-discipline Skills needed for mastery on performance indicators (& possible teaching strategies): • • •
•
Define parts of a circle
•
Angles and Arcs o Central angles o Inscribed angles (include semi-circle) o Interior angles o Exterior angles o Chord/tangent angle
•
Lengths of segments o Intersecting chords, two secants, two tangents, secant & tangent
•
Perpendicular bisectors of chords
•
Lengths of chords relative to distance from center
• Parallel lines intersecting a circle create congruent arcs Key Terms: chords, tangent, secant, arc, arc length, intersecting, central angles, perpendicular, parallel, external point, perpendicular bisector, distance, point of tangency, vertex, segments, inscribed angle, minor arc, major arc, semi-circle
Spencer-Van Etten School District
Revised 5/25/10
Unit: Solids Grade: 10
Number
Performance Indicators to be mastered in this unit:
G.G.10 G.G.11 G.G.12 G.G.13
G.G.14
G.G.15 G.G.16
Performance Indicator Know and apply that the lateral edges of a prism are congruent and parallel Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal Know and apply that the volume of a prism is the product of the area of the base and the altitude Apply the properties of a regular pyramid, including: o lateral edges are congruent o lateral faces are congruent isosceles triangles o volume of a pyramid equals one-third the product of the area of the base and the altitude Apply the properties of a cylinder, including: o bases are congruent o volume equals the product of the area of the base and the altitude o lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base Apply the properties of a right circular cone, including: o lateral area equals one-half the product of the slant height and the circumference of its base o volume is one-third the product of the area of its base and its altitude Apply the properties of a sphere, including: o the intersection of a plane and a sphere is a circle o a great circle is the largest circle that can be drawn on a sphere o two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles o
surface area is 4π r
o
volume is
2
4 3 πr 3
Major Concepts: Solids (rectangular prism, triangular prism, pyramid, cylinder, right circular cone, sphere) Volume of these solids Lateral Area of these solids Essential Understandings: • Area of 2-dimensional shapes • Volume = Area of base times height (comes to a point divide by 3) • Volume and surface area of a sphere • Lateral surface area is the sum of the area of the sides • Planes intersect solids, and the 2 dimensional shapes that are formed Essential Questions: • What shape is formed when a (solid) is intersected by a horizontal/vertical plane. • Given two prisms with the same height, find the missing value of the base to maintain the same volume. • If the height is halved (dilated by a factor of .5), what happens to the base to keep the same volume? • What is the formula for volume/area of _______________. Essential S-VE Exit Behaviors: self-discipline
Spencer-Van Etten School District
Revised 5/25/10
Skills needed for mastery on performance indicators (& possible teaching strategies): •
*Use reference sheet to help with the formulas!!!*
•
Review how to find area of 2-dimensional shapes.
•
Recognize by sight and by name the 6 different solids.
•
Understand that volume is area of base times height.
•
Be able to use the formulas correctly that are on the reference sheet. (B is not just base, but Area of base!!)
•
The solids that come to a point have volumes that are divided by 3.
•
Understand what it means for a plane to intersect a solid.
• Visualize and state the 2-dimensional shapes formed by intersecting planes with a solid. Key Terms: right circular cylinder, sphere, triangular prism, rectangular prism, right circular cone, pyramid, volume, altitude, 3-dimensional shapes, 2-dimensional shapes, lateral area, Comments/notes: Oatmeal containers, toblerone bar, playdough with dental floss, toilet paper roll, foam cone, orages/grapefruit/apples, lateral passes in football, cereal boxes, etc.
Unit 11: Coordinate Geometry: Circles Grade: 10
Number
Performance Indicators to be mastered in this unit:
Performance Indicator
G.G.70
Solve systems of equations involving one linear equation and one quadratic equation graphically G.G.71 Write the equation of a circle, given its center and radius or given the endpoints of a diameter G.G.72 Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer. G.G.73 Find the center and radius of a circle, given the equation of the circle in center-radius form G.G.74 Graph circles of the form ( x − h ) 2 + ( y − k ) 2 = r 2 Major Concepts: • be able to recognize the relationship between a graph of a circle, an equation of a circle, and the center and radius of a circle Spencer-Van Etten School District
Revised 5/25/10
Essential Understandings: • Be able to determine the center and radius of a circle given it’s equation in center radius form • Given a center and radius write an equation for a circle in center radius form • Determine the center given the diameter • to be able to graph a circle given an equation • to be able to write an equation given a graph • graph a quadratic and a linear equation and find point(s) of intersection Essential Questions: • Given a graph, what is the equation? • Given an equation, graph the circle • Given a center and radius, graph the circle it creates and write the equation that is represented • Given a quadratic and a linear equation, find the solution set. Essential S-VE Exit Behaviors: self-esteem Skills needed for mastery on performance indicators, (& possible teaching strategies): • Discuss what a circle is and its basic parts Radius is one half diameter • Graphing circles Determine Center of circle Plot center Determine radius Move north, south, east, west from center Finish sketch • Equations for a circle (center radius form) Extracting center and radius out of equation Putting in center and radius into equation Looking at graph, determine center and radius Given 2 endpoints of a diameter, determine the radius and center (midpoint) • Graph a quadratic • Graph a linear (Get y by itself) • Determine solution set of a system Key Terms: circle, radius, diameter, center, quadratic, linear, center radius equation, integers, system of equations, intersection, solution set
Spencer-Van Etten School District
Revised 5/25/10
Unit 12: Constructions and Loci Grade: 10
Number
Performance Indicators to be mastered in this unit:
Performance Indicator
G.G.17
Construct a bisector of a given angle, using a straightedge and compass, and justify the construction G.G.18 Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction G.G.19 Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction G.G.20 Construct an equilateral triangle, using a straightedge and compass, and justify the construction G.G.22 Solve problems using compound loci G.G.23 Graph and solve compound loci in the coordinate plane Major Concepts: • Create basic constructions and justify them using geometric theorems/properties • Solve problems involving a set of points that satisfy a specific condition Essential Understandings: • Know how to use a compass and straight edge to measure line segments/ angles • Know how to apply the angle sum theorem to construction of an angle whose measure is a specified sum • Be able to use triangle/circle theorems to justify constructions • Apply theorems about parallel lines to justify constructions • Be able to find 5 basic types of loci: 1. A locus that is a fixed distance from a point 2. A locus that is a fixed distance from a line 3. A locus that is equidistant from 2 parallel lines 4. A locus that is equidistant from 2 intersecting lines 5. A locus that is equidistant from 2 points Essential Questions: • What does it mean to bisect? • What type of triangle is formed? • When two parallel lines are cut by a transversal, what do we know about the angles that are formed? • What is the definition of a circle? Essential S-VE Exit Behaviors: life-long learning/self-discipline Skills needed for mastery: 1.
Constructions of line segments, angles, and triangles (use only a compass and straightedge) • Use a compass to measure distance • Copy a line segment • Construct a line segment that is the same length as the sum of two other line segments (segment addition theorem)
Spencer-Van Etten School District
Revised 5/25/10
• • • • • • •
Construct a line segment that is the same length as the difference of two other line segments (segment addition theorem) copy an angle justify copying an angle using congruent triangles (SSS) construct an angle that is equal in measure to the sum of two other angles (angle addition theorem) construct an angle bisector justify angle bisector construction (isosceles triangle properties) construct an equilateral triangle
2. Constructions of perpendicular and parallel lines (use only a compass and straightedge) • Construct a perpendicular through a point on the line (properties of chords intersected by diameters) • Construct a line perpendicular to a given line through a given point not on the line (properties of chords intersected by diameters) • Construct a perpendicular bisector (properties of chords intersected by diameters) • Construct a line parallel to a given line (properties of parallel lines cut by a transversalcorresponding angles are congruent) 3. Loci • Draw and describe a locus that is a fixed distance from a point (circle) • Draw and describe a locus that is a fixed distance from a line (parallel line(s)) • Draw and describe a locus that is equidistant from parallel lines, perpendicular/intersecting lines, and two points. 4. Compound loci • Draw and describe the locus that is equidistant from two parallel lines (line parallel to both lines and ‘halfway’ between both lines) • Draw and describe the locus equidistant from two points • Write equations for lines loci that are a set distance from a given line or point on a coordinate plane • Find the intersection of two loci. Key Terms: Compass, straight edge, construct, angle bisector, equilateral triangle, perpendicular, perpendicular bisector, parallel, locus (plural: loci), equidistant
Spencer-Van Etten School District
Revised 5/25/10
