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Metropolitan Community College COURSE OUTLINE FORM Course Title: Geometry Course Prefix & No.: Math 1260 LEC: LAB: 4.5 0 Credit Hours: 4.5 COURSE DESCRIPTION: This course covers geometric topics of logic, measurement, plane figure relationships and figures in space are presented in this course. COURSE PREREQUISITE (S): Within two years prior to brginning the course, either successful completion of MATH 0930 or higher with a grade of P or C or better or MCC Placement Test RATIONALE: This course is design to provide the geometry skills needed by students weak in geometry. It will give an understanding of geometric figures and properties to students in graphic arts and other programs. REQUIRED TEXTBOOK (S) and/or MATERIALS: Title: Essentials of Geometry for College Students Edition: 2/e 2004 Author: Lial, Brown, Steffensen, Johnson Publisher: Pearson Materials: Scientific calculator, notebook, straight-edge, compass, protractor Attached course outline written by: Math 1260 Committee, Connie Buller Date: 5/15/04 Reviewed/Revised by: Math 1260 Committee Date: 11/SP Effective quarter of course outline: 13/FA Date: 7/29/13 Academic Dean: Brad Morrison Date: ______ Course Objectives, Topical Unit Outlines, and Unit Objectives must be attached to this form. Metropolitan Community College COURSE OUTLINE FORM TITLE: Geometry PREFIX/NO: MATH 1260 COURSE OBJECTIVES: Upon completion of this course, each student will be able to: 1. 2. 3. 4. 5. Give examples of inductive/deductive reasoning Give examples of direct statements, inverses, converses and contrapositives. Define or describe points, lines and angles, postulates and theorems Perform geometric constructions using a straight edge and compass on paper Write basic deductive, direct and indirect proofs for selected problems with parallel lines, triangles, polygons and circles. 6. Classify and apply geometric principles to three-dimensional and two-dimensional objects. 7. Use and apply circle relationships, including arcs, secants and tangents. Do related constructions. 8. Graph points and lines and write the equation of a line. 9. Solve right triangles using trigonometric functions, the Pythagorean Theorem, 30-60-90 and 45-45-90 relationships. 10. Use unit conversions in practical applications TOPICAL UNIT OUTLINE/UNIT OBJECTIVES: Unit 1: Foundations of Geometry: Inductive/Deductive Reasoning, Points, Lines and Angles, Constructions Upon completion of this unit of study, the student will be able to do the following: 1. Define and compare inductive/deductive reasoning, postulates and theorems 2. Write, use and compare direct statements, inverses, converses, and contrapositives. 3. Mathematically define or describe the following: points, lines, segments, rays, half-lines, angles, angle bisectors, midpoints of segments, 4. Construct with compass and straightedge: segments, angles, angle bisectors and midpoints of segments. Unit 2: Classify, use, and do proofs with congruent and similar triangles and quadrilaterals; Construct medians and altitudes; Upon completion of this unit of study, the student will be able to do the following: 1. Identify and construct scalene, isosceles, equilateral, congruent, similar, acute, obtuse and right triangles 2. Construct medians and altitudes of triangles 3. Use basic definitions, postulates and theorems to do deductive proofs with congruent triangles, and apply their properties. 4. Construct parallel and perpendicular lines. Unit 3: Use indirect and deductive proofs, and apply with parallel lines, quadrilaterals, similar triangles and polygons, and use the Pythagorean Theorem Metropolitan Community College COURSE OUTLINE FORM Upon completion of this unit of study, the student will be able to do the following: 1. Use inductive work to make conjectures about parallel and perpendicular lines. 2. Do proofs involving congruent and similar triangles. 3. Construct parallel and perpendicular lines, congruent and similar triangles 4. Use the Pythagorean Theorem. 5. Classify a polygon by the number of sides and tell whether it is regular. 6. Identify the special types of quadrilaterals and their properties. . Unit 4: Use and apply circle relationships, including arcs and secants. Do related constructions. Upon completion of this unit of study, the student will be able to do the following: 1. Define, draw and determine the relationships involved in chords, secants and tangents. 2. Compute the length of arcs, measures of inscribed angles and circumference of circles. 3. Define, draw and determine the relationships involved in chords, secants and tangents 4. Construct circles, and their tangents. Find the center of a circle from a given arc by construction. Unit 5: Use and apply Solid Geometry, Analytic Geometry, Right Triangle Trigonometry Upon completion of this unit of study, the student will be able to do the following: 1. Identify and sketch 3-dimensional geometric solids, finding their lateral areas and volumes 2. Use 30-60-90 and 45-45-90 triangle relationships and the Pythagorean Theorem to write and calculate formulas 3. Graph point and lines. 4. Calculate slopes of lines, including parallel and perpendicular lines. 5. Write the equation of a line, and calculate distance between points. 6. Define a degree. 7. Identify the adjacent side, the opposite side and the hypotenuse of a right triangle. 8. Review the Pythagorean Theorem. 9. Define and use the sin, cos and tan of an angle as ratios of the side lengths of a right triangle 10. Use unit conversions in practical applications opposite (memorize these ratios): sin × = -----------------hypotenuse adjacent cos × = -----------------hypotenuse opposite tan × = -----------------adjacent 11. Find the angle when the trigonometric ratio is given. Metropolitan Community College COURSE OUTLINE FORM 12. Solve right triangles using sine, cosine and tangent. (find all sides and all angles) a. Solve applied word problems using sin, cos and tan. Unit 6. ENRICHMENT These objectives are only to be taught if all of the above required objectives have been taught and tested over following the testing guidelines. 1. Construct a Golden Rectangle 2. Construct the sum, difference, product, quotient and square root of a given line segment; divide it into fractional parts, like thirds or fifths (these Constructible Numbers are all applications of parallel and perpendicular lines and similar triangles) 3. Truth Tables COURSE REQUIREMENTS/EVALUATION: Upon completion of the objectives for this course, each of the objectives will be assessed and measured as follows: COURSE OBJECTIVES/ASSESSMENT MEASURES COURSE OBJECTIVES ASSESSMENT MEASURES Unit 1 Give examples of inductive/deductive reasoning Give examples of direct statements, inverses, converses and contrapositives. Define or describe points, lines and angles, postulates and theorems Perform geometric constructions using a straightedge and compass on paper Unit 2 Identify and construct scalene, isosceles, equilateral, congruent, similar, acute, obtuse and right triangles Construct medians and altitudes of triangles, parallel and perpendicular lines using a straightedge and compass on paper. Use basic definitions, postulates and theorems to do deductive proofs with congruent triangles, and apply their properties. A minimum of 4 in-class, closed-book, nonotes, individual written exams covering ALL the topical unit objectives must be completed with a combined average score of 60% or higher in order to achieve a course grade of C or above. Compass and straightedge will be required on some exams. A scientific calculator may be used on all exams (no cell phone or laptop) Other assessment measures such as homework, papers, portfolios, may be used in addition to the above Metropolitan Community College COURSE OUTLINE FORM Unit 3 Write basic deductive, direct and indirect proofs for selected problems with parallel lines Give examples of direct statements, inverses, converses and contrapositives. Perform geometric constructions using a straight edge and compass on paper Unit 4 Use and apply circle relationships, including arcs, secants and tangents. Construct circles, and their tangents. Find the center of a circle from a given arc by construction. Compute the length of arcs, measures of inscribed angles and circumference of circles. Define, draw and determine the relationships involved in chords, secants and tangents Unit 5 Classify and apply geometric principles to three-dimensional and two-dimensional objects. Use unit conversions in practical applications Graph points and lines and write the equation of a line. Solve right triangles using trigonometric functions, the Pythagorean Theorem, 30-60-90 and 45-45-90 relationships. The basic formulas for area/volume of basic shapes and solids should be on the exam. The basic formulas for 30-60-90 and 45-45-90 triangles should be on the exam. The pi button should be used instead of 3.14. Rounding should be done at the END of the problem. Sketches done by students should be included, particularly for solid shapes. In the case of extenuating circumstances, you have the option of allowing students to retake at most one (1) exam. If you choose to offer this retake, to increase the chances of improving student learning, it is highly recommended that the student complete additional requirements before being allowed to retake. Metropolitan Community College COURSE OUTLINE FORM TO THE INSTRUCTOR: The following are recommendations for instructors teaching MATH 1260. 1. When necessary, topics from prerequisite courses should be quickly reviewed. If time permits at the end of the quarter, enrichment topics may be included in the course as listed in the outline. 2. Scientific calculators should be incorporated into instruction. It is the responsibility of the student to bring a calculator to class. Cell phones are not to be used as calculators. 3. Compass and straightedge constructions should be incorporated into instruction. It is the responsibility of the student to bring a compass and straightedge to class. 4. Assistance is available in the Academic Resource Centers /Math Centers. Solids and rulers are available for you as an instructor, and also available for student use. Check with Academic Resource Centers/Math Center personnel at your site. 5. The Academic Resource Centers/Math Center staff are available to refer students to the tutorial material and to answer some specific questions. However, their presence does not replace the instructor’s responsibility for helping his/her students outside of class. COURSE REQUIREMENTS/EVALUATION: Upon completion of the objectives for this course, the following objectives will be assessed and measured as follows: 1. A minimum of four in-class, no notes, closed book exams must be taken individually covering all the required (not enrichment) topical unit objectives. The combined average of these tests must be 60% or above to achieve a grade of C or above. 2. Other assessment tools such as journals, group work, quizzes, and homework can be used as part of the assessment. TEXTBOOK-SPECIFIC INFORMATION: The following is a listing of specific sections from the Lial/Brown/Steffenson/JohnsonTable of Contents required to be taught during one quarter. The following schedule of lectures can be used by the instructor as a guide when planning the course syllabus. You should follow the timeline listed for each of the sections. This gives you thirty-two 55 minutes periods to teach the material (with one additional day for your discretion), five periods to test review, and five periods to administer tests in a typical 44 day quarter meeting four days a week for 55 minutes. Section 1.1 (1-31 odd) Objective 1: Define Inductive Reasoning Objective 2: Define an Axiomatic System Objective 3: Define Deductive Reasoning Section 1.2 (1-29 odd) Objective 1:. State undefined terms Objective 2: State postulates about points, lines, planes, and real numbers Section 1.1 should be covered in one 55 minute class along with the syllabus. Section 1.2 should be covered in one 55 minute class. Section 1.3 (1-67 odd) Objective 1: Define line segment and ray Objective 2: Define angles, including degrees, minutes, seconds, and borrowing to find complements/supplements Objective 3: Identify special angles (adjacent,complementary, supplementary, acute, obtuse, right) Sections 1.3 should be covered in one 55 minute class. Metropolitan Community College COURSE OUTLINE FORM Section 1.4 (1-28 odd) Objective 1: Define conditional, converse, inverse and contrapositive statements Objective 2: Introduce deductive proofs Note: these terms will be used later in the course, particularly with isosceles triangles and also parallel lines Section 1.5 (1-31 odd) Objective 1: Define the format of a formal geometric proof Objective 2: Describe the thinking process in a formal proof Objective 3: Use the format of the thinking process Objective 4: Define vertical angles; use them with proofs Note: it is very nice to bring wax paper, and have students fold 2 intersecting lines to make vertical angles, and see that they seem to be equal. Section 1.4 should be covered in one 55 minute classes. Section 1.6 (1-29 odd) + 6 Constructions Involving Lines and Angles Objective 1: Define perpendicular lines Objective 2: Define angle bisector Objective 3: Use compass and straightedge to do 6 basic constructions: copy segment; copy angle; bisect segment, bisect angle, drop a perpendicular to a line from a point off the line; construct a perpendicular to a line from a point on the line) Note: The sheet at the end of this course outline is the last sheet of my first Unit exam, and lists the last 5 of the 6 basic constructions (copy segment; copy angle; bisect segment, bisect angle, drop a perpendicular to a line from a point off the line; construct a perpendicular to a line from a point on the line) Note: it is nice to bring wax paper, have students fold a line, mark 2 points on it (can pierce the paper with their compass point), and fold one point over the other, creasing the new line—it is a perpendicular bisector of the segment determined by the two points. Section 2.1 (1-45 odd) ; Review Objective 1: Identify parts of a triangle Objective 2: Classify triangles by their sides; by their angles Objective 3: Define interior and exterior angles of a triangle Objective 4: Find the perimeter of a triangle Section 1.6 should be covered in one 55 minute class; students will need compass and straightedge Unit Exam 1 (Chapter 1 only; constructions required—compass needed) Note: a straightedge may be simply a folded sheet of paper; you may choose to loan students a piece of string to tie around their pencil if they have forgotten their compass) Section 2.2 (1-29 odd) Objective 1: Define congruent segments, angles, and triangles Objective 2: State the SAS, ASA, SSS postulates Objective 3: Use the postulates to show congruence of triangles. Note: it can be helpful to write SAS SAS as the end of a proof instead of simply SAS Note: it is helpful to bring a scissors and two sheets of paper, cutting out One 55 minute class; students will need compass and straightedge Section 1.5 should be covered in one 55 minute class. Section 2.1 and Review should be covered in one 55 minute class. Section 2.2 should be taught in one 55 minute classes. Metropolitan Community College COURSE OUTLINE FORM a shape on both sheets at the same time, and using that to demonstrate the idea of congruence—and then moving the two shapes to let students see the shapes are still congruent even if they have a different orientation. Section 2.3 (1-25 odd + 18,20) construct angle bisectors and segment bisectors Objective 1: Define and use Corresponding Parts of Triangles are Objective 2: Prove the Angle Bisector and Segment Bisector Constructions Note: Although the book uses CPCTE, this is pretty meaningless to students—it is safer to abbreviate word “Corresponding”, and use the triangle and congruence symbols) Note: Problems 17-20 are very nice practical applications of congruence, and one reason we selected this text; the problems in this text are not paired, and so we include the evens on these practical applications. Section 2.4 (1-29 odd + 28) construct medians, altitudes, angle bisectors Objective 1: Prove some properties of isosceles trianges Objective 2: Identify the converse of a statement, and concurrent lines Objective 3: Define and construct medians, altitudes and angle bisectors of triangles Objective 4: Investigate some properties of concurrent lines in triangles Note: Very nice to mention the application of medians to constructing mobiles in art projects; do not make students memorize words like centroid, orthocenter, or incenter, but they should be able to construct these, and understand some of the properties. Section 2.3 should be taught in one 55 minute classes; students will need compass and straightedge Section 2.4 should be taught in one 55 minute classes; students will need compass and straightedge Section 2.5 (1-19 odd) Objective 1: Define the LA theorem for right triangles and prove congruent Objective 2: Define the LL theorem for right triangles and prove congruent Section 2.5 should be taught in one 55 minute class. Section 2.6 (1-29 odd) construct congruent triangles given parts Objective 1: Construct triangles with various given parts Objective 2: Construct altitudes and medians of triangles Review Section 2.6 should be taught in one 55 minute class. Students will need compass and straightedge One 55 minute class Test 2 (chapter 2) Compass and Straightedge required. One 55 minute class Sections 3.1 (1-29 odd) Section 3.2 (1-41 odd) Objective 1: Indirect Proof Objective 2: The Parallel Postulate Objective 3: Define angles formed by parallel lines and a transversal (corresponding, alternate interior, alternate exterior; compare to vertical) Objective 4: Describe ways to prove lines parallel, and prove theorems Note: it is very nice to bring wax paper, and fold parallel lines, crossed by a transversal, and have students mark and note the various angles formed. Dry erase markers are handy to mark on the wax paper. Sections 3.3 (1-31 odd) Section 3.4 (1-41 odd) Objective 1: Define and classify polygons, with perimeter, diagonals, interior and exterior angles Sections 3.1 and 3.2should be taught in one 55 minute class. Sections 3.3 and 3.4 could be taught in One 55 minute class, but also Metropolitan Community College COURSE OUTLINE FORM Objective 2: Construct parallel lines with compass and straightedge Objective 3: Calculate degrees of angles in polygons, both interior and exterior Note: there are formulas, but it is easier to cut polygons into triangles, using the idea that there are 1800 in a triangle; use the idea of zooming out to help students recognize that there will be 360o circling around the dot that the polygon finally shrinks down to, counting only one exterior angle at each vertex (this is a limiting idea from calculus, but you need not mention that). Objective 4: prove AAS, HA and HL theorems (using congruent triangles and corresponding parts of Triangles are Section 4.1 (1-20 odd) and Section 4.2 (1-11 odd) Objective 1: Define and classify parallelograms Objective 2: Use properties of parallelograms, rhombi and kites and do related proofs could be taught in two, depending on your class—watch the time at the end. Students will need compass and straightedge Section 4.3 (1-35 odd, also #24) Objective 1: Define rectangles and squares, with their properties Objective 2: Prove and use the theorem about segments joining the midpoints of two sides of a triangle. Objective 3: Construct a rectangle with compass and straightedge Objective 4: Discuss and use practical applications of the above Section 4.4 (1-31 odd + divide a segment into parts by construction and parallel lines) Objective 1: Define and use properties of trapezoids Objective 2: Divide a segment into n equal parts by construction, using parallel lines. Note: do both by compass and straightedge, but also show students how it can be done with lined paper (see example in book) Section 4.3 should be taught in one 55 minute class. Students will need compass and straightedge Section 4.4 should be taught in one 55 minute class. Students will need compass and straightedge Section 5.1 (1-37 odd) Objective 1: Review ratio and proportion Objective 2: Define means and extremes, and mean proportional (geometric mean) Objective 3: Do multiple proportions, such as the proportions of blending paint is 3:5:2 for red, white and blue Note: Students will be unfamiliar with this practical application—an easy way is 3x + 5x + 2x = the number of ounces needed in the final mixture. Note: an old-fashioned way of writing proportions is 2:3::4:6. The inner two are the means; the outer two are the extremes, and the means-extremes property is what some call “cross-multiplying” for proportions. Section 5.1 should be taught in one 55 minute class. Section 5.2 (1-45 odd) Objective 1: Define similar polygons Objective 2: Prove triangles similar with AA = AA Objective 3: Calculate proportions formed by a line intersecting a triangle that is parallel to one side Objective 4: Calculate proportions formed by a line bisecting one angle of a triangle. Sections 5.2 should be taught in one 55 minute class. Sections 4.1 and 4.2 should be taught in one 55 minute class Metropolitan Community College COURSE OUTLINE FORM Section 5.3 (1-45 odd) and Section 5.4 (1-45 odd) Objective 1: Calculate proportions formed by an altitude to the hypotenuse of a right triangle (it is a mean proportional; construct a mean proportional) Objective 2: Use and apply the Pythagorean Theorem Note: Although the Pythagorean Theorem is presented in its traditional a2+b2 = c2 format, it removes one layer of abstraction if it is presented as leg2 + leg2 = hypotenuse2. This is particularly practical as most triangles in ordinary use do not come pre-labelled a,b,c and in this course there will be much use of 30-60-90 and 45-45-90 triangles, which depend on the Pythagorean Theorem Review; Section 6.1 (1-33 odds)+ Handout Using Circles Practically Objective 1: Define circles, arcs, inscribed and central angles Objective 2: Prove and use the facts that inscribed angles are ½ the measure of their intercepted arcs; central angles are equal to their inscribed arcs Objective 3: Using Circles practically Note: Encourage students to draw circles to illustrate their homework problems in this and following circle sections—an easy way to sketch a circle is to use a small coin, like a penny Note: The handout has the formula for area of a circle, and also has unit conversions. The answers are on the handout. Sections 5.3 and 5.4 should be taught in one 55 minute class. Students will need compass and straightedge Unit Test 3 (Chapters 3,4,5 only) Section 6.2A (1-15 odd) Objective 1: Define and use chords of circles Objective 2: Find angles and segment lengths created by intersecting chords One 55 minute class. Section 6.1A should be taught in one 55 minute class. Section 6.2B (17-33 odd) + worksheet using circles practically Objective 1: Define and use secants of circles Objective 2: Find angles and segment lengths created by intersecting secants Note: do #31, 33 with students in class—the main aim should be that they can find the angles and segment lengths, though understanding where these theorems come from reinforces similar triangle relationships and inscribed and central angle relationships Section 6.3 (1-39 odd) Objective 1: Define and Use Tangents to a Circle Objective 2: Construct a Tangent to a Circle from a point on the circle Objective 3: Construct a Tangent to a Circle from a point off the circle Objective 4: Construct the center of a circle, given an arc from the circle Objective 5: Find lengths of tangents and segments formed by tangents; Objective 6: Find angles formed by tangents Objective 7: Find the distance to the horizon, using tangent properties Note: These applications have much to do with the concepts of 6.1 and 6.2 Section 6.2B should be taught in one 55 minute classes. Review Unit Exam 4 (Chapter 6) scientific calculators; compass/straightedge used Section 8.1 (1-31 odd) + Euler, Hamilton Objective 1: Determine the behavior of lines and planes in space One 55 minute class One 55 minute class Section 6.1 and Review should be taught in one 55 minute class. The test will only cover Chapters 3,4,5 Section 6.3 should be taught in one 55 minute class. Students will need compass and straightedge Section 8.1 should be taught in one 55 minute Metropolitan Community College COURSE OUTLINE FORM Objective 2: Define vertices, edges, faces and polyhedrons; sketch 3-d shapes; Objective 3: Use toothpicks and raisins to model polyhedrons; use them to work with Euler’s formula Note: divide students into small groups of 2 or 3. Give each group some toothpicks and raisins (explain that the raisins have been dropped on the floor, so they won’t eat them) and have them build polyhedra. Toothpicks are edges; raisins are vertices; the polyhedra are hollow (don’t count any toothpicks used just for internal struts). Euler’s circuit: can cover all the edges without retracing any edge (like snow-plowing streets—don’t want to re-trace any streets) and return to start. Euler path: the same, tracing all edges, but not making a complete circuit—end somewhere other than start Hamiltonian circuit: the same, but going to all vertices without visiting a vertex twice, and returning to start Hamiltonion path: the same, but going to all vertices without being able to return to start. Note: this activity does two things: it gives students a glimpse of how geometry is being used today in graduate school projects, and more importantly it REALLY lets them know what is a vertex (raisin) an edge (toothpick) and a face. It also REALLY helps them distinguish between lateral surface area and volume. Then do Euler’s formula for each structure the students have made. Section 8.2 (1-31 odd) Objective 1: Defining prisms, lateral surface area and volume Objective 2: Sketching prisms Objective 3: Finding surface area and volume Objective 4: Using 30-60-90 triangles to find the area of a hexagon Objective 5: Use unit conversions in practical applications Note; Keep memorization to a minimum by having students break complex shapes into familiar triangles and rectangles. Note: Help students understand that volume of a prism is simply how many layers there are of the base. V = Area of base x height. This is true no matter what the shape of the base. Note: in sketching a solid, give a bit of a pinch to the congruent bases (a rectangle looks like a non-rectangular parallelogram; a circle looks like a non-circular ellipse). Let the sides of the rectangles be parallel to each other, and then connect the corresponding vertices, letting the “hidden” sides appear with dashed lines. Section 8.3 (1-19 odd) Objective 1: Defining pyramids, slant height and altitudes Objective 2: Find lateral area and volume Objective 3: Use unit conversions in practical applications Note: Again, break complex shapes into familiar triangles and rectangles rather than using formulas like LA = ½ pl. There is enough for them to memorize without that. Note: A good activity is to take a pointed solid (cone or pyramid) and its related cylinder or cone. Show the bases are congruent, and the heights are the same. Ask the students how many pyramids (cones) it takes to fill class. Teacher: bring toothpicks and raisins Section 8.2 should be taught in one 55 minute class. Teacher: bring geometric solids (may be found in mailroom or also in Math Centers) Section 8.3 should be taught in one 55 minute class. Teacher: bring geometric solids (may be found in mailroom or also in Math Centers; bring a larger container of small beans or rice for volume experiment Metropolitan Community College COURSE OUTLINE FORM the related prism (cylinder). Go around the room, recording guesses. Then fill the pointed one with small beans or rice, and pour it into the larger shape. The conclusion is that it takes 3 of the pointed to equal one of the other. This helps in understanding the formulas for volume. Note: In sketching a pyramid, give a pinch to the base as described earlier. Let a dot stand for the point that is the apex of the pyramid, and connect the dot to each vertex of the base. In this class, all pyramids will be right pyramids, though the set of geometric solids may have some slant pyramids. Section 8.4 (1-39 odd) Objective 1: Define cylinders and calculate volume and surface area Objective 2: Define cones and calculate volume and surface area Objective 3: Use unit conversions in practical applications Note: Be certain to use the pi button on their calculators instead of 3.14 Note: Roll a sheet of paper. The students already know the area of a rectangle to be base x height. Now the base is the circumference of a circle, so the lateral area must be 2 rh. Note: if you didn’t do the volume with cylinders/cones and beans earlier, this would be a good place. I like it earlier so that students can see the overall ideas of Volume upright solid = Area of Base x height; Volume upright cone/pyramid = 1/3 Area of Base x height. Note: to illustrate the effect of the round-off error occurring with use of 3.14 for consider a LARGE cylindrical tank holding corn syrup (like at Cargill). Let the tank be about 40 feet high, and with a diameter of 20 feet. Find how many gallons of corn syrup are in there, and multiply by the cost per quart. See how many gallons are given for free, and the cost, if the seller uses 3.14. This is a nice exercise in unit conversions as well, which employers of our Metro students say that they need to have in employees. Ask the students whether they are rounding down or up in using 3.14, and why they think Cargill might value use of the button Note: To illustrate the area of a circle r2, sketch a circle, and divide it up into sections, with vertex at center, and as narrow a sections as you can. Now take those sections and sketch them lined up wide end, point end, wide, point, so that they now seem to form a rectangle. Students know the area of a rectangle is base times height: if the sections were cut VERY narrow, the height corresponds to the radius, and the base corresponds to ½ a circumference. So it makes the Area of a circle: r (r)= r2 more believable. Note: Linear measure is in 1 dimension, so plain feet/inches/cm; area, involving 2 dimensions and multiplication, is measured in square units; volume, involving 3 dimensions, is measured in solid units, or cubic units. Note: In unit conversion, it is sometimes difficult for students to realize that while 3 feet = a yard, 9 square feet = 1 square yard, and 27 cubic feet = 1 cubic yard. When ordering cement for a project, this matters…. Section 8.5 (1-29 odd) Objective 1: Define a sphere and calculate surface area and volume Objective 2: Break composite shapes into simpler forms like hemispheres and cylinders to find area and volume Objective 3: Use unit conversions in practical applications Section 8.4 should be taught in one 55 minute class. Teacher: bring geometric solids (may be found in Math Centers) Section 8.5 should be taught in one 55 minute class. Teacher: bring geometric solids (may Metropolitan Community College COURSE OUTLINE FORM Note: pronounce sphere as ss-FEER Note: do the even numbered problems in class, particularly #24, 26, 28. Problems 25,27,29 depend on these, and address the general objective of unit conversion Section 9.1 (1-55 odd) and Section 9.2 (1-51 odd)+ worksheet using lines in business Objective 1: Use the Cartesian coordinate system Objective 2: Apply the Pythagorean Theorem to find the distance between two points; use averaging to find the midpoint between two points Objective 3: Define and use slope of a line, including parallel and perpendicular lines Objective 4: Write and graph the equation of a line Objective 5: Use practical applications of using equations to describe lines be found in Math Centers) At the close of this chapter, return the set of solids to where you found them, so another class can use the; the set in the Math Centers is for the students’ use outside of class. Sections 9.1 and 9.2 should be taught in one 55 minute class. Note: Although it is possible to clear fractions and use the traditional slope formula (assuming all points on a line have the same slope, so you can use (x,y) instead of (x2,y2) and get y – y1 = m(x – x2), it is easier for students to derive the equation of a line by substituting the slope and a single known point into y = mx + b and then solving for b. It is also much easier to remember y = mx + b than the other form. Note: Because students are by now VERY familiar with the Pythagorean Theorem, and with parallel and perpendicular lines, and with bisecting segments, this is not as hard as if it were done in an Algebra class. Section 10.1 (1-25 odd) and Section 10.2 (1-35 odd) Objective 1: Define and use Sine, Cosine, and Tangent ratios, including 30-60-90. Objective 2: Find an angle, given its trig ratio Note: These sections will serve as a review of much learned earlier, such as “opposite side” and right triangles, as well as 30-60-90 and 45-45-90 (which come from base angles of isosceles triangles, and diagonals of squares). A way to introduce the trig ratios is to sketch a right triangle, choosing one of the acute angles, and labeling the adjacent, opposite sides as well as the hypotenuse. Let students help ‘discover’ the 6 possible ratios—and then define them as Sine Cosine and Tangent. They do need to know SohCahToa (Sine = opp/hyp; Cosine = adj/hyp; Tan = opp/adj) but not to memorize the reciprocal functions of Cosecant/Secant/Cotangent). They have already used the mnemonic (P)EMDAS for order of operations—this is My Dear Aunt Sally’s mother’s maiden name…) Note: Do not make students rationalize denominators, but show them how to check their answers against the book answers, using their calculators. Sections 10.1 and 10.2 should be taught in one 55 minute class. Metropolitan Community College COURSE OUTLINE FORM Note; Using the inverse function to find the underlying angle is an extension of what they have always done to solve linear equations (do the same to both sides). Students will need help in finding the Sine-1 key on their calculators. Make sure students have their calculators in degree mode, and do several in class. Section 10.3 (1-23 odd) and Section 10.4 (1-19 odd) Objective 1: Solving Right Triangles Objective 2: Practical Applications using trigonometry Note: This again will provide a review of earlier material. The angles of elevation/depression are the alternate interior angles. To avoid making round-off errors worse, use given measurements when possible. Review Test 5 (chapters 8,9,10) Note: on the exam, you may wish to have the following formulas: Use the Sections 10.3 and 10.4 should be taught in one 55 minute class. One 55 minute class. One 55 minute class period. button on your calculator; label units on final answer r 2 Circumference circle: 2 r Vol cylinder: r 2 h 2 Vol prism: lwh Lateral Area cylinder: 2 rh Surface area sphere: 4 r Area circle: 1 1 4 r 2 h Vol sphere: r 3 Vol pyramid lwh 3 3 3 1 bh SohCahToa Area triangle: 2 Vol cone: MATH 1260 Using Circles Practically C. Buller accompanying Section 6.1 Show work Round answers to the nearest tenth, if necessary. Use the π button Note: Area of a circle = πr2; Circumference of a circle = πd 1. A circular garden at the Elkhorn Valley Campus has radius 9 m. If a 1 meter-wide circular walk surrounds it, what is the area of the walk? a).Draw a picture representing the situation. b) What is the area of the walk? 1b) ________ 2. A 12-inch diameter pizza costs $8.50. A 16-inch diameter pizza costs $11.50. Which pizza costs less per square inch? Hint: “per” means to divide by the square inch a) What is the cost per square inch of the smaller pizza? 2a) ________ b) Which one costs less per square inch? 2b)________ Metropolitan Community College COURSE OUTLINE FORM c) What might be a good reason to buy the pizza that costs more per square inch? 3. Mrs. Buller has a patio in the shape of a trapezoid with bases 8.1 yd and 6.7 yd and height 5.8 yd. A circular dining area in the center of the patio has diameter 3.2 yd and is covered with Aztec tile. Assuming no waste, how much will it cost to the nearest dollar to cover the remainder of the patio with outdoor carpeting that costs 80¢ per square foot? Hint: be careful about conversions here a) Draw a picture representing the situation. b) What is the dollar cost of the outdoor carpeting? 3b)_______ Answers: 1a) 59.7 m2 2a) 7.5¢ 2b) The larger, by nearly 2¢ a square inch 3b ) $251 MATH1260 Lines in Business C. Buller accompanying Section 9.2 Show work Round answers to the nearest penny, if necessary 1. Mr. Toline owns a small business that manufactures picture frames. He has determined that the weekly cost in dollars y of making a number of picture frames x is described by the equation y = 6x + 200. a).Find the cost of making 80 picture frames in one week. 1a)_________ b) How many frames were made during a week when the costs were $1100? 1b) ________ 2. Mrs. Buller makes wood-burning kilns. She discovers that 10 kilns can be constructed for $2800, and 24 kilns can be made for $6125. Let y be the cost of making x kilns. a) Find a linear equation describing this relationship. Hint: (x,y) = (kilns, cost) 2a) y = ___________ b) Use your equation to calculate the cost of making 31 kilns 2b) ______________ c) What do you think the $425 in the answer to 2a stands for? 3. In 2004 (call this year 0), the Metropolitan Manufacturing Company had sales of $40,000. In 2009 (call this year 5), the total sales were $170,000. Let y be the total sales in year x. a) Find a linear equation describing this relationship. Hint: (x,y) = (year, sales) 3a) y = ___________ Metropolitan Community College COURSE OUTLINE FORM b) Use your equation to estimate the sales in year 2014 3b) ______________ c) Why might this estimate be wrong, even if you got the answer to 3a right? Answers: 1a) $680 Revised 05-25-12 1b) 150 2a) y = 237.5x + 425 2b) $7787.50 3a ) y = 26000x + 40000 3b) $300,000 Metropolitan Community College COURSE OUTLINE FORM (Page 17 of 17) Five Basic Constructions To be used in Unit 1; on Test 1 I allow 16 of the 100 points for doing any 4 of the 5—my 5 point bonus is to do the fifth one. For the following five constructions. Show compass marks. a) Bisect AB A b) Bisect Angle A B A c) Copy Angle C at P d) Construct a line perpendicular to l from D D C P e) Construct a perpendicular to m at E E ESO Revised 3-13-01