School of Education Lesson Design TemplateGrade 11th

Subject Math SAT Preparation: Geometry Lesson 2 of 2Date 2012-03-073. Learning Targets – What are the objectives for the lesson?3.3 – Cite the EALRs/standards using the numbers and text. Usually limit the lesson to 1 – 2 EALRs.Grade Level: High School - GeometryCore Content: G.2. Lines and Angles (Geometry/Measurement)Description: Students study basic properties of parallel and perpendicular lines, their respective slopes, and the properties of the angles formed when parallel lines are intersected by a transversal. They prove related theorems and apply them to solve both mathematical and practical problems.

Finished up that one at the beginning of class.

Grade Level: High School – GeometryCore Content: G.3. Two- and Three-Dimensional Figures (Geometry/Measurement)Description: Students know and can prove theorems about two- and three-dimensional geometric figures, both formally and informally. They identify necessary and sufficient conditions for proving congruence, similarity, and properties of figures. Triangles are a primary focus, beginning with general properties of triangles, working with right triangles and special triangles, proving and applying the Pythagorean Theorem and its converse, and applying the basic trigonometric ratios of sine, cosine, and tangent. Students extend their learning to other polygons and the circle, and do some work with three-dimensional figures.

Grade Level: High School – GeometryCore Content: G.7. Reasoning, problem solving, and communicationDescription: Students formalize the development of reasoning in Geometry as they become more sophisticated in their ability to reason inductively and begin to use deductive reasoning in formal proofs. They extend the problem-solving practices developed in earlier grades and apply them to more challenging problems, including problems related to mathematical and applied situations. Students use a coherent problem-solving process in which they analyze the situation to determine the question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions that have been made. They examine their solution(s) to determine reasonableness, accuracy, and meaning in the context of the original problem. They use correct mathematical language, terms, symbols, and conventions as they address problems in Geometry and provide descriptions and justifications of solution processes. The mathematical thinking, reasoning, and problem-solving processes students learn in high school mathematics can be used throughout their lives as they deal with a world in which an increasing amount of information is presented in quantitative ways, and more and more occupations and fields of study rely on mathematics.3.4 – Cite the corresponding GLEs/performance expectations using the numbers and text.G.2.B Know…about angles, including angles that arise from parallel lines intersected by a transversal.G.2.D Describe the intersections of lines in the plane …

G.3.C Use the properties of special right triangles (30°-60°-90° and 45°-45°-90°) to solve problems.G.3.D Know… and apply the Pythagorean Theorem and its converse.G.3.I Explain and perform constructions related to the circle.

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G.7.B Select and apply strategies to solve problems.G.7.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.G.7.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.G.7.F Summarize mathematical ideas with precision and efficiency for a given audience and purpose.G.7.G Synthesize information to draw conclusions and evaluate the arguments and conclusions of others.3.5 – Cite the objectives (skills or concepts) for the lesson. What do you want students to think, know and/or be able to do at the end of the lesson?Students will be able to state properties of angles created by the intersection of a transversal with two parallel lines. Namely:

Students will be able to define condition for parallel lines (i.e. never intersect in the plane). Students will be able to state and apply theorem for alternate interior angles. Students will be able to combine alternate interior/exterior angles and vertical angles, plus

properties of supplementary angles to define all angles created by transversals. Students will be able to state and apply facts about triangles:

o Angles in a triangle sum to 180°o Right triangles have only one 90° angle, side opposite that angle is hypotenuse, also the

longest side.

o Ratio of side lengths in a 45°-45°-90° triangle is 1:1:

o Ratio of side lengths in a 30°-60°-90° triangle is 1: Students will gain familiarity with Reference Information that is provided at the beginning of

every SAT math section. Students will be able to interpret and understand a multiple-choice, geometry-related problem

similar to those used on the SAT. Students will be able to choose an answer, defend/explain their answer, and explain why other

choices were not their answer.3.6 – Rephrase your learning targets using student-friendly language.Two (or more) lines are parallel when they don’t intersect in the plane. When parallel lines are cut by another (third, non-parallel) line, then the smaller angles thus created are all equal, as are the larger angles thus created. In the special case that the third line is perpendicular to any one of the parallel lines, it is also parallel to the other line, and all angles thus created are 90°.

Know what a right triangle is, how angles are named, which side the hypotenuse is.

Use the Reference Information given with every math section on the SAT to solve problems related to geometry including:

Sum of angles in a triangle is 180° Pythagoras’ Theorem is a relationship between the length of sides of a right triangle. Area of a triangle, Perimeter of a triangle Degrees in a circle, area of a circle, circumference of a circle area of a triangle, 45°-45°-90° triangles, 30°-60°-90° triangles.

State why you pick certain answers, i.e. why other answer choices were not acceptable.

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4. Lesson Assessment – How will students demonstrate their learning?4.8 – Complete the following table to highlight what the students will do to demonstrate competence specific to learning for this lesson.

Description of assessment activity Evaluative criteria What the assessment is

designed to assess Feedback to students

At the end of class will give a short quiz (10 problems) which are meant to gauge student understanding and recall of learning targets in the lesson.

1.Have students understood goals of lesson

2.Can students interpret and apply “Reference Information” on math section of SAT.

3.Are students proficient at bubblesheets, and recording answers in that way.

Can students interpret information on the “Reference Information” from an SAT math section.

Feedback will be given to students on their answer sheets as to what questions they missed and how they could improve their understanding.

5. Instructing and Engaging Students in Learning – What will happen in the lesson?5.5 – Describe the sequence of steps in the lesson in the following table. General lesson sequences may be more directive (e.g., ITIP) or open (constructivist). Whatever design is used, the lesson should be explicitly outlined.

Complete the following table: Provide an estimate of time. List the sequence of the various learning experiences in the lesson.Time Learning experiences

0:00 – 0:15

[15min]

Slide: two parallel lines and transversal.Motivation: copy one line, and the move the copy slowly away, keeping it parallel.Cold-Calling: Find two angles that are equal, Find two angles that sum to 180°.

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Slide: put up a problem for discussionGo around the

Go around the room and have everyone say something true about the problem.0:15 – 0:30

[15min]

Hand out triangle manipulatives.Remind students of triangle inequality that we discussed previously.Slide: triangle, angles sum to 180°.

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Slide: here’s where you find that on the Reference Information.

Slide: special triangle, right triangle

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Go over how to name angles. How the vertex must be the middle of the triplet.Write another triangle on the board with angles labeled, talk about the “Angle opposite the 30°angle… etc.”

Slide: Pythagoras, very strict condition on something we already have a feel for, that two shorter sides must sum to a length greater than the largest side.A lot of Academic Language in these slides. Linguistic demands are ability to read and interpret geometry notation and naming procedures.

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Slide: Pythagoras on Reference Information.Go over the common mistakes in applying the formula. (squared is not square root is not 2 times).Go over a sample of applying the formula (3-4-5).Go over the algebra involved, i.e. “square both sides of the equation) or

0:30 - 0:40

[10min]

Slide: Go over the area of a triangle.

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Slide: Go over the location of that formula on Reference Information

Slide: Circles have 360°.

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Slide: Circles have 360° on the Reference Information

Slide: Circle circumference, make distinction between radius and diameter.

Discuss what means. The ratio of circumference of a circle to its diameter.

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Slide: Circle area[insert slide here]Slide: Circle area on Reference Information[insert slide here]

0:55- 1:05

[10min]

Break.

1:05 – 1:25

[20min]

Hand out and ask students to do the quiz.5 minutes to hand out and 15 minutes to do. Remind students that they need to get used to doing a problem in 1 minute. Be hard core here.

1:25 -1:30

[5 min]

Wrap up class, go over the highlights of prior content.

5.9; 5.10 – Materials – What materials, including community resources and educational technology, will you need in order to teach this lesson? What materials will students need for this lesson?

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Computer with projection system, and PowerPoint for presentation.

Geometry Manipulatives: two sticks joined with a screw at the middle so they can pivot and illustrate intersecting lines. Multiple sets of sticks (yardsticks cut into interesting lengths with screw holes drilled at the ends for joining into triangles).

Printout of Quiz (Assessment) Answer SheetsPrintout of Quiz

QUIZ

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1. The hypotenuse of a right triangle, , is:

a.

b.

c.

d.

2. The area of a rectangle is:

a.

b.

c.

d.

3. The sum of the measure of the angles in a triangle is:

a. 180°b. 90°

c. 360°d. 45°

4. The number of degrees in a circle is:

a. 180°b. 90°

c. 360°d. 45°

5. The circumference of a circle is:

a.

b.

c.

d.

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6. Given a 45°-45°-90° triangle, if the hypotenuse is , the length of one side is

a.

b.

c.

d.

7. The volume of a cylinder is:

a.

b.

c.

d.

8. A figure on the SAT which should not be used to accurately interpret an answer will be labeled:a. “All numbers used are real numbers”b. “All figures lie in a plane.”c. “Figure is not drawn to scale.”d. “The use of a calculator is permitted.”

9. A 30°-60°-90° right triangle has a hypotenuse which is 30 units long. What is the length of the side opposite the 60° angle?

a. 60 unitsb. 30 unitsc. 15 units

d. 15 units

10. The volume of a rectangular prism is:

a.

b.

c.

d.

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