Teach-Now Unit Plan Template

Unit Plan TemplateTeacher Candidate: Kirk AdairUnit Name: GeometrySubject and Grade Level: Geometry / Secondary Level

Vision for the Unit:

The Mathematical Practices

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.

Learning should look like and what I expect for mastery would entail:

1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

The Mathematical Content Standards

I have developed the unit around the mathematical content standards because it is essential curriculum.

The Mathematical Content Standards (Essential Curriculum) that follow are designed to promote a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the mathematical practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards that set an expectation of understanding are potential “points of intersection” between the Mathematical Content Standards and the Mathematical Practices.

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Teach-Now Unit Plan Template

Lesson Title:  GeometryGrade Level and Course: Secondary High school LevelTime Segment of Lesson: ___50______ minutes  

Student Diversity and Differentiation of Instruction

Student Diversity Differentiation of Instruction

      1. English Language Learners (Non-native speakers of instructional language

Allow them to give shorter answers to questions and pair them with a partner who can help them with the readings and understanding the vocabulary.

2. SPED Same as #1, with also the option of easier questions.

       3. General Ed. students Allow them to read independently or with a partner and answer the given questions as is.

       4. Honors Students Give them extra credit questions and extra reading materials with slight variations on the theme.

 Objectives with Formative and Summative Assessments

Measurable Objectives to be Addressed Formative and Summative Assessment

     1.  Students will understand what a theorem is and how theorems are used in math.

Formative: Notes on main idea, key concepts and new vocabularySummative: Answers to questions in quiz form

     2. Students will prove one theorem about triangles. Formative: List one secondary sourcesSummative: Answers to questions in quiz form

 

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 Sum mative Assessment: What evidence or project will students submit to demonstrate that they have met the standard and objectives? How will you assess these products.

From above: the students will present notes on the main idea, key concepts and new vocabulary, also they will list secondary sources of the proof theorem about triangles.

From above: the students will answer questions in quiz form.

Objectives: Identify the objectives for the unit and a table that shows where they fall on Bloom’s taxonomy.

Objective Level of Bloom’s Taxonomy     1.  Students will understand what a theorem is and how theorems are used in math.

Analyzing/Applying

     2. Students will prove one theorem about triangles. Understanding/Remembering

Lesson Standards:  What mathematical skill(s) and understanding(s) will be developed? Which Mathematical Practices do you expect students to engage in during the lesson?

G.CO.C.10 Prove theorems about triangles.  Theorems include:  measures of interior angles of a triangle sum to 180 degrees.

MP5:  Use appropriate tools strategically.MP7:  Look for and make use of structure.MP8:  Look for and express regularity in repeated reasoning.

Lesson Launch Notes: Exactly how will you use the first five minutes of the lesson?

Have each student draw a triangle with a ruler. Have the students measure the three angles of the triangle.

(Look for evidence of MP5.)

Lesson Closure Notes:  Exactly what summary activity, questions, and discussion will close the lesson and connect big ideas? List the questions. Provide a foreshadowing of tomorrow.

Ask students questions to review the learning from the lesson such as:

What is the sum of the angles of a triangle in degrees?

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How do you think we can use the sum of the angles of a triangle to solve other problems?

What are parallel lines? A transversal?

Lesson Tasks, Problems, and Activities (attach resource sheets):  What specific activities, investigations, problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic connections to appropriate mathematical practices.

1. Have the class discuss the findings from the Lesson Launch.  What were their sums?   What can we conclude about the sum of a triangle?

2. Have students again draw a triangle or give each student a triangle.  Have them tear off the three angles.  Use the three angles and line them up on a line back to back. (Angle to angle) Have the students discuss what happens.  They should discover that the three angles line up to form a straight line. Therefore you can conclude that the angles of a triangle add up to 180 degrees since a line is 180 degrees. (Look for evidence of MP7 and MP8.)

3. Assign students to small groups.  Have groups work through a more formalized activity for proof using dynamic geometry software such as Geometer’s Sketchpad or GeoGebra. Have them begin with a triangle and parallel lines. Students can manipulate the vertices of the triangles and the parallel line to explore all possibilities. See example.G.CO.C12 Triangle Sum.gsp. Sketchpad document. A tutorial on how to show the triangle sum of 180 degrees on sketchpad is located at https://docs.google.com/file/d/0B_2_-NMZ5KYqRzV4MHhOOUdYeHM/edit (Look for evidence of MP7 and MP8.)

4. Have students use the proof blocks to prove the angles of a triangle sum to 180 degrees.  A sample solution using proof blocks is pictured with the proof blocks for triangle sum document. (Look for evidence of MP7 and MP8.)

5. Listed here is an additional example for students to view of how the angles of a triangle equal 180 degrees. Have students visit or watch the video proving that a triangle’s sum is 180 degrees, available at https://www.khanacademy.org/math/geometry/angles/v/proof---sum-of-measures-of-angles-in-a-triangle-are-180

Evidence of Success:  What exactly do I expect students to be able to do by the end of the lesson, and how will I measure student success? That is, deliberate consideration of what performances will convince you (and any outside observer) that your students have developed a deepened and conceptual understanding.

Students will to be able to explain/ prove with words, pictures or numbers why a triangle has a sum of 180 degrees.

Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.

Reflexive congruence

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TransversalParallelSum

Remember to make connections between the conceptual and formal proofs. Ask students to formulate an understanding for 180degree sum of a triangle.

Resources:  What materials or resources are essential for students to successfully complete the lesson tasks or activities?

RulersProtractorsPaperScissorsPatty paperGeometry softwareProof Blocks

Homework: Exactly what follow-up homework tasks, problems, and/or exercises will be assigned upon the completion of the lesson?

To be determined by teacher.

Lesson Reflections:  How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?

Do students have a conceptual understanding of the sum of the measures of a triangle are 180 degrees? Can they explain to you why there is 180 degrees a triangle?

Can students formally prove that the angles in a triangle sum to 180 degrees?

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Teach-Now Unit Plan Template

 Sequencing and Scaffolding: How will you sequence and scaffold the lessons that you will teach for this unit? In what order will you teach the lessons you developed in Units 2-4? What additional lessons will you need to develop to complete the unit?

Please see sequencing chart below with a focus on Geometry:

A Comprehensive List of “High Yield” Strategies that Relate to Effective Teaching that will be used in this LESSONI. CONTENT

1.1.1.1 A. Lessons Involving New Content

1.1.1.1.1 STRATEGY

1. Identifying critical information (e.g., the teacher provides cues as to which in-formation is important) A&S2. Organizing students to interact with new knowledge (e.g., the teacher organizes students into dyads or triads to discuss

small chunks of content) CITW3. Previewing new content (e.g., the teacher uses strategies such as: K-W-L,advance organizers, preview questions) CITW

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4. Chunking content into “digestible bites” (e.g., the teacher presents content in small portions that are tailored to students’ level of understanding) A&S

5. Group processing of new information (e.g., after each chunk of information, the teacher asks students to summarize and clarify what they have experienced) CITW

6. Elaborating on new information (e.g., the teacher asks questions that require students to make and defend inferences) CITW

7. Recording and representing knowledge (e.g., the teacher ask students to summarize, take notes, or use nonlinguistic representations) CITW

8. Reflecting on learning (e.g., the teacher asks students to reflect on what they understand or what they are still confused about) CAGTW

1.1.1.2 B. Lessons Involving Practicing and Deepening Content That Has Been Previously Addressed

1.1.1.2.1 STRATEGY

9. Reviewing content (e.g., the teacher briefly reviews related content addressed previously) CITW10. Organizing students to practice and deepen knowledge (e.g., the teacher organizes students into groups designed to

review information or practice skills) CITW11. Practicing skills, strategies, and processes (the teacher uses massed and distributed practice) CITW12. Examining similarities and differences (e.g., the teacher engages students in comparing , classifying, creating analogies

and metaphors) CITW13. Examining errors in reasoning (e.g., the teacher asks students to examine informal fallacies, propaganda, bias) A&S14. Using homework (e.g., the teacher uses homework for independent practice or to elaborate on information) CITW15. Revising knowledge (e.g., the teacher asks students to revise entries in notebooks to clarify and add to previous

information) CITW

1.1.1.3 C. Lessons Involving Cognitively Complex Tasks (Generating and Testing Hypotheses)

1.1.1.3.1 STRATEGY

16. Organizing students for cognitively complex tasks (e.g., the teacher organizes students into small groups to facilitate cognitively complex tasks) CITW

17. Engaging students in cognitively complex tasks (e.g., the teacher engages students in decision-making tasks, problem-solving tasks, experimental inquiry tasks, investigation tasks) CITW

18. Providing resources and guidance (e.g., the teacher makes resources available that are specific to cognitively complex tasks and helps students execute such tasks) A&S

II. ROUTINE ACTIVITIES

1.1.1.4 D. Communicating Learning Goals, Tracking Student Progress, and Celebrating Success

1.1.1.4.1 STRATEGY

19. Providing clear learning goals and scales to measure those goals (e.g., the teacher provides or reminds students about a specific learning goal) CAGTW

20. Tracking student progress (e.g., using formative assessment, the teacher helps students chart their individual and group progress on a learning goal) CAGTW

21. Celebrating student success (e.g., the teacher helps student acknowledge and celebrate current status on a learning goal as well as knowledge gain)CAGTW, CITW

1.1.1.5 E. Establishing and Maintaining Classroom Rules and Procedures

1.1.1.5.1 STRATEGY

22. Establishing classroom routines (e.g., the teacher reminds students of a rule or procedure or establishes a new rule or procedure) CMTW

23. Organizing the physical layout of the classroom for learning (e.g., the teacher organizes materials, traffic patterns, and displays to enhance learning)CMTW

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III. BEHAVIORS THAT ARE ENACTED ON THE SPOT AS SITUATIONS OCCUR

1.1.1.6 F. Engaging Students

1.1.1.6.1 STRATEGY

24. Noticing and reacting when students are not engaged (e.g., the teacher scans the classroom to monitor students’ level of engagement) CMTW

25. Using academic games (e.g., when students are not engaged, the teacher uses adaptations of popular games to reengage them and focus their attention on academic content) A&S

26. Managing response rates during questioning (e.g., the teacher uses strategies to ensure that multiple students respond to questions such as: response cards, response chaining, voting technologies) A&S

27. Using physical movement (e.g., the teacher uses strategies that require students to move physically such as: vote with your feet, physical reenactments of content) CMTW

28. Maintaining a lively pace (e.g., the teacher slows and quickens the pace of instruction in such a way as to enhance engagement) CMTW

29. Demonstrating intensity and enthusiasm (e.g., the teacher uses verbal and nonverbal signals that he or she is enthusiastic about the content) CMTW

30. Using friendly controversy (e.g., the teacher uses techniques that require students to take and defend a position about content) A&S

31. Providing opportunities for students to talk about themselves (e.g., the teacher uses techniques that allow students to relate content to their personal lives and interests) CMTW

32. Presenting unusual information (e.g., the teacher provides or encourages the identification of intriguing information about the content) A&S

1.1.1.7 G. Recognizing Adherence and Lack of Adherence to Classroom Rules and Procedures

1.1.1.7.1 STRATEGY

33. Demonstrating “withitness” (e.g., the teacher is aware of variations in student behavior that might indicate potential disruptions and attends to them immediately) CMTW

34. Applying consequences (e.g., the teacher applies consequences to lack of adherence to rules and procedures consistently and fairly) CMTW

35. Acknowledging adherence to rules and procedures (e.g., the teacher acknowledges adherence to rules and procedures consistently and fairly)CMTW

1.1.1.8 H. Maintaining Effective Relationships with Students

1.1.1.8.1 STRATEGY

36. Understanding students’ interests and backgrounds (e.g., the teacher seeks out knowledge about students and uses that knowledge to engage in informal, friendly discussions with students) CMTW

37. Using behaviors that indicate affection for students (e.g., the teacher uses humor and friendly banter appropriately with students) CMTW

38. Displaying objectivity and control (e.g., the teacher behaves in ways that indicate he or she does not take infractions personally) CMTW

1.1.1.9 I. Communicating High Expectations

1.1.1.9.1 STRATEGY

39. Demonstrating value and respect for low-expectancy students (e.g., the teacher demonstrates the same positive affective tone with low-expectancy students as with high-expectancy students) A&S

40. Asking questions of low-expectancy students (e.g., the teacher asks questions of low-expectancy students with the same frequency and level of difficulty as with high-expectancy students) A&S

41. Probing incorrect answers with low-expectancy students (e.g., the teacher inquires into incorrect answers with low-expectancy students with the same depth and rigor as with high-expectancy students) A&S

© Robert J. Marzano (2009)

CITW: addressed in Classroom Instruction That Works (Marzano, Pickering, and Pollock 2001). CMTW: addressed in Classroom Management That Works (Marzano, Pickering, and Marzano 2003). CAGTW: addressed in Classroom Assessment

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Teach-Now Unit Plan Template

and Grading That Works (Marzano 2006). A&S: addressed in The Art and Science of Teaching (Marzano 2007) but not addressed in CITW, CMTW, or CAGTW.

Vocabulary

Unit 1Rigid motion: any transformation that preserves shape such as rotations, translations reflections, shifts and

slides. If a figure is dilated or compressed the motion is no longer a rigid motion. Examples: on the graph below the shape of the arrow is unchanged but it’s location has changed through a series of movements.

Horizontal stretch: when graphing functions a horizontal stretch results when the independent variable is multiplied by a number between zero and one. The example below shows the graph of one period of

. The graphs show that the vertical spans of the graphs are the same but horizontal spans are not. The dashed graph represents a horizontal stretch of the solid graph

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Examples of geometric plane figure that has gone through a horizontal stretch

corresponding parts: given two of the same shape, the corresponding parts of the two shapes are the angles and sides of the shapes that are in the same spot in the shape.

Unit 2similarity transformations: a

composition of an isometry (preserves distance) and a dilation;

Density: The mass or weight per unit volume

Two objects may appear identical in size and shape, yet one weighs considerably more than the other. The simple explanation is that the heavier object is denser. An object's density tells us how much it weighs for a certain size. For example, an item that weighs 3 pounds per square foot will be lighter than an object that weighs 8 pounds per square foot. Density is useful in calculating the weight of substances that are difficult to weight. You can determine its weight simply by multiplying the density by the size, or volume, of the item.

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Corresponding angles

Corresponding sides

Teach-Now Unit Plan Template

Some common units used to measure density are and One of the most common uses of density is in how different materials interact when mixed together. Wood floats in water because it has a lower density, while an anchor sinks because the metal has a higher density. Helium balloons float because the density of the helium is lower than the density of the air.

Typographic grid systems: A typographic grid is a two-dimensional structure made up of a series of intersecting vertical and horizontal axes used to structure content. The grid serves as a framework on which a designer can organize text and images in a rational, easy to absorb manner.

Unit 3Finding the area of a circle using a dissection argument along with an informal limit argument

To derive the formula for calculating the area of a circle with radius r, we cut a circle into 4 equal wedges as shown in the picture. Arrange the four wedges in a row, alternating the tips up and down to form a shape that resembles a parallelogram. The reason for changing a circle into a "parallelogram" is because we don't know how to calculate the area of a circle yet. We transform a circle into a shape whose area we know how to compute. As shown, the length of the bumped base (top or bottom) is equal to half of the circumference of the original circle and the length of the other side is equal to the radius r. During this process, no area has been lost or gained so that the area of this newly formed "parallelogram" is the same as that of the original circle. However, this "parallelogram" has bumps on both its top and bottom, so we still don't know how to calculate its area.

To solve this problem, we cut the original circle into a greater number of equal wedges. As we increase the number, the bumps become smoother and the parallelogram looks more and more like a rectangle. As the number approaches infinity, the bumped "parallelogram" becomes a perfect rectangle, with its width equal to and its height equal to

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. As illustrated earlier, The width of this newly formed rectangle equals half of the circumference of the original circle and the height is equal to the radius . As a result,

Area of Circle = Area of Rectangle =

Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly, which is the area of the circle.

This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral:

Cavalier’s principle: A method of finding the volume of any solid for which cross-sections by parallel planes have equal areas. This includes, but is not limited to, cylinders and prisms. The formula shown below represents this principle.

the area of a cross-section

= the height of the solid

Informal limit arguments: see dissection argument

Unit 4

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Directed line segment: A line segment extending from some point P1 to another point P2 in space viewed as having direction associated with it, the positive direction being from P1 to P2. A directed line segment P1P2 corresponds to a vector which extends from point P1 to point P2.

See http://www.ilovemaths.com/3section.asp for an how this standard might be applied

Focus of a Parabola: The focus of a parabola is a fixed point on the interior of a parabola used in the formal definition of the curve.

A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

Directrix of a Parabola: A line perpendicular to the axis of symmetry used in the definition of a parabola. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

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Example:

Unit 5

Complete the square:

Example of how to use the “Completing the Square” technique to find the center and radius of a circle given its equation

Problem

Find the center and radius of the circle given by .

Solution

The equation of a circle can be expressed as

So the task is to take given equation and convert it to this format using the technique referred to as “Completing the Square”.

Step 1: Rearrange the terms using the commutative property along with the addition property of equality

Step 2: Replace the question marks with numbers whose values make each expression inside parentheses a perfect square trinomial. (This process is where this technique gets the name “Completing the

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Square”) To keep the expression equivalent to the previous expression the new amount added to the left side must also be added to the right side of the equation.

Step 3: Write the trinomials that are in parentheses as “Perfect Square” and simplify the right side of the equation.

Step 4: The center of the circle is and the radius of the circle is the square root of 4 or 2.

Unit 6

Conditional probability:

When dealing with two events (usually called A and B), sometimes the events are so related to each other, that the probability of one depends on whether the other event has occurred. When we talk about probabilities based on the fact that something else has already happened we call this conditional probability.

What changes when dealing with conditional probability is that we know for certain that something else has already happened. This means that in our definition of probability that says

where the “total number of ways” is based on the fact that we know something else has already occurred.

There are two ways to approach conditional probability and depends on the type of problem that you are given. In a situation where you are given percentages and probabilities (usually but not always in a table format) we make use of the conditional probability rule (shown below). In situations where you are trying to compute probabilities on your own (instead of them being given to you) most of the time it is easier to not to use the formula.

Conditional Probability Rule:

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Consider events A and B.

The line between A and B is read “given”. So translated, this reads, "the probability of A given that B has happened." The event on the right side of the line is the event that has already happened.

What The Rule Means:

This rule is applied when you have two events and you already know the outcome of one of the events. In doing the computations, you will need to be able to find the probability of A and B, that is, P(A B). Problems of this type make use of the multiplication rule

Geometric Probability Model:

Geometric probability = , so a Geometric

Probability Model might refer to the probability of hitting the center circle on a dart board.

Law of Large Numbers: the more times you do something, the closer you will get to what is supposed to happen. In probability, this means that the more simulations we conduct the closer our experimental probability will get to the theoretical probability. In statistics, it means that the larger sample size you use the closer your sample will represent the entire population. For example if you flip a coin repeatedly you would expect the coin to land with heads up 50% of the time. However what you expect and what actually happens are not likely to be the same if you only carry out the process a few times.

Statistical inference: use of information from a sample to draw conclusions (inferences) about the population from which the sample was taken.

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