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DRAFT-Geometry Unit 2: Similarity, Proof and Trigonometry

Geometry Unit 2 Snap Shot

Unit Title Cluster Statements Standards in this Unit

Unit 2Similarity, Proof and

Trigonometry

Understand similarity in terms of similarity transformations.

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving right triangles.

Apply geometric concepts in modeling situations.

G.SRT.1.a G.SRT.1b G.SRT.2 G.SRT.3 G.SRT.4 G.SRT.5 G.SRT.6 G.SRT.7 G.SRT.8★

G.MG.1★

G.MG.2★

G.MG.3★

PARCC has designated standards as Major, Supporting or Additional Standards. PARCC has defined Major Standards to be those which should receive greater emphasis because of the time they require to master, the depth of the ideas and/or importance in future mathematics. Supporting standards are those which support the development of the major standards. Standards which are designated as additional are important but should receive less emphasis.

OverviewThe overview is intended to provide a summary of major themes in this unit.

Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem.

DRAFT Maryland Common Core State Curriculum Unit Plan for Geometry December, 2012 Page 1 of 38★ Modeling standard

Major


Enduring UnderstandingsEnduring understandings go beyond discrete facts or skills. They focus on larger concepts, principles, or processes. They are transferable and apply to new situations within or beyond the subject. Bolded statements represent Enduring Understandings that span many units and courses. The statements shown in italics represent how the Enduring Understandings might apply to the content in Unit 2 of Geometry.

Objects in space can be transformed in an infinite number of ways and those transformations can be described and analyzed mathematically.

o A dilation is a non-rigid transformation that produces a similar shape.

Representations of geometric ideas and relationships allow multiple approaches to geometric problems and connect geometric interpretations to other contexts.

o Similarity among shapes provides a means of solving geometric problems as well as problems in other contextual settings.

Judging, constructing, and communicating mathematically appropriate arguments are central to the study of mathematics.

o Assumptions about geometric objects must be proven to be true before the assumptions are accepted as facts.o The truth of a conjecture requires communication of a series of logical steps based on previously proven statements. o A valid proof contains a sequence of steps based on principles of logic.



Essential Question(s)A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations. Bolded statements represent Essential Questions that span many units and courses. The statements shown in italics represent Essential Questions that are applicable specifically to the content in Unit 2 of Geometry.

How is visualization essential to the study of geometry?o How does the concept of dilation connect to the concept of similarity?

How does geometry explain or describe the structure of our world?o How does the concept of similarity help to solve problems?

How can reasoning be used to establish or refute conjectures?o What are the characteristics of a valid argument?o What is the role of deductive or inductive reasoning in validating a conjecture?o What facts need to be verified in order to establish that two figures are similar?



Possible Student Outcomes The following list provides outcomes that describe the knowledge and skills that students should understand and be able to do when the unit is completed. The outcomes are often components of more broadly-worded standards and sometimes address knowledge and skills necessarily related to the standards. The lists of outcomes are not exhaustive, and the outcomes should not supplant the standards themselves. Rather, they are designed to help teachers “drill down” from the standards and augment as necessary, providing added focus and clarity for lesson planning purposes. This list is not intended to imply any particular scope or sequence.

G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. (major)

The student will: use inductive reasoning to discover the properties of dilations given by a center and a scale factor. draw a dilation of a given figure using a given scale factor and center.

G.SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. (major)

The student will: understand that corresponding sides that do not pass through the center of dilation of dilated figures are parallel. understand that if a line segment contains the center of the dilation, then the dilated line segment will be on the same line as

the original line segment.

G.SRT.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. (major)

The student will: identify the scale factor of a given dilation. apply the scale factor to determine if the dilation is an enlargement, reduction or isometry.



G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (major)

The student will: define similar triangles as equality of all corresponding pairs of angles and proportionality of all corresponding pairs of sides. describe the connection between dilation and similarity. use the definition of similarity in terms of similarity transformations to determine if two figures are similar.

G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (major)

The student will: establish the AA similarity criterion using similarity transformations.

G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (major)

The student will: prove theorems about triangles. prove the SSS similarity theorem. prove the SAS similarity theorem. prove that a line parallel to one side of a triangle divides the other two sides proportionally and its converse. prove the Pythagorean Theorem using triangle similarity and right triangle trigonometry.



G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (major)(fluency)

The student will: use congruence and similarity criteria for triangles to solve problems. use congruence and similarity criteria to prove relationships in geometric figures.

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (major)

The student will: apply the definitions of similarity to the trigonometric ratios for acute angles of right triangles based on the ratios of the sides. apply the definition of trigonometric ratios to special right triangles.

G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. (major)

The student will: use the sine of one acute angle in a right triangle to determine the cosine of that angle’s complement. use the cosine of one acute angle in a right triangle to determine the sine of that angle’s complement.

G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★

The student will: solve for missing sides and angles of right triangles using the Pythagorean Theorem and trigonometric ratios (sine, cosine

and tangent) in applied problems.

G.MG.1 Use geometric shapes, their measures, and their properties to describe objects. ★ (major)

The student will: use right triangles and/or similar figures to model real world phenomena. use the properties of right triangles and/or similar figures to determine unknown measures of real world objects.



G.MG.2 Apply concepts of density based on area and volume in modeling situations. ★ (e.g., persons per square mile; BTU’s per cubic foot) (major)

The student will: use similar figures and their area/volume to model density concepts. use proportional reasoning to solve density problems.

G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★ (major)

The student will: use right triangles, trigonometry and/or similar figures and their properties to solve design problems.



Possible Organization/Groupings of StandardsThe following charts provide one possible way of how the standards in this unit might be organized. The following organizational charts are intended to demonstrate how some standards will be used to support the development of other standards. This organization is not intended to suggest any particular scope or sequence.

Geometry Unit 2:Similarity, Proof and Trigonometry

Topic #1Similarity Transformations

Major Standard to Address Topic # 1


The standards listed to the right should be used to help develop G.SRT.1.




G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (major)





Topic #2Proving Theorems involving Similarity

Cluster Note: Encourage multiple ways of writing proofs, such as narrative paragraphs, flow diagrams, two column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring variety of formats for expressing that reasoning.

The standard listed to the right should be used to develop Topic #2

G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (major)

Topic #3Trigonometry

Major Standard to Address Topic #3

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (major)

The standard listed to the right should be used to help develop G.SRT.6.




Topic #4Solve Problems Involving Triangles

Cluster Note: Focus on situations well modeled by trigonometric ratios for acute angles.Major Standard to address Topic #4

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (major)

Major Standard to address Topic #4

G.MG.1 Use geometric shapes, their measures, and their properties to describe objects. ★(major)

The standards listed to the right should be used to help develop G.MG.1.


G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★(major)

Major Standard to address Topic# 4

G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile; BTU’s per cubic foot) ★(major)

The standards listed to the right should be used to help develop G.MG.2.


G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ (major)

Major Standard to address Topic #4

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

★(major)The standards listed to the right should be used to help develop G.MG.3.


G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ (major)



Connections to the Standards for Mathematical PracticeThis section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

In this unit, educators should consider implementing learning experiences which provide opportunities for students to:

1. Make sense of problems and persevere in solving them. Determine if sufficient information exists to conclude that two geometric figures are similar. Analyze given information and select a format for proving a given statement pertaining to proportionality and similar polygons. Solve multi-layer problems that make use of basic theorems. Recognize that physical entities can be modeled using geometric figures. For example, a pipe can be modeled using a cylinder and

many states (such as Pennsylvania) can be modeled using a rectangle.

2. Reason abstractly and quantitatively. Use diagrams of specific pairs of triangles, quadrilaterals and polygons as an aid to reason about all such pairs of triangles,

quadrilaterals and polygons. For example, one pair of similar triangles can be used to reason about all pairs of similar triangles. Experiment and make conjectures pertaining to the properties of dilations. Experiment and make conjectures pertaining to the relationships between side ratios and the angle measurements in right triangles. Prove statements about proportionality and similarity using more than one method of proof.

3. Construct Viable Arguments and critique the reasoning of others. Write proofs about proportionality and similar figures in a variety of formats. Complete a proof or find a mistake in a given proof about proportionality and similar figures. Justify the relationship between sine and cosine of complementary angles. Critique the reasoning used by others in writing proofs.

4. Model with Mathematics. Apply theorems about similarity to scaled figures, perspective drawings and isometric drawings.



Recognize that physical entities can be modeled using geometric figures. For example, a pipe can be modeled using a cylinder and many states (such as Pennsylvania) can be modeled using a rectangle.

5. Use appropriate tools strategically. Identify and use an appropriate tool for constructing similar figures using the process of dilation. Use geometric software or internet resources to form conjectures about similar figures and about relationships between the sides and

angles of right triangles. Use appropriate technology for estimating lengths and angles when solving right triangle problems. Use information gathered from a geometric drawing to solve problems. For example, parallel line markings, congruent angle marks, etc.

6. Attend to precision. Determine relationships between sides and angles of similar figures. For example, students establish the AA criterion for two triangles to

be similar and the definitions of sine and cosine in terms of sides of a right triangle. Use appropriate vocabulary and symbolism. For example, sin = ¾ is not an appropriate statement. Realize that the Pythagorean Theorem is a relationship between the sides of a right triangle – not just an algebraic manipulation. Approximate the lengths of sides and the measures of angles to an appropriate level of precision.

7. Look for and make use of structure. Recognize that ratios are the foundation of similarity and trigonometry. Add auxiliary lines to figures in order to develop additional geometric theorems. For example, the Pythagorean Theorem can be

established through drawing an altitude to the hypotenuse and using the geometric mean. Recognize that as an acute angle measure increases from 0 degrees to 90 degrees the sine ratio of the angle increases from 0 to 1 but

that the relationship between the angle and the sine of the angle is not linear. Recognize that as an acute angle measure increases from 0 degrees to 90 degrees the cosine ratio of the angle decreases from 1 to 0

and that the relationship between the angle and the cosine of the angle is not linear.

8. Look for and express regularity in reasoning. Recognize that to create the original figure after an enlargement one would need a reciprocal scale factor for the reduction. Generate Pythagorean Triples. For example, knowing that 6,8,10 is a Pythagorean Triple because it is a multiple of 3,4,5.



Content Standards with Essential Skills and Knowledge Statements, and Clarifications/Teacher Notes The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Geometry framework document. Clarifications and teacher notes were added to provide additional support as needed. Educators should be cautioned against perceiving this as a checklist. Formatting Notes

Red Bold- items unique to Maryland Common Core State Curriculum Frameworks Blue bold – words/phrases that are linked to clarifications



Black bold underline - words within repeated standards that indicate the portion of the statement that is emphasized at this point in the curriculum or words that draw attention to an area of focus

Black bold- Cluster Notes-notes that pertain to all of the standards within the cluster Green bold – standard codes from other courses that are referenced and are hot linked to a full description

Standard Essential Skills and Knowledge

Clarification/Teacher Notes


Properties to include: Collinearity Congruent angle measures Proportional linear dimensions


Ability to connect experiences with dilations and orientation to experiences with lines

The lines “not passing through the center” refer to the edges of the image (the dilated

figure), eg. Segments AB and A’B’ in the sketch below. Notice that

The problem below illustrates both aspects of this standard. The second portion of the standard is illustrated by the fact that the origin is the center of the dilation

and since passes through the center of the dilation both





are contained in .

PROBLEM: Draw the dilation image of pentagon ABCDE with the center of dilation at the origin and a scale factor of 1/3.


Ability to develop a hypothesis based on observations

G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are

Ability to make connections between the definition of similarity and the attributes of two





similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (major)

given figures

Ability to set up and use appropriate ratios and proportions


Ability to recognize why particular combinations of corresponding parts establish similarity and why others do not

G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Ability to construct a proof using one of a variety of methods

Proof of the Pythagorean Theorem using similar triangles

Proof using similar triangles (see next page)


http://en.wikipedia.org/wiki/File:Pythagoras_similar_triangles.PNG

http://en.wikipedia.org/wiki/File:Pythagoras_similar_triangles.PNG




(major)This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:

The first result equates the cosine of each angle θ and the second result equates the sines.These ratios can be written as:

Summing these two equalities, we obtain

which, tidying up, is the Pythagorean theorem:


http://en.wikipedia.org/wiki/Sine

http://en.wikipedia.org/wiki/Cosine

http://en.wikipedia.org/wiki/Parallel_postulate

http://en.wikipedia.org/wiki/Triangle_postulate

http://en.wikipedia.org/wiki/Similarity_(geometry)

http://en.wikipedia.org/wiki/Altitude_(triangle)

http://en.wikipedia.org/wiki/Ratio

http://en.wikipedia.org/wiki/Similarity_(geometry)

http://en.wikipedia.org/wiki/Proportionality_(mathematics)




The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements, and that the theory of proportions needed further development at that time.

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.(major)(fluency)

Ability to use information given in verbal or pictorial form about geometric figures to set up a proportion that accurately models the situation

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.(major)

Ability to generalize that side ratios from similar triangles are equal and that these relationships lead to the definition of the six trigonometric ratios

When creating lessons that address this standard you may want to also discuss the relationships that exist in Special Right Triangles.






In this standard, the idea is to use either Sine or Cosine to find the measure of one acute angle and then use the fact that the acute angles in a right triangle are complementary to find the measure of the other acute angle.

G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★(major)

Cluster Note: Focus on situations well modeled by trigonometric ratios for acute angles.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects. ★

G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile; BTU’s per cubic foot). ★

An example of how this standard might apply would be to estimate the total number of people who are standing in a one square mile region you could count the number of people who are standing in a 3 yard by 3 yard square region and then use proportionality of similar figures to estimate the total number of people in the one square mile region.





G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★



Vocabulary/Terminology/ConceptsThe following definitions/examples are provided to help the reader decode the language used in the standard or the Essential Skills and Knowledge statements. This list is not intended to serve as a complete list of the mathematical vocabulary that students would need in order to gain full understanding of the concepts in the unit.

Term Standard Definitionsimilarity transformations

G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

A similarity transformation is a rigid motion together with a rescaling. In other words, a similarity transformation may alter both position and size, but preserves shape.



Term Standard Definitiondensity G.MG.2 Apply

concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★

The number of inhabitants, dwellings, or the like, per unit area: The commissioner noted that the population density of certain city blocks had fallen dramatically. or

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.

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.



Term Standard Definition

typographic grid systems

G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★

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.

Progressions from the Common Core State Standards in MathematicsFor an in-depth discussion of overarching, “big picture” perspective on student learning of the Common Core State Standards please access the documents found at the site below.

http://ime.math.arizona.edu/progressions/

To see what the Geometry standards in the Common Core State Curriculum Standards for High School mathematics progress from refer to the document below.http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf


http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf

http://ime.math.arizona.edu/progressions/


Vertical AlignmentVertical alignment provides two pieces of information:

A description of prior learning that should support the learning of the concepts in this unit A description of how the concepts studied in this unit will support the learning of other mathematical concepts.

Previous Math Courses Geometry Unit 2 Future Mathematics

Concepts developed in previous mathematics course/units which serve as a foundation for the development of the “Key Concept”

Key Concept(s) Concepts that a student will study either later in Geometry or in future mathematics courses for which this “Key Concept” will be a foundation.

In 6th grade, students: understand the concept of a ratio and

use ratio language to describe a ratio relationship between two quantities.

In 7th grade, students: solve problems involving scale drawings

of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

In 8th grade, students: understand that a two-dimensional

figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using

Similarity Transformations (Dilations), Scale Factor, Similarity, and Similarity Proofs

In Algebra II, students will: study the effect on the graph of f(x) by

replacing f(x) with f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Note that f(kx) becomes a special example of scale factor, which dilates the function.






coordinates.

In Algebra I, students: study the effect of replacing f(x) with

f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Where f(kx) becomes a special example of scale factor, which dilates the function.

In 8th grade, students: explain a proof of the Pythagorean

Theorem and its converse. apply the Pythagorean Theorem to

determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Solving problems using Pythagorean Theorem and Trigonometric Ratios

In Algebra II, students will: study how the unit circle in the coordinate

plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

prove the Pythagorean identity and use it to find sin(θ),

cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.






In 6th grade, students: learn the area of right triangles, other

triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world. and mathematical problems.

learn the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fractionedge lengths, and showed that the volume was the same as would be found by multiplying the edge lengths of the prism.

apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures and apply these techniques in the context of solving real-worldand mathematical problems.

Modeling In Calculus, students will: use integrals to find areas under the curves defined

by the graphs of functions.






In 7th grade, students: learn the formulas for the area and

circumference of a circle and used them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

In 8th grade, students: learn the formulas for the volumes of

cones, cylinders, and spheres and use them to solve real-world and mathematical problems.



Common MisconceptionsThis list includes general misunderstandings and issues that frequently hinder student mastery of concepts regarding the content of this unit.

Topic/Standard/Concept Misconception Strategies to Address Misconception Similarity Transformations/G.SRT.1b

Students may assume triangles are similar and create proportions without just cause.

Require students to write the criteria they used to determine that the triangles were similar before writing any proportions.

Create a diagram for a real world problem (i.e.a bridge spanning a river) that lets the students assume similarity. Let the students use proportions to determine the bridge length. After the students have completed the calculations mention that the triangles were not similar and that the bridge is really not long enough to span the river.

Similarity Transformations/G.SRT.2

Students frequently have difficulty identifying corresponding parts of similar triangles. As a result they set up proportions incorrectly.

Have students color code corresponding sides before setting up a proportion. For example, ask the students to shade the shortest sides of two triangles in red.



Topic/Standard/Concept Misconception Strategies to Address Misconception Students may not recognize that congruent figures are also similar.

First have students list the equality of all of the corresponding parts of two congruent triangles. Ask students to look at the list to see if enough criteria are present to claim that the triangles are also similar. Demonstrate that the ratio of corresponding sides of similar triangles that are also congruent triangles will be a 1 to 1 ratio and therefore satisfy the notion that corresponding sides of similar triangles are proportional.

Similarity Theorems/G.SRT.4

When given the length of a leg and the hypotenuse of a triangle students often use the hypotenuse length as a leg length in the Pythagorean Theorem.

Make students label the hypotenuse of any right triangle with the word “hypotenuse” before setting up an equation based on the hypotenuse. For this to work it is also important to have students learn the Pythagorean Theorem as

and not just as

Always require students to check the reasonableness of the answer by labeling the given triangle with the computed lengths to see if the hypotenuse is the longest side.

Students frequently confuse the terms median and mid-segment.

Have students create a Frayer Model for each term. This type of model requires the student to provide examples and non-examples.

Trigonometry/G.SRT.6

Students often confuse what it means to be the side adjacent to versus opposite of a given angle in a right triangle and therefore use the incorrect trigonometric ratio when setting up an equation to determine a missing value.

First have students label the hypotenuse with the word “hypotenuse” so as to not use it as one of the “sides”.Then give students ample practice in identifying “opposite” and “adjacent” sides referencing various angles. Have them reference both angles of the same triangle. Then write the ratios.



Topic/Standard/Concept Misconception Strategies to Address Misconception Students frequently use improper notation when using trigonometric ratios. For example, students frequently write expressions such as

when they should be

writing .

Trigonometric ratios are usually not treated as functions until Algebra II but it is a good practice to still treat them as such. To

show students how the expression is incomplete, compare

it to using without an input value for the function.

Students may struggle when asked to use trigonometry to determine the measure of one of the acute angles of a right triangle because they use the incorrect trigonometric ratio when looking at a trig table or using a calculator.

Ask students to make note of which angle should have the smallest measure and the largest measure based on what they know about the side lengths of the triangle before completing any computations. They should then examine calculated measurements for reasonableness based on their observations.



Topic/Standard/Concept Misconception Strategies to Address Misconception Trigonometry/G.SRT.8

Students often struggle when identifying angles of elevation and angles of depression.

Define depression and elevation. Have students recognize various contexts in which these words are used.

When asked to draw an angle of elevation or depression, tell students to start by drawing the horizontal side of the angle. Then tell them that if they are drawing an angle of depression that the second side of the angle will be drawn below the horizontal side and if they are drawing an angle of elevation that the second side of the angle will be above the horizontal side.

Describe depression and elevation using the horizon as a reference and then model angles of elevation and depression using your arms. Ask students then to use their arms to model angles as well.



Model Lesson Plan(s)The lesson plan(s) have been written with specific standards in mind. Each model lesson plan is only a MODEL – one way the lesson could be developed. We have NOT included any references to the timing associated with delivering this model. Each teacher will need to make decisions related ot the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed

Title Description/Suggested use

G.SRT.3G.MG.1

AA Triangle Similarity and Indirect Measurement

Students will use the AA Similarity Theorem to find the height of various real world objects.



Lesson SeedsThe lesson seeds have been written particularly for the unit, with specific standards in mind. The suggested activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed

Title Description

G.SRT.2G.MG.3

How Far Can You Go in a New York Minute?

ApplicationThe activity in this lesson seed could be used as a partner activity. This should be completed after lessons on similarity.In this lesson seed, students use proportions and similar figures to adjust the size of the New York City Subway Map so that it is drawn to scale. Students are asked to evaluate whether these changes are necessary improvements.

G.SRT.1G.SRT.1aG.SRT.1b

Exploring Dilations InvestigationThe activities in this lesson seed could be used after introductory instruction on dilations. By completing the activities provided in this lesson seed, students should gain a better understanding of some of the relationships between the properties of the image and pre-image of adilation.

G.SRT.3 AA Similarity InvestigationThe activity in this lesson seed could be used during the direct instruction portion of a lesson to develop the criterion for AA Similarity.This activity introduces the concept that AA is a sufficient condition to guarantee similar triangles.(Note: This method does not address Standard G.SRT.3 but it could be used as another means to establish the AA Similarity criterion.)



Standards Addressed

Title Description

G.SRT.4 Conjecturing a Proportionality Theorem

InvestigationThe activity in this lesson seed could be used during the direct instruction portion of a lesson targeting the first theorem mentioned in G.SRT.4.This lesson seed provides an opportunity for students to discover that a line parallel to one side of a triangle divides the other two sides proportionally.

G.SRT.6 Using Similar Triangles to Discover Trigonometric Ratios

InvestigationThe activity in this lesson seed could be used during the direct instruction portion of a lesson targeting G.SRT.6. The results of this discovery activity leads to the definition of the trigonometric ratios.

G.SRT.8 Right Triangle Trig Practice Problems

PracticeThis lesson seed provides an interactive way for students to reinforce their understanding of using Trigonometric ratios in solving right triangles.

G.MG.3 Handicap Ramp ApplicationThe activity provided in this lesson seed would be used to provide a learning experience for students that would help them to build proficiency with Standard for Mathematical Practice #1 (Make sense of problems and persevere in solving them). This activity makes use of ratio and proportion and the Pythagorean Theorem.The activity in the lesson seed sets up a situation where students need to determine the length of a handicap ramp so that cost is minimized.

G.SRT.8 Indirect Measurement ApplicationThe activity provided in this lesson seed would be used to provide a learning experience for students that would help them to build proficiency with Standard for Mathematical Practice #4 (Model with Mathematics).



Sample Assessment ItemsThe items included in this component will be aligned to the standards in the unit.

Topic Standards Addressed Link Notes

Similar figures

G.SRT.5 http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm#

This task from Smarter Balance could be solved using similarity concepts.

ResourcesThis section contains links to materials that are intended to support content instruction in this unit.


Dilations G.SRT.1 http://illustrativemathematics.org/illustrations/602 This task from Illustrative Math addresses standard G.SRT.1 and asks students to "Verify experimentally" that a dilation takes a line that does not pass through the center of dilation to a line parallel to the original line, and that a dilation of a line segment (whether it passes through the center or not) is longer or shorter by the scale factor. This task gives students the opportunity to verify both of these facts for a specific example. It may be helpful to provide students with rulers so that they can measure and duplicate lengths without having to perform formal constructions.

Similarity Transformations

G.SRT.2 http://illustrativemathematics.org/illustrations/603 In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other focuses them on the work of standard G-SRT.2, using the definition of similarity in terms of similarity transformations. Teachers will need to remind students to show that their sequences of


http://illustrativemathematics.org/illustrations/603


http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm

http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm



similarity transformations have the intended effect; that is, that all parts of one triangle get carried on to the corresponding parts of the other triangle

Similarity G.SRT.5 http://illustrativemathematics.org/illustrations/651 Standard G-SRT.5 calls for students to "use congruence and similarity criteria for triangles to solve problems." This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the problem.

Modeling G.SRT.8G.C.5

http://illustrativemathematics.org/illustrations/607 This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles (G-C.5), using trigonometric ratios to solve right triangles (G-SRT.8), and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found (MP.7). This is an example of an integrative task that blends several standards from a variety of units. This would be a good review task to use in preparation for assessment.

Modeling G.SRT.8G.MG.3

http://illustrativemathematics.org/illustrations/720 This is an example of an integrative task that blends several standards from a variety of units. This would be a good review task to use in preparation for assessment.

Modeling G.SRT.8G.MG.3

http://illustrativemathematics.org/illustrations/710 This is an example of an integrative task that blends several standards from a variety of units. This would be a good review task to use in preparation for assessment.

Similarity of G.SRT.2 http://secc.sedl.org/common_core_videos/grade.php? This site contains videos to support the CCSS. This


http://secc.sedl.org/common_core_videos/grade.php?action=view&id=918







Triangles via Transformations

action=view&id=918 particular link will take you to a video that addresses G.SRT.2

All Topics All standards http://ccssmath.org/?page_id=2287 Additional Resources aligned to the CCSS

All Topics All standards http://www.ccsstoolbox.org/ Check this site periodically for new additionsClick on:

Resources for Implementation PARCC Prototyping Project High School Tasks Name of the task

All Topics All standards http://nsdl.org/commcore/math?id=HS.F This is a link to the NSDL Math Common Core Collection. The collection contains digital learning objects that are related to specific Math Common Core State Standards.

All Topics All standards https://www.desmos.com/ This is a link to an online graphing calculator that has many different types of applications

All Topics All standards http://insidemathematics.org/index.php/high-school-geometry

This link will take you to a collection of many different resources aligned to CCSS Geometry. Check back periodically for new additions to the site.

All Topics All standards http://www.parcconline.org/samples/mathematics/high-school-mathematics

This is a link to the PARCC Prototype items. Check this site periodically for new items and assessment information.

All Topics All standards http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf

This is a link to the PARCC Model Content Frameworks. Pages 39 through 59 of the PARCC Model Content Frameworks provide valuable information about the standards and assessments.


http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf

http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf

http://www.parcconline.org/samples/mathematics/high-school-mathematics

http://www.parcconline.org/samples/mathematics/high-school-mathematics

http://insidemathematics.org/index.php/high-school-geometry

http://insidemathematics.org/index.php/high-school-geometry

https://www.desmos.com/

http://nsdl.org/commcore/math

http://nsdl.org/commcore/math

http://nsdl.org/commcore/math?id=HS.F

http://www.ccsstoolbox.org/

http://ccssmath.org/?page_id=2287

http://secc.sedl.org/common_core_videos/grade.php?action=view&id=918


PARCC Components

Fluency RecommendationsAccording to the Partnership for Assessment of Readiness for College and Careers (PARCC), the curricula should provide sufficient supports and opportunities for practice to help students gain fluency. PARCC cites the areas listed below as those areas where a student should be fluent.

G-SRT.5 Fluency with the triangle congruence and similarity criteria will help students throughout their investigations of triangles quadrilaterals, circles, parallelism and trigonometric ratios. These criteria are necessary tools in many geometric modeling tasks,


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