geometry houston isd planning guide 4 six-weeks€¦ · students concretely determined the area and...

GEOMETRY HOUSTON ISD PLANNING GUIDE 4TH SIX-WEEKS

- English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned to Upcoming State Readiness Standard

- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)

© Houston ISD Curriculum 2012 – 2013 of 5

Planning Guide User Information Unit 11: Measuring Lengths and Area

Time Allocations

Unit 7 lessons (90-minutes each)

or 14 lessons (45-minutes each)

Unit Overview

Measuring Lengths and Area – Students concretely and algebraically determine the perimeter and area of various geometric figures or portions of figures. TEKS/SEs (district clarifications/elaborations in italics)

Ⓡ GEOM.8A Determine areas of regular polygons, circles, and composite figures using the area of triangles,

squares, rectangles, parallelograms, and/or trapezoids.

Ⓢ GEOM.8E Use area models to connect geometry to probability and statistics. Ⓢ GEOM.8F Use conversions between measurement systems to solve problems in real-world situations. Ⓡ GEOM.11D Describe the effect on perimeter, area, and volume when one or more dimensions of a figure are

changed and apply this idea in solving problems. English Language Proficiency Standards

ELPS C.1d Speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known).

ELPS C.2d Monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed.

ELPS C.3f Ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments.

College and Career Readiness Standards

CCRS 1.B1 Perform computations with real and complex numbers.

CCRS 3.A1 Identify and represent the features of plane and space figures.

CCRS 3.C2 Make connections between geometry, statistics, and probability.

CCRS 3.C2 Make connections between geometry, statistics, and probability.

CCRS 4.C1 Find the perimeter and area of two-dimensional figures.

Key Concepts

area circle

dimension measurement

perimeter polygon

Academic Vocabulary

compare explore

Content-Specific Vocabulary

apothem arc length

circumference compound figure

radius sector

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_MTAzODNkODQtZmZiMi00ZjYxLWJhZTMtNmZhY2U5NjFmNjMx&hl=en_US&authkey=CP2_wxA

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_YTdkNTA5NWMtMWU5YS00N2RiLWE0MTItMjA3MTIyNTc2ZWU5&hl=en_US&authkey=CJqE4-IJ





Essential Understandings / Guiding Questions

Measurements of dimensions are used to determine perimeter and area of polygons and circles and their composites.

1. How can area of polygons and composite figures be calculated? 2. How will proportional changes in the lengths of the sides of a figure affect the perimeter or the area of the

figure? 3. Why do scale factors affect perimeter differently than area?

Assessment Connections

Performance Expectation o Students will measure or use given measurements to determine the area and perimeter (circumference) of

polygons, circles, or composite figures. o Students will determine the effect on perimeter and area when one or more dimensions of the figure are

changed. Formative Assessment – Geoboards – students are assessed on calculating perimeter and area on a geoboard

simulation. Extend this assessment by having students convert from different units of measures and across systems of measurement.

Formative Assessment: Exit Ticket – students respond to the following on an index card: o Draw a composite figure and describe in writing how to determine the area of the figure; and o Describe the effects on the area of a rectangle when you double the length and width of the rectangle.

SpringBoard® Geometry – Embedded Assessment #2: “Tile We Meet Again” STAAR Sample Item – Item #10 (GEOM.8F) and Item #15 (GEOM.11D)

Texas English Language Proficiency Assessment System (TELPAS): End-of-year assessment in listening, speaking, reading, and writing for all students coded as LEP (ELL) and for students who are LEP but have parental denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide five writing samples – one narrative about a past event, two academic (from science, social studies, or mathematics), and two others.

Instructional Considerations

Information in this section is provided to assist the teacher with the background knowledge needed to plan instruction that facilitates students to internalize the Key Concepts and Essential Understandings for this unit. It is recommended that teachers thoroughly read this section before implementing the strategies and activities in the Instructional Strategies section. Prerequisites and/or Background Knowledge for Students Students concretely determined the area and perimeter of geometric figures in middle school. (Ⓡ MATH.6.3C,

Ⓢ MATH.7.4A) Students have studied proportionality with similar figures in middle school and recently in Geometry Unit 7. (Ⓡ MATH.6.4A, Ⓢ MATH.7.4B, Ⓡ MATH.8.9B, Ⓢ GEOM.11A) Background Knowledge for Teacher Critical Content Calculate the area of geometric figures; Calculate the perimeter and area of similar figures; Calculate the circumference and area of a circle; Calculate areas of sectors and lengths of arcs; Analyze perimeter (circumference) or area of a figure when proportional changes in the linear dimensions of a figure

are made; and Use conversion factors between measuring systems.

http://www.tea.state.tx.us/student.assessment/ell/telpas/





Instructional Considerations

Introduction

Give measurements in terms of real numbers and in terms of algebraic expressions. Students will need to represent perimeter and area with algebraic expressions (GEOM.8A, GEOM.11D).

Integrate Clarifying Activities and Laying the Foundation to give students more hands-on experiences – see Resources.

Note word variations of “Engage, Explore, Explain, Elaborate, and Evaluate” that imply the 5E Lesson Model.

Circumference and Area

This unit introduces arc length in relation to the circumference, and area of the sector in relation to area of a circle. Future opportunities will be available to study sectors and arc lengths in Unit 14.

References for circumferences and arc lengths are repeated in this unit in order to provide deeper understanding of the concept.

Continue in-depth work with the area of circles and regular polygons. Include composite figures in calculating area. Challenge students to create their own composite design. (GEOM.8A, GEOM.11D).

To meet College and Career Readiness Standards, teachers will need to enhance activities and examples by converting between units of measures using conversion factors and dimensional analysis. (GEOM.8F)

Extensions for Pre-AP Extensions in Laying the Foundation activities should be completed after other area activities.

Instructional Strategies / Activities

The strategies and activities in this section are designed to assist the teacher to provide learning experiences to ensure that all learners achieve mastery of the TEKS SEs for this unit. It is recommended that the strategies and activities in this section be taught in the order in which they appear. Instructional Accommodations for Diverse Learners Cues, Questions, and Advance Organizers

Ask probing questions to engage students in Finding the Rectangle, Application Problems, Find the Missing Dimension, and Circle Area. Whether working as a class or in groups, wait time, exploration time, and monitoring group interaction promote a student-centered environment. In each activity, engage students to report what they have discovered. C.1d, C.2d, C.3f

Include applications to area of rectangular figures, with quadratic expressions that need to be solved, in engagement and assessment problems. Take this opportunity in the discussion of area to review solving quadratic equations using graphing or algebraic methods.

Include application problems, involving figures within a figure for calculating the area of shaded regions and

compound figures (two or three shapes attached) in lessons and assignments. (GEOM.8E)

For additional strategies to assist diverse learners, access Recommendations for Accommodating Special Needs Students: Geometry, Cycle 4, Unit 11. Setting Objectives and Providing Feedback

Anticipation Guide (Turn the Light On) Allow students to work in pairs on the Scale Factor Anticipation Guide to discover information they remember with respect to scale factor. From the discussion, write any questions that students may want answered or clarified and revisit the guide once the unit is completed. (GEOM.8F)

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_YTdkNTA5NWMtMWU5YS00N2RiLWE0MTItMjA3MTIyNTc2ZWU5&hl=en_US&authkey=CJqE4-IJ

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_YzJiYzM3NjgtZWM2OS00YzMzLTg2NTgtNzg3NGFjYjhkY2Q4&hl=en_US&authkey=CPKiw68F

http://houstonisdpsd.org/literacy-routines.html





Instructional Strategies / Activities

Summarizing and Note Taking Review students on circumference and area. Students “popcorn” attributes of circles. Nonlinguistic Representation

Graphic Organizers Circle Area explores circumference and area of a circle by subdividing the circle into sectors and rearranging these

sectors into a parallelogram. (GEOM.8A)

Students should construct a circle and divide it into small “pie” pieces. Follow the guiding questions in the activity to determine the relationship of circumference to area.

Segue to using proportional reasoning for determining the area of a sector (McDougal-Littell, pp. 755 – 761) and the

length of an arc (McDougal-Littell, pp. 746 – 752; SpringBoard® Mathematics with Meaning: Geometry, Activity 4.4 “Segment Lengths in Circles”).

Elaborate by exploring the probability that a particular region is selected at random from a composite figure.

(GEOM.8E) Generating and Testing Hypotheses

Two-Column Notes (Pen/cil To Paper) To develop the area of a regular polygon, ask the students to brainstorm the steps to determine this area. Scaffold,

if necessary, to help students see how to divide the polygon into congruent triangles leading to a formula for area of the regular polygon: Area = (# of sides)(1/2)(side length)(apothem). A student may recognize that the number of sides times the length of a side is actually the perimeter. Help students observe that their process leads to another formula: Area = (1/2)(perimeter)(apothem).

Students record the process in their interactive notebook and clarify the definitions of radius of a polygon, center of a polygon, and apothem (SpringBoard® Mathematics with Meaning: Geometry, Activity 5.4 “Derive and Use Area Formulas”). (GEOM.8A)

MATH.8.14D Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.

Have students hypothesize the effects on area and perimeter of a figure by changing its linear dimensions by a

scale factor. Use numeric value examples such as doubling or tripling the length of a side to determine the resulting effect on the area of a geometric figure. (GEOM.11D)

MATH.8.14C Select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem or working backwards to solve a problem.

Elaborate on examples and problems by asking students to convert measurements from standard unit of measure to metric measurements or vice versa. (GEOM.8F)

Extensions for Pre-AP Generating and Testing a Hypotheses and Reinforcing Effort and Providing Recognition Use coordinate geometry to assist in finding area and perimeter of geometric figures. Students will integrate

algebraic properties with geometric concepts.

In “Using Coordinate Geometry to Find Area of Triangle,” several methods are demonstrated to find the area. Challenge the students to determine a method other than using the area formula. They may work in pairs or individually.

Heron’s Formula and Pick’s Formula are highlighted but do not demonstrate these methods until the students

present their own. Have students critique or vote for the most imaginative or mathematical. Later, expose students to the alternate methods shown in Laying the Foundations.



http://houstonisdpsd.org/literacy-routines.html





Resources

Adopted Instructional Materials McDougal-Littell, Geometry: “Areas of Triangles and Parallelograms,” pp. 720 – 726 “Determine Precision and Accuracy,” p. 727 “Areas of Trapezoids and Kites – Investigation,” p. 729 “Areas of Trapezoids, Rhombuses, and Kites,” pp. 730

– 736 “Perimeter and Area of Similar Figures,” pp. 737 – 742 “Circumference and Arc Length,” pp. 746 – 752 “Area of Circles and Sectors,” pp. 755 – 761 “Area of Regular Polygons,” pp. 762

Laying the Foundation, Connecting Geometry: “Exploration of Rectangles with Set Perimeter,” p. 234 “Using Coordinate Geometry to Find Area of Triangle,”

p. 248 “Using Area to Estimate Probability,” pp. 320 – 327 “Optimization: Area and Volume Applications,” p. 166

SpringBoard®

Mathematics with Meaning: Geometry 4.4 “Segment Lengths in Circles” 5.4 “Derive and Use Area Formulas”

Supporting Resources

Finding the Rectangle Application Problems Find the Missing Dimension Scale Factor Anticipation Guide Circle Area Recommendations for Accommodating Special Needs

Students: Geometry, Cycle 4, Unit 11

Extension Activities:

RUSMP Geometry Module – the activities What is Area?, Investigating Area Formulas and Applying Area Formulas assist students in developing a deeper understanding of the underlying concept of area.

Name: Period:

Anticipation Guide: Scale Factors Directions: On the continuum in front of each of the numbers, place an “x” that indicates where you stand in regard to the statement that follows. Be prepared to defend and support your opinions with specific examples. After the lesson, compare your opinions on those statements with what you learned in the lesson. Never True Sometimes True Always True

1. If each side length of a rectangle doubles, then the perimeter of the rectangle also doubles. 2. If each side length of a rectangle doubles, then the area of the rectangle also doubles.

3. Congruent triangles have the same area one another. 4. Similar triangles have the same area as one another.

5. Changing the linear dimensions of a geometric figure does not change the perimeter of the figure.

6. Changing the linear dimensions of a geometric figure does change the area of a figure. 7. If the area of a square is quadrupled, then the side length of the square must have been quadrupled. 8. If the perimeter of a square is quadrupled, then the side length of the square must have been quadrupled. 9. Tripling the base of a triangle while keeping the height the same will triple the perimeter of the triangle. 10. The same scale factor used to multiply the linear dimensions

of a geometric figure is the one used to multiply the area and the perimeter of the figure.

Geometry HOUSTON ISD PLANNING GUIDE

4th SIX-WEEKS

Recommendations for Instructional Enhancements for

Students with Special Needs

Unit 11: Measuring Lengths and Area

Content-specific Accommodations for this Unit

To review the steps for algebraically determining the perimeter, area, or volume of various geometric figures or portions of figures, give students an example of a song written to assist with a mathematical procedure, similar to the one below.

Geometry, Perimeter, Area and Volume Song (Tune: Farmer In The Dell. Lyrics by Linda Bolin)

You measure along the lines. You measure along the lines. If you want to find perimeter, You measure along the lines.

You cover it up with squares. You cover it up with squares. If you want to find the area, You cover it up with squares.

You fill it up with cubes. You fill it up with cubes. If you want to find the volume, You fill it up with cubes.

Challenge students to work in groups to develop a song or poem that helps them

remember steps to measuring various polygons. They write a song to a common tune, develop a visual aide, and perform the song for the class.

Remind students to attend to the cases of letters that represent unknown variables. For example, the formula for determining the area, A, of a regular n-gon with side length, s, is half the product of the apothem, a, and the perimeter, P, written as A = ½ a(P). Students must differentiate between the meaning of “a” and “A.”

Allow students to use a calculator and provide them with written calculator directions specific to calculating length and area.

General Accommodations for this Unit

Present a set of step-by-step sequential directions for a procedure and ask students to mark the step at which they become confused and unable to proceed.

Teach students to number the pages of their notes so they know if any pages are missing.

Ask students to explain their work prior to “showing” their work. In some cases, the amount of work they are required to show may be modified.

Trainer/Instructor Notes: Area Investigating Area Formulas

Geometry Module 5-7

form a rectangle of equal area, but the group may or may not be able to reach a consensus on the triangle and kite. Briefly describe the activity. Each of the figures on the activity sheet has the same area as the rectangle. Using a colored pencil, trace the parallelogram on patty paper. Then, using the least number of cuts possible, cut the parallelogram and rearrange the pieces to form a rectangle of equal area. The rectangle will help you determine where to cut. Lay the patty paper tracing over the rectangle and slide it around to decide where to cut. Using a different colored pencil, draw the cut line on the patty paper figure and then cut. Assemble the pieces to form a rectangle and glue it next to the original parallelogram. Repeat the process for each of the figures. While the groups are working, assign each of the figures to a different participant to draw on a transparency for use during the group discussion. When most participants have completed the task, reconvene as a large group for discussion. Parallelogram:

Can we express the dimensions of the rectangle in terms of the dimensions of the parallelogram? Yes. The base of the rectangle is the base of the parallelogram. The height of the rectangle is the height of the parallelogram. What does this tell us about the formula for the area of the parallelogram? Since the area of the parallelogram is equal to the area of the rectangle, the area of the parallelogram is b h⋅ .

b

h


Geometry Module 5-8

Obtuse Triangle:

How would you describe the location of the cut lines on the obtuse triangle? The cuts must pass through the midpoints of the sides of the triangle as shown. What do we call the segment that connects the midpoints of the sides of a triangle? The midsegment What do we know about the midsegment of a triangle? The midsegment is parallel to the base and one half the length of the base of the triangle. Can we express the dimensions of the rectangle in terms of the dimensions of the triangle? Yes. The length of the base of the rectangle is equal to the length of the midsegment of the triangle, m. The height of the rectangle, h, is the height of the triangle. Can we use this information to derive the formula for the area of a triangle? Since we know Area of triangle = Area of rectangle = b · h By substitution, Area of triangle = m h⋅ By the definition of a midsegment,

Area of triangle = 1 (2 )2

b h⋅

= 12

(base of triangle) (height)

b= m h

2b


Geometry Module 5-9

Acute Triangle:

How would you describe the location of the cut lines on the acute triangle? One cut line goes through the midsegment of the triangle and one cut line is the altitude joining the midsegment to the opposite vertex of the triangle. Can we use this information to derive the formula for the area of a triangle? Since we know Area of triangle = Area of rectangle

= b h⋅

= ( )1 22

b h⋅

= 12

(base of triangle) (height of triangle)

h

b

2h


Geometry Module 5-10

Kite:

d 1

d 2

How can we use the formula we have derived for the area of a triangle to derive the formula for the area of the kite? d1 lies on the line of symmetry for the kite. The two triangles formed by the line of symmetry, d1, are congruent. 12

(d2) is the length of the altitude of each triangle, and 2( )1 d h2

=

Area of one triangle = 1( )1 d h2

⋅ =12

⎛ ⎞⎜ ⎟⎝ ⎠

(d1)12

⎛ ⎞⎜ ⎟⎝ ⎠

(d2)

Area of kite = ( )( )( )( ) ( )1 2 1 21 1 12 d d d d2 2 2

= ⋅

= 1 21 d d2

⋅

Success in this activity indicates that participants are working at the Relational Level because they must discover the relationship between the area rule for a rectangle and the area rule for a parallelogram, triangle, or kite. While a participant at the Descriptive Level will be able to cut the figures to form the rectangle of the same area, he/she will need prompting to explain “why it works” using informal deductive arguments.

h

b

Activity Page: Area Investigating Area Formulas


Investigating Area Formulas

Trace the parallelogram, triangles, and kite on patty paper. Then cut and arrange the pieces of each figure to form a rectangle congruent to the given rectangle. Glue each new figure next to the original figure from which it was made. Label the dimensions of the rectangle b and h. Determine the area for each figure in terms of b and h.

Activity Page: Area Investigating Area Formulas


Trainer/Instructor Notes: Area What Is Area?

Geometry Module 5-1

Unit 5 − Area

What Is Area? Overview: Participants determine the area of a rectangle by counting the number

of square units needed to cover the region. Group discussion deepens participants’ understanding of area (number of square units needed to cover a given region) and connects the formula for the area of a rectangle to the underlying array structure.

Objective: TExES Mathematics Competencies III.013.D. The beginning teacher computes the perimeter, area, and

volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc length, area of sectors).

V.018.D. The beginning teacher uses formal and informal reasoning to justify mathematical ideas.

Geometry TEKS b.3.D. The student uses inductive reasoning to formulate a conjecture. e.1.A. The student finds area of regular polygons and composite figures.

Background: No prerequisite knowledge is necessary for this activity. Materials: index cards, patty paper, straightedge New Terms: area Procedures: Background information: In order to accurately count the units of a given space, it is necessary to mentally organize the space in a structured manner. However, research referenced in Schifter, Bastable, and Russell (2002), on how children learn the concept of area supports the theory that the structure of the rectangular array is not intuitively obvious to children. When asked to cover a rectangular region, children progress from incomplete or unsystematic coverings to individually drawn units to the use of a row or column iteration. Gradually they will rely less on drawing and move towards multiplication or repeated addition. Covering a rectangular region with unit squares helps children understand area measure, but they must ultimately be able to formally connect area, linear measurement, and multiplication in order to truly understand the area formula A b h= ⋅ . According to Schifter, Bastable, and Russell (2002), the drawing, filling, and counting that children use in this developmental process are both motor and mental actions that coordinate to organize spatial structuring. What implications does this research have for secondary teachers? Many of our students come to us with an understanding of area at the Visual Level of the van Hiele model of


Geometry Module 5-2

geometric development. They recognize figures by their shape and understand area as the blank space within the boundaries. Others have moved to the Descriptive Level, which implies that they can mentally see the rectangular array which overlays the shape. This activity asks participants to draw the grid, rather than giving them a pre-structured grid in order for them to experience the type of activity necessary to move a student from the Visual to the Descriptive Level. Secondary students functioning on the Relational Level are able to compare linear dimensions with grid areas and apply formulas with understanding. Teachers must be aware that although students may become proficient at rote application of formulas, they may not be functioning at the Relational Level. If they have not been given sufficient opportunity to understand the principles behind the formulas, they will have difficulty modifying a procedure to fit a particular situation as is necessary to find areas of composite figures and shaded regions. Distribute index cards to participants and ask them to write a response to the question “What is area?”. Indicate that they will have an opportunity to share and revise their responses at the conclusion of this activity. Then, allow time for participants, working independently or in pairs, to complete the activity using the patty paper or straight edge to determine the number of square units needed to completely cover the rectangular region. Note that the given square units are not convenient measures, such as 1 cm2 or 1 in.2. Consequently, participants will be less likely to simply measure the rectangle and use the area formula without having the experience of drawing the units. Whether participants mark off the units on two adjacent sides of the rectangle and multiply or actually draw in one or more rows and columns of units, they will be counting the units by considering how many rows of squares are needed to cover the region. 1. Determine the number of square units needed to cover this rectangular region.

The rectangle measures 6 ·8 square units. Therefore, it will take 48 square units to cover the rectangle.

1 square unit


Geometry Module 5-3

2. Determine the number of square units needed to cover this rectangular region. (Same rectangle, different square unit)

The rectangle measures 9 · 12 square units. Therefore, it will take 108 square units to cover the rectangle. Was it necessary to draw all 108 square units to determine that it would take 108 units to cover the rectangular region? No. After drawing one row and one column of square units the total number of squares can be obtained by considering how many rows and columns of squares will be needed to cover the entire region. Has the activity caused you to reconsider your definition of area? Some participants may have responded to the question “What is area?” by stating that area is the amount of space covered by a particular region. It is important to make the distinction that area is the number of square units needed to completely cover a particular region. If the same figure is measured in different units, the number representing the area of the region will be different, but the area will remain constant. A more abstract definition of area, provided by Michael Serra (Serra, 2003) states that area is a function that assigns to each two-dimensional geometric shape a nonnegative real number so that (1) the area of every point is zero, (2) the areas of congruent figures are equal, and (3) if a shape is partitioned into sub regions, then the sum of the areas of those sub regions equals the area of the shape. If a figure is rotated so that a different side is considered the base, will the area formula necessarily give the same result? Yes. Surprisingly, the answer to this question is not evident to all students. According to the work of Clements and Battista referenced in Schifter, Bastable, and Russell (2002), orientation, the position of objects in space in relation to an external frame of reference, is for some children a part of their definition of a particular shape. If secondary students have not had adequate experience manipulating by rotating, flipping or sliding shapes, they may be working at the Visual Level with an inadequate understanding of shape. In developing an understanding of area, students should observe that rotations, reflections and translations preserve area while dilations do not.

1 square unit

Activity Page: Area What is Area?

Geometry Module 5-4

What Is Area? 1. Determine the number of square units needed to cover this rectangular

region.

1 square unit

Activity Page: Area What is Area?

Geometry Module 5-5

2. Determine the number of square units needed to cover this rectangular region. (Same rectangle, different square unit)

1 square unit


Geometry Module 5-6

Investigating Area Formulas

Overview: Participants cut and rearrange two triangles, a parallelogram, and a kite to form rectangles with the same areas. Examination of the points at which figures must be cut will lead to a deeper understanding of the formula for the area of each figure.

Objective: TExES Mathematics Competencies III.011.A. The beginning teacher applies dimensional analysis to

derive units and formulas in a variety of situations (e.g., rates of change of one variable with respect to another and to find and evaluate solutions to problems.

III.013.C. The beginning teacher uses geometric patterns and properties (e.g., similarity, congruence) to make generalizations about two- and three-dimensional figures and shapes (e.g., relationships of sides, angles).

V.018.D. The beginning teacher uses formal and informal reasoning to justify mathematical ideas.

Geometry TEKS b.2.A. The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. b.3.C. The student demonstrates what it means to prove mathematically that statements are true. d.2.C. The student develops and uses formulas including distance and midpoint.

Background: Participants should know the formula for the area of a rectangle and be able to identify the base and altitude of a triangle, the base and altitude of a parallelogram, and the diagonals of a kite.

Materials: transparency sheets, colored pencils, glue or tape, patty paper, scissors New Terms: Procedures: Participants sit in groups of 3-4 to work collaboratively. However, each participant should cut and glue his/her own figures. Can any parallelogram, triangle or kite be cut and rearranged to form a rectangle of the same area? Allow time for discussion within groups and then ask two or three groups to share their responses with the entire group. It is fairly obvious that any parallelogram can be cut to

What is it we want all students to learn? Mathematics Geometry Objectives: GEOM.8A; GEOM.11D

© Houston ISD – Curriculum 2011 – 2012

Page 1 of 1

Geometry: Formative Assessment Geoboards for Perimeter and Area HISD Objectives: GEOM.8A Determine areas of regular polygons, circles, and composite figures using the area of triangles, squares, rectangles, parallelograms, and/or trapezoids. GEOM.11D Describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems. GEOM(8.14D) Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems. GEOM.8E Use area models to connect geometry to probability and statistics. GEOM.8F Use conversions between measurement systems to solve problems in real-world situations. Resources: National Library of Virtual Manipulatives: • Geoboards • Geoboards - Circles

Connected Activities in Geometry Planning, Cycle 4: • Find the number of square units in each figure • Finding the Rectangle • Application Problems • Find the Missing Dimension • Circle Area Students may access the above website. Students may take screenshots of their work through the activities or have the students sketch on graph paper. If a classroom interactive whiteboard is available, use this website as an all-class assessment by asking various students to provide different responses to each problems. Extensions: To meet College and Career Readiness Standards, teachers will need to enhance activities and examples by converting between units of measures using conversion factors and dimensional analysis. Elaboration of activities may include asking the probability of hitting a certain area within a composite figure. Access to the rubrics is available at these links: • Region IV Teacher rubric • Region IV 6 – 8 Student Rubric • Region IV H. S. Student rubric

http://www.escweb.net/tx_bm/math/docs/rubric.pdf�

http://www.escweb.net/tx_bm/math/docs/6-8studentrubric.pdf�

http://www.escweb.net/tx_bm/math/docs/rubric.pdf�

http://nlvm.usu.edu/en/nav/frames_asid_279_g_4_t_3.html?open=activities&from=category_g_4_t_3.html

The 5 E Learning Cycle Model

Engage Objects, events, or questions are used to engage students. Connections are

made between what students know and can do.

Explore Objects and phenomena are explored through hands-on activities, with

guidance.

Explain

Students explain their understanding of concepts and processes. New

concepts and skills are introduced as conceptual clarity and cohesion are

sought.

Elaboration Activities allow students to apply concepts in contexts, and build on or

extend understanding and skill.

Evaluation Students assess their knowledge, skills, and abilities. Activities permit

evaluation of student development and lesson effectiveness.

Engage:

Learner Teacher

calls up prior knowledge poses problems

has an interest asks questions

experiences doubt or disequilibrium reveals discrepancies

has a question(s) causes disequilibrium or doubt

identifies problems to solve, decisions to be

made, conflicts to be resolved

assess prior knowledge

writes questions, problems, etc.

develops a need to know

self reflects and evaluates

Explore:

Learner Teacher

hypothesizes and predicts questions and probes

explores resources and materials models when needed

designs and plans makes open suggestions

collects data provides resources

builds models provides feedback

seeks possibilities assesses understandings and processes

self reflects and evaluates

Explain:

Learner Teacher

clarifies understandings provides feedback

shares understandings for feedback asks questions, poses new problems and

issues

forms generalizations models or suggests possible modes

reflects on plausibility offers alternative explanations

seeks new explanations enhances or clarifies explanations

employs various modes for explanation

(writing, art, etc)

evaluates explanations

Elaborate:

Learner Teacher

applies new knowledge asks questions

solves problems provides feedback

makes decisions provides resources

performs new related tasks makes open suggestions

resolves conflicts models when necessary

plans and carries out new project

asks new questions

seeks further clarification

Evaluate:

Learner Teacher

self-assess their own learning and

understanding of new concepts

evaluates effectiveness of the instruction

provide feedback to the teacher on lesson

effectiveness

assesses student learning and understanding

reflect with adults and their peers uses information about student learning to

guide subsequent instruction

communicate, in a variety of ways (e.g.

journals, reporting drawing, graphing,

charting) their level of understanding of

concepts that t hey have developed to date

asks open-ended questions to examine

students’ thinking

create and use quality indicators to assess

their own work

employs a rubric on which to give students

feedback on their learning

report and celebrate their strengths and

identify what they'd like to improve upon

Source: http://faculty.mwsu.edu/west/maryann.coe/coe/inquire/inquiry.htm

http://faculty.mwsu.edu/west/maryann.coe/coe/inquire/inquiry.htm

Notes to the Teacher Appendices Materials One copy Blackline Master B8a

from lesson 2 Patty paper Rulers Scissors Tape or glue sticks Vocabulary: Define, draw or give an example of the following: Area of rectangle Area of parallelogram Area of triangle Area of trapezoid This activity continues to develop

the descriptive van Hiele level of understanding of areas of simple planar figures, and begins to reach into the relational level.

A9 Finding the Rectangle

Students use a clean copy of Blackline Master B8a from the previous lesson. In this lesson students find the minimum number of cuts in each non-rectangular figure to rearrange and form the 15 square unit rectangle. Distribute Blackline Master B8a, patty paper, scissors, and tape or glue sticks. Give whole class directions: • All of the figures on the sheet have the same area which was

determined by counting and estimating unit squares. We can’t draw squares every time we have to find areas, especially when you are given length dimensions, which may not always be friendly. It is easy to find areas for rectangles by multiplying the length and width dimensions, which are the same as the number of units that fit along those sides. In this lesson we are going to see how to cut each figure, using the minimum number of cuts, to form the 3 x 5 square-unit rectangle.

• Trace each figure on patty paper. Experiment by moving the traced figure over the rectangle to see where to make the cuts. Draw in the cut lines. Cut, rearrange and glue the rectangle next to the original figure. Draw dotted lines on the original figure to show the cut lines.

Direct students to work collaboratively, but each does his/her own work for about 30 minutes. Anyone finishing sooner can continue to work on Rectangular Prism Tasks B7b, or B7c. Monitor and facilitate as students work. When most have completed the task call the class together for group discussion. • Project the transparency of Blackline Master B8a. Ask a student to

show where the cut lines go on figure B, the parallelogram. Students must sketch in the rectangle and use the length and height of the rectangle to find the area of the parallelogram. Visualizing or drawing the rectangle is required. Point out that the slanty-side measurement on the parallelogram must not be used in the area calculation. Remind students that we counted rows; this is equivalent to the perpendicular measurement. Students add Area of Parallelograms to their geometric dictionaries, along with appropriate sketches. This can be completed at the end of the lesson.

B B

• Ask another student to show the cut lines for the obtuse triangle, C. A possible diagram follows. The midsegment has been drawn into this diagram even though it is not one of the cut lines.

The student describes the exact location of the cuts, which should pass through the midpoints of the sides of the triangles, as shown. The small triangles then rotate 180o through the midpoints to form the shaded rectangle. Ask the student to draw the segment connecting the midpoints. • What do we name this segment? The midsegment. • What is the length of the midsegment compared to the parallel

side? Remind the class of the parallel line activity where midsegments of triangles were studied. The midsegment is half the length of the parallel side.

• What parts of the triangle represent the dimensions of this rectangle? The length is the midsegment of the triangle, and the width or height is the distance from the vertex to the base, parallel to the midsegment.

• What do we call the “segment from the vertex to the opposite base?” The altitude.

• The area of the rectangle? (length of the midsegment)·(length of the altitude)

• What is a substitution for “length of midsegment?” Half of the base.

• Another way to get the area using this substitution? (half the length of the base)·(length of the altitude) Students add Area of Triangles to their geometric dictionaries, along with appropriate sketches. This can be completed at the end of the lesson. Ask another student to demonstrate the cuts for acute triangle E (see figure in the Notes sidebar). • What is different about the cuts in E compared to C? In figure C

we cut the side triangles off and added them to the top. In figure E we cut the top off, divide it down its altitude and add the pieces to the sides.

• What is the same? The cross cut is also made along the midsegment, between the midpoints of the sides.

• What are the two measurements needed to determine the area of the rectangle? (The length of the base of the triangle).(half the length of the altitude)

midsegmentmidpoint

midpoint

E

midpoint

midsegmentmidpoint

This is an opportunity to review:

TEKS A.b.4.B Use the commutative, associative, and distributive properties to simplify algebraic expressions.

• We have three ways to get the area of a triangle: 1. From the tessellation of the blocks: (Base of triangle · altitude) / 2 2. From figure C: (Length of base / 2) · (Altitude) 3. From figure E: (Length of base) ·(Altitude / 2) Replace “Length of base” with “b”, and “Altitude” with “a”. Why are the following the same?

1.2

b ai 2. 2b ai 3.

2abi

Students can inductively prove that these expressions are equivalent by substituting b = 5 and a = 3, or b = 3 and a = 5. Point out that in each case the numerators (insert /1 where needed) are all b · a, and the denominator is 2. Dividing by 2 can be done to either numerator or to the product of b · a, but only one time. Students add a sketch of this triangle’s rearrangement to the information on Area of Triangles in their geometric dictionaries. This can be completed at the end of the lesson. The trapezoid, figure D, is the most difficult, but students might be able to apply what they know from the triangles to this figure. This time call for a volunteer who has discovered the rectangle within the trapezoid. • How is this similar to figure C? Both figures use the midsegment

for the length of the base of the rectangle. • What is the length of the trapezoid’s midsegment? The average of

the lengths of the two parallel bases. • We name the two bases b1 and b2. These subscripts do not have

anything to do with number values; they are ways of naming two parts that are very similar.

• How do we find the average? 1 2

2b b+

• What is different to figure C? The triangle’s altitude was used, the perpendicular length from the base to the opposite vertex. This figure doesn’t have a vertex like the triangle. Here we use the perpendicular distance between the parallel lines.

• Now find the area of the rectangle using these numbers.

1 2 ( )2

b bdistance between parallel lines

+ i

Students add Area of Trapezoids to their geometric dictionaries, along with appropriate sketches. This can be completed at the end of the lesson or for homework along with the second sheet of prism tasks.

midsegment

midpoint

midpoint

D

B8a Find the Number of Square Units in Each Figure

E

C

A

NOT

like

this

Do it

like

this

1 sq

uare

uni

t

DB

B8b Transparency for Find the Number of Square Units in Each Figure

B8c Additional Guided Practice

Figure # of units along

lower side # of rows Total number of

squares ____ x ____

Area (# of square units)

A

B

C

D

E

1 sq

uare

uni

tB

C

D

E

A

Notes to the Teacher Appendices Materials 1 copy of Blackline Master B11 for

each student. This activity allows students to

determine the areas of non-rectangular figures by superimposing an equivalent rectangle on the figure, without resorting to formulas.

A11 moves students toward the

van Hiele Relational level with respect to area of planar figures between parallel lines. A11 is a carefully guided activity. The accompanying text assignment gives students an opportunity to practice similar problems as well as application problems, which will review: TEKS 8.15.A Communicate mathematical ideas using language, efficient tools, appropriate units and graphical, numerical, physical, or algebraic mathematical models.

A11 Application Problems For an introductory application problem refer to McDougal Littell, or Discovering Geometry, page 410 - 411.Go over the problem during whole class discussion. Relate the tiles to the array of squares drawn in the figures in Lesson 2. Distribute Blackline Master B11 to each student. Review the directions, emphasizing that the cutting lines must be drawn. Students will determine the areas of each figure based on the area of the rectangle superimposed into the figure. Answers: 1. Rectangle RECT, whose length is 9ft and whose width is 8ft.

Area of rectangle = 8 ft · 9 ft = 72 ft2 2. Parallelogram ABCD. AD = 8 cm. The perpendicular distance

between AD and BC is 4cm. Area of rectangle = Area of parallelogram = 4cm · 8cm = 32 cm2.

3. A triangle whose base is 18 inches and whose altitude is 8 inches

18 in.

8 in.

9 in.

18 in.

8 in.

4 in.

18 in.

8 in.

Three possible representations are shown. From left to right: Area of triangle = Area of rectangle = 4 in. · 18 in. = 72 in2. Area of triangle = Area of rectangle = 8 in · 9 in. = 72 in2.

Area of triangle = Area of rectangle2

= 18in. 8in.2⋅ = 72 in2.

4.

.5 m.Rectangle base = 17 m.

Short base =14 m.

Long base = 20 m.

8 ft

9 ft

C

ER

T

4 cm.

8 cm.D

C B

A

Area of trapezoid = Area of rectangle = Average of the bases · 0.5 m.

= 20 142+ m · 0.5 m = 8.5 m2.

If centimeter units are used, the dimensions will be 2000 cm, 1400 cm, and 500 cm.

Area of trapezoid = 2000 14002+ cm · 500 m = 850,000 cm2.

Assign the following text problems. Require that students sketch each figure and draw in the equivalent rectangle. McDougal Littell: Pages 364-6, #1-6; 8; 11-12; 17-22 (these require graph paper); 26-27; 31-33. Alternate Discovering Geometry: Pages 413-415, # 1-14; 16; 19; 23; 24. The corresponding section, 8.1, in Discovering Geometry Practice Your Skills is more challenging: Page 49.

B11 Application Problems For each of the following, sketch the figure and write in the appropriate dimensions. For non-rectangular figures, draw in the cutting lines, which would convert the figure into a rectangle of equivalent area. Shade in the rectangle. Write in the rectangle’s dimensions if they are different from those on the given figure. Determine the area of each figure by using the drawn in rectangle. 1. Rectangle RECT, whose length is 9 ft and whose width is 8 ft. 2. Parallelogram ABCD. AD = 8 cm. The perpendicular distance between AD andBC is 4 cm.

D

C B

A

3. A triangle whose base is 18 inches and whose altitude is 8 inches. 4. A trapezoid with bases of 14 and 20 meters, and a height (the perpendicular distance

between the bases) of 500 centimeters. (Hint: Convert all of the measurements to meters or to centimeters)

Notes to the Teacher Appendices Materials Transparency B19 Two or three large circular objects

(classroom trash can, large circular cartons or jars)

Tape measures or string and meter stick.

One sheet of blank paper for each

student Compasses or disposable plastic

cups Scissors Tape or glue sticks Vocabulary: Define, draw or give an example

of Semicircle Sector The circle area investigation is at

the descriptive van Hiele level, adapted from Discovering Geometry, p.433.

A19 Circle Area

Students must understand the ratio CircumferenceDiameter

π= as pre -

requisite knowledge for circle area. Ask students if they had measured circumferences and diameters of circular objects in middle school. Even if a few have not seen this activity before, or do not remember it, the concept can be reviewed very quickly on the spot. Place the classroom trash can on top of a table. If tape measures are not available use string and then measure the string lengths against a meter stick. Ask two students to measure, in inches and in centimeters, the circumference of the top rim, and the diameter inside the top rim. Write the measurements as a ratio on the board. Calculate the decimal value of the ratio. It should be close to 3. Circumference . .

Diameter . .in cmin cm

= = =

Ask two more students to measure the circumference and diameter of another circular object and write the ratio on the board. • In any circle if we divide the circumference by the diameter, how

many diameters do we find in the circumference? About three. • This ratio, pi, has been known for thousands of years. The ancient

Babylonians and Chinese knew it. All mathematics dealing with curved components incorporates pi.

• If I know the length of the diameter, can I find the circumference? The circumference is about 3 · diameter.

• If we want to be very exact, we use pi instead of 3. • 3Circumference Diameter≈ i orCircumference Diameterπ= i

• Ask a student to stand in front

of the class (same gender as yourself). The student makes a circle with his/her arms above the head as shown. Stand behind and point to the elbows as forming the diameter of the circle. You might stretch string or tape across this diameter. The

student now stretches his/her arms out to the side, representing the circumference laid out in a straight line. Using the diameter string, show that there are about three diameters in the stretched out circumference.

• I am thinking of a circle. Its diameter is 6 inches. Write this on the

board. What is the length of its circumference? About 18 inches, but exactly π·18, or 18 π inches, approximately 56.52 inches if you substitute 3.14 for π.

• I am thinking of another circle. Its radius is 6 inches. Write this on the board. What is the length of its circumference? Its diameter is 12 inches. Its circumference is about 36 inches, but exactly 12π inches, which is closer to 37.68 inches using 3.14 for π.

• Now my circle has a circumference of 30 inches. Write this on the board. What is its radius? Its diameter is about 10 inches, but exactly 36/π inches. Its radius is about 5 inches (and exactly 18/π inches).

• Now my circle has a radius r. What is its diameter? 2r. • What is its circumference? About 6r, but exactly 2 2r rππ =i . Distribute the blank paper, compasses or large disposable plastic cups, scissors and tape or glue sticks. Give the following directions, waiting for each step to be completed by most of the class. If some students are slower, they will be able to catch up by watching their table mates. Use transparency B19. Uncover the figures as you progress through the steps. • Use the compass or the wider end of the plastic cup to make a

large circle on the blank paper. Cut out the circular region. • Fold the circular region in half. Fold it in half a second time, then a

third time and a fourth time. • Measure and write down the radius of your circle to the nearest

centimeter. Calculate and write down the circumference of your circle.

• On the board write radius = r and circumference = 2πr. • Unfold the circle. Cut it into two halves along one of the diameter

folds. • What is the name of the arc for half of the circle? Semicircle. • Calculate and write down the length of the semicircle. If students

have trouble, remind them that we know the circumference, which is two semicircles.

On the board write semicircle = 22

r rπ π=

• Take one half. Pretend you are holding half of a pizza, which has a

very thin crust around the curved edge. Starting from the center, cut the pizza slices along the folds, but do not cut through the crust. This keeps the slices attached to each other along the crust.

• Pretend you are going to hang the crust on a line, with its pizza

wedges hanging down. Tape the ends of the crust to the line.

• In geometry we do not call these figures wedges or pizza slices. They are called sectors. Can anyone think of another context where we use the word sectors? Some examples: Air traffic controllers use radar to keep track of incoming and outgoing aircraft at airports. Airplanes fly through space in named and allocated sectors in a three dimensional coordinate system. Data is stored in sectors on a computer disk.

• Define sector using mathematical terms. A region of a circle that is bounded by two radii and an arc of a circle.

• Is the region bounded by a semicircle and a diameter a sector? Yes, the semicircle is an arc; the diameter is two radii.

• Cut the second half in the same way as the first. Instead of hanging this half on the line, arrange the slices so that they fit into the wedge-shaped spaces between the slices of the first set. The second set of slices points up into the slices of the first set which point down.

• What figure does this approximate? A parallelogram, except that

the longer sides are not exactly straight; they are scalloped. • If we make the sectors very skinny then the sides would be much

closer to a straight line and the parallelogram would be closer to a rectangle. This is what we do in calculus! We consider what happens when we make distances infinitely small. We can think about this figure the same way.

• Label the parallelogram’s length and width. Notice that the width of this parallelogram is the radius.

πr

rr

πr

• Use your numbers to calculate the area of your parallelogram. For the general case on the board Area = πr·r = πr2.

• The original circle was rearranged form this parallelogram. So the area of the circle is πr·r = πr2..

• Add this information to circles in your geometric dictionaries. • I am thinking of a circle. Its diameter is 6 inches. (Write D = 6

inches on the board). What is its area? Its radius is 3 inches, so its area is π (3)2 = 9πsquare inches, approximately 28.3 inches2.

• I am thinking of another circle with a radius of 6 inches. (Write r = 6 inches on the board). What is its area? Its area is π (6)2 = 36π square inches, approximately 113.0 inches2.

• The first circle’s radius was 3 inches and the second circle’s radius was 6 inches, double the first radius. But the area of the second circle, 113.0, is not double the area of the first radius 28.3. Divide 113.0 by 28.3. Four.

• Why four? The radius is used twice in the multiplication for circle area. So if it is doubled, the factor 2 is in there twice, making 2(3) · 2(3), which is a factor of 4 compared to the smaller circle.

• Consider this problem. I have a 100ft length of fence to make an enclosed pen for my dog in the backyard. I can make a circle with circumference 100ft, a square with perimeter 100ft, or a rectangle with width 6ft to make a long pen so he can run up and down. Which pen gives him the most area?

Students work in groups for about 10 minutes. Ask a student to show how to find the area for the circular pen. Students correct their own work. When all questions have been addressed ask another student to find the area for the square, and then for the long rectangle.

Circular pen:

2 2

Circumference = 100ft = 2 6.3100 15.96.3

Area= (15.9) 793.8

r r

r ft

ft

π

π

≈

≈ =

≈

Square:

2

Perimeter 100 . 4(side length)100 .Side length= 25 .

4Area 25 25 625 .

ftft ft

l w ft

= =

=

= = =i i

Rectangle:

( )2

Width 6 . Length 44 . 2( ) 100;( ) 50; 44

Area 6 44 264

ft ft L W L W L

L W ft

= = + = + = =

= = =i i

The circular pen has the largest area.

B19 Transparency for Circle Area

π

r

πr

r

Notes to the Teacher Appendices Materials 1 copy of Blackline Master B13 for

each student. A prerequisite for understanding

Pythagoras is the ability to determine length dimensions if the area is known.

This activity reviews:

TEKS 8.15.A Communicate mathematical ideas using language, efficient tools, appropriate units and graphical, numerical, physical, or algebraic mathematical models.

A13 Find the Missing Dimension Distribute Blackline Master B13. Students sketch the equivalent-area rectangle for each figure, even if they are using a formula to solve the problem. Students work in pairs or small groups for about 30 minutes. Ask students to demonstrate their solutions. Alternatively provide poster sheets and markers and have different groups prepare poster presentations of selected problems for whole group discussion. Answers: 1. Find the number of square centimeters in the area of a square

whose perimeter is 40 cm.

P = 40cm. Each side = 10cm. Area = 10cm · 10cm = 100cm2.

2. The area of a triangle is 24 square units, and its height is 6 units.

How many units are in the length of the base?

Possible solutions are shown. In clockwise order starting from the top left:

Area of triangle = Area of rectangle = 6 · base2

= 24 square units

base2

= 224 units

6 units= 4 units

Base = 2 · 4 = 8 units. Area of triangle = Area of rectangle = 3 · base = 24 square units Base = 24/3 = 8 units Area of rectangle = 2 · Area of triangle = 2 · 24 = 48 square units

Triangle Area = 24 square unitsRectangle Area = 2 x 24 = 48 square units

Area = 24 square units

Area = 24 square units

3

6

36

10cm.

10cm.

P = 40cm.

Base · 6 = 48 square units. Base = 48/6 = 8 units. Using a formula:

Area of triangle = base height base 62 2⋅ ⋅

= square units

= base · 3 square units = 24 square units Base = 24/3 = 8 units. 3. If the area of BDF is 60cm2, BF = 10cm, and CE = 22cm, what

is the area of trapezoid CBDE?

22cm

10cm

KFCE

B D

Area of = 60cm2 BDF

Area of rectangle BDKF = 2 · Area of BDF = 120cm2 = 10 · BD BD = 12 cm. Area of trapezoid CBDE = Area of shaded rectangle

= 102

BD CE+ i

= 12 22 102+ i

= 34 102i

=17 · 10 = 170cm2. 4. A parallelogram and a triangle both have 60cm2 areas, and 15cm

bases. What is the ratio of the height of the parallelogram to the height of the triangle?

7.5cm

15cm

15cm

Area of parallelogram = 15 · height = 60 cm2 Height of parallelogram = 60/15 = 4 cm

Area of triangle = Area of tall rectangle ( = Area of wide rectangle) = 7.5 · height = 60 ( = 15 · 0.5 · height = 60) height = 60/7.5 = 8 cm. ( 0.5 · height = 60/15 = 4) (Height = 4/.5 = 8 cm.)

Height of parallelogram 4 1Height of triangle 8 2

cmcm

= =

5. The point M is the midpoint of the segment BE . What percent of

the area of rectangle BTRE is shaded? (Hint: Assign values to the dimensions of the rectangle).

Assign values: TB = 4 units; BE = 10 units. Then ME = 5 units. Area of BTRE = TB · BE = 4 · 10 = 40 square units. Area of = ME · TB/2 TME = 5 · 4/2 = 10 square units. Shaded Area

Total Area = 10 1

40 4= = 25%

An alternative solution: Area of = ½ of rectangle BTRE. TBE = ½ of TBM TBE = ½ of (½ of rectangle BTRE) = ¼ of rectangle BTRE = 25% of rectangle BTRE. 6. Use the figure in #5. Rectangle BTRE is a target. Assuming there

is an equal chance of hitting any point within the rectangle, what is the probability of hitting the unshaded area?

From #5 the probability of hitting the shaded area is Shaded Area

Total Area= 25%

The probability of hitting the unshaded area is 75%. More practice: Discovering Geometry, Lesson 8.2, #1-15. Alternate assignment: Discovering Geometry Practice Your Skills, Lesson 8.2, p. 50.

M

R

B E

T

B13 Find the Missing Dimension For each of the following, sketch the figure and write in the appropriate dimensions. For non-rectangular figures, draw in the cutting lines, which would convert the figure into a rectangle of equivalent area. Shade in the rectangle. Write in the rectangle’s dimensions if they are different from those on the given figure. Use the given information to find the missing dimensions. 1. Find the number of square centimeters in the area of a square whose perimeter is 40 cm. 2. The area of a triangle is 24 square units, and its height is 6 units. How many units are in the

length of the base? 3. If the area of BDF is 60 cm2, BF = 10 cm, and CE = 22 cm, what is the area of trapezoid

CBDE?

FCE

B D

4. A parallelogram and a triangle both have 60 cm2 areas, and 15 cm bases. What is the ratio of the height of the parallelogram to the height of the triangle?

5. The point M is the midpoint of the segment BE . What percent of the area of rectangle BTRE is shaded? (Hint: Assign values to the dimensions of the rectangle).

M

R

B E

T

6. Use the figure in #5. Rectangle BTRE is a target. Assuming there is an equal chance of

hitting any point within the rectangle, what is the probability of hitting the unshaded area?

Name: Period:

Anticipation Guide: Dimensional Change

Directions: On the continuum in front of each of the numbers, place an “x” that indicates where you stand in regard to the statement that follows. Be prepared to defend and support your opinions with specific examples. After the lesson, compare your opinions on those statements with what you learned in the lesson. Never True Sometimes True Always True

1. If each side length of a rectangle doubles, then the perimeter of the rectangle also doubles. 2. If each side length of a rectangle doubles, then the area of the rectangle also doubles.

3. Congruent triangles have the same area one another. 4. Similar triangles have the same area as one another.

5. Changing the dimensions of a geometric figure does not change the perimeter of the figure.

6. Changing the dimensions of a geometric figure does change the area of a figure. 7. If the area of square is quadrupled, then the side length must have been quadrupled. 8. If the perimeter of square is quadrupled, then the side length must have been quadrupled. 9. Tripling the base of a triangle while keeping the height the same will triple the perimeter of the triangle. 10. The same scale factor used to multiply the dimensions

of a geometric figure is the one used to multiply the area and the perimeter of the figure.

Trainer/Instructor Notes: Area Applying Area Formulas


Applying Area Formulas

Overview: Participants use the problem solving process to find the area of composite figures (composite of triangles, quadrilaterals and circles).

Objective: TExES Mathematics Competencies III.011.B. The beginning teacher applies formulas for perimeter, area,

surface area, and volume of geometric figures and shapes (e.g., polygons, pyramids, prisms, cylinders, cones, spheres) to solve problems). III.013.D. The beginning teacher computes the perimeter, area, and volume of figures and shapes created by subdividing and combining other figures and shapes (e.g., arc length, area of sectors).

V.018.E. The beginning teacher understands the problem-solving process (i.e., recognizing that a mathematical problem can be solved in a variety of ways, selecting an appropriate strategy, evaluating the reasonableness of a solution).

V.019.C. The beginning teacher translates mathematical ideas between verbal and symbolic forms.

V.019.D. The beginning teacher communicates mathematical ideas using a variety of representations (e.g., numerical, verbal, graphical, pictorial, symbolic, concrete).

Geometry TEKS

b.4. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

d.2.C. The student develops and uses formulas including distance and midpoint.

e.1.A. The student finds the area of regular polygons, and composite figures.

Background: Participants need to know the area formulas and how to connect the

formulas to models of composite figures (composite of triangles, quadrilaterals and circles).

Materials: calculator New Terms: composite figures Procedures: Write the word “composite” on the overhead and brainstorm some meanings of this word. The Merriam-Webster Collegiate Dictionary, from http://www.yourdictionary.com, defines composite, when used as an adjective, as “consisting of separate interconnected parts.” Remind the participants to add the new term to their glossaries.



What are some real world examples of composite figures? Possible answers are: Photography: an image or scene made up of two or more original images placed side by side, overlapped, or superimposed. Automotive: any material that consists of two or more substances bonded together for strength, such as fiberglass. Architecture: architectural drawings, consisting of more than one room or combination of shapes. Participants work collaboratively on problems. 1. An interior designer created a plan for the kitchen counters and an island to be located in the middle of the kitchen, as shown below. The opposite sides of the counter are parallel and the intersecting straight lines are perpendicular. The curved part of the countertop is a quarter circle. The island has parallel sides, and the curved end is a semicircle. What is the total area of the tops of the counter space and island? Use the π

key on the graphing calculator instead of 227

or 3.14. Round calculations to the

nearest tenth.

The total areas of the tops of the counter and the island are 6232.8 sq.in.

. .6232.8 43.3 sq ft144

≈

The counter tops and the island in the kitchen will be covered with granite. Granite costs $14.38 per square foot. Find the cost of the countertops and island for this kitchen? Round the cost to the nearest penny.

. . $. .

$14.3843.3 sq ft $622.654 622.65sq ft

⋅ = ≈

452.4 sq.in.

720 sq.in.

864sq.in.

452.4 sq.in.2304sq.in.

1440 sq. in.

24 in

30 in24 in24 in

36 in

120 in

60 in



2. P is a random point on side AY of rectangle ARTY. The shaded area is what fraction of the area of the rectangle? Why? The altitude of the triangle, h, is equal to the height of the rectangle. The base of the triangle is equal to the base of the rectangle. So, the

area of the triangle = 12

bh or 12

area of the rectangle.

3. Kit and Kat are building a kite for the big kite festival. Kit has already cut his sticks for the diagonals. He wants to position P so that he will have maximum kite area. He asks Kat for advice. What should Kat tell him? This problem is taken from Discovering Geometry: An Investigative Approach, Practice Your Skills, 3rd Edition, ©2003, p.49, used with permission from Key Curriculum Press. The location of the point of intersection of the two diagonals

does not change the area, because the area of the kite is 12

d1d2.

4. A trapezoid has been created by combining two congruent right triangles and an

isosceles triangle, as shown. Is the isosceles triangle a right triangle? How do you know? Find the area of the trapezoid two ways: first by using the trapezoid area formula, and then by finding the sum of the areas of the three triangles. This problem is taken from Discovering Geometry: An Investigative Approach, 3rd Edition, ©2003, p.420, 21, used with permission from Key Curriculum Press.

The isosceles triangle is a right triangle because the angles on either side of the right

angle are complementary. If you use the trapezoid area formula, the area of the

trapezoid is 12

(a + b)(a + b). If you add the areas of the three triangles, the area of

the trapezoid is 12

c2 + ab.

b c

a

ca

b

Y T

RA

P



5. The rectangle and the square have equal area. The rectangle is 12 ft by 21 ft 4 in. What is the perimeter of the entire hexagon in feet? The area of the rectangle is 256 ft2. The area of the square is 256 ft2. The length of the side of the square is 16 ft. The perimeter of the composite figure is 98.6 ft. Success in this activity indicates that participants are working at the Relational Level because they must solve geometric problems by selecting known properties of figures or formulas and deductive reasoning to solve problems.

Activity Page: Area Applying Area Formulas


Applying Area Formulas

1. An interior designer created a plan for the kitchen counters and an island to be located in the middle of the kitchen, as shown below. The opposite sides of the counter are parallel and the intersecting straight lines are perpendicular. The curved part of the countertop is a quarter circle. The island has parallel sides, and the curved end is a semicircle. What is the total area of the tops of the counter space and island? Use the π key on the graphing calculator instead of 22

7 or

3.14. Round calculations to the nearest tenth.

The counter tops and the island in the kitchen will be covered with granite. Granite costs $14.38 per square foot. Find the cost of the countertops and island for this kitchen? Round the cost to the nearest penny.

24 in

30 in24 in24 in

36 in

120 in

60 in



2. P is a random point on side AY of rectangle ARTY.

The shaded area is what fraction of the area of the rectangle? Why?

3. Kit and Kat are building a kite for the big kite festival. Kit has already cut his sticks for the diagonals. He wants to position P so that he will have maximum kite area. He asks Kat for advice. What should Kat tell him? This problem is taken from Discovering Geometry: An Investigative Approach, Practice Your Skills, 3rd Edition, ©2003, p.49, used with permission from Key Curriculum Press.

4. A trapezoid has been created by combining two congruent right triangles and an isosceles triangle, as shown. Is the isosceles triangle a right triangle? How do you know? Find the area of the trapezoid two ways: first by using the trapezoid area formula, and then by finding the sum of the areas of the three triangles. This problem is taken from Discovering Geometry: An Investigative Approach, 3rd Edition, ©2003, p.420, 21, used with permission from Key Curriculum Press.

b c

a

ca

b

Y T

RA

P



5. The rectangle and the square have equal area. The rectangle is 12 ft by 21 ft 4 in. What is the perimeter of the entire hexagon in feet?

These released questions represent selected TEKS student expectations for each reporting category. These questions are samples only and do not represent all the student expectations eligible for assessment.

Copyright © 2011, Texas Education Agency. All rights reserved. Reproduction of all or portions of this work is prohibited without express written permission from the Texas Education Agency.

2011 Released Test Questions

Geometry

State of Texas Assessments of

Academic Readiness

STAARTM

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 2

1 The figure below was formed by joining 2 segments of equal length at common endpoint Y .

X

Y

Z

If points X , Y , and Z are non-collinear , which of the following statements regarding XZ must always be true?

A XZ І XY

B XZ І 2( XY)

C XZ > 2( XY)

D XZ < 2( XY )


Page 3

2 A geometry student concluded:

If two sides and a non-included angle of one triangle are congruent to two sides and a non-included angle of another triangle, then the two triangles are congruent.

Which diagram can be used as a counterexample to the student’s conclusion?

A

B

C

D


Page 4

3 Which set of statements represents a valid deductive argument?

A All quadrilaterals have 4 angles. All parallelograms have 4 angles. All quadrilaterals are parallelograms.

B All parallelograms have diagonals that bisect each other. All parallelograms have opposite sides that are parallel. All polygons whose diagonals bisect each other have opposite sides that are parallel.

C All rectangles have 4 right angles. All squares have 4 right angles. All rectangles are squares.

D All parallelograms have 4 sides. All polygons with 4 sides are quadrilaterals. All parallelograms are quadrilaterals.

4 In each of the circles below, four angles are formed by the intersection of 2 secant lines. The measures of two intercepted arcs and one angle are shown for the first three circles.

AC

B

86° 77° 68°

32°

39°

25°

91°100° 82°

(5x + 2)°

(4x + 4)°

Which expression can be used to represent m Є�� ABC in degrees?

1A [(5 2x x+ −) (4 + 4)]2

1 B [(5 2x x+ +) (4 + 4)]2

C 2 5[( x x+ −2) (4 + 4)]

D 2 5[( x x+ +2) (4 + 4)]


Page 5

5 Jake took pictures of Ana’s flag while she was practicing her routine for the football game, as shown below.

1 32 4

Which of the following best describes the movement of the flag from picture to picture?

A Reflection, rotation, translation

B Rotation, translation, translation

C Rotation, translation, dilation

D Reflection, translation, translation

6 When viewed from above, the base of a water fountain has the shape of a hexagon composed of a square and 2 congruent isosceles right triangles, as represented in the diagram below.

40 ft

10 ft

Base ofFountain

Top View

Which of the following measurements best represents the perimeter of the water fountain’s base in feet?

A ( )20 + 20 2 ft C ( )40 + 20 2 ft

B ( )20 + 40 2 ft D ( )40 + 40 2 ft


Page 6

7 A side view of the intersection of a plane and a square pyramid is modeled below.

Plane

Base of pyramid

Which of the following best represents the shape formed by this intersection?

A

B

C

D


Page 7

8 The three-dimensional figure shown is composed of 11 identical cubes.

Front

Which of the following could not represent a top, front, or side view of the figure?

A

B

C

D


Page 8

9 RG is graphed on the coordinate grid below.

−5

−4

−6

−7

−8

−9

−10

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

−1 1−2−3−4−5−6−7−8−9−10 2 3 4 5 6 7 8 9 10

y

x

R

G

Which of the following equations best represents the perpendicular bisector of RG?

1 A y x= − 23

C y x= −3 10

1 B y x= −3 8+ D y x= − + 1 3


Page 9

10 Half of an international basketball court is shown below. The shaded region is composed of an isosceles trapezoid and a semicircle. The diameter of the semicircle is 3.6 meters.

6.0 m

5.8 m

3.6 m

If 1 meter is approximately equal to 3.28 feet, which of the following is closest to the area of the shaded region in square feet?

A 32.9 ftầ C 354 ftầ

B 409 ftầ D 108 ftầ

11 In quadrilateral ABCD , AB � CD, ,∠ A ≅ ∠B and AB CD. Which of the following statements is a reasonable conclusion?

A m A∠ ≅ m∠C

B Quadrilateral ABCD is a rectangle.

C Quadrilateral ABCD is an isosceles trapezoid.

D AD � BC


Page 10

12 Triangles RST and VSU are shown below.

R

S

T

V

U

R V≅ , and RT ≅∠ ∠ VU. Which additional condition is sufficient to prove that RS ≅ SV?

A TS ≅ SU

B VS ⊥ RU

C RS ≅ SU

D ∠VUS ≅ ∠RST


Page 11

13 Triangle RST was dilated to create triangle R ′ S ′ T ′, as shown on the coordinate grid below.

−5

−4

−6

−7

−8

−9

−10

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

−1 1−2−3−4−5−6−7−8−9−10 2 3 4 5 6 7 8 9 10

y

x

R

R'

S S'

T'

T

Which statement appears to be true?

A The center of dilation used to create Ј R ′ S ′ T′ w as (−10, 8).

B Ј RST and Ј R ′ S ′ T′ are congruent.

C The scale factor used to create Ј R′ S ′ T ′ is 2.5.

D Ј RST was reduced in size to create Ј R ′ S ′ T ′.


Page 12

14 A tree’s shadow is 4.8 m long on level ground, as shown in the diagram.

4.8 m

50°

The angle of elevation from the tip of the shadow to the sun is 50°. Based on this information, which of the following is closest to the height of the tree?

A 3.6 m

B 5.7 m

C 3.1 m

D 7.5 m


Page 13

15 A company packages their product in two sizes of cylinders. Each dimension of the larger cylinder is twice the size of the corresponding dimension of the smaller cylinder.

d

h

2d

2h

Based on this information, which of the following statements is true?

A The volume of the larger cylinder is 2 times the volume of the smaller cylinder.

B The volume of the larger cylinder is 4 times the volume of the smaller cylinder.

C The volume of the larger cylinder is 8 times the volume of the smaller cylinder.

D The volume of the larger cylinder is 6 times the volume of the smaller cylinder.

STAAR Geometry 2011 ReleaseAnswer Key

Item Reporting Readiness or Content Student Correct Number Category Supporting Expectation Answer

1 1 Readiness G.2(B) D

2 1 Readiness G.3(C) D

3 1 Supporting G.3(E) D

4 2 Readiness G.5(A) B

5 2 Supporting G.5(C) A

6 2 Readiness G.5(D) D

7 3 Supporting G.6(A) D

8 3 Supporting G.6(C) C

9 3 Readiness G.7(B) A

10 4 Supporting G.8(F) C

11 4 Supporting G.9(B) C

12 4 Readiness G.10(B) B

13 5 Supporting G.11(A) C

14 5 Readiness G.11(C) B

15 5 Readiness G.11(D) C

For more information about the new STAAR assessments, go to www.tea.state.tx.us/student.assessment/staar/.

Page 14

geometry houston isd planning guide 4 six-weeks€¦ · students concretely determined the area and...

Documents