understanding by design - unit plan for polygons and quadrilaterals

Design Topic Polygons & Quadrilaterals Subject(s) Geometry Grade(s) 9 – 11 Designer(s) Marianne McFadden

Source: Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)

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STAGE 1 – DESIRED RESULTS

UNIT TITLE: POLYGONS AND QUADRILATERALS (approximately eleven to twelve days – unit duration)

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Distinguish between and categorize various types of polygons

Apply the properties of polygons Compute the measures of interior and exterior

angles of convex polygons Determine the relationships among sides, angles

and diagonals of parallelograms Define and classify special types of parallelograms Verify and use properties of parallelograms, special

parallelograms, trapezoids, and kites Apply mathematical knowledge, skill, and reasoning

to solve real-world problems that can be represented as polygons and then analyzed through the use of corresponding geometric properties

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Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively. Construct viable arguments and critique the

reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and make sense of regularity in repeated

reasoning

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C.C.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures ASSESSMENT ANCHOR: G.1.2 properties of polygons and polyhedra

C.C.2.3.HS.A.14 Apply geometric concepts to model and solve real-world problems ASSESSMENT ANCHOR: G.2.2 measurements of two-dimensional shapes and figures

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■ …the sum of the angle measures of a polygon depends on the number of sides the polygon has; the sum of the measures of the exterior angles in a polygon does not depend on its number of sides.

■ …all parallelograms have special properties regarding their sides, angles, and diagonals, and a quadrilateral with these certain properties can be proven to be a parallelogram.

■ …the special parallelograms – rhombus, rectangle, and square – have basic properties regarding their sides, angles, and diagonals that help identify them.

■ …the angles, sides, and diagonals of a each type of quadrilateral (which includes parallelograms, trapezoids, and kites) have certain properties that differ from all the other polygon categories within

the quadrilateral classifications.

■ …the quantitative data that the students collect, organize, and display in an organized chart will be evaluated in order to draw conclusions regarding the relationship between number of sides and angle sum (interior and exterior) in polygons.

■ …mathematical concepts and relationships can be represented symbolically, and this expression can be utilized in solving real-world problems that are modeled mathematically, with aid of a diagram or a polygonal representation that corresponds to this unit.

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● How do the different types of polygons compare with each other when identifying the properties of their sides and angles?

● How would you describe the relationship between number of sides and interior angle sum; how is this relationship used to develop the expression (formula) for computing this sum?

● How can you describe the process for computing the sum of the measures of the exterior angles of different polygons? How does the relationship between number of sides and exterior angle sum differ from the relationship between number of sides and interior angle sum?

● What are the three main categories of quadrilaterals and what property determines which category the quadrilateral belongs to?

● How can special quadrilaterals be identified?

● How can the properties of polygons and certain quadrilaterals be utilized in finding missing lengths and angle measures?



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Students will know: ddeeffiinniittiioonnss ooff kkeeyy tteerrmmss: polygon, diagonal, opposite sides,

opposite angles, convex polygon, regular polygon, triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, n-gon, parallelogram, rectangle, rhombus, square, trapezoid, isosceles trapezoid, kite the types of polygons and the properties of convex polygons the formula to compute the sum of the interior angles of a

convex polygon the constant value which represents the sum of the exterior

angles of a convex polygon the three types of quadrilaterals the properties of parallelograms, special parallelograms,

trapezoids, isosceles trapezoids, and kites

Students will be able to: determine and clarify the difference between regular

and non-regular polygons explain the difference between convex and non-

convex polygons construct all possible diagonals from a single vertex

in a polygon and describe how the number of ∆s formed within each type of polygon investigated relates to the number of sides of the polygon

describe the effect the total number of sides in a polygon has on the sum of the interior angles and the sum of the exterior angles

formulate a rule to find the interior angle sum of any

polygon by using the number of ∆s formed compare and contrast the properties of the various

types and categories of quadrilaterals apply the properties of specific quadrilaterals in

solving for missing side, diagonal, or angle values reason abstractly and quantitatively when drawing

conclusions with real-world polygonal applications construct viable arguments and critique the

reasoning of others when analyzing polygonal investigations in attempting to develop a rule that relates number of sides of the polygon to the problem being studied

observe and make use of the structure portrayed in simple polygon investigations in order to apply it to real-life problems that utilize polygon properties

STAGE 2 – ASSESSMENT EVIDENCE

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Students will apply their learning to new and authentic situations as means of assessing their understanding and ability to transfer their learning by: interpreting results from the diagonals investigation and the handshakes problem in order to draw conclusions in determining relationships between number of sides, total diagonals, and interior angle sum explain the relationships (diagonals/handshake) verbally (in a small group activity) and support the explanations by sketching a sample polygon and applying the logic of the explanation to the sample polygon apply the formulas developed to several problems presented on classwork examples, worksheets, homework exercises, quizzes, and tests graphic organizer: display a schematic of how all quadrilaterals are related; categorize according to sides property Venn Diagram: display a compare/contrast among properties of special parallelograms (rectangle, rhombus, square); discuss visual with peers self-assess (formative) through classwork exercises on properties of special quadrilaterals – completed on mini-whiteboards for immediate evaluation

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Students will apply their learning as means of assessing their understanding and ability to transfer their learning by these other methods of evidence: FORMATIVE: teacher observation– mini-whiteboards utilized for quick check for understanding in applying formulas learned student work samples – students complete practice worksheets and their work on sample polygon (diagonals) and simple handshake scenario (three, triangular arrangement) in order to show they can demonstrate concepts as applied to simple example assignments from the text, sections 3-5, 5-1, 5-4, 5-5 (Geometry, Jurgensen, 2000) journal entries – requested entries – students write detailed explanations of relationships studied in the unit journal entries – requested entries (self-assessing) – students express what concepts they need more work on and/or what they don’t understand Khan Academy and Compass Learning – students complete additional lessons (videos and practice problems) to strengthen skills in applying formulas in the unit (geared to LS and/or ELL students, and those who need extra help) SUMMATIVE: quizzes – short, single skill tested (a. quiz taken on finding interior/exterior angle sum and missing angle



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measures in a polygon; b. quiz taken on properties of special parallelograms; c. quiz taken on properties of trapezoids and kites, three total) tests – full-period length (a. test taken on properties of polygons; b. test taken on properties of quadrilaterals and the special types) conference – students will participate in an one-on- one conference with the teacher and present his/her sample polygon and sample quadrilateral in fully explaining relationships and formulas developed in class

STAGE 3 – LEARNING PLAN

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WW:: How will you help your students to know where they are headed, why they are going there, and what ways they will be evaluated along the way?

Students will be given a learning plan including all topics to be covered in the unit as well as an outline of the exercises and the worksheet packets that need to be completed. Activities (main two are: diagonals/interior angle sum and handshakes problem) will be explained as being real-world relevant problems.

HH:: How will you hook and hold students’ interest and enthusiasm through thought-provoking experiences at the beginning of each instructional episode?

Four hooks:

general polygon hunt (at the very beginning of the unit) – students draw as many polygons they can think of that they encounter regularly (eg.: STOP sign, home plate, one-way arrow)

diagonals/angle sum activity (as an introduction to polygons) – students draw all possible diagonals from a single vertex in several polygons; formulate rule to determine interior angle sum

handshakes problem (to be presented when finishing polygons) – simulate small groups in investigating how total diagonals relates to total handshakes occurring with different size groups; generalize to much larger size groups

kites interactive computer exercise (to be presented when beginning kites) – students invited to drag a vertex of a kite and record changing measurements of sides and angles; conclusions to focus on unique properties of kites (mathisfun.com)

EE:: What experiences will you provide to help students make their understandings real and equip all learners for success throughout your course or unit?

BrainPop – video explaining properties of polygons Khan Academy – video demonstrating relationship between diagonals and interior angle sum; second

video demonstrating finding exterior angle sum o extra work required on Khan and/or Compass Learning for LS, ELL, and those who need extra

assistance – additional lessons assigned graphic organizer – schematic showing relationship among all types of quadrilaterals studied advanced organizer – chart summarizing all properties of all quadrilaterals studied demonstrate/model – teacher-led; teacher models one or two diagonal drawings, then student models

a few more before class finishes the activity independently lecture (notes included in packet) – notes on properties of polygons (one packet, with worked

examples) and another packet on quadrilaterals (with worked examples) guided practice – sample problems within packets; co-teacher or classroom aide to help with problems

in the packets RR::

How will you cause students to reflect, revisit, revise, and rethink? journal entries – students to summarize steps discussed in class in formulating rule for angle sum error analysis – students given two differing attempts at a problem – volunteers offer their analysis of



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steps taken (which solution is incorrect and why) journal entries – students to express what concepts they need to work on and what needs to be re-

taught or re-explained journal entry/student demonstration – students to explain errors made in computing value in

problem(s) regarding angle sum and/or applying properties of polygons or quads (teacher to provide feedback in student journals)

EE:: How will students express their understandings and engage in meaningful self-evaluation?

whiteboard exercises small group activities (worksheet exercises, diagonals and handshakes activities) graphic organizer and advanced organizer (diagram, chart) Venn Diagram (special parallelograms) (students to work in small groups on graphic and advanced organizers and Venn Diagram – collaborate as they complete the exercises; check and question each other, then grade each others’ work)

TT:: How will you tailor (differentiate) your instruction to address the unique strengths and needs of every learner?

extra basic practice exercises through Khan Academy, Compass Learning and mathworksheets4kids.com for students who need extra support – small groups to work with aide while rest of class completes more challenging worksheet exercises

selected quiz/test questions modified or removed for those who can only handle basic applications those who need extra help encouraged to visit during study hall/flex time for assistance and

opportunities to complete work with assistance OO::

How will you organize learning experiences so that students move from teacher-guided and concrete activities to independent applications that emphasize growing conceptual understandings as opposed to superficial coverage?

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► polygon types – class discussion on polygons often seen in everyday life (eg: stop sign, one-way arrow, yield sign,…)

o polygon hunt: http://www.old.mathleague.com/help/geometry/polygons.htm#polygon o diagonals investigation/formula development:

http://www.virtualnerd.com/geometry/quadrilaterals/polygons/polygon-interior-angle-sum-examples

o journal reflection (see attachment A) - math.kennesaw.edu/.../Interior%20Angles%20... o BrainPop and Khan Academy videos:

http://www.brainpop.com/math/geometryandmeasurement/polygons/preview.weml, and: http://www.bing.com/videos/search?q=khan+academy+angles+of+a+polygon&qpvt=khan+academy+angles+of+a+polygon&FORM=VDRE#view=detail&mid=CB5108ED23EF0133C8C5CB5108ED23EF0133C8C5

o notes/guided practice/classwork examples (see attachment B) o quiz – polygons (definitions, interior angle sum – Jurgensen text, sec. 3-5: class exercises;

attachment AA is the differentiated quiz version) o journal reflections (see attachment BB – methods 1 & 2, p.315); revisit and re-teach

(if necessary) o test – polygons (interior/exterior angle sum, missing values – sides and angles – see

attachment D)

► quadrilaterals o notes/guided practice/classwork examples (see attachment E) o whiteboard practice (textbook: Jurgensen, section 5.1, class exercises and written exercises) o graphic organizer/advanced organizer/Venn Diagram (see attachments F and G; create Venn

from info in attachment F in journal entry below) o journal entry (create Venn, then continue with attachment BB – methods 3 & 4, p. 316) o quiz – quadrilaterals (properties – Jurgensen text, sec. 5-1 and 5-4: written exercises) o handshakes problem/investigation and worksheet examples– small group:

http://mason.gmu.edu/~jsuh4/impact/Handshake_Problem%20teaching.pdf o kite demo; group discussion:

http://math.kendallhunt.com/x19428.html and http://www.mathsisfun.com/geometry/kite.html o whiteboard practice – trapezoids and kites (see attachment L) - http:// web.bend.k12.or.us o quiz – quadrilaterals (applications – see attachment I; attachment AAA is the differentiated quiz

version) o worksheet problems (algebraic applications in quads – see attachment J)



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o Khan Academy video – special parallelograms – use primarily for LS and ELL and students who need extra practice, use of differentiated instruction method: https://www.khanacademy.org/math/geometry/quadrilaterals-and-polygons/quadrilaterals/v/quadrilateral-properties

o whiteboard practice/journal entry (see attachment BB – class discussion question, p. 317 and extending the problem, pgs. 318, 319)

o test (quadrilaterals – properties, applications, real-world applications – see attachment K)

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Worksheets, quizzes, tests, and graphic organizers are attached.

understanding by design - unit plan for polygons and quadrilaterals

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