# Areas of Quadrilaterals (Lesson 5.5)

###### Unit 1: Reasoning in Geometry

**Day 1:**Creating Definitions**Day 2:**Inductive Reasoning**Day 3:**Conditional Statements**Day 4:**Quiz 1.1 to 1.3**Day 5:**What is Deductive Reasoning?**Day 6:**Using Deductive Reasoning**Day 7:**Visual Reasoning**Day 8:**Unit 1 Review**Day 9:**Unit 1 Test

###### Unit 2: Building Blocks of Geometry

**Day 1:**Points, Lines, Segments, and Rays**Day 2:**Coordinate Connection: Midpoint**Day 3:**Naming and Classifying Angles**Day 4:**Vertical Angles and Linear Pairs**Day 5:**Quiz 2.1 to 2.4**Day 6:**Angles on Parallel Lines**Day 7:**Coordinate Connection: Parallel vs. Perpendicular**Day 8:**Coordinate Connection: Parallel vs. Perpendicular**Day 9:**Quiz 2.5 to 2.6**Day 10:**Unit 2 Review**Day 11:**Unit 2 Test

###### Unit 3: Congruence Transformations

**Day 1:**Introduction to Transformations**Day 2:**Translations**Day 3:**Reflections**Day 4:**Rotations**Day 5:**Quiz 3.1 to 3.4**Day 6:**Compositions of Transformations**Day 7:**Compositions of Transformations**Day 8:**Definition of Congruence**Day 9:**Coordinate Connection: Transformations of Equations**Day 10:**Quiz 3.5 to 3.7**Day 11:**Unit 3 Review**Day 12:**Unit 3 Test

###### Unit 4: Triangles and Proof

**Day 1:**What Makes a Triangle?**Day 2:**Triangle Properties**Day 3:**Proving the Exterior Angle Conjecture**Day 4:**Angle Side Relationships in Triangles**Day 5:**Right Triangles & Pythagorean Theorem**Day 6:**Coordinate Connection: Distance**Day 7:**Review 4.1-4.6**Day 8:**Quiz 4.1to 4.6**Day 9:**Establishing Congruent Parts in Triangles**Day 10:**Triangle Congruence Shortcuts**Day 11:**More Triangle Congruence Shortcuts**Day 12:**More Triangle Congruence Shortcuts**Day 13:**Triangle Congruence Proofs**Day 14:**Triangle Congruence Proofs**Day 15:**Quiz 4.7 to 4.10**Day 16:**Unit 4 Review**Day 17:**Unit 4 Test

###### Unit 5: Quadrilaterals and Other Polygons

**Day 1:**Quadrilateral Hierarchy**Day 2:**Proving Parallelogram Properties**Day 3:**Properties of Special Parallelograms**Day 4:**Coordinate Connection: Quadrilaterals on the Plane**Day 5:**Review 5.1-5.4**Day 6:**Quiz 5.1 to 5.4**Day 7:**Areas of Quadrilaterals**Day 8:**Polygon Interior and Exterior Angle Sums**Day 9:**Regular Polygons and their Areas**Day 10:**Quiz 5.5 to 5.7**Day 11:**Unit 5 Review**Day 12:**Unit 5 Test

###### Unit 6: Similarity

**Day 1:**Dilations, Scale Factor, and Similarity**Day 2:**Coordinate Connection: Dilations on the Plane**Day 3:**Proving Similar Figures**Day 4:**Quiz 6.1 to 6.3**Day 5:**Triangle Similarity Shortcuts**Day 6:**Proportional Segments between Parallel Lines**Day 7:**Area and Perimeter of Similar Figures**Day 8:**Quiz 6.4 to 6.6**Day 9:**Unit 6 Review**Day 10:**Unit 6 Test

###### Unit 7: Special Right Triangles & Trigonometry

**Day 1:**45˚, 45˚, 90˚ Triangles**Day 2:**30˚, 60˚, 90˚ Triangles**Day 3:**Trigonometric Ratios**Day 4:**Using Trig Ratios to Solve for Missing Sides**Day 5:**Review 7.1-7.4**Day 6:**Quiz 7.1 to 7.4**Day 7:**Inverse Trig Ratios**Day 8:**Applications of Trigonometry**Day 9:**Quiz 7.5 to 7.6**Day 10:**Unit 7 Review**Day 11:**Unit 7 Test

###### Unit 8: Circles

**Day 1:**Coordinate Connection: Equation of a Circle**Day 2:**Circle Vocabulary**Day 3:**Tangents to Circles**Day 4:**Chords and Arcs**Day 5:**Perpendicular Bisectors of Chords**Day 6:**Inscribed Angles and Quadrilaterals**Day 7:**Review 8.1-8.6**Day 8:**Quiz 8.1 to 8.6**Day 9:**Area and Circumference of a Circle**Day 10:**Area of a Sector**Day 11:**Arc Length**Day 12:**Quiz 8.7 to 8.9**Day 13:**Unit 8 Review**Day 14:**Unit 8 Test

###### Unit 9: Surface Area and Volume

**Day 1:**Introducing Volume with Prisms and Cylinders**Day 2:**Surface Area and Volume of Prisms and Cylinders**Day 3:**Volume of Pyramids and Cones**Day 4:**Surface Area of Pyramids and Cones**Day 5:**Review 9.1-9.4**Day 6:**Quiz 9.1 to 9.4**Day 7:**Volume of Spheres**Day 8:**Surface Area of Spheres**Day 9:**Problem Solving with Volume**Day 10:**Volume of Similar Solids**Day 11:**Quiz 9.5 to 9.8**Day 12:**Unit 9 Review**Day 13:**Unit 9 Test

###### Unit 10: Statistics and Probability

**Day 1:**Categorical Data and Displays**Day 2:**Measures of Center for Quantitative Data**Day 3:**Measures of Spread for Quantitative Data**Day 4:**Quiz Review (10.1 to 10.3)**Day 5:**Quiz 10.1 to 10.3**Day 6:**Scatterplots and Line of Best Fit**Day 7:**Predictions and Residuals**Day 8:**Models for Nonlinear Data**Day 9:**Quiz Review (10.4 to 10.6)**Day 10:**Quiz 10.4 to 10.6**Day 11:**Probability Models and Rules**Day 12:**Probability using Two-Way Tables**Day 13:**Probability using Tree Diagrams**Day 14:**Quiz Review (10.7 to 10.9)**Day 15:**Quiz 10.7 to 10.9**Day 16:**Random Sampling**Day 17:**Margin of Error**Day 18:**Observational Studies and Experiments**Day 19:**Random Sample and Random Assignment**Day 20:**Quiz Review (10.10 to 10.13)**Day 21:**Quiz 10.10 to 10.13

#### Learning Targets

Use the properties of special quadrilaterals to decompose shapes into rectangles and find their area.

Connect the dimensions of the original shape to the base and height of a rectangle to generate an area formula for parallelograms, trapezoids, and rhombi.

Tasks/Activity | Time |
---|---|

Activity | 25 minutes |

Debrief Activity with Margin Notes | 10 minutes |

QuickNotes | 5 minutes |

Check Your Understanding | 10 minutes |

#### Activity: Cut and Paste

#### Lesson Handouts

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#### Answer Key

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#### Homework

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#### Additional Media

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#### Experience First

Now that students have made important observations about the diagonals of several quadrilaterals, we’re ready to apply these properties to finding their areas. First, we remind students that finding an area is nothing more than determining, by counting, or some other strategy, how many unit squares make up an object. For rectangles and squares this is easy, but what if we don’t have full squares? Students investigate this first on a coordinate grid where they may combine partially shaded boxes to make full boxes or take off the triangular part and arrange it on the other side to make a rectangle. Then students start investigating other shapes.

Each group will need several copies of each shape. Since there are two per page, we recommend printing two copies of the whole document for each group so students have four copies of each shape. They will also need scissors.

Using the constraint that they can only make 2 cuts, we ask students to turn a parallelogram, trapezoid, and rhombus into a rectangle and apply the rectangle area formula of base x height. There are multiple ways students can go about doing this, and seeing the different strategies always proves to be very interesting to students. Below we’ve shown some different ways to rearrange a trapezoid into a rectangle. Though students may use different dimensions to determine the “base” and “height”, their final area will be the same. Are there other methods?

A key question to ask students as you monitor groups is why the triangular piece of the parallelogram, for example, fits perfectly onto the side of the parallelogram. It is important for students to make use of the parallel lines of the bases to identify the congruent corresponding angles. Later in question 6, they will see that if no sides are parallel, the path to making a rectangle is not as clear cut because congruent angles are not guaranteed.

Question 5 is aimed at getting students to see which dimensions on the original shape make up the “base” and the “height” of the new rectangle. We want students to look for the perpendicular measurements, which in the case of a rhombus, are the diagonals.

#### Formalize Later

While finding the area of 2-d shapes is generally its own chapter in a Geometry book, this is a skill students should already have familiarity with from the middle grades. Instead of covering each shape on a different day, we ask students to think creatively about how any trapezoid or parallelogram can be turned into a rectangle by making use of its properties. Since students are already comfortable with the idea of finding the area of a rectangle by multiplying the base by the height and that base and height are perpendicular, we want students to see that they can use this formula for other shapes as well, as long as they clearly define what the base is and what the height is. Furthermore, students review that decomposing and recomposing shapes does not affect area, and is a useful strategy for finding the area of non-rectangles.

The Check Your Understanding questions get at this further by asking students to come up with multiple ways to make certain areas or finding possible dimensions for the two bases given constraints. Instead of memorizing area formulas and applying them forwards and backwards, we want students to be able to make use of the properties of a shape to find its area. We will have an area and volume unit later in the year where we will focus more on real-world applications and reasoning with equations.