Unit 11: Geometry
Designer/School
Friedline / AFBMS; modified by Chibbaro
IA Cycle/Date
IA4, Unit 11, LP8
Standards Where does this lesson lie in the sequence?
6.G.1
Students will decompose a trapezoid into two triangles and find the combined area of the two triangles. Students should
notice that they can apply the distributive property to represent A = ½hb1 + ½hb2 as A = ½h(b1 + b2)
Aim Student friendly, bite-sized objective
SWBAT derive and apply the area formula for a trapezoid.
Assessment and Criteria for Success How will scholars show what they know and can do? Include exemplar responses.
1.) Determine the area of following trapezoids:
A= ½ h (b1 + b2)
A= ½ (6) (8 + 13)
A = 3 (21)
A = 63sqin.
A= ½ h (b1 + b2)
A = ½ (9) (5 + 8)
A = 4.5 (13)
A = 58.5sqcm.
2.) The front of the roof of a house is shaped like a trapezoid. One base of the trapezoid
measures 35 feet and the other base measures 18 feet. The perpendicular distance
between the bases is 7 feet.
Part A: If a couple wants to paint the front of the roof, how many square feet of paint should
they buy?
A = ½ h (b1 + b2)
A = ½ (7) (35 + 18)
A = 3.5 (53)
A = 185.5sqft.
Part B: If each can of paint covers 50 sq. ft., how many cans does the couple need to buy?
If each can holds enough paint to cover 50 sq. ft., then four cans of point would be needed
to cover the entire front for the roof. (Scholars could show this by either making a list or by
dividing 185.5 ÷ 50 = 3.71 which scholars must round up to 4 full cans).
Enduring
Understandings
(big ideas)


Area measures the number of square units that cover the inside space of a 2dimensional figure.
Formulas for calculating the area of shapes can be derived from the composition or
decomposition into shapes for which an area formula is already known.
Vocabulary,
WHAT key
points
HOW key
points
(include
variations, skills,
and processes)
Common
Errors,
Misconceptions
Connected
Standards for
Mathematical
Practice (SMP)
Area – measures the number of square units inside a 2-dimensional figure
Polygon: A closed, 2-dimensional shape composed of straight lines.
Parallelogram: A two-dimensional shape that has four sides and two pairs of parallel
sides.
 Trapezoid: A quadrilateral with one set of parallel lines.
 Compose: To create a shape with a readily determinable area using two or more other
shapes.
 Decompose: To split a shape up into readily determinable areas while maintaining the
original area of the shape
 Dimension: Measurement in length, width and thickness (i.e. length, width, base, height)
 Right Triangle: A triangle with one right angle (90 degrees)
 Obtuse Triangle: Obtuse triangles have one angle that is greater than 90 degrees.
 Acute Triangle: Acute triangles have NO angles greater than or equal to 90 degrees -all their angles are less than 90 degrees
 Rectangle: A two-dimensional shape that has four right angles and opposite side lengths
that are equal
 Base: the lower part or bottom of a triangle
 Height- The vertical distance from the “top” of a shape to the “bottom” of a shape
(varies by shape).
 The Height of a trapezoid is the perpendicular distance between two parallel bases.
 Formula: a fixed mathematical expression.
 Given the dimensions, the formula can be used to find the area of all trapezoids,
regardless of the size of the trapezoid. Students should understand that you can apply
this formula to all trapezoids because you can create two triangles with different bases
but the same height. The area of the trapezoid will be the sum of these two triangles.
Because each triangle has the same height and a factor of ½ in the area formula, the
height and ½ can be factored out from the bases giving the final condensed area
formula of a = ½ h (b1 + b2). (Students can also find the area without the formula- by
finding the area of two triangles and combining.
Students will use what they learned about calculating the area of a triangle during a previous
lesson to derive the formula for the area of trapezoids.



Students will substitute the dimensions of the trapezoid into their derived formula.
* When the dimensions are not already labeled, student will need to measure the height by
finding the perpendicular distance between the two parallel bases of the trapezoid, counting in
units. When not provided with a visual, students should draw a trapezoid and label the
dimensions.
Scholars might not determine the correct height for obtuse triangles when deriving the area of
a trapezoid.
Scholars might not factor the height and factor of ½ correctly to derive the condensed area
formula.
Scholars might not correctly identify the height of a right trapezoid.
Scholars might use the length of the diagonal when substituting for the height.
SMP 1- Students will determine whether or not the area they determined for compound shapes
and triangles by reasoning about the relative size of another polygon to which it closely
resembles and for which the area can be calculated (i.e. triangles for trapezoids)
SMP 4- Students will solve real world problems by applying concepts of area and perimeter and
representing the context with formulas.
SMP5- Students will use pencil and paper to calculate area and perimeter of shapes.
Students will use grid paper and grid transparencies to determine the number of square units
that comprise the area of a shape.
SMP 6 - Students will ALWAYS list appropriate units with quantities.
Students will calculate accurately with decimal, fraction and whole number dimensions
Agenda
 Intro
 INM
 GP
 Discussion/Final CFU
 IP
 Closing/ET
Materials
Lesson Opening (Intro)
State the aim. Connect it to their lives and prior knowledge. Discuss how they will be working on it today (sometimes the discussion is here).
Get them involved through questioning.
Approx. Time Allotted: 5 minutes
Frame: In our past lessons, we have been deriving area formulas and applying them to find new area formulas
of different shapes. We have determined the area of triangles to be ½ bh in the past few lessons and we are
going to use that today to find the area of another polygon, the trapezoid.”
Think About It: Scholars work with partners to try and determine a strategy to find the area of a trapezoid.
 Some students might decompose into a rectangle and a triangle. Have them explain how they
determined the area of each shape by applying the correct formula and then adding them
together. Why does this work to find the area? Because when we decompose shapes, find their
separate areas, and then add them together, we get the total area of the shape.
 Look for a student who broke it up into 2 triangles. Ask them to explain the same exact thing as
the student who decomposed into a rectangle and a triangle.
 Given what this student just did, write an expression/formula/or equation to determine the area.
 Stretch it- what if you wanted to write an expression, formula, or equation with variables instead
of numbers? (1/2 bh) + (1/2 bh)
 Are the bases the same? No What could we do about that? Choose different variable.
Frame the rest of the lesson, we are going to use our knowledge of areas of triangles and the distributive
property from earlier this year to determine a more condensed formula for the area of a trapezoid and then
use this new formula to efficiently find the areas of multiple types of trapezoids.
Intro to New Material
Depending on lesson, may include teacher modeling and/or think aloud. During this time, the criteria for success should be
communicated, and any new vocabulary should be explicitly taught. Plan problems and questions to uncover key points and address
common errors and misconceptions.
Approx. Time Allotted: 15 min
Before we can condense our formula, we need to label some important parts of a trapezoid. A trapezoid is a
special type of quadrilateral that has one set of parallel sides. The set of parallel sides on a trapezoid are called
the bases and every trapezoid has exactly two bases. We label these are b1 and b2. The numbers on the
variables are called subscripts and don’t have any numerical meaning, it just means the first base and the
second base.
The next important feature on trapezoids is the perpendicular distance between the two bases. This distance is
the height of the trapezoid. This height is important because it would be the same distance if we cut the
trapezoid into two triangles. Be careful, the height of a trapezoid is not the length of the diagonal sides.
As we saw in the Think About It, a trapezoid can be cut into two triangles with the same height but different
base lengths. The sum of these triangles is the area of the trapezoid and is represented below (shown in the
classwork). Remember back to when we were applying the distributive property and work with your partner to
try and rewrite this equation in a more condensed form.
Scholars work in pairs for 60 seconds. SMS: area = ½ (hb1 + hb2) or area = h( ½b1 + ½b2). Present these to
scholars and ask “Is there a better or even more condensed way to represent this?” “What terms do both of
the separate areas have in common?” Guide scholars with these questions until they arrive at the formal
equation and post this in the key point / formula box.
Great! Now that we have a formal equation for the area of trapezoid, let’s apply it to our Think About It
trapezoid to ensure that it works. The first thing we have to do is define the variables.
What is the height of the triangle? TT – SMS: 12 inches
What is the length of base 1? – TT – Interesting, I heard some people say 10 inches and some people said 14
inches. Who is correct? Looking at our formula, does it matter? Why or why not? SMS: It doesn’t matter which
base we label as base 1 and base 2 because their areas are still just being added together. In the area
formula, we can apply the commutative property to change the order of the bases to show that the order
doesn’t matter.
Now that we have the bases and the height defined, we can substitute this into the formula and evaluate.
(Evaluate the equation and find the final area). Does this equation work? Why does this work? SMS: The
formula for the area of a trapezoid works for all trapezoids because it is the condensed formula for the area of
two triangles which all trapezoids can be cut into.
Let’s look at example 2. What shape do we have? A trapezoid. How do you know? It looks different. There is
one set of parallel sides which is the definition of a trapezoid.”
What are we looking for here? We are looking for the area because it is asking us for the number square inches
of paint needed to cover the painting.
What do we need to do in order to find the area of this trapezoid? We need to define the bases and the
height first to substitute into our area formula. We know that the bases are the one set of parallel sides of the
trapezoid. What are the bases? SMS: “23 inches and 14 inches.” The height is the perpendicular distance
between the bases. What is the height? Why isn’t the height 19 inches? SMS: “19 inches is the diagonal
distance between the two bases and isn’t the perpendicular distance. A right angle needs to be present
between the two bases for it to be considered the height.” Now that we have the bases and height defined,
we can substitute and evaluate the area. (Work through the equation with scholars).
Have students work with partners on the next two application problems where they first have to divide the area
by 5 and then multiply by 12.5
Guided Practice
Could be workshop, whiteboards, partner work. Plan problems and questions to uncover key points and address common errors and
misconceptions. How will you gradually decline scaffolding? How will you check for understanding? Differentiate up and down.
Approx. Time Allotted: 10 min.
Students practice in partners
 Assessment components:
 Are scholars accurately finding the area of the given trapezoid?
 Are scholars correctly identifying and labeling the dimensions of the trapezoid?
 Are scholars correctly showing all steps?
 Are scholars correctly substituting the correct values into the formula?
 Are students correctly applying the concept of area to solve real world problems?
 Are scholars using the correct units?
 Check for Understanding: Ask a student… “How did you know ______?”
 How did you know to use this formula for area?
 How did you know to substitute these values into the formula?
 How did you know this is the correct area?
 How did you know which units to use?
 Intervention: Just with the given dimensions, have students substitute values into the formula and solve.
 Extension: Students work on solving problems that involve compound shapes.
Discussion/Final CFU
Discuss relevant student work samples to push student thinking to the next level. What questions will you ask to get students to articulate
the big ideas before moving on? Final, whole-class CFU.
Approx. Time Allotted: 5 min.
Show student work on doc camera to drive discussion. Review #3
Whole Class CFU: S should solve the whole class CFU. Scholars “shake it, shake it, show it” for a visual
100% CFU.
Independent Practice


Independent daily practice. Which problems will you review in IP? Where do you anticipate missteps? How/when do you plan
opportunities for error analysis, reasoning, or justification?
Approx. Time Allotted: 20 minutes
 Students work independently on IP.
Closing and Exit Ticket
How can we get scholars to summarize what was learned today and connect back to the aim? Daily exit ticket.
Approx. Time Allotted: 5 minutes


Discuss #7, 9, 11
Students work independently on ET.
Name:____________________________
Date:________________________
Area of Trapezoids
Think About It: Given what you already know about decomposition and finding the area
of shapes, how could you find the area of the shape below? Show your work.
12 feet
12 feet
10 feet
16 feet
Write an expression, formula, or an equation that you could use to find the total area of the
trapezoid.
Parts of a Trapezoid:
Area of a Trapezoid: As shown in the Think About It, the area of a trapezoid can be
decomposed into the sum of two triangles with different bases and the same height.
Area = ½ hb1 + ½ hb2
Finish the Formula: Work with a partner to condense this formula by using the distributive
property in reverse, also known as factoring.
Key Point:
The area formula for a trapezoid can be applied to all trapezoids if their bases and
perpendicular distance between them (height) are known.
Formula:
Example 1)
Find the area of the trapezoid in the Think About It problem applying the new formula:
Example 2) Caleb is painting a picture that has the dimensions of the shape below. How
many square inches of paint will he need to cover the entire canvas?
15 inches
14 inches
23 inches
19 inches
One small bucket of paint covers 5 square inches. How many gallons will he need to cover
the entire canvas?
__________________________
If each can costs $12.50, how much will he spend in total for the paint?
Answer: _________________________________
Partner Practice:
1. Determine the area of the trapezoid.
16 cm
9 cm
7 cm
10 cm
8 cm
Answer: ______________________________________
2.) Mr. Rogers wants to retile the front of his roof. The roof of a house is 8 feet tall. If each tile
is 1 foot by 1 foot, how many tiles does he need to buy in order to cover the entire front of
the roof?
14 feet
20 feet
Answer: _____________________________________________
3.) Error Analysis: Mr. Friedline made a mistake while solving for the area of the trapezoid
and got an answer of 144 sq. in. Find the correct area and describe his error and what he
should have done with two complete sentences.
16 in.
6 in.
8 in.
20 in.
Correct Area: _____________________
Written Response: ________________________________________________________________________
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Final CFU
Uh Oh . . . The following trapezoid has so many dimensions labeled! Be careful and
determine the correct area of the trapezoid.
a.) 1,008 sqcm.
22 cm
b.) 616 sqcm.
18 cm
c.) 560 sqcm.
d.) 504 sqcm.
32 cm
24 cm
20 cm
Finished Early? Pick one of the incorrect answers above and describe the error the scholar
would have had to make to get it.
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Independent Practice
1. What is the area of this trapezoid?
20 feet
14 feet
28 feet
Answer: _______________________________
2.) Cut the trapezoid above into two triangles (refer back to the Think About It problem) and
represent them below with the proper dimensions. Use your diagram to prove that the area
of a trapezoid is the sum of two triangles with the same height.
Triangle 1
Triangle 2
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3.) Which trapezoids have an area of 60 inches2?
a. Base1= 10 in, Base2 = 5 in, Height = 4 in
b. Base1 = 8 in, Base2 = 2 in, Height = 12 in
c. A trapezoid with a height of 6 inches and the sum of its bases equal to 20 inches
d. A trapezoid with the sum of the bases equal to 15 inches and a diagonal length of
8 inches.
Explain how you determined your answer.
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4.) What is the area of the trapezoid below?
16.5 cm
11.6 cm
13.7 cm
25.3 cm
Area: _________________________________
5.) Jabril wants to create a planter on top of AFEMS for a science project. He has 100 sqft of
soil to put into his planter. If he builds the planter in the shape of a trapezoid with bases of 10
feet and 12 feet with a height of 21 feet, will he have enough soil to cover the bottom of the
planter? How much soil will be left over or needed to fill the planter?
Is there enough? _______________________
How much left over, or needed? ______________________
6.) Joseph wants to paint the wall of his bedroom green. His wall is shaped like a trapezoid.
The base along the floor is 20 feet long. The base along the ceiling is 12 feet long and the
distance from the floor to the ceiling is 9 feet.
Part A: How many square feet will he paint?
Answer: __________________
Part B: If one can of paint will cover 75 square feet, how many full cans of paint does he
need to buy?
Answer: __________________
Part C: The paint store is having a sale, 30% off all paint. If paint costs $4.50 per can, how
much will he spend on paint?
Answer: ___________________
7.) Determine the area of trapezoid EBCF
A
4 cm
B
6 cm
C
9 cm
D
F
9cm
G
8 cm
E
10cm
Answer: ____________________________________
8.) How much bigger is trapezoid EBDG than trapezoid EBCF?
Answer: ___________________________________
9.) A trapezoid has a longer base that is three times as long as the shorter base and the
height is twice the shorter base.
Step A: Sketch a diagram of what this trapezoid could look like.
Step B: What is the area of the trapezoid if the length of the longer side is 18 feet?
Answer: ______________________________
10.) Determine the area of the shaded region. The length of AB is the same as CD and is 10
feet. The length of AD is 28 feet:
A
B
C
10 feet
14 feet
Answer: ________________________________
D
11.) Mr. Roble find the area of trapezoid to be 120 sqcm. The problem is, he forgot what the
height was. Determine the height of the trapezoid.
a.) 4 cm
10 cm
b.) 8 cm
c.) 6 cm
d.) 12 cm
12 cm
18 cm
10 cm
12.) Explain how you determined your answer to question 9. How else could you have
determined your answer?
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Name:____________________________
Date:________________________
Exit Ticket
Areas of Trapezoids
1.) Determine the area of following trapezoids:
Answer: _______________________
Answer: ___________________________
2.) The front of the roof of a house is shaped like a trapezoid. The front of the roof of a house
is shaped like a trapezoid. One base of the trapezoid measures 35 feet and the other base
measures 18 feet. The perpendicular distance between the bases is 7 feet.
Part A: If a couple wants to paint the front of the roof, how many square feet of paint should
they buy?
Answer: _________________________
Part B: If each can of paint covers 50sqft, how many cans does the couple need to buy?
Answer: _______________________________