2. Do these problems on the back of your homework.
Warm Up
1) Greg and his sister, Laura, have a snack everyday after school. Yesterday, Greg put these cookies on the table and Laura began crying “that’s not fair! Mamaaaaa!”
To even out the piles, how many cookies
should each kid have? How would you do this?
2) José got back 3 quizzes today at school with the following grades: 80, 65, 95. What is the average grade for all 3 quizzes?
Laura Greg
(6 cookies) (10 cookies)
(4 min) 0 – 4
In-Class Notes
Students will work individually on solving the Warm Up question
Use the words “evening out”, “averaging”, “equalizing” here
Move to the more formal language of “finding the mean” with the next slide (unless a student mentions it now)
Preparation Notes
Students will work individually on solving the Warm Up question, which will involve finding the mean of two numbers in an interesting context (piles of cookies!). This will prepare them for (later in the lesson) finding the mean of a trapezoid’s two bases in order to calculate its area.
video
Agenda
Warm up 1
Warm up 2

3. Finally, Laura stopped crying!
Warm Up #1 .
To even out the piles, how many cookies should each kid have?
How would you do this?
Laura
Greg
8 cookies cookies
(1 min) 5 – 6
In-Class Notes
Did Greg even out the piles of cookies? (Yes)
What are we really doing when we’re “evening out” two piles of cookies? Or “averaging” any two numbers? (We’re finding the MEAN!)
How do you find the mean, again? (Add the numbers, then divide by the number of numbers)
If it feels like the majority of the class could use a quick review of “mean,” “mode,” etc. here are 2 links to websites with text, and a 3rd link to a TeacherTube video with a catchy (corny) song*
*If you cannot connect to the internet during class, or if TeacherTube is blocked at your school, you can download this video ahead of time (either from this site or using
Preparation Notes
Finally, Laura stopped crying!
Agenda
Scaffolding
Questions

4. Watch this video on eHow.com!
Forget how to find the Mean?
Watch this video on eHow.com!
Questions

5. 80 65 +95 240 ÷ 3 = 80 Average = Mean Warm Up #2
José got back 3 quizzes today at school with the following grades: 80, 65, 95. What is the average grade for all 3 quizzes?
Average = Mean
80% = B-
65% = D 80 65
+95
240
95% = A
(1 min) 7 – 8
In-Class Notes
Reiterate that “mean” means “average” and that to find the mean you need to find the sum of the numbers and divide by the number of numbers in the data set
So what do we need to do with Jose’s quiz scores? (Add them up to 240, then divide by 3 to get an average or mean of 80.)
÷ 3 = 80
Agenda
Scaffolding
Questions

6. Agenda:
OBJECTIVE: SWBAT prove whether the area formula for a parallelogram (A = b x h) can also be used to find the area of a trapezoid
Warm Up – Cookies!
2) Launch – Can we just do base x height ???
3) Explore – Partners
4) Summary – Yes or No?
(1 min) 8 – 9
In-Class Notes
Briefly show students the agenda for the day
Repeat, rephrase, reiterate the objective
Preparation Notes
It is helpful for students to know a brief outline of the lesson as well as when are the times they will be working alone, with partners or with the whole class.
5) Mini-Lesson
6) Practice – with Interleaving
7) Assessment
Agenda

7. How would you find the AREA of this 4-sided shape?
RECTANGLE
Launch
Quadrilateral … sided shape
What kind of quadrilateral is shown here?
h = 4
b = 10
(1 min) 9 – 10
In-Class Notes
The last two days we’ve been working with triangles, but today we’re shifting to quadrilaterals
What are quadrilaterals? (4-sided shapes)
Click to show the rectangle
More specifically, what kind of quadrilateral is this? (Click for the Answer -> Rectangle)
How would you find the area of this 4-sided shape?
A = base x height
Preparation Notes
This launch has been written with the intention of moving quickly. Later in the lesson – in the summary and mini-lesson – students are going to need to listen and concentrate. Therefore, this launch needs to be fast so students can start working as soon as possible and save their “listening energy” for later!
Calling sticks could be used to keep the launch moving.
The idea is that you will create momentum by showing a series of quadrilaterals, for which all of their areas could be found using A = base x height. By the time you get to the trapezoid, students will think (or almost get tricked into thinking!) that you can use the same formula. This will set the stage to “launch” them off.
How would you find the AREA of this 4-sided shape?
Area = base x height
Agenda

8. How would you find the AREA of this 4-sided shape?
SQUARE
Launch
Quadrilateral … sided shape
What kind of quadrilateral is shown here?
b = 5
h = 5
How would you find the AREA of this 4-sided shape?
Area = base x height
(1 min) 10 – 11
In-Class Notes
Click to show the square
More specifically, what kind of quadrilateral is this? (Click for the Answer -> Square)
How would you find the area of this 4-sided shape?
A = base x height
Agenda

9. How would you find the AREA of this 4-sided shape?
PARALLELOGRAM
Launch
Quadrilateral … sided shape
How would you find the AREA of this 4-sided shape?
Area = base x height
What kind of quadrilateral is shown here?
b = 12
h = 8
(1 min) 11 – 12
In-Class Notes
Click to show the parallelogram
More specifically, what kind of quadrilateral is this? (Click for the Answer -> Parallelogram)
How would you find the area of this 4-sided shape?
A = base x height
Remind students that you can do this because the parallelogram can be decomposed and recomposed into a rectangle
Agenda

10. Can I use Area = base x height???
TRAPEZOID
Launch
Quadrilateral … sided shape
What kind of quadrilateral is shown here?
How would you find the AREA of this 4-sided shape?
Can I use Area = base x height???
(1 min) 12 – 13
In-Class Notes
Today we’re going to find the area of another quadrilateral called a trapezoid ***QUICK EXPLANATION
How would you find the area of this 4-sided shape?
Can you use A = base x height??
That is what you will be exploring with your partner!
Agenda

11. Launch Count the unit squares Decompose (cut into pieces)
Before you test the formula, what are strategies for finding area that always work?
Count the unit squares
Decompose (cut into pieces)
& compose (put back together)
into a different shape
With your partner, you will use these strategies FIRST,
and THEN test the formula A = b x h !
(1 min) 14 – 15
In-Class Notes
Repeat and paraphrase again what students will be doing for the next 10 MINUTES ONLY (emphasize this explore will be quick so they will need to be super focused!)
The objective is to see whether you can use A = base x height but you will need to use other strategies to “back you up”
Take volunteers (not calling sticks) for strategies we’ve used earlier in the unit
Begin!
Agenda

12. Explore
KEY QUESTION:
Can the formula A = base x height be used to find the area of a trapezoid???
You will prove whether the formula works by…
Counting the unit squares of your trapezoid
Decomposing (cutting into pieces) & composing (putting back together) your trapezoid into a different shape
Using A = base x height
(10 min) 15 – 25
In-Class Notes
This can be projected during the explore to remind students of the directions and the key question
Partners work together to find areas, using all 3 strategies
Teacher circulates
encourages students to use all 3 strategies
pushes students to think about WHY the formula A = b x h gives them a totally different answer (because there are 2 bases…)
Preparation Notes
The explore chunk of the lesson is a constructivist approach to learning. It gives students the chance to “construct” knowledge in a self-directed way as they investigate whether the formula A = base x height will give them the same answer as the other two strategies (counting unit squares and decomposing/recomposing shapes)
…and comparing your results!
Agenda

13. Summary When you counted the unit squares…12 11 1 2 3 4 5 6 7 8 9 10
Area = 12 square units
Whole Summary (5 min) 25 – 30
In-Class Notes
Project this first trapezoid from the Explore and quickly discuss the answer you get from counting unit squares
(A = 12 square units)
Preparation Notes
In order to have enough time for students to learn and practice using the correct formula, this summary is teacher-led. Instead of having students present their various methods for counting, decomposing, etc. (which, of course, would be more interesting), the following slides show the methods your students will most likely use.
Agenda

14. Area = 12 square units Did anyone decompose and (re)compose like this?
height
= 2
base = 6
Whole Summary (5 min) 25 – 30
In-Class Notes
Project this first trapezoid from the Explore and quickly discuss the answer you get from decomposing the trapezoid and composing it into a 2 by 6 rectangle
(A = 12 square units)
Preparation Notes
Most students will think the animations are cool, but to get students even more interested in the various methods for decomposing and composing, you could do a quick poll on each slide, asking students to raise their hands if they used that particular method
Area = 12 square units
Agenda

15. Area = 12 square units Did anyone decompose and (re)compose like this?
height = 2
base = 6
Whole Summary (5 min) 25 – 30
In-Class Notes
Project this first trapezoid from the Explore and quickly discuss the answer you get from decomposing the trapezoid and composing it into a parallelogram with a base of 6 and height of 2
(A = 12 square units)
Preparation Notes
This connects back to the first lesson in the unit about finding the area of parallelograms.
Area = 12 square units
Agenda

16. Did anyone decompose and (re)compose like this?
height = 2
Area = ½ b h
Area = ½ x 8 x 2
base = 4
Area = 8
Area = 4 + 8
Area = ½ b h
Area = 12 square units
height
= 2
Area = ½ x 4 x 2
Area = 4
Whole Summary (5 min) 25 – 30
In-Class Notes
Project this first trapezoid from the Explore and quickly discuss the answer you get from decomposing the trapezoid into two triangles – one with a base of 2 and height of 2, the other with a base of 8 and height of 2
(A = 12 square units)
Preparation Notes
This method will probably be less popular, as it is more difficult, but it makes a nice connection to yesterday’s lesson on finding the area of any triangle.
base = 8
Agenda

17. When you used the formula Area = base x height did you get the same answer of 12?
NO!!
A = base x height
A = x 2
A =
TOO BIG!
A = base x height
A = x 2
A =
TOO SMALL!
Whole Summary (5 min) 25 – 30
In-Class Notes
Did the A = b x h formula give you the same answer as what you got when you counted squares or decomposed/recomposed shapes? (NO!)
Why doesn’t the formula work?
What can’t you use one base or the other?
What happens if you just use the shorter base?
then your rectangle covers less space than the trapezoid does (it’s only the middle rectangular piece)
What happen if you just use the longer base?
then your rectangle covers more space than the trapezoid does (it’s a rectangle around the entire shape)
Refer students back to the objective:
SWBAT prove whether the formula to find the area of parallelograms also applies to trapezoids
So does the formula A = base x height apply to trapezoids? NO!!!!!
Preparation Notes
This is when you can get into the conversation with students about “which base do you use?” because there are two!
If you use the shorter “base”, then your rectangle covers less space than the trapezoid does (it’s only the middle rectangular piece)
If you use the longer “base”, then your rectangle covers more space than the trapezoid does (it’s a rectangle around the entire shape)
Agenda

18. Summary When you counted the unit squares…1 2 3 4 5 6 ?
Area = 24 square units
Whole Summary (5 min) 25 – 30
In-Class Notes
Project this second trapezoid from the Explore and quickly discuss the answer you get from counting unit squares (A = 24 square units)
Acknowledge that this strategy is tricky! Counting unit squares when they are not all whole or (clearly) half squares…
Maybe get a few counts from students
As long as their answers are in the ballpark of 24, they’re good to go!
Agenda

19. Area = 24 square units Did anyone decompose and (re)compose like this?
height = 3
base = 8
Whole Summary (5 min) 25 – 30
In-Class Notes
Project this second trapezoid from the Explore and quickly discuss the answer you get from decomposing the trapezoid and composing it into a 3 by 8 rectangle (A = 24 square units)
Area = 24 square units
Agenda

20. Area = 24 square units Did anyone decompose and (re)compose like this?
height= 3
height= 3
base= 6
base= 4
Area = b x h
6 x 3 18
Area = ½ b x h
½ x 4 x 3
½ x 12 6
Whole Summary (5 min) 25 – 30
In-Class Notes
Project this second trapezoid from the Explore and quickly discuss the answer you get from decomposing the trapezoid and composing it into a 3 by 8 rectangle (A = 24 square units)
Area = 24 square units
Agenda

21. When you used the formula Area = base x height did you get the same answer of 24?
NO!!
A = base x height
A = x 3
A =
TOO BIG!
A = base x height
A = x 3
A =
TOO SMALL!
Whole Summary (5 min) 25 – 30
In-Class Notes
Did the A = b x h formula give you the same answer as what you got when you counted squares or decomposed/recomposed shapes? (NO!)
Why doesn’t the formula work?
What can’t you use one base or the other?
What happens if you just use the shorter base?
then your rectangle covers less space than the trapezoid does (it’s only the middle rectangular piece)
What happens if you just use the longer base?
then your rectangle covers more space than the trapezoid does (it’s a rectangle around the entire shape)
Refer students back to the objective:
SWBAT prove whether the formula to find the area of parallelograms also applies to trapezoids
So does the formula A = base x height apply to trapezoids? NO!!!!!
Preparation Notes
This is when you solidify the idea with students that you CANNOT use the formula A = base x height because “which base do you use? There are two!”
Agenda

22. A = (b1 + b2) x h 2 Mini-Lesson
The formula for the Area of a Trapezoid is…
A = (b1 + b2) x h 2 b2
height or h
Whole Mini-Lesson (5 min) 30 – 35
In-Class Notes
Show the (intuitive, not formal) formula for finding the area of a trapezoid: A= (b1 + b2) x h 2
Let’s break this formula down so we can understand it
Let’s start with the easier question: What is the h? (CLICK for the height to appear)
So what is b1? What is b2? (CLICK for the bases to appear)
Does it matter which measurement is b1 or b2? (No)
Why are both bases included in the formula? (Both are important and affect the area of the shape!)
Why can’t you use just one or the other? (One will give you an area too small, one too large)
What happens to those numbers in the formula? (They get added together, then divided by 2)
What are we really doing here? What is it called when we add numbers and then divide by the number of numbers? (Finding the mean!) b1
Agenda

23. Let’s Test it out using the Trapezoids from the Explore…
A = (b1 + b2) x h 2
Let’s Test it out using the Trapezoids from the Explore…
A = (8 + 4) x 2 2
A = (12) x 2 2
It works!
A = x = sq. units
b2 = 4
Trapezoid A
Whole Mini-Lesson (5 min) 30 – 35
In-Class Notes
Show line by line how to substitute (“plug in”) each value
Did it give us the same answer from when you counted squares and decomposed/recomposed shapes? YES!
Reiterate that both bases are included, they are both important
But look, what are we really doing with these bases?
We’re adding them together, then dividing by two… what are you really doing??
Finding the MEAN!!
h = 2
b1 = 8
Agenda

24. It works! A = (b1 + b2) x h 2 A = (10 + 6) x 3 2 A = (16) x 3 2
Let’s try the
formula again…
A = (10 + 6) x 3 2
A = (16) x 3 2
Trapezoid B
A = x 3
A = 24 sq. units
b2 = 6
Whole Mini-Lesson (5 min) 30 – 35
In-Class Notes
Show line by line how to substitute (“plug in”) each value
Reiterate that both bases are included, they are both important
But look, what are we really doing with these bases?
We’re adding them together, then cutting the answer in half! ***Again show the formula 2nd to last line with the ½ and 2 flipped in order
When you take 2 numbers, add them, then cut the answer in half or divide by 2 – what are you really doing??
Finding the MEAN!!
What is the mean of 6 and 10?
h = 3
It works!
b1 = 10
Agenda

25. HOLD UP! I feel like I’ve seen this trapezoid somewhere
before… it reminds me of something… wait for it…
b2 = 6
Whole Mini-Lesson (5 min) 30 – 35
In-Class Notes
Wait a second…
Click to trigger the animation, which shows the trapezoid being rotated so it more closely resembles the two piles of cookies (6 and 10 each)
Make sure students see the connection between the two piles of cookies and the two bases
What did we do to “even out” or “average out” the number of cookies each person had?
When the piles were evened out, how many cookies did the brother and the sister have each? (8)
When the trapezoid is decomposed and recomposed into a rectangle, what are the new dimensions? 2 by 8!
Pass out second CW
h = 3
h = 3
b1 = 10
Agenda

26. YES. Greg and Laura’s cookies. See what I mean
YES! Greg and Laura’s cookies! See what I mean? The piles of cookies had to even out just like the trapezoid’s bases!!!
b = 8
Whole Mini-Lesson (5 min) 30 – 35
In-Class Notes
Wait a second…
Click to trigger the animation, which shows the trapezoid being rotated so it more closely resembles the two piles of cookies (6 and 10 each)
Make sure students see the connection between the two piles of cookies and the two bases
What did we do to “even out” or “average out” the number of cookies each person had?
When the piles were evened out, how many cookies did the brother and the sister have each? (8)
When the trapezoid is decomposed and recomposed into a rectangle, what are the new dimensions? 2 by 8!
Pass out second CW
Preparation Notes
h = 3
Agenda

27. then multiply by the height to get the Area of a Trapezoid!
So, this part of the formula means….
we take the AVERAGE of the base lengths,
Whole Mini-Lesson (5 min) 30 – 35
In-Class Notes
Wait a second…
Click to trigger the animation, which shows the trapezoid being rotated so it more closely resembles the two piles of cookies (6 and 10 each)
Make sure students see the connection between the two piles of cookies and the two bases
What did we do to “even out” or “average out” the number of cookies each person had?
When the piles were evened out, how many cookies did the brother and the sister have each? (8)
When the trapezoid is decomposed and recomposed into a rectangle, what are the new dimensions? 2 by 8!
Pass out second CW
then multiply by the height to get the Area of a Trapezoid!
Agenda

28. Now that we know the formula for the area of a trapezoid, let’s try to use it together!
In-Class Notes
Pass out Practice CW
INTERLEAVE the first 4 problems
Preparation Notes
The strategy of “interleaving” is included here in the lesson because research shows that students learn new material best when they are able to see an example (either on paper or led by a teacher), and then try a similar problem on their own, see another example, then try another one on their own, etc.

29. Practice #1 A = (b1 + b2) x h 2 A = (9 + 5) x 4 2 A = (14) x 4 2
A = x 4 = 28 sq. units
(1 min) 35 – 36
In-Class Notes
Demonstrate the first problem, showing how to plug in for h, b1 and b2
Explain again why both bases are included
If you just use one, your area will be too big or too small
You need to “even” them out, which is why the formula has you find the mean of the two bases and then multiply that by the height
Preparation Notes
The strategy of “interleaving” is included here in the lesson because research shows that students learn new material best when they are able to see an example (either on paper or led by a teacher), and then try a similar problem on their own, see another example, then try another one on their own, etc.
b1 = 9
Agenda

30. Practice #2 A = (b1 + b2) x h 2 A = (9 + 3) x 10 2 A = (12) x 10 2
Now you try one…
A = (b1 + b2) x h 2
b2 = 3
A = (9 + 3) x 10 2
h = 10
A = (12) x 10 2
A = x = 60 sq. units
(2 min) 36 – 38
In-Class Notes
INTERLEAVE the first 4 problems
Give students a minute to write down the formula, substitute the dimensions, solve
Explain again why both bases are included
If you just use one, your area will be too big or too small
You need to “even” them out, which is why the formula has you find the mean of the two bases and then multiply that by the height
Preparation Notes
The strategy of “interleaving” is included here in the lesson because research shows that students learn new material best when they are able to see an example (either on paper or led by a teacher), and then try a similar problem on their own, see another example, then try another one on their own, etc.
b1 = 9
Agenda

31. Practice #3 A = (b1 + b2) x h 2 A = (2 + 4) x 7 2 A = (6) x 7 2
Let’s try another one together, but make it a little more interesting!
Edward wants make a curtain for his living room window so he can sit in the dark and think of ways to get rid of Jacob! Calculate the area so that he knows how much fabric to buy.
b2 = 4 ft
b1 = 2 ft
h = 7 ft
A = (b1 + b2) x h 2
A = (2 + 4) x 7 2
(1 min) 38 – 39
In-Class Notes
INTERLEAVE the first 4 problems
Point out with this real-world trapezoid that it is “upside-down”
Briefly remind students that it doesn’t really matter what is b1 or b2 (because they get added) so the formula can still be used!
Demonstrate how to plug in for h, b1 and b2
Explain again why both bases are included
If you just use one, your area will be too big or too small
You need to “even” them out, which is why the formula has you find the mean of the two bases and then multiply that by the height
Preparation Notes
The strategy of “interleaving” is included here in the lesson because research shows that students learn new material best when they are able to see an example (either on paper or led by a teacher), and then try a similar problem on their own, see another example, then try another one on their own, etc.
A = (6) x 7 2
A = 3 x 7 = 21 ft2
Agenda

32. Practice #4 A = (b1 + b2) x h 2 A =(18 + 12) x 40 2 A = (30) x 40 2
Now you try another one…
Brian is getting a custom sticker to completely cover the back of his guitar. How much space will it take up?
A = (b1 + b2) x h 2
b2 = 12 in
A =( ) x 40 2
A = (30) x 40 2
h =
40 in
(2 min) 39 – 41
In-Class Notes
INTERLEAVE the first 4 problems
Give students a minute to write down the formula, substitute the dimensions, solve
Explain again why both bases are included
If you just use one, your area will be too big or too small
You need to “even” them out, which is why the formula has you find the mean of the two bases and then multiply that by the height
Have students work on the rest independently
Preparation Notes
The strategy of “interleaving” is included here in the lesson because research shows that students learn new material best when they are able to see an example (either on paper or led by a teacher), and then try a similar problem on their own, see another example, then try another one on their own, etc.
A = x = in2
Agenda
b1 = 18 in

33. Continue practicing on your own…
Independent Practice
Continue practicing on your own…
(9 min) 41 – 50
In-Class Notes
Students should work independently, possibly with a partner check before going over the answers as a class.
Can use calling sticks to go over answers. Do not need to spend much time on this unless many students got stuck on a particular problem. Just allow students to check their answers.
Agenda

34. Independent Practice Answers
5. a) 6 ft2 b) 11 cm2 c) 27.6 sq. units d) 20 yd2 6. Juan’s Area: 90 m2 Cheyanne’s Area: 50 m2
(2 min) 50 – 52
In-Class Notes
Can use calling sticks to go over answers. Do not need to spend much time on this unless many students got stuck on a particular problem. Just allow students to check their answers.
Agenda