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

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 8 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* http://www.purplemath.com/modules/meanmode.htm http://www.mathsisfun.com/mean.html http://www.teachertube.com/viewVideo.php?video_id=132329 *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 www.keepvid.com) Preparation Notes Finally, Laura stopped crying! Agenda Scaffolding Questions

Watch this video on eHow.com! Forget how to find the Mean? Watch this video on eHow.com! http://www.ehow.com/video_4754362_finding-mean.html Questions

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

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

How would you find the AREA of this 4-sided shape? RECTANGLE Launch Quadrilateral … 4-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

How would you find the AREA of this 4-sided shape? SQUARE Launch Quadrilateral … 4-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

How would you find the AREA of this 4-sided shape? PARALLELOGRAM Launch Quadrilateral … 4-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

Can I use Area = base x height??? TRAPEZOID Launch Quadrilateral … 4-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 http://www.mathsisfun.com/definitions/trapezoid.html 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

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

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

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

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

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

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

When you used the formula Area = base x height did you get the same answer of 12? NO!! A = base x height A = 8 x 2 A = 16 TOO BIG! A = base x height A = 4 x 2 A = 8 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

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

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

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

When you used the formula Area = base x height did you get the same answer of 24? NO!! A = base x height A = 10 x 3 A = 30 TOO BIG! A = base x height A = 6 x 3 A = 18 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

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

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 = 6 x 2 = 12 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

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 = 8 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

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

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

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

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.

Practice #1 A = (b1 + b2) x h 2 A = (9 + 5) x 4 2 A = (14) x 4 2 A = 7 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

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 = 6 x 10 = 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

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

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 =(18 + 12) 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 = 15 x 40 = 600 in2 Agenda b1 = 18 in

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

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