Andrea Palmer PROSPECT HILL ACADEMY CHARTER SCHOOL, SOMERVILLE, MA
6th Grade Math : Unit #8 - Geometry : Lesson #4

Area of Composite Shapes

Objective: SWBAT: • Find the area of a parallelogram by using the formula. • Find the area of a trapezoid and composite shapes by decomposing them into triangles, rectangles, squares, and/or parallelograms.
Standards: 6.G.A.1 MP3 MP6 MP7 MP8
Subject(s): Math
60 minutes
1 Do Now - 7 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want students to solve problems where they have to determine when to calculate perimeter and when to calculate area.  When finding perimeter, a common mistake is that students include the height inside the triangle so that they add four measurements together.  For the area, a common mistake is for students to just multiply the base times the height.  Other students may struggle to identify the base and the height.   I look for these mistakes and may present one as my answer if I see multiple students making one of these mistakes while I circulate during the do now.

I call on a student to share one idea.  That student then calls on the next student to share his/her idea.  I encourage students to build on what their classmates have said by using sentence starters like, “I agree/disagree with __________ because…” and “My idea connects with ____________’s idea…”  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

2 More Quadrilaterals - 10 minutes

Notes:

  • Before this lesson make copies of the parallelograms and trapezoids.
  • I give each student a copy of these shapes and a pair of scissors.
  • I remove the words in italics on students’ notes before I print them.

I want students to take time to figure out a strategy to find the area of the parallelograms and trapezoids.  Some students will split a parallelogram into two triangles and a rectangle to find the area.  Other students may count the whole squares and ½ squares.  Other students may recognize they can move one of the triangles to the other side of the shape to create a rectangle to find the area.  Students are engaging in MP8: Look for and make sense of repeated reasoning.

For the trapezoids some students may create two triangles and a rectangle.    Some struggling students may try to estimate the area by counting whole squares and partial squares.  Other students may recognize that with trapezoid III they can move a triangle to create a rectangle, although this strategy does not work with the other trapezoids. 

I choose students to share their strategies under the document camera.  I look for a student who created rectangles and triangles.  I also look for a student who moved a triangle to create a rectangle.  If no one uses this strategy I cut out the parallelogram and show them.

When I define parallelogram I also introduce the formula for area of base times height.  When I introduce the trapezoid I do not introduce a formula.  I find that many students get confused with this formula.  Instead, I encourage students to break trapezoids up into triangles and a rectangle/square.

3 Problem - 5 minutes

Students work on breaking the composite shapes into shapes they can use to find area.  Most students will break this shape into 2 right triangles and a rectangle.  Some students may create triangles that are not right triangles and a rectangle.  They will struggle to calculate the area because the grid will not allow them to accurately calculate the area.  

4 Area of Composite Shapes - 3 minutes

We quickly review the strategy of breaking composite shapes that we know.  There is more than one way to do this with many composite shapes.  I emphasize to students that they need to find a way that is efficient and works with the measurements that are given.  Sometimes they will have to do detective work to figure out missing measurements.

Some students look at the composite shape and immediately say, “I don’t know how to find the area of that kind of shape.”  I have two students come and show their different strategies for breaking up the shape from the previous section.

5 Practice - 15 minutes

Note:

  • Before this lesson I Post A Key.

I have a student read over the directions.  I review expectations and students start working independently.  Students are engaging in MP6: Attend to precision and MP7: Look for and make use of structure.

As students work I walk around to monitor student progress and behavior.  If students are struggling, I may ask them one or more of the following questions:

  • How can you break this shape up into shapes you know how to deal with?
  • What are the base and height of the triangle?  How do you know?
  • How can you find the area of this shape?  Why does that work?

When students complete their work, they raise their hands.  I quickly scan their work.  If they are on track, I send them to check with the key.  If there are problems, I tell students what they need to revise.  If students successfully complete the chart they can work on the challenge questions. 

6 Closure and Ticket to Go - 10 minutes

I ask students to turn to problem 7.  I ask students to share their thinking about finding the area of this shape.  Students participate in a Think Pair Share.  I want students to recognize that they can break this shape into a parallelogram, a rectangle, and a triangle.  Some students may struggle to identify the base of the triangle.  By comparing the two parallel lines, we can see that the base of the triangle is 7 yards.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others and MP7: Look for and make use of structure.

I pass out the Ticket to Go and the Homework.

Making Progress from Triangles to Composite Shapes
Intervention and Extension

Some students struggled on the ticket to go from the previous lesson where students learned about finding the area of triangles.  At the beginning of this lesson, I briefly went over the previous night’s homework.  I presented answers that made some of the common mistakes students had made on their tickets to go and had them correct me.  Here are a few examples of students who struggled on the Area of Triangles ticket to go and demonstrated progress on the Area of Composite Shapes Ticket to Go.

Student A:

On her triangles ticket to go, she struggled to differentiate and calculate perimeter and area.  To calculate the perimeter of the triangle, she multiplied 6.2 inches by 8.4 inches.  She mistakenly thought that this was a right triangle, instead of identifying the base and height as 10 inches and 5 inches.  She also struggled to multiply with decimals, forgetting to count the 1 she carried and forgetting how to put back her decimal in her final answer.  For area, she first doubled each measurement, including the height inside the triangle.  Then she added these numbers up.  My guess is that she applied her strategy of finding perimeter of a rectangle, where there are 2 pairs of matching sides, to this triangle.  Needless to say she ended the lesson with many gaps in her understanding.

On her composite shapes ticket to go the next day, this student demonstrated a significant amount of growth.  She was able to correctly calculate the area of the rectangle as well as the triangle.  She is still struggling on identifying the base of a triangle.  She labeled the hypotenuse as 20 feet, instead of the base that is one of the sides of the rectangle. 

Student B:

On his triangles ticket to go, he struggled to find the perimeter of the triangle.  He mistakenly added the base and the height together to get 15 inches.  For area, he correctly identified the base and height.  He multiplied them together to get 50 and then divided it by 2.  He made a mistake in his long division, writing that 5 – 4 = 0.  The student should have checked his work and recognized that 2 square inches cannot be ½ of 50.

On his composite shapes ticket to go the next day, he was able to correctly found the area of the rectangle and the triangle. He also accurately divided 160 by 2.  Like Student A, he did not correctly identify the base of the triangle.  He also forgot to include units with his answer.

Student C:

On her triangles ticket to go she confused perimeter and area.  She was able to correctly identify the base and height of the triangle.  She used the formula to find that the area of the triangle was 25 square inches, but she thought that was the perimeter.  For the perimeter she simply listed the height of the triangle.

On her composite shapes ticket to go the next day’s ticket to go she was able to correctly calculate the area of the rectangle as 160 square feet.  She was able to find the area of the triangle by multiplying 20 x 8 and then divide 160 by 2 to get 80 square feet.  I am unsure why she placed “64” inside of the triangle.  Instead of adding the area of the rectangle and the triangle together she just wrote down the area of the triangle.  She needs to make sure she carefully reads the problem to make sure she answers the correct question.

Next Steps:

Although these students showed growth, each of them still has areas that they need to continue to work on.  At the beginning of the next lesson I will pass back these tickets to go so students can see their growth and correct mistakes.  I will also continue to check a couple homework problems as a class before passing them in.  This gives me another opportunity to commit common mistakes and have students correct them out loud.  I will also take this data into consideration when I make my small groups for the Area, Perimeter, and Circumference lesson.