Area and Combining Like Terms

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Objective

SWBAT: • Calculate the area of a rectangle and square • Write an expression to represent the area of multiple shapes • Create and identify equivalent expressions by combining like terms

Big Idea

How do you represent the area of different blocks? What’s the difference between 2x and x squared? Students use their knowledge of area to combine like terms using algebra tiles.

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 review finding the perimeter and area of a rectangle.  A common mistake is for students to confuse perimeter and area.  I walk around to monitor student progress to see if they are committing this mistake.

I ask students, “How do you find the perimeter of a rectangle? How do you find the area of a rectangle?”  Some students may say that they add up all of the sides of a rectangle to find the perimeter.  Other students may say that they double the width and the double the length and then add those together.  I push students to explain why this works.  For area, some students may say that you can count the number of square units inside of the rectangle.  I ask, “How can you find the area of a rectangle when the square units are not drawn inside of the rectangle?”  I want students to explain that they can multiply the width by the length to find the area.  It is important that students are using precise language and are including the units for the measurements.

Vocabulary Review

5 minutes

Note:

  • When I print out the notes for students I will delete the words in parentheses.  I want students to fill in these words on their own notes.

We review the vocabulary words together and fill in the words.  Later in the lesson they will be identifying and creating equivalent expressions and I want them to use the correct vocabulary words to describe what they are doing.  For each word I ask for students to come up with examples.  I ask the class whether an example matches the definition.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

Algebra Tiles

12 minutes

Notes:

I show students a sample set of algebra tiles.  I have a student read the directions out loud.  I explain that students only need to take out one of each tile for this part of the lesson.  I briefly go over expectations for working with the tiles.  I call on a student to repeat the directions and the expectations.   I pass out the algebra tiles and students work independently.

We come together and share ideas.  I want students to realize that the yellow block shares the same width of the green block.  I want students to see that the green block shares the same length as the blue block.  Some students may say that 3 or of the small blocks make up the medium block.  I use my tiles under the document camera to show that the medium block cannot be created using the unit tiles.

I tell students that the yellow block is called the unit block.  It has a length and width of 1 unit.  Students label their drawings.  I ask students to calculate the area of the yellow block and to explain how they did it.  I write Area = length x width, 1 x 1 = 1 units squared. 

I label the green block as have a width of 1 unit and a length of x units.  We are going to use x to represent the unknown length.  I ask students to find the area of the green block.  Some students may resist and say that it is impossible.  I push students to use the formula of length x width.  We represent the area of the block by multiply 1 times x.  I ask, “What do we get when we multiply any number by 1?”  I call on students to share out.  We show students that we can represent this area by writing 1x or just x, since 1 times x equals x.  The area of the green block is x units squared.

I ask students, “What is the length and width of the blue block?  What is the area of the blue block?”  Students participate in a Think Pair Share.  Students are engaging in MP5: Use appropriate tools strategically, MP6: Attend to precision, and MP7: Look for and make use of structure.

 I call on students to share out their thinking.  I want students to realize that if the length of the green block is x, then the length and width of the blue block is x units.  A common mistake is for students to say that the area of the blue block is 2x.  I ask students to repeat the formula for area of a square or rectangle.  Length times width for the blue block represents x times x.  Is this the same as 2x?  I push students to use their blocks to prove that these two expressions are not equivalent.  An area of 2x or 2 times x can be represented by putting two green blocks together.  Each of the green blocks has an area of x so together they have an area of 2 times x or 2x.  These blocks do not take up the same area as one of the blue blocks, so they are not equal.

 If students are still struggling, I draw a square and label the sides 3.  I ask students to find the area of the square.  What is another way we could write the area of the square?  I push students to see that we can represent the area of 3 x 3 with 3 squared.  I ask students to apply this knowledge to the blue block.  I want students to see that the area of the blue block can be represented by x squared units squared.

Calculating Area

16 minutes

We work through problem 1 and 2 together.  I want students to recognize that the x’s in Jeremy’s drawing refer to the area of the tiles while the x’s and 1’s in Harold’s picture are labeling the length and width of each tile.   I want students to realize that both students are correct in their area expressions.  These expressions are equivalent.  I explain that when we combine measurements that have the same variable raised to the same power it is called combining like terms.    I substitute numbers in for x to demonstrate the equivalence.

 A common mistake for problem 2 is for students to combine the two blocks, even though they have different variables.  I want students to recognize that the sum of x squared and x is not equivalent to 2x or 2 x squared.  Again I have students substitute numbers for x to prove that they are not equivalent.  The most simplified way we can represent the area is x squared + x or x + x squared.

Students work on problems 3-5 independently.  I walk around and monitor student progress and behavior.  Students are engaging in MP1: Make sense of problems and persevere in solving them, MP5: Use appropriate tools strategically, and MP6: Attend to precision, and MP7: Look for and make use of structure.

If students are struggling, I ask them one or more of the following questions:

  • What is the problem asking?
  • What do you know?
  • What do the labels on the tiles represent here?
  • Can you combine any terms?  Why or why not?
  • How can you represent the same area in a different way?

If students successfully complete the problems they can move on to the challenge question.

Closure and Ticket to Go

10 minutes

I tell students to flip to their work on problem 3.  I ask them, “Why can’t we just count the total number of tiles to find the area?”  and “Why can’t we combine the x and x-squared tiles?”  Students participate in a Think Pair Share.  I call on students to share their ideas.  I want students to realize that we can’t count up all of the tiles because the different sized tiles represent different quantities.  If I have time I ask students whether they agree or disagree with Jeremiah in 3b and why.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others

I pass out the Ticket to Go and the HW Area and Combining Like Terms.