Area and Expressions

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SWBAT write expressions to find the area of compound figures.

Big Idea

Conceptual understanding of finding area of compound figures using expressions.


10 minutes

In a previous lesson, students learned about the commutative, associative, and identity properties (Working With the Properties Lesson).  Today, I would like them to be able to identify the properties and explain how they know they are correct.(SMP 3)  We will be continuing our lesson today by talking about and applying the distributive property.  I wanted the students to be able to recall the properties plus, they will be using the commutative property during this lesson.

Tools:  Identifying the properties problems. 

Distributive Property

15 minutes

My students have already learned about the distributive property.  They’ve used it with mental math and also for finding the sum of two numbers by pulling out the greatest common factor. Today they will be using the distributive property to find the area of compound figures. 

The beginning part of this lesson will be review for the students.  I will be talking to them and refreshing their memories about the distributive property. Once I have done this, I’m going to have the students complete 4 problems using the distributive property.  Two problems will be with multiplication and two will be for addition.  My only concern with these problems is that the students will forget the place values for the multiplication and that they need to pull the greatest common factor out for the sum.  Students may need a reminder that this property helps make math easier.  They will need to break the problem down into simpler parts so we can use mental math.(SMP 6) Call on random students to show and explain their work on the board.

Tools:  Area, Expressions and Distributive Property power point and notes. 

Area and Expressions

40 minutes

This part of the lesson was adapted from the Laws of Arithmetic as part of the mathematic assessment project.  I’ve chosen to use some of the project, but separated it into two days because of the intensity of the math.  I felt it may be a little too much of a stretch for my students, but didn’t want them to miss out on the conceptual understanding of expressions and area. Students may need a brush up on finding the area and finding the area of compound figures.  I will be talking to them about the expression used to find area.  I will be asking them what the word compound means?  In hopes they will tell me that it means to put together.  Then I will be asking them what operation joins numbers?  Students should recognize this as addition.  I will continue to explain that compound figures are two or more figures joined together which means we will add the areas.

There are 4 problems that require the students to write expressions to show the area of compound figures.  For each problem, I’m going to do a think-pair-share.  I want students to think about the expressions or what they would look like.  Then pair up, and share with their partner their thoughts about the solution.  Once they have done the think pair share, have them look more deeply in to the problem to come up with their solutions (SMP 1, 2 and  3)

Problem 1:  This problem has them choosing from 3 different expressions to represent the area of the rectangles.  The first two choices should be easy for students to see.  They may have to work it out to find the solution, if they don’t recognize the distributive property at work.  The last expression is also a solution.  Students will probably be confused by this, but if they do the math, they will see it still is the area of the compound figure.  This problem has them dividing the rectangles in ½ so the expression would look something like this: 1 x (3+5) + 1 x (3+5) which is the same as (3+5) + (3+5).  All 3 expressions are a solution for this problem.

Problem 2: This problem has them looking at an expression and deciding which area diagram matches the expression.  Students should try and find the solution to the expression first by using orders of operations.  Watch for this as many students will forget to use it.

Solution:  Figure A

Problem 3: This problem has them writing two expressions.  The trick here is that the figures are squares and this will require them to make a connection to exponents in order to write both expressions.  Some possible solutions:  5 x 5 + 2 x 2 or 5² + 2²

Problem 4:  This problem is asking the students to use the distributive property to write the expression to show the area.  If students get stuck, have them write the whole expression out 3 x 6 + 3 x 4 and then encourage them to re-write this using the distributive property.  If they still need help, have them use their notes as a resource. (SMP 5)

Tools: 4 area problems


25 minutes

Students will be working on an informal assessment from the Laws of Arithmetic project.  I liked this assessment because it’s informally getting an understanding of where the students are in their learning.  There are problems that each student can answer and there are problems that students will struggle with.  It is important to let students know that they should try their best.  If they don’t understand a problem, they can skip it. 

This informal assessment should NOT be graded, however, comments can be made to let students know their progress.  According to the project, teachers should make comments on the students’ papers using questions.  For example, if the student checks an expression that does not fit both figures, you could say “ Does this expression represent the area of both diagrams?”.  Giving feedback instead of a grade will be much more productive. If students are frustrated with their progress, explain to them that by the end of the next lesson, they should be able to answer these questions with less difficulty. 

**The project suggest you give this informal assessment before working through any of the area model problems.  I felt it would be too difficult and the students would reach a frustration level that would be too high. So, I chose to give them a little background knowledge to help alleviate some of the frustration. **

Tools: Expressions and Diagrams informal assessment