I'll begin this lesson with the essential question: How can you write equations to represent and solve perimeter problems? Writing equations for perimeter problems is mathematical practice 4 - model with mathematics.
On the board I will have the drawing of a right triangle with sides labeled 3, 4, and 5. I will also have a rectangle with a length of 12 and a width of 7. I will ask student to take a moment and think to themselves the perimeter of each shape. After a moment I will have students share answers.
As students share I will write an expression on the board to represent the perimeter of each shape. Students will help provide the expression or expressions.
For the triangle I will have: 3 + 4 + 5 = 12 units
For the rectangle I will have: 7 + 7 + 12 + 12 = 38 units or 2(7+12) = 38 or some other variation.
I will then say that the lesson today will be the same expect our lengths will be represented as algebraic expressions.
I will go through the notes filling in the words "distance" and "sum" for the vocabulary.
I will then go through the steps for each problem: 1) Draw the shape and label its sides; 2) Write an equation to represent the perimeter; 3) Solve the equation
In the first example we will first solve the problem using a tape diagram. This will provide a visual model to tie into the equation.
When reading the problem, I will annotate the word consecutive adding that it means "one after the other". I will ask students if they know this before providing the answer.
Students are expected to draw a triangle. The triangle is just a representation to serve as a graphic organizer; the sides do not have to be to scale. I will make sure my students know this.
The equation will be written as : x + (x + 1) + (x + 2) = 45.
We will simplify the equation to 3x + 3 = 45 and then solve for x = 14.
Students will then find the side lengths of 14, 15, and 16 units.
For the guided practice problems, I will have students go step by step through the problems. First I will ask them to draw the shape and label the sides. I will walk around to help as needed. Many students will say "I don't know what the smallest side is". I'll provide the hint - "How do we represent unknown numbers in math?" Hopefully they realize they can use a variable for this. Once I see students have the correct side lengths, I will ask them to write the equation to represent the perimeter. Once I see this is done, I will ask them to solve the equation. Finally, we will use the solution to find the lengths of the missing sides.
The second problem says the triangle sides are consecutive even numbers. Students still may need help. I will refer them to the annotation in the notes and make sure they know what even numbers are. (Yes, a few of my students forget this!). If they are stuck on writing the expression I'll ask them to name an even number. Then I'll ask them to name the next greater even number. Then, I'll ask what can be done to the first number to get the next number? When they see that each consecutive even number is two more than the previous they'll know to add two. So the sides are x, (x+2), and (x+2) + 2 or (x+4).
The independent section has 8 problems. The first four all involve regular polygons. These result in one-step equations. If students do not know what the term "regular" means they should carefully read problem 1.
Problems 5-7 are similar to the problems from the guided problem solving section of the lesson. Students should refer to these problems first.
Problem 8 is perhaps the hardest problem because the length is described as 4 units less than twice the width. Students often confuse this with 4 - 2w or even 4 < 2w! For kids who write the 4 as the minuend ask "What is 4 less than 10?". Usually they know immediately that this is 6 or 10 - 4 = 6. This should help to have a width of w and a length of 2w - 4.
The exit ticket resembles problem GP1 for guided problem solving and problem 5 from independent problem solving. I will expect students to sketch a triangle and label the sides. As far as the equation goes, any equivalent equation will be accepted.
x + 2x + (x+10) = 80 or 4x + 10 = 80 etc....
For part B students should find the lengths either by solving the equation or any other means - tape diagram, guess and check, etc.
I will score this as 2 points: 1 point for a correct equation and 1 point for the correct side lengths of 17.5, 27.5, and 35.