Chapter+7,+Substitution

Graph (1) x – 2y = 5 (2) 4x + 3y = 9
 * Warm- Up**

Note this problem is used later.


 * Model with a Raffle**

Have two prizes you'll raffle at the end of the period. One you are charging $5 for and one you are charging $1 for. (provide students manipulative money). Collect money in an envelope with its contents. Talk about the contents....What are all the things they know in their group. Chart answers going around the room, each group adding something.

Tell them you are going to model these two situations with mathematics....

Ask students to create variables for each of the two kinds of tickets sold.

Let _________________________ Let_________________________

what do we know about the total number of sales? write an equation for this.

Do we know the total cost. Let's write an equation for this.

____________________________________________________________

All the while we are emphasizing modeling with mathematics and using think pair share, and asking for student thinking, holding students accountable for each other's thining......What did Jack just say about the Total Sales, Tim? Oh, you don't know...I'll ask someone else and come back to you and see if you can explain it, etc.

(1) x – 2y = 5 (2) 4x + 3y = 9 The first step is to write the first equation in terms of x: x = 2y + 5 On one side of the index card, the students will write x. On the other side, they will write 2y + 5 in parentheses. Then, the students write Equation 2 (4x + 3y = 9) in very large letters across the 8.5” x 11”paper lengthwise. First, the students will place the index card over the equation on the large paper so that the x on the index card is covering the x in 4x + 3y = 9 equation. Then, the students will flip over the index card. The large paper now should read: 4(2y + 5) + 3y = 9 Now demonstrate the distribution property and solve for y. 8y + 20 + 3y = 9 Distributive Property 11y + 20 = 9 Combine like terms 11y = -11 Divide by the coefficient y = -1 Now that the students have solved for y, the students will find x by substituting -1 for y in both equations. (Side 1 of the classroom uses Equation 1 and Side 2 of the classroom evaluates Equation 2.) Side 1: x – 2y = 5 x – 2(-1) = 5 x + 2 = 5 x = 3 Side 2: 4x + 3y = 9 4x + 3(-1) = 9 4x – 3 = 9 4x = 12 x = 3 Since both sides of the room have solved x to be equal to 3, explain that after one variable has been solved, the other variable can be found using either starting equation. Since the students understand from the previous lesson that the solution to a system of equations is a coordinate, explain that that the solution is to be written as **(3,-1)**. Next, show students how to check their answers by substituting (3, -1) for the variables in each equation and verify that the answer is a true statement. x – 2y = 5 4x + 3y = 9 3 – 2(-1) = 5 4(3) + 3(-1) = 9 3 + 2 = 5 12 – 3 = 9 5 = 5 9 = 9 Explain that each problem should be checked in this way to confirm the solution.
 * Chalkboard Example/Activities: The first example should actually correlate to the one we just did in the raffle, but this is a nice explanation of how to use the index cards.**

From here, many options exist....a another chalkboard example.....work in groups to solve examples...it would be nice to have a graphing calculator screen shot of all of the problems that do next to the problems they are now using substitution to solve.

Maybe the rest of the day on calculations, summing up the day of what we are modeling conversation. Assuming two days allotted to this, second day could be more devoted first writing, then solving systems using substitution.