3.4+Writing+Quadratic+Equations+0910

Oliver Peacock Block F April 26, 2010

__ Big Idea __ : Equations can be used to model real things. __ Essential Question __: What are the distributive and commutative properties useful for? __ Notes __ : Was done at home because of partner quiz so no notes. **__Investigation 3: Some Important Properties __**

__3.4 - Writing Quadratic Expressions __ __Problem 3.4 __ In parts A-C, you will explore three ways of thinking about this question: If a league has //n// teams and each team plays each of the other teams twice, how many games are played in all?

**A. ** **Figure out how many games would be played for leagues with 2, 3, 4, 5, and 6 teams. Record your results in a table similar to the one below. **
 * Look for a pattern in your table. Use the pattern to write an expression for the number of games played by a league with //n// teams. **
 * **Number of teams ** ||  || 2 ||   || 3 ||   || 4 ||   || 5 ||   || 6 ||


 * **Number of games ** ||  || <span style="font-family: 맑은 고딕; font-size: 12pt;">2 ||   || <span style="font-family: 맑은 고딕; font-size: 12pt;">6 ||   || <span style="font-family: 맑은 고딕; font-size: 12pt;">12 ||   || <span style="font-family: 맑은 고딕; font-size: 12pt;">20 ||   || <span style="font-family: 맑은 고딕; font-size: 12pt;">30 ||

<span style="font-family: 맑은 고딕; font-size: 12pt;">Number of Games=(//n^2)//-//n// B. ** **<span style="font-family: 맑은 고딕;">Suppose a sports reporter wants to attend exactly one game in the schedule of an //n//-team league. ** <span style="font-family: 맑은 고딕;">1. <span style="font-family: 맑은 고딕;">The reporter has //n// choices. <span style="font-family: 맑은 고딕;">2. She has //n//-1 choices.
 * <span style="font-family: 맑은 고딕;">
 * <span style="font-family: 맑은 고딕;">1. ** **<span style="font-family: 맑은 고딕;">How many choices does the reporter have for the home team for the game she attends? **
 * <span style="font-family: 맑은 고딕;">2. ** **<span style="font-family: 맑은 고딕;">Once she has chosen a home team, how many choices does she have for the visiting team? **

<span style="font-family: 맑은 고딕;">3. Total number of games= (//n^2)-n = n*(n-1)// **<span style="font-family: 맑은 고딕;">
 * <span style="font-family: 맑은 고딕;">3. ** **<span style="font-family: 맑은 고딕;">Use your answers from parts 1 and 2 to write an expression for the total number of games the reporter can choose from. **

C. Suppose you made a table similar to the one on page 41 to record wins and losses for an //n//-team league. ** <span style="font-family: 맑은 고딕;">1. My table would have //n//2 cells. <span style="font-family: 맑은 고딕;">2. //n// cells in the table would not be used for W or L entries. <span style="font-family: 맑은 고딕;">3. Total number of games played= //n//2-//n// D. In parts A-C, you wrote expressions for the number of games played by an //n//-team league. Show that these expressions are equivalent. ** <span style="font-family: 맑은 고딕;">d. All three expressions are equivalent because they are have the same form when you change them into expanded form, //n//2-//n//.
 * <span style="font-family: 맑은 고딕;">1. How many cells would your table have? **
 * <span style="font-family: 맑은 고딕;">2. How many cells in the table would not be used for W or L entries? **
 * <span style="font-family: 맑은 고딕;">3. Use your answers from parts 1 and 2 to write an expression for the total number of games played. **
 * <span style="font-family: 맑은 고딕;">

__ Problem 3.4 Follow-Up __ 1. ** <span style="font-family: 맑은 고딕;">** x(-4x+3) = ** -4x2+3x 7. ** **<span style="font-family: 맑은 고딕;">Look at your reasoning and your results from questions 1-6. ** <span style="font-family: 맑은 고딕;">a. (a+b)(c+d)= ac+ad+bc+bd. First, do the multiplications as the expression, then add the like terms together. <span style="font-family: 맑은 고딕;">b. ax2+bx= x(ax+b). Divide x from the expression, then put it next to the parenthesis. And write the left over terms in the parenthesis. Like the example expression.
 * <span style="font-family: 맑은 고딕;">In 1-6, use the distributive property to find an expression that is equivalent to the given expression. Check each answer by using your calculator to compare tables and graphs. **
 * <span style="font-family: 맑은 고딕;">
 * <span style="font-family: 맑은 고딕;">2. ** <span style="font-family: 맑은 고딕;">** (4x+3)(x+7) = ** 4x2+10x+21
 * <span style="font-family: 맑은 고딕;">3. ** <span style="font-family: 맑은 고딕;">** (x-4.5)(x+6.5) = ** x2+2x-29.25
 * <span style="font-family: 맑은 고딕;">4. ** <span style="font-family: 맑은 고딕;">** (2x-5)(5x-2) = ** 10x2-21x-10
 * <span style="font-family: 맑은 고딕;">5. ** <span style="font-family: 맑은 고딕;">** x2-6x = ** x(x-6)
 * <span style="font-family: 맑은 고딕;">6. ** <span style="font-family: 맑은 고딕;">** 12x-3x2 = ** 3x(4-x) or x(12-3x)
 * <span style="font-family: 맑은 고딕;">
 * <span style="font-family: 맑은 고딕;">a. ** **<span style="font-family: 맑은 고딕;">Describe any general rules you discover for multiplying two linear factors. **
 * <span style="font-family: 맑은 고딕;">b. ** **<span style="font-family: 맑은 고딕;">Describe any general rules you discover for factoring a quadratic expression. **