3.4+Writing+Quadratic+Equations+0809

March 25, 2009 JSM  __ Big Idea __ : Equations can be used to model real things. __ Essential Question __ : What are the distributive and commutative properties used for?  __ Notes __ : Worked on Problem 3.3 and 3.4. Other notes were not taken, I was not in class. **__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. ** a.  Number of Games=//n//2-//n// **<span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">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="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">1. <span style="font-family: 맑은 고딕;">The reporter has //n// choices. **<span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><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;">2. She has //n//-1 choices. <span style="font-family: 맑은 고딕;">3. Total number of games= //n x// (//n//-1) = //n//2-//n// <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// <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;">
 * <span style="font-family: 맑은 고딕;">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. **
 * **<span style="font-family: 맑은 고딕;">Number of teams ** || <span style="font-family: 맑은 고딕; font-size: 12pt;">2 || <span style="font-family: 맑은 고딕; font-size: 12pt;">3 || <span style="font-family: 맑은 고딕; font-size: 12pt;">4 || <span style="font-family: 맑은 고딕; font-size: 12pt;">5 || <span style="font-family: 맑은 고딕; font-size: 12pt;">6 ||
 * **<span style="font-family: 맑은 고딕;">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="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><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="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><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. **
 * <span style="font-family: 맑은 고딕;">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. 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;">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. **

__ Problem 3.4 Follow-Up __ **<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="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">2. **<span style="font-family: 맑은 고딕;">** (4x+3)(x+7) ** 2a. 4x2+10x+21 **<span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">3. **<span style="font-family: 맑은 고딕;">** (x-4.5)(x+6.5) ** 3a. x2+2x-29.25 **<span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">4. **<span style="font-family: 맑은 고딕;">** (2x-5)(5x-2) ** 4a. 10x2-21x-10 **<span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">5. **<span style="font-family: 맑은 고딕;">** x2-6x ** 5a. x(x-6) **<span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">6. **<span style="font-family: 맑은 고딕;">** 12x-3x2 ** 6a. 3x(4-x) or x(12-3x) **<span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">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;">
 * <span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">1.  **<span style="font-family: 맑은 고딕;">** x(-4x+3) ** 1a. -4x2+3x
 * <span style="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><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="mso-list: Ignore; msobidifontfamily: '맑은 고딕'; msolist: Ignore;"><span style="font-family: 맑은 고딕;">b.  ****<span style="font-family: 맑은 고딕;">Describe any general rules you discover for factoring a quadratic expression. **