Slide 1 / 138 Slide 2 / 138 6th Grade Equations & Inequalities 2015-12-01 www.njctl.org Slide 3 / 138 Slide 3 (Answer) / 138 Table of Contents Table of Contents Click on a topic to Click on a topic to Equations and Identities Equations and Identities · · go to that section. go to that section. Tables Tables · · Vocabulary Words are bolded Teacher Notes Determining Solutions of Equations Determining Solutions of Equations · · in the presentation. The text Solving an Equation for a Variable Solving an Equation for a Variable · · box the word is in is then Solving One Step Addition & Subtraction Equations Solving One Step Addition & Subtraction Equations · · linked to the page at the end Solving One Step Multiplication & Division Equations Solving One Step Multiplication & Division Equations · · of the presentation with the Writing Equations Writing Equations · · word defined on it. Writing Simple Inequalities Writing Simple Inequalities · · [This object is a pull tab] Solutions to Simple Inequalities Solutions to Simple Inequalities · · Graphing Solution Sets to Simple Inequalities Graphing Solution Sets to Simple Inequalities · · Glossary & Standards Glossary & Standards · · Slide 4 / 138 Slide 5 / 138 Equations and Identities What is an equation? How is it different than an identity? Discuss in your groups. Equations and Identities Return to Table of Contents
Slide 5 (Answer) / 138 Slide 6 / 138 Equations and Identities Equations and Identities This activity addresses MP3. Math Practice Additional Questions to Ask: What is an equation? An equation is created when two expressions are set equal - Do you agree with their reasoning? to one another such that they are equal for some values of Why or why not? their variables, but not for all. How is it different than an identity? - Can you repeat to someone else what they said? If they are equal for all values of their variables, then that is Discuss in your groups. - Is there anything you could add on to an identity, not an equation. their answer? So, not all mathematical statements which include an equal [This object is a pull tab] sign are equations...some are identities. Slide 7 / 138 Slide 8 / 138 Equations and Identities Equations and Identities Here are some equations: Here are some identities : x = 5t 2 + 3 = 5 v = 7 + x 9 - 2 = 7 a = x - 8 2x = 2x In all these cases, the variables are interdependent. They are only true for certain sets of variables. 9/3 = 3 Changing the value of the variable on the right side of the These are always true...there are no values that can be equation, changes the possible values of the variables on the assigned to the variables for which these would be untrue. left side...and vice versa. These are equations (not identities) since knowing the value of one variable changes the possible value(s) of the other(s). Slide 9 / 138 Slide 10 / 138 Writing Equations Write an equation in words. Then translate that into a mathematical equation. Tim is eight years older than Kathy. Write an equation for Tables Kathy's age. click for equation in words Tim's age is equal to Kathy's plus 8 T = K + 8 click for mathematical equation Return to Table of Contents
Slide 10 (Answer) / 138 Slide 11 / 138 Writing Equations Writing Equations Slides 10-12 addresses MP2 Write an equation in words. Write an equation in words. Math Practice Then translate that into a mathematical equation. Then translate that into a mathematical equation. Additional Questions to Ask: - How could you represent the problem with symbols & numbers? - Why would you choose certain Tim is eight years older than Kathy. Write an equation for Tim is eight years older than Kathy. Write an equation for variables to represent a certain value? Kathy's age. Kathy's age. - Is there more than one way to represent an expression? an equation? click for equation in words click for equation in words Tim's age is equal to Kathy's plus 8 [This object is a pull tab] Tim's age is equal to Kathy's plus 8 T = K + 8 T = K + 8 click for mathematical equation click for mathematical equation Slide 11 (Answer) / 138 Slide 12 / 138 Writing Equations Writing Equations Write an equation in words. Write an equation in words. Slides 10-12 addresses MP2 Then translate that into a mathematical equation. Then translate that into a mathematical equation. Math Practice Additional Questions to Ask: - How could you represent the problem with symbols & numbers? Tim is eight years older than Kathy. Write an equation for Bob is 6 inches less than twice the height of Fred. Write an - Why would you choose certain Kathy's age. equation for Bob's height. variables to represent a certain value? - Is there more than one way to represent an expression? an equation? Bob's height equals double Fred's height less 6 Tim's age is equal to Kathy's plus 8 click for equation in words click for equation in words [This object is a pull tab] T = K + 8 click for mathematical equation click for mathematical equation B = 2F - 6 Slide 13 / 138 Slide 14 / 138 Writing Equations Tables and Expressions Write an equation in words. Let's use this table to find some solutions to the equation s = d/t; where s represents speed (in meters/second), d represents Then translate that into a mathematical equation. distance (in meters) and t represents time (in seconds). Math Practice Speed is equal to the distance traveled in an amount of time. s = d/t We've entered the distance t (s) d (m) s (m/s) traveled and the time it took 30 2 to travel that distance in two click for equation in words Speed equals distance divided by time of the columns. 60 4 90 6 Use the equation (s = d/t) to 120 2 find the speeds and fill in the click for mathematical equation s = d/t blank column. 240 4 360 6
Slide 15 / 138 Slide 16 / 138 Tables and Expressions Note that in the first three sets of answers, the object was moving at a speed of 15 m/s. The final three sets of answers are for an object traveling four Determining Solutions of times faster, at 60 m/s. s = d/t Equations But, in all cases, knowing d (m) t (s) s (m/s) the value of two of the three 30 2 15 variables determines the values of the third. 60 4 15 90 6 15 120 2 60 240 4 60 Return to Table 360 6 60 of Contents Slide 17 / 138 Slide 18 / 138 Determining the Solutions of Equations Determining the Solutions of Equations Which of the following is a solution of the equation from the A solution to an equation is a number that makes the equation solution set ? true. x + 7 = 9 {2, 3, 4, 5} In order to determine if a number is a solution, replace the variable with the number and evaluate the equation. Write the equation four times. Each time replace x with one of the possible solutions and simplify to see if it is true. If the number makes the equation true, it is a solution. If the number makes the equation false, it is not a solution. 2 + 7 = 9 3 + 7 = 9 4 + 7 = 9 5 + 7 = 9 9 = 9 10 = 9 11 = 9 12 = 9 Yes No No No Answer: 2 is the solution to x + 7 = 9 Slide 19 / 138 Slide 20 / 138 Determining the Solutions of Equations 1 Which of the following is a solution to the equation? Which of the following is a solution of the equation? x + 17 = 21 {2, 3, 4, 5} y - 12 = 8 {17, 18, 19, 20} Write the equation four times. Each time replace y with one of the possible solutions and simplify to see if it is true. 17 - 12 = 8 18 - 12 = 8 19 - 12 = 8 20 - 12 = 8 5 = 8 6 = 8 7 = 8 8 = 8 No No No Yes Answer: 20 is the solution to y - 12 = 8
Slide 20 (Answer) / 138 Slide 21 / 138 1 Which of the following is a solution to the equation? 2 Which of the following is a solution to the equation? x + 17 = 21 {2, 3, 4, 5} m - 13 = 28 {39, 40, 41, 42} Answer 4 [This object is a pull tab] Slide 21 (Answer) / 138 Slide 22 / 138 2 Which of the following is a solution to the equation? 3 Which of the following is a solution to the equation? m - 13 = 28 {39, 40, 41, 42} 12b = 132 {9, 10, 11, 12} Answer 41 [This object is a pull tab] Slide 22 (Answer) / 138 Slide 23 / 138 3 Which of the following is a solution to the equation? 4 Which of the following is a solution to the equation? 12b = 132 {9, 10, 11, 12} 3.5 + d = 7.5 {2.5, 3, 3.5, 4} Answer 11 [This object is a pull tab]
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