Chapter 2 –Properties of Numbers (2-1) (2-2)
WORD BANK:
commutative
associative
additive identity
multiplicative identity
distributive property
Commutative (think: communicate. It doesn’t matter who talks first.)
a+b = b+a; a*b = b*a
Associative (think: associate. It doesn’t matter who sits with whom.)
(a+b) + c = a + (b+c); (a*b) *c = a* (b*c)
Distributive Property (think: distribute, or share. You can break up numbers to make them easier to work with for addition and multiplication.)
a(b + c) = ab + ac; a(b-c) = ab - ac
When you add any number and 0, the sum equals the original number. The additive identity is 0. When you multiply any number and 1, the product equals the original number. The multiplicative identity is 1.
Practice: (pp 69+70) Use the associative property to write two equations for each and solve:
#2) Add: 5, 91, and 11 Multiply: 5, 91, 11
Practice: (pp 69+70) Name that property and solve:
#6) (6 • 15)2 = 6(15 • 2)
#19) – 0.25 + 4.88 + 3.25 = 4.88 + 3.25 – 0.25
Practice: (pp 74+75) Use the distributive property to simplify:
#19) (v- 3)4
#21) - 2(7z + 3)
Answers: 2) 5+ (91+ 11) or (5+ 91) + 11 equals 107; 5(91 • 11) or (5 • 91)11 equals 5005; 6) Associative property, 180; 19) Commutative Property, 7.88; p74 #19) 4v – 12; 21) -14z -6
Chapter 2 –Properties of Numbers (2-1) (2-2)
Practice: (pp 74+75) Use the distributive property to simplify:
#39) – 7(- 3n + 2)
#40) (4 – t) (- 7)
#45) Suppose your friend wrote 7(2m +t) = 14m + t. What error did your friend make?
Answers: 39) +21n – 14; 40) – 28 + 7t; 45) My friend forgot to multiply 7 with t. She should have gotten 14m + 7t.
Chapter 2 –Simplifying Variable Expressions (2-3) (2-4)
WORD BANK:
term
constant
like terms
coefficient
simplify
deductive reasoning
equation
open sentence
solution
A______is a number or the product of a number and a variable. A ______is a term that has no variable ( just a number). ______have identical variables. A ______is a number that multiplies a variable. You ______a variable expression by replacing it with an equivalent expression that has as few terms as possible.
______is the process of reasoning logically from given facts to a conclusion. As you justify your steps in solving equations, you are using deductive reasoning.
An expression does not have an equal sign.
v Label this example: 4X + 5
Practice: (pp. 78+79) Simplify each expression:
#25) – 4 (a + 3) – a
#27) 3 (g + 5) + 2g
#31) 3 (2n + 4) - 2 (3n + 6)
Answers: 25) – 5a – 12; 27) 5g + 15; 31) 0.
Chapter 2 –Simplifying Variable Expressions and
Solving Variable Equations (2-3) (2-4)
Practice: (pp. 79+) Simplify each expression:
#41) 18u – 6 (9k – 7 – 10u) + 4k
(P 82) State whether each equation is true, false, or an open sentence:
#30) – 2 (3 – 8) = 2 [- 3 – (- 8)]
#31) 6 [- 3 – (- 5)] = 2 (- 4 + 10)
#32) 4 [2 + (- 6)] = 2[ x – (- 12)]
#37) Could (– 2) be the solution to: c/2 – 8 = 3(- 3)?
#40) Could (12) be the solution to: 3b ÷ 18 = 2?
Answers: 41) – 42u – 50k -42; 30) false -10 does not equal +10; 31) true 12 = 12; 32) open sentence – 48 = 2x + 24; 37) yes, (- 2) is the solution to this open sentence because (- 9) = (- 9); yes (12) is the solution to this open sentence because 36 ÷ 18 = 2.
Chapter 2 –Solving Equations (2-5) (2-6)
WORD BANK:
inverse operations
subtraction property of equality
addition property of equality
division property of equality
multiplication property of equality
When you solve an equation, your goal is to get the variable alone on one side of the equation. You use ______, which undo each other, to get the variable alone. The ______states that you can subtract the same amount from each side of an equation. The ______states that you can add the same amount to each side of an equation. The ______states that you can divide the same amount from each side of the equation. The ______states that you can multiply the same amount to each side of the equation.
Practice: (pp 90+94-95+98)
#42) – 215 + e + (- 43) = - 145
#45) 183 + k – 20 = - 15
#2) 108 = 9x #29) 39 = c • 3
#21) 6 = a / 7 #23) n/15 = 7
#36) r/- 9 = - 18 #39) – 18/k = - 6
Answers: 42) +113; 45) – 178; 2) 12; 29) 13; 21) 42; 23) 105; 36) 162; 39) 3.
Chapter 2 –Solving Equations (2-5) (2-6)
Practice: (pp 98+120)
#6) Trains leave New York for Boston every 40 min. The first train leaves at 5:20 A.M. What departure time is closest to 12:55 P.M.?
#10) The sum of the page numbers on two facing pages is 245. The product of the numbers is 15,006. What are the page numbers?
#13) Ron puts three pennies in a jar. His father offers to triple the total amount of money in Ron’s jar at the end of each day. How much is in the jar at the end of one week?
#15) t/- 5 = 15 #26) 5 = s/- 7
Answers: 6) 12:40 P.M.; 10) pp. 122+123; 13) 37 or 3 • 3 • 3 • 3 • 3 • 3 • 3 or $65.61; 15) – 75; 26) – 35.
Chapter 2 –Problem Solving (2-7)
Solving word problems
ü Rewrite the information in your own words.
ü Translate as much as possible into mathematical expressions and/ or equations: Make a Verbal Model.
ü (Let x=…) Then write any other unknown quantity in terms of x. (Ex: Let x= hats; shirts= 4x)
ü Start with what you know.
Make Verbal Models:
§ A veterinarian weighs 140 lb. When she steps on a scale while holding a dog, the scale shows 192 lb. Does the dog weigh 52 lb?
§ A family’s expenses are $1,200. One parent makes $850. Must the other earn $400 for them to meet their expenses?
§ Together Mike and Amy weigh 350 lb.s Amy weighs 50 lb.s less than Mike. How much does Mike weigh?
§ A recipe calls for 4 cups of flour. You have 20 cups. How many batches could you make?
Answers:
Chapter 2 –Problem Solving (2-7)
§ Larissa ran 15 miles per week before she decided to train for the marathon. The first week of training she ran 17 miles. The second week she ran 19 miles. If she continues this pattern, how far will she run in the 12th week?
§ Fencing: Thirty-six sections of fencing, all the same length, are joined to form a fence 180 m long. How long is each section of fencing?
§ Brian bought a used bike for $25 less than its original price. He paid a total of $88.00 for the bike. What was the original price of the bike?
§ Mara ordered 5 bags of seed for $7 each and 3 wildflower seeds for $9 each. She also paid a $13 shipping fee. How much did she spend?
§ Kim ran 2.76 miles on Monday, 2.86 miles on Tuesday, and 2.96 miles on Wednesday. If this pattern continues, how far will she run by the end of Saturday?
Answers:
Chapter 2 –Problem Solving (2-7)
§ Pets: Rico’s dog has a litter of 4 puppies. The puppies weigh 2.33 lb., 2.73 lb., 2.27 lb., and 2.64 lb. How much did they weigh altogether? What are two ways that you could solve this problem?
§ Geography: Lake Superior has an area of about 31,760 square miles. Lake Erie is 21,840 square miles smaller. How big is Lake Erie?
Answers:
Chapter 2 –Graphing and Solving Inequalities (2-8) (2-9)
WORD BANK:
inequality
subtraction property of inequality
addition property of inequality
division properties of inequality
multiplication properties of inequality
An ______is a mathematical sentence that contains <, >, ≤, or . Any number that makes an inequality true is a solution of the inequality. You can graph the solutions of an inequality on a number line.
Example:
Solving an inequality is similar to solving an equation. You want to get the variable alone on one side of the inequality. The ______states that you can subtract the same number from each side of an inequality. The ______states that you can add the same number to each side of an inequality.
The ______states that if you divide each side of an inequality by the same positive number, you leave the inequality symbol as it is. If you divide each side of an inequality by the same negative number, you must reverse the inequality symbol. The ______states that is you multiply each side of an inequality by the same positive number, you leave the inequality symbol as it is. If you multiply each side of an inequality by the same negative number, you must reverse the inequality symbol.
Practice: (pp 104+105) Write an inequality for each graph:
Chapter 2 –Graphing and Solving Inequalities (2-8) (2-9)
Practice: (pp 104+105, 108+109) Write an inequality and make a graph for each situation:
#35) A student pays for three movie tickets with a twenty-dollar bill and gets change back. Let t be the cost of a movie ticket.
#33) Explain why graphing the solutions of an inequality is more efficient than listing all the solutions of the inequality.
Solve each inequality then graph the solutions:
#6) – 7 < 5 + x #8) p + 22 - 10
#18) 12 y - 5 #27) a – 0.5 < 2.5
#37) Which of the inequalities m > -2, m < - 2, - 2 < m, and - 2 > m are solutions to
m + 4 > 2? Explain.
#15) m/6 - 18 #32) ½ x - 3
#43) j – 7 24 #19) r/13 3
Answers: 35) 3t < 20; 6) x> -12; 8) p - 32; 18) y 17; 27) a < 3; 37) m > 2; 15) m -108; 32) x - 6; 43) j 31; 19) r 39
Reflections: Chapter 2 (7th grade)
Make a graphic organizer showing all of the operations and which properties apply to each. Be sure to include the communicative, associative, and distributive.
What do the subtraction property of equality, the addition property of equality, the multiplication property of equality, and the division property of equality have in common? Why do we use each property?
How do you know which property to use to solve an equation?
Reflections:
Choose one or more of the following:
Big Ideas / Creative Learning Personalities:
§ Illustrate the most important terms / vocabulary words included in this chapter. How did you decide which terms / words to illustrate?
§ Draw a comic strip to demonstrate one or more of the properties or mathematical terms included in this chapter. Why did you choose this property or term?
§ Write a poem about one or more of the concepts included in this chapter. Why did you choose this concept?
§ Create a game to review the most important skills included in this chapter. How did you choose what to include in your game?
Sequential / Verbal Learning Personalities:
§ Summarize and explain how to solve an equation. Which properties would you use and why would you use them?
§ Make a graphic organizer showing all of the operations and which properties apply to each. How do the communicative, associative, and distributive properties help us to solve equations?
§ Create and solve 3 of your own problems similar to those found in this chapter. Which of your problems is easiest to solve and why?
§ Design a crossword of the most important terms / vocabulary words included in this chapter. Which terms will be the hardest to guess and why?
§ Write a letter to a classmate who was absent describing the most important aspects of this chapter. How can you be sure that you didn’t miss anything?
§ Make a proof of study for this chapter. What do you think is most important and why?