Download Grade 8- Chapter 2

Chapter Two –Solving Equations (2-1)
WORD BANK:
solution
isolate
equivalent equations
inverse operations
reciprocal
An equation is like a balance scale because it shows that two quantities are equal. To
find the ______________ to an equation you must do the same thing to both sides of the
equation.
_______________ _____________ are equations that have the same solution (or
solutions).
To determine the value of a variable you must ________________ the variable.
One way to solve equations is to use _____________ _______________, or undo that
which has been done.
When the coefficient is a fraction, you can use a _________________ to solve the
equation.
The following problems are copied from Prentice Hall Algebra I:
Practice: (pp. 77 +78)
#9) c – 7.88 = 9.24
#23) m/3 = - 6
#26) 10 = a/-2
#27) – c/7 = 35
#28) – r/3 = -101
#35) – 1/6 m = 12
Answers: solution; equivalent equations; isolate; inverse operations; reciprocal; 9) 17.12; 23)-18; 26) -20; 27) -245; 28)
303; 35)-72
Steps for solving a multi-step equation: (2-2) and (2-3)
1. Clear the equation of fractions and decimals so that you can work with whole #s.
2. Use the distributive property to remove parentheses on each side.
3. Combine like terms.
4. Undo operations.
Practice: (p. 84)
#1) 1 + a/5 = - 1
#4) y/2 + 5 = -12
#14) 41 = 2/3 x – 7
#46) 1/2 = 2/5 c – 3
#48) 0.4x + 9.2 = 10
(p. 92)
# 42) 0.5(x – 12) = 4
#46) 1/4(m- 16) = 7
Answers: 1) - 10; 4) – 34; 14) 120; 46) 8 3 / 4; 48) 2; 42) 20; 46) 44
Solving Equations with Variables On Both Sides (2-4)
WORD BANK:
identity
An equation that is true for every value of a variable is an ________________.
An equation has no solution if your answer is not true. (i.e. -5 = 7 is not true so the
equation 6m – 5 = 7m + 7 - m has no solution.)
Steps for solving a multi-step equation:
1. Clear the equation of fractions and decimals so that you can work with whole #s.
2. Use the distributive property to remove parentheses on each side.
3. Combine like terms.
4. Undo operations.
Practice: (p 93)
#64) A work crew has two pumps, one new and one old. The new pump can fill a
tank in 5 hours. The old pump can fill the same tank in 7 hours.
a. How much of a t5ank can be filled in 1 hour with the new pump? With the
old pump?
d. Write and solve an equation for the time it will take the pumps to fill one
tank if the pumps are used together.
#71) The gas tank in Royston’s car holds 12 gal of gasoline. The car averages 29 mi/gal.
Royston filled up the tank and then drove 140 mi. About how many gallons of gasoline
are left in the tank?
Answers: identity; 64) a)1 / 5, 1 / 7; d) 1 / 5 t + 1 / 7 t= 1; 2 11 / 12 h; 71) 8 gal.
Solving Equations with Variables on Both Sides
To solve an equation that has variables on both sides, use the addition or subtraction
properties of equality to get the variables on one side of the equation.
Practice: (p. 98)
#4) 5y – 3 = 2y + 12
#10) 6b + 14 = - 7 – b
#13) 30 – 7z = 10z - 4
#15) – 36 + 2w = - 8w + w
Solve each equation. Determine whether it is an identity or has no solution.
#28) 18 x – 5 = 3(6x – 2 )
#30) 3(x – 4) = 3x – 12
#33) 0.5y + 2 = 0.8y – 0.3y
Answers: 4) 5; 10) -3; 13) 2; 15) 4; 28) no solution; identity; no solution
Equations and Problem Solving (2-5)
WORD BANK:
consecutive integers
________________
uniform motion
_________________ differ by one.
Formulas are special types of equations, containing 2 or more variables. You solve for
one variable in terms of the others.
An object that moves at a constant rate is said to be in _______________
______________. The formula d = r*t gives this relationship.
Uniform motion problems may involve vehicles moving in the same direction, opposite
directions away from each other, or towards each other, or round trips.
Example:
A train and a bus traveled to the same destination, arriving at the same time. The bus
left 1 hr before the train. What was the distance they traveled?
time 1
2
3
4
5
6
7
hr hr
hr
hr
hr
hr
hr
Train 60 120 180 240 300 360 420
mi mi
mi
mi
mi
mi
mi
Bus
50 100 150 200 250 300 350
mi mi
mi
mi
mi
mi
mi
A table can help you to solve these problems by illustrating the relationship between the
two vehicles. The distance was 300 miles. This is the first common multiple.
Practice: (p. 107)
#5) The sum of the two consecutive integers is -35. If n = the first integer, which
equation best models the situation? How do you know?
a. n(n + 1) = -35
b. n +2n = -35
c. n + (n +1) = -35
d. n + (2n + 1) = -35
#6) The sum of two consecutive even integers is 118. What are the integers?
#7) The sum of two consecutive odd integers is 56. What are the integers?
Answers: consecutive integers; uniform motion; 5) c; 6) 58 and 60; 7) 27 and 29.
Equations and Problem Solving (2-5)
Practice: (pp. 108 and 109)
#10) A moving van leaves a house traveling an average rate of 35 mi/hr. The
family leaves the house ¾ hour later following the same route in a car. They travel
at an average rate of 50 mi/hr.
Vehicle
Rate
Time
Distance Traveled
Van
35 mi/hr.
t
35t
Car
50 mi/hr.
t – 3/4
50(t – ¾)
How long did it take the car to catch up with the van?
#12) Juan drives to work. Because of traffic conditions, he averages 22 miles per
hour. He returns home averaging 32 miles per hour. The total time is 2 ¼ hours.
How long did it take Juan to drive to work?
Trip
Rate
Time
Distance
To work
22 mi/hr.
x
Same
From work
32 mi/hr.
2 ¼ hr. – x
Same
#14) John and William leave their home traveling in opposite directions on a
straight road. John drives 20 mi/hr. faster than William. After 4 hours they are
250 miles apart. How fast were they each going?
Person
Rate
Time
Distance
John
x + 20 mi/hr
4 hrs.
250 miles
William
x
same
same
Answers: 10) 1 ¾ hr; 12) 1 1/3 hr.; 14) John’s rate was 41 ¼ mi/hr; William’s rate was 21 ¼ mi/hr.
Formulas (2-6)
WORD BANK:
literal equation
A ________________ _______________ is an equation involving two or more
variables. Formulas are special types of literal equations. To transform a literal equation,
you solve for one variable in terms of the others. This means that you get the varable
you are solving for alone on one side of the equation.
Example: d= r t, solve for r so d ÷ t = r
(p. 108)
#15) Two bicyclists ride in opposite directions. The speed of the first bicyclist is 5 miles
per hour faster than the second. After 2 hours they are 70 miles apart. Find their rates.
second= x mi/hr
t = 2 hours
first= x+5 mi/hr
d = 70 miles
so 70 miles ÷ 2 hours = x + (x+ 5 mi/hr) or 35 = 2x + 5
30 = 2x so x = 15
The second rider= 15 mi/hr and the first rider = 20 mi/hr. They are 35 miles further apart
each hour.
Practice: (p 108)
#11) A jet leaves the Charlotte, North Carolina, airport traveling at an average
rate of 564 km/h. Another jet leaves the airport one half hour later traveling at
744km/h in the same direction. How long will the second jet take to overtake the
first?
Remember: d = r*t
The distance of each jet is equal, since they meet.
(Time of the first jet =
)* (Rate of the first jet =
) is _____________
(Time of the second jet =
) *(Rate of the second jet =
) is ______________
#13) An airplane flies from New Orleans, Louisiana, to Atlanta, Georgia, at an average
rate of 320 miles per hour. The airplane then returns at an average rate of 280 miles per
hour. The total travel time is 3 hours. How long did it take to get from New Orleans to
Atlanta?
Remember: d = r*t
The distance is equal since it was a round trip.
(Time of the first trip =
) *(Rate of the first trip =
) is _________________
(Time of the second trip =
) *(Rate of the second trip =
Answers: #11) 1 17 / 30 hours; #13) 1 2 / 5 hours
) is _________________
Using Measures of Central Tendency (2-7)
WORD BANK:
measures of central tendency
mean
median
outlier
mode
range
stem-and-leaf plot
To organize and summarize a set of data, you look for the ____________ of
__________ ____________.
The ____________ is the average value. (sum/ total # of items)
The _____________ is the middle-most piece of data.
The ______________ is the most frequent piece of data.
An ______________ is a data value that is much higher or lower than the other data
values in the set.
The ______________ is the difference between the highest and the lowest data values,
and tells you how big a spread there is.
A ____________ ______ ___________ __________ helps you to organize numerical
data by separating the tens from the ones and listing them in rows.
You can compare two data sets by creating back-to-back stem-and-leaf plots with the
tens listed in the center column and the ones to the right and the left.
Practice: (p. 122) Make stem-and-leaf plots:
#24) A wildlife manager working at the Everglades National Park in Florida measured and
tagged adult make crocodiles. The data he collected are below.
Crocodile Lengths (centimeters)
240 250 250 230 280 240 230 240 210 220 250 270
a) What are the mean and median lengths of the crocodiles?
b) The wildlife manager captured another crocodile. Its length was 3.3m. What
is the mean with this new piece of data? What is the median? Round to the
nearest tenth.
Using Measures of Central Tendency (2-7)
#25) Two manufacturing plants create sheets of steel for medical instruments. The backto-back stem-and-leaf plot below shows data collected from the two plants.
Manufacturing Plant A
87442
431
means 6.1
1
4
5
6
7
6
Manufacturing Plant B
359
27
34
2
3
means 6.3
a) Find the mean, median, mode, and range of each set of data.
b) Which measure of central tendency best describes each set of data? Explain your
thinking.
c) Which plant has the better quality control? How do you know?
Answers: measures of central tendency; mean; median; mode; outlier; range; stem-and-leaf plot; 24) 2.425, and 2.4;
25) Plant A 5.79, 5.75, 5.4, 1.2 and Plant B 5.56, 5.45, none, 2.9, and 25b) Plant A-mean. There are no outliers. Plant Bmedian. The mean is thrown off by high outliers. ; and 25c) Plant A has better quality control because there is a smaller
range.
Grade 8- Chapter 2
Goals: Key Understandings: Students will understand that…

Expressions and equations are fundamental tools for modeling situations and solving
problems.
 The distributive, associative, commutative, and identity properties can be used to
simplify variable expressions, equations, inequalities, and/or solve problems.
 We use the subtraction/ addition/ division/ and multiplication properties of equality to
solve equations and inequalities.
Essential Questions:




How do we use algebraic expressions or equations to model situations and solve
problems?
How do we express solutions; what do the solutions mean; are the solutions
reasonable?
How do we use properties to help solve equations?
How can data be organized and summarized using visual representations (stemand-leaf plots and box-and-whisker plots)?
Students will know…
Students will be able to…
 You can solve equations by undoing the
 Solve one-step equations using the
operations involved to get the variable
addition, subtraction, multiplication,
alone on one side of the equal sign.
and division properties of equality. (21)
 These are called ‘inverse’ properties. For
 Write equations to solve word
example, addition is the inverse of
problems; then solve.
subtraction and division is the inverse of
multiplication.
______________________________________





First you use the addition or
subtractions property of equality to get
the term with the variable alone on one
side of the equal sign.
Then you use the multiplication or
division property of equality to solve for
x.
You can clear an equation of decimals
by multiplying each term on both sides
by a power of ten.
You can clear an equation of fractions
by multiplying each term on both sides
by the reciprocal.
You sometimes have to use the
_______________________________________________



Solve two-step equations using the
addition, subtraction, multiplication, and
division properties of equality. (2-2)
Multiply by -1 to find the value of b if
the value of –b is known.
Justify each step as they solve
equations.
_________________________________________________

Solve multi-step equations by
clearing the equation of fractions and
decimals; using the distributive
property if necessary; combining like
terms; undo addition or subtraction;
then undo multiplication or division.
(2-3)
Grade 8- Chapter 2
distributive property before you can
solve the equation.
 Like terms must have the same variable
and the same exponent. (Ex: 5y and 6y
are like terms while 5y and 6y2 are not.)
_____________________________________
 An equation has no solution if no value
of the variable makes the statement
true. Ex: 4x ≠ x + x
 An equation is an ‘identity’ if every value
of the variable makes the statement
true. Ex: 2x= x + x
_____________________________________
 First describe what each variable
represents through ‘let’ statements.
Then express one unknown in terms of
the other. Ex: Let n= the first # and n+1
= the second #. If n + n+1 = 5 then n= 2
and n+1 = 3.
 There are 3 types of rate/ time/ distance
problems: opposite directions; round
trip; and same direction. You can use
tables to organize your information;
write your equation; then solve for x.
_____________________________________
 A literal equation is an equation
involving two or more variables and their
relationship(s). Ex: P= 2l + 2w
_________________________________________________


Solve equations with variables on both
sides by adding or subtracting the
variable term, thus isolating it on one
side of the equation. (2-4)
Translate word problems into
equations then find the solution.
_________________________________________________

Define one variable in terms of
another such as with rectangles,
consecutive numbers, and rate / time/
distance problems. (2-5)
__________________________________________________
____________________________________
 Different measures of central tendency
(mean, median, and mode) are
appropriate for various situations.
 There are a variety of ways to represent
data.

_________________________________________________


______________________________________
Transform geometric formulas by
solving for one variable, by getting it
alone on one side of the equal sign, in
terms of others. (2-6)
Use measures of central tendency
(mean, median, mode, and range). (2-7)
Organize, interpret, and extend data sets
through stem-and-leaf plots.
__________________________________________________
Grade 8- Chapter 2
Learning Plan
Day
Section / Objectives
Homework



Q1
Q2
Q3
Capstone Activity- Performance Task
(Project)
12
Chapter Review
13
Chapter Test
Goals: Key Understandings: Students will understand that…
 Equations are fundamental tools for modeling situations and solving problems.
 All numbers have a distinct position on the real number line.
 The absolute value of a number and its opposite will be the same.
Essential Questions:
 How can we use algebraic expressions or equations to solve problems?
 Why is it important to use precise mathematical vocabulary and symbols?
 How can we use a number line to solve problems?

How can we write algebraic expressions to model patterns of numbers?