1.1 expressing rational numbers as decimals eq: how...
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1.1 Expressing Rational Numbers as Decimals
EQ: How do you rewrite rational numbers and decimals, take square roots and cube roots and approximate irrational numbers?
What is a terminating decimal?
A decimal number that has digits that do not go on forever.
.025 3.0375
What is a terminating decimal?
A decimal number that has digits that repeat forever.
.0333… 0.142857142857…
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1.1 Expressing Decimals as Rational Numbers
EQ: How do you rewrite rational numbers and decimals, take square roots and cube roots and approximate irrational numbers?
Decimal to Fraction
Look at the last digit and determine the place value.
.375 = 375 = 75 = 31000 200 8
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1.1 Expressing Decimals as Rational Numbers
EQ: How do you rewrite rational numbers and decimals, take square roots and cube roots and approximate irrational numbers?Repeating Decimal to Fraction
1. Let x = the number2. Identify the place value of
the last repeating digit.3. Multiply (by 10, 100,
1000)4. Subtract x 5. Divide
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1.1 Finding Square Roots and Cube Roots
EQ: How do you take square roots and cube roots?
Square Root There are two square roots for every positive number.
62 = 36 and (−62) = 36
36 = ±6
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1.1 Finding Square Roots and Cube Roots
EQ: How do you rewrite rational numbers and decimals, take square roots and cube roots and approximate irrational numbers?
Cube Root There is one cube root for every positive number.
The cube root of 8 is 2 because . 23 = 8
38 = 2
2 ⋅ 2 ⋅ 2 = 8
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1.2 Classifying Real Numbers
EQ: How can you describe relationships between sets of real numbers?
Real Numbers
Set of rational numbers and irrational numbers.
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1.3 Comparing Irrational Numbers
EQ: How do you order a set of real numbers?
Compare Real Numbers
Use perfect squares to estimate the square root.
2 12 = 122 = 4
A number between 1 and 2.
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2.1 Simplifying Expressions with Powers
EQ: How can you develop and use the properties of integer exponents?
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2.2 Scientific Notation with Positive Powers of 10
EQ: How can you scientific notation to express very large quantities?
Move the decimal behind the first number and drop the zeros.
Count the number of places the decimal was moved.
Scientific Notation
1.2300000000 = 1.23 x 1011
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2.3 Scientific Notation with Negative Powers of 10
Move the decimal after the first nonzero number.
Count the number of places the decimal was moved.
Scientific Notation
EQ: How can you use scientific notation to express very small quantities?
When the number is between 0 and 1 the exponent is negative.
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3.1 Representing Proportional Relationships with Equations
Ratio of one quantity to the other is constant.
ProportionalRelationship
EQ: How can you use tables, graphs and equations to represent proportional situations?
Described by one of these equations:
Constant of Proportionality
𝑘 =𝑦
𝑥𝑦 = 𝑘𝑥
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Representing Proportions with Equations
Step 2 For the number of hours, write the relationship of the amount earned and the number of hours as a ratio (another word for ratio: _____) in simplest form.
amount earned
number of hours
12
1
24
48
96
Is the relationship proportional? Yes / No Explain.
𝑘 =𝑦
𝑥
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Representing Proportions with Equations
Step 3 Write an equation
Let x = ____________________
Let y = ____________________
Use the equation y kx
The equation is:
amount earned
number of hours
12y x
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3.1 Representing Proportional Relationships with Graphs
Graph will be a line that goes through the origin (0,0).
ProportionalGraph
EQ: How can you use tables, graphs and equations to represent proportional situations?
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Representing Proportions with Graphs
Earth weight
(lbs)6 12 18 30
Moon weight
(lbs)
The graph shows the relationship between the weight of an object on the Moon and its weight on Earth. Write an equation for this relationship.
Step 1 Use the points on the graph to make a table.
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Representing Proportions with Equations
Step 2 Find the constant of proportionality.
Moon weight
Earth weight
1
6
2
3
5
The constant of proportionality is:
𝑘 =𝑦
𝑥
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Representing Proportions with Graphs
Step 3 Write an equation
Let x = ____________________
Let y = ____________________
Use the equation y kx
The equation is:
weight on Earth
weight on Moon
1
6y x
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3.2 Rate of Change
Rate of Change
EQ: How do you find a rate of change?
constant rateThe cost is $0.75 per Snickers.
.75y x b
Example
Dependent variable Independent
variable
The cost depends on the number you buy.
chg. in dependent variable ( )
chg. in independent variable ( )
y
x
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Rate of ChangeEve keeps record of the number of lawns she has mowed and the money she has earned. Tell whether the rates of change are constant or variable.
Step 1 Identify the independent and dependent variables.
Independent Variable _____________________
Dependent Variable _______________________
Explain ________________________________
# of lawns
$ earnedThe amount earned depends on the number of lawns mowed.
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Rate of ChangeStep 2 Find the rates of change
Day 1 to Day 2: change in $
change in lawns
45 15
3 1
30
2 15
Day 2 to Day 3:
Day 3 to Day 4:
change in $
change in lawns
change in $
change in lawns
90 45
6 3
120 90
8 6
45
3
30
2
15
15
The rates of change are / are not constant. Price per lawn is _________.
The rates of change are constant. Price per lawn is $15.
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3.2 Slope
Slope (m)EQ: How do you find the slope of a line?
2 1
2 1
rise
run
y ym
x x
The steepness of a line.
x x
x x
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Method 1
Label ordered pairs
1 1 2 2
1,4 3, 5
, ,x y x y
Write down formula 2 1
2 1
y ym
x x
Substitute values 5 4 9 9
3 1 4 4m
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Method 2
Write ordered pairs
Subtract y values
and x values
1 , 4
3, 5
4 5 9
1 3 4
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Method 3
Plot the points
Find the rise / run
1 , 4
3, 5
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3.3 Interpreting Unit Rate as Slope
Rate
EQ: How do you interpret the unit rate as slope?
dollars
oranges
Ratio comparing quantities measured in different units.
Unit Rate Ratio where the second quantity is 1.
Example $ per orange oranges per $1oranges
dollars
$2 10 $0.20
10 10 1
10 2 5
$2 2 $1
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A storm is raging on Misty Mountain. The graph shows the constant rate of change of the snow level on the mountain.
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4.2 Determining Slope & Y-intercept
Slope Intercept Form of an Equation
EQ: How can you determine the slope and y-intercept of a line?
Rate of Change
𝑦 = 𝑚𝑥 + 𝑏
slope y-int
Start value
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Find the slope (m) and Y-intercept (b).
change in 𝑦
change in 𝑥=
change in 𝑦
change in 𝑥=
change in 𝑦
change in 𝑥=
The rate of change (m) is ____
The start value (b) is ____
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4.2 Determining Slope & Y-intercept
Step 1
EQ: How can you graph a line using the slope and y-intercept?
𝑦 = 𝑚𝑥 + 𝑏
slope y-int
Identify m and b
Step 2 Plot the point of the y-intercept.Use the slope to find a second point.
Step 3
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4.4 Proportional and Nonproportional
Proportional
EQ: How can you distinguish between proportional and nonproportionalsituations?
𝑦 = 𝑚𝑥
Nonproportional 𝑦 = 𝑚𝑥 + 𝑏
Line goes through the point (0,0)
Starting point of line is the y-intercept