Chapter 1 Integers Idea and Examples Chart

Copyright © Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Red 1

Chapter 1 Integers

Idea and Examples Chart

1–6. Sample answers are given.

1. Integers: … , −3, −2, −1, 0, 1, 2, 3, …

Example

−586

Example

0

Example

16

2a. Adding integers with the same sign:Add the absolute values of the integers.Then use the common sign.

Example

16 + 17 = 33

−5 + (−4) = −9

−55 + (−45) = −100

Example

Example

2b. Adding integers with different signs:Subtract the lesser absolute value from thegreater absolute value. Then use the sign ofthe integer with the greater absolute value.

Example

−8 + 2 = −6

−97 + 19 = −78

8 + (−2) = 6

Example

Example

3. Additive Inverse Property: The sum of an integer and its additive inverse, or opposite, is 0.

5 + (−5) = 0

Example

−100 + 100 = 0

Example

16 + (−16) = 0

Example

4. Subtracting integers: To subtract an integer,add its opposite.

1 − 4 = 1 + (−4) = −3

Example

−1 − (−4) = −1 + 4 = 3

Example

1 − (−4) = 1 + 4 = 5

Example

2 Big Ideas Math Red Copyright © Big Ideas Learning, LLC. All rights reserved.

Chapter 1 (continued)

5a.

Example

−6 · (−3) = 18

6 · 3 = 18

−18(−16) = 288

Example

Example

Multiplying integers with the same sign: The productof two integers with the same sign is positive.

5b.

Example

−6 · 3 = −18

6 · (−3) = −18

18(−16) = −288

Example

Example

Multiplying integers with different signs: The productof two integers with different signs is negative.

6a. Dividing integers with the same sign: Thequotient of two integers with the same signis positive.

100 ÷ 4 = 25

Example

−100 ÷ (−4) = 25

Example

−98−7

Example

= 14

6b. Dividing integers with different signs: Thequotient of two integers with different signsis negative.

100 ÷ (−4) = −25

Example

−100 ÷ 4 = −25

Example

−987

Example

= −14


Chapter 2 Rational Numbers

Process Diagram


1.

ab

Use long division to findthe quotient a ÷ b.

Write the rational numberas a fraction if necessary.

Writing rational numbersas decimals

as a decimal.Write −5 15

= −5.2.So, −5 15

Example

If the remainder is 0,the rational number is a

terminating decimal.

=−5 15

− 265

26.05.2

52 5

1 0−

1 00

−

as a decimal.Write 215

215

= 0.13.So,

Example

If the remainder repeats,the rational number is a

repeating decimal.

2.0000. 1 33

151 5

5045

−

−5045

5−

2. Subtracting rational numbers

=− 14

−− 32

− 14

+ 32

= − 14

+ 64

= 54

, or 1 14

Add the opposite of the rationalnumber being subtracted.

if the addends havedifferent signs

Subtract the lesser absolute valuefrom the greater absolute value.

Write the sum using the sign ofthe rational number with the

greater absolute value.

9.03 − 5.41 = 9.03 + (−5.41)= 3.62

Examples

=−2 58

−2 58

1 38

−1 38

− +

= −4

2.58 − (−3.67) = 2.58 + 3.67= 6.25

Examples

if the addends havethe same sign

Add the absolute valuesof the rational numbers.

Write the sum usingthe common sign.



3. Multiplying rationalnumbers

If the factorshave the same sign

If the factorshave different signs

The product is positive. The product is negative.

(––) (––) = (––) (––)Examples Examples

= —

34

29

34

29

= – 16

–3.7× –2.4

1 48+ 7 4

8.88

1

2

1

3

The product is 8.88.

12

(–2–) (––)= —

35

13

35

73

= ––, or –1– 75

25

–2.64 5

14

–1.1

1

1

The product is –1.17.

23

4.

(− )

Dividing rationalnumbers

If the dividendand divisor havethe same sign

If the dividendand divisor havedifferent signs

The quotient is positive. The quotient is negative.

–2.45 ÷ (–0.7) = 3.5

Examples Examples

–6 ÷ (– ) = – ÷ (– )14

14

15

254

1551

= – 254

= –

= 1254 , or 31

–2.45 ÷ (0.7) = –3.515

310

23

15

32

= –

= –


Chapter 3 Expressions and Equations

Four Square


1.

5x2 + 6x – 3x2 + 8 – x= 5x2 + 6x + (–3x2) + 8 + (–1x)= 5x2 + (–3x2) + 6x + (–1x) + 8= [5 + (–3)]x2 + [6 + (–1)]x + 8= 2x2 + 5x + 8

Definition

An algebraic expressionis in simplest formwhen it has: 1. no like terms and 2. no parentheses.

WordsTo write an algebraicexpression in simplestform:Step 1: Rewrite as a sum.Step 2: Use theDistributive Property onparentheses, if necessary.Step 3: Rearrange terms.Step 4: Combine like terms.

Simplest form

Example9 – 3(–m ––) + 3m

= 9 + (–3)(–m + (––)) + 3m

= 9 + (–3)(–m) + (–3)(––) + 3m

= 9 + (–2m) + 1 + 3m= (–2m) + 3m + 9 + 1= (–2 + 3)m + (9 + 1)= m + 10

Example13

23

23

13

13

23

2.

(7 – w) + 3(–2w + 4)= 7 + (–1w) + 3(–2w) + 3(4)= 7 + (–1w) + (–6w) + 12= (–1w) + (–6w) + 7 + 12= –7w + 19

Definition

An algebraic expressionin which the exponentof the variable is 1

Examples

–7x, 2x + 3, 8 – –x

Non-examples:

x3, –5x2 + x, x7 – 9

Linear expressionExampleAdding linear expressions:

(4y + 7) – (y – 8) 4y + 7–(1y – 8)

4y + 7(–1y) + 8 3y + 15

Subtracting linearexpressions:

Example

14

+

3.

–r = – r

– = –

–r + – = – r + –

= –(r

Words

Write the expression asa product of factors.You can use theDistributive Property.

Example

Factor 12athe GCF.12a a

612a 6(2a 6 = 6(2a

ExampleFactor – out of –r + –.

–21p = –7 p 28 = –7–21p + 28 = –7 p = –7 p

Factor –7 out of –21p + 28.

Example

1

1

1 1 1

1

1

1 1

4. WordsTwo equations areequivalent equations ifthey have the samesolutions. You can usethe Addition, Subtraction,Multiplication, andDivision Properties ofEquality to write equivalentequations.

Algebra

a = b and a + c = b + ca = b and a – c = b – ca = b and a c = b c

a = b and – = –, c ≠ 0

Equivalent equationsExamplesx – 7 = 2 andx – 7 + 7 = 2 + 72d + 5 = –7 and2d + 5 – 5 = –7 – 524 = – and–4 24 = –4 –3c = –12 and – = –

x + 7 = 2 andx + 7 – 7 = 2 + 73c = –4 and – = (–4)7 = m + 3 and7 – 7 = m + 3 – 33x + 7 = 3 and 3x = 7 + 3

Non-Examples

y–4 y

–43c3

3c3

–123

ac

bc



5. WordsTo undo addition, use theSubtraction Property of Equality:subtracting the same numberfrom each side of an equationproduces an equivalent equation.To undo subtraction, use theAddition Property of Equality:adding the same number to eachside of an equation produces anequivalent equation.

Algebra

If a + b = c,then a + b – b = c – b.

If a – b = c,then a – b + b = c + b.

Solving equations usingaddition or subtraction

Example

x – 4 = 12 +4 +4 x = 16

Check x – 4 = 1216 – 4 = 12 12 = 12

Example

?x + 4.1 = 12 –4.1 –4.1 x = 7.9

Check x + 4.1 = 127.9 + 4.1 = 12 12 = 12

?

6. WordsTo undo multiplication, use theDivision Property of Equality:dividing each side of an equationby the same number producesan equivalent equation.To undo division, use theMultiplication Property ofEquality: multiplying each side ofan equation by the same numberproduces an equivalentequation.

Algebra

If ab = c,then – = –, b ≠ 0.

If – = c,then – = b c, b ≠ 0.

Solving equations usingmultiplication or division

Example

–2x = 3–2x = 3 –2 –2 x = ––, or –1–

Check –2x = 3–2(––) = 3 3 = 3

Example

?

abb

cb

ab a

b

32

12

32

– = –3

– = (–3) x = –6

Check – = –3

– = –3

–3 = –3

?

x2x2

x2

–62

7. WordsUndo the operations in thereverse order of the orderof operations: 1. Undo addition or subtraction. 2. Undo multiplication or division.After solving for the variable,check your solution.

3x + 4 = 1 –4 –4 3x = –3 3x = –3 3 3 x = –1

Solving two-stepequations

Example

Example

– – – = ––

–– ––

–– = –1

– (––) = – (–1) a = 3

Example37

Check – – – = ––

– – – = ––

– + (–1) = ––

– + (––) = ––

–– = ––

?

?

?

37

a3

47

37

33

47

37

47

37

77

47

47

47

37

37

a3

a3

a3

47

Check 3x + 4 = 13(–1) + 4 = 1 –3 + 4 = 1 1 = 1

?

?

4(x – 2) = 124x + (–8) = 12 +8 +8 4x = 20 4 4 x = 5

Check4(x – 2) = 124(5 – 2) = 12 4(3) = 12 12 = 12

?

?


Chapter 4 Inequalities

Y Chart


1.

The sign between two expressions is an equal sign, =.

One number is the solution.

Writing equations Writing inequalities

The sign between two expressions is an

inequality symbol: <, >, < or > . More than one number

can be a solution.

Write one expression on the left and one expression on the right. Look for key phrases to determine which operation(s)

to use: +, −, ×, or ÷. Look for key phrases to determine where to place the

equal or inequality sign.

2.

A solution is represented by a closed circle, .


Graphing the solution of an equation

Graphing the solution of an inequality

The endpoint of the graph can be an open circle, ,

or a closed circle, . An arrow pointing to

the left or the right shows that the graph continues

in that direction. More than one number can

be a solution.

Solve for the variable. Graph the solution on a number line.

3.

For x greater than a number, use an arrow pointing to the right.

Graphing inequalities that use >

Graphing inequalities that use <

For x less than a number, use an arrow pointing to the left.

Use a number line to graph the solution. Use an open circle, . Use an arrow to show that the graph continues. Solution usually includes many numbers.



4. Graphing inequalitiesthat use > or <

Graphing inequalitiesthat use ≥ or ≤

Use an open circle, .

Use a number line.Use an arrow to show that the graph continues.

Use a closed circle, .

5. Solving inequalitiesusing addition

Solving inequalitiesusing subtraction

Use the Addition Property of Inequality: When you add the same number to each side of an inequality, the inequality remains true.

Use inverse operations to group numbers on one side.

Use inverse operations to group variables on one side.

Solve for the variable.

Example: x – 5 > 8 +5 +5 x > 13

Use the Subtraction Property of Inequality: When you subtract the same number from each side of an inequality, the inequality remains true. Example: x + 5 > 8 –5 –5 x > 3



6.

Use the Multiplication Property of Inequality(Case 1): When you multiplyeach side of an inequality bythe same positive number, the inequality remains true.

Use the Multiplication Property of Inequality(Case 2): When you multiplyeach side of an inequality bythe same negative number, the direction of the inequality symbol must be reversed for the inequality to remain true.

Use the Division Property of Inequality (Case 1): When

you divide each side of an inequality by the same

positive number, the inequality remains true.

Use the Division Property of Inequality (Case 2): When

you divide each side of an inequality by the same

negative number, the direction of the

inequality symbol must be reversed for the

inequality to remain true.

Solving inequalities using multiplication

Solving inequalities using division

Use inverse operations to group numbers on one side. Use inverse operations to group variables on one side. Solve for the variable.

Example: – > 10

– 5 > 10 5 x > 50

x5

x5

Example: ––x ≤ 10

–– (––x) ≥ –– 10 x ≥ –20

12

21

12

21

Example: 5x > 10

–– > –– x > 2

5x5

105

Example: –3x ≥ 9

–– ≤ –– x ≤ –3

–3x–3

9–3

7. Solving two-step equations

Solving two-stepinequalities

Use inverse operations to isolate the variable. Group numbers on one side, variables on the other side,

and solve for the variable.

The sign between two expressions is an equal sign, =.


The sign between two expressions is an

inequality symbol: <, >, < or > . More than one number

can be a solution.

8. Adding integerswith the same sign

Adding integerswith different signs

Sums can be negative or positive.

Add the absolute values of the integers. Then use the common sign.

Subtract the lesser absolute value from the

greater absolute value. Then use the sign of

the integer with the greater absolute

value.


Chapter 5 Ratios and Proportions

Information Wheel


1.

Rate

A ratio of two quantities

with different unitsExample:

Unit rate: A rate with

a denominator of 1

63 feet12 seconds

Example: 8 hits25 at bats

Example:$135

3 hours

Example of a unit rate:60 miles1 hour

2.

Unitrate

A rate with a denominator of 1Example: 5.25 feet

1 second

Example:You can write rates as unit rates.

=400 units

8 hours

50 units

1 hourExample:

You can write unit rates as

rates with denominators other than 1.=

0.5 mile1 hour 1 mile2 hours

Example: 60 miles1 hour

Example:72 words

1 minute

3.

Proportion

An equation stating that

two ratios are equivalentExample: =

You can use a table to write a

proportion. Example: See page 180.

The cross products of a proportion are equal. Example: 3 · 8 = 4 · 6

To solve a proportion,

you can use mental math.

Two quantities that form a proportion

are proportional.

34

68



4. The products a · d and b · c

in the proportion = Cross Products Propert

y

In words: The cross products

of a proportio

n are equal.

You can use t

he Cross Products

Property t

o solve a proportio

n. Example: =

2 · x = 3 · 4

x = 6

Numbers: = The cross productsare 3 · 8 and 4 · 6. Note that they

are both equal to 24.Cross products

Cross Products Property usingalgebra: ad = bc in = ,

where b ≠ 0 and d ≠ 0

ab c

d

34

68 a

bcd

23 4

x

5. The graph has a constant rate

of change.

The graph passes th

rough the o

rigin.

Example: Find the unit r

ate.

– = –32

1.51

The graph is a line that passes

through the origin. So, x and y are in

a proportional relationship.

Example:

Graphingproportionalrelationships

Example:

xy

23

46

69

+2

+3 +3

+2

0

2

4

6

8

y

0 2 4 6 8 x

6. Method 1: Use mental math.

Example: =

5 · 2 = 10

, so x = 1 ·

2 = 2

Method 3: U

se the C

ross

Products Propert

y.

Example: =

1 · 10 = 5 · x

10 = 5x

2 = x

Method 2: Use the MultiplicationProperty of Equality. Solving

proportions

Example: = 10 · = 10 ·1

515

x10

x10

15 x

10

15

x10

2 = x



7. Rate of change between

any two points on a line

A measure of the

steepness of a line

Through (0, 0) and (4

, 3):

slope =

Through (2, 3) and (−6, −3):

slope = =

vertical changehorizontal change Slope

change in ychange in x

34

34

−6−8

8. Two quantities x and y show

direct variation when y = kx,

where k is a number and k ≠ 0. The graph of y

= kx is a lin

e that

passes through th

e origin.

y = 4x

k = 4

y = xk =

“y varies directly with x.” Directvariation

“x and y are directly proportional.”

14

14


Chapter 6 Percents

Summary Triangle


1.

Writing a decimal

as apercent

Multiply by 100, or just move the decimal point two

places to the right. Then add a percent symbol.

n = (n 100)%

Example: 0.231 = 23.1%

2.

Write the numbersas all fractions, all decimals,

or all percents. Then compare.Use a number line if necessary.

Comparingand ordering

fractions,decimals, and percents

Examples:Which is greater, 5.1% or ? = = 5%. So, 5.1% is greater.

Which is greater, 3.3% or 0.03?3.3% = 0.033. So, 3.3% is greater.

1201

205

100

3.

Example: What percent of 40 is 25?

So, 62.5% of 40 is 25.

Use a proportionto represent “a is p

percent of w.”

aw

Thepercent

proportion

p100

2540

p100

=

part

whole

percent

– –62.5 = p

4.

Example: What number is 35% of 60?

So, 21 is 35% of 60.

Use an equationto represent “a is p

percent of w.”

a = p w

Thepercent

equation

35100

percent in fractionor decimal form

part of the whole

whole

a = –

= 21



5.

Example: Find the percent of change from 15inches to 24 inches. Because 24 > 15, thepercent of change is a percent of increase.

The percent thata quantity changes

from the original amount

percent of change =

Percentof

change

24 – 1515

915

amount of changeoriginal amount

percent of increase = — = – = 0.6, or 60%

6.

Discount

A decrease inthe original price

of an item

Example: An item that originally sellsfor $25 is on sale for 20% off.

What is the sale price?a = 0.2 · 25 = 5 Sale price = 25 − 5 = 20

The sale price is $20.

Discount: a = p · w

= original price − discountSale price

7.

Markup

To make aprofit, stores

charge more thanwhat they pay. The

increase from what thestore pays to the sellingprice is called a markup.

Example: The wholesale price of an itemis $15. The percent of markup is 12%.

What is the selling price?a = 0.12 · 15 = 1.8 Selling price = 15 + 1.8 = 16.8

The selling price is $16.80.

Markup: a = p · wSelling price = cost to store + markup

8.

Money paid orearned only onthe principal

Simpleinterest

Simpleinterest

Example: You deposit $1000 in an accountthat earns 3% simple interest per year.

How much interest will you earn after 4 years?I = Prt = 1000(0.03)(4) = $120

Annual interest rate(in decimal form)

Principal Time (in years)

I = Prt


Chapter 7 Constructions and Scale Drawings

Example and Non-Example Chart


1. Adjacent angles

Examples Non-Examples

1 23

1 and 22 and 3

41 2

34

1 and 32 and 4

3 and 44 and 1

ADB and BDCBDC and CDACDA and ADB

AB

C

D

2. Vertical angles


1 234

1

6 789

5

234

1 and 22 and 33 and 44 and 1

1 and 32 and 4

5 and 8only

AGB and DGEAGF and DGCBGC and EGF

A BC

DEF

G

3. Supplementary angles


1 2

3 456

3 456

3 and 54 and 6

3 and 44 and 55 and 66 and 3

1 and 2

50˚ 130˚

117˚, 63˚ 104˚, 62˚

4. Quadrilaterals


5. Scale factor


5 : 11 : 2001 : 13 : 2

1 cm : 2 mm1 mm : 20 cm12 in. : 1 ft3 mi : 2 in.


Chapter 8 Circles and Area

Word Magnet


1. Semicircle

One half of a circle

C = one-half the circumference of a circle with a diameter of d

= –– d2

Perimeter = one-half circumference + diameter = –– + d d

2diameter = 2r

2. Compositefigure

Perimeter = 3 + 3 + 5 + 7 = 18 feet

Example: Example: semicircle

Perimeter = –––– + 3 + 6 + 3 = 12 + 3 21.42 feet

(6)2

Made up of triangles, squares, rectangles, semicircles, and

other two-dimensional figures

7 ft

3 ft

5 ft 6 ft3 ft

3 ft

Perimeter: sum of the lengths around the figure

3. Perimeter

Perimeter = 5(5) = 25 centimeters

Example: Example:

Perimeter = (5) + 5 + 5 = 10 + 5 25.7 feet

Distance around a figure

Sum of the lengths around figure

Measured in linear units, such as feet

The perimeter of a circle is called its circumference.

5 cm

5 cm5 cm

5 cm5 cm

semicircles

5 ft

5 ft



4. Area of acircle

Example:

times the radius squared A = r²

Diameter = 2 radius d = 2r A = – ²= ––– d²

4d2

A = (14)² = 196 616 square meters

14 m

Example:

(2)²4A = ––––– =

3.14 square meters

2 m

5. Area of acomposite figure

Example: Example:

A = ––––– + 5² = –––– + 25 44.625 square feet

(5)²4

25 4

Made up of triangles, squares, rectangles, semicircles, and

other two-dimensional figures

7 ft

3 ft

5 ft3 ft

To find area, split up into figures with areas you know how to find.

Then add the areas of those figures.

A = 3² + – (3)(4) = 15 square feet

12

semicircles

5 ft

5 ft


Chapter 9 Surface Area and Volume

Information Frame


1.

Surface areasof prisms

Visual

ExampleRectangular prism:

Example

S = 2 w + 2 h + 2wh = 2(4)(2) + 2(4)(3) + 2(2)(3) = 16 + 24 + 12 = 52 square inches

S = 2( ·3·4) + 3·6 + 5·6 + 4·6 = 12 + 18 + 30 + 24 = 84 square feet

3 in.

4 ft

4 ft4 ft

5 ft

5 ft5 ft

6 ft

6 ft

3 ft

3 ft

4 in. 2 in.

4 ft

5 ft

6 ft3 ft

12

Words

The surface areaS of a prism is thesum of the areasof the bases andlateral faces.

Triangularprism:

lateralface

lateralface

lateralface

base

base

Triangularprism:

2.

Surface areasof pyramids

Visual

Algebra

S = +

12

Words

The surface areaS of a pyramid isthe sum of theareas of the baseand lateral faces,which aretriangles.

Example

S = 6² + 4 – 6 9 = 36 + 4 27 = 36 + 108 = 144 square inches

12

6 in.6 in.

9 in.

Slantheight

Slantheight

Lateralfaces

Lateralfaces

Base

area of a lateralface (triangle):

A = –bh

area ofbase

areas oflateralfaces

edge ofbase

slantheight

3.

Surface areasof cylinders

Example

5 cm

4 cm

Visual

S = 2πr² + 2πrh = 2π(4)² + 2π(4)(5) = 32π + 40π = 72π ≈ 226.08 square centimeters

WordsAlgebraThe surface

area S of acylinder is thesum of the areasof the bases andthe lateralsurface.

S = 2πr² + 2πrh

Areaof bases Area of

lateralsurface

Base

Base

2πr

h h

r r

r

Lateral surface



4.

Volumes ofprisms

Example

5 in.

4 in.3 in.

Visual

V = Bh

= 60 cubic inches

Words

AlgebraThe volume Vof a rectangularprism is theproduct of the areaof the base and theheight of theprism.

V = BhArea

of base Heightof prism

area of base, B

height, h area ofbase, B

height, h

5.

Volumes ofpyramids

Example

9 m

5 m4 m

Visual

Words

AlgebraThe volume Vof a pyramid isone-third theproduct of the areaof the base and theheight of thepyramid.

V = –Bh

Areaof base

Heightof pyramid

area of base, B

height, hheight, h

13

V = –Bh131313


Chapter 10 Probability and Statistics

Notetaking Organizer

1– 7. Sample answers are given.

1.

How can I find the probability of an event without doing an experiment?

Experimental Probability

Probability that is based on repeated trials of anexperiment

P(event) = ––––––––––––number of timesthe event occurs

total number of trials

Example: You flip a coin 100 times. You flip heads 52 times and tails 48 times. The experimental probabilites are

P(heads) = ––– = 0.52 = 52%,

and

P(tails) = ––– = 0.48 = 48%.

52100

48100

Experiment: aninvestigation or a procedure that has varying results

Outcomes: the possible results of an experiment

Event: a collection of one or more outcomes

2.

What if the possible outcomes are not equally likely?



Favorable outcomes: the outcomes of a specific event

Theoretical Probability

The ratio of the number of favorable outcomes to the number of possible outcomes, when all possible outcomes are equally likely

P(event) = ––––––––––––––number of

favorable outcomesnumber of

possible outcomes

Example: You flip a coin. The theoretical probability of flipping heads and the theoretical probability of flipping tails isP(heads) = –, and

P(tails) = – .

12

12

3.

How do I use the Fundamental Counting Principle to find the probability of more than two events?



Sample space: the set of all possible outcomes of one or more events

Fundamental Counting Principle

A way to find the total numberof possible outcomes of an event;can use a table or a tree diagram.

Example:The total number of possibleoutcomes of rolling a numbercube and flipping a coin is6 × 2 = 12.

numbercube

coin total

eventM

eventN

M followedby N

An event M has m possibleoutcomes. An event N has npossible outcomes. The totalnumber of outcomes of eventM followed by event N is m × n.

m × n = total



4.

What if the occurence of one event does affect the likelihood of the other?

P(A and B and C) = P(A P(B P(C)

A compound event consists of two or more events.

Choosing with replacement means the events are indepenent.

Independent Events

Events are independentevents if the occurence ofone event does not affect the likelihood that the other event(s) will occur.

P(A and B) = P(A) P(B)

The probability of two or moreindependent events is the product of the probabilitiesof the events.

P(1 and 1) = P(1) P(1)

=

=

probabilityof bothevents

probabilityof firstevent

probabilityof second

event

13

13

19

Example: A spinner hasthree equal sectionsnumbered 1, 2, and 3. Youspin it twice. What is the probability of spinning a 1 both times?

5.

How do you use probability to make predictions?

A compound event consists of two or more events.

Choosing without replacement means the events are dependent.

Events are dependentevents if the occurence ofone event does affect the likelihood that the other event(s) will occur.

P(A and B) = P(A) P(B after A)

The probability of twodependent events A and B is the probability of A times the probability of Bafter A occurs.

probabilityof bothevents

probabilityof firstevent

probabilityof second

event afterfirst event

occurs

Dependent Events

P(red and red)

= P(red) P(red after red)

=

= 29

49

510

Example: You have 5 red cardsand 5 blue cards. You randomlychoose one card. Withoutreplacing the first card, yourandomly choose a secondcard. What is the probabilitythat both cards are red?



6.

How do you select an unbiased sample from a largepopulation?

Sample

Part of a population

The results of an unbiased sample are proportional to the results of the population. So, you can use unbiased samples to make predictions about the population.

Biased samples are not representative of the population. So, you should not use them to make predictions about the population because the predictions may not be valid.

Example:Population: All of the seventh-grade students in your schoolUnbiased sample: 100 seventh-grade students selected randomly during lunchBiased sample: The seventh-grade students at your lunch table

An unbiased sample is representative of a population. It is selected at random and is large enough to provide accurate data.

A biased sample is not represen-tative of a population. One or more parts of the population are favored over others.

Population

Sample

Interpretation

Inference

7.

How can you tell whether a distribution is skewed or symmetric?

Population

An entire group of people or objects

To compare two populations, use the mean and the MAD when both distributions are symmetric. Use the median and the IQR when either one or both distributions are skewed.

You do not need to have all the data from two populations to make comparisons. You can use random samples to make comparisons.

An inference about a population is more precise if you use multiple samples.

Mean: the sum of the data divided by the number of data values

Median: For a data set with an odd number of ordered values, the median is the middle value. For a data set with an even number of ordered values, the median is the mean of the two middle values.

Interquartile range (IQR): the difference between the third quartile and the first quartile

Mean Absolute Deviation (MAD): an average of how much data values differ from the mean

Chapter 1 Integers Idea and Examples Chart

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Transcript of Chapter 1 Integers Idea and Examples Chart