5.1 & 5.2 Study GuideFractions and Decimals

Learning Goals5.1 given a decimal through thousandths, round to the nearest whole number, tenth, or hundredth.

5.2 - a) recognize and name fractions in their equivalent decimal form and vice versa; andb) compare and order fractions and decimals in a given set from least to greatest and greatest to least.

VocabularyPlace value – the value a digit represents depending on its place in the number Example: the digit 6 represents the hundredths placeValue – how much a digit is worth according to its place in a number Example: the value of the 6 is 0.06 (six hundredths)Digit – there are 10 digits: - 0 , 1, 2, 3, 4, 5, 6, 7, 8, and 9Rounding – reducing the digits in a number while trying to keep the value similarComparing – seeing whether two numbers are greater than, less than, or equal to each otherWhole Number – the counting numbers plus zero (0, 1, 2, 3, 4…)Greater than - >Less than - <Equal to - =Practice at Home

Rounding decimals: Underline the place that you are rounding to. Look to the place to the right to determine if you should round up or down.

Example: 5.829 rounded to the nearest whole (ones): 5.829 = 6

rounded to the tenths place 5.829 = 5.8

rounded to the hundredths place 5.829 = 5.83

Converting fractions to decimals: Some fractions are easy to remember….

= 0.50 or 0.5 = 0.25 = 0.20 or 0.2 0.1

Other fractions can be converted to decimals by looking at the fraction as a division equation. Students may use a calculator to perform these equations.

= 0.33 (repeating decimal) = 0.125

Ordering fractions and decimals: First convert all of the fractions to decimals. Line them up vertically to compare them. Determine the correct order (least to greatest or greatest to least).

, 0.56, , = 0.50 least to greatest: , , 0.560.56 = 0.56

= 0.375 greatest to least: 0.56, ,

Sample Questions

1 Round 5.693 to the nearest whole.

A 5B 5.7C 5.69D 6

2. Which decimal is equal to ?A 0.8B 0.45C 0.08D 0.045

3. Which set of numbers is ordered from least to greatest?

, , , 0.35

A , , , 0.35

B , , 0.35,

C , , 0.35,

D , 0.35, ,

4. Which statement below is true?

A 25.34 > 25.6B 25.34 < 25.021C 25.34 < 25.314D 25.34 > 25.334

5. When rounded to the nearest tenth, which of the decimal numbers below rounds to 546.7?

A 546.772B 546.64C 546.681D 546.75

6. Which set of decimals is in order from greatest to least?

A 2.002, 2.02, 2.220, 2.2B 2.002, 2.02, 2.2, 2.220C 2.220, 2.2, 2.02, 2.002D 2.220, 2.002, 2.02, 2.2

5.3 Study GuidePrime/Composite & Odd/Even

Learning Goals

5.3 a) identify and describe the characteristics of prime and composite numbers; andb) identify and describe the characteristics of even and odd numbers.

Vocabulary

Divisible - capable of being divided by another number without a remainderEven - any number ending in 0, 2, 4, 6, or 8Odd -any number ending in 1, 3, 5, 7, or 9Natural Number - the counting numbers Prime Number - a natural number with exactly two factors (itself and one)Composite Number - any natural number with more than two factorsProduct – the answer in multiplicationFactor - a number that is multiplied by another number to find a product.

Example: factor x factor = productMultiple - the product of that number and any other whole number Example: the multiples of 3 are 3,6, 9, 12…

Practice at Home

Practice identifying prime and composite numbers (to 100) by using the following steps to determine if a number is prime or composite:

Prime #

Composite Numbers Divisibility Rule

2 Multiples of 2 Even numbers > 23 Multiples of 3 If the sum of the digits of a

number is divisible by 3, then the number is divisible by 3. Example: 78 - 7+8=15 (a multiple of 3) so 78 is divisible by 3.

5 Multiples of 5 Numbers ending in 0 and 57 Multiples of 7 This rule is quite

complicated. Students can use calculator to determine if it is divisible by 7.

** The number 1 is neither prime nor composite.

To determine if a number is odd or even, look at the ONES place.

786 is EVEN because there is a 6 in the ones place.483 is ODD because there is a 3 in the ones place

Sample Questions

1. Which is a prime number?

A 2B 4

C 18D 27

2. Which is a composite number?

A 7B 9C 17D 19

3. Which number is not prime?

A 31B 49C 79D 97

4. Which number is not composite?

A 45B 67C 86D 91

5. Which number is neither prime nor composite?

A 1B 2C 3D 4

6. Which number is even?

A 27B 34C 41D 63

7. Which number is odd?

A 36B 43C 50D 78

5.4 Study GuideWhole Number Operations

Learning Goals

5.4 The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers

Vocabulary

Sum - the answer in an addition equationDifference - the distance between 2 numbers on a number line; the answer to a subtraction equationProduct – the answer in a multiplication equationFactor - a number that is multiplied by another to Quotient – the answer in a division equationDivisor – the number that divides into another numberDividend - the number being dividedRegrouping – an equal exchange from one place to the next (the terms trading/borrowing and carrying are also often used when regrouping)Whole Numbers - a number from the set {0, 1, 2, 3…}Estimate – Find a value that is close enough to the correct answer (key word: about)

Practice at Home

Continue to practice basic addition and subtraction facts (flashcards). Many mistakes are often made when incorrectly adding or subtracting two digits. A great FREE resource is to download the interactive game Timez Attack from http://www.bigbrainz.com/ . This program individualizes for each child based on the facts that he/she already knows!

Use real life situations to create story problems for your child to solve.

When estimating whole numbers, round to largest place of the smallest number.

2367 2400 4972 5000243 200 5241 5000

+ 1186 + 1200 + 1082 +1000 3800 11,000

Key words can help students identify a story problem as an addition or subtraction problem. If the words about or closet to are used, the solution should be an estimate.

Addition key words – all together, together, both, combined, in all, sum, total

Subtraction key words – fewer, left, left over, difference, how many more, how much more, comparing words that end in –er (higher, faster, older)

Multiplication key words – each (used when searching for a total)

Division key words – each (used when sharing or grouping), equally; same amount

Sample Questions

1. The farmer’s market sold 307 pumpkins last weekend. On Saturday they sold 179. How many did they sell on Sunday?

A 128

B 272C 486D 487

2. Susan went shopping with her mom. She spent $32 on jeans, $38 on a sweatshirt and $18 on a t-shirt. Which is the best estimate of how much money she spent?

A about $80B about $90C about $100D about $110

3. Michael earned $159 by mowing lawns all summer. If he mowed 39 lawns, about how much did he make per lawn?

A $3.00B $4.00C $30.00D $40.00

4. Katie is placing her seashell collection in a special case. There are 5 compartments in the case. She can fit 12 shells in each compartment. She has already put 25 shells in the case. How many more shells can she fit?

A 32B 35C 45D 60

5. Mr. Hodell is buying pencils for his students. He has 98 students in his classes. The pencils come in packages of 7. How many packages of pencils should Mr. Hodell buy?

A 11B 12C 13D 14

6. Forty-five students are attending a play at the Barksdale Theatre. Each ticket costs $26. How much money will it cost for all the students to attend the play?

F $71G $1,070H $1,170J $1,440

5.5 Study GuideDecimal Operations

Learning Goals

5.5 a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and

b) create and solve single-step and multistep practical problems involving decimals.

Vocabulary

*See vocabulary on the 5.4 Study Guide

Practice at Home

Continue to practice basic addition and subtraction facts (flashcards). Many mistakes are often made when incorrectly adding or subtracting two digits. A great FREE resource is to download the interactive game Timez Attack from http://www.bigbrainz.com/ . This program individualizes for each child based on the facts that he/she already knows!

Addition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. The easiest way to do that is to line up the decimal points.

When multiplying numbers with decimals, we first multiply them as if they were whole numbers. Then, the placement of the decimal in the result is equal to the sum of the number of decimal places of the numbers being multiplied.

*We use the partial product method to multiply. See the study guide on Partial Product Multiplication

Dividing decimal numbers is like dividing whole numbers. We just need to remember to bring the decimal up to the quotient. We can also “count over” the number of decimal places.

When estimating decimal numbers, round to largest place of the smallest number.

23.67 24.00 49.72 50.002.43 2.00 52.41 50.00

+ 11.86 + 12.00 + 10.82 +10.00 38.00 110.00

Sample Questions

1. Claire’s sells bracelets for $4.99 each. Michelle bought 4 yellow bracelets and 5 pink bracelets. How much did Michelle spend at Claire’s?

A $19.96

Add a 0 here so that you can regroup

-2 00 35 - 35

B $24.95C $44.91D $49.90

2. What is 56.834 – 3.497

A 53.337B 53.463C 60.331D 91.804

3. 56.89 X 45.2 =

A 2,571,428B 25,714.28C 2,571.428D 257.1428

4. Andrew and his two brothers want to put their money together to buy a new gaming system. If the gaming system costs $168.69, how much money will each boy need to contribute so they put in equal amounts?

A $56.23B $84.34C $171.69D $506.07

5. 98.34 + 61.7

A 36.64B 92.17C 104.51D 160.04

5.6 Study GuideAdding and Subtracting Fractions and Mixed Numbers

Learning Goals

5.6   The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form.

Vocabulary

Multiple of a Number – The product of the number and a natural (counting) numberFactors - The numbers you multiply together to get another number; the divisors of a numberCommon Factor – given two or more numbers, a factor of all the numbersGreatest Common Factor (GCF) - The largest of the common factors of a set of given numbers Least Common Multiple – the smallest common multiple of two or more numbersLeast Common Denominator (LCD) – the least common multiple of the denominators of two or more fractions Simplest Form – A fraction whose numerator and denominator have no common factors other than one

Examples and Explanations

The Greatest Common Factor (GCF) of two numbers is found by writing all the factors of each number and determining the largest common factor.

8 1,2,4,8The GCF of 8 and 12 is 4. 12 1,2,3,4,6,12

The Least Common Multiple (LCM) of two numbers if found by recording all of the multiples of each number and determining the smallest common multiple. This is also referred to as the Least Common Denominator (LCD).

The LCM/LCD of 4 and 6 is 12.

When adding or subtracting like fractions, only the numerator changes. The denominator stays the same.

+ = - =

When adding or subtracting unlike fractions, first find equivalent fractions with the least common denominator.Then, multiply each fraction to find an equivalent fraction.

=

+ =

4 68 1

212

18

16 24

All answers must be in simplest form.

Step 1: Improper fractions must be converted to mixed numbers (when appropriate).Step 2: Find the GCF of the numerator and denominator.Step 3: If the GCF is 2 or more, divide both the numerator and denominator by the GCF to create an equivalent fraction.

Example: Rewrite in simplest form

Step 1: = 1 Step 2: The GCF of 2 and 6 is 2. Step 3: 2 ÷ 2 = 1 Final answer: 1 6 ÷ 2 = 3

Sample Questions

1. Ryan ate of the pizza and Mark ate of the pizza. How much of the pizza is left?

A C

B D

2. Mark walked 3 miles on Saturday and 1 on Sunday. If he wants to walk a total of 6 miles, how much farther does he have to walk?

A 2 C 2

B 4 D 1

3. Susan has of a yard of yellow fabric and of a yard of blue fabric. How much fabric does she have in all?

A C

B D

5.7 Study GuideOrder of Operations

Learning Goals

5.7 The student will evaluate whole number numerical expressions, using the order of operations limited to parentheses, addition, subtraction, multiplication, and division.

Vocabulary

Parentheses – a pair of symbols used to enclose sections of a mathematical expression Example: (4 x 5) + 3

Strategies (www.mathgoodies.com)

Rule 1: First perform any calculations inside parentheses*Exponents are not included in fifth grade, however, we feel it is important for students to know that this will be the next step in sixth grade and beyond.Rule 2: Next, perform all multiplication and division, working from left to rightRule 3: Lastly, perform all addition and subtraction, working from left to right

Example 1:   Evaluate 6 + 7 x 8 using the order of operations.

Solution:   Step 1:   6 + 7 x 8 - 2 MultiplicationStep 2:   6 + 56 - 2 AdditionStep 3:   62 - 2 Subtraction   60

Example 2:   Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of operations.

Solution:   Step 1:  

3 + 6 x (5 + 4) ÷ 3 - 7

Parentheses

Step 2:  

3 + 6 x 9 ÷ 3 - 7 Multiplication

Step 3:  

3 + 54 ÷ 3 - 7 Division

Step 3 + 18 - 7 Addition

*PMDASPlease Make Donuts After SupperP= Parentheses M= MultiplicationD= DivisionA= AdditionS=Subtraction

MD Together whichever one comes firstAS Together whichever one comes first

4:  Step 5:  

21 - 7 Subtraction

14

http://lhsmath.wikispaces.com/file/view/pemdas.gif/30655651/pemdas.gif

Practice at Home

Continue to practice basic addition, subtraction, multiplication and division facts. It is vital that students know the basic facts in order to solve these equations.

Create expressions for your child to solve.

Sample Questions

1. What is the first step when following order of operations?

A multiplicationB divisionC parenthesesD addition

2. Evaluate the following expression using order of operations:

8 (7 + 2) - 3

A 48B 69C 86D 111

3. Evaluate the following expression using order of operations:

6 ÷ 2 + (8 – 3) x 7

A 38B 56C 87D 92

4. Evaluate the following expression using order of operations:

67 – 3 (5 x 2)

A 37B 45

C 54D 76

5. What is the first equation that is solved in the expression 7 + 9 ÷ 3 x 2 + 5 ?

A 7 + 9B 9 ÷ 3C 3 x 2D 2 + 5

Math Study Guide • SOL 5.8Goal: a) find perimeter, area, and volume in standard units of measure; b) differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation; c) identify equivalent measurements within the metric system; d) estimate and then measure to solve problems, using U.S. Customary and metric units; and e) choose an appropriate unit of measure for a given situation involving measurement using customary and metric units.

Dates we studiedParent Initials

Mastered or Struggling?

Please be specific if struggling

Words I Need to Know!Perimeter - a measure of the distance around a polygon; found by adding the measures of the sides.Area - the number of square units needed to cover a surface; found by multiplying length x widthVolume - a measure of capacity; measured in cubic units; found by multiplying length x width x height

How to Practice at HomeTo find the perimeter, add the measure of each side. Square: side+side+side+side or 4xside; Rectangle: L+L+W+W or 2xL+2xW;Triangle: side+side+side

3 ft

6+6 = 12 9+7+5+6 = 27 3+3+3+3 =12

3+3 = 6

The perimeter is 18 in. The perimeter is 27 cm. The perimeter is 12 ft.

To find the area of a rectangle, multiply the length times the width (A= l x w). The

18

area of a square can be found by multiplying one side by another. (A = s x s)

3 ft.

The area of the rectangle The area of the square isis 18 square inches or 18 in2. is 9 square feet or 9 ft2.

A right triangle is half of a rectangle. To find the area of a right triangle, multiply the base times the height, then divide the number in half. A= ½ (b x h) or A= (b x h)/2

3 x 4 = 12 6 x 9 = 63 cm ½ of 12 = 6 square cm. ½ of 63 =

31.5 square ft. or 6 cm2 9 ft. or 31.5 ft2

cm

cm base 6 ft. base

Volume of a solid figure is found by counting the cubic units. Be sure to account for any “hidden” cubes. Volume of a rectangular solid can be found by multiplying the base x height x width.The volume of an irregular solid can be found by determining the number of cubes in each layer.

4 cubes

4 cubes (1 is hidden)

2 rows Volume = 8 cubic cm. 4 cubes or 8 cm3

4 x 4 x 2 = 32 cubic cm. or 32 cm3

Equivalent units of measure – MetricStudents will be required to convert Metric units of measure. For example there are 2000 mm in 2 meters.Length:

heig

ht

heig

ht

4 la

yers

Length: inches (to the nearest 1/8 of an inch), feet, yards, millimeters, centimeters, meters, kilometersWeight: ounces, pounds, tonsMass: grams, kilogramsLiquid volume: cups, pints, quarts, gallons, milliliters, litersArea: square unitsTemperature: Celsius, Fahrenheit

1 cm = 10 mm cm = centimeter 1 m = 100 cm m = meter1 m = 1000 mm km = kilometer1 km = 1000 m mm = millimeter

Mass:1 kg = 1000 g g = gram kg = kilogram

Liquid Volume:1 L = 1000 mL L = liter mL = milliliter

Sample Questions

1.

2.

3. Andrea is buying a rectangular rug that is 3 feet wide by 4 feet long. What is the total area that the rug will cover?

a) 12 square feetb) 14 square feetc) 24 square feetd) 28 square feet

4. Seth ran 4.65 kilometers on Saturday. How many meters are equivalent to 4.65 kilometers?a) 465 metersb) 4,650 meters

c) 46.5 metersd) 12 meters

5. 3,500 milliliters = liters

6. 15 grams = kilograms

7. Which of the following is not an appropriate tool for measuring length?a) rulerb) yardstickc) scaled) measuring tape

5.9 Study GuideCircles

Learning Goals

5.9 The student will identify and describe the diameter, radius, chord, and circumference of a circle.

Vocabulary

Circle - A set of points on a flat surface (plane) with every point an equal distance from a given point called the center.

Chord - A line segment that extends between any two unique points of a circleCircumference – The distance around the edge of a circleRadius - A line segment that extends between the center and the circumference of the circleDiameter - A special chord that goes through the center of a circle.

Examples and Explanations

Circles are named for the center point. The circumference of a circle is the perimeter

of the circle. It is equal to about three timesthe diameter. (c ≈3d) lt is equal to about six times the radius. (c≈6r)

A chord can connect any two points a circle.Chords are line segments. This chord is .

The diameter is a special chord. It passes through the center of the circle. The

diameter of this circle AC.

The radius line segment that extends between the center and the circumference of the circleThe radii in this circle are DS , DG and DO The diameter is equal to twice the radius. (d=2r)

The radius is half the diameter. (r= )

Sample Questions

1. Point B is the center of the circle shown.

Which of the following best describes

A ChordB RadiusC DiameterD Circumference

2. Which statement about circles is true?

A The radius is twice the diameterB The circumference is three times the radiusC The diameter is twice the radiusD The circumference is twice the diameter

3. Which line segment represents the diameter of the circle below? Record your answer in the box.

a

cb

4. What point is used to name the circle below?

A point AB point BC point FD point D

5.10 Study GuideElapsed Time

Learning Goals 5.10 The student will determine an amount of elapsed time in hours and minutes within a 24-hour period.

Strategy

Use a T-chart: Start time – 5:15 am End time – 8:05 am

1. First, figure out the hours 3. Begin the minutes where the hours left off.

2. Since we can’t add another

whole hour, switch to 4. Add the total hours and minutes minutes.

2 hours, 50 minutes

Sample Questions

1. What time will it be in 45 minutes?

A 10:15 amB 10:20 amC 11:20 amD 7:35 am

2. What time is 12 hours and 20 minutes after 11:15 pm?

7:15 min7:30 157:45 158:00 158:05 5

502

A 12:30 amB 12:30 pmC 11:35 pmD 11:35 am

5.11/5.12 Study GuideAngles and Triangles

Learning Goals

5.11 The student will measure right, acute, obtuse, and straight angles.5.12 The student will classify

a) angles as right, acute, obtuse, or straight; andb) triangles as right, acute, obtuse, equilateral, scalene, or isosceles.

Vocabulary

Protractor - An instrument used in measuring or drawing anglesVertex - A point where two or more straight lines meetDegree – A measure for angles. There are 360 in a full rotation.Right Angle - An angle which is equal to 90°.Straight Angle - An angle that looks like a straight line; It measures 180°Acute Angle - An angle which measures less than 90°Obtuse Angle - An angle which measures more than 90°Right Triangle - A triangle that contains one right angle.Scalene Triangle - A triangle that has no congruent sides.Acute Triangle - A triangle that contains three acute anglesObtuse Triangle - A triangle that has one obtuse angleIsosceles Triangle - A triangle that contains two congruent sidesEquilateral Triangle - A triangle in which all sides are congruent

Examples and Explanations An angle is two line segments or rays that meet at a common endpoint (vertex). Angles are classified into four categories:

Acute Right Obtuse Straight< 90° = 90° >90° =180°

www.MathIsFun.com

A Protractor is a tool used to measure angles. Outer scale

zero lineTo measure an angle:

1. Determine the type of angle. This will help you in deciding which scale to use.2. Place the center mark of the protractor on the vertex of the angle.3. Rotate the zero line of the protractor to line up with one ray of the angle. 4. Read the measure of the angle using the ray that crosses the protractor scale.

HINT: Be sure to read the scale carefully and pay attention to which direction is it going. Knowing what type of angle you are measuring can help you determine if your measure is reasonable.

This acute angle measures 50°. This obtuse angle measures 110°.

Triangles are classified by their angles.

Acute Triangle Obtuse Triangle Right Triangle (three acute angles) (contains an obtuse angle) (contains a right angle)

Triangles can also be classified by their line segments.

A scalene triangle has no congruent sides. None of the sides are of equal length.

Right Scalene Triangle Obtuse Scalene Triangle

An equilateral triangle sides are An isosceles triangle is a triangle thatall congruent. They are all the same length. contains two congruent sides.

Inner scale

center mark

Acute Right Obtuse Isosceles Isosceles Isosceles Acute Equilateral Triangle Tirangle Triangle Triangle

Sample Questions

1. Which of the figures can be classified as both an acute and an isosceles triangle?

2. Which angle is closest to 90°?

3. Which figure appears to be an equilateral triangle?

A C

B D

3. Measure the angle. Record your responses in the boxes below.

Type of angle Measure

Learning Goals: Develop definitions of square, rectangle, triangle, parallelogram, rhombus, and trapezoid Investigate and Describe the results of combining and subdividing plane figures

Definitions : Examples: A triangle is a polygon with three sides. Triangles may be

classified according to the measure of their angles, i.e., right, acute, or obtuse. Triangles may also be classified according to the measure of their sides, i.e., scalene (no sides congruent), isosceles (at least two sides congruent) and equilateral (all sides congruent).

A quadrilateral is a polygon with four sides.

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Properties of a parallelogram include the following: A diagonal (a segment that connects two vertices of a

polygon but is not a side) divides the parallelogram into two congruent triangles.

The opposite sides of a parallelogram are congruent. The opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other.

Definitions : Examples: A rectangle is a parallelogram with four right angles. Since a

rectangle is a parallelogram, a rectangle has the same properties as those of a parallelogram.

A square is a rectangle with four congruent sides. Since a square is a rectangle, a square has all the properties of a rectangle and of a parallelogram.

A rhombus is a parallelogram with four congruent sides. Opposite angles of a rhombus are congruent. Since a rhombus is a parallelogram, the rhombus has all the properties of a parallelogram.

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the non-parallel sides are called legs. If the legs have the same length, then the trapezoid is an isosceles trapezoid.

A kite is a quadrilateral with two distinct pairs of adjacent congruent sides.

Two figures can be combined to form a new shape. Students should be able to identify the figures that have been combined.

A polygon may be subdivided into two or more figures. Students should understand how to divide a polygon into familiar figures.

5.14 Study Guide

Probability

Learning Goals

5.14 The student will make predictions and determine the probability of an outcome by constructing a sample space.

Vocabulary

Probability: the chance of an event occurringLikelihood: the probability of an event occurringOutcome: result of an experimentImpossible: an event is impossible if it has a probability of 0Unlikely: not likely to occur As likely as: equally likelyEqually likely: outcomes that have the same probabilityLikely: seeming like certainty Certain: an event is certain to occur if it has a probability of 1Sample Space: A sample space represents all possible outcomes of an experiment. The sample space may be organized

in a list, chart, or tree diagram.

Examples and ExplanationsThe possible outcomes of the spinner are GREEN, YELLOW, PURPLE, ORANGE, and RED. There are 5 possible outcomes. The probability of the spinner landing on a particular color can be expressed in words and as a fraction.

All of the possible outcomes of an experiment are called the sample space. A tree diagram can be used to determine the sample space. Here is a tree diagram for an experiment involving flipping a coin three times. The tree diagram shows all of the possible outcome

There are 8 possible outcomes. This is the sample space.An organized list or chart can also show the sample

space

SOL 5.15 Study Guide

Event ProbabilityWord Fraction

Landing on GREEN, YELLOW, PURPLE, ORANGE, or RED.

Certain 1

Landing on any color except GREEN

Likely

Landing on GREEN as related to landing on RED

Equally likely and

Landing on ORANGE Unlikely

Landing on BROWN Impossible

0

1st Roll 2nd Roll 3rd RollHeads Heads HeadsHeads Heads TailsHeads Tails HeadsHeads Tails TailsTails Tails TailsTails Tails HeadsTails Heads TailsTails Heads Heads

Stem and Leaf Plots / Line Graphs

What you need to know how to do: collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots and line graphs.

.Key Vocabulary:

Stem-and-Leaf Plot – a data display that organizes data points by separating each into a leaf (last digit) and a stem (remaining digits).

Line Graph - a type of graph in which points representing data pairs are connected by line segments.Data – information, facts, or numbers that describe something.Vertical Axis – the y-axis in the coordinate plane.Horizontal Axis – the x-axis in the coordinate plane.

How to do it: A stem-and-leaf plot is a visual representation of data. It is organized by place value and is very helpful when finding mean, median, mode, and range.

There are three steps for drawing a stem-and-leaf diagram.

1. Split the data into two pieces, stem and leaf. 2. Arrange the stems from low to high. 3. Attach each leaf to the appropriate stem.

Be sure to include a title and a key.

A line graph is used to show how two variables are related. It may also be used to show how one variable changes over time. By looking at a line graph, it can be determined whether the variable is increasing, decreasing, or staying the same over time.To create a line graph:

Examples:

1. Label the vertical axis (Y) with equal increments (scale).

2. Label the horizontal axis (X) with data related to time.

3. Plot each data point. (In 5th grade, we have no more than 6)

4. Give your graph a title.

1. Mr. Murphy asked 24 of his students how many miles they live from the school. The stem-and-leaf plot shows the data. Which of the following is a true conclusion based on the data in the stem-and-leaf plot?

A Five students live 1 mile from the school.B Three students live 4 miles from the school.C Exactly half the students live 7 or more miles from the school.D The maximum distance a student lives from the school is 15 miles.

2. This graph shows the number of people who used the Internet each year in the United States from 1997 to 2003.

Based on the information in the graph, between which two years did the smallest increase occur?

A 1997 and 1998B 1998 and 1999C 2001 and 2002D 2002 and 2003

3. Julian made the following list of all his math quiz scores.

77, 85, 86, 88, 88, 89, 89, 91, 93, 94, 97, 99, 99

Which stem-and-leaf plot correctly shows Julian’s quiz scores?

5.16 Study GuideStatistics

Learning Goals

5.16 a) describe mean, median, and mode as measures of center;b) describe mean as fair share;c) find the mean, median, mode, and range of a set of data; andd) describe the range of a set of data as a measure of variation.

Vocabulary

Mean – The sum of the values in a data set divided by the number of values. Also known as “average”.Median – The middle value or the average of the two middle values in an ordered set.Mode – The value in a data set that occurs most often.Range – The difference between the greatest and least values in a set of data.

Skills and Strategies

The mean is computed by adding all of the numbers in the data together and dividing by the number elements contained in the data set. [Mean represents a fair share concept of the data.]

Example :Data Set = 2, 5, 9, 3, 5, 4, 7Number of Elements in Data Set = 7Mean = ( 2 + 5 + 9 + 7 + 5 + 4 + 3 ) / 7 = 5

The median of a data set is dependant on whether the number of elements in the data set is odd or even. First reorder the data set from the smallest to the largest then if the number of elements are odd, then the Median is the element in the middle of the data set. If the number of elements are even, then the Median is the average of the two middle terms.Example : Odd Number of Elements Example : Even Number of ElementsData Set = 2, 5, 9, 3, 5, 4, 7 Data Set = 2, 5, 9, 3, 5, 4Reordered = 2, 3, 4, 5, 5, 7, 9 Reordered = 2, 3, 4, 5, 5, 9Median = 5 Median = ( 4 + 5 ) / 2 = 4.5

The mode for a data set is the element that occurs the most often. It is not uncommon for a data set to have more than one mode. This happens when two or more elements accur with equal frequencyin the data set. A data set with two modes is called bimodal. A data set with three modes is called trimodal.

Example : Single Mode Example : Bimodal Example : TrimodalData Set = 2, 5, 9, 3, 5, 4, 7 Data Set = 2, 5, 2, 3, 5, 4, 7 Data Set = 2, 5, 2, 7, 5, 4, 7Mode = 5 Modes = 2 and 5 Modes = 2, 5, and 7

The range for a data set is the difference between the largest value and smallest value contained in the data set. First reorder the data set from smallest to largest then subtract the first element from the last element.

Example :Data Set = 2, 5, 9, 3, 5, 4, 7Reordered = 2, 3, 4, 5, 5, 7, 9Range = ( 9 - 2 ) = 7

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Sample Questions

1. What is the mean (average) for the following set of data?

6, 4, 22, 21, 37A 18B 20C 24D 31

2. What is the mode for the following set of numbers?

6, 10, 8, 7, 9, 8, 9, 9, 7

A 6B 7C 8D 9

3. What is the range for the following set of numbers?

21, 12, 13, 12, 24, 11, 19, 20

A 11B 12C 13D 24

4. What is the median for the following set of numbers?

6, 10, 9, 4, 8, 7

A 7B 7.5C 8D 8.5

5. Use the stem-and-leaf plot below to determine the mode for the following data.

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Key: 1|2 = 12

A 12B 22C 24D 32

5.17 Study GuidePatterns

Learning Goals

5.17 The student will describe the relationship found in a number pattern and express the relationship.

Vocabulary

numerical – expressed as numberspattern – a sequence that follows a rule or rules

Examples and Explanations

To determine the rule of numerical patterns, use a caret (V) between numbers. Increasing number patterns use addition or multiplication in the rule.Decreasing number patterns use subtraction or division in the rule.

2, 6, 10, 14 In this pattern, the numbers are increasing, or getting larger. v v v We know the operation will be either addition or multiplication.

+4 +4 +4 2 x 3 = 6 2 +4 = 6 x3 Each possible rule must be tested to see if all of the numbers

follow the pattern. 6x3 does not equal 10. The rule for this pattern is Add 4.

Patterns can also be in the form of a table

Compare the IN column to the OUT column. In this pattern, the numbers are increasing, or getting larger. We know the operation will be either addition or multiplication.

2 x4 = 8 2 +6= 8 Each possible rule must be tested to see if all of the numbers follow the pattern.

3+6 does not equal 12. The rule for this pattern is x4.

Use the rule to complete the missing boxes.

Rules for patterns can also be expressed with variables.

Determine the pattern between the IN (Pizza) and OUT (Number of Slices) column. The pattern is x8. The rule can be expressed as n x 8.

IN OUT2 83 12? 165 ?

Pizzas

Number of

Slices2 164 325 40n

5.18 Study GuideAlgebraic Expressions

Learning Goals

5.18 a) investigate and describe the concept of variable;b) write an open sentence to represent a given mathematical relationship, using a variable;c) model one-step linear equations in one variable, using addition and subtraction; andd) create a problem situation based on a given open sentence, using a single variable.

Vocabulary

Variable Expression - mathematical phrase which can contain numbers, operators (add, subtract, multiply, divide), and at least one variable. Expressions do not contain an = sign. (Also known as algebraic expressions)

example: n + 7Variable – a letter or symbol representing a varying quantity

example: n + 6 = 8Open Sentence – a mathematical sentence which contains one or more variable and an = sign

n + 7 = 9

Examples and Explanations

Problem Situation Variable Expression

Mike had some baseball cards and his brother gave him four more

b + 47 boxes each contained the same number of apples

7aThe cookies were split evenly among 3 friends

c/4

Problem Situation Open SentenceMike had six baseball cards, his brother gave him four more, and now he has 18 cards.

b + 4 = 18

7 boxes, each containing the same number of apples, totaled 84 apples in all.

7a = 84

How many cookies were baked if four friends each received three cookies?

c/4 = 3

Phrases can also be used to represent expressions

Phrase Variable Expression

four divided by a number 4 ÷ neight more than a number t + 8seven less than a number k – 7six times a number 6n

Multiplication and Division can be represented in different forms

Multiplication Division8 x 48 ∙ 48(4)8n

8 ÷ 48/4

Practice at Home

Continue to practice basic addition, subtraction, multiplication and division facts. It is vital that students know the basic facts in order to solve these equations.

Use simple phrases and real life problem situations to create open sentences or expressions.

Sample Questions

1. Which word describes the letter “p” in the expression below?

9 x p

A whole numberB operationC algebraD variable

2. Which of these could be solved using the open sentence 7 – 2 = h?

A Ryan had 7 hamsters. Each hamster had two babies. How many hamsters does Ryan have now?B Ryan had 7 hamsters. The bought two more at the pet store. How many does he have altogether?C Ryan had 7 hamsters. He gave each of them 2 treats? How many treats did he give them

in all?D Ryan had 7 hamsters. He gave 2 to a friend. How many does he have left?

3. Kathleen has three times as many seashells in her collections as her friend Gabby. Gabby has 15 seashells. Which number sentence could be used to find k, the number of seashells that Kathleen has?

A k = 15 - 3B k = 15 + 3C k = 15 x 3D k = 15/3

4. Which of the following phrases represents the phrase “4 less than a certain number”?

A 4 - nB 4 + nC n - 4D n + 4

5.19 Study GuideDistributive Property

Learning Goals

5.19 The student will investigate and recognize the distributive property of multiplication over addition.

Vocabulary

Distributive Property – multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products.

Examples and Explanations

8 x 23 = 8 x (20 + 3) First, break apart larger numbers using expanded notation= (8 x 20) + (8 x 3) Second, distribute multiplication over each addend= 160 + 24 Third, solve each equation in the parentheses= 184 Finally, add the products

Additional examples:

6 x 53 n x 98 74 x 9 = 6(50 + 3) =n(90 + 8) =(70 + 4) x 9

=(6 x 50) + (6 x 3) =(n x 90) + (n x 8) =(70 x 9) + (4 x 9)

Practice at Home

Continue to practice basic addition, subtraction, multiplication and division facts. It is vital that students know the basic facts in order to solve these equations.

Have your child practice solving simple multiplication equations, such as the ones above, using the distributive property

Sample Questions

1. Which of the following makes the statement below true?

9 x 67 =

A (9 x 60) + (9 x 7)B 9 + (60 + 7)C (9 x 60) + 7D 9 (60 x 7)

2. Which of the following makes the statement below true?

8 (p + 6) =

A (8 x p) + 6B 8 (p – 6)C p (8 + 6)D (8 x p) + (8 x 6)

3. Which does NOT show the distributive property?

A 8(n + 6) = (8 x n) + (8 x 6)B (5 x 6 ) x 4 = 5 x (6 x 4)C 7 x (50 + 3) = (7 x 50) + ( 7 x 3)D (5 + 4) x 7 = (5 x 7) + (4 x 7)

4. Which of the following shows the distributive property?

A 7 x 32 = 7 + 32B 5 + 6 = 6 + 5C 3 x 92 = 3 (90 + 2)D (3 + 4) + 9 = 3 + (4 + 9)