Sixth Grade Math State Test Review Ms. Mikalauskas - St. Ann School

Topic 1: Ratios and Proportional Relationships Standards: 6.RP.1, 6.RP.2 & 6.RP.3 Ratio Tables Ratio Relationships between 2 quantities Equivalent Ratios Unit Rate, Rate Unit, Rate Percent of quantity using rate per 100 Ratio reasoning to convert measurement units

What is a ratio? A relationship in which for every x units of one quantity there are y units of another quantity -Or- How one quantity compares to another quantity Can be written as x to y x:y Fraction: x/y

Ratio Examples For every 3 girls at the concert there is 1 boy; What is the ratio of girls to boys? 3 to 1; 3:1; 3/1 What is the ratio of boys to girls? 1 to 3; 1:3; 1/3 For every 4 red apples there is 2 green 4 to 2; 4:2; 4/2

Ratio Tables Ratio tables can be used to find equivalent ratios. Red Apples Green Apples 4 2 8 12 6 16 20 10 Ratio tables can be used to find equivalent ratios. An equivalent ratio expresses the same relationship to a given ratio. It can be found by dividing or multiplying both numbers by the same number x 2 x 2 x 3 x 3 x 4 x 4 x 5 x 5

Equivalent Ratio Examples To make equivalent ratios you can make ratio tables or multiply or divide the given ratio by the same number Ex: 4:16 Multiply both numbers by 2 (4x2 and 16x2) Equivalent ratio= 8:32 Divide both numbers by 2 (4/2 and 16/2) Equivalent ratio= 2:8

Comparing Ratios You can use ratio tables to compare ratios when one of the corresponding terms are the same Example: Who uses more sugar in their recipe? Frank Glen Flour tbs 2 4 6 Sugar tbs 5 10 15 Flour tbs 3 6 9 Sugar tbs 7 14 21 Frank uses more sugar in his recipe because every time there are 6 tbs of flour he uses 15 tbs of sugar while Glen uses 14tbs.

What is a Rate? A rate is a special type of ratio that compares the quantities with unlike units of measure. Example: A racecar can travel 10 km in 3 minIf the race car continues to travel at the same rate, how long will it take to travel 25 km. Solve by making a ratio table until 25 km is reached. You can also solve like a proportion . What must we multiply 10 by to get 25? Whatever you do to the top you do to the bottom! 10 km 25 km X = 3 x 3.5 = 7 ½ minutes km Minutes 5 1 ½ 10 3 15 4 ½ 20 6 25 7 ½ X 2.5 3 mins X 2.5 = x mins

Rate Examples: Find the Value of X and missing values Miles 45 90 Hours 4 x Miles 45 135 Hours 4 x Fish 16 48 Bowls 2 x Pages 9 Minutes 18 1 10 15 Miles 25 125 Gallons 3 5 12 Look for relationships between each quantity. There is always half the amount of pages per minute. How does miles compare to gas?

Unit Rates, Rate Units, and RAte Rate- One quantity per another quantity $10 for every 4 packs of pepsi Unit Rate- The value of the ratio $2.50 Rate Unit- Word form of your two quantities dollars/packs of Pepsi

Rate unit: Miles/Hour

Using Unit Rates to make Comparisons Example: Ethan swam 11 laps in the pool in 8 minutes. Austin swam 7 laps in the pool in 5 minutes. Which boy swam at a faster rate? You must find out how many laps they can swim with a denominator of 1 7 laps /5 1.4 5 minutes / 5 = 1 Austin can swim 1.4 laps in 1 minute. 11 laps /8 1.375 8 minutes / 8 = 1 Ethan can swim 1.375 laps in 1 minute *we divide by 8 to get a denominator of 1 Austin swims at a faster rate.

Unit Prices A unit price is a unit rate that gives the price of 1 item Example: A mega-lunch special serves 3 tacos for $2.40; a extra-lunch serves 4 tacos for $3.40. Which has a better value? Divide the dollar amounts by the amount of tacos to get the price for one taco Mega: $2.40 / 3 = .80 per taco < Better value Extra: $3.40 / 4 = .85 per taco

Constants If an object is moving at a constant rate of speed for a certain amount of time, it is possible to find how far the object went by multiplying the rate and the time. In mathematical language, we say, 𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞 = 𝐫𝐚𝐭𝐞 x 𝐭𝐢𝐦𝐞. Examples of expression Distance: Miles, meters, yards, kilometers, feet Rate: miles per hour, km/minute, feet per second (distance per time) Time: Hours, seconds, etc.

Example I drove my car on cruise control at 65 miles per hour for 3 hours without stopping. How far did I go? STEP 1: WRITE YOUR FORMULA! Distance= rate x time STEP 2: PLUG IN WHAT YOU KNOW Distance = 65 x 3 STEP 3: SOLVE Distance = 195 miles

Converting Units Conversion factor- rate that compares equivalent measures Dimensional analysis- convert measures by measurement units when you multiply by a conversion factor Example: how many inches is a 4.5ft sidewalk? 4.5 ft x 12in. 1 1ft = 54 inches Example: There is 15 gallons of water on a camping trip. How many quarts of water are there? (4qt= 1 gal) 15 gal x 4 qt. 1 1 gal = 60 quarts

Common Conversions There will be a reference sheet with some of these conversions for you. However, it does not hurt to be familiar with them.

HOW TO CONVERT When converting draw an arrow from what you have to what you are going to. The amount of columns you count is the amount of times you move your decimal. Example: 128.5 meters converted into: Centimeters (2 columns to the right of meters) - move decimal twice to the right = 128.5 cm Kilometers (3 columns to the left of meters) - move decimal three times to the left = .1285 km Example 2: 48 centigrams Kilograms (5 columns to the left of centi) - move decimal 5 times to the left = .00048 Milligrams (1 column to the right of centi) - move decimal once to the right = 480 mg

Topic 2: Percentages A rate in which the first term is compared to 100. The percent is the number of hundredths that represents the part of the whole

Example STEP 1: WRITE YOUR FORMULA! STEP 2: Plug in what you know Part 80% 64,000 100% STEP 3: Cross-Multiply and Solve 100 x = 80 x 64,000 100 x = 5,120,000 100 100 X = 51,200 = =

Writing Percents, decimals, and Fractions

Changing a Fraction To a Decimal

Changing a Percent to a Decimal

Changing a Percent to a Fraction

Change a Fraction into a Percent

Topic 3: Fractions Adding Subtracting Multiply Dividing Standards: 6.NS.1 Adding Subtracting Multiply Dividing Equivalent Fractions Simplifying GCF & LCM (6.NS.2, 6.NS.3, & 6.NS.4)

Least Common Multiple Least common multiples are used to find the Least common denominator in order to solve fraction addition and subtraction problems

Greatest Common FActor GCF is used when simplifying fractions

Adding and Subtracting Fractions Least Common Multiple: 3 5 4 | 4 8 12 16 20 6| 6 12 18 24 STEP 1: Find the LCD STEP 2: Stack Fractions and convert to LCD STEP 3: Add or subtract numerators ONLY STEP 4: Denominator stays the same STEP 5: Reduce Fraction + 4 6 1 1 4 x 3 12 3 9 + 6 x 3 12 10 Greatest Common Factor: 12 10 | 1 2 5 10 12 | 1 2 3 4 6 12 10 / 2 5 12 / 2 = 6

Multiplying Fractions STEP 1: Put any whole numbers over 1 STEP 2: Multiply the numerators STEP 3: Multiply the denominators STEP 4: Simplify using the GCF

Mixed Numbers -> Improper Fractions

Improper Fractions -> Mixed Numbers

Multiplying Mixed Numbers STEP 1: Change Mixed numbers into improper fractions STEP 2: Multiply numerators and denominators STEP 3: Change improper fraction to mixed number STEP 4: Simplify

Dividing Fractions STEP 1: Keep the first STEP 2: Change division to multiplication STEP 3: Flip the second fraction STEP 4: Multiply STEP 5: Simplify

Dividing Fractions with Whole Numbers

Topic 4: Decimals Adding and subtracting Multiplying Dividing Standards: Computing multi-digit numbers 6.NS.2, 6.NS.3, & 6.NS.4 Adding and subtracting Multiplying Dividing

Adding and Subtracting Decimals

Multiplying Decimals

Dividing Decimals

Topic 5: Integers Rational Numbers Positive and Negative Numbers Standards: 6.NS.5, 6.NS.6, 6.NS.7 & 6.NS.8 Rational Numbers Positive and Negative Numbers Number Lines Order Absolute Value

Rational Numbers and their opposites Given a nonzero number, 𝑎, on a number line, the opposite of 𝑎, labeled −𝑎, is the number such that 0 is in between and the distance between 0 and 𝑎 is equal to the distance between 0 and −𝑎 The opposite of 0 is 0

Positive and Negative Number Examples Gain Added Above sea level Deposit (Money ^) Temperature Negative Loss Debt Below sea level Withdrew (Money v) Temperature

Number Lines Not all number lines increase or decrease by intervals of one Follow these steps to correctly label number lines:

Integer order Listing numbers in least to greatest or greatest to least can be found by using a number line. Order from least to greatest: A, E, D, B, C Order from greatest to least: C, B, D, E, A

Absolute Value Always positive Explains a numbers distance from zero Written as |x|

Comparing Absolute Value Change to value first |7| = |-7| 7 = 7 |-6| > |-4| 6 > 4 |-3| > |0| 3 > 0 -|-2| < |2| -2 < 2 |1| < -|-5| 1 < -5

Topic 6: Graphing on the Coordinate Plane Quadrants X and y Axis Plotting Locating

Axis The x-axis goes left to right across the coordinate plane (horizontal) The y-axis goes up and down the coordinate plane (vertical) Coordinates are always plotted (x,y) (*alphabetical) Start at origin - move along x and then move along y

Example

Topic 7: Order of Operations LEFT TO RIGHT Used when solving multi-step problems LEFT TO RIGHT LEFT TO RIGHT

Topic 8: Properties Associative Property. 5 + (2 + 4) = (5 + 2) + 4. Distributive Property. 6(3 + 5) = (6 x 3) + (6 x 5) Multiplicative Identity Property. 34 x 1 = 34. Additive Identity Property. 95 + 0 = 95. Commutative Property. 8 + 3 = 3 + 8. Zero Property of Multiplication. 19 x 0 = 0.

Associative Property Addition: Parenthesis can move without affecting the outcome of your problem Multiplication: Parenthesis can move without affecting the outcome of your problem

Distributive Property

Commutative Property The order may change when only one operation is present and the outcome will not change.

Topic 9: Exponents Additional Rules: 4 = 4 Anything to the power of zero = 1 Anything to the power of 1 is itself 4 = 4 1

Topic 9: Word Problems Choosing Operations Step-by-Step Write out your answer with appropriate units Underline the question your are solving for SHOW ALL WORK!

Topic 10: Algebraic Expressions Standards: 6.EE.1-6.EE.9 Writing expressions Using substitutions to solve for unknown variables Combining like terms Evaluating Expressions

Definitions Term- separated by addition or subtraction Coefficient- Number before a variable Variable- Any unknown number represented by a letter Constant- A number that is not affected by a variable

Writing Algebraic Expressions Use math key words to determine the correct operation Use the variable given in the problem Write as an algebraic expression Example: Miss M. has 5 less pencils (p) than Mrs. Brass. Write an expression to show how many pencils Ms. M has. p-5

More Examples

Using Substitution

What is a “Like Term”?

Combining Like Terms

Combining Like Terms - Example:

Combining Like Terms - Example:

Topic 11: Geometry Shapes Perimeter Area Volume

Perimeter of Shapes The length of it’s outside

Area of A Rectangle

Area of Parallelograms

Rhombus

Finding area on a coordinate plane In order to find the area of an object on a coordinate plane- a base of the object must line up with the grid lines. If it does not, follow these steps: Connect points to form a rectangle around the object. Find area of the shapes that were formed and subtract Giant rectangle Area Add areas of blue, green, and orange triangles Subtract to get area of the inside object

Topic 12: Statistic Variability and Distributions Standards: 6.SP.1, 6.SP.2,6.SP.3, 6.RP.4 & 6.RP.5 When given a list of numbers you can find the following: Mean (Average) Median (Middle) Mode (Most common) Range (Greatest – Least) Distributions

Box Plots A box plot is a graph that is used to summarize a data distribution A box plot is a graph made using the following five numbers: the smallest value in the data set, the lower quartile, the median, the upper quartile, and the largest value in the data set. To make a box plot: Find the median of all of the data. Find Quartile 1: the median of the bottom half of the data Find Quartile 3: the median of the top half of the data Draw a number line, and then draw a box that goes from Q1 to Q3. Draw a vertical line in the box at the value of the median. Draw a line segment connecting the minimum value to the box and a line segment that connects the maximum value to the box.

Example: “Box and Whiskers” The mean daily temperatures in degrees Fahrenheit for the month of February for a certain city are as follows: 𝟒, 𝟏𝟏, 𝟏𝟒, 𝟏𝟓, 𝟏𝟕, 𝟐𝟎, 𝟑𝟎, 𝟐𝟑, 𝟐𝟎, 𝟑𝟓, 𝟑𝟓, 𝟑𝟏, 𝟑𝟒, 𝟐𝟑, 𝟏𝟓, 𝟏𝟗, 𝟑𝟗, 𝟐𝟐, 𝟏𝟓, 𝟏𝟓, 𝟏𝟗, 𝟑𝟗, 𝟐𝟐, 𝟐𝟑, 𝟐𝟗, 𝟐𝟔, 𝟐𝟗, 𝟐𝟗 STEP 1: Find median by putting number in order: 4, 11, 14, 15, 15, 15, 15, 17, 19, 19, 20, 20, 22, 22, 23, 23, 23, 26, 29, 29, 29, 30, 31, 34, 35, 35, 39, 39 = 22.5 STEP 2: Find median for bottom half (Q1) 4, 11, 14, 15, 15, 15, 15, 17, 19, 19, 20, 20, 22 22 = 16 STEP 3: Find median for bottom half (Q3) 23, 23, 23, 26, 29, 29, 29, 30, 31, 34, 35, 35, 39, 39= 29.5 STEP 4: Lowest and highest : 4 and 39 Median Q3 Median Q1 Median Highest # Lowest #