Praxis 7803

equivalent decimals

Equivalent decimal fractions are unlike fractions which are equal in value. Let us consider the following examples: (i) 0.4, 0.40, 0.400, 0.4000 Each of the above is equal to 0.4 or 4/10. (ii) 1.9, 1.90, 1.900, 1.9000 Each of the above is equal to 1.9 or 19/10. (iii) 10.14, 10.140, 10.1400, 10.14000 Each of the above is equal to 10.14 or 1014/100. Equivalent decimal fractions are unlike fractions which are equal in value. (iv) 0.94, 0.940, 0.9400, 0.94000 Each of the above is equal to 0.94 or 94/100. (v) 9.1, 9.10, 9.100, 9.1000 Each of the above is equal to 9.1 or 91/10. (vi) 60.49, 60.4900, 60.490 Each of the above is equal to 60.49 or 6049/100. Similarly, we have 0.300 = 0.30 = 0.3 0.700 = 0.70 = 0.7 0.200 = 0.20 = 0.2 Thus, by adding any number of zeros after the extreme right digit in the decimal part of a decimal number does not change the value of the number.

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a unique product of prime numbers.

Regroup

Exchange equal amounts of tens and ones, hundreds and tens, thousands and hundreds, etc. 10 ones = 1 ten 1,000 = 10 hundreds

The difference between and equation and expression

Expression: 4y + 2 Equation: 4y + 2 = 14

Finding fractions of whole number

For finding a fraction of a whole number, we multiply the numerator of the fraction by the given number and then divide the product by the denominator of the fraction. Find 2/5 of 15. To find 2/5 of 15, we multiply the numerator 2 by the given whole number 15 and then divide the product 30 by the denominator 5. 2/5 × 15 = 2 × 15/5 = 30/5 = 6 So, 2/5 of 15 = 6.

congruent

Having the same size and shape

division clues (fractions)

How many times greater, How many times less, quotient, split into groups.

ARRAYS2, AREA3-GROUP SIZE UNKNOWN ("HOW MANY IN EACH GROUP?" DIVISION)

If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it?

ARRAYS2, AREA3-NUMBER OF GROUPS UNKNOWN ("HOW MANY GROUPS?" DIVISION)

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it?

Equal groups GROUP SIZE UNKNOWN ("HOW MANY IN EACH GROUP?" DIVISION)

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be?

Equal groups NUMBER OF GROUPS UNKNOWN ("HOW MANY GROUPS?" DIVISION)

If 18 plums are to be packed 6 to a bag, then how many bags are needed? Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have?

benchmark numbers

Numbers that help you estimate objects without counting them, such as 25, 50, 100

A grocery store sells both green grapes and red grapes for a regular price of $2.89 per pound. Nelson buys 1.5 pounds of green grapes and 2.25 pounds of red grapes at the store on a day when the regular price is reduced by $0.75 per pound. Which of the following expressions represents the amount, in dollars, that Nelson will pay for the grapes? 1.5+2.25×2.89−0.75 (1.5+2.25)×2.89−0.75 1.5+2.25×(2.89−0.75) (1.5+2.25)×(2.89−0.75)

Option (D) is correct. To find the amount, in dollars, that Nelson will pay for the grapes, the total weight of the grapes, in pounds, needs to be multiplied by the reduced price of the grapes, in dollars. The total weight of the grapes, in pounds, is 1.5+2.25, and the reduced price of the grapes, in dollars, is 2.89−0.75, so the amount, in dollars, that Nelson will pay for the grapes is (1.5+2.25)×(2.89−0.75). The parentheses must be included in the expression as shown so that the total weight of the grapes will be multiplied by the reduced price of the grapes.

Mr. Varela asked his students to define a square in terms of other two-dimensional geometric figures. Which two of the following student definitions precisely define a square? A square is a rectangle that has 4 sides of equal length. A square is a parallelogram that has 4 angles of equal measure. A square is a parallelogram that has 4 sides of equal length. A square is a rhombus that is also a rectangle. A square is a rectangle that is not a rhombus.

Options (A) and (D) are correct. A square is a quadrilateral with 4 sides of equal length and 4 angles of equal measure, whereas a rectangle is a quadrilateral with 4 angles of equal measure, a rhombus is a quadrilateral with 4 sides of equal length, and a parallelogram is a quadrilateral where opposite sides are parallel. Therefore, a rectangle that has 4 sides of equal length is a square, and a rhombus that is also a rectangle is a square, so options (A) and (D) are both precise definitions of a square. Option (B) describes a rectangle that is not necessarily a square, option (C) describes a rhombus that is not necessarily a square, and option (E) describes a rectangle that is not a square.

Which three of the following word problems can be represented by a division equation that has an unknown quotient? A.) Ms. Bronson works the same number of hours each day. After 8 days of work, she had worked 32 hours. How many hours does Ms. Bronson work each day? B.) Mr. Kanagaki put tape around 6 windows before painting a room. He used 7 feet of tape for each window. How many feet of tape did he use? C.) Micah used the same number of sheets of paper in each of 5 notebooks. He used 45 sheets of paper in all. How many sheets of paper did Micah use in each notebook? D.) Each shelf in a school supply store has 8 packs of markers on it. Each pack has 12 markers in it. How many markers are on each shelf in the store? E.) Trina gave each of 7 friends an equal number of beads to use to make a bracelet. She gave the friends a total of 63 beads. How many beads did she give to each friend?

Options (A), (C), and (E) are correct.

Click on your choices. Question: Mr. French's students are working on finding numbers less than 100 that are multiples of given one-digit numbers. When Mr. French asks them how they know when a number is a multiple of 6, one student, Crystal, says, "Even numbers are multiples of 6!" Mr. French wants to use two numbers to show Crystal that her description of multiples of 6 is incomplete and needs to be refined. Which of the following numbers are best for Mr. French to use for this purpose? Select two numbers. 15 16 20 24 27 30

Options (B) and (C) are correct. The best numbers for Mr. French's purpose are even numbers that are not multiples of 6, and 16 and 20 are both even numbers, but they are not multiples of 6. Options (A) and (E) are incorrect because 15 and 27 are not even numbers, and options (D) and (F) are incorrect because 24 and 30 are both multiples of 6.

Mr. Benner places a row of 5 cubes on a student's desk and asks the student, Chanel, how many cubes are on the desk. As Chanel points at the cubes one by one from left to right, she counts, saying, "One, two, three, four, five." Then she says, "There are five cubes!" Mr. Benner then asks Chanel to pick up the third cube in the row. As Chanel points at three cubes one by one from left to right, she counts, saying, "One, two, three." She stops, then picks up the three cubes, and gives them to Mr. Benner. Chanel has demonstrated evidence of understanding which two of the following mathematical ideas or skills? A. Using numerals to describe quantities B. Recognizing a small quantity by sight C. Counting out a particular quantity from a larger set D. Understanding that the last word count indicates the amount of objects in the set E. Understanding that ordinal numbers refer to the position of an object in an ordered set

Options (C) and (D) are correct. Chanel first counts the cubes one by one and then she states that there are 5 cubes.

Which two of the following inequalities are true? 0.56>0.605 0.065>0.56 0.56>0.506 0.605<0.056 0.506<0.65 0.65<0.605

Options (C) and (E) are correct. To compare these decimal numbers, first compare the digits in the tenths place—the decimal number with the greater digit in the tenths place will be the greater number. If the digits in the tenths place are the same, compare the digits in the hundredths place to determine which decimal number is greater. This process can be continued as needed.

Vertex

Point at which the rays meet on an angle

polygon

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Congruent Polygons

Polygons with same size and shape

Transversal

If a third line intersects two intersecting lines at the same point of intersection the third intersecting line is called a transversal.

Congruent

If the measure of two angles are the same they are ______________.

quotative division

If the number in each group is known, and you are trying to find the number of groups, then the problem is referred to as a quotative division problem. Quotative division may also be called measurement, or repeated subtraction. You are, in effect, counting or measuring the number of times you can subtract the divisor from the dividend. Long division (remember long division?!) uses this concept.

partitive division

If the number of groups is known, and you are trying to find the number in each group, then the problem is referred to as a partitive division problem. Partitive division may also be called equal groups, or sharing and distribution. You are, in effect, partitioning the dividend into the number of groups indicated by the divisor and then counting the number of items in each of the groups.

Vertical Angles

If two lines intersect they form two pairs of vertical angles.

Decimal division by powers of 10

If you are dividing by 10, 100, 1000 then move the decimal to the left the same number as there are zeros. If you are dividing by 0.1, 0.01, or 0.001 then move the decimal to the right the same number as there are zeroes. Example 94.56/100=0.9456

Regrouping

In addition, a process once called carrying. it is used in problems such as 26+ 6, 16+7. In multiplication occurs in problems such as 268 x 26 = 6968. In subtraction a process once called borrowing used in problems such as 23-7.

Base 10 numerals

In math, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are base ten numerals. We can only count to nine without the need for two numerals or digits. All numbers in the number system are made by combining these 10 numerals or digits.

Ordinal Number

Indicate the order of things, ex: first, second

Dividing Fractions:

Invert (reciprocal) the second fraction to the right of the division symbol (cancel if possible and multiply

Addition

Is a binary operation. meaning it combines only two number at a time to produce a third unique number. adding two whole numbers always results in a whole number. An operation that when performed two numbers results in a sum

Multiplication

Is a binary operation. the result of the operation is a product. 4 x 9 = 36. the product is 36

Numeral

Is a symbol used to represent a number.

Subtraction

Is the inverse of addition. Is a binary operation.

Supplementary Angles

Is the sum of two angles is 180 degrees

Complementary Angles

Is when the sum of two angles is 90 degrees.

Adding Fractions same denominator

Just add the numerators. Keep the same denominator.

metric units of length

Kilometer (km)1000 Meter (m) Decimeter (dm) Centimeter (cm) Millimeter (mm) Micrometer (um) Nanometer (nm)

area

Length x Width The number of square units required to cover a surface.

Area

Length x Width. Measure of the surface inside of the perimeter -- in square units

Like terms in algebraic expression

Like terms are terms that contain the same variables raised to the same power. Only the numerical coefficients are different.

Subtracting Decimals

Line up the decimal points and subtract normally (ex. 20.02-1.36=18.66)

Pre-Number Concepts

Matching, sorting, comparing, ordering

Natural Numbers

Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject. Include the set of counting numbers 1,2,3,4,5.... and the set of whole numbers 0,1,2,3,4,5....

Take From: Start Unknown

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?? -2 = 3

Add to: Start Unknown

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 =5

Additive Identity/Identity Element of Addition

States that the number 0, when added to any number the sum is that of the other number. 2 + 0=2. therefore 0 is the additive identity or identity element of addition

Dividing fractions by whole numbers

Step 1. Multiply the bottom number of the fraction by the whole number Step 2. Simplify the fraction (if needed)

Joshua walks the length of each of three trails on a hike. The first trail is 3.6 kilometers long. The second trail is 3.7 kilometers long. The third trail is 600 meters shorter than the sum of the lengths of the first two trails. Joshua walks at an average speed of 3 kilometers per hour over the course of the entire hike. How many minutes does it take Joshua to complete his hike?

The correct answer is 280 minutes. Since 600 meters is equivalent to 0.6 kilometers, the third trail is 3.6+3.7−0.6=6.7 kilometers long. Therefore, Joshua walked a total distance of 3.6+3.7+6.7=14 kilometers on his hike. Since Joshua walks at an average speed of 3 kilometers per hour and there are 60 minutes in an hour, the proportion 3 kilometers/60 minutes=14 kilometers/x minutes can be used to find how many minutes it takes Joshua to complete his hike. Based on the proportion, 3x=(14)(60), and since (14)(60)=840, x=8403=280, which means that it takes Joshua 280 minutes to complete his hike.

Decimals and their Equivalent Fractions

The decimal equivalent of any fraction can be found by dividing the numerator of the by the denominator. For example: 2/100 is equivalent to 0.02, 20/100 and 2/10 are equivalent to 0.2

range

The difference between the highest and lowest scores in a distribution.

absolute value

The distance a number is from zero on a number line ( absolute value is always the positive version of the number). 0 is neutral neither positive or negative.

perimeter

The distance around a figure

Greatest common factor

The largest factor that two or more numbers have in common.

lowest common denominator

The lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of fractions. It is the smallest positive integer that is a multiple of each denominator in the set.

Cardinal number

Tells how many. Example: 9 players on a baseball team

Properties of Operations Rules

That help us add, subtract, multiply and divide effectively and efficiently

properties of triangle

The 3 interior angles of a any triangle add up to 180 degrees. An exterior angle of a triangle is equal to the sum of the remote, that is, non-adjacent, interior angles.

Area of a shape

The amount of space inside an object Area of a Square = side of square times itself Ex: 6x6 Area of a Rectangle = width x height = w x h Area of a Triangle =1/2(base⋅height) Area of a Semi-Circle = (π x radius2)/2 = (π x r2)/2

Exterior Angles

The angles outside two lines that are crossed by a transversal

A chef at a restaurant uses 1/5 liter of lemon juice and 3/10 liter of teriyaki sauce to make a marinade for 2 kilograms of salmon. How many liters of marinade does the chef use per kilogram of salmon? Give your answer as a fraction.

The correct answer is 1/4. This means that the chef uses 1/2÷2=1/4 liter of marinade per kilogram of salmon.

Property of Reciprocals

The product of any number multiplied by its reciprocal is one. 5/1 x 1/5= 1 the fives cancel out

Commutative Property

The property that says that two or more numbers can be added or multiplied in any order without changing the result.

Ratio

The quantitative relation between two amounts showing the number of times one value contains or is contained within the other.

Place Value

The value of a digit based on its position within a number

place value

The value of each digit in a number based on the location of the digit

Equal Groups - Unknown Product

There are 3 bags with 6 plums in each bag. How many plums are there in all? Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether?

ARRAYS2, AREA3-UNKNOWN PRODUCT

There are 3 rows of apples with 6 apples in each row. How many apples are there? Area example. What is the area of a 3 cm by 6 cm rectangle?

Acute Triangle

Triangle with exactly three acute angles

Isosceles Triangle

Triangle with two sides of the same length

Adjacent

Two angles are _____________ if they share a common vertex.

Add to: Result Unknown

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ?

Add to: Change Unknown

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ? = 5

perpendicular lines

Two lines that intersect to form right angles

Changing a Mixed Number into an Improper Fraction

When carrying out mathematical operations, it is usually necessary to work with improper fractions rather than mixed numbers. To change a mixed number into an improper fraction: 1. Multiply the whole number by the denominator. 2. Add the numerator to the product. 3. Place the sum in the numerator over the original denominator.

Value of decimals to the right of the decimal

When comparing decimals, begin on the left and compare the digits in each place. Example: Compare 0.11 and 0.12. In the tenths place the digits are the same. Look at the hundredths. 2 is greater than 1, so 0.12 > 0.11. Compare 0.02 and 0.120. The ones are the same. 1 is greater than 0 in the tenths place, so 0.120 > 0.02. Compare 2.17 and 0.99. The ones are different. Since 2 is greater than 0, 2.17 > 0.99. Remind students that when there are non-zero digits on both sides of the decimal point, they should say, "and," where they see the decimal point. For example, 2.17 is read, "two and seventeen hundredths." Use models on a 10 x 10 grid as necessary to guide the class in comparing decimals numbers using > and <.> 1. 0.1 (>) 0.01 2. 0.51 (>) 0.509 3. 0.183 (>) 0.083 4. 1.003 (>) 0.339 5. 1.06 (>) 1.007

Rule for Multiplication of Fractions

When multiplying fractions, simply multiply the numerators together and then multiply the denominators together. Simplify the result.

Multiplying Fractions

When multiplying fractions, simply multiply the the numerators together and multiply the denominators together. It is good practice to see if and numbers can cancel. Canceling is done when the numerator and denominator can be divided evenly by the same number. Note: canceling happen top to bottom and or diagonally but never across.

Addition words

sum add all together or altogether and both combined how many in all how much in all increased by plus sum together total

the sum of the areas of all sides of a three-dimensional object

surface area

volume measurement: customary units

teaspoons, tablespoons,cups, pints, quarts and galloons; 1 gallon=4quarts, 1qt=2pints, 1pint=2cups,

10 ones make up.....

tens

creating patterns through the tiling of polygons

tessellations

extrapolation

the act of estimation by projecting known information

quotient

the answer to a division problem

median

the middle score in a distribution; half the scores are above it and half are below it

mode

the most frequently occurring score in a distribution

divisor

the number you divide by in a division problem

Order of Operations (PEMDAS)

the sequence in which operations are performed when evaluating an expression: Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction. Parentheses (simplify inside 'em) Exponents Multiplication and Division (from left to right) Addition and Subtraction (from left to right)

hypotenuse

the side opposite the right angle in a right triangle

Least common multiple

the smallest multiple that is exactly divisible by every member of a set of numbers (of 12 and 18 , the least common multiple is 36)

scalene triangle

three unequal sides

two sets if data that show a pattern

trends

having length ad width

two dinebioal

vertex

union of two segments or point of intersection of two sides of a polygon

a fraction where the numerator is 1

unit fractions

the ratio of two measurements in which the second term is 1

unit rates

Scientific notation

using powers of 10 to express large numbers (43,700 is 4.37x10 to the 4)

the amount of space that an object occupies as measured in cubic units

volume

counting numbers, including 0, that are not fractions or decimals

whole numbers

Negative of an expression

x-3 would be 3-x

horizontal position on a graph where y=0

x-axis

vertical position on a graph where x=0

y-axis

slope

y=mx+b

When you have more excess flats, rods, and etc.

you regroup them

ASA angle side angle

if 2 pairs of angles of 2 triangles are equal and the included sides are equal then the triangles are congruent

SSS side side side

if 3 pairs of sides of 2 triangles are equal, then the triangles are congruent

AAS angle angle side

if two pairs of angles of two triangles are equal and a pair of corresponding sides are equal in length, the two triangles are congruent

SAS side angle side

if two pairs of sides of two triangles are equal and the included angles are equal then the triangles are congruent

decimal multiplication powers of 10

if you are multiplying by 10, 100, 1000 then move the decimal to the right the same numbers as there are zero. if you are multiplying by 0.1, 0.01, or 0.001 then move the decimal to the left the same number there are zeros. Example: 6.3 x 1000=6.300.0

Multiplying Decimals

ignore decimal points and multiply the two numbers, then count the digits to the right of the decimal points in the original numbers and place the decimal so there are the same number of digits to the right of the decimal

linear measurement: customary units

inches, feet, yards and miles: 3ft=1yd, 1760yds=1miles, 5280ft=1mile

value that determine the value of other varibles

independent variables

two mathematical quantities that are not equal to each other

inequalities

positive or negative whole numbers that are not fractions or decimals

integer

an operation that reverse another operation

inverse operations

Cardinality

is a measure of the "number of elements of the set". For example, the set contains 3 elements, and therefore has a cardinality of 3.

measurement of something from one end to end

length

acute angle

less than 90 degrees

ray

like a line segment except it extends forever in one direction

a graph that uses points connected by lines to show data

line graphs

a part of a line that connects two points

line segements

and equation that results in a straight line when graphed

linear equations

one-dimensional geometric shape that is infinitely long

lines

the amount of matter in an object

mass

the average

mean

include mean, median, and mode

measures of center

the number in the middle when the data set is arranged from least to greatest

median

a term that is used for any calculation that you can do in your head

mental math

units of measurement based on the metric system

metric units

millimeter, centimeter, meter, kilometer

metric units of length

volume measurement; metric

milliliters and liters, liter is slightly larger than a quart; fewer than 4 liters to make a galloon

linear measurement: metric units

millimeters, centimeters, meters, kilometers; 2.5cm to 1in; 1.5k to 1mile,

the most frequency

mode

a mathematical representation of the real world

moels

obtuse angle

more than 90 degrees, less than 180

the product of two whole numbers

multiples

repeated addition of the same number itself

multiplication

numbers used when counting; do not include 0, fractions, or decimals

natural numbers

the shape of a flattened three-dimension object

nets

probability of a particular even occurring

number of ways the event can occur/total number of possible events

Prime

number that has no divisors other than one and itself

the top number in a fraction

numerator

multiplication clues (fractions)

of, product, times (especially after a number), repeated addition

only having length

one -demensianl

centimeter to meter

one hundredth of a meter

Reciprocal

one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2

ten hundreds make up.........

one thousands

the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction

order of operation

A pair of numbers that can be used to locate a point on a coordinate plane.

order pairs

The point where the two axes intersect at their zero points

orgin

mass measurement: customary units

ounces, pounds, tons: 1ton=2000pounds, 1lb=16oz

an extreme deviation from the mean

outlier

lines that remain the same distnance apart over their entire length and never cross

parallel lines

line segment

part of a line with two endpoints

a part of a whole coneyed per 100

percentages

lines that cross at a 90 degree anle

perendicular lines

distance around a two dimensional shape

perimeter

The value of a digit based on its position within a number

place value

Direction in Place Value

place value increases ten times with each shift to the left in a multi-digit number

TWO-DIMENSIONAL SHAPES that have three or more STRAIGHT SIDES

polygons

equivalent fractions

You can make equivalent fractions by multiplying or dividing both top and bottom by the same amount. You only multiply or divide, never add or subtract, to get an equivalent fraction. Only divide when the top and bottom stay as whole numbers.

To find the value of a set of fractions

You can turn the fraction into a decimal by diving the numerator by the denominator.

a number which can only be divided by itself and 1

prime numbers

the likelihood of something happening

probability

the results of multiplying two or more numbers

product

histogram

a bar chart representing a frequency distribution, heights of bars represent frequency

weighted average

a forecasting model that assigns a different weight to each period's demand according to its importance.

quadrilateral

a four-sided polygon.

scatterplot

a graphed cluster of dots, each of which represents the values of two variables. The slope of the points suggests the direction of the relationship between the two variables. The amount of scatter suggests the strength of the correlation (little scatter indicates high correlation). (Also called a scattergram or scatter diagram.) (Myers Psychology 8e p. 031)

edge

a line segment where two faces of a 3d figure meet

algorithm

a methodical, logical rule or procedure that guarantees solving a particular problem

Equivalent forms

a number can be represented in more than one way (2, 4/2...or...1/2,%50,.5...)

mixed number

a number made up of a whole number AND a fraction 2 1/3 -Divide the numerator by the denominator. -Write down the whole number answer -Then write down any remainder above the denominator.

Base 10 Number System

a number system in which all numbers are expressed using the digits 0-9

addend

a number that is added to another number

Ordinal number

a number used to tell order or sequence, 1st 2nd, 3rd....

spread

a numerical summary of how tightly the values are clustered around the "center"

tree diagrams

a organized list of all possible outcomes for an events

rhomboid

a parallelogram that is neither a rhombus nor a rectangle A=bh

Distributive property

a property indicating a special way in which multiplication is applied to addition of two or more numbers in which each term inside a set of parentheses can be multiplied by a factor outside the parentheses, such as a(b + c) = ab + ac

Distributive Property

a property indicating a special way in which multiplication is applied to addition of two or more numbers in which each term inside a set of parentheses can be multiplied by a factor outside the parentheses, such as a(b + c) = ab + ac The distributive property lets you multiply a sum by multiplying each addend separately and then add the products.

direct relationship

a relationship between two factors in which the factors move in the same direction

Digit

a symbol used to show a number 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

metric system

a system of measurement based on the number 10

equilateral triangle

a triangle with three equal sides

isosceles triangle

a triangle with two equal sides

Pascal's triangle

a triangular arrangement of numbers in which every row starts and ends with 1 and each other number is the sum of the number above it

Composite number

a whole number greater than 1 that has more than 2 factors. (opposite of prime numbers...4, 6, 8...)

Prime number

a whole number that has exactly two factors, 1 and itself. (ex: 2, 3, 5...)

the four areas of dividing one number into another

quadrants

the difference between the highest number and the lowest number in a data set

range

a number that can be made by dividing two integers; includes fractions and terminating or repating decimals

rational numbers

a comparison of two things

ratios

a shape that starts at one point and goes infinitely in ne direction

rays

making common sense

reasonables

Draw Base 10 blocks

are a set of four different types of blocks that, when used together, can help you to see what a number looks like and understand its value. Additionally, base 10 blocks can be used to help understand addition, subtraction, multiplication, division, volume, perimeter, and area.

the size of a surface measured in square unites

area

(also called the box method) a nontraditional approach to multiplication that promotes understanding of place value

area model

a pictorial representation of a multiplication problem

arrays

Division Key words

as much cut up each group has equal sharing half (or other fractions) how many in each parts per percent quotient of ratio of separated share something equally

in multiplication and addition, the way numbers are grouped in parentheses does not matter (a+b)+c=c+(a+b)

associative property

water boils

at 212 F or 100C

water freezes

at 32F or 0C

a graph that uses lengths of rectangles to show data

bar graph

the number that is left over when one number does not divide evenly in another

remainder

the numbering system where each digit is worth 10 times as much as the digit to the right of it

base 10

are two algebraic expressions of two terms. The FOIL method is one way to multiply binomials.

binomials

(also called box and whiskers plots) data is shown using the mdian and range of a data set

box plots

Multiplication key words

by (dimension) double each group every factor of increased by multiplied by of product times triple twice

Interior Angles

When two lines are crossed by a transversal they form eight angles. the four angles that lie between the two lines are called the _____________ _____________. "inside the lines"

1/0

can't divide a number by zero, doesn't equal a number

Subtractions clue words

change decreased by difference fewer or fewer than how many are left (or have left) how many did not have how many (or much) more how much longer (shorter, taller, heavier, etc.) less or less than lost minus need to reduce remain subtract take away

Associative property

changing the grouping does not change their sum or product (a+b)c=(c+a)b

Rule for dividing fractions

When you divide two fractions, take the reciprocal of the second fraction and multiply. (Taking the reciprocal of a fraction means to flip it over.)

a pie chart where each "piece" demonstrates a quantity

circle graphs

lines of symmetry

circles have an infinite number of lines of symmetry, squares have 4, rectangles have 2.

in multiplication and addition,the order of numbers, the order of the numbers on each side of the equaition does noth matter: ab=ba

commutative property

a natural number by greater than 1

composite numbers

Modeling operations

concerte method- working with real objects semiconcrete- the students work with visual reps semiabstract method-the students work with a symbol (tally marks) to represent objects

the plane containing the x axis and y axis

coordinate plane

units of measurements used in the United states

customary units

a graph of plotted points that compares two data sets

rounding

simplifying a number to any given place value

rounding

Congruent

same

Numbers to the left of the decimal point are whole numbers; numbers to the right of the decimal point are fractions with denominators of only 10, 100, 1,000, 10,000, etc

decimals

variables whose value depends on other varibles

dependent variables

van Hiele theory

describes levels of thinking and the notion of stages of learning as means by which the learner may be assisted to use higher level thinking skills

any numbers 0-9

digits

a(b + c) = ab + ac multiplication over addition

distributive properties

a number that is being divided by another number

dividends

splitting a number into equal parts

division

the numbers by which another number is divided

divisors

A GRAPHICAL DISPLAY DATA USING DOTS

dot plots

bar graphs that present more than one type of data

double bar graphs

a line graph tjat presents more than one type of data

double line graphs

Strategies for solving mathematical problems

drawing a picture, working backwards, finding a pattern, adding lines to a geometric figure...)

Base 10

each place represents ten times the value of the place to the right.

octagon

eight sided polygon

Supplementary angle

either of two angles whose sum is 180

Complementary angle

either of two angles whose sum is 90

to the zero power

equal to one

0/1

equal to zero

Diving a number by 0

equals 0

Zero multiplied or divided by any number

equals 0

algebraic expressions that use an equal sign

equations

a close prediction that involves minor calulations

estimation

breaking up a number by the value of each digit

expand form

The number that is small and raised to show how many times to multiply the number by itself.

exponents

equivalent expressions

expressions that have the same value

numbers that are multipy with each other

factors

gemotric forms made of points, lines or planes

figures

dividing decimals by whole numbers

send decimal straight up above, then divide like normal numbers

pentagon

five sided polygon

Factor tree

showing the prime factors of a number in a simple diagram

hexagon

six sided polygon

three-dimensional objects

solids

algebraic inequality

statement that is written using one or more variables and constants that shows a greater than or less than relationship 2x+8>24

the study of data

statistics

An arithmetical expression used to calculate values

formulas

parallelogram

four sided polygon with two pairs of parallel sides. A=bh

a number that expresses parts of a whole or a group

fractions

Equivalent Fractions

fractions that have the same value but may look different. Ex, 1/2, 2/4, 3/6,etc.

a relationship between input and output

functions

mass measurement: metric

grams and kilograms; 28 grams to 1 ounce

bar graphs showing continuous data over time

histogram

Tens 10's make up.......

hundreds

Compare: Bigger Unknown

(Version with "more"): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Version with "fewer"): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 2 + 3 = ?, 3 + 2 = ?

Compare: Smaller Unknown

(Version with "more"):Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?(Version with "fewer"): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 - 3 = ?, ? + 3 = 5

Writing Fractions in it's simplest form

-Fractions can be simplified when the numerator and denominator have a common factor in them. If both the numerator and denominator have common factors, then we can cancel these factors out. For example, in the fraction 8/12, 4 is a common factor of both 8 and 12. -We can simplify the fraction by canceling the 4 from both the numerator and denominator of the fraction. Canceling is equivalent to dividing both the numerator and denominator by the same number.

Value of a fraction

-When the numerator stays the same, and the denominator increases, the value of the fraction decreases. -When the denominator stays the same, and the numerator increases, the value of the fraction increases.

1/10000

0.0001

1/1000

0.001 (0.1%)

1/100

0.01, 1%

1/10

0.1 10%

miles to yards

1 mile = 1760 yards

yard to feet

1 yard = 3 feet

time

1 year=365 days, 1 year=52 weeks

kilo

1,000

1 liter = ____ milliliters

1,000 milliliters

yards in a mile

1,760 yards in a mile

Multiplying mixed numbers

1. Change the mixed number into an improper fraction by multiplying the denominator by the whole number. Then add the numerator in. Your denominator stays the same. 2. Multiply the numerators. 3. Multiply the denominators. 4. Simplify

The rules for adding and subtracting fractions can be broken down into several steps:

1. Determine whether the fractions have the same denominator. If the denominators are the same, move to step 4. 2. If the denominators are different, find the LCD for the fractions being added or subtracted. 3. For each of the original fractions, find its equivalent fraction with the LCD in the denominator. 4. Add or subtract the numerators of the fractions. 5. Simplify the resulting fraction.

possible place value strategies

1. Draw base 10 blocks 2. draw jumps on an empty number line 3. Partial Sums (Expanded form layout): Each addend is represented using expanded notation. Like place values are added or subtracted. 4.Partial Sums: Expanded Form layout 5. Partial Differences: 6. Use multiplication facts and place value to multiply by multiples of ten: 7. Use the distributive property to multiply within 100. 8. Area model 9. Partial Quotients 10. Partial Dividends 11. add or subtract decimals on empty number line

Considerations in place value

1. Recognize that in a multi-digit whole number, a digit in one place represents 10 times what it represents in the place to its right. 2.Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 3. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10.

proportion

1. The relationship of one thing to another in size, amount, etc. 2. Size or weight relationships among structures or among elements in a single structure.

centi

1/100, .01

the little square represents

1/1000 of the whole

milli

1/1000, .001

area of a trapezoid

1/2(b1+b2)h

millimeter to centimeter

10 centimeters

A flat with decimals Represents

10/1000 or 1/10 of a whole

Rods with decimal represents

10/1000 or 1/100 of a whole

1 kilogram (kg)

1000 grams (g)

meters to kilometers

1000 m = 1 km

kilometer

1000 meters

kilometer to meter

1000 meters

1 liter

1000 milliliters

Straight Line

180-degree angle

Algorithms

2 addend + 3 addend= 5 sum 5 minuend- 3 subtrahend= 2 difference

supplementary angles

2 angles that add up to 180

complementary angles

2 angles that add up to 90

1 pint

2 cups

1 quart

2 pints or 4 cups

diameter of a circle

2 x radius

1 gallon

4 quarts 8 pints 16 cups 128 fluid ounces

regular quadrilateral

4 sided polygon. All sides and angles are equal. (Square)

distributive property

4(8+3)=(4x8)+(4x3)

Define a square in terms of other two-dimensional geometric figures

A square is a quadrilateral with 4 sides of equal length and 4 angles of equal measure, whereas a rectangle is a quadrilateral with 4 angles of equal measure, a rhombus is a quadrilateral with 4 sides of equal length, and a parallelogram is a quadrilateral where opposite sides are parallel. Therefore, a rectangle that has 4 sides of equal length is a square, and a rhombus that is also a rectangle is a square,

stem and leaf plot

A system used to condense a set of data where the greatest place value of the data forms the stem and the next greatest place value forms the leaves

frequency table

A table for organizing a set of data that shows the number of times each item or number appears.

net

A two-dimensional pattern that can be folded to make a three-dimensional prism or pyramid

compare-UNKNOWN PRODUCT

A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?

algebraic expression

A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

Inequalities

A mathematical sentence involving <, >, or = plus < or >

millimeter to meter

A metric unit of linear measure equal to 1/1000 of a meter.

dividend

A number that is divided by another number.

Integer

A number that is not a fraction in anyway

subtrahend

A number that is subtracted.

Multiplicative Inverse

A number times its multiplicative inverse is equal to 1; also called reciprocal

Decimal

A number written on the basis of powers of ten 53.109

Base system

A place-value system in which each place value is 10 times larger than the place value to its right.

Regular Rectangle

A polygon in which all sides have equal lengths and all angles have equal measures.

Regular Polygon

A polygon with all sides and all angles are equal

compare-NUMBER OF GROUPS UNKNOWN ("HOW MANY GROUPS?" DIVISION)

A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first?

compare-GROUP SIZE UNKNOWN ("HOW MANY IN EACH GROUP?" DIVISION)

A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost? Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first?

inverse relationship

A relationship in which one variable decreases when another variable increases

Acute Angle

An angle that measures less than 90 degrees

Obtuse Angle

An angle that measures more than 90 degrees but less than 180 degrees

Decomposing Numbers

Break down numbers into their sub-parts. (Taking a standard number and changing it into expanded form). process of breaking a number into smaller units to simplify problem solving ex. 15 can be 10+5 or 10 can be 6+ 4

Associative Property

Changing the grouping of numbers will NOT change the value. For example: (7 + 4) + 8 = 7 + (4 + 8) also works with multiplication

counting on

Counting up from the lesser number to the greater number.

Cardinal Number

Counting, indicate quantity

pints in a quart

2

feet to yards

3 ft = 1 yd

1 yard = _____ inches

36 inches

cube

3d solid figure, has 6 faces, 12 edges and 8 vertices

quarts in a gallon

4

Whole number exponents to denote powers of 10

Each decimal to the left of the decimal point increases progressively in powers of 10. Each digit to the right of the decimal point decreases progressively in powers of 10.

put together/take apart both addends unknown

Grandma has five flowers. How many can she put in the red vase and how many in her blue vase? 5 = 0 + 5, 5 + 0 5 = 1 +4, 5 = 4 +1, 5 = 2 + 3, 5 = 3 + 2

Intergers

An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero.

Alternate Exterior Angles

Angles that lay outside the parallel lines and are on opposite sides of the transversal; They are congruent

Corresponding Angles

Angles that lie on the same side of the transversal in corresponding positions

Consecutive Exterior Angles

Angles which share the same side of the transversal and are outside the lines.

Compare Difference Unknown

("How many more?" version):Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?("How many fewer?" version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have then Julie? 2 + ? = 5, 5 - 2 = ?

ounce in a pound

16 ounces in a pound

Match each fraction with its equivalent decimal number. 20/100 2/10 20/10 200/100 2/100 Categories: 0.02 0.2 2

2/100 = 0.02 20/100 and 2/10 = 0.2 20/10 and 200/100 = 2

pounds in a ton

2000 pounds in a ton

Coin flip

2^number of

Circumference

2pir

Area of a cylinder

2pirh+2pir^2

circumference of a circle

2πr

Properties of a circle

= radi means congruent, different radi are similar

Polygons

A closed figure, all straight lines, and no intersecting lines.

algabraic expression

A combination of variables, numbers, and at least one operation.

Polygon (convex)

A convex polygon is defined as a polygon with all its interior angles less than 180°. This means that all the vertices of the polygon will point outwards, away from the interior of the shape. Think of it as a 'bulging' polygon. Note that a triangle (3-gon) is always convex.

Zero

A digit representing the absence of quantity. Zero is necessary in holding place value. Ex: 402,005

Decimal point:

A dot noting the change from positive powers of ten (left of point) to negative powers of ten (right of point) Ex: 53.109

Plane

A flat two-dimensional surface that extends infinitely in all directions.

Common Fractions

A fraction where both the top and bottom are whole numbers.

Mr. Kirk asked his students to compare 0.196 and 0.15. Four of his students correctly answered that 0.196 is greater than 0.15, but they gave different explanations when asked to describe their strategies to the class. Indicate whether each of the following student explanations provides evidence of a mathematically valid strategy for comparing decimal numbers. A. 0.196 is larger because there is one in the tenths, and then nine hundredths is more than five hundredths. And then I'm done. B. 0.196 is greater because in the thousandths place six is greater than five, and in the hundredths place nine is greater than one. C. 0.196 is bigger than 0.15 because if it is three numbers long, it will always be bigger than if it is two numbers long. D. 0.196 is more than 0.15 because nineteen hundredths is bigger than fifteen hundredths.

A. Provides Evidence B. Does NOT Provide Evidence C. Does NOT Provide Evidence D. Provides Evidence

Ayana's banana bread recipe uses 3 bananas to make 2 loaves of banana bread. Natalie's banana bread recipe uses 4 bananas to make 3 loaves of banana bread. Whose recipe results in a greater amount of banana in each loaf of banana bread? Mr. Ma asked his class to solve the word problem shown. Three students correctly answered that Ayana's recipe results in a greater amount of banana in each loaf, but they gave different explanations when describing their strategies to the class. Indicate whether each of the following student explanations provides evidence of a mathematically valid strategy for determining whose recipe results in a greater amount of banana in each loaf. A. In Ayana's recipe there are 3 bananas for 2 loaves, so there is a whole banana for each loaf and you split the last banana in half. In Natalie's recipe there is one banana for each loaf and the fourth banana is split in 3. So in Ayana's loaf there are 1 and a half bananas, and in Natalie's there are 1 and a third, and a half is more than a third. B. In Ayana's recipe the bananas are split between only 2 loaves, while in Natalie's recipe the bananas are split between 3 loaves. If I have to split a cookie, I would rather split it in two because I get more, so Ayana's loaves contain more bananas. C. Ayana makes only 2 loaves and Natalie makes 3 loaves. If they made the same number of loaves, like 6, then Ayana would use 9 bananas and Natalie would use 8. So Ayana's loaves have more because 9 is more than 8.

A. Provides Evidence B. Does NOT Provide Evidence C. Provides Evidence

Area of a rectangle formula

A= length x width

volume of a rectangular solid

A=lwh

Area of a circle

A=pir^2

area of a triangle

A=½bh

area of a circle

A=πr²

Fraction Rules

Addition and Subtraction = common denominator multiplication = straight across Division = To divide one fraction by another, invert (turn upside-down) the second fraction, then multiply.

Commutative Property

Addition and multiplication the order of addends does not determine the sum or product. example: a+b=b+a 6x9 and 9x6 both equal 54. Subtraction and division are not ________________.

adjacent angles

Adjacent angles are two angles that have a common vertex and a common side. The vertex of an angle is the endpoint of the rays that form the sides of the angle. When we say common vertex and common side, we mean that the vertex point and the side are shared by the two angles. Here's an example of adjacent angles:

Rational Numbers

Can be written as a ratio or fraction. in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. 1/2. 3/4. 5/6.

contain numbers, variables, and mathematical expression

Algebraic Expression

A methodical, logical rule or procedure that guarantees solving a particular problem.

Algorithm

Adding Decimals

Align the decimal points and add as you normally would.

Regular Triangle

All congruent sides and angles

Irrational Numbers

All numbers that are not rational are considered ______________. Can be written as a decimal, but not as a fraction. An _______________ number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers: 0, Pi - 3. 14...

rational numbers

All positive and negative integers, fractions and decimal numbers.

Real Number

All rational numbers and all irritational numbers, basically everything

Converting Between Improper Fractions and Whole/Mixed Numbers

An improper fraction is one in which the numerator is larger than the denominator. For example, if you were told you had six-fourths (6/4) of a pie left, you would know that you had one whole pie (4/4) plus one-half of a pie (2/4)

Inverse property

Any number multiplied by its inverse =1

Real Numbers

Are all numbers that can be represented by points on the number line. zero, irrational, rational, and negative or positive, decimals

Integers

Are like whole numbers, but they also include negative numbers ... but still no fractions allowed! So, integers can be negative {-1, - 2,-3, -4, -5, ... }, positive {1, 2, 3, 4, 5, ... }, or zero {0}

Prime Numbers

Are numbers with only two whole number factors. 1 and itself. the first few are. 2, 3, 5,7,11,13, 17, 19...1 x 2 = 2, 1 x 3 = 3... they are always odd numbers.

Whole Numbers

Are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on) ; are the counting numbers. 0,1,2,3,4,5...

Consecutive Interior Angles

Are the interior angles that lie on the same side of the transversal are called ______________ ______________ ________________.

Permutation

Arrangement where order matters

Decimals Explained

As you move right from the decimal point, each place value is 1/10 the value of the number to its left. The first number to the right of the decimal point is in the tenths place. Tenths are tenths of one whole. Ten tenths are equal to one whole. The second number to the right of the decimal point is in the hundredths place. A hundredth is one hundredth of a whole. One hundred hundredths make up a whole. The third number to the right of the decimal point is in the thousandths place. A thousandth is one thousandth of a whole. One thousand thousandths make up a whole.

Comparing fractions strategies

Comparing Fractions Example: 12/32 and 28/40 Do you really need to find a common denominator in order to compare these two fractions? I think not. The first fraction is clearly less than one-half, while the second is greater than one-half. Case closed. Comparing fractions using a benchmark of one-half is just one of the strategies students should have in their toolbox.

formed by two rays with a common endpoint

angles

Perimeter (circumference) of a circle

C=2πr

congruent angles

Congruent Angles have the same angle (in degrees or radians). That is all.

Covert a decimal to a fraction

Convert decimal 0.05 to a fraction 0.05 = 1/20 as a fraction Step by Step Solution To convert the decimal 0.05 to a fraction follow these steps: Step 1: Write down the number as a fraction of one: 0.05 = 0.05/1 Step 2: Multiply both top and bottom by 10 for every number after the decimal point: As we have 2 numbers after the decimal point, we multiply both numerator and denominator by 100. So, 0.05/1 = (0.05 x 100)/(1 x 100) = 5/100. Step 3: Simplify (or reduce) the fraction: 5/100 = 1/20 when reduced to the simplest form.

Dividing Decimals

Move the decimal to the right on both numbers until there is nothing to the right of the decimal

Multiplying by decimals-10

Moves every digit one place to the left

The bottom number in a fraction

DENOMINATOR

Place Value Decimals

Decimals are a shorthand way to write fractions and mixed numbers with denominators that are powers of 10 , like 10,100,1000,10000, etc. If a number has a decimal point , then the first digit to the right of the decimal point indicates the number of tenths. For example, the decimal 0.3 is the same as the fraction 3/10 . The second digit to the right of the decimal point indicates the number of hundredths. For example, the decimal 3.26 is the same as the mixed number 3 26/100 . (Note that the first digit to the left of the decimal point is the ones digit.) You can write decimals with many places to the right of the decimal point. For example, this is a representation of the mixed number 51480531000000 , with the place values named:

Composite NUmber

Dividible by more than one

Multiples

Of any whole numbers are the results of multiplying that whole number by the counting numbers. Every whole number has an infinite number of multiples. example 7x1=7, 7x2=14,7x3=21,7x4=28. the multiples of 7 are 7,14,21,28 and so on.

Distributive Property

Of multiplication over addition example: 6 x 47 gives the same result as multiplying 6 x 40 and then 6 x 7. and then adding the products. 6 x 47 = (6 x 40) + (6 x 7). notation is a(b+c) = (a x b) + (a x c)

Variance Sum

Of the squares quantity divided by the number of items

Subitizing

The ability to instantly "see" the number of objects in a small set without having to count them.

convert a decimal to a fraction

First, convert the decimal to fraction using tenths, hundredths, thousandths, etc. depending on the number of decimal places. e.g. 1.75 = 1 75/100. Next, simplify the fraction part to the lowest common term. e.g. 75/100 = 3/4.

Take From: Change Unknown

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?5 - ? = 3

Take From: Result Unknown

Five apples were on the table. I ate two apples. How many apples are on the table now?5-2 = ?

Rational numbers

Have a distinct end and is a quotient/fraction

Alternate Interior Angles

The interior angles that lie on opposite sides of the transversal are _____________ _____________ ______________.

Make 10 strategy

Make Ten Strategy for Addition Step 1: The first addend and what make ten? Step 2: Write the number below the second addend Step 3: The number below the second addend and what make the second addend? Step 4: Add the rest to 10

Composite Numbers

Most of our numbers are ________________ because they are composed of several whole number factors

Decimal multiplication powers of 10

Move the decimal point right as many places as there are 0's in the power. If there are not enough digits, add on 0's.

Identity Property

Multiply by one and get original

Which of the following word problems can be represented by the equation 4×n+8=16? A set of 5 baskets holds a total of 16 apples. The first basket has 8 apples and the other baskets each hold an equal number of apples. How many apples are in each of the other baskets? There are 12 baskets, 8 of which are empty. There are 16 apples, with an equal number of apples in each of the other 4 baskets. How many apples are in each of the 4 baskets? There are 16 baskets, 8 of which are empty. Each of the other baskets contains 4 apples. How many apples are there in all? There are 8 baskets with 4 apples in each basket and 16 apples that are not in a basket. How many apples are there in all?

Option (A) is correct. If there are 5 baskets and one basket holds 8 apples, the rest of the apples are split evenly among the other 4 baskets. Therefore, to find the number of apples in each of the 4 baskets, the equation 4×n+8=16 can be set up, where n is the number of apples in each of the 4 baskets.

A rectangular message board in Aleyah's dormitory room has a length of 30 inches and a perimeter of 108 inches. A rectangular bulletin board in the hallway outside Aleyah's room is twice as long and twice as wide as the message board. Which of the following statements about the bulletin board is true? The bulletin board has a width of 48 inches. The bulletin board has a length of 96 inches. The area of the bulletin board is twice the area of the message board. The perimeter of the bulletin board is four times the perimeter of the message board.

Option (A) is correct. Since the message board has a length of 30 inches and a perimeter of 108 inches, the width of the message board can be found by solving the equation 2(30)+2w=108 for w. To solve the equation for w, subtract 60 from both sides of the equation and then divide both sides of the equation by 2 to find that w=24. This means that the length and width of the bulletin board are 60 inches and 48 inches, respectively, and it can be concluded that the area of the message board is 720 square inches, the perimeter of the bulletin board is 216 inches, and the area of the bulletin board is 2,880 square inches. Thus, the only true statement is that the bulletin board has a width of 48 inches.

Ms. Garrett has been working on verbal counting with her students. She wants them to be more aware of patterns in the way number names are typically constructed. Which of the following number names LEAST reflects the typical pattern in the way number names are constructed in the base ten system? Eleven Sixteen Twenty-five Ninety

Option (A) is correct. The number name "eleven" does not follow any pattern of number-name construction with reference to the tens and ones. Option (C) is not correct because "twenty-five" follows the most typical structure of how number names are constructed for whole numbers, since the number of tens in the number is referred to first, followed by the number of ones. Although the numbers in options (B) and (D) do not follow the most typical structure like "twenty-five" does, where the tens are called out specifically, the numbers in these options do follow a structure of the number of ones being named, followed by "teen," which refers to the ten in the number. Therefore, these numbers follow a pattern, unlike "eleven."

The scenario in a word problem states that an office supply store sells pens in packages of 12 and pencils in packages of 20. Which of the following questions about the scenario involves finding a common multiple of 12 and 20 ? In one package each of pens and pencils, what is the ratio of pens to pencils? How many packages of pens and how many packages of pencils are needed to have the same number of pens as pencils? If the store sells 4 packages each of pens and pencils, what is the total number of pens and pencils sold in the packages altogether? How many gift sets can be made from one package each of pens and pencils if there are the same number of pens in each set, the same number of pencils in each set, and all the pens and pencils are used?

Option (B) is correct. The least common multiple of 12 and 20 is 60, and 5 packages of pens and 3 packages of pencils are needed to have 60 of each writing utensil

Mr. Walters asked his students to order 89, 708, 37, and 93 from least to greatest, and to be ready to explain the process they used to order the numbers. One student, Brianna, ordered the numbers correctly, and when Mr. Walters asked her to explain her process, she said, "The numbers 89, 37, and 93 are less than 100, so they are all less than 708, since that is greater than 100. Also, 37 is the least because it comes before 50 and the other two numbers are close to 100. Then 89 is less than 90, but 93 is greater than 90." Which of the following best describes the strategy on which Brianna's explanation is based? A counting strategy A benchmarking strategy An estimation strategy A place-value strategy

Option (B) is correct. Brianna first indicates that 708 is the greatest number because it is greater than 100, while 37, 89, and 93 are all less than 100. Next, Brianna indicates that 37 is the least number because it is less than 50, while 89 and 93 are greater than 50. Finally, Brianna recognizes that 89 is less than 93 because 89 is less than 90, while 93 is greater than 90. Thus, over the course of her explanation, Brianna used 100, then 50, and then 90 as points of reference for comparisons, which is exactly what benchmark numbers are—points of reference for comparison. Brianna did not count between any of the numbers, estimate the numbers, or use the place values in any of the numbers to make her comparisons, so the other options do not describe the strategy on which Brianna's explanation is based.

Ms. Roderick asked her lunch helper in her kindergarten class to get one paper plate for each student in the class. Which of the following counting tasks assesses the same mathematical counting work as this task? Having students line up according to the number of the day of the month in which they were born Showing students 10 pencils and asking them to get enough erasers for all the pencils Showing students a row of 12 buttons and asking them to make a pile of 8 buttons Asking students to count the number of triangles printed on the classroom rug

Option (B) is correct. Getting one paper plate for each student in the class assesses whether students can determine when the number of objects in one set is equal to the number of objects in another set, and the task described in option (B) involves a similar determination. The task in option (A) assesses whether students can compare and order numbers. The task in option (C) assesses whether students can count a subset of objects from a larger set. The task in option (D) assesses whether students can count the number of objects in a set.

Ms. Simeone is working with her first-grade students on writing two-digit numerals. She wants to use an activity to assess whether her students are attending to the left-to-right directionality of the number system. Which of the following activities is best aligned with Ms. Simeone's purpose? Asking students to read the numbers 20 through 29 Asking students to represent the numbers 35 and 53 using base-ten blocks Asking students how many tens and how many ones are in the number 33 Showing students 23 cubes and 32 cubes and asking them which quantity is greater

Option (B) is correct. Having the students represent 35 and 53 using base-ten blocks will help Ms. Simeone assess whether students know which place is the tens place and which place is the ones place or whether students have reversed the ones place and the tens place, thinking the ones place is on the left and the tens place is on the right. Representing the numbers provides more information about students' understanding of place value than just reading numbers.

In word problems that have a multiplicative comparison problem structure, two different sets are compared, and one of the sets consists of multiple copies of the other set. Which of the following best illustrates a word problem that has a multiplicative comparison problem structure? There are 4 shelves in Joaquin's bookcase, and there are 28 books on each shelf. How many books are in Joaquin's bookcase? Marcus drives 3 times as many miles to get to work as Hannah does. Hannah drives 16 miles to get to work. How many miles does Marcus drive to get to work? A football field is 360 feet long and 160 feet wide. A soccer field is 300 feet long and 150 feet wide. The area of the football field is how many square feet greater than the area of the soccer field? An ice cream parlor sells 29 different flavors of ice cream and 4 different types of cones. How many different combinations consisting of an ice cream flavor and a type of cone are available at the ice cream parlor?

Option (B) is correct. In the problem in option (B), the two values being compared are the number of miles that Marcus drives to get to work and the number of miles that Hannah drives to get to work, and the number of miles that Marcus drives is 3 times the number of miles that Hannah drives. The problem in option (A) has an equal-groups problem structure, the problem in option (C) has a product-of-measures problem structure (since the product is a different type of unit from the factors in the problem), and the problem in option (D) has a combinations problem structure.

Answer the question below by clicking on the correct response. Question: The figure presents two congruent squares. Each square is divided into 4 equal sections. During a lesson in her second-grade class, Ms. Costa draws two squares of the same size, each representing the same whole. She then divides and shades the squares as represented in the figure. Her students consistently identify the area of each shaded region as one-fourth, but when they are asked if the areas are equal, some students say no. Which of the following statements most likely explains why the students see the areas as not being equal? The students think that the areas are not equal because the wholes are different sizes. The students think that the areas are not equal because the shaded regions are different shapes. The students have difficulty determining the size of geometric figures that include diagonal lines. The students have difficulty determining the part-to-whole relationship when working with visual models of fractions.

Option (B) is correct. One misconception that students often have when first beginning to work with area models of fractions is that the parts of the whole must be congruent for the areas of the parts to be equal, and this misconception explains the responses described in the question. The statement in option (A) is not correct because students at this level do not normally attend to the size of the whole when working with visual models of fractions, and the question states that Ms. Costa draws two squares of the same size. The statement in option (C) is not correct because students will make a similar error even when figures do not include diagonal lines. The statement in option (D) is not correct because it does not explain the situation presented, since the students consistently identify the area of each shaded region as one-fourth.

Answer the question below by clicking on the correct response. Question: 1/3=3/9=6/18 1/4=4/16=3/12 Ms. White's students are working on generating equivalent fractions like the ones shown. She asks her students to write a set of instructions for how to generate equivalent fractions. One student writes, "You have to multiply the bottom and the top of the fraction by a number." Which of the following revisions most improves the student statement in terms of validity and generalizability? You have to multiply both denominator and numerator by the same number. You have to multiply both denominator and numerator by the same nonzero number. You have to multiply both denominator and numerator by the same whole number. You have to multiply both denominator and numerator by the same positive whole number.

Option (B) is correct. To generate an equivalent fraction, it is not necessary to multiply the numerator and denominator of the original fraction by a whole number, but it is necessary to multiply the numerator and the denominator by the same number and for that number to be a number other than zero. The revision in option (B) is the only sentence that restates the student conjecture, makes it valid, and generalizes it by including all fractions.

Which of the following fractions has a value between the values of the fractions 7/9 and 8/11? 1/2 2/3 3/4 4/5

Option (C) is correct.

Ms. Rodriguez is working with her kindergarten students to develop the skill of counting on. Which of the following tasks is best aligned with the goal of having students count on? The teacher gives each student a number book with a different number on each page. The students must count out and glue the same number of pictures to match the given number on each page. The teacher gives each student a 10-piece puzzle, disassembled with a single number written on each piece. The students must put the puzzle together with the numbers in order. The teacher gives each student a shuffled deck of 10 cards, each with a single number from 1 to 10. When the students draw a number card, they must count to 20, starting from the number on the card they drew. The teacher gives each student 8 blocks and a number cube, with the sides of the number cube numbered from 3 to 8. When the students roll the number cube, they must count out the same number of blocks as the number rolled and create a tower with that number of blocks.

Option (C) is correct. A student would begin with the number drawn and count on from that number until 20 is reached. For example, if the student draws a card with 15 on it, the student would count on from 15, saying, "15, 16, 17, 18, 19, 20." The other tasks described do not require students to count on.

One of Mr. Spilker's students, Vanessa, incorrectly answered the addition problem 457+138 as represented in the work shown. The figure presents the work the student did to add the numbers. The problem is written vertically. The work shows 457 + 138= 585. Mr. Spilker wants to give Vanessa another problem to check whether she misunderstands the standard addition algorithm or whether she simply made a careless error. Which of the following problems will be most useful for Mr. Spilker's purpose? 784+214 555+134 394+182 871+225

Option (C) is correct. In the work shown, after adding the ones and recording the 5 in the ones place, Vanessa did not record that the additional 10 ones were 1 ten, nor did she add the regrouped ten in the tens place. The problem in option (C) will be most useful for Mr. Spilker's purpose because it requires regrouping from the tens place to the hundreds place. The problems in options (A) and (B) do not require any regrouping, and Vanessa may just record 10 hundreds without thinking about regrouping when answering the problem in option (D).

Ms. Cook's class was discussing strategies to compare two fractions. One student, Levi, said, "When the top numbers are the same, you know that the one with the smaller number on bottom is bigger." Ms. Cook asked her students to explain why Levi's claim is true. After giving the class time to work, she asked another student, Maria, to present her explanation. Maria said, "It's just like Levi said. For 1/4 and 1/2, they both have ones on top, and 4 is greater than 2, so 1/4 is less, just like 1/4 of a pizza is less than 1/2 of a pizza." Which of the following statements best characterizes Maria's explanation? It clearly explains why Levi's claim is true. It clearly explains why the converse of Levi's claim is true, but it does not explain why his actual claim is true. It shows that Levi's claim is true for one example, but it does not establish why his claim is true in general. It assumes that Levi's claim is true, but it does not establish why his claim is true in general.

Option (C) is correct. Maria explains why 1/4 is less than 1/2, which provides one example for which Levi's claim is true, but it does not explain why whenever two fractions have the same numerator, the fraction with the smaller denominator will always be the greater fraction. A general explanation would point out that when a whole is broken into a greater number of pieces of equal size, then each of those pieces will be smaller than the pieces when the whole is broken into fewer pieces of equal size.

Ms. Duchamp asked her students to write explanations of how they found the answer to the problem 24×15 One student, Sergio, wrote, "I did 24 times 10 and got 240, then I did 24 times 5 and that's the same as 12 times 10 or 120, and then I put together 240 and 120 and got 360." Ms. Duchamp noticed that four other students found the same answer to the problem but explained their strategies differently. Which of the following student explanations uses reasoning that is most mathematically similar to Sergio's reasoning? Since 24 is the same as 12 times 2 and 15 is the same as 5 times 3, I did 12 times 5 and got 60, then I did 2 times 3 and got 6, and 60 times 6 is 360. To get 24 times 5, I did 20 times 5 and 4 times 5, which is 120 altogether, and then I needed 3 of that, and 120 times 3 is 360. 15 times 20 is the same as 30 times 10, and that gave me 300, and then I did 15 times 4 to get 60, and 300 plus 60 is 360. 24 divided by 2 is 12, and 15 times 2 is 30, so 24 times 15 is the same as 12 times 30, and so my answer is 360.

Option (C) is correct. Sergio first uses the distributive property to think of 24×15 as 24×(10+5), or 24×10+24×5. After Sergio multiplies 24 and 10 to get 240, he multiplies 24 and 5 using a doubling and halving strategy. Since 24=12×2, 24×5=(12×2)×5=12×(2×5)=12×10, so the product of 24 and 5 is equal to the product of 12 and 10, which is 120. The explanation in option (C) also uses the distributive property but in a different way. This student thinks of 24×15 as (20+4)×15, or 20×15+4×15. After the student multiplies 20 and 15, the student uses the doubling and halving strategy to find the product of 4 and 15. Therefore, this explanation uses reasoning that is most mathematically similar to Sergio's reasoning. The explanations in options (A) and (B) do not use the doubling and halving strategy, and the explanation in option (D) does not use the distributive property.

Answer the question below by clicking on the correct response. Question: Ms. Shaughnessy is working with her class on measuring area using nonstandard units. While the students are finding the area of the surface of their desks using rectangular note cards, one student says, "I can just measure the long side of the desk with the long side of the card, then measure the short side of the desk with the short side of the card, and multiply them." Which of the following best describes the validity of the student's strategy? The strategy is not valid because the same unit must be used to measure each side of the desk. The strategy is valid only if the note cards are squares. The strategy is valid and the unit of measurement is square units. The strategy is valid and the unit of measurement is note cards.

Option (D) is correct. Area can be measured using any two-dimensional unit that covers a surface, but the label of the area must reflect that unit. In this case the student has used note cards as the unit to measure the area of the desk. When using square units, one counts how many times the side of the square unit fits on each side of the rectangle whose area is to be measured. When using a unit that is not a square, like a note card, it is important to keep the orientation of the unit constant to cover the area without overlapping. This method results in one dimension of the rectangle being measured with the long side of the note card and the other dimension of the rectangle being measured with the short side of the note card.

Ms. Howe's students are learning how to use models to help them answer word problems. The models use bars to represent the relationships between the given quantities and the unknown quantity. In each model, the unknown quantity is represented with a question mark. The quantities given in the word problem occupy the other boxes. Ms. Howe shows the following model to her students. Which of the following word problems best corresponds to the model shown? Max had $24. He gave $18 to Olivia and the rest to Sarah. How much money did Max give to Sarah? Max had $24. He gave 1/3 of his money to Sarah and the rest to Olivia. How much money did Max give to Olivia? Max gave $24 to his friend Sarah and $18 to his friend Olivia. What is the total amount of money Max gave to his two friends? Max has $24 in his piggy bank, which is 23 of the amount of money that Max has altogether. How much money does Max have altogether?

Option (D) is correct. In the model shown, the total amount is the unknown quantity, and the quantity of $24 given in the problem is 2/3 of the total quantity. Since the problem in (D) asks for the total amount of money that Max has and states that $24 is 2/3 of the total, it is the problem that best corresponds to the model.

Ms. Carter shows one of her students, Brandon, a set of cubes. She tells Brandon that there are 13 cubes in the set and asks him to take 1 cube away from the set. Ms. Carter then asks Brandon, "How many cubes do you think are in the set now?" Brandon quickly answers, "Twelve." Brandon has demonstrated evidence of understanding which of the following mathematical ideas or skills? A. Using numerals to describe quantities B. Counting with one-to-one correspondence C. Recognizing a small quantity without counting D. Knowing that each previous number name refers to a quantity which is one less

Option (D) is correct. In the scenario, Ms. Carter shows Brandon a set of cubes, explicitly tells him how many cubes are in the set, and asks him to take one cube away from the set. This process allows Ms. Carter to ensure that Brandon knows that there is now one less cube in the set. When Ms. Carter asks how many cubes are in the set after one cube is removed, Brandon readily states, without counting the cubes, that there are 12 cubes. This provides evidence that Brandon knows that 12 is the number name that precedes 13 and that 12 refers to a quantity that is one less than 13; it can also be assumed that Brandon has the same understanding for other whole numbers. Brandon did not use written numerals in the scenario, so option (A) is not correct. Also, Brandon is told how many cubes are in the set, so there is no evidence that he can count with one-to-one correspondence or recognize a small quantity without counting, so options (B) and (C) are not correct.

A student incorrectly answered the problem 305.74×100 . The student's answer is represented in the work shown. 305 point 74 times 100 equals 305 point 7 4 0 0 Which of the following student work samples shows incorrect work that is most similar to the preceding work? 246.7 X 100= 2,467 13.05 X 100= 13,500 46.13 X 10 = 460.130 94.03 X 10 = 94.030

Option (D) is correct. In the work shown, when the student multiplied 305.74 by 100, the student rewrote 305.74 and added two zeros at the end. The work sample that is most similar to this is the sample in option (D), since this sample shows that when the student multiplied 94.03 by 10, the student rewrote 94.03 and added one zero at the end.

Answer the question below by clicking on the correct response. Question: Last Tuesday, a group of 5 researchers in a laboratory recorded observations during a 24-hour period. The day was broken into 5 nonoverlapping shifts of equal length, and each researcher recorded observations during one of the shifts. Which of the following best represents the amount of time each researcher spent recording observations last Tuesday? Between 4 and 4 1/4 hours Between 4 1/4 and 4 1/2 hours Between 4 1/2 and 4 3/4 hours Between 4 3/4 and 5 hours

Option (D) is correct. Since the 24-hour period is broken into 5 overlapping shifts of equal length, the problem is solved by finding 24/5

Ms. Fisher's students are working on identifying like terms in algebraic expressions. When Ms. Fisher asks them how they know when terms are like terms, one student, Coleman, says, "Like terms have to have the same variable in them." Ms. Fisher wants to use a pair of terms to show Coleman that his description of like terms is incomplete and needs to be refined. Which of the following pairs of terms is best for Ms. Fisher to use for this purpose? 9d and 5 8xy and xy 5a^4 and 2a^4 4h^2 and 7h^3

Option (D) is correct. The best pair of terms for Ms. Fisher's purpose should contain the same variable but should not be like terms. The only option that shows such a pair is option (D), in which the variables are the same but the terms are not like terms because they have different exponents.

Ordinal

Ordinal numbers indicate the order or rank of things in a set rank, order 1st, 2nd, 3rd......

Perimeter of a square

P = 4s

Perimeter of a rectangle

P= 2L + 2w

Perimeter formula for a rectangle

P=2l+2w

Order of Operations

PEMDAS

Order of operations

PEMDAS, Please excuse my dear aunt sally; parenthesis, exponents, multiplication, division, addition, subtraction

Value

Quality of a digit 2 = 2 ones 39 = 3 tens and 9 ones

Proportion

Says that two ratios (or fractions) are equal

Composing Numbers

Subparts into a whole. (Taking an expanded number and changing it to standard notation).

Round

Substitute an approximate value (usually to the nearest 10, 100, 1,000, etc.)

Fahrenheit to celsius

Subtract 32 and divide by 1.8

Sum of Squares

Sum of the squares of the differences between each item and the mean

minuend

The number from which another number is subtracted; the first number in a subtraction problem

Commutative property

The order of two numbers may be switched around and the answer is the same. (a+b=b+a)

using a grid to find perimeter and area

The perimeter of a figure is the total length of its outline or boundary and this is found by counting the number of unit lengths along the boundary of the figure. The area of a figure on grid is found by counting the total number of unit squares it occupies (inside the figure).

Place

The position of a digit relative to the decimal Ones, tens, hundreds, etc.

How to calculate the least common denominator

The same fraction can be expressed in many different forms. If the ratio between numerator and denominator is the same, the fractions represent the same rational number. The least common denominator of a set of fractions is the least number that is a multiple of all the denominators: their "least common multiple". Method of calculating the LCD is the same as calculation least common multiple of fractions denominators. The simple method for computing common denominator (not least at all) is multiply all denominators.

least common multiple

The smallest multiple (other than zero) that two or more numbers have in common. The least common multiple, or LCM, is another number that's useful in solving many math problems. Let's find the LCM of 30 and 45. One way to find the least common multiple of two numbers is to first list the prime factors of each number. 30 = 2 × 3 × 5 45 = 3 × 3 × 5 Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs. 2: one occurrence 3: two occurrences 5: one occurrence 2 × 3 × 3 × 5 = 90 <— LCM

Standard Deviation

The square root of the variance

Put Together/Take Apart, Total Unknown

Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ?

Comparing fractions when the denominators are different

To compare fractions that have different denominators, convert them all to a set of fractions that have the same denominator. There are three steps to comparing fractions when the denominators are different. 1. Find the Least Common Denominator (LCD) for the group of fractions you are comparing. 2. Find the multiplier for each fraction. Multiply both the top and bottom of each fraction by that multiplier. 3. Compare and order the numerators of each fraction.

Fractions as ratios

To convert a fraction to a ratio, first write down the numerator, or top number. Second, write a colon. Thirdly, write down the denominator, or bottom number. For example, the fraction 1/6 can be written as the ratio 1:6.

Decimal division by tenths

To divide decimal numbers: If the divisor is not a whole number, move decimal point to right to make it a whole number and move decimal point in dividend the same number of places. Divide as usual. Keep dividing until the answer terminates or repeats. Put decimal point directly above decimal point in the dividend. Check your answer. Multiply quotient by divisor. Does it equal the dividend?

Area of an Irregular shape

To find the area of irregular shapes, the first thing to do is to divide the irregular shape into regular shapes that you can recognize such as triangles, rectangles, circles, squares and so forth...

Steps of Problem Solving

Understand the problem, devise a plan, carry out the plan, look back

volume of sphere

V=4/3πr³

volume of a cylinder

V=area of the base x h

volume of a prism

V=area of the base x h

volume of cube

V=s³ or lxwxh

volume of cylinder

V=πr²h

Length vs. Width

Width- the shortest side of a shape Length-the longest side of a shape

Expanded notation:

Writing a number and showing the place value of each digit EX: 40,000 + 2,000 + 100 + 3

Word form

Writing a number using words. Forty-two thousand, one-hundred three.

Standard notation

Writing a number with one digit in each place value example: 42,103

Pythagorean theorem

a2+b2=c2 applies to right triangles

Pythagorean Theorem

a^2 +b^2=c^2

the process of combining two or more numbers

addition

permutations

all possible arrangements of a given number of items in which the order of the items makes a difference (placement of books on a shelf)

reflection

also called a flip, reflect an object or to make the figure appear to be backwards or flipped, provides the mirror image of a figure

translation

also called a slide, simply means moving, not rotated or reflected

rotation

also called a turn, a transformation that means a rotation or to turn the shape around

right angle

an angle that measures 90 degrees

Line

an arrow at each end to show that it extends infinitely.