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Assessment of these competencies and skills will use real-world problems when feasible.

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1. Associate multiple representations of numbers using word names, standard numerals, and pictorial models for real numbers (whole numbers, decimals, fractions, and integers).

Term: Example:

Whole Numbers Counting Numbers (1, 2, 1000)

Factions a/b where a is the numerator and b is the denominator (2/3)

Dividing Fractions;" The number you're dividing by, Turn upside-down and multiply "

and integers).

Decimals fractions written as powers of 10 (25/100 = 0.25)

Integers +7, -10

Word Names One, Two, Five-Hundred

Standard Numerals 1, 2, 500

Pictorial Models Using pictures to represent numbers

Write an example for each real number: 11, 35, 8

Whole number
Decimal
Integer
Word names
Standard numerals
Pictorial

2. Compare the relative size of integers, fractions, and decimals, numbers expressed as percents, numbers with exponents, and/or numbers in scientific notation.

Fractions, Decimals, Percents

Converting from fractions to decimals to percents:

Operation / Explanation / Example
Convert a decimal to a percent / Move the decimal point 2 places to the right and add a percent (%) sign. If you need to, add a zero on the back to get the second decimal place. / .123 = 12.3%
Convert a percent to a decimal / Move the decimal point 2 places to the left. If you need to, put a zero on the front. / 5% = .05
Convert a fraction to a decimal / Divide the numerator by the denominator, using your calculator. / 1/8 = .125
Convert a percent to a fraction /
First turn the number into a decimal. Then turn the number into a fraction by putting it over 10, 100, 1000, or whatever number is big enough to have enough zeroes for each place after the decimal. / 18% = .18 = 18/100 = 9/50

See http://www.marthalakecov.org/~math/fr_dec_pct.html for additional examples.

Exponents

Exponential notation is shorthand for repeated multiplications:

103, 36, and 18 as an exponent, in factor form and in standard form:
Exponential
Form / Factor
Form / Standard
Form
103 / 10 x 10 x 10 / 1,000
36 / 3 x 3 x 3 x 3 x 3 x 3 / 729
18 / 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 / 1

Rules for numbers with exponents of 0, 1, 2 and 3:

Rule Example

Any number (except 0) raised to the zero power is equal to 1. 1490 = 1

Any number raised to the first power is always equal to itself. 81 = 8

If a number is raised to the second power, we say it is squared. 32 is three squared

If a number is raised to the third power, we say it is cubed. 43 is four cubed

Scientific Notation

Scientific notation is a way of writing very large and very small numbers more easily.

5,234 = 5.234 x 103

View this quick video on Scientific Notation from United Streaming.

To login, use your B.E.E.P. username and password:

Video: http://player.discoveryeducation.com/index.cfm?guidAssetId=8E415256-A36C-4D02-A45A-D06E54030CC7&blnFromSearch=1&productcode=US

3. Apply ratios, proportions, and percents in real-world situations.

Ratio is the relationship of one number to another

to

4:3

Proportion refers to the relationship between two equivalent ratios

Quick video on proportions:

http://viewer.nutshellmath.com/?solution=67-33-71-41-77-31

Write each as a decimal, fraction, percent, exponent, in Scientific Notation and Ratio: 12, -240

Decimal / 12.0 / -240.0
Fraction
Percent
Exponent
Scientific Notation
Ratio

Use Virtual Manipulatives to help solve or convert:

http://nlvm.usu.edu/en/nav/topic_t_1.html

4. Represent numbers in a variety of equivalent forms, including whole numbers, integers, fractions, decimals, percents, scientific notation, and exponents. Equivalent Forms:

Write the numbers 25, and -1090 in each equivalent form:

Whole number / 25 / -1090
Integer
Fraction
Decimal
Percent
Scientific Notation
Exponent

Solve.

In her science class, Leanne was doing an experiment that involved weighing small objects in grams and then converting those weights into ounces. Her teacher told her that each gram is equal to 0.035 ounces. Which is equivalent to 0.035?

a. 7 b. thiry-five thousandths

20

c. 35% d. 35

100

5. Recognize the effects of operations on rational numbers and the relationships among these operations (i.e., addition, subtraction, multiplication, and division). Rational numbers

Rational numbers a real number that can be written as

a ratio of two integers (fraction) excluding zero as a denominator

a/b (a/0 is not allowed)

a repeating or terminating decimal

0.33333

an integer -24

Operations:

Addition (sum)

Subtraction (difference)

Multiplication (product)

Division (quotient)

Order of Operations PEMDAS (Please Excuse My Dear Aunt Sally)

Parenthesis

Exponents

Multiplication*

Division*

Addition**

Subtraction**

When multiplying and dividing, solve whichever comes first from left to right.

Example:

5 x 9 / 3 x 4 15 ÷ 5 / 3 x 4

45 / 3 x 4 3 / 12

45/12

When adding and subtracting, solve whichever comes first from left to right.

Example:

5 + 9 / 3 + 4 9 – 5 / 3 + 4

14 / 3 + 4 4 / 7

14/ 7

Relationships of operations using integers:

Adding Integers:

Adding with Same signs (find the SUM):

(+) + (+) = + (+4) + (+5) = (+9)

(-) + (-) = - (-3) + (-6) = (-9)
Adding with Different signs (find the DIFFERENCE):

Find the difference (subtract) and take the sign of the larger number

(+) + (-) = Find difference, take sign of larger number

(+5) + (-7) = (-2)

(-) + (+) = Find difference, take sign of larger number

(-4) + (+6) = (+2)

Subtracting Integers

To subtract an integer, add it’s OPPOSITE

5 - 2 = 3

5 + (-2) = 3 (Change sign to +)
- 3 – 4= -7

-3 + (-4) = -7

Multiplying Integers

When you multiply two integers with the same signs, the result is always positive.

+ x + = + (Same signs = Positive)

x - = + (Same signs = Positive)

When you multiply two integers with different signs, the result is always negative.

x + = - (Different Signs = Negative)

Dividing Integers

When you divide two integers with the same sign, the result is always positive.

+ / + = + (Same signs = Positive)

2 = 5

/ - = + (Same signs = Positive)

-10 / -2 = 5

When you divide two integers with different signs, the result is always negative.

/ + = - (Different Signs = Negative)

-10 / 2 = -5

10 / -2 = -5

6. Select the appropriate operation(s) to solve problems involving ratios, proportions, and percents and the addition, subtraction, multiplication, and division of rational numbers.

Ratios:

A ratio is a comparison of two numbers.

Example:

Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.

1) What is the ratio of books to marbles?

Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4.
Two other ways of writing the ratio are 7 to 4, and 7:4.

2) What is the ratio of videocassettes to the total number of items in the bag?
There are 3 videocassettes, and 3+4+7+1=15 items total.
The answer can be expressed as 3/15, 3 to 15, or 3:15.

Proportions:

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal (3/4=6/8 is an example of a proportion).

When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Variables are frequently used in place of the unknown number.

Example:

Solve for n: 1/2=n/4.
Using cross products we see that 2×n=1×4=4, so 2×n=4. Dividing both sides by 2, n=4÷2 so that n=2.

7. Use estimation in problem-solving situations. Estimation:

Estimation is determining an approximate or rough calculation based on rounding.

Example:

Problem:

You ate at a restaurant and the bill you will pay is $28.90. You want to leave a tip of 15% of the bill. What is the equivalent amount of the 15% tip that you want to leave?

Solution:

1) You can round off $28.90 to $30 since it is very close to that amount.

2) Using 10 as our reference number, divide $30/10 = $3. You will retain the $3.

3) Divide $3/2 = $1.50.

4) Then add $3 + $1.50 = $4.50, which is the estimated tip you will pay for a $28.90 bill.

5) If you use a calculator to compute the answer for the 15% tip from a $28.90 bill, you will get $4.33, which is close to our estimate of $4.50.

8. Apply number theory concepts (e.g., primes, composites, multiples, factors, number sequences, number properties, and rules of divisibility). Prime A number that has exactly two factors, 1 and itself

Composite A number with more than 2 factors

Multiples A number added to itself a number of times

Factors A whole number that divides exactly into another number

Number sequences Numbers expressed in a pattern in the nth term

Number properties Commutative, Associative, Distributive,

Rules of divisibility A number is divisible by two if it is even.

A number is divisible by three if the sum of the digits adds up to a multiple of three

A number is divisible by four if it is even and can be divided by two twice.

A number is divisible by five if it ends in a five or a zero

A number is divisible by six if it is divisible by both two and three.

A number is divisible by nine if the sum of the digits adds up to a multiple of nine. This rule is similar to the divisibility rule for three.

A number is divisible by ten if it ends in a zero. This rule is similar to the divisibility rule for five

Write two examples for each category:

Prime / Composite / Multiples / Factors / Number Sequence / Number Properties / Rules of Divisibility

9. Apply the order of operations. (Please Make Deliveries After Sunset) (Patrick Makes Delicious Apple Smoothies)

Order of Operations

The precedence set for solving math equations.

In what order should the operations be performed in the expression below? Solve.

6 x (3 + 2) ÷ 3 - 1

a. +, x,Π,- c. +,-, x,Π

b. +, x,-,Π d. x,+,Π,-

What is the value of the expression 2 + 5 X 32 ?

32 C.227

47 d. 441

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K / 1st / 2nd / 3rd / 4th / 5th
add
after*
before
between (M)
equal*
even numbers*
fourth(s)*
group* (M)
half (halves)*
hundred(s)*
less than* (M)
more than
(not) equal
number* (M)
number line*(M)**
odd number*
one(s)*
opposite
order
pattern* (M) **
small(est)
subtract
ten(s)*
zero* (M) / add*
addition (M)
after*
before*
between* (M)
count (on)
difference (M)
double (plus one)
equal*
estimate* (M)
even numbers*
fact family
fourth(s)*
group * (M)
half (halves)*
hundred(s)*
less than* (M)
more than*
near (M)
(not) equal*
number* (M)
number line* (M) **
odd number*
one(s)*
opposite*
order*
pattern*(M)**
rule
skip-counting*
small(est)*
subtract*
subtract(ion)* (M)
sum
ten(s)*
zero* (M) / addition* (M)
after*
count backward
count forward
count (on)*
difference* (M)
double (plus one)*
equal*
estimate* (M)
even numbers*
fact family*
fourth(s)*
fraction
greater than (M)
greatest
group* (M)
half (halves)*
hundred(s)*
less than* (M)
near* (M)
number* (M)
number line* (M) **
odd number*
one(s)*
ordinal number (M)
pattern* (M) **
regroup
rule*
set (M)
skip-counting*
small(est)*
subtract(ion)* (M)
sum*
ten(s)*
whole number (M) **
zero* (M) / addend*(M)**
array*(M)
denominator*
difference(M)
digit*
estimation*(M)**
expanded
form*(M)
factor*(M)**
fraction*(M)**
minuend*
multiplicand*
multiplication(M)
multiples*(M)**
multiplier*
number line**
numerator*
place value* **
product*(M)**
regroup*
relative
size*(M)**
rounding*(M)
solution*
subtrahend*
sum(M)**
whole
numbers*(M) / addend*(M)**
array*
calculate*
cardinal
number(M)
composite
number*(M)**
consecutive*
decimal
number*(M)**
denominator*
digit*
divisible* **
divisor**
estimation(M)**
expanded
notation*(M)
extraneous
information**
equivalent*
factor(M)**
fraction(M)**
inverse
operation**
million*
minuend*
multiples(M)**
multiplicand*
multiplier*
natural numbers*
numerator*
operation* **
ordinal
number*(M)
percent*(M)**
perfect number*
place value* **
prime
number*(M)**
product*(M)**
quotient*(M)**
regroup*
relative
size*(M)**
remainder*(M)
rounding*(M)
similarity*(M)**
simplify*
solution*
subtrahend*
value*
whole
numbers*(M)** / billion
calculate*
cardinal number*(M)
composite
number*(M)**
consecutive*
decimal number*(M)**
decimal point
digit*
dividend(M)
divisible* **
divisor* **
equivalent
extraneous
information* **
equivalent*
greatest common
factor (GCF)
inverse operation* **
least common
denominator (LCD)
least common
multiple(LCM)
million*
natural numbers*
operation* **
ordinal number*(M)
percent*(M)**
perfect number*
prime factorization(M)
prime number*(M)**
quotient*(M)**
reduce
remainder*(M)
similarity*(M)**
simplify*

1. Apply given measurement formulas for perimeter= 2xL+2xW , circumference =pi(3.14) x diameter, area= length x width, volume = length x width x height or displacement, and surface area = the sum of the areas of all sides in problem situations. Perimeter The distance around a shape.

Circumference The distance around a circle.

Area The size a surface takes up, measured in square units.

Volume Amount of space occupied by a 3D object, measured in cubic units.

Surface Area Total area of a surface of a 3D object, measured in squared units.

Perimeter and Area:

Find the area and perimeter of each rectangle.

7 ft wide, 37 ft long 10 ft wide, 15 ft long

Perimeter = Perimeter =

Area = Area =

Sachi is building a brick patio and needs to determine its total area. The dimensions of the patio are shown in the diagram below.

Circumference:

The radius of a circle is 2 inches. What is the circumference? = 3.14

The diameter of a circle is 3 centimeters. What is the circumference?

Volume:

Find the volume of the rectangular prism to the nearest tenth.

Length 8.8 feet

Width 6 feet

Height 11.5 feet

F. 210.0 feet3

G. 380.3 feet3

H. 596.2 feet3

I. 607.2 feet3

Surface Area:

a=27 cm