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Name:________________________Chapter 1 Test Date:____
Chapter 1: Numbers, Expressions, and Equations
Lesson 1.1 Place Value
Warm-Up:
1. 7 x 100 =
2. 6 x 1,000 =
3. 9 x 10,000 =
4. 3 x 1,000,000 =
Place Value: There are several ways to read and write a number
Places: There are 3 places in each period ~ H T O.
Periods: Trillions, Billions, Millions, Thousands, Ones (Units).
Standard form: 67,240,000 (commas are used to show periods)
Word form: sixty-seven million, two hundred forty thousand
Short-word form: 67 million, 240 thousand
Expanded form: 60,000,000 + 7,000,000 + 200,000 + 40,000
OR
(6 x 10,000,000) + (7 x 1,000,000) + (2 x 100,000) + (4 x 10,000)
***place value chart***
Example 1: Write the place and value of each underlined digit:
4,536,021,985______________________________
_________________________________________
_________________________________________
Example 2: Write each number in short-word form and expanded
form.
6,354,000,120,000__________________________
________________________________________
77,537,000,320_______________________________
___________________________________________
Example 3: Write each number in standard form and word form.
20,000 + 5,000 + 60 + 2 __________________________
_____________________________________________
15 trillion, 450 billion, 89 million, 34__________________
_____________________________________________
Example 4: What does the zero in 62,778,304,525 represent?
_____________________________________________
Lesson 1.2 Exponents
Warm-Up:
1. 3 x 3 x 3 x 3 =
2. 8 x 8 =
3. 10 x 10 x 10 =
Exponents are another way to write repeated factors.
base: is the repeated factor
34
exponent
base
exponent or power: tells how many times the base is used as a factor
exponential form: 27 (a base with an exponent)
expanded form:
2,324 = (2 x 1000) + (3 x 100) + (2 x 10) + (4 x 1)
2,324 = (2 x 103) + (3 x 102) + (2 x 101) + (4 x 100)
2,324 = (2 • 103) + (3 • 102) + (2 • 101) + (4 • 100)
*expanded form using exponents- A number written in
expanded form with the place values written in exponential form
(using exponents):
squared: a number squared is the second power of a number.
72 = 7 squared or 7 to the second power
cubed: A number cubed is the third power of a number.
53 = 5 cubed or 5 to the third power.
Example 1: What are two ways to read 6³?______________________
______________________________________________________
Example 2: Write 6³as a product and then evaluate._______________
______________________________________________________
Example 3: Write 15 x 15 x 15 x 15 x 15 x 15 in exponential form______
______________________________________________________
Example 4: Write 35,402 in expanded form using exponents._________
______________________________________________________
Example 5: Are 25 and 52 equal? Explain why or why not? Justify your
answer._________________________________________________
_______________________________________________________
_______________________________________________________
Lesson 1.3 Comparing and Ordering Whole Numbers
Warm-Up: Give the place of each underlined digit.
1. 43,005 =___________________________________________
2. 578 = _____________________________________________
3. 201,459 = __________________________________________
4. 7,655 = ____________________________________________
>
<
You can compare two numbers by using “ greater than” or “
Numbers can also be = equal to.
Example 1: Use
less than.”
< or > to compare.
783 ____ 2,001
201,053 ____ 200,855
4,619 ____ 4,618
1,211 ____ 1,121
569,120 ____596,120
5,856 ____5,860
Lesson 1.3
You can order numbers from greatest to least and least to greatest.
Read directions carefully.
Example 1: Order these numbers from greatest to least:
26,750;
26,810;
29,035;
28,208
______________________
______________________
______________________
______________________
Order these numbers from least to greatest:
26,750;
26,810;
29,035;
28,208
______________________
______________________
______________________
______________________
Example 2: Write 3 numbers that are between
23,455,700 and 23,455,789.
23,455,700
______________________
______________________
______________________
23,455,789
Lesson 1.4 Rounding Whole Numbers
Warm-Up: Give the value of each underlined digit.
1. 28,709 _______
2. 468
_______
3. 833,406 _______
4. 1,398
_______
There are times when an exact answer is not needed therefore allowing you
to round.
round: To give an approximation for a number to the nearest one, ten,
hundred, thousand, and so on
Use these four steps to round numbers:
1. Find the rounding place and underline that digit.
2. Look at the digit to the right of this place and circle that digit.
3. If the circled digit is less than 5 leave the underlined digit the
same. If the circled digit is greater than or = to 5 round the
underlined digit up one number.
4. Change all digits to the right of the underlined digit (rounding
digit) to zeros.
Example 1: Round 285,610,627 to the nearest ten million then nearest
ten thousand.
285,610,627
____________________________
285,610,627
____________________________
Example 2: Round each number to the underlined place.
6,808,386 ______________________
52,799,335 ______________________
3,101,922
______________________
45,699,825 ______________________
895,774
______________________
Example 3: Write three numbers that would round to 5,600 when
rounded to the nearest hundred.
_____________;_____________;_____________
Lesson 1.5 Estimating Sums and Differences
Warm-Up: Round 634,998,050 to the indicated places.
1. hundreds ____________________________
2. 10-thousands__________________________
3. 100-thousands_________________________
4. 100-millions___________________________
You can estimate the sums and differences of whole numbers in a variety
of ways:
1. round: To give an approximation for a number to the nearest one,
ten, hundred, thousand, and so on.
Example: 668 + 4,239
2. front-end estimation: To find a number that is close to an exact
answer. You use front-end estimation to get a rough estimate.
Add only the first digits that have the same place value.
Example: 7,924 + 3,168 + 209
3. front-end estimation with adjusting: A method of front-end
estimation that adjusts the result based on the remaining digits
of each addend. Use this method to get a better answer.
Example:
Add the front end digits.
6,829 + 3,401
Add the hundreds.
Add the front end estimate by adding the
estimate from the front end digits.
So, 6,829 + 3,401 ≈ _______________
4. clustering: An estimation method where numbers that are
approximately equal are treated as if they were equal.
*You can use clustering when to estimate when the numbers are
close together.
Example:
412 + 398 + 385 + 409
So, 412 + 398 + 385 + 409 ≈ _________.
List four estimating methods for adding (sums) and subtracting
differences.
1.___________________________
2.___________________________
3.___________________________
4.___________________________
Lesson 1.6 Estimating Products and Quotients
Warm-Up: Round each number to one nonzero digit.
1. 5,880
__________________________
2. 30,998 __________________________
3. 998
__________________________
4. 4,500
__________________________
You can estimate products and quotients or whole numbers in a variety of
ways:
1. round: To give an approximation for a number to the nearest one,
ten, hundred, thousand, and so on.
6,192 x 11 = 6,000 x 10 ≈
2. compatible numbers: Numbers that are easy to compute mentally.
24 x 41
25 x 40 =
3. range: The difference of the greatest and the lowest number in a
set of data.
underestimate: When two factors are rounded down, the
product is an underestimate.
Example:
342 x 687
300 x 600 = ____________
overestimate: When two factors are rounded up, the
product is an overestimate.
342 x 687
400 x 700 = _____________
Example: Estimate each answer. Tell which method you used.
1. 562 x 9,031 ≈ ______________, ______________
2. 638 ÷ 72 ≈ ______________, _______________
3. 5,893 ÷ 301 ≈ ______________, ______________
4. 489 x 2,970 ≈ ______________, ______________
Estimating products means __________________________________
______________________________________________________.
Estimating quotients means__________________________________
______________________________________________________.
Lesson 1.7 Read and Understand
What steps can help you get started with solving a problem?
Step 1: What do you know?
Step 2: What are you trying to find?
**Read and understand is the first phase of the
problem-solving process.
Lesson 1.8 Order of Operations
Warm-Up:
1. 12 + 5 – 3
2. 72 ÷ 8
3. 4 x 5 x 8
4. 34
numerical expression: An expression, consisting of numbers and
operations to be performed, that represents a particular value.
order of operations: The order in which to perform operations when
evaluating expressions with more than one operation.
P E M D A S
Lesson 1.8 Order of Operations
Example 1:
18 – 3 x 2
Example 2: 40 ÷ (2 x 4)
Example 3:
9 + 23
Example 4: 4 x (5 + 5) ÷ 20 + 6
Example 5: (3 x 2) + 32
Example 6: 10 – 6 + 4 x 1
A plan for order of operations Lesson 1.8
A plan would be to first look for parenthesis, then look for exponents,
multiply or divide which ever comes first going from left to right, next look
for subtraction or addition which ever comes first from left to right.
After you perform each step you bring down whatever is left. It should look
like an upside down triangle.
Tell which operation you would perform first to evaluate 14 – 9 + 2 – 3.
______________________________________________________.
Explain why 5 x 3 + 42 does not equal 95.________________________
______________________________________________________
How would you generate a plan to solve an order of operations problem?
_______________________________________________________
_______________________________________________________
Lesson 1.9 Properties of Operations
Warm-Up:
Decide if statement if each statement if true or false.
1. 28 + 678 = 678 + 28
__________
2. 24 ÷ 4 = 4 ÷ 24
__________
3. (18 – 6) + 12 = 18 – (6 + 12)
__________
4. 14 + (6 + 37) = (14 + 6) + 37
__________
Properties of Addition and Multiplication
Commutative Properties: The properties that state the order of the
addends or factors does not affect the sum (+) or product (x).
9 + 15 = 15 + 9
4 x 12 = 12 x 4
Associative Properties: Properties that state the way in which addends or
factors are grouped does not affect the sum (+) or product (x).
4 + (5 + 6) = (4 + 5) + 6
(3 x 2) x 4 = 3 x (2 x 4)
Identity Properties: The properties that state the sum of any number and
zero is that number and the product of any number and 1 is that number.
567 + 0 = 567
422 x 1 = 422
Multiplication Properties of Zero: The property that states the product of
any number and zero is zero.
389 x 0 = 0
Find the missing number. Then tell what property or properties are shown.
Example 1: 15 + (48 + 5) = 15 + ( __ + 48) ____________________
Example 2: (78 + 29) + __ = 78 + 29
____________________
Example 3: 84 ÷ 12 = 84 ÷ 12 x __
_____________________
Example 4: (26 x __) x 30 = 30 x (26 x 4)
_____________________
Show two different ways to compute 5 x 6 x 3.
____________________
____________________
Do you get the same answer each time? Why or why not?
_________________________________________________________
_________________________________________________________
Lesson 1.10 Mental Math: Using the Distributive Property
Warm-Up:
Tell which property is shown.
1. 47 + 68 = 68 + 47
2. 568 = 568 x 1
3. (39 x 5) x 6 = 39 x (5 x 6)
_____________________
_____________________
_____________________
Distributive Property: The Distributive Property states that multiplying a
sum by a number produces the same result as multiplying each addend by the
number and adding the products.
8 x (20 + 4)
(8 x 20) + (8 x 4)
160 + 32
192
**another way to solve using pencil and paper: (add first then multiply)
8 x (20 + 4)
8x
24
192
break apart: using the Distributive property to compute mentally
Example 1: Find 47 x 6 using mental math.
47 x 6
(40 + 7)6
Break apart 47.
40(6) + 7(6)
Use the Distributive Property.
240 + 42
282
Example 2: Find (9)45 + (9)5 using mental math.
(9)45 + 9(5)
Each product has a 9 as one of its factors.
9(45 + 5)
Use the Distributive Property.
(9)50
450
Find each missing number.
1. 8(40 + 2) = 8( __ ) + 8(2)
2. 5(8 + 12) = 5(8) + __ (12)
3. 9( __ + 7) = 9(80) + 9(7)
Use the Distributive Property to multiply mentally.
1. (35 + 7)2
____________________
2. 7(150)
15(40 + 4)
Does the Distributive Property work with subtraction?
Try it with 9(50 – 1) and (20 – 4)5.
Lesson 1.11 Mental Math Strategies
Warm-Up: Solve mentally
1. 25 + 89 + 275 = _______
2. 20 x (40 + 8) = _______
3. 2 x 137 x 5 = _______
4. 350 ÷ 10 = __________
You can use the properties of operations to compute mentally.
Here are three different strategies you can use:
Strategy A: Break apart the numbers.
What You Think: Find 38 + 77 using mental math.
Break apart the numbers into tens and ones.
Add the tens: 30 + 70 = 100
Add the ones: 8 + 7 = 15
Add the sums: 100 + 15 = 115
So, 38 + 77 = 115
Why It Works:
38 + 77
(30 + 8) + (70 + 7) Commutative and Associative
(30 + 70) + (8 + 7) Properties of Addition
100 + 15
115
Strategy B: Look for multiples of 10 or 100.
Find 20 x 6 x 5 using mental math
Multiply number pairs having a product of 10 or 100. Multiply the
other number.
20 x 5 = 100
100 x 6= 600
So, 20 x 5 x 6 = 600.
Strategy C: Use compensation.
compensation: Choosing numbers close to the numbers in a problem,
and then adjusting the answer to compensate for the numbers chosen.
Add an amount to one number and subtract the same amount from the
sum.
39 + 88
l
39 + 1
Add 9 to 31
l
40 + 88 = 128
128 – 1 = 127
Subtract 1 from the answer to compensate.
equal additions: Adding the same number to two numbers in a
subtraction problem does not affect the difference.
742 – 295
(742 + 5) - (295 + 5)
Adding 5 to both numbers does not affect
the difference.
747 – 300
447
Practice Problems:
Compute mentally
1. 20 x 39 x 5 __________
2. 973 – 645 __________
3. 159 + 328 ___________
4. 4 x 23 x 250________
5. Explain the steps you can use to find 794 – 439 mentally.
_________________________________________________________
_________________________________________________________
_________________________________________________________
Lesson 1.12 Plan and Solve
What steps can help you find a solution to a problem?
Step 1: Choose a strategy
•Show what you know
Draw a picture
Make an organized list
Make a Table
Make a Graph
Act It Out
Use Objects
•Look for a Pattern
•Try, Check, and Revise
•Write an Equation
•Use Logical Reasoning
•Solve a Simpler Problem
•Work Backward
Step 2: Stuck: Don’t give up.
Try these suggestions:
•Reread the problem
•Tell what you know
•Identify key facts and details
•Tell the problem in your own words
•Show the main idea
•Try a different strategy
•Retrace your steps
Step 3: Answer the question in the problem.
Lesson 1.13 Variables and Expressions
Warm-Up: Use symbols to write the expression.
1.
sum of 16 and 29
2.
difference of 216 and 89
variable: A quantity that can change or vary, often represented with a letter
algebraic expression: A mathematical phrase containing variables, numbers
and operational symbols.
evaluate: To find the number that an algebraic expression names by
replacing a variable with a given number. (Solve)
substitution: To put something in another’s place or replace the variable
with a number.
Writing Expressions
The following common words and phrases indicate addition, subtraction,
multiplication, and division.
Addition
plus
the sum of
increased by
total
more than
added to
Subtraction
minus
the difference
decreased by
fewer than
less than
subtracted
Multiplication
times
the product of
multiplied by
of
Operation
Word Phrase
Addition
the sum of 9 and a number n
a number m increased by 8
six more than a number t
add eighteen to a number h
seventy-seven plus number r
the difference of 12 and a number n
seven less than a number y
ten decreased by a number p
the product of 4 and a number k
fifteen times a number t
two multiplied by a number m
the quotient of a number divided by five
twenty-five divided by a number m
Subtraction
Multiplication
Division
Division
divided by
the quotient of
per
Algebraic
Phrase
9+n
m+8
t+6
h + 18
77 + r
12 – n
y–7
10 – p
4k
15t
2m
a/5
25/m
Practice Problems:
1.
5 less than k ___________________________________________
2.
n increased by 7
______________________________________
3.
twice b divided by 9 _____________________________________
4.
Evaluate each expression for x = 5.
45/x ____________
20x – 1 ___________
5.
Write two different word phrases for 5 – d.___________________
____________________________________________________
____________________________________________________
x2 – 19 ________
Lesson 1.14 Properties of Equality
Warm-Up:
1. 12 + 19 – 19
2. 15 – 8 + 8
3. 6 x 7 ÷ 7
4. 18 ÷ 3 x 3
______________________
______________________
______________________
______________________
Lesson 1.15 Solving Equations with Whole Numbers
Warm-Up: Explain how to get the variable alone
1. 12x = 60
______________________________
2. d – 10 = 10
______________________________
3. 32 = 8 + a
______________________________
equation: A mathematical sentence stating that two expressions are equal
properties of equality: Properties that state performing the same operation
to both the same operation to both sides of an equation keeps the equation
balanced
inverse operations: Operations that “undo” each other, such as addition and
subtraction, or multiplication or division
Steps to Solving Equations with Integers:
1. Locate the side of the equation with the variable and determine the
operation being used.
2. Use the inverse operation on the side with the variable.
The inverse of addition is subtraction.
The inverse of subtraction is addition.
The inverse of multiplication is division.
The inverse of division is multiplication.
3. Use the properties of equality to balance the equation. Whatever you do
to one side of the equation you must do to the other side of the
equation. Keep the balance!
4. Solve the equation to find out the answer to the variable.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Steps for Checking your Algebraic Equations:
*** The check must be written as follows:
Check:
F
1. Write the original formula/equation.
S
2. Substitute the value of the variable for the variable. Prove it by
doing the work.
S
3. Solve…check to make sure the equation is =.
Lesson 1.16 Problem-Solving Skill: Look Back and Check
workbook page 52