______- Estimate That Is Less Than the Exact Answer
Vocabulary
______- answers that are close to the exact answer but are easier and faster to find
______- numbers that are close to the numbers in the problem that work well together to help you find the answer mentally
______- estimate that is less than the exact answer
______- estimate that is more than the exact answer
Example 1 Estimating a Sum or Difference by Rounding
Estimate the sum or difference by rounding to the place value indicated.
A. 12,345 + 62,167; ten thousands:
Round 12,345 ______.
Round 62,167______.
The sum is about ______.
- 4,983 – 2,447; thousands:
Round 4,983 ______.
Round 2,447______.
The difference is about ______.
Example 2Estimating a Product by Rounding
Chelsea is planning the annual softball banquet for the 8 teams in the region. Each team has 18 members. Estimate how many plates she will need to buy if all the members attend.
8 x 18 ______; this is an ______of the number of plates.
The actual number of plates is ______than 160.
If Chelsea buys ______cups, she will have enough for each person.
Example 3Estimating a Quotient Using Compatible Numbers
Mr. Dehmel will drive 243 miles to the fair at 65 mi/h. About how long will his trip take?
243 ÷ 65 ______240 and 60 are ______numbers.
Underestimate the speed.
Because he underestimated the speed, the actual time will be ______
_____ hours.
Think and Discuss
- Suppose you are buying items for a party and you have $50. Would it be better to overestimate or underestimate the cost of the items?
- Suppose your car can travel between 20 and 25 miles on a gallon of gas. You want to go on a 100-mile trip. Would it be better to overestimate or underestimate the number of miles per gallon your car can travel?
Vocabulary
______- tells how many times a number called the base is multiplied by itself
______- number that is multiplied by itself
______- a number form written with a base and an exponent
Example 1Writing Numbers in Exponential Form
Write each expression in exponential form.
A.
______5 is a factor 4 times.
B.
__________ is a factor of ____ times.
Example 2Find the Value of Numbers in Exponential Form
Find each value.
A.
B.
Example 3: Problem Solving Application
A phone tree is used to contact families at Paul’s school. The secretary calls 4 families. Then each family calls 4 other families, and so on. How many families will be notified during the fourth round of calls?
Step 1: Understand the Problem
The answer will be the number of families called in the 4th round.
List the important information:
• The secretary calls 4 families.
• Each family calls 4 families.
Step 2: Make a Plan
You can draw a diagram to see how many calls are in each round.
Step 3: Solve
Notice that in each round, the number of calls is a power of 4.
1st round: 4 calls = 4 = 41
2nd round: 16 calls = 4 x 4 = 42
So during the 4th round, there will be 44 calls. 44 = 4 x 4 x 4 x 4 = 256
During the 4th round of calls, 256 families will be notified.
Step 4: Look Back
Drawing a diagram helps you see how to use exponents to solve the problem.
Think and Discuss
- Read each number: .
- Explain which has the greater value, oror neither.
Lesson Resources:
- Interactivity
Vocabulary
______- mathematical phrase that includes only numbers and
operation symbols (no = sign)
______- find the value
______- the order in which you must do the
operations in an expression (PEMDAS)
Example 1: Using Order of Operations
Simplify each expression.
A. There are no parentheses or exponents, so divide first.
subtract
______final answer
B. Perform operations within ______.
______
9 + ______add
______final answer
Example 2: Using the Order of Operations with Exponents
Simplify each expression.
A. Find the value of numbers with ______.
____________
16 + ______add
______final answer
B.Perform operations within ______.
Find the value of numbers with ______.
______
______
subtract
______final answer
Example 3: Consumer Application
A. Mr. Kellett bought 6 used CDs for $4 each and 5 used CDs for $3 each. Simplify the following expression to find the amount Mr. Kellett spent on CDs.
6 x 4 + 5 x 3
______
______
B. Ms. Nivia bought 4 new CDs for $8 each and 6 used CDs for $4 each. Simplify the following expression to find the amount Ms. Nivia spent on CDs.
4 x 8 + 6 x 4
______
______
Think and Discuss
- Explain why but.
- Tell how you can add parentheses to the numerical expression so that 27 is the correct answer.
Vocabulary
Example 1: Using Properties to Add and Multiply Whole Numbers
- Simplify .
Look for sums that are multiples of ______.
Use the ______Property.
Use the ______Property to
______+ ______make groups of ______numbers.
______Use ______math to add.
B.Simplify . Find numbers that are compatible.
Use the ______Property.
Use the ______Property to group
______numbers.
______Use ______math to multiply.
Example 2: Using the Distributive Property to Multiply
A.
“Break apart” 35 into ______+ ______.
Use the ______Property.
= ______+ ______Use ______math to multiply.
= ______Use ______math to ______.
B.
“Break apart” 87 into ______+ ______.
Use the ______Property.
= ______+ ______Use mental math to ______.
= ______Use mental math to ______.
Check It Out: Example 2A
Use the Distributive Property to find the products:
4 x 276 x 43
______
______
______
______
Vocabulary:
Computation methods:
paper and pencil (done with paper and pencil)
mental math (done in your head)
calculator (done with a calculator)
Examples:
Simplify the expression and state the method of computation you used.
4 + 3 + 2 + 10 + 8 + 2 + 5 + 1 method: ______
______final answer
Simplify the expression and state the method of computation you used.
4,562 – 397 method: ______
______final answer
Simplify the expression and state the method of computation you used.
9,288 ÷ 24method: ______
______final answer
Think and Discuss
- Give an example of a situation in which you would use mental math to solve a problem.
When would you use pencil and paper?
- Tell how you could use mental math to solve 867 + 59.
Vocabulary
______- an ordered set of numbers
______- a number in a sequence
______- a sequence with terms that change by the same amount
each time
Example 1: Extending Arithmetic Sequences
Identify the pattern in each sequence and then find the missing terms.
48, 42, 36, 30, ____, ____, ____, … ______from each term to get the next term.
So, ______, ______, and ______will be the next three terms.
Example 1B: Arithmetic Sequences in a Table
______from each term to get the next term.
So, ______and ______will be the next two terms.
Example 2: Completing Other Sequences
In nonarithmetic sequences, look for patterns that involve multiplication or division. Some sequences may even be combinations or different operations.
Identify a pattern in each sequence. Name the missing terms.
24, 34, 31, 41, 38, 48, ____, ____, ____, … ______to one term and
______from the next
So, ______, ______, and ______will be the next three terms.
Example 2B: Other Sequences in a Table:
______one term and ______the next
So, ______and ______are the missing terms.
Think and Discuss
- Tell how you could check whether the next two terms in the arithmetic sequence 5, 7, 9, 11, … are 13 and 15.
- Explain how to find the next term in the sequence 16, 8, 4, 2, …
- Explain how to determine whether 256, 128, 64, 32, … is an arithmetic or nonarithmetic sequence.
Lesson Resources:
- Interactivity:
Example 1A: Extending Geometric Patterns
Identify a possible pattern. Use the pattern to draw the next figure.
next figure: ______
Example 1B: Extending Geometric Patterns
Identify a possible pattern. Use the pattern to draw the next figure.
next figure: ______
Example 2A: Completing Geometric Patterns
Identify a possible pattern. Use the pattern to draw the missing figure.
missing figure: ______
Example 2B: Completing Geometric Patterns
Identify a possible pattern. Use the pattern to draw the missing figure.
missing figure: ______
Check It Out: Example 3
Nancy is designing a plate. Identify a pattern that Nancy is using and draw what the finished plate might look like.
finished plate: