ARITHMETIC

Whole Numbers (students may test out in part or whole)

Overview of Objectives, students should be able to:

1. Find the place value of a digit in a whole number

2. Write a whole number in words and in standard form and

expanded form.

3. Add whole numbers

4. Subtracting whole numbers

5. Round whole numbers

6. Use rounding to estimate sums and differences

7. Solve problems by estimating

8. Use the properties of multiplication

9. Divide whole numbers

10. Perform long division

11. Solve problems that require adding, subtracting,

multiplying and/or dividing whole numbers

ARITHMETIC

Main Overarching Questions:

1. How do I know if my answer is reasonable?

2. How do I remember the rules for all the different operations?

Objectives: Activities and Questions to ask students:

3. What is the relationship between adding and subtracting? Multiplying and dividing?

• Find the place value of a digit in a whole number • What is the value of the 6 in each of these numbers?

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• Write a whole number in words and in standard form and

expanded form.

(12,634} (162,345) (1,236)

• Why is it important to be able to write numbers as words?

• How do you write numbers in expanded form? What does 1000 + 200 + 30+ 6 = ?

• How can you relate place value in the expanded form?

• Add whole numbers

• Why do I need to line up place values when I add or subtract?

o Addition property of 0 (a + 0 = a)

•

•

What do these properties allow me to do?

What happens if I don’t line the numbers up according to their place value?

o Commutative Property of Addition

• If I have nothing and add $6 to my bank account, how much do I have? That’s the addition

property of zero.

o Associative Property of Addition

• Make a game that will show the addition properties. (Like a worksheet that shows the

patterns – 3 or 4 column notes where they create their own)

• Subtracting whole numbers • How do I borrow and what does that mean?

• When you have $20 in the bank, and you deposited $30, but then take out $30 the next day

– how much do you have?

•

•

What is the relationship between adding and subtracting?

• Round whole numbers • When rounding whole numbers how do I know which place value to stop at?

•

•

What is a reasonable rounding answer?

• Use rounding to estimate sums and differences • When you buy groceries, how do you estimate what the bill should be?

• What is a reasonable answer?

• Solve problems by estimating • How does estimating relate to rounding?

• Use the properties of multiplication

o Multiplication property of 0 ( 0 * a = 0)

• When and what should you estimate?

• Does your answer make sense?

• How does rounding figures before computing the answer effect your answer??

• What are the properties related to multiplication?

• How do they allow me to manipulate the numbers?



o Multiplication property of 1 ( 1 * a = a)

o Commutative property (2 * 9 = 9 * 2 = 18)

o Associative Property ((2*3)*4 = 2 *(3*4))

o Distributive Property (2(3+4) = 2*4 + 3*4)

• Divide whole numbers

o Division property of 1 (a/a = 1; a/1 = a)

o Division property of 0 (0/a = 0, but a/0 is undefined)

• Why do I need to line up place values when I multiply?

• What do these properties allow me to do?

• What happens if I don’t line the numbers up according to their place value?

• Make a game that will show the multiplication properties. (Like a worksheet that shows the

patterns – 3 or 4 column notes where they create their own)

• What does it mean to divide?

• Ask students to divide 10 pencils into 10 groups, how many in each group? (a/a = 1)

• Ask students to divide 10 pencils into groups of 1, how many groups? (a/1 = a)

• What skills does it take to be able to divide?

• What is the relationship between dividing and multiplying?

• Why can you not divide by zero? Ask students to take 10 pencils and divide them into groups

of zero. Can they? (If they somehow can, tell them they have a doctoral dissertation waiting

for them)

• Perform long division • Take to students about what are the quotient, divisor, and dividend in a long division problem.

• What are different ways to express a division problem?

• Ask student to 2 9 . How would you divide 9 into groups of 2? Can you? Why or why not? What

• Solve problems that require adding, subtracting,

multiplying and/or dividing whole numbers

happens when you try to divide into groups of 2. Students should notice they have 4 groups with 1

remaining – talk about the remainder.

• Then move onto larger digit numbers.

• Worksheet

• Worksheet



Exponents (1 day);


1. Write repeated factors using exponential notation

2. Evaluate expressions containing exponents


1. How can you rewrite numbers in exponential form? 4 = 2 2

2. How can you rewrite 2*2*2*2*2*2*2 using exponents?

3. What is a prime number?


• Write repeated factors using exponential notation • How do you make a factor tree?

• When you make a factor tree, how do you express the result using prime numbers and

exponents?

• Factor 12, 24, 36 and 64

• Evaluate expressions containing exponents

• How do you evaluate an expression with exponents? Ask students to evaluate 2 3 *5 2

• What does this mean? Have students express it as 2*2*2*5*5 = 200

• Have students work in groups working on several examples of this type in both directions.

• Have students discuss their answers and make sure there is consensus on the answers.

Integers (2 days)


1. Graph integers on a number line.

2. Compare integers

3. Find the absolute value of a number


1. How can a digit on the negative side of zero get larger, but the value gets smaller?

2. What is absolute value?

3. How do you adding/subtract/multiply/divide signed numbers?



4. Find the opposite of a number

5. Adding integers with the same sign

6. Adding integers with different signs

7. Subtracting integers

8. Adding and subtracting integers

9. Solve problems by subtracting integers

10. Multiplying integers with like signs and different signs

11. Dividing integers with like signs and different signs


• Graph integers on a number line.

• Compare integers

• Find the absolute value of a number

• Find the opposite of a number

• Adding integers with the same sign

• Adding integers with different signs

• Use football fields or having and owing money

• On a number line, ask students to graph gaining $10 and then have graph being overdrawn

$10 in the bank.

• Ask students to graph opposite numbers and ask which one is smaller.

• Why is the negative number smaller than it’s positive counterpart? It looks the same, why is

it smaller?

• How can a digit on the negative side of zero get larger, but the value gets smaller?

• What is the absolute value? What is absolute value used for in real life? You lose 10 yards in

a football game. See if students can make the distinction between positive and negative

direction

• Ask students if 2 + 4 = 6, what do you think -2 + -4 equals? What rule can you write to help

you remember it should be -6?

• So, if 2 + 4 = 6 and -2 + -4 = -6, what do you think -2 + 4 equals? Do the same rules apply?

What rule can you write to help you remember this one?

• So, if 2 + 4 = 6, -2 + -4 = -6, and -2 + 4 = 2, what do you think 2 + -4 equals? What is the



• Subtracting integers

• Adding and subtracting integers within the same

expression

• Solve problems by subtracting integers

difference between these examples? Why is your answer negative?

• What does it mean to have like signs or different signs?

• How do adding signed numbers differ than adding positive numbers?

• As an activity, use a number line to help you add signed numbers.

• Why is 6 - 2 the same as 6 + -2?

• What do you think 10 – (-4) is equal to? Try changing it to an addition problem? Why would

you do that? What makes that the same expression?

• Why is it important when you subtract integers that you change them to addition problems?

• Multiplying integers with like signs and different signs • Show students a pattern to show why a negative times a positive is equal to a negative

number.

3*2 = 6

2*2 = 4

1*2 = 2

0*2 = 0

-1*2 = ?

• What happens? What is the pattern? What would the next term become? Can you think of

a rule that would apply to multiplying a negative * a positive?

• Now to show the negative*negative:

2* (-3) = -6

1* (-3) = -3

0 * (-3) = 0

-1 * (-3) = ??

• What is the pattern? What would the next term become? Can you think of a rule that would

apply to multiplying negative * negative?

• What are the rules for multiplying and dividing integers?

• Dividing integers with like signs and different signs • Since multiplying (-)(-) = +, what do you think +/- = ?

• Since multiplying (-)(+) = -, what do you think -/+ = ?

• Are the rules the same?



Order of Operations


1. Simplify expressions by using the order of operations

2. Evaluate an algebraic expression


1. What is the order of operations?

2. Why is it important that everyone use the order of operations?


• Simplify expressions by using the order of operations • Ask this type of question: If (-3)(-3) = 9, what does (-3) 2 = ?

• How does that differ from -3 2 ?

• PEMDAS – Please Excuse My Dear Aunt Sally

• What happens in this equation if I add first and you subtract first? 5-3+2, who is right?

• Point out that with PEMDAS – multiplication and division has the same priority so just go

from left to right. Same with addition/subtraction. Left to right.

• Please go over several examples, a lot of examples like -3[-3 + 2(-1+6)] 2 – 5

• Evaluate an algebraic expression • If x = 3, what does 3x + 1? (10)

• If x = 3 and y = -2, what does 3y 3 – xy + 2y = ?

Fractions and Mixed Numbers (3 days)

Overview of Objectives, students should be able to: Main Overarching Questions:

1. What is a fraction?



1. Identify the numerator and the denominator of a fraction

2. Write a fraction to represent parts of figures or real-life

data.

3. Graph fractions on a number line.

4. Write mixed numbers as improper fractions

5. Write a number as a product of prime numbers

6. Write a fraction in simplest form

7. Determine whether two fractions are equivalent

8. Solve problems by writing fractions in simplest form

9. Multiply fractions

10. Evaluate exponential expressions with fractional bases

11. Divide fractions

12. Solve applications that require multiplication of fractions

13. Adding and Subtracting Like Fractions, Least Common

Denominator, and Equivalent Fractions

a. Add or subtract like fractions

b. Solve problems by adding or subtracting like fractions

c. Find the least Common Denominator of a list of fractions

d. Write equivalent fractions

e. Write ratios as fractions

14. Adding and Subtracting Unlike Fractions

2. What are the parts of a fraction?

3. What are the rules for simplifying and performing operations on fractions?

4. How do you write a fraction as a decimal or percentage?



a. Add or subtract unlike fractions

b. Write fractions in order

c. Solve problems by adding or subtracting unlike fractions

15. Order of operations involving fractions

16. Operations on Mixed Numbers

a. Graph positive and negative fractions and mixed numbers

b. Multiply or divide mixed or whole numbers

c. Add or subtract mixed numbers

d. Perform operations on negative mixed numbers


• Identify the numerator and the denominator of a fraction • What is a fraction? Any part of a whole is called a fraction.

• What does a fraction look like?

Numerator Part

Denominator Whole

The fraction bar separates the numerator from the denominator. It serves as a dividing line.

• Based on the previous definition given for a fraction bar, it represents what mathematical

operation? Division

e.g. One divided by two is ½.

One-half written in fractional form is ½.

• Which number in a fraction is the numerator?



• Write a fraction to represent parts of figures or real-life

data.

The number on top is the numerator.

• What is the numerator’s purpose?

This number represents how many parts of the whole are present or being described.

e.g. ¼ describes 1 part of a 4 part whole (1 is the numerator)

• Which number in a fraction is the denominator?

The number on bottom is the denominator.

• What is the denominator’s purpose?

This number represents the number of equal sized groups or parts that make up a whole.

e.g. In the above example, ¼, the denominator is 4.

• *Key for remembering:

The Denominator is Down below the fraction bar/dividing line.

(“D” – “Denominator Down”)

• What types of numbers can be fractions?

Every integer can be written as a fraction by writing the integer with a denominator of 1.

Review the definition for integer.

As a rule, any integer n can be written as n/1, which is a quotient of the two integers.

• What types of real-life applications use fractions? Have students give their own examples.

Some can be:

¾ of a personal pan pizza – 3 of 4 slices remain in a whole pie that was divided into 4

pieces when it was cut



9/10 – 9 wins in a 10 game season

3/5 – 3 losses in a 5 game season

It is the top of the sixth inning at a baseball game, which has 9 innings total. How many

innings have been completed thus far? 5/9

• Graph fractions on a number line. • Have students draw a number line on their paper. Instruct them to place zero in the

“middle” of the line.

• On a number line, numbers to the right of 0 are positive. Numbers to the left of 0 are

negative. Given any two numbers on a number line, the one on the right is always larger,

regardless of its sign (positive or negative). Fractions may be placed on a number line in

addition to integers. Fractions may be negative as well as positive. Use the integers 0 and 1.

Show where ½,1/3, and ¼ would lie on the number line. Do the same process using the

opposites of those numbers.

• Are these fractions equal in value? - 5/7, (-5)/7 or 5/(-7)

• Now graph -5/7.

• Now attempt to graph 3 1/2 and -2 ¼.

• Write mixed numbers as improper fractions • Is 9/2 the same as 4 ½? Why or why not? 4 ½ would be considered a mixed number.

• Use the concept of a “single” serving microwaveable pizza being cut in half. If you have

nine of these halves, how many whole pizzas would that be? Would there be a “remainder”

or part of a whole remaining?

• So, what is a mixed number?

A number that is a combination of an integer and a proper fraction is “mixed”.

e.g. -4 ½, 9 ¾



• 9/2 would be considered an “improper” fraction. So, what is an improper fraction?

A fraction is improper if the numerator is equal to or greater than the denominator.

e.g. -5/2, 14/11, 7/7, 1/1

• Write a number as a product of prime numbers • What can you divide 2 by to get a whole number? What can you divide 7 by to get a whole

number? How about 23? What do you notice? These are considered prime numbers.

• So, help me define prime numbers. What are they?

• Any whole number with only two whole number factors, 1 and itself

e.g. 2,3,5,7,11,13,17,19,etc.

• How can you rewrite 7 as a product of 2 numbers? 1*7

• Review divisibility rules.

• So, think about ways to rewrite 25 in terms of a product of prime numbers.

• Now, think about ways to rewrite 63 in terms of prime numbers. Have students discuss their

answers. Make sure they give you answers in terms of prime numbers.

• In groups, write 828 and 836 in terms of primes. Have students share their answers. Make

sure there is a consensus on the correct answer. Ask students to tell how they came up with

their answers.

• Show students the prime factorization tree. Ask if anyone else did it differently and to show

the method (perhaps the “ladder” method).

• Illustrate prime factorization using the ladder method.

• Write a fraction in simplest form • What is another way to write 2/4? ½ is equivalent to 2/4. That is called simplifying.

• How did you simplify 2/4? What did you do to get ½? So, would you conclude that the same



method would work with 8/10? What is equivalent to 8/10? How about 10/20? If students

come back with 5/10, ask if there is another equivalency.

• You can simplify a fraction to its lowest terms by dividing the numerator and denominator by

their greatest common factor. The fraction has been simplified to lowest terms when its

numerator and denominator have no common factor other than 1 and -1.

• What are common factors, let alone greatest common factors?

Factors are what you call the integers (whole numbers and their opposites) that are

multiplied together to form a product. You could also describe them as integers that

divide cleanly into the given number without leaving a remainder. Once you have the

common factors, the greatest (or largest) factor shared by your numbers is your

greatest common factor.

e.g. With 12/20, both 12 and 20 share the factor 4. Therefore, dividing both by 4 gives

you 3/5. Both 12 and 20 are even numbers and divisible by 2; however, 4 is the GCF. It

is the largest number that divides evenly into both of the two numbers.

• One way to try to determine the common factor is to subtract the numerator from the

denominator. Try the difference that you find as the GCF. It will not be the GCF in every

instance.

e.g. 2/4… 4-2=2 …. 2÷2 = 1… 4÷2 = 2 … ½

8/10… 10-8 = 2 …. 8÷2 = 4 … 10÷2 = 5 … 4/5

5/10… 10-5 = 5 …. 5÷5 = 1 … 10÷5 = 2 … 1/2

12/20… 20-12 = 8 …. 8 will not divide evenly into 12 or 20

This is an example that the difference will not be the GCF in every instance.

• Determine whether two fractions are equivalent • What does it mean for two fractions to be equivalent?



Two fractions are considered equivalent when they represent the same value, even

though they may look different.

• Is there more than one way to decide if two fractions are equal to each other? What are

they?

Method 2

Some methods may be:

Method 1

Convert both fractions to have a common denominator.

e.g. 8/10 and 4/5

8/10 and 8/10

If the numerators are equivalent after the conversion of the denominators, then the

fractions are equivalent.

You can set the two fractions equal to each other (form a “proportion”) and use

cross multiplication (multiplying each numerator with the opposite denominator). If

the products are equivalent, then the fractions are equivalent.

In the fractions a/b and c/d, multiply ad and bc. The product of ad will be the same

as the product of bc if the fractions are equivalent.

e.g. Are 7/35 and 4/20 equivalent?

Cross multiply as follows…

7*20 =140

4*35=140



• Solve problems by writing fractions in simplest form Cooperative learning activity

Since the products are equivalent, the fractions are equivalent.

Special note: When you are asked to simplify a fraction, the “reduced” fraction is an equivalent

fraction to the one with which you began. Therefore, the term “reduced” is not accurate to use since

the overall value is equivalent and has not decreased. The correct term to use is “simplify”.

Have students play a card game.

• Make ahead of time multiple identical decks of 52 math cards, with proper and improper

fractions as well as mixed numbers. Divide the class into groups of four, five, or six players.

Provide each group with a deck of cards. Deal 5 cards to each player. The remaining cards are

to be placed face down to be drawn from later.

• Each player should sort his/her hand by putting the equivalent cards together.

• Each player may keep the hand that he/she was dealt or discard up to three cards and draw

from the “deck”.

• After the new cards are drawn, each player lays down his/her hand and receives points for the

following combinations:

One pair = 1 point

Two pairs = 2 points (pair of one kind and pair of another)

Three of a kind = 3 points

Four of a kind = 4 points

Five of a kind = 5 points

Full House (three of one kind and two of another) = 6 points

• Play for 10 minutes. Have a reward for each group’s winner.

To play as an entire class, have 2 identical decks of cards. Give each student 4 or 5 cards from one

deck. Hold up a card from your deck and ask the students to look through their cards to see if they

have an equivalent match. If they do, ask them to hold up the card. Take away the matching cards.

The first student out of cards wins.

Alternative activity:

Using a similar deck of cards (only 24 cards needed). Have students work in groups of 2 – 4. Give

each group a set of cards. Have the students lay out the cards facing down. Have each student,

taking turns, pick two cards and turn them over looking for a match. If the cards are a match



they get to keep the cards.

• Multiply fractions • What do you think 4/3 * 3/5 = ? Why?

• What does it mean to multiply fractions?

• What are you doing pictorially? Visualize this : What is ½ of ½ a dollar? ½ * ½ = ¼ One

half of a half of a dollar is one quarter.

• How do you multiply fractions?

Simply multiply across the top (numerators) and across the bottom (denominators).

Simplify if possible.

The rules for multiplying signed numbers apply.

What is a way to keep from having to do the step of simplifying?

Use the “cancellation” method. You may “cancel” when multiplying fractions and that

eliminates the need to simplify your answer. To cancel, find a factor that divides evenly

into one numerator and one denominator. This process may only be used when

multiplying fractions.

e.g. 4/3 * 3/5 12/7 * 14/3

• How do you multiply mixed numbers?

e.g. 1 5/7 * 4 2/3

• Evaluate exponential expressions with fractional bases • How would you “square” ½ ?

First, convert any mixed number to an improper fraction. Then, follow the previously

mentioned rules for multiplication of fractions.

You would multiply it times itself. Therefore, you should follow the steps for multiplying

fractions.

e.g. ( ½ ) 2 … ½ * ½ = ¼



• You could look at it a different way though. You could raise the numerator to the exponent,

and then do the same with the denominator. Finally, simplify.

e.g. ( ½ ) 2 … 1 2 = 1 and 2 2 = 4 … 1/4

• Divide fractions • To what operation is division equivalent?

Multiplying by the reciprocal

• What is a reciprocal?

When you invert (flip) the fraction, the numerator becomes the denominator and viceversa.

e.g. 4/3 and 3/4

• Can this be done to a whole number? Why? (Last point for objective 1)

Yes, the reciprocal of 5 is 1/5.

• What is “special” about dividing by a fraction?

You multiply by the reciprocal (refliprocal) of the second fraction.

The second fraction is the one that is your divisor. You simply invert (flip) it and

multiply. Simplify your answer if needed.

The rules for signed numbers apply here as well.

e.g. 4/3 ÷ 1/5

4/3 ÷ 5

• Solve applications that require multiplication of fractions • A recipe that makes 8 servings needs to be changed to a recipe that serves 4. All of the

ingredients will need to be halved. The recipe calls for ¼ tbsp of EVOO. How much EVOO



• Adding and Subtracting Like Fractions, Least Common

Denominator, and Equivalent Fractions

o Add or subtract like fractions

o Solve problems by adding or subtracting like fractions

o Find the least Common Denominator of a list of fractions

o Write equivalent fractions

o Write ratios as fractions

• Adding and Subtracting Unlike Fractions

o Add or subtract unlike fractions

o Write fractions in order

o Solve problems by adding or subtracting unlike fractions

will be needed now? ¼ * ½ = 1/8.

• To perform the operation of addition or subtraction, keep the common denominator and

simplify according to the operation. The rules for signed numbers apply to fractions as

well.

e.g. 1/3 + 2/3 7/9 – 4/9

• When would a denominator be referred to as common?

A denominator is “common” when it is a whole number (except zero) that is divisible by

all the denominators that are being compared.

e. g. ½, 1/5, and 1/10 all have a common denominator of 10, 20, or 30 (among the

multiples of 10) because 10, 20, and 30 are divisible by 2, 5, and 10.

• What would be an equivalent fractions for ½ with denominators of 4, 8, 10, and 24?

• Reinforce divisibility rules covered when discussing how to write a number as a product of

primes.

• How many students are in the class? How many females are there compared to males in

the class? What did we just do? We made a comparison.

• In math how can we write comparisons? as ratios

e.g. females to males, females : males, females/males, 1 to 2, 1:2, 1/2

• To add or subtract unlike fractions, you must first have a common denominator.

• Go back to the number line established with objective 3. “Order” those fractions based on

the drawing.

• Now find a common denominator for all of the fractions. Order according to these

equivalent fractions formed. Does it match the ordering produced by the number line?

• What if the denominators are the same in the beginning? List in ascending order of

numerators.

• What if the numerators are the same in the beginning? List in descending order of



denominators.

• Order of operations involving fractions • Review the order of operations (PEMDAS). Stress that a fraction bar is a grouping symbol,

• Operations on Mixed Numbers

o Graph positive and negative fractions and mixed numbers

o Multiply or divide mixed or whole numbers

o Add or subtract mixed numbers

o Perform operations on negative mixed numbers

Decimals (2 days)


1. Know the meaning of place value for a decimal number.

2. Write decimals in standard form

3. Write decimals as a fraction

falling under “P”.

• The order of operations remains the same no matter the form of the numbers.

• Already done with objective 3.

• What form would make multiplying or dividing mixed numbers possible? Improper

fractions

e.g. 1 ¼ * 2 3/5; 1 ¼ ÷ 2 3/5 ; 5 * 1 ½ ; 3 1/3 ÷ 5

• Would it be necessary to convert mixed numbers to improper fractions in order to subtract?

e.g. 1 ½ + 1 ½ 3 ¾ - 1 ¼

• What if the fraction parts of the mixed numbers do not have common denominators? How

would you begin the process of performing the operation?

e.g. 2 3/5 + 7 ½ 8 6/7 – 4 3/5

• Would sign rules apply to fractions as they do whole numbers?

e.g. 1 ¼ * -- 2 3/5; 1 ¼ - 2 3/5


1. What is the “place value chart”? What is the base ten number system? How does

understanding these two concepts allow me to work with decimal numbers?

2. What does a decimal point do?

3. When a number is called a decimal number, what is it?

4. What are the types of decimals?

5. What are the rules for operations with decimals?

6. What are the rules for changing decimals to fractions?



4. Compare decimals

5. Round decimals to a given place value.

6. Adding and Subtracting Decimals

a. Add and subtract decimals

b. Estimate when adding or subtracting decimals

c. Evaluate expression with decimal replacement values

d. Simplify expressions containing decimals

e. Solve problems that involve adding or subtracting

decimals

7. Multiplying Decimals

a. Multiply decimals

b. Estimate when multiplying decimals

c. Multiply decimals by powers of 10

d. Solve problems by multiplying decimals

8. Dividing Decimals

a. Divide decimals

b. Estimate when dividing decimals

c. Divide decimals by powers of 10

d. Solve problems by dividing decimals

7. What are the rules for changing decimals to percents?


• Know the meaning of place value for a decimal number. • Have the students read the fractions aloud as a group.



Fraction 3/10 257/100 512/1000 19/10000

Denominator 10 1 = 10 10 2 = 100

Places right of

the decimal

point

10 3 = 1000 10 4 = 10000

1 2 3 4

Decimal 0.3 2.57 0.512 0.0019

• What is the “place” of each denominator?

• Draw the connection between the “place” and the power of 10 that it represents.

• Write decimals in standard form • “Standard form” is what we commonly refer to as scientific notation (when a number is

written in two parts):

the digits are written with the decimal point just after the first nonzero digit on the left

those digits are followed by X 10 to a power

e.g. 5326.6 is 5.3266 x 10 3 in standard form 0.4123 is 4.123 x 10 -1

• Write decimals as a fraction • How do you change a decimal to a fraction?

Look at the last digit in the decimal number. What “place” is it? The denominator will

be a power of 10. The power is determined by how many places are to the right of the

decimal point.

e.g. 0.7 = 7/10 and 0.13 = 13/100

The fraction must be written in simplest form

e.g. 0.25 = 25/100 = ¼.

• Can decimals be “mixed” numbers?



• Let’s expand 1.25 in to its “parts”: 1 + 0.25 = 1 25/100 = 1 ¼

• Compare decimals Method 1

• When you compare two decimals, start from left to right using your fingers to cover up all of

the digits except the first one after the decimal point. (Those digits are not important unless

the ones being compared are equivalent.)

• If one of the numbers is not the same, then the number that is larger indicates your larger

number.

• If the numbers are the same, then move your finger one place to the right. Check the next

digit the same way you checked the previous digits. Again if one of them is larger, then you

have found the larger number.

• If they are still the same, continue this process until you find a number that is larger.

e.g.

9.46 ________ 9.141

9.42 ________ 9.459

Method 2

• Write your numbers one underneath the other, with the decimal points lined up as well as

the “places” .

• Vertically mark through “matches” to the right of the decimal until you find a discrepancy.

The larger number in the column belongs to your larger decimal.

e.g.

9.46 9.42

9.141 9.459

• Naturally, if there are whole numbers to the left of the decimal point that are different, you

already know which number is larger.

e.g. 9.43 is larger than 5.43 because 9 is larger than 5

• Round decimals to a given place value. • Review place values

• Using the following examples, have the students circle the digit to which the place is being

rounded and underline the digit directly behind the circle.

e.g.

$1.769 rounded to the nearest cent

7.42 rounded to the nearest tenth

$8.95 rounded to the nearest dollar



• Adding and Subtracting Decimals

o Add and subtract decimals

o Estimate when adding or subtracting decimals

o Evaluate expression with decimal replacement values

o Simplify expressions containing decimals

o Solve problems that involve adding or subtracting decimals

• Multiplying Decimals

o Multiply decimals

o Estimate when multiplying decimals

o Multiply decimals by powers of 10

o Solve problems by multiplying decimals

• If the underlined digit is 5 or larger, the circled digit goes up one value while all the following

digits are dropped. If the underlined digit is less than 5, the circled digit remains the same

while all of the following digits are dropped.

• How do you add or subtract money in your banking records?

• So, how do you think you add and subtract decimals?

You arrange the numbers so that the decimal points line up directly.

You may add zeros only to the end of the numbers if that aides you in lining up the

numbers.

Then, add or subtract as usual.

Finally, pull the decimal point directly down to the answer.

e.g. 1.42 + 5.073, 10.09 – 4.912

• If a = 13.35, then evaluate a + 7.25 and 20 – a.

• What do you get when you double $10.50?

• How do you think you would get the product of $10.50 and 2.5?

You simply multiply as normal forgetting that the decimals are in the factors.

Next, count how many total digits are behind (to the right of) the decimal factors.

Finally, starting at the right end of the product count the digits moving left one place for

each number of digits that were previously behind the decimal.

e.g. 10.50 * 2.5 = 26.250

• You may drop the “0”, because it is all the way to the right after the decimal. It has no effect

on the number’s value.

e.g. 26.250 = 26.25 17.040 = 17.04

• How does moving a decimal place right or left in a number affect the value?

e.g. 12.4 124 (if decimal moved right) 1.24 (if decimal moved left)

Larger Number Smaller Number



• Dividing Decimals

o Divide decimals

o Estimate when dividing decimals

o Divide decimals by powers of 10

o Solve problems by dividing decimals

Percent, Fractions, Decimals, and Order of Operations (2 days)


1. Write fractions as decimals

2. Compare fractions and decimals

• Decimals are all based on the number 10.

• Moving a decimal place to the right is the same as multiplying by 10 (12.4 * 10 = 124)

• Moving a decimal place to the left is the same as dividing by 10 (12.4 ÷ 10 = 1.24)

• You go out to eat with two friends and the total bill is $24.96. What is your cost if the bill is

divided evenly among the three of you?

• How did you get your answer? $24.96 ÷ 3 = $8.32

• When dividing into decimals by a whole number, just pull the decimal point of the dividend

directly up to the “roof of the house”. Add zeros where needed and divide normally.

e.g. You have $36.88. How much would you be able to spend on four gifts, if each gift

was to be of equal value?

• If you have $1.50 in quarters how many quarters do you have?

• How did you get your answer? $1.50 ÷ .25 = 6 quarters.

• When dividing by decimals move the decimal point of the divisor to the right so that it

becomes a whole number. Then, move the decimal point the same number of places to the

right in the dividend. Be sure to pull the decimal point in the dividend directly up to the

“roof of the house”. Divide normally.

e.g. If your jump drive only has 154.8 MB of space available, how many songs can you

download if each one takes up about 26.4 MB?


1. What is a percent?

2. What are the rules for changing percents to fractions?

3. What are the rules for changing percents to decimals?

4. How do the order of operations apply to problems involving any or all of the following types



3. Simplify expressions containing decimals and fractions

using order of operations

4. Solve area problems containing fractions and decimals

5. Evaluate expressions given decimal replacement values

6. Understand percent

7. Write percents as decimals or fractions

8. Write decimals or fractions as percents

9. Applications with percents, decimals, and fractions

of numbers: fractions, decimals, or percents?


• Write fractions as decimals • How do you change a fraction to a decimal?

You divide the numerator (dividend) by the denominator (divisor). Use long division.

e.g. 2/5 is 2 ÷ 5 = 0.4 5/2 is 5 ÷ 2 = 2.5

• What happens to the fraction 1/3 when it is converted to a decimal number?

It becomes 0.333333333… and never ends (goes to infinity which is an amount with no

limit or boundaries).

• How else could you describe 0.33333 …?

• What is a repeating decimal?

It is a decimal whose digits repeat endlessly, as seen in 1/3 being converted to

0.333… or 0.3 where the 3 has a bar over it to represent that it repeats.

• When converting fractions to decimals, what is interesting about ½ and 1/5? ½ converts to

0.5 and 1/5 converts to 0.2



• How could you describe 0.5 compared to 0.3333…?

• What is a terminating decimal?

It is a decimal whose digits end (terminate/stop). It usually stops pretty quickly such as

½ to 0.5 or ¼ to 0.25. Sometimes it extends further before stopping, such as 1/8 is

0.125.

• Compare fractions and decimals • With which form of a number do you feel more comfortable working, fraction or decimal?

• Simplify expressions containing decimals and fractions

using order of operations

• When comparing two numbers where each is in a different form, convert the one that is in

your less favorable form to your more favorable form by following the steps that we have

previously discussed.

e.g. compare ¼ and 0.3

As fractions:

Change 0.3 to 3/10. Get a common denominator. Compare the new numerators… so

5/20 is less than 6/20 or ¼ is less than 0.3.

As decimals:

• Change ¼ to a decimal (0.25). Compare 0.25 to 0.3. Compare place value digits. “2” is less

than “3”. ¼ is less than 0.3.

• e.g. ¼ (.25 + 3/10) ÷ 1/2 2 – 1/4

• Solve area problems containing fractions and decimals • e.g. If a garden measures 14 feet in length and its width is 2/5 of that, then what is the area

of the garden?

• 0.75 of a house is going to be recarpeted. The area of the house is 1850 ft 2 . How much of

the house will receive new carpet?



• Evaluate expressions given decimal replacement values • e.g. If a = 0.213, then evaluate a +1.46 and a(1 1/8)

• Understand percent • What does percent mean?

It means hundredths, out of 100, or per 100. The symbol for percent is %.

• Where do we encounter percents in every day life?

Loan rates, sales at stores, income taxes

• A dollar bill is equivalent to 100 cents (pennies).

• Write percents as decimals or fractions • How do you convert a percent to a decimal or fraction in order to make it easy to use?

To convert a percent to a decimal move the decimal point two places to the left and

drop the % symbol.

To convert a percent to a fraction, write the percent as the numerator and 100 as the

denominator without the % symbol. Then, write the fraction in its simplest form.

e.g. 100% = 1.00 = 1 or 100/100 = 1

43% = 0.43 or 43/100

• Write decimals or fractions as percents • Can you “reverse” the process and change decimals or fractions to percents?

• How can you go about making the change?

A decimal can be written as a percent by moving the decimal point two places to the

right and adding the symbol % to the end.

e.g. 0.81= 81%

0.123 = 12.3 %



1 = 100%

4.9 = 490%

Add zeros if necessary to make the two moves.

• A fraction can be written as a percent using one of these methods:

Method 1

You could convert the fraction to a decimal and use the previously discussed method.

e.g. 4/5 = 0.8 = 80%

Method 2

You could set up a proportion.

Numerator = X

Denominator 100

Solve for X, and add a % symbol. This “is/of” equals “X/100” is quite useful for word problems.

e.g. 4/5 = X/100

4(100) = 5X

400 = 5X

80 = X

80%

Cooperative Learning Activity: Math card game, same rules as fraction’s objective 8, but with a

different deck which would have fractions, decimals, and percents for recognizing equivalent

quantities.

• Applications with percents, decimals, and fractions • Vary applications for conversions.



Symbols and Sets of Real Numbers (2 days)


1. Define the meaning of symbols =≠≤≥ , , , , ,

2. Translate sentences into mathematical statements

3. Identify integers, rational numbers, irrational numbers,

and real numbers

4. Review/connection between real and whole number

properties

a. Use the commutative properties

i. Addition: a + b = b + a

ii. Multiplication: a * b = b * a

b. Use the associative properties

i. Addition: (a + b) + c = a + (b + c)

ii. Multiplication: (a*b)*c = a*(b*c)

c. Use the distributive property

• Cooperative Learning Activity: Provide a MATHO card for each student, along the concept of

BINGO. The squares should be filled with the answers to various problems that include

percents, decimals, and fractions. Work as many problems as time allow and provide prizes.

Problems could be pulled from course textbook so that a listing could be provided to the

students of problems worked with answers before they leave class to aide in studying.

Main Underlying Questions:

1. What are the meanings of the equality and inequality symbols: =, ≠, , ≤ , ≥ ?

2. How do you identify the individual sets of real numbers including integers, rational and

irrational?

3. Do the commutative, associative, distributive, identity and inverse property apply to all

mathematical operations? If not, which ones do they apply to and when are they used.



i. a(b+c) = a*b + a*c

d. Use the identity property

i. 0 for Addition: a + 0 = a and 0 + a = a

ii. 1 for Multiplication: 1*a = a and a*1=a

e. Use the inverse properties

i. Additive: a + (-a) = 0

ii. Multiplicative:

1

b⋅ = 1

b


• Define the meaning of symbols =≠≤≥ , , , , ,

• What is the difference between = and ≠?

• What is the difference between “less than” “ symbols and what is the

significance of the direction in which the sign is pointing?

• What happens when you add the line under the inequality symbol, for example, e.g. ≤ and ≥?

Activities

Give students a worksheet with both fill in the blank and true/false statements concerning

equality and inequality statements.

• Translate sentences into mathematical statements • How do you use the equality and inequality symbols to translate sentences into mathematical

• Identify integers, rational numbers, irrational numbers,

and real numbers

statements? e.g. “Negative two is less than or equal to zero” is the same as -2 ≤ 0.

• What numbers make up the set of natural numbers? Natural numbers: {1, 2, 3, 4, 5, 6, …}

• What number is added to the natural numbers to make up the set of whole numbers? zero

Whole Numbers: {0, 1, 2, 3, 4, 5, 6, …}

• Which group of numbers is added to the whole numbers to form the integers?



• Review/connection between real and whole number

properties

o Use the commutative properties

o Use the associative properties

o Use the distributive property

o Use the identity property

o Use the inverse properties

a

• What is the definition of a rational number? { | a and b are integers but b ≠0}

b

• How can rational numbers be written as decimals? (By dividing their numerator by their

denominator).

• What is the difference between rational and irrational numbers? (If the resulting decimal number

is terminating or repeating then the number is a rational number and if it is non-terminating or

non-repeating then it is irrational.)

• What numbers make up the real numbers?

Activities: Students will be ask to construct a diagram of the sets of numbers, given examples of

each group.

Students will be given worksheets and ask to determine which set or sets that given numbers

belong to.

• What is the commutative property of addition and multiplication?

Addition: a + b = b + a

Multiplication: a • b = b • a

• What is the associative property of addition and multiplication?

Addition: (a + b) + c = a + (b + c)

Multiplication: (a • b) • c = a • (b • c)

• Do these properties also apply to subtraction and division, if not, why?

• How do you use the distributive property to simplify an expression? (By multiplying the term

in front of the parentheses by each term within the parentheses: a(b + c) = ab + ac)

• What is the Identity Properties for Addition and Multiplication?

0 is the identity element for addition : a + 0 = a and 0 + a = a



Geometry (4 days)


1. Use formula to find perimeter

2. Use formulas to find circumferences

3. Find the area of plane regions

4. Find the volume and surface area of solids

5. Linear Measurement (*)

a. Define US units of length and convert from one unit to

another

b. Use mixed units of length

c. Perform arithmetic operations on US units of length

d. Define metric units of length and convert from one

unit to another

e. Perform arithmetic operations on metric units of

length

Activities

1 is the identity element for multiplication: a • 1 = a and 1 • a = a

• Students will be given worksheets where they will complete statements using the

commutative and associative properties, use the distributive property to write each

expression without parentheses, and name each statement that is used to make the true

statement.

Main Underlying Questions:

1. Define perimeter, area, volume and surface area of geometric shapes and objects.

2. How can formulas be used to find the perimeter, area, volume, and surface area of

geometric shapes and objects.

3. What is the US units of length, weight, mass and capacity and how can they be converted

from one to another?

4. How do you perform arithmetic operations on US units of lengths, weight, mass and

capacity?

5. What is the metric units of length, mass, weight, and capacity and how are they converted

from one to another?

6. How are arithmetic operations performed on metric varying units of length, mass, weight,

and capacity?

7. How do we convert temperatures from degrees Celsius to degrees Fahrenheit and from

degrees Fahrenheit to degrees Celsius?



6. Weight and Mass(*)

7. Capacity(*)

a. Define US units of weight and convert from one unit to

another

b. Perform arithmetic operations on units of weight

c. Define metric units of mass and convert from one unit

to another

d. Perform arithmetic operations on units of mass

a. Define US units of capacity and convert from one unit

to another

b. Perform arithmetic operations on US units of capacity

c. Define metric units of capacity and convert from one

unit to another

d. Perform arithmetic operations on metric units of

capacity

8. Temperature and Conversions between the US and Metric

System(*)

a. Convert between the US and metric systems

b. Convert temperatures from degrees Celsius to degrees

Fahrenheit

c. Convert temperatures from degrees Fahrenheit to

degrees Celsius


• Use formula to find perimeter • What is the perimeter of a polygon and how do you find it? (The distance around the

polygon; by adding all the measurements of the sides.)



• How is perimeter measured? (Perimeter is measured in units.)

• What is the formula used to find the perimeter of a rectangle? (2•length + 2•width =

perimeter)

• Use formulas to find circumference • What is the distance around a circle called? Circumference

• What is the formula used to find the circumference of a circle? C = 2πr or C = πd ; where

d=diameter, r = radius, π ≈ 3.14

• Find the area of plane regions • What is the definition of area of a plane region?

• How is area measured and why? (square units)

• What is the formula used to find the area of a rectangle? ( Area = length x width)

1

• What is the formula used to find the area of a triangle? (Area = ●base●height)

2

• What is the formula used for finding the area of a circle? (Area = πr 2 )

• Find the volume and surface area of solids • What does volume measure? (The measure of space of a given region)

• Linear Measurement (*)

o Define US units of length and convert from one unit to

another

• How is volume measured and why? (Volume is measured in cubic units)

• What is the formula for the volume of a rectangular solid? V = lwh

• What is the definition of surface area of a solid? (The sum of the areas of the faces of the

solid.)

• How is surface area measured and why? (Surface area is measured in square units)

• What is the formula for the surface area of a rectangular solid? (SA = 2lh + 2lw + 2wh)

Activity: Begin with U.S. system of measurements: inch, foot, yard, and mile. Have students prepare

an equivalency chart for memorization, for example:

U.S. Units of Lengths

12 inches = 1 foot



o Use mixed units of length

o Perform arithmetic operations on US units of length

o Define metric units of length and convert from one unit

to another

o Perform arithmetic operations on metric units of length

3 feet = 1 yard

36 inches = 1 yard

5280 feet = 1 mile

• Using the above chart, how could you write “12 inches = 1 foot” as a unit fraction?

6 ft 12in.

• How can you convert 6 feet to inches using the unit fraction above? • = 72 in.

1 1 ft



12in.

1 ft

• How can you convert 68 inches into feet using unit fractions? (Hint: Since it is the opposite

68in 1 ft 68 17 2

conversion, perhaps use the reciprocal unit fraction). { ● = = = 5 ft }

1 12in

12 3 3

• How do you add and subtract units of lengths? Should you simplify the answer when

necessary, if so, how?

• How are units of length in the metric system measured? All the units in the metric system are

based on the meter.

Activity: Have students build a conversion chart for memorization of the metric unit of

length using the summary of prefixes:

Metric Unit of Length

1 kilometer (km) = 1000 meters (m)

1 hectometer (hm) = 100 m

1 deckameter (dam) = 10 m

1 meter (m) = 1 m

1 decimeter = .1 m

• Weight and Mass(*)

o Define US units of weight and convert from one unit to

another

o Perform arithmetic operations on units of weight

o Define metric units of mass and convert from one unit

to another

o Perform arithmetic operations on units of mass

1 centimeter = .01 m

1 millimeter = .001 m

km hm dam m dm cm mm

• How can you convert metric units by moving the decimal point? Why can this method be

used in the metric system conversion and not in U.S. system conversions?

• What does the weight of an object refer to and what are its units of measure?

Activity: Have students write The weight units of measure as unit fractions:

16oz

1lb

e.g. = ;

1lb

16oz

2000lb

1ton

=

1ton

2000lb

• Using the above unit fractions, how can you convert 6500 pounds to tons?

6500lbs 1ton

6500 13 1

● = = = 3 tons

1 2000lbs

2000 4 4

• Can you perform arithmetic operations on units of weights the same way you perform them

on units of lengths? If so, what must first be done on the example below before you can

perform the subtraction?

8 tons 100 lbs = 7 tons 2100 lbs (after borrowing)

• 5 tons 1200 lbs = - 5 tons 1200 lbs

2 tons 900 lbs

• What is the basic unit of mass in the metric system? (the gram)

Activity: Have students build a conversion chart for memorization of the metric unit of mass

using the following table:



• Capacity(*)

o Define US units of capacity and convert from one unit

to another

o Perform arithmetic operations on US units of capacity

o Define metric units of capacity and convert from one

unit to another

o Perform arithmetic operations on metric units of

capacity

Metric Units of Mass

1 kilogram (kg) = 1000 grams (g)

1 hectogram (hg) = 100 g

1 deckagram (dag) = 10 g

1 gram (g) = 1 g

1 decigram (dg) = .1 g

1 centigram (cg) = .01 g

1 milligram = .001 g

kg hg dag g dg cg mg

• How do you convert metric units by moving the decimal point? Why can this method be


• What is used to measure liquids? capacity

Activity: Have students build a conversion chart for U.S. Units of Capacity:

U.S. Units of Capacity

8 fluid ounces (fl oz) = 1 cup (c)

2 cups = 1 pint (pt)

2 pints = 1 quart (qt)

4 quarts = 1 gallon (gal)

• Can the unit fraction method be used to convert from one capacity unit to another?

• How can we use unit fractions to convert 26 quarts to cups?

cups

26qts

●

1

• What is the basic unit of capacity in the metric system? (the liter)

4cups

= 26●4= 104

1qt



• Temperature and Conversions between the US and Metric

System(*)

o Convert between the US and metric systems

o Convert temperatures from degrees Celsius to degrees

Fahrenheit

o Convert temperatures from degrees Fahrenheit to

degrees Celsius

Activity: Have students build a conversion chart for memorization of the metric unit of capacity

using the following table:

Metric Units of Capacity

1 kiloliter (kl) = 1000 liter (L)

1 hectoliter (hl) = 100 L

1 deckaliter (dal) = 10 L

1 liter (L) = 1 L

1 deciliter (dl) = .1 L

1 centiliter (cl) = .01 L

1 milliliter = .001 L

kl hl dal L dl cl ml

• How can you convert metric units by moving the decimal point? Why can this method be


• When given a conversion chart, how can you use unit fractions to convert metric units to US

system units? How can you convert 12 cm to inches ?

12cm

●

1

1in

= 12/2.54 ≈ 4.72 in.

2.

54cm

• How can you convert temperatures from degrees Celsius to degrees Fahrenheit using one of

the following formulas:

9

F= C + 32 or F = 1.8C+ 32

5

• How can you convert temperatures from degrees Fahrenheit to degrees Celsius using the

following formula:

5

C = (F -32)

9



ARITHMETIC

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