[PDF] - Factor Vocab Word 2 Fraction Division Its meaning (As it is used PDF Document

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6th Grade Fraction and Decimal Computation

www.njctl.org

2014-10-16

Slide 3 / 315 Fraction and Decimal Computation Unit Topics

· Long Division Review · Adding Decimals · Subtracting Decimals

Click on the topic to go to that section

· Multiplying Decimals · Dividing Decimals

Common Core Standards: 6.NS.1, 6.NS.2, 6.NS.3

· Glossary · Fraction Division · Distributive Property & Product of Decimals · Greatest Common Factor · Least Common Multiple · GCF and LCM Word Problems · Divisibility Rules for 3 & 9 · Even and Odd Numbers

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Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. How many thirds are in 1 whole? How many fifths are in 1 whole? How many ninths are in 1 whole?

Vocabulary words are identified with a dotted underline.

The underline is linked to the glossary at the end of the

Notebook. It can also be printed for a word wall.

(Click on the dotted underline.)

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Back to Instruction

Factor

A whole number that can divide into another number with no remainder.

15 3 5

3 is a factor of 15

3 x 5 = 15

3 and 5 are factors of 15

16 3 5 .1

R 3 is not a factor of 16

A whole number that multiplies with another number to make a third number.

The charts have 4 parts.

Vocab Word

1

Its meaning

2

Examples/ Counterexamples

3

Link to return to the instructional page.

4

(As it is used in the lesson.)

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Fraction Division

Return to Table of Contents

SLIDE 2

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Modeling Division

Recall from 5th grade: When we are dividing, we are breaking apart into equal groups. The model below represents: 8 4 = 2 2 groups of 4 Dividend Divisor = Quotient

Slide 8 / 315 Applying to Fractions

The previous example used whole numbers and grouped the dividend according to the divisor. The same strategy can be applied when dividing with fractions. Use the model below to demonstrate: 8 8 The pink rectangle represents . See how many you can fit in the 8 squares. Teacher Notes

Slide 9 / 315 Example

Use the model below to demonstrate 2 2

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1 Evaluate the following problem using the model

below:

3 3 Answer

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2 Evaluate the following problem using the model

below:

5 5 Answer

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3/5 ÷ 4 = 3/20

Click to Reveal

A fraction can be divided by a whole number using the following visual model. 3/5 ÷ 4 Divide into 4 groups 1 2 3 4

SLIDE 3

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The previous expression can be represented by the following word problem: How much will each person receive if 4 friends share a 3/5 pound bag of popcorn? 1 2 3 4 Each friend will receive 3/20 lb. of popcorn.

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3 Evaluate the following problem using the model below.

Drag the line

Answer

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4 Evaluate the following problem using the model below.

Drag the line

Answer

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Create a story to represent the problem and use a visual model to show the quotient.

Slide 17 / 315 Fraction Divided by a Fraction

The same strategy we utilized for the previous examples can also be applied when dividing a fraction by another fraction. In this example our division problem is: We need to determine how many 's there are in

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Use the model below to evaluate:

SLIDE 4

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5 Evaluate the following problem using the model

below:

Answer

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6 Evaluate the following problem using the model

below:

Answer

Slide 21 / 315 Vocabulary Review

Complex Fraction: A fraction with another fraction in the numerator, denominator or both Reciprocal: The inverse of a number/fraction

Original Number 4 Reciprocal 2

Slide 22 / 315 Patterns

Do you notice a pattern between the division of fractions and their solution? Teacher Notes

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If you think about it, we are dividing by a fraction which creates a complex fraction. You need to eliminate the fraction in the denominator in order to solve the problem. To do this, multiply the numerator and denominator of the complex fraction by the reciprocal of the denominator (making the denominator = 1). You can then simplify the fraction by rewriting it without the denominator of 1 and solve the new multiplication problem.

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source - http://www.helpwithfractions.com/dividing-fractions.html

There are rules that can be applied to fraction division problems to eliminate steps from this lengthy procedure. 1 2 x 3 2 = 1 2 2 3 = 1 2 2 3 = 1 2 2 3 x 3 2 x 3 2 = 1 2 x 3 2

1

Original Problem Complex Fraction Multiply by Reciprocal Simplify Denominator Rewrite Without 1

Example

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Step 1: Leave the first fraction the same. Step 2: Multiply the first fraction by the reciprocal of the second fraction. Step 3: Simplify your answer.

Dividing Fractions Algorithm

1 5 x 2 1 = 1 x 2 5 x 1 = 2 5 1 5 1 2 =

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Some people use the saying "Keep Change Flip" to help them remember the algorithm. 3 5 x 8 7 = 3 x 8 5 x 7 = 24 35 3 5 7 8 = Kept Changed Flipped Keep Change Flip

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Evaluate: Kept Changed Flipped Keep Change Flip x = = =

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Checking Your Answer

To check your answer, use your knowledge of fact families. 3 5 7 8 24 35

÷

=

3 5

=

24 35 7 8 x 3 5 is 7 8

f

24 35

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7

True False 8 10 = 5 4 x 8 10 4 5 Answer

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8

True False 2 7 = 3 4

2 7

8 Answer

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9

1

A

39 40

B C

8 10 = 4 5 40 42 Answer

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10

Answer

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11

Answer

Slide 34 / 315 Sometimes you can cross simplify prior to multiplying.

without cross simplifying with cross simplifying

3 1 2 5

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12 Can this problem be cross simplified?

Yes No Answer

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13 Can this problem be cross simplified?

Yes No Answer

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14 Can this problem be cross simplified?

Yes No Answer

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15 Can this problem be cross simplified?

Yes No Answer

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16

Answer

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17

Answer

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18

Answer

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19

Answer

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A mixed number can be divided by a mixed number using the following visual model. First find the least common denominator (LCD) which is 6. If every 6 lines represents a whole, then how many lines should we draw to make sure both mixed numbers fit?

18

Click to Reveal

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Since our LCD is 6, every 6 lines is considered a whole. 1 1/2 is equivalent to 9 sections on the number line. 1 1/2 2 2/3 is equivalent to 16 sections on the number line. So 1 1/2 ÷ 2 2/3 = 9/16 2 2/3

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1/2 2 2/3 1 1/2 What if the problem were written as ? How many times does 1 1/2 divide into 2 2/3?

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Step 1: Rewrite the Mixed Number(s) as an improper fraction(s). (write whole numbers / 1) Step 2: Follow the same steps for dividing fractions

Dividing Mixed Numbers Algorithm

6 1 x 2 3 = 12 3 =

6

1 2

1

6 1 3 2 = = 4

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5 3 x 2 7 = 10 21 2 3 =

1

1 2

3

5 3 7 2 =

Example

Evaluate:

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20

= 1 2

2 2

3

1

Answer

SLIDE 9

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21

= 1 2

5 2

Answer

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22

= 2 5

5 1

2

4

Answer

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23

= 1 2

2 3

8

3

Answer

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Winnie needs pieces of string for a craft project. How many 1/6 yd pieces of string can she cut from a piece that is 2/3 yd long? 1 6 2 3 ÷ 2 3 x 6 1 12 3 = = 4 pieces 4 1

r

2 3 x 6 1 = 1 2 4 1 = 4 pieces

Application Problems - Examples

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One student brings 1/2 yd of ribbon. If 3 students receive an equal length of the ribbon, how much ribbons will each student receive? 1 2 ÷ 3 1 2 x 1 3 1 6 yards of ribbon =

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Kristen is making a ladder and wants to cut ladder rungs from a 6 ft

board. Each rung needs to be 3/4 ft long. How many ladder rungs

can she cut? 6 ÷ 3 4 6 1 ÷ 3 4 6 1 x 4 3 = 24 3 8 1 8 rungs = =

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A box weighing 9 1/3 lb contains toy robots weighing 1 1/6 lb

apiece. How many toy robots are in the box?

9 1 3 1 1 6 ÷ 28 3 7 6 ÷ 6 7 28 3 x 1 4 1 2 = 8 1 8 robots =

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24 Robert bought 3/4 pound of grapes and

divided them into 6 equal portions. What is the weight of each portion?

A 8 pounds B

4 1/2 pounds

C

2/5 pounds

D

1/8 pound

Answer

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25 A car travels 83 7/10 miles on 2 1/4

gallons of fuel. Which is the best estimate

f the number miles the car travels on
ne gallon of fuel?

A 84 miles B

62 miles

C

42 miles

D

38 miles

Answer

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26 One tablespoon is equal to 1/16 cup. It is

also equal to 1/2 ounce. A recipe uses 3/4 cup of flour. How many tablespoons of flour does the recipe use?

A 48 tablespoons B

24 tablespoons

C

12 tablespoons

D

6 tablespoons

Answer

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27 A bookstore packs 6 books in a box. The

total weight of the books is 14 2/5

pounds. If each book has the same

weight, what is the weight of one book?

A 5/12 pound B

2 2/5 pounds

C

8 2/5 pounds

D

86 2/5 pounds

Answer

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28 There is gallon of distilled water in the

class science supplies. If each pair of students doing an experiment uses gallon of distilled water, there will be gallon left in the supplies . How many students are doing the experiments?

Answer

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Long Division Review

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Some division terms to remember.... · The number to be divided into is known as the dividend · The number which divides the other number is known as the divisor · The answer to a division problem is called the quotient

divisor 5 20 dividend

4 quotient

20 ÷ 5 = 4 20

__

5 = 4

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When we are dividing, we are breaking apart into equal groups EXAMPLE 1 Find 132 3 Step 1: Can 3 go into 1, no so can 3 go into 13, yes 4

12

1 3 x 4 = 12 13 - 12 = 1 Compare 1 < 3 3 132 3 x 4 = 12 12 - 12 = 0 Compare 0 < 3

12

Step 2: Bring down the 2. Can 3 go into 12, yes 2 4

Click for step 1

Click for step 2

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Step 3: Check your answer. 44 x 3

132

Click to Reveal

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Estimating Your Answer Before any calculations, estimate your answer to make sure you are on the right track. What place value should we round to? Round to the largest place value. 357 rounds to ____ 15 rounds to ____ Our answer should approximately be ...

20

357 ÷ 15

click

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EXAMPLE 2 (change pages to see each step) Step 1: Can 15 go into 3, no so can 15 go into 35, yes 2

30

5 15 x 2 = 30 35 - 30 = 5 Compare 5 < 15 15 357

SLIDE 12

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2

30

5 15 357 15 x 3 = 45 57 - 45 =12 Compare 12 < 15 7

45

12 Step 2: Bring down the 7. Can 25 go into 207, yes 3 EXAMPLE 2 (change pages to see each step)

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2

30

5 15 357.0 7

45

120

120

3 Step 3: You need to add a decimal and a zero since the division is not complete. Bring the zero down and continue the long division. 15 x 8 = 120 120 - 120 = 0 Compare 0 < 15 .8 EXAMPLE 2 (change pages to see each step) Is our answer close to our estimate?

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Check your answer. 23.8 x 15

357

Click to Reveal

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Estimate the following problems. Discuss your answers with your group. 1. 2. 3. 4. 35 300 15 20

Click to Reveal Click to Reveal Click to Reveal Click to Reveal

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Complete the following problems. Discuss your answers with your group. 1. 2. 3. 4. 41 324 19.5 23.2

Click to Reveal Click to Reveal Click to Reveal Click to Reveal

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29 Estimate the quotient.

Answer

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30 Compute.

Answer

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31 Estimate the quotient.

Answer

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32 Compute.

Answer

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33 Estimate the quotient.

Answer

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34 Compute.

Answer

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35 The school concert hall contains 312 chairs in 12

rows. Estimate how many chairs are in each row.

Answer

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36 The school concert hall contains 312 chairs in 12

rows. How many chairs are in each row?

Answer

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37 Compute.

Answer

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38 The local Italian restaurant receives the same number

f visitors every day. If 343 people visit the restaurant
ver the course of one week, how many visitors visit

each day?

Answer

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39 Compute.

Answer

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40 Compute.

Answer

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Adding Decimals

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If you know how to add whole numbers then you can add

decimals. Just follow these few steps.

Step 1: Put the numbers in a vertical column, aligning the decimal points. Step 2: Add each column of digits, starting on the right and working to the left. Step 3: Place the decimal point in the answer directly below the decimal points that you lined up in Step 1.

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When adding or subtracting decimals, always remember to align the decimals vertically... 0.25 0.25 0.25 0.25 1.00 +

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Estimating Your Answer Before any calculations, estimate your answer to make sure you are on the right track. What place value should we round to? Round to the nearest whole number. 5.1 rounds to ____ 1.25 rounds to ____ 0.04 rounds to ____ 1.99 rounds to ____ Our answer should approximately be ...

8

5.1 + 1.25 + 0.04 + 1.99

click

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Now, try this - Don't forget - LINE 'EM UP 5.1 + 1.25 + 0.04 + 1.99 5.10 1.25 0.04 1.99 + 8.38

You can add a zero as a place holder to help line your numbers up.

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TRY THESE. Estimate the following sums in your notebook. Check with the rest

f your group. To check your answer, click the box.

1) 8.23 + 4.125 + 0.1189 2) 3.178 + 12.28 + 9 3) 17.009 + 2.965 + 8.4 4) 9.999 + 3.1567 + 4.5656 8 + 4 + 0 = 12 3 + 12 + 9 = 24 17 + 3 + 8 = 28 10 + 3 + 5 = 18

Click to Reveal Click to Reveal Click to Reveal Click to Reveal

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TRY THESE. Complete in your notebook then check with the rest of your group. To check your answer, click the box. 1) 8.23 + 4.125 + 0.1189 2) 3.178 + 12.28 + 9 8.23 3.178 4.125 12.28 + 0.1189 + 9. 12.4739 24.458 3) 17.009 + 2.965 + 8.4 4) 9.999 + 3.1567 + 4.5656 17.009 9.999 2.965 3.1567 + 8.4 + 4.5656 28.374 17.7213

Click to Reveal Click to Reveal Click to Reveal Click to Reveal

SLIDE 16

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41 Add the following: 0.6 + 0.55 = A 6.1 B 0.115 C 1.15 D 0.16

Answer

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42 Joanne and Peter are working together to solve the problem 0.6 + 0.55. Joanne says that the sum should be approximately 2. Peter disagrees and says the sum should be approximately 0. Who is correct? Why?

A

Joanne

B

Peter

Answer

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43 Find the sum 1.025 + 0.03 + 14.0001 =

Answer

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44 Franco went to buy new video games. He bought MaxRush for $19.95, Duplo Race for $23.95 and Garage Mate for $21.95. Estimate how much Franco spent on the video games.

Answer

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45 Franco went to buy new video games. He bought

MaxRush for $19.95, Duplo Race for $23.95 and Garage Mate for $21.95. How much did he spend on video games?

Answer

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46 What is the sum of

12.034 and 0.0104? A 12.1344 B 12.0444 C 12.138 D 1.20444

Answer

SLIDE 17

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47 Estimate the sum 8.5 + 0.042 + 12.31

A

20

B

21

C

22

D

23

Answer

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48 Find the sum 8.5 + 0.042 + 12.31 = A 13.58 B 21.23 C 20.852 D 20.14

Answer

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49 Five students collected paper to be recycled. Shelly's stack was .008 cm. thick; Ken's stack was .125 cm. thick; Joe's stack was .150 cm. thick; Betty's stack was .185 cm. thick; Mary's stack was .005 cm. thick. What was the thickness of the papers collected to be recycled?

A

.561 cm.

B

.452 cm.

C

.480 cm.

D

.473 cm.

Answer

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50 Find the sum:

5 + 100.145 + 57.8962 + 2.312 =

Answer

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Let's go to Cool Math and practice addition:

Cool Math Link

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Subtracting Decimals

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SLIDE 18

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If you know how to subtract whole numbers then you can subtract decimals. Just follow these few steps. Step 1: Put the numbers in a vertical column, aligning the decimal points. Step 2: Subtract the numbers from right to left using the same rules as whole numbers. Step 3: Place the decimal point in the answer directly below the decimal points that you lined up in Step 1. 1.1

0.3

1.1

0.3

0.8 0 1

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Estimating Your Answer Before any calculations, estimate your answer to make sure you are on the right track. 21.7 - 8.21 What place value should we round to? Round to the nearest whole number. 21.7 rounds to ____ 8.21 rounds to ____ Our answer should approximately be ...

14

click

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What do we do if there aren't enough decimal places when we subtract? 21.7 - 8.21 Don't forget...Line 'em Up! 21.7 8.21 What goes here? 21.70 8.21 13.49

6 1 1 1

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TRY THESE. Estimate the following differences in your notebook. Then check with the rest of your group. To check your answer, click the box. 1) 8.23 - 0.1189 2) 12.283 - 9.025 3) 17.009 - 8.4 4) 9.999 - 4.5656 8 - 0 = 8 12 - 9 = 3 10 - 5 = 5 17 - 8 = 9 Click to Reveal Click to Reveal Click to Reveal Click to Reveal

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TRY THESE. Complete in your notebook then check with the rest of your group. To check your answer, click the box. 1) 8.23 - 0.1189 2) 12.283 - 9.025 8.23 12.283

0.1189
9.025

8.1111 3.258 3) 17.009 - 8.4 4) 9.999 - 4.5656 17.009 9.999

8.4
4.5656

8.609 5.4334

Click to Reveal Click to Reveal Click to Reveal Click to Reveal

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51

5 - 0.238 =

Answer

SLIDE 19

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52

12.809 - 4 =

Answer

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53 Sally won $25.00 for her science fair project. Her project cost $12.57 to prepare. What is the estimate of Sally's profit? A $20 B $18 C $13 D $12 Answer

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54 Sally won $25.00 for her science fair project.

Her project cost $12.57 to prepare. How much did Sally actually make as a profit? A $37.57 B $12.43 C $13.57 D $12.00 Answer

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55

1897.112 - 0.647 =

Answer

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56 The Johnson twins raced each other in the 200-meter

dash. Jordan finished in 23.48 seconds, and Max

finished in 26.13 seconds. How much faster was Jordan than Max?

Answer

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57 Timothy is working on the problem 4.1 - 0.094. He estimates his answer before solving and rounds the numbers to the nearest tenths. He uses 4.1 and 0.1 to estimate the answer. Is he correct in doing so? Why or why not?

Yes No Answer

SLIDE 20

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58

4.1 - 0.094 =

Answer

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59

17 - 13.008 =

Answer

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60 Which problem below would give you two different estimates when you either round to the nearest whole or round to the nearest tenths?

A

27.85 - 12.91

B

14.17 - 8.2

C

7.9 - 3.88

D

21.25 - 18.16

Answer

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61

If you buy two movie tickets for $8.25 each, what will your change be from $20?

Answer

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Cool Math Link Let's go to Cool Math and practice subtraction:

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The Distributive Property and the Product of Decimals

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SLIDE 21

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If you know how to multiply whole numbers then you can multiply decimals. Just follow these few steps. Step 1: Ignore the decimal points. Step 2: Multiply the numbers using the same rules as whole numbers. Step 3: Count the total number of digits to the right of the decimal point. Put that many digits to the right of the decimal point in your answer.

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Evaluate 200 x 41.5 8,300 We can also use the distributive property to calculate the product. 200 x 41.5 200 x (41 + 0.5) (200 x 41) + (200 x 0.5) 8,200 + 100 = 8,300 Separate 41.5 into an addition expression with two addends Apply the distributive property Apply the order of operations

click

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Try this! Evaluate 400 x 18.33 400 x ( ________ + ________ ) (400 x ________ ) + (400 x ________ ) ________ + ________ = ________ 7332 This method is known as partial products.

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How can we use partial products to calculate the area of the rectangle shown below? 58 ft 200 ft 0.6 ft 200 x 58.6 200 x ( ________ + ________ ) (200 x ________ ) + (200 x ________ ) ________ + ________ = ________ 11,720 Click to reveal 58.6 ft 200 ft Click on the yellow box to show the visual representation of partial products.

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62 12(43) = 12(40) x 12(3)

True False Answer

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63 Use the distributive property to rewrite the expression: 3(76.8)

Students type their answers here

Answer

SLIDE 22

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64 Calculate the product using partial products. 5(48)

Answer

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65 Calculate the product using partial products. 13(5.2)

Answer

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66 Calculate the product using partial products. 300(7.4)

Answer

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67 Calculate the product using partial products. 200(6.5)

Answer

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68 Calculate the area of the rectangle using partial products.

300 43.9 Answer

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Multiplying Decimals

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SLIDE 23

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Convert the following decimal numbers into fractions. 0.7 x 0.09 What is the product? 63 1000 We multiplied seven tenths by nine hundredths. What place value will the last digit in the product be in if we convert it into a decimal number?

Thousandths

click click

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Try these! What place value will the last digit be in for the following problems? Don't forget to convert them to fractions first. Fractions Product Place Value 1) 0.3 x 0.7 2) 0.2 x 0.13 3) 0.08 x 0.231

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Do you notice a pattern for multiplying decimals? 3.5 x 1.72 3 5 10 1 72 100 x 35 10 x 172 100 6020 1000 Where does the decimal point go? Drag the decimal point.

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If you know how to multiply whole numbers then you can multiply decimals. Just follow these few steps. Step 1: Ignore the decimal points. Step 2: Multiply the numbers using the same rules as whole numbers. Step 3: Count the total number of digits to the right of the decimal points in both numbers. Put that many digits to the right of the decimal point in your answer.

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Multiplying Decimals

3.21 x .04 .1284 There are a total of four digits to the right of the decimal points. There must be four digits to the right

f the decimal point in the answer.

}

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Estimating Your Answer Before any calculations, estimate your answer to make sure you are on the right track. 23.2 x 4.04 What place value should we round to? Round to the nearest whole number. 23.2 rounds to ____ 4.04 rounds to ____ Our answer should approximately be ...

92

click

SLIDE 24

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23.2 x 4.04 928 92800

0000

93.728

}

There are a total of three digits to the right of the decimal points. There must be three digits to the right of the decimal point in the answer. Estimating helps us recognize where the decimal point belongs! EXAMPLE

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Estimate your answer for the following problem by rounding the numbers to the nearest whole number. 9.5 x 0.05 9.5 rounds to _____ 0.05 rounds to _____ What is your estimate? For problems like these, use your number sense! You are multiplying 9.5 by 0.05 which means you are taking a part (fraction) of 9.5. So your answer must be smaller than 9.5!

click

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smaller than 3.214 TRY THESE. Estimate the following products in your notebook then check with the rest of your group. To check your answer, click the box. 1) 14.512 2) 8.31 x 4.21 x 1.008 3) 7.0045 4) 3.214 x 5.2 x 0.0034

Click to Reveal

15 x 4 = 60 8 x 1 = 8 7 x 5 = 35

Click to Reveal Click to Reveal Click to Reveal

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TRY THESE. Complete in your notebook then check with the rest of your group. To check your answer, click the box. 1) 14.512 2) 8.31 x 4.21 x 1.008 14512 290240 5804800 61.09552 3) 7.0045 4) 3.214 x 5.2 x 0.0034 140090 3502250 36.42340

Click to Reveal Click to Reveal

6648 0000 00000 831000 8.37648

Click to Reveal

12856 96420 0.0109276

Click to Reveal

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69 Estimate the product 0.42 x 0.032

A

The product will be less than 1

B

The product will be equal to 1

C

The product will be greater than 1

Answer

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70 The product of 0.42 x 0.032 will have 4 digits to the right of the decimal point.

True False Answer

SLIDE 25

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71 Multiply 0.42 x 0.032

Answer

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72 Multiply 3.452 x 2.1

Answer

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73 You need to buy 6 notebooks that cost $0.87 each. If you have $5, do you have enough money? Estimate to determine your answer. Do not solve.

Yes No Answer

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74 You need to buy 6 notebooks that cost $0.87 each. How much will this cost?

Answer

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75 Multiply 53.24 x 0.089

Answer

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76 The regular price of a pair of jeans is $29.99.

Mrs. Jones has four children

for whom she must buy new jeans. The jeans are on sale for $22.50. What would the total cost be of four pairs of jeans on sale?

A

$119.96

B

$90.00

C

$86.00

D

$52.49

Answer

SLIDE 26

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77 How many digits will be to the right of decimal point in the product for the problem 4.0156 x 7.8?

A

2

B

3

C

4

D

5

Answer

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78 Multiply 4.0156 x 7.8

Answer

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79 Multiply 0.012 x 0.21

Answer

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Dividing Decimals

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Slide 155 / 315 Divide Decimals by Whole Numbers

56.08 28 04 2 Step 1: Use long division. Step 2: Bring the decimal point up into the quotient.

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Try this! Answer

SLIDE 27

Slide 157 / 315 The Power of Ten

10

Multiplying by a power of ten makes dividing by decimals easier! 1) 13 x 10 = _______ 2) 94 x 100 = _______ 3) 28 x 1000 = _______ 4) 6.2 x 10 = _______ 5) 4.78 x 100 = _______ 6) 51.293 x 1000 = _______ Do you see a pattern for multiplying by a power of ten? The decimal point moves to the right depending

n the number of zeros in the power of ten!

Click to Reveal

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Dividend Divisor Step 1: Change the divisor to a whole number by multiplying by a power of 10. Step 2: Multiply the dividend by the same power of 10. Step 3: Use long division. Step 4: Bring the decimal point up into the quotient.

Divide by Decimals

Quotient

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15.6 6.24 Multiply by 10, so that 15.6 becomes 156 6.24 must also be multiplied by 10 156 62.4 .234 23.4 Multiply by 1000, so that .234 becomes 234 23.4 must also be multiplied by 1000 234 23400 Try rewriting these problems so you are ready to divide!

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6789.21 09 415 25020 Rewrite each problem after multiplying by a power of 10. 1) 2) 3) 4) 4.15 250.2 .008 0.9 68.342 4.2 678.921 2.2

Click to Reveal

4200 008

Click to Reveal Click to Reveal

22 683.42

Click to Reveal

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Estimating Your Answer Before any calculations, estimate your answer to make sure you are on the right track. 23.2 ÷ 4.04 What place value should we round to? Round to the nearest whole number. 23.2 rounds to ____ 4.04 rounds to ____ Our answer should approximately be ...

5

click

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4.04 23.2 Try this! Be sure to round your answer to the thousandths. 5.743

click

SLIDE 28

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Estimate your answer for the following problem by rounding the numbers to the nearest whole number. 9.5 ÷ 0.05 9.5 rounds to _____ 0.05 rounds to _____ What is your estimate? For problems like these, use your number sense! If you are dividing 9.5 by 0.05, then does that mean the quotient will be smaller than 9.5 or greater than 9.5? Your answer must be greater than 9.5!

click

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80 Divide

0.78 ÷ 0.02 =

Answer

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81 Use estimation to figure out if the quotient will be A less than 4.866 B around 4.866 C greater than 4.866

Answer

4.866 ÷ 0.6

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82

0.6 4.866

Answer

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83

10 divided by 0.25 =

Answer

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84

12.03 ÷ 0.04 =

Answer

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85

0.012 24.6

Answer

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86 Estimate.

Answer

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87 Evaluate.

Answer

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88 Estimate.

Answer

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89 Evaluate.

Answer

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There are two types of decimals - terminating and repeating. A terminating decimal is a decimal that ends. All of the examples we have completed so far are terminating. A repeating decimal is a decimal that continues forever with

ne or more digits repeating in a pattern.

To denote a repeating decimal, a line is drawn above the numbers that repeat. However, with a calculator, the last digit is rounded.

SLIDE 30

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Let's consider the following... Click to Reveal

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Continue dividing from the problems you set up earlier. 3) 4)

6600 2342 2200 14200 13200 10000 8800 12000 11000 10000 8800 12000 11000 63 48 45 39 36 32 27 51 45 60 54 6

Click to Reveal Click to Reveal

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90

Answer

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91

Answer

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92

You need to put some gas in your car. Regular gasoline is $3.59 per gallon. You only have a $20 bill on you. How many gallons can you buy?

Answer

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93

A

2.27

B

22.73

C

22.7

D

22.72

Answer

SLIDE 31

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94

Answer

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95

If 6 people are on an elevator and together they weight 931.56 pounds, find the average weight of each person.

Answer

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96

Answer

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97

Heather has 5.5 lbs of jelly beans. She will put them in 8.5 bags. How much will be in each bag? Answer

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98

Answer

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99

Answer

SLIDE 32

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100

Texas suffered through a heat wave in August 2011. The highest four temperatures (in degrees Fahrenheit) were 103.4, 102.8, 101.9 and 102.5. What was the average temperature for those four days? Answer

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101

For your sewing project at school, you need to purchase 3.5 yards of fabric. You spend $9.10 on one pattern and $8.40 on

another. How much does one yard cost?

Answer

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102

A 40.9 B 40.90 C 40.91 D 40.9 Answer

Slide 190 / 315 Even and Odd Numbers

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Warm-Up Exercise Think about the following questions and write your answers in your notes. 1) What is an even number? 2) List some examples of even numbers. 3) What is an odd number? 4) List some examples of odd numbers. Answer

Derived from

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What happens when we add two even numbers? Will we always get an even number?

SLIDE 33

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Drag the paw prints into the box to model 6 + 8 + Circle pairs of paw prints to determine if any of the paw prints are left over. Will the sum be even or odd every time two even numbers are added together? Why or why not?

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Drag the paw prints into the box to model 9 + 5 + Circle pairs of paw prints to determine if any of the paw prints are left over. Will the sum be even or odd every time two odd numbers are added together? Why or why not?

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Drag the paw prints into the box to model 7 + 8 + Circle pairs of paw prints to determine if any of the paw prints are left over. Will the sum be even or odd every time an odd and even number are added together? Why or why not? If the first addend was even and the second was odd, then would your answer change? Why or why not?

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103 The product of two even numbers is even.

True False Answer

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Explain your answer.

104 The product of two odd numbers is

A

dd

B

even

Multiplication is repeated addition. If you add an odd number

ver and over, then the sum will switch between even and
dd. Since you are adding the number an odd number of times,

your product will be odd. Click to Reveal Answer

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105 The product of 13 x 8 is

A

dd

B

even

Explain your answer. 13 x 8 is equivalent to saying 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13. Since you are adding it an even number of times, the product will be even. Click to Reveal Answer

SLIDE 34

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106 The sum of 32,877 + 14,521 is

A

dd

B

even

Explain your answer. If you model the numbers using dots and circle all the pairs, the single dots leftover from each number will create a pair and none will be leftover making the sum an even number. Click to Reveal Answer

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107 The product of 12 x 9 is

A

dd

B

even

Explain your answer. 12 x 9 is equivalent to 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12. No matter how many times you add 12, since it is even the sum will always be even. Click to Reveal Answer

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108 The sum of 8,972 + 1,999 is

A

dd

B

even

Explain your answer. If you model the problem using dots and circle all the pairs, then there will be one dot leftover since one of the addends is odd. Click to Reveal Answer

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109 The sum of 9 + 10 + 11 + 12 + 13 is

A

dd

B

even

Answer Explain your answer. The first two addends will result in an odd number. By adding another odd number, the sum is even. Adding an even number will result in an even number. Since the last addend is odd, the final answer will be odd. Click to Reveal

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110 The product of 250 x 19 is

A

dd

B

even

Explain your answer. The product of an odd and even number will always result in an even number. Click to Reveal Answer

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111 The product of 15 x 0 is

A

dd

B

even

Explain your answer. 0 is an even number and the product of any even number and

dd number is always even.

Click to Reveal Answer

SLIDE 35

Slide 205 / 315 Divisibility Rules for 3 and 9

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Let's review! Below is a list of numbers. Drag each number in the circle(s) that is a factor of the number. You may place some numbers in more than one circle. 2 8 4 10 5 24 36 80 115 214 360 975 4,678 29,785 414,940

Derived from

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Divisibility Rules 2: If and only if its last digit is 0, 2, 4, 6, or 8. 4: If and only if its last two digits are a number divisible by 4. 5: If and only if its last digit is 0 or 5. 8: If and only if its last three digits are a number divisible by 8. 10: If and only if its last digit is 0.

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Divisibility Rule for 3 What factor do the numbers 12, 15, 27, and 66 have in common? They are all divisible by 3. Now, take each of those numbers and calculate the sum of its digits. 12 1 + 2 = 3 15 ________ 27 ________ 66 ________ What do all these sums have in common? They are all divisible by 3!

Click Click

A number is divisible by 3 if the sum of the number's digits is divisible by 3.

Click

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Divisibility Rule for 9 What factor do the numbers 18, 27, 45, and 99 have in common? They are all divisible by 9. Now, take each of those numbers and calculate the sum of its digits. 18 1 + 8 = 9 27 ________ 45 ________ 99 ________ What do all these sums have in common? They are all divisible by 9!

Click Click

A number is divisible by 9 if the sum of the number's digits is divisible by 9.

Click

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Try these! Check if the numbers in the chart are divisible by 3 or 9. Put a check mark in the box in the correct column.

Divisible by 3 Divisible by 9 228 531 735 1,476

SLIDE 36

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112 Is 135 divisible by 3?

Yes No Answer

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113 129 is divisible by 9.

True False Answer

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114 Is 54 divisible by 3 and 9?

Yes No Answer

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115 Which of the following numbers is divisible by 3, 4 and 5?

A

45

B

54

C

60

D

80

Answer

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116 120 is divisible by: (choose all that apply)

A

2

B

3

C

4

D

5

E

8

F

9

G

10

Answer

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117 126 is divisible by: (choose all that apply)

A

2

B

3

C

4

D

5

E

8

F

9

G

10

Answer

SLIDE 37

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118 468 is divisible by: (choose all that apply)

A

2

B

3

C

4

D

5

E

8

F

9

G

10

Answer

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119 Is any number divisible by 9 also divisible by 3? Explain.

Yes No Answer

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120 Any number divisible by 3 is also divisible by 9.

True False Answer

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121 Is 24,981 divisible by 3? If it is, type the quotient. If it is not, type 00.

Answer

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122 Is 15,516 divisible by 9? If it is, type the quotient. If it is not, type 00.

Answer

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Greatest Common Factor

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SLIDE 38

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Review of factors, prime and composite numbers Interactive Website Play the Factor Game a few times with a partner. Be sure to take turns going first. Find moves that will help you score more points than your partner. Be sure to write down strategies or patterns you use or find. Answer the Discussion Questions.

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Player 1 chose 24 to earn 24 points. Player 2 finds 1, 2, 3, 4, 6, 8, 12 and earns 36 points. Player 2 chose 28 to earn 28 points. Player 1 finds 7 and 14 are the only available factors and earns 21 points. (Rows and Columns can be adjusted prior to starting the game)

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Discussion Questions

1. Make a table listing all the possible first moves, proper

factors, your score and your partner's score. Here's an example:

2. What number is the best first move? Why?
3. Choosing what number as your first move would make you

lose your next turn? Why?

4. What is the worst first move other than the number you

chose in Question 3?

First Move Proper Factors My Score Partner's Score 1 None Lose a Turn 2 1 2 1 3 1 3 1 4 1, 2 4 3

Least Common Multiple

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SLIDE 42

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Text-to-World Connection

1. Use what you know about factor pairs to evaluate George

Banks' mathematical thinking. Is his thinking accurate? What mathematical relationship is he missing?

2. How many hot dogs came in a pack? Buns?
3. How many "superfluous" buns did George Banks remove from

each package? How many packages did he do this to?

4. How many buns did he want to buy? Was his thinking correct?

Did he end up with 24 hot dog buns?

5. Was there a more logical way for him to do this? What was he

missing?

6. What is the significance of the number 24?

(Click for Link to Video Clip) Teacher Notes

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A multiple of a whole number is the product of the number and any nonzero whole number. A multiple that is shared by two or more numbers is a common multiple. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ... Multiples of 14: 14, 28, 42, 56, 70, 84,... The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is 42.

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There are 2 ways to find the LCM:

1. List the multiples of each number until you find the first
ne they have in common.
2. Write the prime factorization of each number. Multiply

all factors together. Use common factors only once (in

ther words, use the highest exponent for a repeated

factor).

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EXAMPLE: 6 and 8 Multiples of 6: 6, 12, 18, 24, 30 Multiples of 8: 8, 16, 24 LCM = 24 Prime Factorization: 6 8 2 3 2 4 2 2 2 2 3 23 LCM: 23 3 = 8 3 = 24

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Find the least common multiple of 18 and 24. Multiples of 18: 18, 36, 54, 72, ... Multiples of 24: 24, 48, 72, ... LCM: 72 Prime Factorization: 18 24 2 9 6 4 2 3 3 3 2 2 2 2 32 23 3 LCM: 23 32 = 8 9 = 72

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133 Find the least common multiple

f 10 and 14.

A 2 B 20 C

70

D

140

Answer

SLIDE 43

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134 Find the least common multiple

f 6 and 14.

A 10 B 30 C

42

D

150

Answer

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135 Find the least common multiple

f 9 and 15.

A 3 B 45 C 60 D 135 Answer

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136 Find the least common multiple

f 6 and 9.

A 3 B 12 C

18

D

36

Answer

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137 Find the least common multiple

f 16 and 20.

A 80 B

100

C

240

D

320

Answer

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138 Find the LCM of 12 and 20.

Answer

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139 Find the LCM of 24 and 60.

Answer

SLIDE 44

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140 Find the LCM of 15 and 18.

Answer

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141 Find the LCM of 24 and 32.

Answer

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142 Find the LCM of 15 and 35.

Answer

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143 Find the LCM of 20 and 75.

Answer

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Uses a venn diagram to find the GCF and LCM for extra practice.

Interactive Website Slide 264 / 315 GCF and LCM Word Problems

Return to Table of Contents

SLIDE 45

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How can you tell if a word problem requires you to use Greatest Common Factor or Least Common Multiple to solve?

Slide 266 / 315 GCF Problems

Do we have to split things into smaller sections? Are we trying to figure out how many people we can invite? Are we trying to arrange something into rows

r groups?

Slide 267 / 315 LCM Problems

Do we have an event that is or will be repeating over and over? Will we have to purchase or get multiple items in order to have enough? Are we trying to figure out when something will happen again at the same time?

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Samantha has two pieces of cloth. One piece is 72 inches wide and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips? What is the question: How wide should she cut the strips? Important information: One cloth is 72 inches wide. The other is 90 inches wide. Is this a GCF or LCM problem? Does she need smaller or larger pieces? This is a GCF problem because we are cutting or "dividing" the pieces of cloth into smaller pieces (factor) of 72 and 90.

click

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90 inches Use the greatest common factor to determine the greatest width possible. The greatest common factor represents the greatest width possible not the number of pieces, because all the pieces need to be of equal length. 72 inches 18 inches

Bar Modeling

click

Teacher Notes

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Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel both exercised today. How many days will it be until they exercise together again? What is the question: How many days until they exercise together again? Important information: Ben exercises every 12 days Isabel exercises every 8 days Is this a GCF or LCM problem? Are they repeating the event over and over or splitting up the days? This is a LCM problem because they are repeating the event to find out when they will exercise together again.

click

SLIDE 46

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Ben exercises in: Isabel exercises in:

Bar Modeling

Use the least common multiple to determine the least amount of days possible. The least common multiple represents the number of days not how many times they will exercise. 12 Days 8 Days Teacher Notes

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144 Mrs. Evans has 90 crayons and 15 pieces of paper to give to her students. What is the largest number

f students she can have in her class so that each

student gets an equal number of crayons and an equal number of paper?

A

GCF Problem

B

LCM Problem

Answer

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Answer

145 Mrs. Evans has 90 crayons and 15 pieces of paper

to give to her students. What is the largest number

f students she can have in her class so that each

student gets an equal number of crayons and an equal number of paper?

A 3 B

5

C

15

D

90

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146 How many crayons and pieces of paper does each

student receive if there are 15 students in the class?

A 30 crayons and 10 pieces of paper B

12 crayons and pieces of paper

C

18 crayons and 6 pieces of paper

D

6 crayons and 1 piece of paper

Challenge problems are notated with a star.

Answer

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147 Rosa is making a game board that is 16 inches by

24 inches. She wants to use square tiles. What is the largest tile she can use?

A GCF Problem B

LCM Problem

Answer

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148 Rosa is making a game board that is 16 inches by

24 inches. She wants to use square tiles. What is the largest tile she can use?

Answer

SLIDE 47

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149 How many tiles will she need? Answer

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150 Y100 gave away a $100 bill for every 12th caller.

Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $100 bill and a concert ticket?

A GCF Problem B

LCM Problem

Answer

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151 Y100 gave away a $100 bill for every 12th caller.

Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $100 bill and a concert ticket?

A 36 B

3

C

108

D

6

Answer

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152 There are two ferris wheels at the state fair. The

children's ferris wheel takes 8 minutes to rotate

fully. The bigger ferris wheel takes 12 minutes to

rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris

wheel. If they both start at the bottom, how many

minutes will it take for both of them to meet at the bottom at the same time?

A GCF Problem B

LCM Problem

Answer

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153 There are two ferris wheels at the state fair. The

children's ferris wheel takes 8 minutes to rotate

fully. The bigger ferris wheel takes 12 minutes to

rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris

wheel. If they both start at the bottom, how many

minutes will it take for both of them to meet at the bottom at the same time?

A 2 B

4

C

24

D

96

Answer

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154

How many rotations will each ferris wheel complete before they meet at the bottom at the same time? (Input the answer for the small ferris wheel.)

Answer

SLIDE 48

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155

Sean has 8-inch pieces of toy train track and Ruth has 18-inch pieces of train track. How many of each piece would each child need to build tracks that are equal in length?

A GCF Problem B

LCM Problem

Answer

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156

What is the length of the track that each child will build?

Answer

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157

I am planting 50 apple trees and 30 peach trees. I want the same number and type of trees per row. What is the maximum number of trees I can plant per row?

A GCF Problem B

LCM Problem

Answer

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Glossary

Return to Table of Contents

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Algorithm

A step-by-step process to find a solution.

It's like a cooking recipe for mathematics.

24 + 12 =

Add the ones then add the tens

How to...

Step 1: Step 2: Step 3:

Back to Instruction

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Average

3 + 4 + 2 = 9

= 9 3 = 3

The value/amount of each item when the total is distributed across each item equally.

Back to Instruction

SLIDE 49

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Back to Instruction

Bar Model

A diagram that uses bars to show the relationship between two or more numbers.

Whole

Part Part

Part + Part = Whole Whole - Part = Part Larger Amount

Smaller Amount

Difference

Large - Difference = Small Large - Small = Difference

Part Whole One part Whole # of parts

x

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Back to Instruction

Complex Fraction

A fraction whose numerator or denominator or both contain fractions.

3

1 5 1 5 2 3

=3

1 5

=

1 5 2 3

Must be written as a fraction.

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Back to Instruction

Composite Number

A number that has more than two factors. 12

1 x 12 2 x 6 3 x 4

6 factors

3 x 5 = 15

Any number with factors other than

ne and itself is

composite.

13

1 x 13

Only 2 factors. Slide 292 / 315

Cross Simplify

Used to make operations with fractions easier. Divide the numerator of one fraction and the denominator of another fraction by their GCF.

1 5 15 20 + = 3 20

1+

1 5 15 20 +

1 3

GCF of 5 and 15 is 5.

Back to Instruction

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Back to Instruction

Distributive Property

5 (3 + 2) 3

3x5=3(3+2)

2(3+4)= (2x3)+(2x4)

2

3 4

a(b+c)=ab+ac

Multiplying a sum by a number is the same as multiplying each addend in the sum by the same number and then adding the products.

a(b-c)=ab-ac

also applies to subtraction

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Back to Instruction

Dividend

24 8 = 3

24 8 3 24 8 = 3

Dividend Dividend Dividend

The number being divided in a division equation.

SLIDE 50

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Back to Instruction

Divisor

24 8 = 3

24 8 3

25 8 = 3 R1

Divisor Divisor

The number the dividend is divided by. A number that divides another number without a remainder.

Must divide evenly.

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Back to Instruction

Exponent

A small, raised number that shows how many times the base is used as a factor.

3

2

Base

Exponent

3

2= x

3 3

3 = x x 3 3 3

3

2 x

2 3 3

3 x

3 3

"3 to the second power"

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Back to Instruction

Factor

A whole number that can divide into another number with no remainder.

15 3 5

3 is a factor of 15

3 x 5 = 15

3 and 5 are factors of 15

16 3 5 .1

R 3 is not a factor of 16

A whole number that multiplies with another number to make a third number.

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Back to Instruction

Greatest Common Factor (GCF) The largest number that will divide two or more numbers without a remainder.

12: 1, 2, 3, 4, 6, 12 16: 1, 2, 4, 8, 16 Common Factors

are 1, 2, 4

GCF is 4

12 = 2 x 2 x 3 16 = 2 x 2 x 2 x 2

GCF = 2 x 2

GCF is 4

Using Prime Factorization

1 and 2 are common factors, but not the greatest common factor.

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Back to Instruction

Least Common Multiple (LCM)

The smallest number that two

r more numbers share as a

multiple.

9 = 3 x 3 15 = 3 x 5

LCM = 3 x 3 x 5

LCM is 45

Using Prime Factorization

9: 9, 18, 27, 36, 45 15: 15, 30, 45

LCM is 45

2: 2, 4, 6, 8 4: 4, 8

4 is the LCM, not 8

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Back to Instruction

Multiple

The product of two whole numbers is a multiple of each of those numbers.

3 x 5 = 15

15 is a multiple of 3.

2 x 6 = 12

Factors Product /

Multiple

4 x 5 = 20

5 and 4 are factors of 20, not multiples.

SLIDE 51

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Back to Instruction

10

1 10

=

Power of 10

Any integer powers of the number ten. (Ten is the base, the exponent is the power.

10

2 100

=

10

3 1,000

=

10x10 = 10 = 10x10x10 = Slide 302 / 315

Prime Factorization

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A number written as the product

f all its prime factors.

18 = 2 x 3 x 3

18 = 2 x 3

2

18 = 1 x 2 x 3 x 3

Only prime numbers are included in prime factorizations.

There is

nly one

for any number.

r

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Back to Instruction

Prime Number

A positive integer that is greater than 1 and has exactly two factors, one and itself.

1

One is not a prime number, because it has

nly one factor.

2

Two is the only even prime number.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Prime #s to 30

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Back to Instruction

Profit

The difference between the amount earned and the amount spent.

Earned Spent Profit

______

$30 Washing

Cars

$12

______

$18

Supplies Profit

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Proper Factor

Back to Instruction

All of the factors of a number

ther than one and itself.

6: 1, 2, 3, 6

Proper Factors: 2 and 3

9: 1, 3, 9

Proper Factor: 3

7: 1, 7

The number 7 does not have any proper factors.

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Back to Instruction

Quotient

The number that is the result of dividing one number by another.

12 4 3 =

Quotient

12 4 3

Quotient

12

4 3 =

Quotient

SLIDE 52

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Reciprocal

One of two numbers whose product is one. 1 x 1 = 1

1 is the reciprocal of 1.

2 x 1

2 = 1

Number Reciprocal

r x r = 1

Back to Instruction

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Back to Instruction

Relatively Prime

Two numbers who only have 1 as a common factor.

8: 1, 2, 4, 8 15: 1, 3, 5

Only Common Factor is 1 All prime numbers are relatively prime to every other number.

9: 1, 3, 9 15: 1, 3, 5, 15

Common Factors: 1 and 3

Slide 309 / 315

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Repeating Decimal

A decimal with a digit or group

f digits that repeats endlessly.

3 1.0

.3

9

___

1 3

___

9

1

3 9

___

1

...

1 3 = .3

__

7 33 = .21

__

(.212121...)

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Terminating Decimal

A decimal that ends and doesn't go

n forever.

3 1.0

.3

9

___

1 3

___

9

1

3 9

___

1

...

1/2 = .5

3/8 = .375

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Vertical

In an up-down position. vertical horizontal diagonal Slide 312 / 315

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