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A Basic Arithmetic Exam Trainer Starter Set

© Andre Lanzaro, modemics.com, 2016

The success of math course gaming rests upon the effectiveness of Exam

Trainers to prepare “mathletes” to score exceedingly well on math course exams.

The central issue of Exam Trainers is whether terrible math students, who are

terrible math “studiers,” can master critical math skills by methods which mimic

sport “training” Sport training requires short quick trials and attempts followed by

immediate correction, rapid retry, practice, practice, and more practice. Sport

training cannot be generally successful without the people power of a motivating

infrastructure. A motivating infrastructure, similarly, cannot be successful without

effective training methodologies—which further emphasizes the critical role Exam

Trainers play in gaming math courses.

The importance of Exam Trainers necessitates that interested developers get to

see some real life examples that might suffice as a starting guide. What follows is

offered to serve such a purpose, and to address a second important related issue

dealing with weakness in basic math skills.

By ”starter set” is meant a set of Exam Trainers that is bare bones in the sense

that it is marginally sufficient to convey conceptual understanding, but definitely

too meager to provide that all important intensive training effect to assure math

exam success. Mathletes and their coaching support must “bulk up” a starter set by

greatly increasing the number of its test questions and problems in order to put

“training” power into each Exam Trainer. The additional problems needed to

provide an adequate training effect should be taken from a math text appropriate

for the student’s math course as well as the student’s level of need. The Exam

Trainer Starter Set, below, is offered to also provide useful insight into Exam

Trainer production.

The student will quickly observe that for the more complex problem

“executions” the starter set Exam Trainers presented below provide “enhanced

executions.” Enhanced executions go beyond executing a problem’s prescription

as specified by the solution’s Exam Trainer’s “prescription” section. Exam

Trainer Start Set executions are “enhanced,” where useful, by appending

illustrative commentary (replete with pointing arrows) demonstrating how the steps

shown carry out the problem’s specified prescription. Execution enhancements

are suitable for a starter set of Exam Trainers but do not provide a benefit worth

the additional effort for the larger number of problem statement and solution pairs

needed to produce Exam Trainers that deliver a potent training effect.

As typical for Exam Trainers, questions and problems are posed on odd

numbered pages. Their respective answers and solutions are posed on the

succeeding even number page maintaining flashcard alignment. If each odd

numbered page were to have its succeeding even numbered page printed on its

backside, then every question/problem would be flashcard aligned with its

answer/solution. The presentation below can be converted into flashcards using its

Microsoft Word rendition. [Obtain flashcard page sizing by going to Microsoft

Word’s “Page Layout” ribbon, clicking on its “Size” icon, and then selecting “3 X

5 index” sheet size.]

Pages that present CSU questions/problems are referenced by the particular

CSU code number they test. However, the verbal title of the particular CSU code

under test is posted on the answer side of its Exam Trainer sheet. The restriction to

reference a CSU by its coding (math text chapter number “dot” section number) on

2

its Exam Trainer’s question/ problem statement page title aims to minimize context

clues such as the problem’s source chapter title or section title.

If a student is enrolled in a math course that demands greater mathematical

prowess then the student possesses, the student’s math course knowledge space

must be expanded to incorporate the student’s math course prowess deficit. The

expansion of the student’s math course knowledge space is necessary to

realistically represent, track, and guide the student’s math course learning progress.

It clearly illustrates the simple but often hard to see fact that to pass the course the

student has to learn more than what the course teaches. Identifying and

developing the required remedial CSU Exam Trainers would generally require

working with the course teacher, professional tutors and coaches. Here is where a

Math Corps group can lend some timely and highly efficient assistance.

The starter set below starting with absolutely basic arithmetic CSUs, and

progressing as far as “percents,” can serve to strengthen basic arithmetical skills by

supplying additional remedial knowledge space to motivate and track the needed

training. The starter set as mentioned above must be “bulked up” with sufficient

skill practice problems to provide the desired skill engendering training effect.

A listing of our starter set Exam Trainers CSUs is given below. The subject

matter associated with each CSU code is also indicated. The starter set follows the

CSU code listing commencing on the next odd numbered page maintaining

flashcard alignment between its question/answers, and their respective

answer/solutions on the following even numbered page.

The reader is encouraged to contact us through www.modemics.com or via

email at [email protected] to resolve confusions and offer suggestions.

3

mailto:[email protected]

http://www.modemics.com/

Basic Arithmetic Starter Exam Trainers Set CSU Codes.

1. CSU 1.1: Whole Numbers/Place Value Numbering

2. CSU 1.2 : Rounding Place Value Numbers

3. CSU 1.3: Addition of Whole Numbers

4. CSU 1.4: Subtraction of Whole Numbers

5. CSU 1.5: Multiplication of Whole Numbers

6. CSU 1.6: Division of Whole Numbers

7. CSU 1.7: Whole Numbers to Exponential Power

8. CSU 1.7: Fractions/Fundamentals

9. CSU 2.2: Fractions/Prime Numbers

10. CSU 2.3: Fractions/ Multiplying

11. CSU 2.4: Fractions/Division

12. CSU 2.5: Fractions/Adding and Subtracting

13. CSU 2.6: Fractions/ Mixed Numbers

14. CSU 2.7: Fractions/Multiplying and Dividing Mixed Numbers

15. CSU 2.8: Fractions/ Adding and Subtracting Mixed Numbers

16. CSU 3.1: Decimal Numbers

17. CSU 3.2: Decimals/Adding and Subtracting

18. CSU 3.3: Decimals/ Multiplication

19. CSU 3.4: Decimals/Division

20. CSU 3.5: Decimals/Fractions

21. CSU 4.1: Percents

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CSU 1.1

What is a natural number?

What is a whole number?

What is a digit?

What is a numeral?

We use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9: each of which represents a whole number.

How do we represent whole numbers that are bigger than 9 using only these digits?

What is the name of this system that allows us to represent numbers bigger than 9 without inventing a new symbol or digit?

How do you represent the next whole number after nine using two digits?

How do you represent eleven using place valued digits?

How do you represent twelve using place value?

If our number system is called “place value” what is the “value” of each numerical place?

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CSU 1.1: Whole Numbers/Place Value Numbering

A natural number is numbers we use for counting. We start with one and go to two, three, etc. So 1,2,3, and so on are the natural numbers. They were first called natural because it was seen that they occur in nature.

A whole number is a natural number but includes and starts with zero. Zero was not considered natural when it first came out.

A digit is one of the first ten whole numbers. The ten digits we use are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and no more.

A numeral is the written form of a number. Note: numbers are concepts that exist only in our heads. Numerals we can see.

Putting one digit in front of another allows us a way to represent whole numbers bigger than 9 without having to create a new symbol.

This system of representing numbers larger than 9 is called “Place Value.”

Adding digit “1” to digit “9” gives us a whole number called ten. o We can write ten using two digits one in front of the other as “10.”

Adding “1” to “10 gives us the whole number eleven.o We write eleven using two digits one in front of the other as “11.” represented as “11.”

Adding “1” to “11” gives us twelve.o Twelve is represented as “12.”

The value of the first place is “1.” The value of second digit in front of the first going from right to left is 10. The value of place in from 10 is, 100. In front of 100 the place value is, 1000, followed by 10,000, etc.

6

CSU 1.1

Show how 347 can represent a number much bigger than 9 using the symbols (digits) 3, 4, and 7. Express it is a sum of numbers times their place values in 347.

It is easy write a big number using place value but how do we say big numbers in words?

Light travels approximately 5,865,696,000,000 miles in one year.

How do we say that distance in English words?

Sometimes we are given a big number specified in words. To do math with this number we need to translate the word form of the number into its digit form. Explain through Prescription and Execution how to do the following.

Write each number with digits .

a. Three million, fifty–one thousand, seven hundredb. Two billion fivec. Seven million, seven hundred seven

7

CSU 1.1: Place Value Numbering

347 is really 300 + 40 + 7 due to the place value of its digits

3 is in the 100s place4 is in the 10s place7 is in the 1s place

300 40

347 = 300 + 40 +7 = + 7 347

Prescription and Execution:

1. Break down the number into groups of three places by putting a comma after every third number going from right to left5,865,696,000,000 <<< number groups

2. Know the name of each number group of three place values as for example:trillions, billions, millions, thousands, ones <<< number group names

3. Match the number groups with their respective number group names

trillions, billions, millions, thousands, ones Every 3 places is a 5, 865, 696, 000, 000 named number group

4. Then say the big number by telling the number in each number group name going from left to right as follows.5 trillion, 865 billion, 696 million

Diagnosis: Whole Numbers/Place ValuePrescription:

1. Write down or think of the number group names as followstrillions, billions, millions, thousands, ones

2. Write the digits given in the worded number under the respective number group name. Make sure that under each number group name three digits are used after the first digit is placed at the left

Execution: Number Group NamesHere are the solutions to a. b. and c,

trillions, billions, millions, thousands, onesa. 3, 051, 700…………..……ans: 3,051,700

Note: In this one you had to use 3 digits to fill thousands and ones b. 2, 000, 000, 005………………..ans: 2,000,000,005

Note: You had to use zeros to fill unused number groups c. 7, 000, 707…………………ans: 7,000,707

8

CSU 1.1

Give the place value of each digit in the number 2, 781, 609, 453.

Show 24,781,670,499 as a vertical sum of its digits times their respective place values.

When you see a place value number what mathematical operation are you also seeing?

What is meant by writing a number in “expanded form?”

9


Diagnosis: Place Value Problem / specifying place value of each digit.

Prescription:1. To find the place value of a digit in a large number write down 1 and put as many zeros after it as there are places to the right of the

digit.Execution:

1. The place value of 2 in 2, 781, 609, 453 is 1,000,000,000 or 1 trillion2. The place value of 7 in 2, 781, 609, 453 is 100,000,000 or 1 hundred million3. The place value of 8 in 2, 781, 609, 453 is 10,000,000 or 10 million4. The place value of 1 in 2, 781, 609, 453 is 1,000,000 or 1 million5. The place value of 6 in 2, 781, 609, 453 is 100,000 or one hundred thousand6. The place value of 0 in 2, 781, 609, 453 is 10,000 or ten thousand7. The place value of 9 in 2, 781, 609, 453 is 1,000 or one thousand8. The place value of 4 in 2, 781, 609, 453 is 100 or one hundred9. The place value of 5 in 2, 781, 609, 453 is 10 or ten10. The place value of 3 in 2, 781, 609, 453 is 1 or one

Diagnosis: Place Value Problem/ specifying place value of each digit.

Prescription:1. Build a vertical sum of numbers where each number in the sum is a digit of the large number multiplied by its place value

Execution:1. 20,000,000,000

4,000,000,000 700,000,000 80,000,000 1,000,000 600,000 70,000 0,000

400 90

+ 9 24,781,670,499

When you see a place value number you are seeing a sum of numbers.

Writing a number in expanded form means writing it as sum of its digits times their respective place values.

10

All of this is compressed into this.

Isn’t that amazing!!

CSU 1.1

Write 3942 in expanded form.

Also put 517, 644, 073 in expanded form.

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1. Diagnosis: Whole Numbers/ Place Value. Here we represent the number as a sum of its digits times their respective place values. Prescription/Execution:

1. Take the first digit in the number starting from the far left and put as many zeroes after it as it has number places after it.In 3,942, 3 is first digit from the left and has three place after it so we have

30002. Add (“+”) this number to it the next digit to its right with as many zeroes after it as it has number places after it

3000 + 9003. Repeat until all digits in the number have been incorporated in the sum.

3000+900 + 40 + 24. The number3,942 is put it into expanded form as follows

3, 942 = 3000 + 900 + 40 + 2

2. For example, the number 517, 644, 073 in expanded form would be:517, 644, 073 = 500,000,000 + 10,000,000 + 7,000,000 + 600,000 + 40,000 + 4,000+ 0 + 70 +3

CSU 1.2

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With large numbers like the distance light travels in a year, we do not want to say or work with all of its digits. This is tiresome and not really that useful. We would like to say only two or three digits to give a close approximation of how big it is.

Light travels approximately 5,865,696,000,000 miles in one year.

How do we best approximate that number using only three significant digits, and the rest all zeros?

How would we state the abbreviated number verbally?

Round the following to the indicated place value.

Round 6,487 to the nearest hundredth

Round 84 to the nearest tenth

Round 56, 406, 797 to the nearest millionth

Round 969 to the nearest hundredth

13

CSU 1.2: Rounding Place Value Numbers

1. Diagnosis: Whole Numbers/rounding. Prescription:

1. “Round” to the last significant digit you want to keep.2. “Rounding” to a digit in a number means increasing it by one only if the next digit to its right is five or

greater.3. All digits to right of the rounded digit are then set to zero.

Execution:1. To round 5, 865, 696, 000, 000 to three significant places look at the third digit from left2. Next digit is greater than 5

Digit to round to Rounded digit increased by 1The next digit

5, 865, 696, 000, 000 rounds to 5, 870, 000, 000, 000

Remaining digits set to zeros

3. Verbally this number would be stated as “5 trillion, 870 billion.”

Diagnosis: Whole Numbers/rounding. Prescription:

1. Increase the digit you want to round to by if the next digit is 5 or greater. If not leave it unchanged2. Set all digits to the right of the rounded digit to zero.

Execution:1. In 6, 487 round to 4 in the hundredths place. The next digit the right of 4 is 8, which is bigger than 5.

Therefore we add 1 to 4 and replace all digits to its right with zero.2. The answer is: 6,487 to the nearest hundredth is 65003. In 84 the digit to round to is 8 in the tens place. Rounding to 8 we look at the 4 to its right. It is less than

five therefore 8 remains and all other digits are set to zero.4. The answer is: 84 to the nearest tenth is 80

Diagnosis: Whole Numbers/rounding. Prescription:

1. Increase the digit you want to round to by if the next digit is 5 or greater. If not leave it unchanged2. Set all digits to the right of the rounded digit to zero.

Execution:1. In 56,406,797, the millionth place digit is 6. The next digit to its right is 4, which is less than five hence no

change on the 6 and we replace all digits to right of 6 with zeros. 56,406,797 to the nearest millionth is 57,000,000

2. In 969, 9 is in the hundredths place. The next digit to the right of 9 is 6. Therefore to round to the nine we need to add 1 to it and make it 10, all digits after this number are set to zero.

The answer is : 969 to the nearest hundredth is 1000

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CSU 1.3

Draw a larger version of the table below. Within 3minutes put in each box the sum of its column digit and its row digit. See given examples.

1. The result of adding two numbers is called…?

2. What is the mathematical symbol indicating addition?

3. Give another word that means to add.

1. How do we use single digit addition facts to add two numbers that are two digits each?

2. What number do you get to when you add 64 to 23?

CSU 1.3: Addition of Whole Numbers

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+ 7 3 8 4 5 1 2 6 9 081 43 5742590 06

Diagnosis: Whole Numbers/Single digit addition facts.Prescription:

1. Pick an empty box, then quickly enter in the sum of its column digit and its row digit. Execution:

Answers to Questions:

1. A “sum” is the number that is the outcome of adding two or more numbers.

2. The symbol used to indicate addition is a plus sign: “+.”

3. “Plus” also means to add.

Diagnosis: Whole numbers/Addition. Prescription:

1. Line up numbers to be added vertically keeping same place value under same place value.2. Add digits in each place value column using addition facts3. Place sum of each place value column under underline at bottom.4. Bottom place value sums are the sum of numbers to be added.

Execution: Place Value Columns 64

+ 23 6 +2 =8 87 4 + 3 = 7: sum of same place value digits

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+ 7 3 8 4 5 1 2 6 9 08 15 11 1

612 13 9 1

014 17 8

1 8 4 9 5 6 2 3 7 10 13 10 6 1

17 8 4 5 9 12 3

7 14 10 15

11 12 8 9 13 16 7

4 11 7 12

8 9 5 6 10 13 4

2 9 7 10

6 7 3 4 8 11 2

5 12 8 13

9 10 6 7 11 14 5

9 16 12 17

13 14 10 11

15 18 9

0 7 3 8 4 5 1 2 6 9 06 13 9 1

410 11 7 8 12 15 6

CSU 1.3

1. How do you add two numbers of more than two digits each?

2. What is the sum of 246 and 513 (viz. 246 + 513)?

Addition of multi digit numbers involves adding single digits in each place value column.

This works easily so long as the sum of the single digits in a place value column is also a single digit. But, what do we do if the sum of place value digits in place value column is a two digit number?

1. How do we find the sum of two numbers when it contains place value digit sums that are two digit numbers?

2. Add (find the sum of) 46,789 + 2,490 + 864

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CSU 1.3: Addition of Whole Numbers

Diagnosis: Whole numbers/Addition.

Prescription:1. Put one number on top of the other aligning same place value digits2. Place a horizontal line under the columns of digits3. Add the digits in each column using single digit addition facts and enter the sum below the horizontal line

and in the column.Execution: 246

513759 Adding of the digits in the one’s place Adding of the digits in the ten’s place

Adding of the hundred’s place digits

Diagnosis: Whole numbers/ Addition/ /Use of “carrying” Prescription:

1. If the sum of place value digits in a place value column is a two digit number enter the one’s digit of that number under the column and also enter ten’s digit of that number to the top of the place value digits column to the left. This is called “carrying.”

2. Add place value digits with carried digits to complete sum as shown below.

Execution: For example 1 2 1 1 Carried over from the adjoining column5 7, 6 7 6 3, 7 5 0 7 4 66 2, 1 7 2 Sum is 12, 2 is entered here, 1 carried

Sum is 17, 7 is entered here, 1 carried Enter 1 here and carry the 2 to handle 21 Enter2 and carry the 1 to handle the 12 Enter 6, no carry necessary

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CSU 1.4

A sum is the outcome of addition.

1. What is the word meaning the outcome of subtraction?

The operation sign or symbol indicating addition is referred to as “plus” and is written symbolically as “+.”

2. What is the operation sign indicating subtraction and what is it called? What are two other words to indicate subtraction?

The order of the numbers to be added does not change the sum.

Does the order of the numbers to be subtracted make a difference?

Draw the table below. Within 3 minutes put in each box the difference of its column digit and its row digit. See given examples.

- 7 3 8 4 5 1 2 6 9 081 23 17 342590 06

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CSU 1.4: Subtraction of Whole Numbers

Diagnosis: Whole Numbers/ Subtraction

Answers to questions:

1. The outcome of subtraction is referred to as a “difference.” We say that the difference between 8 and 3 is five.

2. The operational sign for subtraction is the minus sign “-“. Minus also means subtraction as in “8 minus 3 is five.” Another way to verbally indicate subtraction is to use the phrase “take away.” For example 8 take away 3 is five. We can also use the word “less.” For example 8 less 3 is five

Diagnosis: Whole Numbers/Subtraction

Answers to questions:

When adding two numbers say 8 and 2 to get ten, the order of adding does not matter. That is 8 + 2 = 10.

We can change the order of adding to 2 + 8 and the sum is also ten.

The order of addition does not change the sum, which means that the order of adding does not matter.

Order of subtraction does matter. Eight minus two is six, but two minus eight is not.

Diagnosis: Single digit subtraction Facts.Prescription:

1. Pick an empty box, then quickly enter in the difference of its column digit and its row digit. Note put the larger of the two numbers first in taking the difference.

Execution:

- 7 3 8 4 5 1 2 6 9 08 1 5 0 4 3 7 6 2 1 81 6 2 7 3 4 0 1 5 8 13 4 0 5 1 2 2 1 3 6 37 0 4 1 3 2 6 5 1 2 74 3 1 4 0 1 3 2 2 5 42 5 1 6 2 3 1 0 4 7 25 2 2 3 1 0 4 3 1 4 59 2 6 1 5 4 8 7 3 0 9

20

0 7 3 8 4 5 1 2 6 9 06 1 3 2 2 1 5 4 0 3 6

21

CSU 1.4

Use subtraction facts to do the following subtractions.

1. Find 457 – 234

2. Subtract 704 from 9,746

The bottom digit is subtracted from the top digit within the same place value column to get the place value digit for that column when subtracting two whole numbers.

What do you do when the lower digit is larger than the upper digit? How can you subtract the larger lower number from a smaller upper number?

Find 83 – 47

Find the difference of 766 and 278

22

CSU 1.4: Subtraction of Whole Numbers

Diagnosis: Whole Number/Subtraction

Prescription:1. Write the larger whole number on top of the smaller. 2. Align digits of the same place value in the same column.3. Place a line under the columns4. Subtract the lower digit from the upper digit and enter this difference in the same column below the line.

Execution: 1. The solution to problem 1. is developed as follows. 457

- 234 223

2. Problem 2’s solution is: 9, 7 4 6 - 7 0 4

9, 0 4 2 = 6 – 4 = 4 – 0 = 7 – 7= 9 – 0

Diagnosis: Whole Number/Subtraction /borrowing

Prescription:1. Write the larger whole number on top of the smaller. Align digits of the same place value in the same column.2. Place a line under the columns3. Subtract the lower digit from the upper digit and enter this difference in the same column below the line. 4. If the lower digit is larger than the upper digit add ten to the upper digit and subtract from it the lower digit and enter this difference

in the same column below the line.5. If you had to add 10 to the upper digit, subtract (borrow)1 from the upper digit’s neighboring digit to its left. This is called

“borrowing.”Execution: For example: 8 3 7 13

- 4 7 - 4 72 – 5 = ? 3 6

Diagnosis: Whole Number/ Subtraction/borrowing

Prescription:1. Write the larger whole number on top of the smaller. Align digits of the same place value in the same column.2. Place a line under the columns3. “Borrow” as needed for each digit starting from the right

Execution: Here is a diagram of this operation solving 766 - 278

Borrowing 1 from tens Borrowing 1 from hundredths gives 10 to ones gives 10 to tens

7 6 6 7 6 – 1 6 + 10 7 5 16 6 15 16 - 2 7 8 2 7 8 2 7 8 2 7 8

4 8 8

The borrowing of 1 from one column and the giving of 10 to the column on its right is to be done mentally when doing the column by column subtraction starting from the right most column and moving to succeeding columns to the left.

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Borrowing one from the ten’s column is worth 10 in the one’s column.Hence:83 – 47 = 36

CSU 1.4

Borrowing works great, but what do you do when you have to borrow from a zero digit? How can you borrow from zero?

Find the difference

1. 700 - 536

24

CSU1.4: Subtraction of Whole Numbers

Diagnosis: Whole Numbers/ Subtraction/ multiple borrowing.

Prescription:1. Write the larger whole number on top of the smaller. Align digits of the same place value in the same column.2. Place a line under the columns3. “Borrow” as needed for each digit starting from the right 4. When borrowing from a zero digit , borrow one its digit to the left 5. If the next digit to the left is also a zero borrow one from its digit on the left…and so on.

Execution: Here is the sequence of borrowing needed to find 600 – 437. -1 -1 Two borrowings to cover borrowing from a zero digit 7 0 0 (7 -1) 10 0 6 (10 – 1) 10 6 9 10 - 5 3 6 - 5 3 6 - 5 3 6 - 5 3 6

1 6 4The sequence shown above needs to be “seen “mentally, such that the answer digits are derived with a minimum of additional markings on the problem sheet.

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CSU 1.5

1. How is multiplication related to addition?

2. What is another way to “say” multiply?

1. What is the outcome of multiplying two numbers called?

2. What are the numbers that are multiplied called?

Which mathematical symbols indicate multiplication?

26

CSU 1.5: Multiplication of Whole Numbers

Diagnosis: Whole Numbers/Multiplication

Answer to questions:

1. Multiplication is repeated addition.

For example 3 X 5 = 5 + 5 + 5 , or 5 X 3 = 3 + 3 + 3 + 3 + 3

2. To indicate multiplication one can say “times”.


Answer to questions:

1. The outcome of multiplying two numbers is called a “product.”The product of three times five is fifteen.

2. The numbers that are multiplied together to make a product are called “factors.”The factors of fifteen are three and five.


Answer to question:

The mathematical symbols indicating multiplication are:a. “x”: such as 3 X 5 b. “ ∙”: such as 3∙5 = 3 X 5c. 3(5), (3)5, (3)(5) are all ways of writing 3 x 5d. Similar to adding and subtracting there is

the following 3

X 5 15

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CSU 1.5

Draw the table below. Within 3 minutes put in each box the product of the box’s column digit times the box’s row’ digit. See given examples.

With multiplication facts we can multiply single digit numbers.

What property of multiplying numbers do we use to multiply a single digit number times a double digit number?

Give an example of this property

Show how to use this property to multiply 3 times 21.

Also determine 7 times 65.

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X 1 2 3 4 5 6 7 8 91234 205 1067 4289


Diagnosis: Whole Numbers/ Multiplication/ Facts.

Prescription: 1. Pick an empty box, then quickly enter in the box product of its column digit and its row digit.

Execution:

Diagnosis: Whole Numbers/Multiplication/Distributive property. Answers to Questions:

1. The property referred to is called the “Distributive Property” of numbers.2. For example we know that 3(4) = 12

We can take the “(4)” and replace it with “(1 + 3)”That means 3(4) = 3(1 + 3)The distributive property says we can go inside the parenthesis as follows. 3(4) = 3(1 + 3) = 3(1) + 3(3) = 3 + 9 = 12

Here we are “distributing” the outside 3 to the inside numbers. We can also do 3(4) = 3(2 +2)= 6 + 6 = 12 to further illustrate the truth of the distributive property.

Diagnosis: Whole Numbers/Multiplication/Distributive property.


1. Doing 3(21) we get 3(21) = 3( 20 + 1) = 3(20) + 3(1) = 60 + 3 = 63

Here we used that 20 is 2 tens and 3 times 20 is the same as 3 times 2 tens. Three times 2 tens is six tens, which is 60.

2. Similarly doing 7 times 65 means doing 7(65) which becomes, using the distributive property, 7(60 + 5).

Using multiplication facts we get that 7(60 +5) = 420 + 35 = 455

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X 1 2 3 4 5 6 7 8 91 1 2 3 4 5 6 7 8 92 2 4 6 8 10 12 14 16 183 3 6 9 12 15 18 21 24 284 4 8 12 16 20 24 28 32 365 5 10 15 20 25 30 35 40 456 6 12 18 24 30 36 42 48 547 7 14 21 28 35 42 49 56 638 8 16 24 32 40 48 56 64 729 9 18 27 36 45 54 63 72 81

CSU 1.5

We know we can do 7(65) horizontally by going to 7(60 + 5).

How can we do 7(65) vertically, using place value columns?

Multiplying a multi digit number by a one digit number using place value columns shows us the power of place value number representation.

We need a shorthand compact way of multiplying multi-digit numbers by a single digit. What is that shorthand method favored for this? Use 7 X 65 to illustrate this method. Is 7 times 65 equal to the same number as 65 times 7?

If you had to do the following products mentally, “in your head,” what would your sequence of thoughts be?

1. 8(54)2. 6(65)

30


Diagnosis: Whole Numbers/Multiplication/Distributive property.

Answer to Question:

1. Vertically 7(65) is 65 X 7

2. In terms of place value columns >> 6 5X 7

If the column product is two3. Multiply upper column digits by 7 >> 6 5 digits, the tens digit is placed

X 7 in the column to the left.

7 X 5 puts 3 in the ten’s column >> 3 5 << 7 X 5 puts 5 in the ones column

7 X 60 puts 4 in the hundreds column >> 4 2 0 <<7 X 60 puts 2 in the tens column Now sum the columns >> 4 5 5

4. The answer is 455.

Diagnosis: Whole Numbes/ Multiplication/Shorthand method

Answer to Question:1. Write the multiplication problem vertically

65X7

2. Multiply the ones digit of 65, 5, by 7 and get 353. Put the 5 in the one’s column and carry the 3 to the tens column as shown below.

3 3 carried over from the tens digit in doing 7 X 5 = 3565 5 from ones digit in doing 7 X 5 = 35

X7 455

4. Now add the 3 that is carried over to the product of 7 X 6. This is then 42 + 3 = 455. Now enter45 in hundreds and tens columns as shown,6. Now do this mentally, without writing the 3 above the six

Note that 7 X 65 = 65 X 7; or 7(65) = 65(7)

Diagnosis: Whole Numbers/Multiplication/Single Digit

Answer to Questions:1. 8(54) done mentally, in your head, becomes:

a. Eight times 4 is 32. Write down the 2 and carry the 3 in your head. b. Then do 8 times 5, which is 40. Add the carry of 3 to 40 to get 43. c. Write down 43 next to the 2 written earlier and get 432 as 8(54).

2. 6(65) in your head becomes:Six times the 5 is 30, which means we write 0 down and carry 3. Six times 6 is 36. We add the carry of 3 to 36 to make it 39. We write down the 39 next to the 0, to get the final answer of 390.

31

CSU 1.5

If you had to do the following products mentally, “in your head,” what would your sequence of thoughts be?

1. 3(52)2. 30(52)

How do we do two digit multiplication of a multi digit whole number?

Show how with 73(56).

Find the product 368(347)

The product 368(347)) is a whole number.What will we know about that number that we did not know before we found the product 368(347)?

32


Diagnosis: Whole Numbers/Multiplication/Single Digit

Answer to Questions:

1. 3(52) in your head becomes:Three times 2 is 6. We write down 6 and carry nothing. Next we do 3 times 5 to get 15. There is no carry so we directly write down 15 next to the 6 to get 156 as the answer.

2. 30(52) in your head becomes:We know that 30 is 10 times 3, so 30(52) is 10 times 3(52). We know that 3(52) is 156, so 30(52) is times 156, which is 1560.

Diagnosis: Whole Numbers/ Multiplication/Two Digits

Prescription/Execution:1. Write down the factors vertically as follows.

73 X 56

2. Enter the products of 6(73) and 50(73) under the underline keeping place value alignment. Then add these products and put the result under a second underline.

73 X 56 438 < Enter the product 6(73) = 438 (done mentally)

+ 3650 < Enter the product 50(73) = 3650 (done mentally)

4088 < The sum of the two products3. Notice how place value is maintained through place value column alignment of the digits. The answer is found to be 4088.

Diagnosis: Whole Numbers/ Multiplication/Multi-Digits

Prescription:1. Set up the multiplication vertically2. Under the underline enter each single digit’s product (done mentally) keeping place value alignment as shown below.3. Put an underline after all products that have been entered4. The product of the original two whole numbers is the sum of the single digit products. This sum is placed under a second underline.

Execution: 368

X 347 Multiply the upper number by each of these single digits.

2, 576 Enter7(368) = 2,576 done mentally

14,720 Enter 40(368) = 14720 done mentally

+ 110,400 Enter 300(368) = 110, 400 done mentally

127,696 Sum the entered products to get the product sought.

We now know that 127,696 is 347 times 368. How many know that?

33

CSU 1.6

1. What number will you reach to if you add 368 to itself 347 times?

2. What is the most efficient way to answer this question?

1. How many times can you subtract 368 from 127,696?

2. What mathematical process is to be used to answer this question quickly without repeated subtractions?

1. How many times can you subtract 7 from 28, 5 from 30, 9 from 63?

2. What is the fast way to answer these questions instantly?

34

CSU 1.6: Division of Whole Numbers

Diagnosis: Whole Numbers/Division/Fundamentals

Answers to Questions:1. Adding 368 to itself 347 times produces the number 127,696.

2. Instead of painstakingly repeatedly adding368 to itself 347 times, we can use the method of whole number multiplication and find out what 368 multiplied by 347 to find out what is 368 added to itself 347 times is.

Recall: repeated addition is multiplication. When 368 is multiplied by 347 the product is 127,696. Do this procedure to be sure.



1. You can subtract 368 from 127,696 exactly 347 times.We can infer this from knowing that 368 x 347 = 127,696

2. Whole number multiplication tells us the result of repeated addition without doing repeated additions.

The process of whole number division tells us the result of repeated subtraction without doing the needed subtractions.



1. You can subtract 7 from 28 four times. Five can be subtracted from 30 six times. And, 9 can be subtracted from 63 seven times.

2. By knowing that 7 times 4 is 28; 5 times 6 is 30; and 9 times 7 is 63, we can quickly find out the number subtractions needed.

35

CSU 1.6

1. How is finding the number of times you can subtract one number from a larger number, the same as dividing the larger number into equal parts?

1. Explain how we can use the factors of 28 to divide 28 into parts of equal size.

2. Why is repeated subtraction and important way to understand whole number division?

1. If multiplication is a form of repeated addition, what then is division?

2. Give four ways to represent the division of one number by another.

36


Diagnosis: Whole Numbers/ Division/Fundamentals


1. The number of subtractions that are possible tells us how many times the smaller number can “fit” into the larger number.

2. The number smaller equal parts you can divide a whole into is also the number of times you can subtract a smaller equal part from the whole.



1. The factors of 28 are 2, 4, 7, and 14. That means we can divide 28 into the following equal parts. From 2 x 14 = 28 2 parts of 14 each; or 14 parts of 2 each

From 4 x 7 = 28 4 parts of 7 each; or 7 parts of 4 each

2. The computational process to divide one number into another uses the repeated subtraction of multiples. That is why repeated subtraction is key to understanding how long division works.

Diagnosis: Whole Numbers/Division

Answers to Questions:1. Division may be called repeated subtraction.2. Four symbols representing division :

10 ÷ 5 Uses the “division symbol”

105

Uses the fraction bar

10/5 Uses the slash

5)10 Uses the long division sign

37

CSU 1.6

1. What is the outcome of division called?

2. The number that is being divided into is called what?

3. The number that is doing the division is called what?

4. The number that is left over after all possible subtractions were taken is called what?

1. Explain how would you find out the number of times can you subtract 35 from 770, without repeatedly subtracting 35 from 770 until 770 is reduced to zero, and without using the methods of long division.

2. Is this the same as finding out how to divide 770 by 35?

Show a way to determine that 465 ÷ 5 is 93 using repeated subtractions of multiples of the divisor, 5.

a. Do not use the long division sign. b. Do not use the methods of long divisionc. Choose various multiples of the divisor (divisor multiples) to quickly reduce the dividend

through repeated subtractions.d. Keep track of multipliers used to form divisor multiples. Note: divisor multiples in this

case are multiples of the divisor 5.

38


Diagnosis: Whole Numbers/Division/Basics


1. The outcome, result, of division is called a quotient.

2. The number that is being divided into is called a dividend

3. The number that is doing the dividing is called the divisor.

4. The number left over after all possible subtractions is called the remainder.


Answers to Questions:1. It would take too long to repeatedly subtract 35 from 770.

It is easier to subtract multiples of 35 from 770 and keep track of the multipliers used.770

20 x 35 = 700 - 700 1st subtraction multipliers used 70 1st remainder

+ 2 x 35 = 70 - 70 2nd subtractionSum of multipliers 22 x 35 = 770 00 2nd remainder = 0The sum of the multipliers is the number of times 35 can be subtracted from 770.

This is a way to divide 35 into 770, but not the most efficient way

Diagnosis: Whole Numbers/ Division/Basics

Prescription:1. Set up for repeated vertical subtractions of divisor multiples. 2. Choose large enough divisor multipliers to reduce dividend and remainders with a few subtractions.3. The sum of the divisor multipliers used is the quotient, which should be 93.

Execution:

Divisor 465 Dividend 40 x 5 =200 - 200 1st Subtraction Chosen Divisor 265 1st Remainder Multipliers 40 x 5 =200 - 200 2nd Subtraction

65 2nd Remainder 13 x 5 = 65 - 65 3rd SubtractionSum of multipliers 93 x 5 = 465 00 No Remainder

39

CSU 1.6

Given that 370 ÷ 5 = 74.1. How many repeated subtractions of 5 would be required to reduce 370 to zero?2. How can we solve 370 ÷ 5 using less repeated subtractions?3. What divisor multiple would result in the least repeated subtractions?4. What divisor multiples having only single non-zero digits would produce the least number of

repeated subtractions?

Show a way to determine 370 ÷ 5 = 74 that minimizes the number of repeated subtractions by finding the 7 and the 4 digits in two successive repeated subtractions

a. In doing this use the long division sign. b. Dot in place value columns to keep track of the place value size of each digit.c. Use repeated subtractions of divisor multiples to determine the quotientd. Keep track of multipliers used to form divisor multiples.

We want to be able to do division of whole numbers, just as efficiently as we do multiplication of whole numbers. This means we want to minimize repeated subtractions and take advantage of place value to save writing zeros.

Compare doing 370 ÷ 5 respecting place value and the quicker way of “long division” that takes a short cut and arrives at the correct answer.

40

CSU 1.6: Division with Whole Numbers

Diagnosis: Whole Numbers/ Division

Answers to Questions:1. It would take 74 subtractions of 5 from 370 to reduce to zero.2. Instead of repeatedly subtracting the divisor, 5, we can subtract multiples of the divisor. This would

produce fewer subtractions and require summing the divisor multipliers used.3. The divisor multiple of 74 x 5 = 370 would allow only one subtraction.4. Since 74 is the total number of 5s in 370, we can say that 70 and 4 are the two divisor multipliers that use

one single non-zero digits and would allow 370 to be subtracted to zero.

Diagnosis: Whole Numbers/ Division/ Long Division

Prescription:1. Set up the division problem using the long division sign: 5)4652. Place the largest digit in the place of the largest place value above the dividend that produces the largest divisor multiple less than

the dividend. 3. Multiply the divisor by this digit times its place value and subtract from the dividend4. Repeat the two steps above on remainders, placing divisor multipliers as shown.5. The quotient is the sum of the numbers collected above the dividend.

Execution:Place Value Columns 4 < 4 X 5 produces largest multiple of 5 not more than 20

7 0 < 70 X 5 produces largest multiple of 5 not more than 3705)3 7 0 < Need to find largest multiple of 5 not more than 370 - 3 5 0 < 1st subtraction based on the multiple 5 X 70 = 350 2 0 < Need to find largest multiple of 5 not more than 20

- 2 0 < 2nd subtraction based on the 5 X 4 = 20 0 < Less than 5 means no further subtractions possible

The solution is the sum of multipliers entered above the division bar = 74

Diagnosis: Whole Numbers/ Division/ Long Division

Answer to Question:1. To make the process more efficient use digit placement to represent multiples of ten and eliminate

unnecessary use of zeros and mathematical signs.2. Compare the two ways

4 70 74 < 93 means 90 + 35)370 5)370

Basic - 350 Shorter 35 < Place value of 45 means 450

20 20 - 20 20 Use down arrow to bring

0 0 down 5 from 465

41

CSU 1.6

How do we divide by a two digit number?

Explain using 7920 ÷ 45

If we use the method of long division we find that 58 does not divide evenly into 1860. There is a small remainder after the last 58 is subtracted from 1860.

Divide 1860 by 58 using the method long division and indicate the amount of remainder left over.

Show how the divisor, the quotient, and the remainder connect to the dividend.

If you divided the number 20, 800 into 23 parts, how big would each part be and what would be the remainder, if any?

How can you use multiplication to check the correctness of your division?

42

CSU 1.6 Division with Whole Numbers

Diagnosis: Whole Number Long Division/Multiple digits with no remainder

Prescription:1. Set up the problem using the long division set up.2. Repeatedly subtract multiples of the divisor from the dividend placing quotient digits above dividend as shown. 3. Place starting quotient digit above the place in dividend that picks off digits just larger than divisor.

Execution: 176 Quotient digits Quotient digit (1) X Divisor (45)

Divisor 45) 7,920 Dividend 4 5 First divisor multiple is 1 X 35 = 35 1st subtraction = 34 3 4 2 Bring down 2 to set up 2nd subtraction 3 1 5 Second divisor multiple is 7 X 45 =315 2nd subtraction = 27 2 7 0 Bring down dividend digit 0 to set up 3rd subtraction

2 7 0 Third divisor multiple is 6 X 45 = 270 3rd subtraction = 0 0 “0” indicates that the complete solution = 176

Diagnosis: Whole Number Long Division/by two digits with remainderPrescription:

1. Set up for long division and do repeated subtractions of divisor multiples until the remainder is less than divisor.Execution and answer to questons:

1. 32 Quotient digits

58) 1,8601st subtraction 1 74 Quotient digit 2 times 67 = 134

120 Bring down 0 to set up 1st remainder2nd subtraction 116 Quotient digit 5 times 67 = 335

4 4 is let over. This is the Remainder 2. After two successive subtractions 15 remained. Since 67 cannot be subtracted from 15, the long division stops here.3. The divisor, quotient, & remainder, connect to dividend (67 X 25) + 15 = 1, 6904. The 4 main numbers relate (Divisor) x (Quotient) + (Remainder) = (Dividend)

Diagnosis: Whole Number Long Division/by two digits with remainder

Prescription:1. Set up a long division process and execute as follows.

Execution: 1st quotient digit starts here to work with “208.”2nd quotient digit is “0” because “10” is too small3rd quotient digit starts here to work with “100.”

90423) 20, 800

1st subtraction 20 7 9 X 23 = 207 is 1st divisor multiple subtractedNeed 2nd zero, 10 too small 100 Two zeros need to come down. 10 too small. 2nd subtraction 92 4 X 23 = 92 is 2nd divisor multiple subtracted 8 Remainder = 8

2. Dividing 20, 800 by 23 gives us a quotient of 904 with a remainder of 8 3. We can write (904 X 23) + 8 = 20, 800

Answer to Question: To check division:, multiply 904 by 23 and add 8 to the result you should get 20, 800

43

CSU 1.7

What is an exponential number?

1. What is the “base” of an exponential number?

2. What is the “exponent” of an exponential number?

1. When we say 2 to the fourth power, what do we mean?

2. So what is another term for exponent?

44

CSU 1.7: Whole Numbers to Exponential Power

Diagnosis: Whole Numbers/ Exponential Numbers/Fundamentals


An exponential number is a number of the form 25 which represents the number formed by multiplying 2 by itself 5 times, which is 2 ∙2 ∙ 2∙2∙2=32. Here we see that exponential number 25 is equal to 32.

Diagnosis: Whole Numbers/Exponential Numbers/Fundamentals


1. In an exponential number the base is the number that is multiplying itself. In the case of 25 the base is the number 2.

2. In an exponential number the exponent is the number that indicates the number of self multiplications. In the case of 25 the exponent is the number 5.



1. Two to the fourth power refers to the exponential number 24, which means that two is the base and the exponent is 4.

2. We can use the word “power” to represent exponent when referring to exponential numbers. For example: five to the eight power would the exponential number 58. Here 5 is the base and 8 is the exponent.

45

CSU 1.7

If multiplication is repeated addition what then is raising a number to a power?

1. Is 3 squared an exponential number?

2. What is 3 squared as a number?

1. Is 3 cubed an exponential number?

2. What is 3 cubed as a number?

46

CSU 1.7: Whole Numbers/Exponentials


Answer to Question:

Whereas multiplication is the result of repeated addition, we can say that exponential numbers are the result of repeated multiplication.

For example two raised to the fifth power, written as 25, means 2 multiplied by itself 5 times, which is 2 x 2 x 2 x 2 x 2 bringing us to the number 32.



1. Three squared means 32 = 3 x 3. So, three squared does refer to an exponential number.

2. Since three squared is 32 = 3 x 3, and noting that 3 x 3 = 9 we can conclude that 3 squared is a number and that number is 9.



1. Three cubed means 33 = 3 X 3 X 3. So three cubed does refer to an exponential number.

2. Since three cubed is 33 = 3 X 3 X 3, and 3 X 3 X 3 = 27 we can conclude that 3 cubed is a number and that number is 27.

47

CSU 1.7

1. What is the exponent on a number that has no exponent or is not raised to a power?

2. If a number has an exponent of zero what number does it equal?

Evaluate the following exponential numbers.a. 34

b. 42

c. 24

d. 91

e. 90

48

CSU 1.7: Whole Numbers/ Exponential Numbers



1. An exponent of 1 is assumed on a number that does not carry an exponent For example 5 is considered to have an exponent of 1, as in 51 = 5

2. A number is raised to an exponent of zero equals one, for example 50 = 1.



a. 34 = 3 x 3 x 3 x3 = 27 x 3 = 81

b. 42 = 4 x 4 =16

c. 24 = 2 x 2 x 2 x 2 = 16

d. 91 = 9

e. 90 = 1

49

CSU 2.1

1. What kind of number is a fraction? Compare it with whole numbers.

2. Why do we need to know how to compute with fractions?

How are fractions represented using whole numbers?

Name all the parts of a fraction.

Write the fractions that represent the following fractional quantities.a. Three fourthsb. Seven-eighthsc. Nine-fifths

50

CSU 2.1: Fractions/Fundamentals

Answers to Questions:1. A fraction is the opposite of a whole number.

Whole numbers quantify collections of whole things Fractions quantify portions of a whole.

2. Business, science, government and many other areas of human civilization cannot function unless we have a method to compute with numbers that represent parts of a whole.

Fractions allow us to do math with parts of whole units and numbers.

Answers to Question:Math uses one whole number divided by another whole number to represent a number that is called a fraction.

The top whole number represents a number of whole units. The bottom whole number is the number of parts the top number is divided into.

A fraction is the number obtained when its top number is divided by its bottom number.

Answer to Question:

The top number in a fraction is called the numerator.

The bottom number of a fraction is called a denominator.

The numerator and the denominator are known as the terms of the fraction.

The bar separating the numerator and the denominator is called the fraction bar.

Answers to Questions:a. Three-fourths means three whole units divided into 4 parts. The numerator is 3 and the denominator

is 4.

So, three fourths is represented by 3/4 or 34 .

b. Seven-eighths is represented by 7/8 or 78 .

c. Nine-fifths is represented by 9/5 or 95 .

51

CSU 2.1

1. Can the whole number 7 be represented as a fraction? If so how?2. Is there a whole number that cannot be used as the denominator of a fraction?3. What is a “proper fraction”?4. What is an “improper fraction”?

Can two fractions be equal even though they are made up of different number pairs? Give an example to explain.

1. What do you call fractions that are equal to each other but are made up of different numbers?

2. When you reduce a fraction to its smallest possible numerator denominator pair: what is this process called?

1. If the denominator of a fraction is one what is the fraction equal to?

2. If the denominator of a fraction is equal to its numerator what is the fraction equal to?

52


Answers to Questions:1. The whole number 7 can be represented as a fraction by the terms 7/1.2. The denominator in a fraction cannot be zero. Division by zero is undefined.3. A proper fraction is one where the numerator is smaller than the denominator.4. An improper fraction is one where the numerator is larger than the denominator.

Answer to Question:

Yes, two fractions that are made up of different numbers can produce the same net fraction.

For example, the fraction 1/2 is the same as 2/4. This is so because dividing 1 into 2 parts produces the same part size as dividing two units into four.

Similarly, 4/8 is equal to 6/12, which is numerically equal to ½.


1. Fractions that are made up of different terms but are numerically equal to each other are called “equivalent.”

2. The process that determines the smallest denominator-numerator pair for a given fraction is called “simplification.”


1. If the denominator is one, then the fraction is equal to whatever the numerator is. For example the fraction 7/1 is equal to 7.

2. If the denominator of a fraction is the same as its numerator, then the fraction is equal to 1. For example 7/7 is the same as 1.

53

CSU 2.1

Show the two ways to construct equivalent fractions.

Write 3/4 as an equivalent fraction with a denominator of 24.

Write 12/18 as an equivalent fraction with a denominator of 6.

Write the following four fractions in order of size from smallest to largest.

58

, 56

, 34

, 23

54


Answer to Question:

1. Given the fraction ab

, the two ways to generate equivalent fractions are called multiplication way and

the division way.

2. The Multiplication Way is to multiply the numerator, a, and the denominator, b, by the same number, c to get an equivalent fraction.

ab=¿

a ∙ cb ∙ c

Equivalent fractions

3. The Division Way is to divide the numerator, a, and the denominator, b, by the same number, c to get

ab=a÷ c

b÷ c Equivalent fractions

Diagnosis: Fractions/ Equivalent Fractions/Multiplication WayPrescription:

1. Going to an equivalent fraction with a larger denominator means multiplying both the numerator and denominator by the same number that yields the desired denominator

Execution: 1. To go from a denominator of 4 to 24 using multiplication means we need a multiplier of 6. Since 4 time 6

is 24. So we multiply the terms of the fraction by 6.2.

34=3∙ 6

4 ∙ 6=18

24 Equivalent Fractions

Diagnosis: Fractions/ Equivalent Fractions/Division WayPrescription:

1. Going to an equivalent fraction with a smaller denominator means dividing both the numerator and denominator by the same number that yields the desired denominator.

Execution:1. To go from a denominator of 18 to 6 using division means we need to divide 18 by 3. Since 18 divided by

3 is 6. 2. Divide numerator and denominator by 3.

1218

=12 ÷ 318 ÷ 3

=46

Equivalent Fractions

Diagnosis: Fractions/Equivalent FractionsPrescription:

1. Put all four fractions into equivalent fractions with the same “common denominator.” 2. Compare numerators to compare fraction size.

Execution:1. The common denominator needed is 24.2. Writing each given fraction as an equivalent fraction with denominator 24.

58=5∙ 3

8 ∙3=15

24; 5

6=5 ∙ 4

6 ∙ 4=20

24; 3

4=3 ∙6

4 ∙ 6=18

24; 2

3=2 ∙8

3 ∙ 8=16

24

3. Put the equivalent fractions in numerical order based on numerator size

55

1524

< 1624

< 1824

< 2024

4. Replace the equivalent fractions with the original fractions 58< 2

3< 3

4< 5

6

5. The order shown is the solution

CSU 2.2

1. What is a prime number?a. Give the first twelve prime numbersb. How many prime numbers are there?

2. What is a composite number?

1. How do composite numbers use prime numbers?

2. What is a fraction reduced to lowest terms?

Identify each of the following numbers as either prime or compositea. 47b. 18

The number 72 is a composite

Factor 72 into a product of prime numbers.

56

CSU 2.2: Fractions/Prime Numbers

Answers to Questions:1. A number that is not a multiple of any other number except 1 and itself is a prime number.

a. The first 12 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.b. There are an infinite number of prime numbers

2. A composite number is any whole number greater than 1 that is not a prime number. A composite number has at least one factor (or divisor) that is not itself or 1.


1. Every composite number can be broken down into factors which are all prime numbers.

2. A fraction is reduced to lowest terms if the numerator and the denominator have no factors in common other than the number 1.

Diagnosis: Fractions/Prime Numbers Prescription:

1. A number that is not a multiple of another number is a prime number2. A number that is a multiple of another number is a composite number.

Execution: a. 47 is a prime because it has no factors or numbers that can divide even into it except itself and 1.

b. 18 is not a prime because it is equal to 3 times 6. It is also equal to 2 times 9. In fact 18 has 2, 3, 6, and 9 as its factors (or as divisors). Hence, 18 is a composite number.

Diagnosis: Fractions/Prime NumbersPrescription:

1. Decompose the composite number into at least two of its factors.2. Decompose any of its composite factors 3. Repeat until all composite factors have been decomposed into prime factors.

Execution:1. 72 = 6 ∙ 12 Two composite numbers

72 = 2∙3 ∙3∙ 4 Decomposing of each composite72 = 2∙3 ∙3∙ 2∙ 2 Only prime factors remain

72=2 ∙ 2∙2∙3 ∙3 Solution. Primes ordered by size2. Could have been done with a different starting decomposition.

72 = 3 X 24 = 3 X 4 X 6 =3 X 2 X 2 X 3 X 2 = 2 X 2 X 2 X 3 x 3

57

CSU 2.2

1. How do you determine if fractions are reduced to lowest terms?2. Are the fractions 1/2, 2/3, 3/4, 3/5 reduced to their lowest terms?

3. How do you reduce a fraction to its lowest terms?

1. Write 6/8 into an equivalent fraction at its lowest terms.

2. Write 12/15 into an equivalent fraction at its lowest terms.

58

CSU 2.2: Fractions/Prime Numbers

Diagnosis: Fractions/Reducing to Lowest terms

Prescription:1. Check numerator and denominator for common factors.2. If no common factors exist then fraction is reduced to lowest terms.3. To reduce a fraction to its lowest terms remove all factors common to the numerator and the

denominator.Execution:

1. In each of the following fractions there are no factors common to both numerator and denominator, so these fractions are reduced to lowest terms: 1/2, 2/3, 3/4, 1/5, 3/5

Diagnosis: Fractions/Reducing to Lowest Terms

Prescription:1. To reduce a fraction to its lowest terms remove all factors common to the numerator and the

denominator.

Execution:

1. 1218

=6∙ 26 ∙ 3

=23

Reduced to lowest terms by removing common factor of 6

2.1215

=4 ∙35 ∙3

=45

Reduced to lowest terms by removing common

factor of 3

59

CSU 2.3

A fraction is a number. How do you multiply fractions?Explain with the following multiplications

1. Multiply, 35

∙ 25

2. Multiply, 37

∙4

Solve two ways.

1. Solve by working within parenthesis first

2. Solve by using the associative law of multiplication

Multiply 13 ( 4

5∙ 17 )

Multiply and then reduce to lowest terms129

∙ 58

60

CSU 2.3: Fractions/ Multiplying

Diagnosis: Fractions/MultiplyingPrescription:

1. The numerator of the product of fractions is the product of their numerators.2. The denominator of the product of fractions is the product of their denominators

Execution: Product of numerators

1.35

∙ 25= 3.2

5 ∙5= 6

25

Product of denominators

2. 37

∙ 41=3 ∙ 4

7 ∙ 1=12

7 [A whole number has 1 for a denominator


1. Find the product within the parenthesis first,2. The numerator of the product of fractions is the product of their numerators.3. The denominator of the product of fractions is the product of their denominators. 4. The associative law of multiplication, a(b∙c) = (a∙b)c, applies.

Execution: 13 ( 4

5∙ 17 )=1

3 ( 4 ∙15 ∙7 )=1

3 ( 435 )= 4

105Using the associative law

13 ( 4

5∙ 17 )=( 1

3∙ 45 )∙ 1

7=( 4

15 ) ∙ 17= 4

105


1. The numerator of the product of fractions is the product of their numerators.2. The denominator of the product of fractions is the product of their denominators. 3. Reduce to lowest terms by removing factors common to numerator and denominator.

Execution:

129

∙ 58=12∙5

9 ∙8=60

72 R educe to lowest terms by…

…removing common factors6072

=12 ∙ 58 ∙ 9

=4 ∙3∙54 ∙2∙9

= 2 ∙2∙3 ∙52∙ 2 ∙2 ∙ 3∙ 3

= 2 ∙ 2∙3 ∙52∙ 2 ∙2 ∙3 ∙ 3

=56

61

CSU 2.3

Multiply and reduce to lowest terms.

83

∙ 916

Multiply and reduce to lowest terms.

( 34 )

2

∙ 12

1. Find 1/3 of 3/7.

2. What is 2/3 of 18?

62

CSU 2.3: Fractions/Multiplying



Execution:

83

∙ 916

= 8 ∙ 93 ∙ 16

=(2∙ 2∙ 2 ) ∙ (3 ∙3 )3∙ (2 ∙2 ∙ 2∙ 2 )

=32

…or a faster way… 3

83

∙ 916

= 8 ∙93 ∙ 16

=32

2



Execution:

( 34 )

2

∙ 12=3

4∙ 34

∙ 12= 3 ∙3

4 ∙ 4 ∙2= 9

32

No common terms


1. The word “of” means multiply with respect to fractions.2. The numerator of the product of fractions is the product of their numerators.3. The denominator of the product of fractions is the product of their denominators. 4. Reduce to lowest terms by removing factors common to numerator and denominator.

Execution:

1.13

of 37

is the same as : 13

∙ 37=1∙3

3 ∙7=1

7

2.23

of 18 isthe same as : 23

∙18=2∙ 183

=2∙ 3 ∙63

=12

1

63

CSU 2.4

Divide 1/2 by 1/4.

Divide 4/25 by 12/5.

Do the following divisions

1427

÷ 7 ;21 ÷ 32

64

CSU 2.4: Fractions/Division

Diagnosis: Fractions/DividingPrescription:

1. Invert divisor and multiply. Multiply fractions as follows.2. Product of fractions is product of numerators over product of denominators.3. Reduce to lowest terms by removing common factors between numerator and denominator

Execution: Invert and multiply divisor

Divide 12

by 14

means : 12

÷ 14=1

2∙ 41=4

2=2

1=2



Execution: Cancel common factors

4

25÷ 12

5= 4

25∙ 512

= 4 ∙525 ∙12

= 4 ∙ 5(5 ⋅5 ) ⋅ (3∙ 4 )

= 4 ∙55⋅5 ⋅3 ⋅ 4

= 115



Execution: Cancel common factor

1. 1427

÷ 7=1427

∙ 17=14 ∙ 1

27 ∙ 7=2∙7 ∙1

27 ∙7= 2

27

21 ÷ 32=21 ∙ 2

3=21⋅2

3=3 ∙7 ⋅2

3=7 ⋅2=14

65

CSU 2.5

Add the two fractions 312

+ 512

Subtract the two fractions:57−3

7

How can you change the denominator of a fraction to form an equivalent fraction?Change the denominator of the following fraction to an equivalent fraction with a denominator of 9 instead of 3.

23

66

CSU 2.5: Fractions/Adding and Subtracting

Diagnosis: Fractions/Adding with same denominatorPrescription:

1. The sum of fractions with same denominator has a numerator equal to the sum all numerators.2. The sum of fractions with same denominator keeps its denominator the same.3. Reduce to lowest terms

Execution:Sum of numerators

Numerators

312

+ 512

=3+512

= 812

=23 Reduced to lowest terms

Diagnosis: Fractions/Subtracting with same denominatorPrescription:

4. The difference of two fractions with same denominator has a numerator equal to the difference of the two numerators.

5. The difference of fractions with same denominator keeps its denominator the same.6. Reduce to lowest terms

Execution:Difference of numerators

Numerators

57−3

7=5−3

7=2

7

Diagnosis: Fractions/Equivalent fractionsPrescription:

7. Find the number you have to multiply the denominator by to change it to the desired value.8. Multiply both numerator and denominator by the number found above in (1.)

Execution:Multiplying the denominator by 3 will convert it to 9. So multiply both numerator and denominator by 3.

23=2∙3

3 ∙3=6

9

67

CSU 2.5

Find the sum of these fractions with differing denominators.Hint: Convert fractions to equivalent fractions based on their Least Common Denominator, LCD.

38+ 5

12

Find the difference, by converting to equivalent fractions based on the Least Common Denominator, LCD.

67−1

3

Find the sum:

1

16+ 1

8+ 1

4

CSU 2.5: Fractions/Addition and Subtraction

Diagnosis: Fractions/Adding Subtracting/Least Common DenominatorPrescription:

1. Change fractions to equivalent fractions such that denominators equal to their Least Common Denominator, LCD.

2. The LCD is found by multiplying the largest denominator by the smallest number which will allow the smaller denominators to divide into it evenly.

3. Add or subtract fractions easily since denominators are the same.Execution: Equivalent fractions

68

38+ 5

12=3⋅ 3

8 ∙ 3+ 5 ∙2

12∙ 2= 9

24+ 10

24=19

24 LCD=24

12 ∙2=24∧24 ÷ 8=3 [24, the LCD, is smallest multiple of 12 divisible by 8] Smallest multiple of largest denominator divisible by smaller denominators Largest denominator


1. Change fractions to equivalent fractions such that denominators equal to their Least Common Denominator, LCD.


3. Add or subtract fractions easily since denominators are the same.Execution: Equivalent fractions

67−1

3=6 ⋅3

7 ∙ 3−1 ∙ 7

3 ∙ 7=18

21− 7

21=11

21 LCD=21

7 ∙3=21∧21÷ 3=7 [21, the LCD, is smallest multiple of 7 divisible by 3] Smallest multiple of largest denominator divisible by smaller denominators Largest denominator


1. Change fractions to equivalent fractions such that denominators are equal to their Least Common Denominator, LCD.


3. Add or subtract fractions easily since denominators are the same.Execution: Find LCD: 8(highest) x 3(lowest)= 24(divisible by 6 &4) LCD=24

116

+ 18+ 1

4= 1 ∙2

16 ⋅2 + 1 ∙ 48⋅4

+ 1∙ 84 ⋅8 = 2

32+ 4

32+ 8

32=14

32= 7

16

Producing LCD Equivalent Fractions LCD Fractions

CSU 2.6

What is a proper fraction?What is an improper fraction?What is a mixed number?

69

How is a mixed number like an improper fraction?

Change 234 to an improper fraction by treating this as the addition of two fractions, as

follows.

338=3+ 3

8=3

1+ 3

8

70

CSU 2.6: Fractions/ Mixed Numbers

Diagnosis: Fractions/Mixed Numbers/Vocabulary


1. A proper fraction is a fraction where the numerator is less than the denominator.

2. An improper fraction is a fraction where the numerator is more than the denominator.

3. A mixed number is a whole number that is immediately followed by a fraction, such as5 38 , which does

not imply multiplication but does imply addition as follows. 5 38=5+ 3

8

Mixed Number

Diagnosis: Fractions/Mixed NumbersAnswer to Question:

A fraction is one number divided by another number. In an improper fraction the denominator divides into the numerator a whole number of times plus a fractional remainder because the numerator is larger than the denominator. Since an improper fraction can be seen as a whole number plus a fraction, it can also be seen as a mixed number.

Likewise, a mixed number can be converted into an improper fraction.

Diagnosis: Fractions/Mixed numbersPrescription:

1. Convert the fraction with denominator 1 into an equivalent fraction having a denominator equal to the denominator of the fractional part of the mixed number.

2. Add the equivalent fractions easily since denominators are the same.

Improper fractionExecution:

3 38=3+ 3

8=3

1+ 3

8=3 ∙8

1∙ 8+ 3

8=24

8+ 3

8=24+3

8=27

8

Mixed Number Whole # + Fraction Whole# into LCD = 8 Whole # into Sum

a Fraction equivalent fraction Numerators

71

CSU 2.6

Change 438 to an improper fraction using a “short cut” method that quickly produces

the correct answer.

Change the improper fraction 133 to a mixed number.

Find the mixed number representation of 10718 .

72

CSU 2.6: Fractions/Improper Fractions

Diagnosis: Fractions/Mixed numbers

Prescription:1. Multiply the whole number part of the mixed number by the denominator of the fraction part.2. Add the product in (1.) to the numerator of the fraction part to make the numerator of the improper

fraction3. The denominator of the improper fraction is the same as the denominator of the fraction part of the

mixed number.Execution:

4 38=4 ∙ 8+3

8=32+3

8=35

8

Denominator use

Diagnosis: Fractions/Improper fractionsPrescription:

1. Divide the numerator by the denominator using methods of whole number division.2. The quotient part of the division is the whole number part of the mixed number. 3. The remainder part of the division is the numerator of the fraction part of the mixed number4. The denominator is the same as the denominator of the improper fraction.

Execution: 4 Quotient

3)13133

=4+13=4 1

3 12 1 Remainder

Diagnosis: Fractions/Improper fractionsPrescription:

1. Divide the numerator by the denominator using methods of whole number division, obtaining a quotient with a remainder.

2. The quotient is the whole number part of the mixed number. 3. The remainder is the numerator of the fraction part of the mixed number4. The denominator of the improper fraction is the denominator of the fraction part of the mixed number. 5. Reduce the fraction part of the mixed number to lowest terms.

Execution: Quotient 5

18)10710718

=5+ 1718

=5 1718

9017 Remainder Divisor

73

CSU 2.7

Multiply 234

∙3 15

Divide 135

÷2 45

74

CSU 2.7: Fractions/Multiplying and Dividing Mixed Numbers

Diagnosis: Fractions/Multiplying mixed numbers

Prescription:1. Change the mixed numbers to improper fractions2. Multiply the improper fractions as you would proper fractions.3. Reduce to lowest terms.4. Reduce to mixed number Reduce improper fraction

Execution: to mixed number

(3 23 ) ∙(2 1

5 )=( 3 ∙3+23 ) ∙(2 ∙5+1

5 )=113

∙ 115

=12115 = 8

115

Convert to improperfractions using short cut method

Diagnosis: Fractions/Dividing mixed numbers

Prescription:1. Convert the mixed numbers to improper fractions.2. Divide the improper fractions by inverting the divisor and multiplying.3. Reduce fraction to lowest terms and convert to mixed number if improper.

Execution: Invert and multiply

(1 25 )÷ (4 3

8 )=( 75 )÷ (35

8 )=(75 )∙ ( 8

35 )

¿ 7 ∙ 85∙ 35

= 7 ∙ 85∙7 ∙ 5

= 825

Cancel common factors before multiplication to reduce to lowest terms

75

CSU 2.8

Add 214+3 1

6

Add 458+7 7

12

Subtract 479−2 5

9

76

CSU 2.8: Fractions/ Adding and Subtracting Mixed Numbers

Diagnosis: Fractions/Adding mixed numbersPrescription:

1. Add the whole number portions of the mixed numbers in the sum.2. Add the fraction parts of the mixed numbers in the sum reducing to mixed number if fraction sum results

in improper fraction.3. Combine all whole numbers and fraction sum into mixed number.

Execution:

2 14=2 1 ∙3

4 ∙3=2 3

12

+ 3 16=3 1 ∙2

6 ∙2=4 2

12

Equivalent fractions 6 512

Sum of fractions

with LCD as denominator Sum of whole numbers


1. Add the whole number portions of the mixed numbers in the sum.2. Add the fraction parts of the mixed numbers in the sum reducing to mixed number if fraction sum results

in improper fraction.3. Combine all whole numbers and fraction sum into mixed number.

Execution:

4 58=4 5 ∙3

8 ∙3=4 15

24Improper fraction

+ 7 712

=7 7 ∙212 ∙2

=9 1424 = Mixed number

Equivalent fractions 13( 2924 )=13+(1+ 5

24 )=14 524

with LCD as denominator Sum of whole numbers


1. Subtract the whole number portions of the mixed numbers 2. Subtract the fraction parts of the mixed number, borrowing a 1 from the whole number if necessary.3. Combine above differences into mixed number.4. Reduce fraction part to lowest terms

Execution:

4 89−2 2

9=2 6

9=2 2 ∙3

3 ∙3=2 2

3

Reduce to lowest terms by removing common factors

77

CSU 2.8

In subtraction of mixed numbers we sometimes have to subtract a larger fraction part from a smaller fraction part.

Explain using the subtraction:3 38−1 5

8 , how we can subtract the 58 part from the

38 part of the

mixed numbers?

Explain in detail showing every step needed.

In subtracting mixed numbers we sometimes have to borrow a 1 from the whole number part and give it to the fraction part.

State a quick rule to do this in your head using a few steps.

Apply this quick way to the subtraction:3 38−1 5

8

Subtract 434−1 5

6

78

CSU 2.8: Fractions/ Adding and Subtracting Mixed Numbers

Diagnosis: Fractions/Subtraction/Borrowing

Answer to Question:

We have to borrow 1 from the whole number “3” and add it to the 38 to make it large enough to subtract

58 from

it.Borrow 1 from “3” and give it to the fraction

3 38=3+ 3

8=2+1+ 3

8=2+(1+ 3

8 )=2+( 88+ 3

8 )

338=2+( 8

8+ 3

8 )=2+ 118

=2 118

−1 58

1 68=1 3 ∙2

4 ∙2=1 3

4

Diagnosis: Fractions/Subtraction/Borrowing

Answer to Question:

The quick way to “borrow 1 from the whole number part and give it to the fraction part” is: reduce the whole number part by 1 and add the denominator to the numerator of the fraction part.

In your head you subtracted 1 from 3 to get 2, and added the denominator, 8, to the pre-existing numerator, 3, to get 11 .

(3 38 )−1 5

8=(2 11

8 )−1 58=1 6

8=1 3∙2

4 ∙2=1 3

4

Reduce to lowest terms

Diagnosis: Fractions/Adding mixed numbers

Prescription:1. Subtract the whole number portions of the mixed numbers 2. Subtract the fraction parts of the mixed number, borrowing a 1 from the whole number part if necessary.3. Combine above differences into a mixed number.4. Reduce fraction part to lowest terms

Execution: Borrowing 1 from “5”…

538=5 3 ∙ 5

8 ∙ 5=5 15

40=4 55

40 adds4040 to fraction

1540

−1 45=−1 4 ∙ 8

5 ∙8=−1 32

40=−1 32

40 1540

+ 4040

=15+4040

=5540

79

3 2340 Sum of numerator and denominator

CSU 3.1

We know the power of using a place value based whole number system. 1. What is the name of the system that uses place value digits to represent fractions?

2. What is the symbol that divides whole number place value digits from fractional number place value digits?

1. What is the value of place values to the right of the decimal point?

2. Draw a table of place values showing place values to the left and to the right of the decimal point.

Write 423.576 in expanded form.

80

81

CSU 3.1: Decimal Numbers

Diagnosis: Decimals/Place Value


1. The name of the system that can represent fractions with place value digits is called “decimal.”

2. The symbol that divides whole number place value digits from fractional number place value digits is called “the decimal point.”


Answer to Questions:1. Digits to right of the decimal are given place values of powers of one tenth. Every place to the right of the

decimal point has a place value of one-tenth the place value of the place to its left.

2. Table of values showing place values relative to the decimal point.

Fractional place values of places past the decimal point


Prescription:1. Write a sum of products made up of each digit times its place value.2. Starting from the left of the number and work toward the right.

Execution:Digits times their respective place values

721.459 = 700 + 20 + 1 + 4

10+ 5

100+ 9

1000 Digits occupying places Each place has its place value

CSU 3.1

82

Thou-sands

Hun-dreds

tenths ones DecimalPoint

Tenths Hund-reths

Thou-sanths

TenThou-sandths

1,000 100 10 1 ∙ 110

1100

11000

110 ,000

Write each number in words

1. .072. .0073. 3.0074. 3.875. 34.5762

Write each number as a fraction or a mixed number. Do not reduce to lowest terms.1. 0.007

2. 3.87

3. 34.5762

And, explain how to construct fractions or mixed numbers

1. Round 7, 146.879 to nearest hundred

2. Round 0.00387 to the nearest ten thousandth.

83

CSU 3.1: Decimal Numbers



1. .07 in words is “seven hundredths.”

2. .007 in words is “seven thousandths.”

3. 3.007 in words is “three and seven thousandths.”

4. 3.87 in words is “three and eighty seven hundredths.”

5. 34.5762 is written as “thirty four and five thousand, seven hundred and sixty two ten thousandths”.



1. 0.007= 71,000

3 digits after the decimal 3 zeros

Same digits here as after decimal

2. 3.87=3+ 810

+ 7100

=3+ 80100

+ 7100

=3+ 87100

=3 87100

2 digits after decimal Same number of zeros as digits after decimal

Expanded form equivalent fraction with LCD

3. 25.4936=25+ 493610,000

=25 493610,000

Same digits here as after decimal

4 digits after decimal means 4 zeros after 1 in denominator

Diagnosis: Decimals/Place Value/RoundingPrescription:

1. The digit in the place chosen to be round to is increased by 1 if the following digit is five or greater, otherwise left unchanged.

2. All digits to right of the rounded place then set to zero.

Execution: This digit unchanged all others to right of it set to zero.

1. 7,146.879≅ 7,100 Next digit is less than 5 The hundreds place to be rounded to

2. 0.00387≅ 0.0039 Digit 8 increased by 1 all others set to zeroNext digit is greater than 5The ten thousandths place to be rounded to

84

CSU 3.2

Add 32.31 + 4.298 + 554.7

Subtract: 24.914 – 13.658

Subtract: 6.7 – 3.0728

85

CSU 3.2: Decimals/Adding and Subtracting

Diagnosis: Decimals/Adding and Subtracting

Prescription:1. Vertically align the numbers to be added according to place value keeping the decimal under decimal.2. Fill in zeros where necessary to supply digits to the right of the decimal of each number.3. Sum the digits in each place value column carrying or borrowing digits as required to compute the sum or

difference.Execution: 1 1 1 Carry digits added to place value columns

32.310 4. 298 Zeros filled in to fill out columns past decimal

+ 554.700 591. 308 Sum of digits in each column including

carry digits



difference.Execution: Place value columns -1 10 Borrowing 1 from adjoining

-1 10 column adds 10 to the next24.9 1 4 24.9 1 4

- 13.6 58 - 13.6 5 8 Subtract lower from sum of upper 11.2 5 6 plus upper carries and borrows



difference.Execution: Place value columns -1 10 Fill in zeros -1 10 Borrowing 1 from adjoining

- 1 10 column adds 10 to the next6.7000 6. 7 0 0 0

- 3.0728 - 3. 0 7 2 8 Subtract lower from sum of upper 3 .6 2 7 2 plus upper carries and borrows

86

CSU 3.3

Multiply 0.7 x 0.3Do this by converting decimals to their fractional equivalents.

Multiply 0.07 x 0.004Determine a short cut based on counting places past the decimal.

Multiply 3.2 x 0.08Do this by converting decimals to their fractional equivalents.

Do this using the short cut.

87

CSU 3.3: Decimals/ Multiplication

Diagnosis: Decimals/multiplication

Prescription:1. Convert decimal numbers into fractions (proper or improper). 2. Multiply numerators to get product numerator. 3. Multiply denominators to get product denominator.4. Put resulting product fraction into equivalent decimal form.

Execution:

0.7 ∙ 0.4= 710

⋅ 410

= 7⋅410 ⋅10

= 28100

=0.28


Answer to Question:The normal procedure is as follows

1. Convert decimal numbers into fractions (proper or improper). 2. Multiply numerators to get product numerator. 3. Multiply denominators to get product denominator.4. Put resulting product fraction into equivalent decimal form.

This leads to:

0.07 ∙ 0.004= 7100

⋅ 41000

= 7 ⋅ 4100 ⋅1000

= 28100,000

=0.00028

We simply multiply 7 x 4 to get 28 and then move the decimal to the left the sum of the number of places past the decimal 0.07 and 0.004 use.

0.07 ∙ 0.004=0.00028 28 comes from 7 X 42 places past decimal + 3 places = 5 places past decimal


Prescription:1. Convert decimal numbers into fractions (proper or improper). 2. Multiply numerators to get product numerator. 3. Multiply denominators to get product denominator.4. Put resulting product fraction into equivalent decimal form.

Short cut:1. Multiply as whole numbers then move decimal point to left by the total number of decimal places used

by the two factorsExecution:

1. 3.2 ⋅0.08=3210⋅ 8

100= 32⋅8

10⋅100= 256

1000=0.256 3 places =

11000

Short cut:1. 32 ⋅8=256 0.256 After moving decimal point 3 places to left

88

89

CSU 3.3

Multiply 7.14 x 6.52

Do this problem as a multiplication of whole number.

Place the decimal point in the correct position after completing the whole number multiplication.

90

CSU 3.3: Decimals/Multiplication

Diagnosis: Decimals/Multiplication via decimal point placement

Prescription:1. Multiply the whole numbers vertically, ignoring the decimal points.2. The number of decimal places in the final product is the total number of decimal places used by the two

factors.

Execution:7.14 2 decimal places used

x 6.52 2 decimal places used 1428 3570 4284 465528 Moving 2 + 2 = 4 decimal places moved to left. 4.65528

91

CSU 3.4

Divide: 7167 ÷ 30 Convert the remainder fraction to a decimal.

Show how you can obtain the same result by merely adding a decimal place to the dividend and continuing the long division to an additional place

Divide 2085.3 ÷ 45

Add additional zeros past the last digit to 1,138.9 to continue the long division process until there is no remainder.

Where do you put the decimal point in the quotient when both divisor and dividend have decimal points?

Prove your answer with the problem:

18.21÷ 5.3

92

CSU 3.4: Decimals/Division

Diagnosis: Decimals/DivisionAnswer to Question:

Adding decimal place picks up remainder

238 .9 30) 7167.0 Added decimal place

60 116

90 267 240

270 270

0 No remainder

Diagnosis: Decimals/DivisionPrescription:

1. Set up and run a long division process ignoring decimal point.2. Place decimal in quotient digits above decimal point in dividend.3. Compute more quotient digits by adding zeros to dividend

Execution: 46.34 Quotient digits have decimal above decimal 45)2085.30 Added zero allows extending division to remove

180 remainder of 18 285 270 153 135

18 0 A remainder of 18 would persist if another zero 18 0 was not added to the dividend. 0

Diagnosis: Decimals/Division with two decimal numbersAnswer to Question:

The decimal point in the quotient is to be placed above the decimal point in the dividend after it has been moved the number of decimal places used by the divisor.

Starting divisor’s decimal moved to end of divisor

18.21÷ 5.3=18.215.3

=18.21⋅105.3⋅10

=182.153

=182.1÷ 53

Moving decimal Move dividend’s decimalpoint to right is like by number of decimal multiplying by 10 places used by starting

divisor.

93

CSU 3.4

Divide, and round the answer to the nearest hundredth.

0.4561 ÷ 0.21

94

CSU 3.4: Decimals/Division

Diagnosis: Decimals/Division/RoundingPrescription:

1. Divide using long division moving decimal of dividend to right the number of decimal places used by divisor2. Compute quotient to one more place than required for rounding.3. Round quotient to required decimal place.

Execution: 2.1719 0.4561 ÷ 0.21=45.61 ÷ 21 21) 45.6100 Rounding 2.1719 to 42 thousandths place = 2.172

36 21 Add extra zeros as needed

28 151 to compute quotient to one 147 place past rounded place.

40 21 190 189

95

Divisor uses 2 decimal places.Move decimal in dividend 2 to right.Move decimal in divisor 2 to rightDo long division.

CSU 3.5

Write 68 as a decimal.

Write 512 as a decimal to the thousandths place.

Write 511 as a decimal.

96

CSU 3.5: Decimals/Fractions

Diagnosis: Decimals/Fractions

Prescription:1. Use long division to divide the numerator by the denominator.

Execution: .75

68

means 6÷ 8, going to long division 8)6.00

5 6 40 40 0

68

is equivalent ¿ .75

Diagnosis: Decimals/Fractions/RoundingPrescription:

1. Use long division to divide the numerator by the denominator2. Compute quotient to one more place than required to round to.3. Then round quotient to the required place.

Execution: .4166 .4166 rounded to thousandths place is .417 12)5.0000

4 8 Thousandths Next digit 20 digit = 6 is greater than 5

12 so add 1 to thousandths 80 digit, 6, to become 7 72 80

72

Diagnosis: Decimals/Fractions unending decimals.Prescription:

1. Long divide denominator into numerator2. Add zeros to dividend to allow computing more quotient digits 3. When quotient digits repeat in a fixed pattern stop and indicate repeated digit pattern with an overhead

bar (shown below).

Execution: .454545 Repeating decimal written as . 45=.4545... 11)5.000000 “ … “ = “and so on.”

4 4 60 55 50

97

44 60 55 …and so on

98

CSU 3.5

Convert 0.48 to a fraction in lowest terms.

Convert 0.064 to a fraction in lowest terms.

Write 21.8 as a mixed number.

Do this using a short cut and the prove it by also doing it the long way.

99

CSU 3.5: Decimals/Fractions

Diagnosis: Decimals/Converting decimals to fractions

Prescription:1. The numerator of the fraction is the decimal number’s digits2. The denominator is 1 with as many zeros after it as there are digits after the decimal.3. Reduce resulting fraction to lowest terms.

Execution:Decimal number’s digits Reduce to lowest terms

.48= 48100

= 12∙425 ⋅4 =12

25

2 digits here…means “1 with 2 zeros” here


Prescription:1. The numerator of the fraction is the decimal number’s digits2. The denominator is 1 with as many zeros after it as there are digits after the decimal.3. Reduce resulting fraction to lowest terms.

Execution: Same digits except zeros Reducing to lowest terms

0.0064= 6410000

= 16 ⋅4100 ⋅100

= 16 ⋅ 4100⋅25⋅ 4

= 16100 ⋅25

= 4 ⋅ 425⋅ 4 ⋅25

= 425 ⋅25

= 4625

4 digits after decimal means 1 with 4 zeros for denominator


Prescription:1. Convert the digits after the decimal to a fraction 2. Combine the fraction with the whole number digits to the left of the decimal to form a mixed number

Execution:From 21.8 we take .8 and convert it to a fraction

.8= 810

= 4 ∙25 ⋅2=4

5

Combine the whole number part 21 with 45 to get the mixed number 21 4

5 Proof using the long way:

21.8=21+ .8=21+ 810

=15+ 4 ∙25⋅2=21+ 4

5=21 4

5

100

CSU 4.1

1. What is the mathematical meaning of percent?

2. What does percent tell us about a number?

3. What kind of number is a percent number?

4. What symbol is used to indicate percent?

What is the difference between a regular number (such as a whole number, a mixed number, a fraction, a decimal number) and percent?

Convert the following percents into numbers.1. 50%

2. 75%

3. 25%

4. 33%

5. 17%

6. 150%

101

CSU 4.1: Percents

Diagnosis: Percents/FundamentalsAnswer to Questions:

1. Percent means “per hundred.”

2. Percent tells us how big a quantity is relative to 100.

3. A percent number is a kind of fraction based on a denominator of 100.

4. The symbol used to indicate percent is the percent sign “%.”

Diagnosis: Percent/Fundamentals

Answer to Question:

Any number is relative to the number one.

A percent is relative to the number 100.

For example the number 50 means a quantity that is 50 times the quantity of one.

However, 50% means 50 relative to 100, which is the same as 50

100

Since 50

100=.5=1

2 , we see that 50% means one half.

Diagnosis: Percents/Converting to numbersPrescription:

1. To convert a whole number percent to a regular number remove the percent sign, “%,” and divide the number by 100.

Execution:

50%=50

100

75%=75

100

25%=25

100

33%=33

100

17%=17

100

150%=150100

102

103

CSU 4.1

Change 58.6% to a decimal number.

Do this the long way and then indicate a short cut.

Change the following percents to decimals.

1. 50%2. 13.8%3. 4.42%4. .7%5. .006%

Show how to change decimals into percents.Change the following:

1. 0.422. 2.53. 0.94. 0.0075. 15

104

CSU 4.1: Percents

Diagnosis: Percents/Converting to numbers

Prescription:1. To convert a percent number to a regular number remove the percent sign, “%,” and divide the number

by 100.Execution:The long way:

58.6 %=58.6100

=58+0.6100

= 58100

+ 0.6100

=.58+0.006=0.586

Dividing by 100 is the same as moving the decimal 2 places to the left.

Short cut: Remove % sign and move decimal point 2 places to left.

Diagnosis: Percents/Converting percents to numbers

Prescription:1. Remove % sign and move decimal point two places to the left.

Execution:1. 50% = 0.50

2. 13.8% = .138

3. 4.42% = .0442

4. 0.7% = 0.007

5. 0.006% = 0.00006

Diagnosis: Percents/Converting numbers to percents

Prescription:1. Move the decimal point 2 places to the right and affix a % sign.

Execution: 1. 0.42 = 42%2. 2.5 =250 %3. 0.9 = 90%4. 0.007 = .7%5. 15 = 1500%

105

CSU 4.1

Change the following percent numbers to fractions.

1. 27%2. 55%3. 54%

Change the following percent number to a fraction in lowest terms.

6.5%

Change the following percent number to a fraction in lowest terms.

16 14

%

106

CSU 4.1: Percents

Diagnosis: Percents/Converting percents to fractions

Prescription:1. Create a fraction by dropping the % sign and dividing by 100.2. Reduce fraction to lowest terms.

Execution:

1. 27 %= 27100

2. 55 %= 55100

=11 ∙520 ∙ 5

=1120

Remove common factors

to reduce to lowest terms

3. 54 %= 54100

=27 ∙250 ∙2

=2750


Prescription:1. Create a fraction by dropping the % sign and dividing by 100.2. If decimal is in numerator, multiply numerator and denominator by the factor of 10 that eliminates the

decimal.3. Reduce fraction to lowest terms.

Execution: To eliminate this decimal, multiply top and bottom by 10

6.5 %= 6.5100

= 6.5 ∙10100 ∙10

= 651000

=13∙ 5

200 ∙5= 13

200

Reduce to lowest terms


Prescription:1. Create a fraction by dropping the % sign and dividing by 100.2. If the numerator is a mixed number convert it to an improper fraction 3. To divide the denominator into the improper fraction multiply the numerator (improper fraction) by the

inverted denominator. 4. Reduce fraction to lowest terms.

Execution: Forming improper fraction. Inverting and multiplying

107

16 14

%=16 1

4100

=

16 ∙ 4+14

100=

654

100=65

4∙ 1100

= 65400

=13 ∙580 ∙5

=1380

CSU 4.1

Change 58 to a percent.

Fill in the blanks in the table below connecting simple fractions to both their decimal and percent equivalents.

Fraction Decimal Percent

12143413231525

108

Fill in the blanks in the table below connecting simple fractions to both their decimal and percent equivalents.


183858

CSU 4.1: Percents

Diagnosis: Percents/ Fractions to PercentsPrescription:

1. Divide numerator by denominator using long division, adding zeros after the decimal of the dividend to get more places past decimal.

2. After above division move decimal two places to left to get %.Execution:

.625 8)5.0000 After long division move decimal two place to right to get 4 8 percent.

20 5/8=.625 = 62.5% 16 40 Fraction as a percent

40 Fraction as a decimal

Diagnosis: Percents/Fractions to Decimals to Percents

Answer to Questions:Fraction Decimal Percent

12

0.5 50%

14

0.25 25%

34

0.75 75%

13

. 3 33 13

%

23

. 6 66 23

%

15

0.2 20%

109

25

0.4 40

Diagnosis: Percents/ Fractions to Decimals to Percents



18

.125 12.5 %

38

.375 37.5%

58

.625 62.5%

CSU 4.1

Change 712 to a percent.

Change 2 34 to a percent

110

111

CSU 4.1: Percents

Diagnosis: Percents/Changing fractions to percentsPrescription:

1. Divide numerator by denominator using long division, adding zeros after the decimal of the dividend to get more quotient places.

2. After above division move decimal two places to left to get %.3. Reduce repeating digits in quotient to equivalent fraction.

Execution: .5833 Repeating digit 6 can be replaced by 3

12)7.0000 Added zeros Note that 3=.333 …=13

6 0 1 00 96 40 Move decimal 2 places to right 36 to convert to %. 40

36 .5833…= .5833 …=58.33 …%=58. 3%=58 13

%

Diagnosis: Percents/Changing fractions to percents

Prescription:1. When possible write down decimal equivalent to given fraction or given mixed number.2. Move decimal point 2 places to right.

Execution:

2 34=2.75=275 %

112

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