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### ..Multiplying and Dividing Fractions Without Using a CalculatorUsing Factoring

Reminder:

The Syllabus Quiz is due next session at the start of class.

You can review/redo this quiz as many times as you want to get 100%.

Only your best score counts towards your grade.

PLEASE HELP US OUT WITH THIS:When you go to the open lab next door in 203, please make sure you sign in on the log sheet and enter your instructor’s name and your section number. We need to collect this information to document lab usage and ensure future funding for tutors.

This is Math 010 section number 00 ???

My instructor’s last name is ???

(Your instructor will write this information on the whiteboard; please copy it into your notebook.)

- Monday through Thursday
- 8:00 a.m. – 6:30 p.m.

CLOSE

YOUR LAPTOPS

(You may reopen them when I finish the lecture, at which time I will take roll for today’s attendance points.)

NOTE:

There are TWO online homework assignments due at the start of the next class period:

1. Gateway Homework #1 – This one has a required paper worksheet on which you must show all the steps of your work for all 8 problems. (This worksheet was handed out on Day 1, but if you didn’t get a copy, just pick one up from a TA before you leave class today.) This worksheet must be turned in at the start of the next class period in order to get full credit for the assignment. (You still need to enter the answers online, AND show work on the worksheet.)

2. HW 1.3A – This one is completely online. You should still show work for these 20 problems in your class notebook, but you don’t need to turn any work in, just enter the answers online.

Lecture Slides:

- Power Point lecture slides are always posted in the “PowerPoint Lecture Slides” section for each day, so you can copy, view, or print them at any time.
- The slides might not always be projected during the lecture, but they will give you a good overview/ preview and will also help you review for tests and quizzes.
- If you have trouble keeping up with the pace of the lecture, you can print the slides as a handout (3 or 4 to a page) before you come to class and then take notes on them during the lecture.

Other study aids available in the “Assignments” section each day:

- Additional video lectures are also available in the Assignments section for each book section we cover in class.
- An on-line version of the textbook is also provided for each section, with interactive practice problems and video clips.
- Practice tutorial exercises can be tried before starting the regular homework assignment.

How to get to the online version of the textbook: 1. Click on the “Online Study Aids” button 2. Now click on “Multimedia Library”

Gateway Quiz Information:

- The Gateway Quiz covers material from sections 1.3-1.7 on fractions and the order of operations.
- The first Gateway Quiz will be given on the day before Quiz 1 (in the fourth week of the semester).
- You will have the chance to take several practice Gateway Quizzes before then, and there will also be Gateway-type problems covered in each day’s class lecture and on each day’s homework assignment.
- One-on-one or small group review/help/study sessions for the Gateway Quiz are available with the specially trained tutors and teaching staff in the Math TLC open lab in 203 JHSW from 8:00 a.m. until 6:30 p.m. every Monday, Tuesday, Wednesday and Thursday. No appointments are necessary.

Gateway Quiz Overview:

8 questions on working with fractions and order of operations.

No calculators are allowed for this quiz, so don’t use one when you take the practice versions.

Gateway Quiz must be passed with 100% score before you can pass Math 010.

You may take the quiz multiple times until you receive 100%, with different versions of the quiz given each time.

A worksheet for a preview Gateway homework assignment will be handed out on the first day of class (due on the third day of class.

Practice versions of the Gateway Quiz, both online versions and printed versions, can be found in the “Gateway Quiz Info” menu area of the Math 010 CourseCompass web site.

HELPFUL HINT:

Make sure you always take written notes during lecture (and as you do your homework assignments). Notes will be a big help as you study for quizzes and tests.

(And they might even earn you extra credit points some days as the TA and I walk around the class after lecture...)

Section 1.3A

Factoring a number means writing it as a product of prime numbers.

A prime number is a natural number (other than 1) whose only factors are 1 and itself.

The first few prime numbers are:

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

(It will help you on the Gateway Quiz

if you learn these by heart!)

Question: Why isn’t 6 a prime number?

Answer: Because it can be written as 2x3 (or 2·3)

Why isn’t 8 a prime number? 9? 10? 12?

What’s the next prime number after 29?

Check 30: Can you divide 30 by anything other than 1 and 30?

Yes, so it’s NOT prime.

(Numbers that are not prime are called composite numbers.)

Check 31: Try dividing it by all of the prime numbers up to half of 31. If none of them work, then 31 is prime.

Can you divide 31 by 2? NO By 3? NO By 5? NO

By 7? By 11? By 13? By 17?

( 17 is more than half of 31, and none of the primes up through 17 divide into 31, so we conclude that 31 is a prime number.)

Factor the number 44 into a product of primes.

Solution:

First, think of some number that divides into 44.

How about 2?

Then write 44 as 2·22. (Because 44÷2 = 22)

2 is prime, but 22 can be divided further, into 2·11

So 2·22 = 2·2·11 (NOTE: We could also write this as 22·11)

These are now all prime numbers, so we’re done.

(Always arrange the numbers in order from smallest to largest in your final answer).

Factor the number 150 into a product of primes.

Solution:

First, think of some number that divides into 150.

How about 10?

Then write 150 as 10·15. (Because 150÷10 = 15)

Both of these can be divided further:

10·15 = 2·5·3·5

These are now all prime numbers, so we’re done, except forarranging the numbers in order from smallest to largest.

Final answer:2·3·5·5(or 2·3·52)

Sample problem from today’s homework:

ANSWER: 2•2•5

Recall that afraction is a quotient of two numbers.

(“quotient” means you’re dividing the top number by the bottom number)

- The numerator is the top number.
- The denominatoris the bottom number.
Simplifying fractions (reducing to lowest terms) involves factoring numerator and denominator into prime numbers and then canceling any primes that appear on both top and bottom

- Fundamental Principle of Fractions
- Can cancel common factors in numerator and denominator.
- If a, b, c are real numbers such that b and c 0.

Why this works:

as long as c 0.

Simplify the following fractions.

Since there are no common terms, the original fraction is already simplified.

NOTE: Gateway Quiz problems 3 & 5 on multiplying and dividing fractions canboth be done using the similar stepsthat will be covered in the following slides. There will be some problems in the online homework for sections 1.3A & B that also use these techniques.

Basic strategy for multiplying and dividing fractions:

These kinds of problems DO NOT require finding a common denominator. They can be most easily done by factoring both the numerator (top number) and denominator of both fractions into a product of prime numbers, and then canceling any common factors (numbers that appear on both the top and the bottom.)

Sample Problem: Multiplying fractions

Step 1: Factor both the numerators and denominators into

prime factors, then write each fraction in factored form:

First fraction: 39= 3∙13 and50 = 2∙5∙5

Second fraction: 15= 3∙5 and26 = 2∙13

So you can write 39 • 15 as 3∙13 • 3∙5.

50 26 2∙5∙52∙13

Sample Problem: Multiplying fractions (continued)

Step 2: Now just cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer.

3∙13 • 3∙5 = 3∙3= 9 .

2∙5∙5 2∙13 2∙5∙2 20

/ /

/ /

NOTE: It is much easier to factor first and then cancel, rather than multiplying out the numerators and denominators and then trying to simplify the answer (especially if you aren’t using a calculator!) If you multiplied first, you’d have gotten

585, which would be nasty to simplify by hand…

1300

Sample Problem: Dividing fractions

Step 1: Multiply the first fraction by the 45÷21 =45•26

reciprocal of the second fraction.13 2614 21

(i.e. flip the second fraction upside down and change ÷ to • .)

Step 2: Factor both the numerators and denominators into

prime factors, then write each fraction in factored form:

First fraction: 45 = 3∙3∙5 and 13 = 13 (prime)

Second fraction: 26 = 2∙13 and21 = 3∙7

So you can write 45 • 26 as 3∙3∙5• 2∙13 .

13 21 13 3∙7

Sample Problem: Dividing fractions (continued)

Step 3: Now just cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer.

3∙3∙5 • 2∙13 = 3∙5∙2 = 30

13 3∙7 7 7

/ /

/ /

NOTE: Once again, it is much easier to factor first and then cancel, rather than multiplying out the numerators and denominators and then trying to simplify the answer (especially if you aren’t using a calculator!) If you multiplied first, you’d have gotten 1170, which would be pretty hard to simplify by hand.

273

REMINDER:

There are TWO online homework assignments due at the start of the next class period:

1. Gateway Homework #1 – This one has a required paper worksheet on which you must show all the steps of your work for all 8 problems. (This worksheet was handed out on Day 1, but if you didn’t get a copy, just pick one up from a TA before you leave class today.) This worksheet must be turned in at the start of the next class period in order to get full credit for the assignment. (You still need to enter the answers online, AND show work on the worksheet.)

2. HW 1.3A – This one is completely online. You should still show work for these 20 problems in your class notebook, but you don’t need to turn any work in, just enter the answers online.

Now open your laptops

- Open your browser.(http://pearsonmylabandmastering.com)
- Sign in to MyLab and Masteringusing the id and password you created when you registered on day 1.

3. Click on the name of this course.

Here’s what you should now be seeing:

- Now click the “Assignments” button at the left side of the screen.
- Then click on today’s lesson (Section 1.3A).

Here’s what you should now be seeing:

- The top link will always take you to the PowerPoint lecture slides that are available for you to print before class to take notes on or to look at after class to review any points that went by too fast during the lecture.
- At the bottom of the Assignments screen for each day you’ll find a link to the homework assignment that’s due before the next class. (Click where it says “Click hereto do the assigned homework…”)
- Today there are TWO assignments; remember that the Gateway HW #1 part has a required worksheet

NOW: Click on the 1.3A homework:

- Click on the 1.3A homework assignment that’s due before the next class. (Click where it says “Click here to do the assigned homework…”)
- Then select HW 1.3A from the list that appears:

- Now click on Exercise 1

Here’s what you should now be seeing:

- After you enter the answer to a problem, click “Check Answer” to see if it’s correct. For most problems, you’ll get three tries to get it right.
- Even if you get a problem wrong, you can do it over until you get it right (the computer will generate a new version for you to try.)

- You DO NOT need to save or print your homework or email me your results – as soon as you click “check answer” on a problem, your results are automatically saved and recorded in the online gradebook (yours and mine.)
- If you are having trouble with a problem, check the on-line help available for each problem:
- Help Me Solve This
- Textbook Pages
- Animation(for some problems)

IMPORTANT:Even if you get a problem wrong on each of your three tries, you can still go back and do it again by clicking “similar exercise”at the bottom of the exercise box. You can do this nine times, for a total of 30 tries (3 tries at each of 10 different problems. You should always work to get 100% on each assignment!

You can go back in and work on an assignment even after submitting it, provided the deadline has not expired, so again, you should always work to get 100% on each assignment.

Another thing to remember:

Take notes as you do each homework problem. Write down all steps (show your work!). Again, this helps tremendously when you’re studying for tests.

(Some days I might do a random “notebook check” after lecture, and will give extra credit attendance points to those who’ve taken legible notes.)

If there’s still time left in the class session after lecture, you should stay in the classroom to work on your homework till the end of the session. If you have finished the homework already, or if you get it finished before the end of the class period, show your on-screen 100% score to the teacher or TA and you may then leave class early.

Reminder lecture, you should stay in the classroom to work on your homework till the end of the session. :

Today’s homework assignment on section 1.3A is due at the start of our next class session, along with Gateway HW #1 (if you didn’t get a worksheet for that on Day 1, get one before you leave class today).

- Monday through Thursday
- 8:00 a.m. – 6:30 p.m.

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