Math 010 online work that was due today at the start of class:

• Gateway Homework (turn in worksheet now, while I take roll)

• Section 1.3A Online Homework

• Please log in to MyLabandMastering • Click on “Gradebook”• Click “Review” by HW 1.3A• Notice than you can look at each problem and

answer, and even work similar problems again for practice, even after the deadline, without affecting your grade.

Any questions on any of the problems from Section 1.3A?

Some comments about entering answers for fractions:

1. Enter an answer like 12/1 just as 12. (If the denominator is 1, just write the answer as a whole number without the denominator.)

2. Check to make sure all fraction answers are completely simplified.

Example: How would you write the answer if it comes out to

42/66? (Answer: 7/11)

• 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.

• Once you’ve clicked “check answer” on a problem, that problem’s result is stored on the system and will be retained even if you don’t click “Save”.

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!

Remember: Homework counts for about 25% of your entire course grade, so scoring 100% on each assignment really makes a difference.

Things 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.

• If you are having trouble with a problem, check the on-line help available for each problem:

• Help Me Solve This• Textbook Pages• Video Clip (for some problems)

THIS IS SO IMPORTANT I’M SAYING IT ONE MORE TIME:

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

REMINDER: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.

Monday through Thursday 8:00 a.m. – 7:30 p.m.

NOW

CLOSE

YOUR LAPTOPS(You may reopen them when I finish the lecture, at which time we will help you

get started on the homework assignment.)

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.

You should be keeping both your lecture notes and written work for each problem in every homework assignment in a notebook.

Section 1.3, Part BLast time we covered:• Factoring numbers into prime factors• Simplifying fractions using prime factors• Multiplying fractions• Dividing fractions

Problems in Section 1.3A homework used these concepts, as do problems #3 & #5 in the Gateway Homework due today and on all regular Gateway Quizzes.

Today we will explore how to work with mixed fractions and how to add and subtract fractions.

• In this course, we will usually need to change mixed numbers into improper fractions in order to do our calculations.

• Also, fraction answers with a numerator larger than the denominator should be entered into the computer in that form (improper fraction) rather than as a mixed number.

NOTE: Gateway problems 4 & 6 and several of today’s homework problems

using mixed numbers all start with the same step.

A mixed fraction (mixed number) consists of an integer part and a fraction part. We want to covert the mixed number into an improper fraction, This is done by multiplying the integer part by the denominator of the fraction part, then adding that product to the numerator of the fraction and putting that sum over the original denominator.

Example: Convert the mixed number into an improper fraction:

Solution: First, note that

Then:

51 1 14 4 1 45 5

5 5 201 1 1 211 4 1 4 4 4

44 4

Converting a Mixed NumberInto an Improper Fraction:

145

Converting a Mixed NumberInto an Improper Fraction

Another way to look at it:

To convert 5 ¼: 1. Multiply the denominator of the fraction part (4)

by the whole number part (5) 5 4 = 20∙

2. Add the numerator of the fraction part (1) to this result: 1 + 20 = 21

3. Write this number over the denominator of the original fraction : ANSWER: 21/4

Sample Gateway Problem #4: Multiplying mixed numbers

Step 1: Convert the mixed number into an improper fraction: (Note that ) .

52 2 23 3 1 35 5

235

5 5 15 172 2 21 3 1 3 33 3 3

3

So becomes , which we can then solve the same way we did our fraction multiplication problems in the last lecture and in HW 1.3A that was due today.

623 75 17 6

3 7

Sample Problem #4 (continued)

Step 2: Factor both the numerators and denominators into prime factors, then write each fraction in factored form:

First fraction: 17 and 3 are both prime Second fraction: 6 = 2 3∙ and 7 is prime

So you can write 17 ∙ 6 as 17 ∙ 2 3∙ . 3 7 3 7Step 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.

17 63 7

//

17 ∙ 2 3∙ = 17 2∙ = 34 . 3 7 7 7

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Example from today’s homework:You try it now – you can work with others at your table, and we’ll come around and give you help, too.

ANSWER: 3 2

Sample Gateway Problem #6: Dividing with mixed numbers

Step 1: Convert the mixed numbers into improper fractions:

77 7 49 501 1 1 1 1

7 7 1 7 1 7 7 7 777 7

251 1 12 1 12 22

1 24 12 2 1 2 1 2 2 2 212 12

50 251 1

7 2 7 27 12 50 25 50 27 2 7 25

Now we can rewrite the problem as:

Then convert from division to multiplication by using the reciprocal of the second fraction:

Sample Problem #6 (continued)

Step 2: Factor both the numerators and denominators into prime factors, then write each fraction in factored form:

First fraction: 50 = 2 5 5 ∙ ∙ and 7 is prime Second fraction: 2 is prime and 25 = 5 5∙So you can write 50 • 2 as 2 5 5∙ ∙ • 2 . 7 25 7 5 5∙Step 3: Now just cancel any common factors that appear in Both numerator and denominator. Once you multiply out anyremaining factors, the result is your simplified answer.

2 5 5∙ ∙ • 2 = 2 2∙ = 4 7 5 5 7 7∙

50 25 501 1 27 2 7 2 7 257 12

/ / / / 19

Now we’ll move on to adding and subtracting fractions, which is usually a little more work than multiplying or dividing fractions, because before you add or subtract, both fractions have to be converted so they have the same denominator.

If your two fractions already have the same denominator, just add (or subtract) the numerators and put the result over that denominator:

b

ca

b

c

b

a

b

ca

b

c

b

a

Example Add the following fractions.

20

9

20

3

20

12

522

322

5

3

Adding or subtracting fractions with the SAME denominator:

1. Add (or subtract) the numerators together and write the sum (or difference) over the common denominator.

2. Simplify the fraction.

Example from today’s homework:You try it now – you can work with others at your table, and we’ll come around and give you help, too.

ANSWER: 2 3

Steps to follow for finding the least common denominator (LCD) of two fractions:

1. Factor both denominators into primes.2. List all the primes in the first denominator (with

multiplication signs between each number)3. After these numbers, list any NEW primes that appear

in the second denominator but not the first.4. Multiply this whole list of primes together. This is your

LCD.

But what if you need to add fractions in which the two denominators are different? •Then you have to find a COMMON (same) DENOMINATOR before you can add the numerators together. Simplifying your answer will be MUCH easier if you use the smallest possible (“least”) denominator that works for both fractions.

Finding the least common denominator (LCD) of two fractions:

Example: Find the LCD of 3/4 and 7/18: 1. Factor both denominators into primes.

4 = 2*2 18 = 2*9 = 2*3*32. Start with all the primes in the first denominator (with multiplication

signs between each number). If any prime number appears more than once in the first denominator, include each one in the LCD.

2*23. After these numbers, list any NEW primes that appear in the second

denominator but not the first.

2*2*3*34. Multiply this whole list of primes together. This is your LCD.

2*2*3*3= 4*9 = 36

Now you try:

Example: Find the LCD of 5/6 and 2/15: 1. Factor both denominators into primes.

2. Start with all the primes in the first denominator (with multiplication

signs between each number)

3. After these numbers, list any NEW primes that appear in the second

denominator but not the first.

4. Multiply this whole list of primes together. This is your LCD.

ANSWER: 2*3*5 = 30

NOTE: Gateway problems 1 & 2 on adding and subtracting fractions as well as many of the problems on today’s homework assignment can all be

done using the same set of steps.

Adding fractions and subtracting fractions both require finding a least common denominator (LCD), which as we just saw is most easily done by factoring the denominator (bottom number) of each fraction into a product of prime numbers (a number that can be divided only by itself and 1).

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Sample Gateway Problem #1: Adding Fractions

Step 1: Factor the two denominators into prime factors, then write each fraction with its denominator in factored form:

10 = 2 5 ∙ and 35 = 5 7∙ , so 3 + 2 = 3 + 2 . . 10 35 2 5 ∙ 5 7∙

Step 2: Find the least common denominator (LCD): LCD = 2 5∙ 7∙

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Sample Problem #1 (continued)Step 3: Multiply the numerator (top)and denominator of each fraction by the factor(s) needed to turn each denominator into the LCD.

LCD = 2 5∙ 7∙ 3 7 ∙ + 2 2 ∙ . 2 5∙ 7∙ 5 7∙ 2∙

Step 4: Multiply each numerator out, leaving the denominators in factored form, then add the two numerators and put them over the common denominator. 21 + 4 = 21 + 4 = 25 (note that 5 7 2 = 2 5 7 ∙ ∙ ∙ ∙

by 2 5 7 5 7 2 2 5 7 2 5 7 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ the commutative

property)

Step 5: Now factor the numerator, then cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer.

= 25 = 5 5 ∙ = 5 5 ∙ = 5 = 5 . 2 5 7 2 5 7 2 5 7 2 7 14∙ ∙ ∙ ∙ ∙ ∙ ∙

/ / 28

Example from today’s homework:You try it now – you can work with others at your table, and we’ll come around and give you help, too.

ANSWER: 13 15

Sample Gateway Problem #2: Subtracting Fractions

Step 1: Factor the two denominators into prime factors, then write each fraction with its denominator in factored form:

14 = 2 7 ∙ and 35 = 5 7∙ , so 5 - 2 . 2 7 ∙ 5 7∙

Step 2: Find the least common denominator (LCD): LCD = 2 7∙ 5∙

30

Sample Problem #2 (continued)Step 3: Multiply the numerator and denominator of each fraction by the factor(s) needed to turn each denominator into the LCD: form:

LCD = 2 7∙ 5∙ 5 5 ∙ - 2 2 ∙ 2 7∙ 5∙ 5 7∙ 2∙

Step 4: Multiply out the numerators, leaving the denominators in factored form, then add the two numerators and put them over the common denominator. 25 - 4 = 25 - 4 = 21 .

2 5 7 5 7 2 2 5 7 2 5 7∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙

Step 5: Now factor the numerator, then cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer.

21 = 3 7 ∙ = 3 7 ∙ = 3 = 3 . 2 5 7 2 5 7 2 5 7 2 5 10∙ ∙ ∙ ∙ ∙ ∙ ∙

/ / 31

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.

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You may now OPEN your LAPTOPS.

Reminder:The homework assignment

on section 1.3B is dueat the start of our next

class session.

Monday through Thursday 8:00 a.m. – 7:30 p.m.