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November 4th Lecture on Multiples, Least Common Multiples, and Comparing/Ordering Fractions.

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- 1. Chapter 5: Section 1 Comparing and Ordering Fractions November 4 th , 2008. Remind Your Folks to Vote.
- 2. Multiples <ul><li>A multiple of a number is the PRODUCT of that </li></ul><ul><li>number and any nonzero number. </li></ul>12 is a multiple of 4
- 3. <ul><li>Multiples of 5: </li></ul><ul><li>1 x 5 = 5 </li></ul><ul><li>2 x 5 = 10 </li></ul><ul><li>3 x 5 = 15 </li></ul><ul><li>etcetera </li></ul><ul><li>Multiples of 10: </li></ul><ul><li>1 x 10 = 10 </li></ul><ul><li>2 x 10 = 20 </li></ul><ul><li>3 x 10 = 30 </li></ul><ul><li>etcetera </li></ul>5, 10, 15, 20, 25, 30, etc. 10, 20, 30, 40, 50, 60, etc. What is the LEAST COMMON MULTIPLE of 5 and 10? What are the COMMON MULTIPLES of 5 and 10?
- 4. Word Problem <ul><li>Today both the school baseball and school soccer teams had games. The baseball team plays every 6 days. The soccer team plays every 5 days. When will both teams have games on the same day again? </li></ul>Finding the COMMON MULTIPLES of 5 and 6 will tell you on which days, from TODAY , that the two teams play on the SAME day .
- 5. <ul><li>Find the COMMON MULTIPLES of 5 and 6. </li></ul>Word Problem Multiples of 5: 5, 10, 25, 30, 35, 40…etc. Multiples of 6: 6, 12, 18, 24, 30…etc. Go no further! What is the LEAST COMMON MULTIPLE ? The next time the Baseball team and Soccer team will play on the same day is 30 days from today!
- 6. Find the Least Common Multiple: LCM <ul><li>3, 4: </li></ul><ul><li>4, 5: </li></ul><ul><li>3, 4, 5: </li></ul>
- 7. LCM with Prime Factorization <ul><li>Find the LCM of Nasty Numbers . </li></ul><ul><li>1) Write the Prime Factorizations ( Factor Tree ) </li></ul><ul><li>2) Use the greatest power of each factor. </li></ul><ul><li>3) Multiply . </li></ul>Multiples of 12 = 2 2 • 3 (2 • 2 • 3) = 12 Multiples of 40 = 2 3 • 5 (2 • 2 • 2 • 5) =40 Multiply: 2 3 • 5 • 3 = 2 • 2 • 2 • 5 • 3 = 8 • 15 =120
- 8. <ul><li>9, 15: </li></ul><ul><li>12, 15, 18: </li></ul>Find the Least Common Multiple: LCM
- 9. Use Prime Factorization for Variables. <ul><li>Find the LCM for 6a 2 and 18a 3 : </li></ul><ul><li>6a 2 : 2 • 3 • a 2 </li></ul><ul><li>18a 3 : 2 • 3 2 • a 3 </li></ul>2 • 3 2 • a 3 = 2 • 3 • 3 • a 3 = 9 • 2 • a 3 = 18a 3 18a 3 is the Least Common Multiple
- 10. Comparing Fractions <ul><li>You can use a number line to compare fractions. </li></ul><ul><li>Comparing fractions means INEQUALITIES . </li></ul>4/9 and 2/9 -4/9 and -2/9 -4/9 and 2/9
- 11. Least Common Denominator <ul><li>Fractions can have different denominators. </li></ul><ul><li>1) Rewrite the fractions with a common denominator. </li></ul><ul><li>2) Compare the numerators. </li></ul><ul><li>3) The Least Common Denominator (LCD) of two ore more fractions is the Least Common Multiple of the denominators. </li></ul>
- 12. Quick Reminder on Finding Common Denominators. <ul><li>Multiply the fraction by 1 (or a fractional form of 1) to make an equivalent fractions with a usable denominator. </li></ul><ul><li>A usable denominator would be the Least Common Denominator, which can be found through Prime Factorization. </li></ul>
- 13. Word Problem Example <ul><li>The math team won 5/8 of its competitions and the debate team won 7/10 of its competitions. Which team won the greater fraction of competitions? </li></ul>Step 2) Write equivalent fractions with a denominator of 40.
- 14. Word Problem Example <ul><li>The math team won 5/8 of its competitions and the debate team won 7/10 of its competitions. Which team won the greater fraction of competitions? </li></ul>Step 1) Find the LCM of 8 and 10: 8 = 2 3 . 10 = 2 times 5. LCM = 2 3 • 5 = 2 • 2 • 2 • 5 = 8 • 5 = 40
- 15. Word Problem Example <ul><li>The math team won 5/8 of its competitions and the debate team won 7/10 of its competitions. Which team won the greater fraction of competitions? </li></ul>Step 3) Compare the fractions. The debate team won the greater fraction of competitions.
- 16. Ordering Fractions = Inequalities <ul><li>Do the previous three steps for ordering fractions. </li></ul><ul><li>1) Find the LCM of the denominators. </li></ul><ul><li>2) Write equivalent fractions with the LCM of the denominators. </li></ul><ul><li>3) Compare the fractions with inequalities. </li></ul>
- 17. Chapter 5, Section 2: Fractions and Decimals.
- 18. Remember Long Decimal Division? <ul><li>Convert Fractions to Decimals by Dividing the Numerator into the Denominator . </li></ul>When there is no remainder to a division problem, then the quotient is called a TERMINATING DECIMAL . 5/8 = ? There may be several decimal places before you get to a remainder of zero.
- 19. Write each fraction or mixed number as a decimal. <ul><li>¼ </li></ul><ul><li>1 7/8 </li></ul><ul><li>3 3/10 </li></ul><ul><li>3/5 </li></ul>
- 20. Repeating Decimal <ul><li>Repeating Decimal : When the same block of digits are repeated without end in the quotient. </li></ul>Examples: 2/3 and 15/11, try them…
- 21. Comparing/Ordering Fractions <ul><li>When Comparing or Ordering Fractions, it might help to convert the Fractions to Decimals. </li></ul>Order from least to Greatest: 0.2, 4/5, 7/10, 0.5
- 22. Writing Decimals as Fractions <ul><li>Reading a decimal correctly provides a way to write a fraction. </li></ul>0.43 is read as “ forty-three hundredths ”, which is the same as 43/100 . Write 1.12 as a mixed number in simplest form: Write 2.32 as a mixed number in simplest form:
- 23. Algebra and Repeating Decimals as Fractions. <ul><li>Write 0. 7272 (repeating) as a fraction in simplest form. </li></ul>N = 0. 72 Let the variable n equal the decimal . 100N = 72. 72 Because 2 digits repeat, multiply each side by 10 2 , or 100. 100N = 72. 72 - N - 0.72 The subtraction property of Equality allows you to subtract an equal quantity from each side of the equation. So, subtract to eliminate 0. 72 .
- 24. Algebra and Repeating Decimals as Fractions. <ul><li>Write 0. 7272 (repeating) as a fraction in simplest form. </li></ul>100N = 72. 72 - N - 0.72 99N = 72 Divide each side by 99. 99 99 N = 72/99 Divide the numerator and denominator by the GCF, 9. N = 8/9 Simplified.
- 25. Assignment #33 <ul><li>Page 227: 15-43 odd. (15 problems). </li></ul><ul><li>Page 232: 27-49 odd. (12 problems). </li></ul><ul><li>25 Problems total. </li></ul>

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