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Fractions

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Fractions

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  1. Fractions http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.FRA&lesson=html/video_interactives/fractions/fractionsSmall.html

  2. A Review • Remember that an equivalent fraction is a fraction that when reduced or enlarged has the same value • 6 = 12 = 1 24 48 4 Try 2130 14 72

  3. A Review • Remember that a mixed number is a combination of a whole number and proper fraction • Remember that an improper fraction is when the numerator is larger than the denominator • 13 = 7(Take the denominator and multiply it by the whole number, add the 4 4 numerator and express over the denominator) • 9 = 21 (Take the denominator and multiply it by the whole number, add the 4 4numerator and express over the denominator) Try #2 and 3 and in workbook pg 48

  4. A Review • When adding and subtracting fractions, you need to ensure that you have a common denominator • It is best to use the least common multiple as the common denominator – less simplifying later • There are 3 Simple Steps to add fractions: • Step 1: Make sure the bottom numbers (the denominators) are the same • Step 2: Add the top numbers (the numerators). Put the answer over the same denominator. • Step 3: Simplify the fraction (if needed). • 1 + 1 3 6 • 2  +  1 = Step 1 6 6 • 2 + 1 = Step 2 6 • 3 = 1 Step 3 6 2

  5. Adding Mixed Fractions I find this is the best way to add mixed fractions: Step 1: convert them to Improper Fractions Step 2: then add them (using the steps for adding) Step 3: then convert back to Mixed Fractions: Example: What is 2 3/4 + 3 1/2 ? • Convert to Improper Fractions: 2 3/4 = 11/4 3 1/2 = 7/2 • Common denominator of 4: 11/4 stays as 11/4 7/2 becomes 14/4 • Now Add: 11/4 + 14/4 = 25/4 • Convert back to Mixed Fractions: 25/4 = 6 1/4

  6. Another way to Add Mixed Fractions To add mixed fractions add the whole parts of the mixed fractions firstand then the fraction parts.. • Example: • 21 + 32 77 • Solution: • 21+32 = (2 + 3)+ (1 + 2) = 5 +3=53 777 77 7 • 38 + 55 96 • Solution: • 38+55 = (3 + 5)+ (8 + 5) = Must first make the denominators equal  9696 • 8+ (16 + 15) = 831=(8 + 1)13=913 1818181818

  7. Subtracting Proper Fractions • There are 3 simple steps to subtract fractions • Step 1. Make sure the bottom numbers (the denominators) are the same • Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator. • Step 3. Simplify the fraction. 3 – 1 4 4 3  –  1  = Step 1 4 4  3 – 1  =  Step 2 4 2 = 1 Step 3 4 2

  8. Subtracting Mixed Numbers • Just follow the same method, but subtract instead of add: Example: What is 15 3/4 - 8 5/6 ? • Convert to Improper Fractions: 15 3/4 = 63/4 8 5/6 = 53/6 • Common denominator of 12: 63/4 becomes 189/12 53/6 becomes 106/12 • Now Subtract: 189/12 - 106/12 = 83/12 • Convert back to Mixed Fractions: 83/12 = 6 11/12

  9. To subtract mixed fractions subtract the whole parts of the mixed fractions first and then the fraction parts. But  what to do if the left side fraction part is smaller than the right side fraction part? • Example: • Find 72 - 24 55 • Solution: • 72-24 = (7 - 2)+ (2 - 4) = 5+ (2 - 4) = Our problem   is   2<4 55555555 • 4+ (5+2 - 4) =      Take 1 from 5 and add it to 2 5555 (Remember?  1 = 5) 5 • 4+(7-4) =43 555

  10. 3.1 Using models to Multiply fractions and whole numbers

  11. Fraction Circle Model • 7 × ¾ 5 ¼

  12. Rectangle Model • 7 × ¾ draw a rectangle that has a base of 7 units and height of 1 unit, divide the height into fourths. Shade ¾ of the rectangle. • The area of the shaded rectangle is base x height = 7 x ¾ Each small rectangle has an area of ¼ so you have 21 x ¼ = 21/4 = 5 ¼

  13. Number Line • 7 × ¾ • You can use a number line – for this example, you would break the line into fourths. Make 7 jumps of ¾ and see where you land – 5 ¼ 5 3 4 7 1 2 6

  14. Practice • 3 x 7/12 • 20 x ¾ • 2/3 x 18 • 4/9 x 10 • 6 x ¾ • 5/8 x 9 • Complete workbook pg 50 and 51

  15. 3.2 Using Models to Multiply Fractions

  16. Counters • 2/3 x 6/8 can also be read 2/3 of 6/8 • Model one whole set of 8ths with eight counters. You have 6 of 8 counters • To model thirds, arrange the 6 counters into equal groups. Each group of 2 counters represents 1/3. So 2/3 of 6 counters is 4 counters. So 4 counters represents 4/8 or ½ of the original whole.

  17. Counters • ¼ of 2/5 • Because you cannot take the 2 of the 5 and break into even groups of 4, you need to make groups of 2/5 until you can break it into 4 even groups. • So ¼ of 2/5 is 1/10 because you have 1 of the 10 whole. • Always refer back to the whole

  18. Paper • ¾ x ½ • Take a piece of paper and fold it in 4. Colour 3 sections. • Take the same piece of paper and fold it the other direction in 2. Colour 1 section. • How many sections are coloured twice? • How many sections are there in total? • 3/8

  19. Area Model • ¾ of ½ • Draw a rectangle that is and break it into 2 even sections, colour 1 of 2 sections • Break that same rectangle into 4 sections and colour 3 of 4 sections • See how many sections are coloured twice and express that over the whole number of sections. ¾ x ½ = 3/8

  20. Area Model • 6/7 x 1/3 • 6 of 21 are coloured so 6/7 x 1/3 = 6/21 = 2/7

  21. Practice • 1/5 x ¼ • ¾ x 2/5 • 3/5 x 5/8 • 4/7 x 2/3 • Complete workbook pg 52 - 53

  22. 3.3 Multiplying Fractions

  23. A Question For You • One quarter of Canada’s 20 ecozones are marine ecozones, which include parts of the ocean. The rest of Canada’s ecozones are terrestrial ecozones. They include parts of the land, and may contain rivers, lakes, and wetlands. • A) How many marine ecozones does Canada have? • B) How many terrestrial ecozones does Canada have?

  24. Remember to Reduce • Multiplying fractions is easy, you simply multiply the numerators than multiply the denominators, however, this can lead to huge numbers that need to be simplified • 13 x 3 = 13 x 3 = 39(Yikes who wants to simplify this?!) 9 26 9 x 26 234

  25. Remember to Reduce • Instead you can reduce the question before you multiply, this needs to be done across the question 13 x 3 = 9 26 1 1 1 x 1 = 1 3 x 2 6 2 3

  26. Remember to Reduce • Remember to reduce the question before you multiply, this needs to be done across the question 7 x 9 = 3 14 3 1 1 x 3 = 3 = 1 ½ 1 x 2 2 2 1

  27. Practice • 5/6 x 2/7 • 3/2 x 1/6 • 8/9 x 9/8 • ¾ x 2/9 • Complete workbook pg 54 and 55

  28. 3.4 Multiplying Mixed Numbers

  29. Area Model • You can model the multiplication of tow mixed numbers of improper fractions using partial areas of a rectangle. • 2 ½ x 1 ¼ 2 ½ 2 x 1 = 2 ½ x 1 = ½ 2 x ¼ = ½ ½ x ¼ = +1/8 3 1/8 1 2 x 1 ½ x 1 ¼ 2 x ¼ ½ x ¼

  30. Another Example • 1 ¾ x ½ ½ + 3/8 = 7/8 ½ 1 ½ x 1 = ½ ¾ ½ x ¾ = 3/8

  31. A Rule for Multiplying Mixed Numbers • Express them as improper fractions and then multiply the numerators and multiply the denominators • 1 1/5 x 3 1/8 = • 6 x 25 = 5 8 5 3 3 x 5 = 1 x 4 15 = 3 ¾ 4 1 4

  32. Practice • http://nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=grade_g_3.html • Try Workbook pg 56 and 57

  33. 3.5 Dividing whole numbers and fractions

  34. Number Line =8 • 6 ÷ ¾ • This is asking how many three-quarters there are in 6. Take the 6 wholes, break each into quarters, make jumps of size 3, how many jumps 6 1 2 3 4 5 7 8

  35. When the Fraction is First? • 1/6 ÷ 2 • End on 1/6 and break it into 2 equal groups, how large is each group 1/12 6 6 6 6 6 6 6 6

  36. Manipulatives, Whole Number First • 5 ÷ 3/4 • Think how many 3/4 are in 5 wholes. 1 1 3 4 2 3 2 2 1 3 4 4 5 5 6 6 2/3 5 6 6

  37. A game • Fraction 50 • Player 1 chooses a square on the board, and places a counter on that square. They calculate the answer and record • Player 2 chooses a square on the board, and places a counter on that square. They calculate the answer and record. • Play until one player reaches 50 • Another round – get exactly 50

  38. Practice • Try Workbook pg 58 and 59

  39. 3.6 Dividing fractions

  40. Common Denominators • One way to divide fractions is to find common denominators 4 ÷ 1 = 8 ÷ 1 = 5 10 10 10 This is asking how many 1 tenths are there in 8 tenths? On a number line, how many jumps does it take to make it to 8 tenths? 8 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10

  41. Number Line • To use a number line, you need to find common denominators • 3/5 ÷ 1/4 = 12/20 ÷ 5/20 • This is asking how many jumps of 5 twentieths can you make to get to 12 twentieths 2 2/5 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 6 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 2 2/5

  42. Multiplication is the key 3/1 = 3 12/4 = 3 2/1 = 2 6/3 = 2 4/1 = 4 12/3 = 4 6/1 = 6 6/1 = 6 2/1 = 2 48/24 = 2

  43. A Game • Reciprocal Concentration • Shuffle the set of cards and place face down into two rows • Player 1 turns over a card and tries to match the pairs of reciprocals – no match turn them back over • Player 2 takes a turn • If a match is found, the player keeps those cards and has another turn. • The player with the most cards wins

  44. Multiplication • What pattern did you notice • To divide by a fraction, you can multiply by its reciprocal (remember if it is a mixed number, to convert to an improper fraction and reduce, before multiplying)

  45. Practice • Try workbook pg 60 and 61

  46. 3.7 Dividing Mixed numbers

  47. Remember Write the mixed number as an improper fraction first

  48. Use a Number Line Divide 2 2⁄3 ÷ 1 3⁄4 = Make improper fractions 8/3 ÷ 7/4 Write each fraction with a common denominator 32/12 ÷ 21/12 How many 21 twelfths are in 32 twelfths Use a number line and divide into twelfths, each group of 1 will be 21 segments 1 11/21 1 11/21 32/12

  49. Use Common Denominators Divide 2 2⁄3 ÷ 1 3⁄4 = Make improper fractions 8/3 ÷ 7/4 Write each fraction with a common denominator 32 ÷ 21 = 32 ÷ 21 = 111 12 12 1 21

  50. Use Multiplication Divide 2 2⁄3 ÷ 1 3⁄4 = Make improper fractions 8/3 ÷ 7/4 Write the reciprocal 8 x 4 = 32 = 111 3 7 21 21