MULTIPLYING AND DIVIDING FRACTIONS

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1 MULTIPLYING AND DIVIDING FRACTIONS Build fractions from unit fractions. CCSS.Math.Content..NF.B. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. CCSS.Math.Content..NF.B..a Understand a fraction a/b as a multiple of /b. For example, use a visual fraction model to represent / as the product (/), recording the conclusion by the equation / = (/). CCSS.Math.Content..NF.B..b Understand a multiple of a/b as a multiple of /b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express (/) as (/), recognizing this product as /. (In general, n (a/b) = (n a)/b.) Apply and extend previous understandings of multiplication and division. CCSS.Math.Content..NF.B. Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret / as the result of dividing by, noting that / multiplied by equals, and that when wholes are shared equally among people each person has a share of size /. If 9 people want to share a 0-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? CCSS.Math.Content..NF.B. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. CCSS.Math.Content..NF.B..a Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. For example, use a visual fraction model to show (/) = 8/, and create a story context for this equation. Do the same with (/) (/) = 8/. (In general, (a/b) (c/d) = ac/bd.) CCSS.Math.Content..NF.B..b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. CCSS.Math.Content..NF.B. Interpret multiplication as scaling (resizing), by: CCSS.Math.Content..NF.B..a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. CCSS.Math.Content..NF.B..b Explaining why multiplying a given number by a fraction greater than results in a product greater than the given number (recognizing multiplication by whole numbers greater than as a familiar case); explaining why multiplying a given number by a fraction less than results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n a)/(n b) to the effect of multiplying a/b by. CCSS.Math.Content..NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. CCSS.Math.Content..NF.B.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (/), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (/) = / because (/) = /. CCSS.Math.Content..NF.B.7.b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, Barbara Boschmans of 0

2 create a story context for (/), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (/) = 0 because 0 (/) =. Exploration # : Multiplying Fractions with Pattern Blocks Example: Use TWO yellow hexagons as your whole to solve:. When multiplying fractions we are taking a fraction OF a fraction: we are finding OF. So first we need to find of the whole and shade it, then we find of the shaded part and color this with a different color (or pattern). Our answer will be what fraction the DOUBLESHADED area is of the whole. Shade of the whole: Then find of the shaded part: What fraction of the whole is DOUBLESHADED? of the whole is double shaded so Note: It s OK to show all steps on one picture, just make sure you show BOTH areas that are shaded and indicate your final answer. Barbara Boschmans of 0

3 Multiply using two yellow hexagons as your whole: a) b) c) d) Barbara Boschmans of 0

4 Exploration # : Multiplying Fractions Using Rectangles We can use a rectangle model to multiply fractions. Example: Multiply and. Remember that means taking OF! Cut the rectangle into fourths: Then divide the fourths into thirds and doubleshade : What fraction of the whole is DOUBLESHADED? of the whole is double shaded so Note: It s OK to show all steps on one picture, just make sure you show BOTH areas that are shaded and indicate your final answer. Use the rectangle method to multiply: a) b) c) Barbara Boschmans of 0

5 Exploration # : Dividing Fractions with Pattern Blocks Example: Use TWO yellow hexagons as your whole to solve: QUESTION: HOW MANY s GO INTO? Shade of the whole: Then find of the whole: Then see how many s go into ANSWER =! There are s that go into! a) Use TWO yellow hexagons as your whole to solve: Barbara Boschmans of 0

6 Example: Use TWO yellow hexagons as your whole to solve: QUESTION: HOW MANY s GO INTO? Shade of the whole: Then find of the whole: Then see how many s go into ANSWER = Since is bigger than, only part of will fit. This part is doubleshaded in the third picture. What part of goes into? of goes into, so the answer is! b) Use TWO yellow hexagons as your whole to solve: Barbara Boschmans of 0

7 Example: Use TWO yellow hexagons as your whole to solve: QUESTION: HOW MANY s GO INTO? Shade of the whole: Then find of the whole: Then see how many s go into Half of / fits One / fits ANSWER. fits into one whole time and then half of it also fits. So the answer is! c) Use TWO yellow hexagons as your whole to solve: Barbara Boschmans 7 of 0

8 Exploration # : Dividing Fractions Using Rectangles We can use a rectangle model to divide fractions. Example: Divide. QUESTION: HOW MANY s GO INTO? Shade of the whole: Then find of the whole: Now we divide the thirds into fourths vertically and the fourths into thirds horizontally: Note that is equivalent to 8 and is equivalent to. One fits: A second fits: :and another part of fits: two out of three parts of also fit ANSWER: s goes into whole times and then two out of three also fit. So the answer is. This is the EXACT answer. We can also express our answer in a whole number answer with a remainder: s goes into whole times and then two out of twelve remain. So the answer is remainder. Note: it s OK to do the first two steps at once (i.e. divide your two rectangles into fourths and thirds and then use the same rectangles to go to twelfths). You ll still need to show your last step. Use the rectangle method to divide: Barbara Boschmans 8 of 0

9 a) QUESTION: HOW MANY s GO INTO? Exact answer: Whole number with remainder answer: b) QUESTION: HOW MANY s GO INTO? Exact answer: Whole number with remainder answer: Barbara Boschmans 9 of 0

10 Multiplication and Division of Fractions: Applications CCSS.Math.Content..NF.B..c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat /8 of a pound of roast beef, and there will be people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? CCSS.Math.Content..NF.B. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. CCSS.Math.Content..NF.B.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if people share / lb of chocolate equally? How many /-cup servings are in cups of raisins? Apply and extend previous understandings of multiplication and division to divide fractions by fractions. CCSS.Math.Content..NS.A. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (/) (/) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (/) (/) = 8/9 because / of 8/9 is /. (In general, (a/b) (c/d) = ad/bc.) How much chocolate will each person get if people share / lb of chocolate equally? How many /-cup servings are in / of a cup of yogurt? How wide is a rectangular strip of land with length / mi and area / square mi?.. In a cost-saving measure, Chuck s company reduced all salaries by /8 of their present salaries. If Chuck s monthly salary was $00, what will he now receive? If his new salary is $800, what was his old salary?. Mrs. Martin bought 0 ¼ yards of material to make bridesmaid dresses and dress for the flower girl. The flower girl s dress needs only half as much material as a bridesmaid dress. How much material is needed for a bridesmaid dress? For the flower girl s dress?. An analysis of MAT 0 students revealed that of the students were from homes where both parents were professionals. Of these students, had at least one parent who is an educator. What fraction of MAT 0 students is from homes with two professional parents and at least one educator? If there are 8 students from homes with two professional parents and at least one educator, how many were enrolled in MAT 0? Barbara Boschmans 0 of 0