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Assembly Line Grade 7 Ratios Clarification Standard 6.0 Knowledge of Number Relationships and Computation/Arithmetic Topic C. Number Computation Indicator 3. Analyze ratios, proportions, or percents Objective b. Determine or use ratios, unit rates, and percents in the context of a problem Clarification of Math Discussion Terms A RATIO is a comparison of two numbers or quantities. The ratio of 3 to 5 can be written as 3 out of five, 3:5, or as a fraction where the first number becomes the 3 numerator and the second, the denominator: . A PROPORTION is a statement that two 5 ratios are equivalent, an equation that illustrates that the two ratios are the same 3 18 proportional amounts of a whole thing. For example, the proportion = indicates that 4 24 if a whole thing were divided into 4 equal parts, taking 3 of the 4 parts would be the same proportional amount as taking 18 parts out of the same thing divided into 24 equal parts. EQUIVALENT RATIOS can be generated by using the number 1 as the identity element for both multiplication and division. That is, any number multiplied or divided 2 2 1 × 1= , ÷1 = by 1 gives the identical original number : 8 × 1 = 8, 57 ÷ 1 = 57, 9 9 2 1 2 673 10,000 etc. The number 1, however, can be written in infinitely, such as , , , 2 2 673 10,000 37.5 n abc , , etc. Therefore, multiplying or dividing a fraction or ratio by a version of 37.5 n abc 2 7 2 × 7 14 = . This illustrates that 2 out of 1 keeps its identical proportional value: × = 3 7 3 × 7 21 3 is the same proportional part as 14 out of 21. 2 14 can also be thought of as reduced to lowest terms by dividing by the common 3 21 14 7 are divided by 1, written as , the factor, 7. If the numerator and denominator of 21 7 14 ÷ 7 2 result is called the reduced, or simplified form: = If the numerator and 21 ÷ 7 3 denominator of the reduced form have no common factors, the ratio is said to be in lowest terms. Since there are an infinite number of fractions that are equivalent to a given 2 ratio such as , when comparing ratios, it is convenient to write the fractions in reduced, 3 Copyright 2008, Maryland Public Television, Learning Games to Go Project or simplified form: 6 2 = , 9 3 8 2 = , 12 3 10 2 = , 15 3 22 2 = , 33 3 17 2 = , 51 3 42 2 = 63 3 2,000,000 2 = etc. 3,000,000 3 Classroom Example 1 7 42 Is equal to ? 13 78 Method 1: Using multiplication 7 42 is equal to if there is a common factor by which 7 and 13 are multiplied to get 13 78 42 : 78 7×? 42 = , 13 × ? 78 7 × 6 42 7 42 = , therefore does equal . 13 × 6 78 13 78 Method 2: Using divisors 42 Starting with , we can find divisors, or factors of 42 and 78 that would simplify, or 78 42 7 42 ÷ ? 7 42 ÷ 6 7 reduce to : = . = . 78 13 78 ÷ ? 13 78 ÷ 6 13 Often students will do the simplifying, or reducing to lowest terms in several steps. For example, a student might recognize that 42 and 78 are both even numbers, making them 42 ÷ 2 21 divisible by 2. Dividing 42 and 78 by 2 would yield an equivalent ratio: = , 78 ÷ 2 39 however if we want the ratio in lowest, or simplest terms, as is desirable for comparison 21 purposes, would have to be reduced again, since 21 and 39 have a common factor, 3 : 39 21 ÷ 3 7 = . Although successive divisions will give the same final reduced form, it is 39 ÷ 3 13 more efficient to simplify by dividing by the GREATEST COMMON FACTOR, as 42 when dividing by 6 rather than 2 to reduce . The fraction will be in lowest or 78 simplified form if the numerator and denominator of the fraction have no common factors, which is to say that they are RELATIVELY PRIME. Copyright 2008, Maryland Public Television, Learning Games to Go Project Method 3: Using Cross-Products A useful characteristic of proportions is the CROSS- PRODUCT PROPERTY, which a c 7 42 states that if = , then a × d = b × c. Therefore, if = , then 7 × 78 = 13 × b d 13 78 42. 7 × 78 = 546 and 13 × 42 = 546, so yes, 7 42 does equal . 13 78 Classroom Example 2 4 Write five ratios that are equal to . 9 Answers will vary. Some possible answers would include: 8 4 = , 18 9 12 4 = , 27 9 20 4 = , 45 9 24 4 = , 54 9 36 4 = , 81 9 40 4 = , 90 9 44 4 = , 99 9 48 4 = , 108 9 28 4 = , 63 9 52 4 = , 117 9 32 4 = , 72 9 72 4 = 162 9 Students might be encouraged to check their answers by reducing or simplifying their 4 fractions to see if they do, in fact, reduce to . This would also offer the opportunity to 9 compare and contrast the efficiency of dividing by the greatest common factor, such as 40 dividing by 10 when reducing , compared with successive divisions in cases when the 90 72 72 ÷ 2 36 36 ÷ 9 4 : = , = greatest common factor is not obvious, as in 162 162 ÷ 2 81 81 ÷ 9 9 The Math in the Puzzle In the Assembly Line puzzle, players must demonstrate their knowledge of ratios and proportion by using gear ratios to label premium proportion of premium cans. To achieve the correct cans to total cans in the collection bins, players must study visual clues. Clues include the “can counter” array of small square lights at the top of the screen that matches the layout of the collection bin, the numerical relationship between the cans and the total number of cans in the destination bin, and the appearance of Copyright 2008, Maryland Public Television, Learning Games to Go Project each division on the conveyor belt. The visual clues on the belt are especially useful when the player is presented with two possible sets of gears that meet the required gear ratio. In the screen shot above, when the collection bin has place-holders for 3 premium cans out of the 15 total spots reserved for all cans, the target ratio is 3 1 or . Given the 15 5 choice of left gears with 4, 5, or 6 gear teeth, and right gears with 20, 25, and 26 teeth, players must choose between two acceptable combinations to get the required 1 ratio. 5 The markings on the belt will give the clue as to the correct gears. Note also that in all cases, both original gears on the opening screen of the puzzle must be replaced, even if one is being replaced by a gear with the same number of teeth. Copyright 2008, Maryland Public Television, Learning Games to Go Project