Chapter 5: Ratios, Rates & Proportions Section 3

Anticipatory Set

California Standards Number Sense 1.2: Interpret and use ratios in different contexts. Number Sense 1.3: Use proportions to solve problems. Use Cross-Multiplication as a method for solving such problems.

Language of the Discipline PROPORTION: An equation stating that two RATIOS are EQUAL. Examples: 1/2 =2/4 a/b = c/d, where b and d CANNOT equal ZERO CROSS PRODUCTS PROPERTY: When given two ratios, this property states that the CROSS PRODUCTS will EQUAL each other. If the two ratios have EQUAL cross products, they form a PROPORTION. Example: 6/7 = 12/14 Are these a PROPORTION? -Using CROSS PRODUCTS, we take opposing values and multiply. *Remember to use the Numerators and Denominators on the diagonal from each other. a/b = c/d mean (a)(d) = (b)(c) (6)(14) = (7)(12) 84 = 84 -CROSS PRODUCTS proves that these two RATIOS are a PROPORTION

What is a PROPORTION? (Input) A PROPORTION is an EQUATION stating that 2 RATIOS are EQUAL. Some people think of EQUIVALENT Fractions as PROPORTIONAL. Another way to test for PROPORTIONALITY is to use the Cross Products Property. Here, 2 ratios are set equal, values are multiplied diagonally, if BOTH resulting products are EQUAL you have a PROPORTION. If not EQUAL, the ratios are NOT PROPORTIONAL.

Writing Ratios in Simplest Form (Input/Modeling) Once again, we realize that new skills build off of previous ones. In this case, we are revisiting EQUIVALENT Fractions. When given RATIOS, you can determine PROPORTIONALITY if when simplified to the lowest terms, they share the same simplified form. Example: Determine whether the given ratios are PROPORTIONAL. 21/24 and 77/88 Here BOTH ratios need to be simplified to LOWEST Terms. 21/24 can have its Numerator and Denominator divided by 3. 21/24 simplifies down to 7/8 77/88 can have its Numerator and Demonolater divided by 11. 77/88 simplifies down to 7/8. 21/24 and 77/88 are PROPORTIONAL.

CROSS PRODUCTS PROPERTY (Input/Modeling) With RATIOS and PROPORTIONALITY, a Mathematic Property will come in handy. Remember that properties come in handy because that give the RULE or GUIDELINE on how to attack a problem. The CROSS PRODUCTS PROPERTY states that if two ratios form a proportion, the CROSS PRODUCTS are EQUAL. If two ratios have EQUAL Cross Products, they form a PROPORTION. There are two ways to look at PROPROTIONS. ARITHMETIC: 5/7 = 25/35 (5)(35) = (7)(25) 175 = 175 ALGEBRAIC: a/b = c/d b and d CANNOT equal ZERO (0). ad = bc

Using the CROSS PRODUCTS Property (Input/Modeling) Example #2: Do the pairs of ratios form a proportion? 4/11 and 12/44 Use Cross Products (4)(44) = (11)(12) 176 = 132 Note how the Cross Products do not equal the same product. The pair of ratios is NOT PROPORTIONAL Example #3 Do the pairs of ratios form a proportion? 12/19 and 60/95 Use Cross Products Property (12)(95) = (19)(60) 1,140 = 1,140 Note how the Cross Products equal the same product. The pair of ratios is PROPORTIONAL

The Big Idea PROPORTIONS CROSS PRODUCTS PROPERTY A pair of ratios that equal one another. There are two ways to determine whether the pairs of ratios are a true PROPORTION. One method is to simplify both ratios to the lowest terms. If the result is the same, they are a PROPORTION. CROSS PRODUCTS PROPERTY Cross Products Property states that a pair of Ratios are a PROPORTION when their cross products equal the same value. Remember that you are taking the NUMERATOR from one Ratio and MUTLIPLYING it by the DENOMINATOR of the other. Do this two times on the diagonal and you are done. Once the cross products are determined, check the result for equality.

Check for Understanding Please determine the BEST answer for the following expression. Carry out ALL work and calculations in your NOTES for later reference Please write your answer on your wipe boards and wait for the teacher’s signal. On the count of 3, hold up your wipe boards.

Checking for Understanding Question #1 Which of the following pairs of ratios form a proportion? Select the BEST answer: A. 9/13 = 36/42 B. 1/7 = 7/56 C. 13/15 = 39/45 D. 8/15 = 24/60

Checking for Understanding Question #2 Which of the following pairs of ratios form a proportion? Select the BEST answer: A. 4/7 = 16/35 B. 3/13= 9/39 C. 21/29 = 42/57 D. 12/21 = 120/201

Checking for Understanding Question #3 The pair of ratios can be simplified down to: A. 7/24 B. 4/9 C. 3/5 D. 2/7

Checking for Understanding Question #4 What is the resulting product when the Cross Products Property is used? Select the BEST answer: A. 684 B. 646 C. 628 D. 696

Guided Practice/Independent Practice Textbook on pg. Work carefully, show your problem solving process, and double check all calculations. Use scratch paper to carry out your work. Once you have completed the assigned problems, please raise your pencil. When you get a stamp from Ms. Graham, continue on to Independent Practice. If you receive an “R” on your paper go to the back table. Independent Practice Textbook pg.