Chapter 3: Operations with Fractions Section 2

Chapter 5: Ratios, Rates & Proportions Section 4Solving Proportions

1Anticipatory SetCalifornia StandardsNumber 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 DisciplinePROPORTION: An equation stating that two RATIOS are EQUAL.Examples: 1/2 =2/4a/b = c/d, where b and d CANNOT equal ZEROUNIT RATE: The RATE of ONE UNIT for a given quantity.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 = 12/14Are 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)PROPORTION: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.Solving Proportions Using Unit Rate(Input/Modeling)Once again, we realize that new skills build off of previous ones. In this case, we are revisiting UNIT RATES & applying them to a new situation.When given a proportion, you can use UNIT RATE to solve a proportion.First, find the UNIT RATE, then MULTIPLY to solve the problem.Example: A store sells 4 Champion Candy bars for $3.00. You plan on purchasing 10. How much will the candy bars cost you?4 Champion Candy Bars cost $3.00. $3.00 for 4. $3.00/4 = $0.75 a candy bar.The UNIT RATE is $0.75 for ONE Champion Candy Bar.You plan on purchasing 10 candy bars. (Unit Rate)(Number of Bars) = COST.($0.75)(10 Candy Bars) = $7.5010 Champion Candy Bars will cost you $7.50.

Solving Proportions Using Unit Rate(Input/Modeling)Example #2A store charges $8.40 for a dozen Hello Kitty pencils.You only want to purchase 7 pencils. What is your total cost?$8.40 for a DOZEN pencils means $8.40/12.Unit Rate is $0.70 a pencil.(Unit Rate)(Number) = COST($0.70)(7) = $4.907 Hello Kitty Pencils will cost you EXACTLY $4.90. Example #3At a bakery, you can purchase 20 croissants for $30.00.You would like to purchase 6 croissants for your family. How much will you be charged?$30.00 for 20 croissants means $30.00/20.Unit Rate is $1.50 a croissant.(Unit Rate)(Number) = COST($1.50)(6) = $9.00The bakery will charge you $9.00 for the 6 croissants.

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 = 175ALGEBRAIC: a/b = c/db and d CANNOT equal ZERO (0). ad = bcSolving Using the Cross Products Property(Input/Modeling)Referring back to CROSS PRODUCTS PORPERTY, you can use PROPORTIONS and ALGEBRA to solve.Use PROPORTIONAL setup & Cross Products Property to solve the problem.Example: Solve 28/35 = X/175Remember the Cross Products Property. You use proportion and the property to create a math equation where Algebra can solve for the One unknown value.28/35 = X/175(28)(175) = (35)(X)4,900 = 35X4,900/35 = 35X/35140 = X28/35 = 140/175 BOTH simplify down to 4/5. CPP yields 4,900 on BOTH SIDESSolving Using the Cross Products Property(Input/Modeling)Example #2:Solve 2/5 = E/86.52/5 = E/86.5Use Cross Products(2)(86.5) = (5)(E)173 = 5E173/5 = 5E/534.6 = EDOUBLE CHECK2/5 = 34.6/86.5(2)(86.5) = (34.6)(5)173 = 173Answer is CORRECTExample #3Solve 1.4/5.7 = 28/H1.4/5.7 = 28/HUse Cross Products (1.4)(H) = (5.7)(28) 1.4H = 159.61.4H/1.4 = 159.6H = 114DOUBLE CHECK1.4/5.7 = 28/114(1.4)(114) = (5.7)(28)159.6 = 159.6Answer is CORRECT

The Big IdeaPROPORTIONSA pair of ratios that equal one another.Proportions can be solved using multiple methods.Using UNIT RATES to Solve Use the original rate to determine a UNIT RATE.Multiply the UNIT RATE by the NUMBER of Units to determine the Cost.Using CROSS PRODUCTS PROPERTY to SolveCross 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. Use this property and ALGEBRA to solve the missing value.Once the missing cross product is determined, DOUBLE CHECK to make certain it works in the original proportion.Check for UnderstandingPlease 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 teachers signal. On the count of 3, hold up your wipe boards.

Checking for Understanding Question #1Question #1:

A store sells 5 pairs of socks for only $15.00. What is the Unit Rate?Select the BEST answer:A. $15.00 a pairB. $1.50 a pairC. $3.00 a pairD. $5.00 a pair

Checking for Understanding Question #2Question #2:

A 36 candies cost $10.80. How much would you be charged for 11 pieces of candy? Select the BEST answer:A. $39.60B. $64.80C. $118.80D. $10.80

Checking for UnderstandingQuestion #3Question #3:

A boutique sells 5 pairs of DESIGNER jeans for $650. How much would 3 pairs of jeans cost?

The pair of ratios can be simplified down to:A. $1.950.00B. $195.00C. $675.00D. $390.00Checking for Understanding Question #4Question #4:

Solve 45/55 = D/440

The pair of ratios can be simplified down to:A. 180B. 360C. 280D. 320

Checking for Understanding Question #5Question #5:

Solve 2.7/10.8 = R/75.6

Select the BEST answer:A. 18.9B. 22.8C. 21.7D. 19.4

Guided Practice/Independent PracticeGuided 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 PracticeTextbook pg. Card SharksVarious ArtistsTV Theme SongsiTunes 10.6.3Dating GameVarious ArtistsTV Theme SongsiTunes 10.6.3