Ratio & Rate Reasoning PRESENTED BY MR. LAWS 6 TH GRADE MATH

Transcription

1 Ratio & Rate Reasoning PRESENTED BY MR. LAWS 6 TH GRADE MATH JCMS

2 Common Core State Standard (CCSS) 6.RP.3 -Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

3 Essential Question: How do I use ratio and rate reasoning to solve real-world and mathematical problems?

4 Key Vocabulary Chart, Equivalent, Graphs, Proportion, Rate, Ratio, Table. Unit Rate.

5 What is a Ratio? 1. A ratio is a comparison of two quantities by division. 2. A ratio can be written as the following: a. By writing the word to between two quantities. Ex. 2 to 1 or $1 to 3lbs etc. b. By placing a colon : make in between two quantities. Ex. 2:1; 1:3 etc. c. As a fraction. Ex. 2/1; ¾ etc.

6 What is a Rate? 3. A rate is a ratio that compares two quantities measured in different units. Ex.: 1 in to 50 miles; 1 mile per gallon etc. 4. A unit rate is a rate in which the second quantity in the comparison is one unit. Ex.: 3 books for $18; the unit rate is 1 book per $6.00.

7 How Do I Find Ratios from Tables? Lets take a look at the table below. 5. What is the answer for the missing yards at the end of the table? Feet Yards ? There is 1 yard to 3 feet(unit rate). The ratio is 1yd:3ft. Note: Always find the unit rate first! You can use multiplication - multiply 3 feet for each yard and you will get the correct number of feet. Ex. 3 feet x 2 yards = 6 feet. So 3 feet times what will give me 24 feet? You can also use division - divide 24 feet by 3 feet, and it will give you the number of yards. 24 feet/3 feet = 8 yards; Check! 3 feet x 8 yards = 24 feet.

8 How Do I Find Ratios from Tables? Example # 1 : At books unlimited, 3 paperback books cost $18. a. What would 7 books cost? b. How many books could be purchased with $54? # of Books ? Cost $18 $24 $30? $54

9 How Do I Find Ratios from Tables? Example # 1 : At books unlimited, 3 paperback books cost $18. a. What would 7 books cost? $42 ($6 x 7 books) b. How many books could be purchased with $54? 9 books ($54/6) # of Books ? Cost $18 $24 $30? $54 The ratio is $18 to 3 books = $6 per 1 book. Why?

10 How Do I Use Ratios to Plot the Pairs of Values on a Coordinate Plane? Example # 2: A shower uses 12 gallons of water in 3 minutes. a. Complete the table below. b. Plot the values on a coordinate Plane. Time (min) Water used (gal) What is the unit rate? 4 gallons of water to 1 min. Why? Note: Multiply the time by 4 to get the gallons used. How do I find the time for 20 gallons used? 20 Divide 20 gallons by 4 to get the time. 20/4 = 5 26

11 How Do I Use Ratios to Plot the Pairs of Values on a Coordinate Plane? b. Plot the values on a coordinate Plane. Time (min) Water used (gal)

12 What is Proportion? 6. A proportion is a statement that two ratios or rates are equivalent (equal). Example: 1 2 and 2 4 are equivalent. Therefore, is a proportion or is proportional.

13 How Do I Use Equivalent Ratios to Solve Proportions? Example # 3. In trail mix, the ratio of cups of peanuts to cups of chocolate candies is 3 to 2. How many cups of chocolate candies would be needed for 9 cups of peanuts? cups (peanuts) cups (chocolate) cups (peanuts) cups (chocolate) = = x 3 x 3 9 cups (peanuts)? cups (chocolate) 9 cups (peanuts) cups (chocolate) Answer: 9 cups of peanuts will need 6 cups of chocolate to make trial mix. 6 Step 1: Write a proportion. Step 2: Find the equivalent top proportion by finding how many times 3 cups of peanuts will give you 9 cups of peanuts. 3 x 3 = 9 Step 3: Multiply 2 cups of chocolate by 3, and you will get 6 cups of chocolate. What is the Unit Rate?

14 How Do I Use Equivalent Ratios to Solve Proportions? Example # 4. The ratio of nitrogen to potassium in a sample soil is 12: 9. The sample has 36 units of nitrogen. How much potassium does the sample have? (nitrogen) (potassium) (nitrogen) (potassium) = x 3 Answer: 27 units of potassium. 36 (nitrogen)? (potassium) = 36 x 3 27 (nitrogen) (potassium) Step 1: Write a proportion. Step 2: Find the equivalent top proportion by finding how many times 12 unit of nitrogen will give you 36 units of nitrogen. Step 3: Multiply 9 units of potassium by 3 and you will get 27 units of potassium. What is the Unit Rate?

15 How Do I Use Equivalent Ratios to Solve Proportions? Your Turn: a. One serving of Mike s crackers has 150 calories and a mass of 30 grams. How many calories are in 6 grams of the crackers? b. To clean a tank, ¾ cup of disinfectant is needed for every 2 gallons of water. How many cups of disinfected are needed for 20 gallons of water?

16 Summary What have you learned about this lesson? What are some important steps to remember when using ratios and rates to solve problems? Do you have any more questions about this lesson? Make sure you review your notes, add additional questions or notes, and write a summary or reflection.

17 How Do I Use Equivalent Ratios to Solve Proportions? Your Turn: Answers a. One serving of Mike s crackers has 150 calories and a mass of 30 grams. How many calories are in 6 grams of the crackers? 30 calories b. To clean a tank, ¾ cup of disinfectant is needed for every 2 gallons of water. How many cups of disinfected are needed for 20 gallons of water? 7 ½ or 7.5 cups