Using the CCSS to Improve Student Achievement

Transcription

1 Using the CCSS to Improve Student Achievement Grade 6 Participant Handouts Ratio and Rate Problems Defining Rates and Ratios Use of Representations to build Ratio Applications of Ratios. Use of Variables in Expressions Equivalent expressions Linking expressions to words Representing expressions MD School Solutions, Inc. Page 1 Dr.Michele Douglass

2 GRADE 6 Common Core State Standards Critical Areas In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. (1) Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates. (2) Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems. Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. (3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one step equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities. (4) Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. Students recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected. Bolded text represents the content that will be explored in district trainings. MD School Solutions, Inc. Page 2 Dr.Michele Douglass

3 Defining Rates For each of the problems below, define the rate or rates given in each problem. What representation would you use to show each rate? Recipes 1. Thomas is making fruit punch for his friends. One tablespoon of mix makes 1 pint of fruit punch. He wants to make 2 gallons of fruit punch. How many tablespoons of mix will he need? 2. An old fashioned smoke bomb is made up of 4 parts potassium nitrate and 7 parts sugar. You can fill a tube that contains 44 ounces. How much potassium nitrate and sugar do you use? Measurement 3. Tyra s family went on a trip to Canada. Tyra had $25 in United States currency to exchange for Canadian dollars. For every one U.S. dollar, Tyra received 1.17 Canadian dollars. How many Canadian dollars did Tyra receive for her 25 U.S. dollars? 4. If you know that 1 inch is approximately the same as 2.5 centimeters, how long will a 15 inch ribbon be if you measure it in centimeters? 5. How long with a 3 yard spool of ribbon be if you measure it in centimeters? Time/Distance 6. You just traveled 21 miles in 7 minutes. How would you describe your rate? About how long will it take you to travel 100 miles? How do you know? 7. You realize that you just drove 200 miles on 12 gallons of gas. At this rate, how many gallons of gas would you use to drive 500 miles? How many miles will you be able to travel on 3 gallons of gas? MD School Solutions, Inc. Page 3 Dr.Michele Douglass

4 Defining Rates Costs 8. Lorenzo s Candy Shop sells gumballs at the following rates: 3 gumballs for $.20 OR 16 gumballs for $ Barry bought 4 candy bars for $3.75. How much would a dozen candy bars cost? 10. Which is the better deal for you to buy at the grocery store: a 1 pound bag of apples for $2.15 or a 3 pound bag of apples for $5.76? How can you show your solution with a representation? Miscellaneous 11. Daniel painted 9 planks in 6 minutes. If he continues to paint at the same rate, how many planks will he paint in 18 minutes? In 45 minutes? 12. Compare the number of stars to circles. If the ratio remains the same, how many stars will there be when there are 33 circles? 13. Fritz is trying to write the new article for tomorrow s newspaper. The storm delivered 6 inches of rain in 4 hours to Fairfax and 8 inches of rain in 5 hours to DePaul. Which city had the greatest rate of rainfall during the storm? Show your answer in multiple ways. Percentages 14. Coretta buys a pair of jeans that is on sale for 20% off. The regular price is marked as $ What is the sale price of the pair of jeans? 15. The original price of a jacket is $ It is on sale this week for 30% off. What is the final purchase price (without sales tax)? 16. Your uncle deposited $600 in a special account for your car when you turn 16. How much money will be in your account after 1 year if it earns 5% interest per year? MD School Solutions, Inc. Page 4 Dr.Michele Douglass

5 Equivalent and nonequivalent ratios The school is working on a poster to show the number of students that are participating in school activities. They know that for every 4 students that are playing soccer, there is 1 student playing volleyball. Which of the following would be accurate ways to show this information? Justify your thinking. How could you change the ones that don t align to show the school participation accurately? A. B. C. D. E. F. You are going to order 2 bottles of soda for every 5 students that will attend camp. Which of the following show this information accurately? A. B. C. D. E. F. G. H. I. How would you help students understand the multiplicative thinking in this problem? 1. Which class has more girls? Why? a. b. MD School Solutions, Inc. Page 5 Dr.Michele Douglass

6 Is this Equality? Write the number sentence for each balance using the appropriate symbol (=, >, < ) 4 x x 8 (8 x 4) x 6 4(2 x 7) + 3 (2 + 9)6 7 If the following expressions are equivalent, what would the value of x and y have to be? (5x) + 2 (y 2)9 x + 3 y x 2 MD School Solutions, Inc. Page 6 Dr.Michele Douglass

7 Writing Expressions You will use the defined variables to write expressions and to evaluate the validity of symbolic expressions. M = number of men (over 18) attending a baseball game W = number of women (over 18) attending a baseball game Y = number of youth (18 and under) attending a baseball game A = adult ticket price for a single game T = youth ticket price for a single game N = number of home game for this season H = cost for a hot dog F = cost for an order of french fries C = total number of seats in the stadium Write the expression for each description. 1. The total number of people attending a game 2. Income from adult tickets 3. Income from youth tickets 4. Number of empty seats in the stadium Which of these expressions are actually possible? Why? Which are impossible? Why? 5. Y + H 6. YH 7. MF 8. N(M+W) 9. CA(M+W) MD School Solutions, Inc. Page 7 Dr.Michele Douglass

8 Writing Expressions 1. On Thursday Chris drove 167 miles, on Friday he drove 68 miles, and on Saturday he drove 73 miles. Approximately how many miles did Chris drive in the three days? Write an equation that you could use to figure this out. Define your variable. 2. Jonathan read 541 pages during his summer reading program. In order to reach his goal of 650 pages, how many more pages does he need to read? Write an equation that you could use to figure this out. Define your variable. 3. Louie made 17 bag lunches for the school outing. If Louie had made 4 more lunches, he would have made exactly 3 times as many bag lunches as Marc did. How many bag lunches did Marc make? Write an equation that you could use to figure this out. Define your variable. 4. Four people on the same team each ran 100 meters of a relay race. The team finished the total 400 meter race in 50.8 seconds. The first three runners ran their leg of the relay in 12.6 seconds, 13.2 seconds, and 12.8 seconds. Write an equation that would find the time, t, of the fourth runner. 5. In a party room, 20 workers will decorate 70 tables. Each table will be decorated with 10 silver balloons and 15 gold balloons. Write an equation to find t, the total number of silver and gold balloons needed to decorate all the tables? 6. A sixth grade class is having a book sale. The students earn $6 for each book they sell. To determine how many books they need to sell to reach their goal of $144, they use the equation 6b = 144 where b represents a certain number of books. Is this equation going to help them solve their problem? Why or why not? 7. It costs $5 for bowling shoes and $8 for each game you bowl. Write an expression to find the cost for any number of games (g) you bowl. How many games would you bowl if you paid a total of $38? 8. You got paid $7.50 per hour for a summer job and also earned a $20 bonus for doing excellent work. Write an expression that would represent how much money you earned. How many hours did you work if you were paid $50? How many hours did you work if you were paid $65? MD School Solutions, Inc. Page 8 Dr.Michele Douglass