CCGPS Frameworks Student Edition. Mathematics. 6 th Grade Unit 4: One Step Equations and Inequalities

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1 CCGPS Frameworks Student Edition Mathematics 6 th Grade Unit 4: One Step Equations and Inequalities These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. The contents of this guide were developed under a grant from the U. S. Department of Education. However, those contents do not necessarily represent the policy of the U. S. Department of Education, and you should not assume endorsement by the Federal Government.

2 TABLE OF CONTENTS Unit 4 One Step Equations and Inequalities Overview...3 Key Standards...4 Standards for Mathematical Practice...5 Enduring Understandings...6 Concepts & Skills to Maintain...6 Selected Terms and Symbols...7 Misconceptions...8 Formative Assessment Lesson (FAL) Overview...9 Tasks Set It Up...10 Building with Toothpicks...13 Fruit Punch...15 Picturing Proportions...16 Making Sense of Graphs...17 Analyzing Tables...19 When is it Not Equal...20 It is on sale!!...22 The Catering Job...24 Converting metrics...26 July 2013 Page 2 of 28

3 OVERVIEW In this unit students will: Determine if an equation or inequality is appropriate for a given situation Solve mathematical and real world problems with equations. Represent real-world situations as inequalities. Interpret the solutions to equations and inequalities Represent the solutions to inequalities on a number line Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs Beginning experiences in solving equations will require students to understand the meaning of the equation as well as the question being asked. The use of illustrations, drawings, and balance models to represent and solve equations and inequalities will help students to develop this understanding. Solving equations will also require students to develop effective strategies such as fact families, and inverse operations. As effective strategies are developed students will revisit rate and proportional reasoning problems and solve them using strategies developed in solving similar one-step equations. Students will represent and model equations and inequalities that are based on mathematical and real world problems. Presented with these situations, students must determine if a single value is required as a solution or if the situation allows for multiple solutions will be included. This creates the need for both equations (single solution for the situation) and inequalities (multiple solutions for the situation). When working with inequalities, students will work with situations in which the solution is not limited to the set of positive whole numbers but includes positive rational numbers. As an extension to this concept, certain situations may require a solution between two numbers. Therefore, the exploration with students as to what this would look like both on a number line and symbolically will be explored. The process of translating between mathematical phrases and symbolic notation is essential in the writing of equations and inequalities for a situation. This is a two-way process and students will be able to write a mathematical phrase for an equation. The goal is to help students connect the pieces. This is done by having students use multiple representations for mathematical relationships. Students will translate freely among the story, words (mathematical phrases), models, tables, graphs and equations/inequalities. Given any one of these representations students should be able to develop the others. July 2013 Page 3 of 28

4 STANDARDS ADDRESSED IN THIS UNIT KEY STANDARDS Reason about and solve one-variable equations and inequalities. MCC6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. MCC6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. MCC.6.EE.8 Write an inequality of the form x < c or x > c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x < c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Represent and analyze quantitative relationships between dependent and independent variables. MCC6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Understand ratio concepts and use ratio reasoning to solve problems. MCC.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. MCC.6.RP.3a 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. July 2013 Page 4 of 28

5 MCC.6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed. MCC.6.RP.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent. MCC.6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. RELATED STANDARDS Apply and extend previous understandings of multiplication and division to divide fractions by fractions. MCC6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Compute fluently with multi-digit numbers and find common factors and multiples. MCC6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. MCC6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. MCC6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor. STANDARDS FOR MATHEMATICAL PRACTICE: 1. Make sense of problems and persevere in solving them. Students choose the appropriate algebraic representations for given contexts and can create contexts given equations or inequalities. 2. Reason abstractly and quantitatively. Students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations. July 2013 Page 5 of 28

6 3. Construct viable arguments and critique the reasoning of others. Students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, graphs, and tables. 4. Model with mathematics. Students model problem situations in symbolic, graphic, tabular, and contextual formats. Students form expressions, equations, and inequalities from real world contexts and connect symbolic and visual representations. 5. Use appropriate tools strategically. Students use number lines to graph equations and inequalities. Students use tables to organize information to write equations and inequalities. 6. Attend to precision. Students precisely define variables. 7. Look for and make use of structure. Students seek patterns or structures to model and solve problems using tables, and equations. Students seek patterns or structures to model problems using tables and inequalities. Students apply properties to generate equivalent expressions (i.e x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality, c = 6 by division property of equality). 8. Look for and express regularity in repeated reasoning. Students generalize effective processes for representing equations and inequalities based upon experiences. Students will be solving for equations. ENDURING UNDERSTANDINGS Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules. Relate and compare different forms of representation for a relationship. Use values from specified sets to make an equation or inequality true. Develop an initial conceptual understanding of different uses of variables. Graphs can be used to represent all of the possible solutions to a given situation. Many problems encountered in everyday life can be solved using proportions, equations or inequalities. Students will solve one-step equations. CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. Using parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols. Write and interpret numerical expressions. Generating two numerical patterns using two given rules. Interpret a fraction as division Operations with whole numbers, fractions, and decimals July 2013 Page 6 of 28

7 SELECTED TERMS AND SYMBOLS The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The websites below are interactive and include a math glossary suitable for middle school students. Note Different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. This web site has activities to help students more fully understand and retain new vocabulary Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students. Addition Property of Equality: Adding the same number to each side of an equation produces an equivalent expression. Constant of proportionality: The constant value of the ratio of two proportional quantities x and y; usually written y = kx, where k is the constant of proportionality. In a proportional relationship, y = kx, k is the constant of proportionality, which is the value of the ratio between y and x. Dependent variable- A variable that depends on other factors. For example, a test score could be a dependent variable because it could change depending on several factors such as how much you studied, how much sleep you got the night before you took the test, or even how hungry you were when you took it. July 2013 Page 7 of 28

8 Direct Proportion (Direct Variation): The relation between two quantities whose ratio remains constant. When one variable increases the other increases proportionally: When one variable doubles the other doubles, when one variable triples the other triples, and so on. When A changes by some factor, then B changes by the same factor: A=kB, where k is the constant of proportionality. Division Property of Equality: States that when both sides of an equation are divided by the same number, the remaining expressions are still equal Equation: A mathematical sentence that contains an equal sign Independent variable- A variable that stands alone and isn't changed by the other variables you are trying to measure. For example, someone's age might be an independent variable. Inequality: A mathematical sentence that contains the symbols >, <,, or. Inverse Operation: A mathematical process that combines two or more numbers such that its product or sum equals the identity. Multiplication Property of Equality: States that when both sides of an equation are multiplied by the same number, the remaining expressions are still equal. Proportion: An equation which states that two ratios are equal. Solution - the set of all values which, when substituted for unknowns, make an equation true. Substitution - the process of replacing a variable in an expression with its actual value. Subtraction Property of Equality: States that when both sides of an equation have the same number subtracted from them, the remaining expressions are still equal. Term: A number, a variable, or a product of numbers and variables. Variable: A letter or symbol used to represent a number or quantities that vary MISCONCEPTIONS The equal sign represents that the two sides of the equation balance, have the exact same value. Some students have developed the notion that the equal sign means the answer July 2013 Page 8 of 28

9 is, based often on limited experiences in elementary school where the answer is always on the right side of the equal sign. Students need practice translating verbal expressions into expressions and equations, and also translating expressions and equations into verbal expressions. The wording must dictate the order of the terms. For example, three more than a number is transcribed x + 3, not 3 + x even though those expressions are equivalent. In the example, three less than a number, it is imperative that the order be x - 3. Students often confuse the statement five less than a number ( x 5 ) with the inequality, five is less than a number (5 < x). The difference in an expression and an inequality needs to be clearly made. FORMATIVE ASSESSMENT LESSONS (FALs) Formative Assessment Lessons are intended to support teachers in formative assessment. They reveal and develop students understanding of key mathematical ideas and applications. These lessons enable teachers and students to monitor in more detail their progress towards the targets of the standards. They assess students understanding of important concepts and problem solving performance, and help teachers and their students to work effectively together to move each student s mathematical reasoning forward. More information on types of Formative Assessment Lessons may be found in the Comprehensive Course Guide. July 2013 Page 9 of 28

10 TASK: SET IT UP Adapted from Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Part I. Marcus has 6 pet rabbits. He keeps them in two cages that are connected so they can go back and forth between the cages. One cage is orange and the other cage is blue. 1. Show all the ways that 6 rabbits can be in two cages. 2. Write an equation that represents the rabbits. 3. Write a different equation that represents the rabbits. 4. Write a different equation that represents the rabbits. July 2013 Page 10 of 28

11 Part II. Find the weight of the pair of shoes and pair of socks. = 13.9 ounces 1. Write an equation that represents the above balance scale. 2. What does 13.9 represent in the equation? 3. What do you notice about the shoes if the pair of socks weighs 0.8 ounces? How can you find the weight of the pair of shoes if the pair of socks weighs 0.8 ounces? 4. How can you find the weight of the pair of socks if the pair of shoes weighs 13.1 ounces? July 2013 Page 11 of 28

12 5. + = 13.9 ounces a. Select a variable to represent the athletic shoes (tennis shoes). b. Select a variable to represent the socks. c. Write an equation that represents the above equations using variables instead of pictures. d. Write an equation in terms of athletic shoes. e. Write an equation in terms of socks. July 2013 Page 12 of 28

13 TASK: BUILDING WITH TOOTHPICKS The shapes shown below are made with toothpicks. Look for patterns in the number of toothpicks in the perimeter of each shape. Shape 1 Shape 2 Shape 3 Shape 4 1. Use a pattern from the shapes above to determine the perimeter of the fifth shape in the sequence. Show or explain how you arrived at your answer. 2. Graph the relationship between the shape number and the perimeter. Based on your graph identify the dependent and independent variable. July 2013 Page 13 of 28

14 3. What is the perimeter for shape n? 4. Write a formula that you could use to find the perimeter of any shape n. Explain how you found your formula. 5. What is the shape number if it had a perimeter of 128? 6. What is the perimeter for shape 10? 7. Is there a figure with a perimeter of 62? Explain your reasoning. July 2013 Page 14 of 28

15 TASK: FRUIT PUNCH This task was taken from ACHIEVE John was making fruit punch for a party using crystals that you mix with water. He mixed four scoops of crystals with nine cups of water and it tasted just right. His sister, Sarah, who likes sweet drinks, walked by and dumped another scoop of crystals into the pitcher. How much water does John need to add so that the fruit punch will taste exactly the same as it did before? July 2013 Page 15 of 28

16 TASK: PICTURING PROPORTIONS Draw a picture and write an explanation to convince someone that = Triangle ABC has side lengths 3 units, 4 units, and 5 units. Draw two triangles, one smaller than Triangle ABC, and one larger than Triangle ABC so that the sides of your triangles vary proportionally with the sides of Triangle ABC. 3. Isabella takes 4 2 of a candy bar and gives her little brother Lucas 2 1 of a different candy bar. Look at the picture representing the two candy bars. Isabella explains to Lucas that 1 2 =, so she is giving him the same amount of candy that she takes for herself. Is 2 4 Isabella s explanation correct or is she trying to trick her little brother? Explain your reasoning. Isabella s portion Lucas s portion July 2013 Page 16 of 28

17 TASK: MAKING SENSE OF GRAPHS The graph below shows the amount of money required to buy gasoline if the cost per gallon is $2.00. a. What two quantities vary proportionally in this situation? July 2013 Page 17 of 28

18 b. What is the value of the constant of proportionality? What does this value represent in the context of the problem? How is the constant of proportionality represented on the graph? c. Write an equation to represent this situation. d. Suppose gas prices rose to $3.00 per gallon. How would the graph change? Explain your reasoning. e. Write an equation to represent the situation in part d. July 2013 Page 18 of 28

19 TASK: ANALYZING TABLES Consider the tables below where the x- and y-values represent two quantities. For each table, do the following: a. Do the quantities vary proportionally? Explain how you know. b. Write a rule for each table in words. c. Write the rule as an equation. Table 1 x y Table 2 x y Table 3 x y July 2013 Page 19 of 28

20 TASK: WHEN IS IT NOT EQUAL Write an inequality for the following statements. 1. You need to earn at least $ You can spend no more than $ The trip will take at least 4 hours. 4. The car ride will be no more than 8 hours. 5. Four boxes of candy contained at least 48 pieces total. 6. With John s 7 marbles and mine, we had less than 20 marbles together. 7. Seven buses can hold no more than 560 students. Graph the following inequalities: 8. p > b < t < r > k < m > d > 2 July 2013 Page 20 of 28

21 Circle the numbers that are part of the solution for the inequalities below. 15. x + 2 > 5 ( ) 16. v 4 < 10 ( ) 17. 4b < 15 ( ) 18 1/3r > 3½ ( ) w > 2.3 ( ) 20. t < 3.6 ( ) Write an inequality for each situation. 21. What is the minimum number of 80-passenger buses needed to transport 375 students? Choose and justify a solution (4, 4 11 / 16, 5) 22. What is the minimum speed needed to travel at least 440 miles in 8 hours? Choose and justify a solution (54 mph, 55 mph, 56 mph) 23. What is the least number of boxes are needed to package 300 candies if each box will hold 16 candies? Choose and justify a solution (18, 18 ¾, 19 ) July 2013 Page 21 of 28

22 TASK: IT IS ON SALE Name Part 1 Dishwashers on sale for 45% off. Original price is $310. Washer and Dryer combo for $2450. This is 40% off the original price. Refrigerator was $2450 and is now $1, Logan went to the local appliance store and bought a washer/dryer set, a refrigerator, and a dishwasher. She was thrilled to discover that all the items were on sale. A. How much did the dishwasher cost at the sale price? B. What was the original price of the washer/dryer set? C. The refrigerator is on sale. Logan wants to know what percent of the original price she paid for this item. D. What is the total amount she spent on all the items she purchased? Explain how you found her total cost. July 2013 Page 22 of 28

23 Part 2 Solve the given problems and show your work. 2. In the annual car show, 40% of the cars in the main display area are used cars. Given that 294 cars are used cars, what is the total number of cars in the show? 3. Rachel is 40 cm taller than Lucy. Rachel is 25% taller than Lucy. How tall is Lucy? 4. Mike and Chris went to dinner. Chris insisted that they leave a 20% tip for the waitress. Mike left a $17.20 tip. How much was their bill before the tip? 5. Jenny and Tim run a bed and breakfast in Crawford, Georgia. Jenny painted ¼ of her rocking chairs red to match her new door. A. What percent of her chairs did she paint red? B. Jenny has 12 chairs on her front porch. How many chairs are painted red? C. If she wants to paint all her chairs red, what percent still need to be painted? How many chairs need to be painted? July 2013 Page 23 of 28

24 TASK: THE CATERING JOB Name Mrs. Smith works for a catering company. She is in charge of buying the food for a special event. It is very important to her boss that she buy the supplies at the best unit rate. 1. Below are the prices Mrs. Smith found on shrimp: Kroger sells a package of 15 shrimp for $8.99. Publix sells a package of 12 shrimp for $8.34. The Bluffton Oyster Company sells a package of 20 shrimp for $ Where should she buy the shrimp? Show your work and WRITE an explanation for Mrs. Smith to give to her boss to show him that she chose the best deal on shrimp. 2. Below are the prices she found on potatoes: Kroger sells 10 pound bags of potatoes for 3.69 Publix sells 5 pound bags of potatoes for $2.45 Wal-Mart sells 20 pound bags of potatoes for $7 Where should she buy potatoes? Show your work and WRITE an explanation for Mrs. Smith to give to her boss to show him that she chose the best deal on potatoes. July 2013 Page 24 of 28

25 3. Below are the prices Mrs. Smith found on corn on the cobb: Snook s sells 25 pieces of corn for Publix sells 3 pieces of corn for $1 The Farmer s Market sells 12 pieces of corn for $4.80 Where should she buy corn on the cob? Show your work and WRITE an explanation for Mrs. Smith to give to her boss to show him that she chose the best deal on corn on the cob. July 2013 Page 25 of 28

26 TASK: CONVERTING METRICS Name Use the chart as a resource to solve the problems. 1. Instant Meals sent out free samples to introduce its new product, Sesame Pasta. Each sample weighs 40 ounces. They send out 24 samples. How many pounds did they send through the mail? 2. The track used by the first successful steam locomotive was 25.3 kilometers long. How many meters long was the track? July 2013 Page 26 of 28

27 3. There is a jar on the cabinet by the refrigerator. Kelly pours 144 ounces of water into the jug three times to fill it completely. A. How many cups does it take to fill the jar? B. How many pints? C. How many quarts? 4. Brad s father asked an engineer to survey the field behind their house. He wanted to plant some orange and tangerine trees there. According to the survey the field is 8.5 yards long and six yards wide. What are the dimensions of the field in feet? 5. The St. Gotthard Tunnel in Switzerland is the world s longest tunnel. It is 16.3 kilometers long. What is the tunnel s length in meters? 6. Titus saw a raccoon in his backyard last weekend. He put out some food and a bowl with 3 cups of water. The raccoon ate all the food and drank half the water. How many ounces of water did the raccoon drink? 7. The average male elephant drinks about 120,000 milliliters of water each day. How many liters of water do most male elephants drink each day? July 2013 Page 27 of 28

28 8. Kelsey has a new baby brother. He weighs 8 pounds. The doctor said the baby should gain about 4 ounces per week. At this rate, how many pounds should the baby weigh in 3 months (12 weeks)? 9. One ruler is 30 centimeters long. A different ruler is 200 millimeters long. Which ruler is longer? 10. An official hockey puck is 2.54 centimeters thick. What is the official thickness of the hockey puck in millimeters? 11. An official hockey goal is meters tall. What is the height of a hockey goal in centimeters? 12. The top layer of trees in a tropical forest has trees that can reach 6,096 centimeters in height. How many meters tall are these trees? 13. An official hockey puck can weigh no more than 170 grams. What is the puck s maximum weight in kilograms? July 2013 Page 28 of 28