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Fulton County Schools Parent Guide Coordinate Algebra: Unit 1 – Relationships between Quantities (4 weeks) Unit Overview: By the end of eighth grade students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. The first unit of Coordinate Algebra involves relationships between quantities. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation. Students will be provided with examples of real-world problems that can be modeled by writing an equation or inequality. Students will develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They will also master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations (limited to integer exponents). Skills from this unit will also be embedded in other units in this course. Content Standards: Reason quantitatively and use units to solve problems. MCC9-12.N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★ MCC9-12.N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. ★ MCC9-12.N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. ★ Interpret the structure of expressions. MCC9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. ★ (Emphasis on linear expressions and exponential expressions with integer exponents.) MCC9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. ★ (Emphasis on linear expressions and exponential expressions with integer exponents.) MCC9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. ★ (Emphasis on linear expressions and exponential expressions with integer exponents.) Create equations that describe numbers or relationships. MCC9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and exponential functions.★ (excluding quadratic functions and simple rational) MCC9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★ (Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.) MCC9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.★ (Limit to linear equations and inequalities.) MCC9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.★ (Limit to formulas with a linear focus.) .★Represent modeling standards Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 6. Attend to precision. Standards for Mathematical Practice (#1, 6) Essential Question: How do mathematically proficient students approach problem solving? Learning Targets: I can … use Polya’s four steps of problem solving to work through word problems. (MP1) demonstrate and organize my work using Polya’s four steps of problem solving (MP1) state the meaning of the symbols I choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. (MP6) calculate accurately and efficiently, expressing numerical answers with a degree of precision appropriate for the problem context . (MP6) Concept Overview: SMP 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Fulton County Schools Parent Guide solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. High school students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. In his book, How to Solve It, George Polya describes his study of effective problem solvers. He surmised that if he discovered commonalities among good problem solvers, that he could teach others to become good problem solvers. His hypothesis turned out to be correct. He found that by explicitly teaching students the four steps of problem solving, their problem solving abilities actually improved. The four steps are as follows: 1. Understand the problem What are you asked to find out or show? Can you draw a picture or diagram to help you understand the problem? Can you restate the problem in your own words? Can you work out some numerical examples that would help make the problem clearer? 2. Devise a plan A partial list of Problem Solving Strategies include: Guess and check Solve a simpler problem Make an organized list Experiment Draw a picture or diagram Act it out Look for a pattern Work backwards Make a table Use deduction Use a variable Change your point of view 3. Carry out the plan Carrying out the plan is usually easier than devising the plan Be patient – most problems are not solved quickly nor on the first attempt If a plan does not work immediately, be persistent Do not let yourself get discouraged If one strategy isn’t working, try a different one 4. Look back (reflect) Does your answer make sense? Did you answer all of the questions? What did you learn by doing this? Could you have done this problem another way – maybe even an easier way? SMP 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Resources: Teaching Problem Solving Strategies Problem Solving Rubric Make Sense of Problems – Inside Mathematics Website Attend to Precision – Inside Mathematics Website Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Fulton Reasoning in Problem Solving QuantitativeCounty Schools Parent Guide Reason quantitatively and use units to solve problems. MCC9-12.N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★ MCC9-12.N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. ★ MCC9-12.N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. ★ Essential Question: In what ways can the choice of units, quantities, and levels of accuracy impact a solution? Learning Targets: I can … N.Q.1 convert units of measure using dimensional analysis. (N.Q.1) label units through multiple steps of a problem. (N.Q.1) select and use appropriate units of measurement for problems with and without context. (N.Q.1) given a graph draw conclusions and make inferences. (N.Q.1) choose the appropriate scale and the orIgin to create linear and exponential graphs. (N.Q.1) determine from the labels on a graph what the units of the rate of change are (e.g., gallons per hour). (N.Q.1) determine the accuracy of values based on their limitations in the context of the situation. (N.Q.1) N.Q.2 choose appropriate measures and units for problem situations. (N.Q.2) create a relationship among different units (i.e., feet per second, bacteria per hour, miles per gallon). (N.Q.2) N.Q.3 determine whether whole numbers, fractions, or decimals are most appropriate. (N.Q.3) determine the appropriate power of ten to reasonably measure a quantity. (N.Q.3) use precision of initial measurements to determine the level of precision with which answers can be reported. (N.Q.3) determine what level of rounding should be used in a problem situation. (N.Q.3) Concept Overview: In real world problems, the answers are usually not numbers but quantities: numbers with units, which involves measurement. In their work in measurement up through Grade 8, students primarily measure commonly used attributes such as length, area, and volume. In high school, students encounter a wider variety of units in modeling, e.g., acceleration, currency conversions, derived quantities such as person-hours and heating degree days, social science rates such as per-capita income, and rates in everyday life such as points scored per game or batting averages. They also encounter novel situations in which they themselves must conceive the attributes of interest. For example, to find a good measure of overall highway safety, they might propose measures such as fatalities per year, fatalities per year per driver, or fatalities per vehicle-mile traveled. Such a conceptual process is sometimes called quantification. Quantification is important for science, as when surface area suddenly “stands out” as an important variable in evaporation. Quantification is also important for companies, which must conceptualize relevant attributes and create or choose suitable measures for them. In real-world situations, quantities are usually represented by numbers associated with units. Units involve measurement and often require a conversion. Measurement involves both precision and accuracy. Estimation and approximation often precede more exact computations. Students need to develop sound mathematical reasoning skills and forms of argument to make reasonable judgments about their solutions. They should be able to decide whether a problem calls for an estimate, for an approximation, or for an exact answer. To accomplish this goal, teachers should provide students with a broad range of contextual problems that offer opportunities for performing operations with quantities involving units. These problems should be connected to science, engineering, economics, finance, medicine, etc. Include word problems where quantities are given in different units, which must be converted to make sense of the problem. For example, a problem might have an object moving 12 feet per second and another at 5 miles per hour. To compare speeds, students convert 12 feet per second to miles per hour: 1min 1hr 1day 24000 sec which is more than 8 miles per hour 60 sec 60 min 24hr Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Fulton County Schools Parent Guide Some contextual problems may require an understanding of derived measurements and capability in unit analysis. Keeping track of derived units during computations and making reasonable estimates and rational conclusions about accuracy and the precision of the answers help in the problem-solving process. For example, while driving in the United Kingdom (UK), a U.S. tourist puts 60 liters of gasoline in his car. The gasoline cost is £1.28 per liter The exchange rate is £ 0.62978 for each $1.00. The price for a gallon of a gasoline in the United States is $3.05. The driver wants to compare the costs for the same amount and the same type of gasoline when he/she pays in UK pounds. Making reasonable estimates should be encouraged prior to solving this problem. Since the current exchange rate has inflated the UK pound at almost twice the U.S. dollar, the driver will pay more for less gasoline. By dividing $3.05 by 3.79L (the number of liters in one gallon), students can see that 80.47 cents per liter of gasoline in US is less expensive than £1.28 or $ 2.03 per liter of the same type of gasoline in the UK when paid in U.S. dollars. The cost of 60 liters of gasoline in UK is In order to compute the cost of the same quantity of gasoline in the United States in UK currency, it is necessary to convert between both monetary systems and units of volume. Based on UK pounds, the cost of 60 liters of gasoline in the U.S. is The computation shows that the gasoline is less expensive in the United States and how an analysis can be helpful in keeping track of unit conversions. Students should be able to correctly identify the degree of precision of the answers which should not be far greater than the actual accuracy of the measurements. Graphical representations serve as visual models for understanding phenomena that take place in our daily surroundings. The use of different kinds of graphical representations along with their units, labels and titles demonstrate the level of students’ understanding and foster the ability to reason, prove, self-check, examine relationships and establish the validity of arguments. Students need to be able to identify misleading graphs by choosing correct units and scales to create a correct representation of a situation or to make a correct conclusion from it. Graphical representations and data displays include, but are not limited to: line graphs, circle graphs, histograms, multi-line graphs, scatterplots, and multi-bar graphs. Discuss misconceptions in resulting calculations involving measurement, e.g., you cannot increase accuracy through calculation, only through more accurate measurement. Compare the difference between rounding at difference places in a calculation and discuss which yields the best result. Discuss how mathematicians maintain precision through the representations that they use. For example, determining the price of gas by estimating to the nearest cent is appropriate because you will not pay in fractions of a cent $3.479 even though the cost of gas is . gallon Vocabulary: scale – the ratio between two corresponding measurements origin – point of intersection of the two axes in a coordinate plane descriptive model – definition based on CCSS documents accuracy – the degree of correctness of a quantity, expression significant figures – definition based on CCSS documents Sample Problem(s): The price of copper fluctuates. Between 2002 and 2011, there were times when its price was lower than $1.00 per pound and other times when its price was higher than $4.00 per pound. Copper pennies minted between 1962 and 1982 are 95% copper and 5% zinc, and each weighs 3.11 grams. At what price per pound of copper does such a penny contain exactly one cent worth of copper? (There are 454 grams in one pound.) (N.Q.1) (N.Q.1 solution) Create a scenario you have encountered involving two changing quantities and determine appropriate units to describe the relationship between the quantities. (N.Q.2) What type of measurements would one use to determine their income and expenses for one month? (N.Q.2) Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Fulton County Schools Parent Guide How could one express the number of accidents in Georgia? (N.Q.2) Julio went to Germany to watch an international soccer tournament. He first watched Argentina play Germany in Berlin, Germany. The next day Julio went to Frankfurt, Germany to watch Brazil play France. To get from Berlin to Frankfurt for this second game, Julio took a bus from Berlin to Erfurt (303 km); then he rented a car and drove from Erfurt to Frankfurt (254 km). Julio drove on German highways, called autobahns, which have no general speed limit for passenger vehicles; however, buses have an enforced speed limit of 80 km/hr. If the bus drove 80 km/hr from Berlin to Erfurt and Julio drove 130 km/hr. from Erfurt to Frankfurt, what was the total amount of time it took Julio to travel from Berlin to Frankfurt (not counting the transfer time between the bus and the car)? Give your answer to a reasonable level of accuracy. (N.Q.3) (N.Q.3 solution) Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Structure of Expressions Fulton County Schools Parent Guide Interpret the structure of expressions. MCC9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. ★ (Emphasis on linear expressions and exponential expressions with integer exponents.) MCC9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. ★ (Emphasis on linear expressions and exponential expressions with integer exponents.) MCC9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. ★ (Emphasis on linear expressions and exponential expressions with integer exponents.) Essential Question: Why structure equations in different ways? Learning Targets: I can … make connections between symbolic representations and proper mathematics vocabulary. (SSE.1) identify parts of an expression such as terms, factors, coefficients, etc. (SSE.1) interpret and apply rules for order of operations. (SSE.1) Concept Overview: Extending beyond simplifying an expression, this cluster addresses interpretation of the components in an algebraic expression. A student should recognize that in the expression 2x + 1, “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Development and proper use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored. For example, suppose the cost of cell phone service for a month is represented by the expression 0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the number of cell phone minutes used in the month. Similar real-world examples, such as tax rates, can also be used to explore the meaning of expressions. Vocabulary: Expression – mathematical phrase that contains operations, numbers and/or variables Term – a number, a variable, a product, or a quotient of numbers and variables Factor – the quantities being multiplied are factors Coefficient – the numerical part of a term Equivalent – have the same solution or value Sample Problem(s): Consider the formula for Surface Area = 2B + Ph. What are the terms of this formula? What are the coefficients? Commentary: A term is a number, variable, product or quotient of numbers and/or variables. In an equation or an expression, terms are separated by additional or subtraction operations or symbols (+) or (-). Coefficients are the numerical part of a term and are traditionally written in front of the variable. Remind students that if the coefficient of a term is the number one, it will not be written in front of the term. Solution: Surface Area = 2B + Ph “2B” is a term “Ph” is a term In the term “2B” the number “2” is the coefficient. In the term “Ph” the number “1” is the coefficient. Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Fulton County Schools Parent Guide 2 Interpret the expression: 5- 3(x – y) . (SSE.1) Commentary: Students should be able to see algebraic expressions as single objects or as being composed of several objects. Students should recognize mathematical operators and their processes, positive and negative number values and the implied meaning of quantities. Solution: 2 5 – 3(x – y) should be understood by students as “5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.” A candy shop sells a box of chocolates for $30. It has $29 worth of chocolates plus $1 for the box. The box includes two kinds of candy: caramels and truffles. Lita knows how much the different types of candies cost per pound and how many pounds are in a box. She said, If x is the number of pounds of caramels included in the box and y is the number of pounds of truffles in the box, then I can write the following equations based on what I know about one of these boxes: x+y=3 8x+12y+1=30 Assuming Lita used the information given and her other knowledge of the candies, use her equations to answer the following: a. How many pounds of candy are in the box? b. What is the price per pound of the caramels? c. What does the term “12y” in the second equation represent? d. What does “8x+12y+1” in the second equation represent? Commentary: This task assumes students are familiar with mixing problems. This approach brings out different issues than simply asking students to solve a mixing problem, which they can often set up using patterns rather than thinking about the meaning of each part of the equations. Students have a difficult time distinguishing between the coefficients (dollars per pound) and the terms (total dollar value of a given amount of particular kind of candy) in the second equation. Solution: a. The box contains 3 pounds of chocolates, since the total number of pounds of the caramels and truffles, represented by x+y, equals 3. b. It appears that the second equation is based on the cost of a box, since everything equals 30. If that is true, then the caramels cost $8 per pound; you can tell because 8 is multiplied by the number of pounds of caramels in the equation that relates the number of pounds of each kind of candy to the cost of a box. c. 12y represents the value of the truffles. Since 12y is in the equation that relates the number of pounds of each kind of candy to the total value of the box, the truffles must cost $12 per pound, and that multiplied by y, the number of pounds of truffles, will give their dollar value. d. This represents the total value of the box of chocolates: the value of the caramels added to the value of the truffles added to the fixed cost of $1. Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Fulton County Schools Parent Guide Creating Equations Create equations that describe numbers or relationships. MCC9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and exponential functions.★ (excluding quadratic functions and simple rational) MCC9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★ (Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.) MCC9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.★ (Limit to linear equations and inequalities.) MCC9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.★ (Limit to formulas with a linear focus.) Essential Question: How can algebra describe the relationship between sets of numbers? Learning Targets: I can … CED.1 create one-variable linear equations and inequalities from contextual situations (stories). (CED.1) create one-variable exponential equations and inequalities from contextual situations (stories). (CED.1) solve and interpret the solution to multi-step linear equations and inequalities in context. (CED.1) use properties of exponents to solve and interpret the solution to exponential equations and inequalities in context. (CED.1) CED.2 write and graph an equation to represent a linear relationship. (CED.2) write and graph an equation to represent an exponential relationship. (CED.2) model a data set using an equation. (CED.2) choose the best form of an equation to model linear and exponential functions. (CED.2) CED.3 determine whether a point is a solution to an equation or inequality. (CED.3) determine whether a solution has meaning in a real-world context. (CED.3) write and graph equations and inequalities representing constraints in contextual situations. (CED.3) CED.4 extend the concepts used in solving numerical equations to rearranging formulas for a particular variable. (CED.4) Concept Overview: Equations can represent real world and mathematical problems Provide examples of real-world problems that can be modeled by writing an equation or inequality. Begin with simple equations and inequalities and build up to more complex equations in two or more variables that may involve exponential functions. Discuss the importance of using appropriate labels and scales on the axes when representing functions with graphs. Examine real-world graphs in terms of constraints that are necessary to balance a mathematical model with the real-world context. Explore examples illustrating when it is useful to rewrite a formula by solving for one of the variables in the formula. For example, the formula for the area of a trapezoid can be solved for h if the area and lengths of the bases are known but the height needs to be calculated. This strategy of selecting a different representation has many applications in science and business when using formulas. Provide examples of real-world problems that can be solved by writing an equation, and have students explore the graphs of the equations on a graphing calculator to determine which parts of the graph are relevant to the problem context. Use a graphing calculator to demonstrate how dramatically the shape of a curve can change when the scale of the graph is altered for one or both variables. Give students formulas, such as area and volume (or from science or business), and have students solve the equations for each of the different variables in the formula. Vocabulary: Linear – equation whose graph is a straight line, can be written in the form y = mx + b where m and b are constraints Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents Fulton County Schools Parent Guide 2 Quadratic – equation of the form ax + bx + c = 0, where a = 0, where a, b and c are real numbers Rational – fraction whose numerator and denominator are polynomials x Exponential – involves the expression b where the base b is a positive number other than 1. Coordinate axes – one of two perpendicular number lines, called the x-axis and the y-axis, used to define the location of a point one of the fixed reference lines of a coordinate system Labels – descriptive term Constraints – the linear inequalities that form a system System of Equations – two or more equations that can be written in one of the following forms: Ax + By = C, Dx + Ey = F where x and y are variables, A and B are not both zero, and D and E are not both zero System of Inequalities – two or more inequalities that can be written in one of the following forms: Ax + By < C, Ax + By < C, Ax + By > C, or Ax + By > C Solutions – replacement for the variable in an open sentence that results in a true statement Sample Problem(s): Juan pays $52.35 a month for his cable bill and an additional $1.99 for each streamed movie. Gail pays $40.32 a month for her cable bill and an additional $2.49 for each streamed movie. Who has the better deal? Justify your choice. (CED.1) 2 Given that the following trapezoid has area 54 cm , set up an equation to find the length of the base, and solve the equation. (CED.1) CED.1 Solution Jeanette can invest $2000 at 3% interest compounded annually or she can invest $1500 at 3.2% interest compounded annually. Which is the better investment and why? (CED.2) Iced tea costs $1.50 a glass and lemonade costs $2. If you have $12, what can you buy? (CED.3) CED.3 Solution A club is selling hats and jackets as a fundraiser. Their budget is $1500 and they want to order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat costs $5 and each jacket costs $8. Write a system of inequalities to represent the situation. Graph the inequalities. If the club buys 150 hats and 100 jackets, will the conditions be satisfied? What is the maximum number of jackets they can buy and still meet the conditions? (CED.3) The Pythagorean Theorem expresses the relation between the legs a and b of a right triangle and its hypotenuse c with the equation 2 2 2 a + b = c . (CED.4) CED.4 Solution o Why might the theorem need to be solved for c? o Solve the equation for c and write a problem situation where this form of the equation might be useful. 4 3 Solve V r for radius r. (CED.4) CED.4 Solution 3 Motion can be described by the formula below, where t = time elapsed, u=initial velocity, a = acceleration, and s = distance traveled: 2 s = ut+½at Why might the equation need to be rewritten in terms of a? Rewrite the equation in terms of a. (CED.4) CED.4 Solution Paul just arrived in England and heard the temperature in degrees Celsius. He remembers that c = 5/9(F – 32). How will Paul find the temperature in Fahrenheit? (CED.4) CED.4 Solution Adapted from Arizona Dept. of Education, Ohio Dept. of Education, Utah State Office of Education, and CCSS Progression Documents