Algebra 1, Quarter 3, Unit 3.1 Solving Word Problems Using ...curriculum.lincolnps.org/sites/default/files/Math/nocollab-math-gr... · Essential questions • How does graphing a

Cumberland, Lincoln, and Woonsocket Public Schools, with process support from the Charles A. Dana Center at the University of Texas at Austin

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Algebra 1, Quarter 3, Unit 3.1

Solving Word Problems Using Systems of Equations

Overview Number of instructional days: 10 (1 day = 45–60 minutes)

Content to be learned Mathematical practices to be integrated • Connect real-world problem scenarios to a

linear mathematical representation.

• Interpret the solution to a system of linear equations in the context of a problem situation.

• Use an efficient method (graphing, substitution, elimination) to solve a system of linear equations formed by a problem scenario.

Make sense of problems and persevere in solving them.

• Use systems of equations to plan a solution pathway for a problem situation.

• Monitor and evaluate progress in solving a problem and change course if necessary.

Use appropriate tools strategically.

• Determine appropriate tools to use for modeling systems of linear equations.

• Detect possible errors by strategically using estimation and other mathematical knowledge.

Essential questions • What are examples of real-world situations that

can be modeled by a system of linear equations?

• How can you determine which method for solving a system of linear equations is best to use given a problem scenario?

• What is the meaning of a solution of a system of linear equations in the context of a problem scenario?

Algebra 1, Quarter 3, Unit 3.1 Solving Word Problems Using Systems of Equations (10 days)


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Written Curriculum

Common Core State Standards for Mathematical Content

Creating Equations★ A-CED

Create equations that describe numbers or relationships [Linear, quadratic, and exponential (integer inputs only); for A.CED.3 linear only]

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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Reasoning with Equations and Inequalities A-REI

Solve systems of equations

A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Quantities N-Q

Reason quantitatively and use units to solve problems. N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

Common Core Standards for Mathematical Practice

2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.



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4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 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.



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Clarifying the Standards

Prior Learning

In grade 8, students used systems of linear equations to represent, analyze, and solve a variety of problems. They understood that a solution to a system of two linear equations in two variables corresponds to a point of intersection of a graph. They also solved simple systems of two linear equations in two variables algebraically and by estimating solutions by graphing the equations. In the previous unit of this course, students solved systems of linear equations by graphing, substitution, and linear combinations.

Current Learning

This unit is a continuation of the content from the previous unit, focusing on solving word problems that can be solved with systems of linear equations. Students demonstrate conceptual understanding of equality by solving problems involving algebraic reasoning about equality. Students provide graphical interpretations of solution(s) in problem-solving situations and solve problems involving systems of linear equations in a context (using equations or graphs) or using models or representations.

Future Learning

Solving systems is a skill that students will continue to use in subsequent course work as they solve systems of nonlinear equations and inequalities. In grades 11, 12, and advanced math, students will demonstrate conceptual understanding of equality by solving equations and systems of equations or inequalities and interpreting the solutions algebraically and graphically. They will use methods of solving systems of equations and inequalities in algebra 2 when studying linear programming.

Additional Findings

A Research Companion to Principles and Standards for School Mathematics (NCTM), pp. 250–261, indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities.

Principles and Standards for School Mathematics notes that in high school, students should build on their prior knowledge, learning more varied and more sophisticated problem-solving techniques. In addition, improving fluency with algebraic symbolism helps students represent and solve problems in many areas of the curriculum. Students should be able to operate fluently on algebraic expressions, combining them and re-expressing them in alternative forms (p. 288).


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Systems of Linear Inequalities


Content to be learned Mathematical practices to be integrated • Graph single linear inequalities in two

variables.

• Graph systems of linear inequalities on the coordinate plane.

• Represent constraints as linear inequalities in order to solve real-world situations with linear programming.

Make sense of problems and persevere in solving them.

• Use systems of equations to plan a solution pathway for a problem situation.

• Monitor and evaluate progress in solving a problem and change course if necessary.

Use appropriate tools strategically.

• Determine appropriate tools to use for solving systems of linear inequalities.

• Detect possible errors by strategically using estimation and other mathematical knowledge.

Essential questions • How does graphing a system of linear

inequalities differ from graphing a system of linear equations?

• What does the graph of a system of inequalities actually show?

• How can graphing a system of linear inequalities be used to model real-world situation?

Algebra 1, Quarter 3, Unit 3.2 Systems of Linear Inequalities (15 days)


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Written Curriculum


Reasoning with Equations and Inequalities A-REI

Represent and solve equations and inequalities graphically [Linear and exponential; learn as general principle]

A-REI.12 Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality) and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Creating Equations A-CED A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and./or

inequalities, and interpret solutions as viable of non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.


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 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. Older 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. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. 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.



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Prior Learning

In grade 8, students used systems of linear equations to represent, analyze, and solve a variety of problems. They understood that a solution to a system of two linear equations in two variables corresponds to a point of intersection of a graph. They also solved simple systems of two linear equations in two variables algebraically and by estimating solutions by graphing the equations. In the previous unit of this course, students solved systems of linear equations by graphing, substitution, and linear combinations.

Current Learning

Students solve linear inequalities (symbolically and graphically) and express the solution set symbolically or graphically. They provide graphical interpretations of solution(s) in problem-solving situations and solve problems involving systems of linear inequalities in a context (using inequalities or graphs) or using models or representations.

Future Learning

Solving systems is a skill that students will continue to use in subsequent course work as they solve systems of non-linear equations and inequalities. In grades 11, 12, and advanced math, students will demonstrate conceptual understanding of equality by solving equations and systems of equations or inequalities and interpreting the solutions algebraically and graphically. They will use methods of solving systems of equations and inequalities in algebra 2 when studying linear programming.

Additional Findings

Principles and Standards for School Mathematics notes that in high school, students should build on their prior knowledge, learning more varied and more sophisticated problem-solving techniques. In addition, improving fluency with algebraic symbolism helps students represent and solve problems in many areas of the curriculum. Students should be able to operate fluently on algebraic expressions, combining them and re-expressing them in alternative forms (p. 288).



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A Research Companion to Principles and Standards for School Mathematics (NCTM), pp. 250–261, indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities.


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Exponents and Exponential Functions


Content to be learned Mathematical practices to be integrated • Distinguish between graphs, tables, and real-

life situations that can be modeled with linear functions and exponential functions.

• Use the properties of exponents to write and interpret exponential growth and decay functions.

• Rewrite expressions involving radicals and rational exponents using the properties of exponents.

• Identify geometric sequences.

• Write, graph and interpret exponential growth and decay functions.

Model with mathematics.

• Identify the important quantities and their relationships using mathematical models.

Look for and make use of structure.

• Apply prior learning to new situations.

• Identify patterns and structures.

Essential questions • What are examples of real-world situations that

can be modeled by exponential functions?

• How are the graph and table related for an exponential function?

• How do the tables for exponential functions differ from those for linear and quadratic functions?

• What are the characteristics of an exponential graph?

Algebra 1, Quarter 3, Unit 3.3 Exponents and Exponential Functions (15 days)


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Written Curriculum


Seeing Structure in Expressions A-SSE

Write expressions in equivalent forms to solve problems

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

c. Use the properties of exponents to transform expressions for exponential functions.

Interpret the structure of expressions

A-SSE.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

The Real Number System N-RN

Extend the properties of exponents to rational exponents. N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the

properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Quantities N-Q

Reason quantitatively and use units to solve problems. N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

Interpreting Functions F-IF

Analyze functions using different representations

F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.



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F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y-(1.01)2t, y=(1.2)t/10, and classify them as representing growth or decay.

Linear, Quadratic, and Exponential Models F-LE

Construct and compare linear, quadratic, and exponential models and solve problems

F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).★

F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★

Interpret expressions for functions in terms of the situation they model [Linear and exponential of form f(x) = bx + k]

F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.★


4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.



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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.


Prior Learning

Beginning in grade 3, students ordered and compared whole numbers, fractions, decimals, and benchmark percents with number formations (fractions to fractions, decimals to decimals, and percents to percents). Starting in sixth grade, students compared numbers with exponents, rational numbers, and absolute value. They also compared numbers in scientific notation, irrational numbers, and square roots of numbers.

Also, beginning in the third grade, students mentally added and subtracted whole numbers, working with two-and three-digit whole numbers (multiples of 10). Students also mentally calculated change from $1.00, $5.00, and $10.00. By grade 6, students calculated change (denominations up to $100) and used benchmark percents (1%, 10%, 25%, 50%, and 75%). In seventh grade, students mentally calculated benchmark perfect squares and related square roots.

Students began estimating in kindergarten; however, they began to make estimates and use various techniques for estimating numbers in grade 3. In fifth grade, students determined accuracy and analyzed the effects of estimation. In later grades, students used estimation to determine tips, discounts, and tax, as well as estimated values of non-perfect square roots.

In grade 4, students began using symbols to represent unknown quantities to write simple linear algebraic expressions. In fifth grade, students began evaluating linear algebraic expressions by using whole



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numbers. In sixth grade, students worked with four operations, writing and evaluating linear algebraic expressions with more than one variable. By eighth grade, students wrote and evaluated linear algebraic expressions with rational numbers and exponents.

Current Learning

Students learn the properties of exponents, extending to rational exponents and radical equivalents Students distinguish between graphs, tables, and real-life situations that can be modeled with linear and exponential functions, and they use the properties of exponents to write and interpret exponential growth and decay functions. Students rewrite expressions involving radicals and rational exponents. They also identify geometric sequences.

Future Learning

In grades 11, 12, and advanced mathematics, students will represent and analyze functions in several ways, analyze characteristics of functions (exponential, logarithmic, and trigonometric), and apply knowledge of functions to interpret situations.

Additional Findings

Principles and Standards for School Mathematics notes that high school students should be able to interpret functions in a variety of formats. “Students should solve problems in which they use tables, graphs, words and symbolic expressions to represent and examine functions and patterns of change. Students should learn algebra both as a set of concepts and competencies tied to the representation of quantitative relationships and as a style of mathematical thinking for formalizing patterns, functions, and generalizations” (p. 287).

By the completion of algebra 1, high school students should have substantial experience in exploring the properties of different classes of functions.



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Documents

Algebra 1, Quarter 3, Unit 3.1 Solving Word Problems Using ...curriculum.lincolnps.org/sites/default/files/Math/nocollab-math-gr... · Essential questions • How does graphing a