Model with mathematics.

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1 1 of 6 10/7/ :27 AM Math Model Unit 1.3 Algebra East Bay Overview Overview Number of instructional days: 15 (1 day = minutes) Content to be learned Use multiple representations (algebraic, table, graph and verbal) linear and exponential equations to solve problems. Solve linear inequalities graphically and algebraically. Graph a system of inequalities in two variables and identify the solution. Understand the definition of a graph of an equation in two variables. Solve literal equations (linear only). Rearrange formulas to isolate a variable in real-life situations. Mathematical practices to be integrated Model with mathematics. Use hands-on manipulatives to represent formulas and solve for a specific variable. Use real-world situations to create and solve inequalities. Graph systems of inequalities from a real-world situation, using graphing technology. Look for and express regularity in repeated reasoning. Use real-world situations to create equations. Content to be Learned Content to be learned Use multiple representations (algebraic, table, graph and verbal) linear and exponential

2 2 of 6 10/7/ :27 AM equations to solve problems. Solve linear inequalities graphically and algebraically. Graph a system of inequalities in two variables and identify the solution. Understand the definition of a graph of an equation in two variables. Solve literal equations (linear only). Rearrange formulas to isolate a variable in real-life situations. Mathematical Practices to be Integrated Mathematical practices to be integrated Model with mathematics. Use hands-on manipulatives to represent formulas and solve for a specific variable. Use real-world situations to create and solve inequalities. Graph systems of inequalities from a real-world situation, using graphing technology. Look for and express regularity in repeated reasoning. Use real-world situations to create equations. Essential Questions Essential questions What does the graph of a linear equation in two variables represent? How can rearranging formulas assist in solving a real-world problem? Why is the range of an exponential What are the similarities of and differences between equations and inequalities? Why is it important to understand inequalities?

3 3 of 6 10/7/ :27 AM function always one unless a transformation has happened to the function? What are the similarities and differences between exponential decay and growth? Why is the concept of opposite operations important in the study of rearrangement of formulas? How do the properties of exponents transform exponential expressions to solve for a given variable? Written Curriculum Written Curriculum Common Core State Standards for Mathematical Content Algebra 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.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities A-REI Solve equations and inequalities in one variable [Linear inequalities; literal that are linear in the variables being solved for; quadratics with real solutions] A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

4 4 of 6 10/7/ :27 AM Represent and solve equations and inequalities graphically [Linear and exponential; learn as general principle] A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 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. Common Core State Standards for Mathematical Practice 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. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

5 5 of 6 10/7/ :27 AM Clarifying the Standards Prior Learning In grade 1, students worked with addition and subtraction equations. In grade 2, they represented and solved problems involving addition and subtraction. In grade 3, students solved two-step word problems using all four operations. In grade 4, students solved multistep word problems with whole numbers and using all four operations. In grade 5, students graphed points on the coordinate plane to solve real-world mathematical problems. In grade 6, they reasoned about and solved one-variable equations and inequalities. In grade 7, students solved real-life and mathematical problems using numerical and algebraic expressions and equations. In grade 8, students analyzed and solved linear equations and pairs of simultaneous linear equations. Current Learning Algebra 1 students create equations and inequalities from real-world problems. They use equations and inequalities to solve real-world problems. Students graph and solve linear equations and inequalities. They also show and describe the solution of a system of linear inequalities on a graph. Students calculate equation and inequality solutions graphically and algebraically. Students solve literal equations that are linear only, and rearrange formulas to solve for a specific variable. Students create and solve quadratic and exponential functions limited to integer exponent inputs. These concepts are to be mastered in algebra 1. Future Learning Students will use this knowledge again in algebra 2 when they further investigate all types of equations including simple rational or radical. Students will be able to apply this concept to real-life situations involving linear programming, business, finance, and construction. Additional Findings Additional Findings

6 6 of 6 10/7/ :27 AM It is important that students gain experience using multiple representations to deepen their understanding. Students should recognize connections among different representations, thus enabling them to use these representations flexibly. (Principles and Standards for School Mathematics, p. 309) Students are challenged when they are asked to distinguish among different kinds of functions by looking at the functions equations: As high school students study several classes of functions, they should begin to see that classifying functions as linear, quadratic, or exponential makes sense because the functions in each of these classes share important attributes. (Principles, p. 300) Students have difficulty understanding what the solution means in the context of the problem. This can be clarified by connecting different representations of the problem. Students translate among verbal, tabular, graphical, and algebraic representations of functions, and they describe how such aspects of a function as slope and y-intercept appear in different representations. (Curriculum Focal Points, p. 20) Resources (0) Standards (6) Other Info

Extending Place Value with Whole Numbers to 1,000,000

Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Recognize that a digit