Subject: MathematicsGrade: 7 Timeline: 20 daysUnit 5 Title: Moving Straight Ahead Part II
Building on core concepts introduced and explored in a previous unit, Variables and Patterns, Moving Straight Ahead continues to help students develop an understanding of linear relationships. They will observe a constant rate of change in tables, graphs and equations and describe how this rate manifests in each representation.
In Moving Straight Ahead, students are presented with opportunities to observe the relationship between a dependent and an independent variable, and will begin to recognize that the constant rate of change is the variable, m, in the linear equation, y=mx+b. In the fourth investigation, the slope and the y-intercept are more explicitly defined and formalized, as are the processes for “finding” both of these terms when given specific tables, graphs, equations and situations. Students will also write equations using variables and will find solutions to equations.
Pearson/Prentice-Hall, the publishers of the Connected Math series, developed a Transition Kit and Common Core Additional Investigations in response to implementation of the Common Core State Standards for Mathematics (CCSS-M). These resources contain additional units of study for districts using the Connected Math series to weave into their current curricula. Three of these units of study were created to provide more rigorous and extensive application of the standards addressed in Moving Straight Ahead, as well as to provide new instructional content for the concepts of “inequalities” and “simplifying algebraic expressions,” neither of which were previously addressed to the same extent in grade seven. These three investigations are integrated into those from the Moving Straight Ahead book, and the order in which each investigation is taught is based on a progressive sequence with respect to content, standards focus, and level of rigor.
At the end of this unit, all students must:
- Recognize problem situations in which two or more variables have a linear relationship to each other
- Describe the patterns of change between the independent and the dependent variables for linear relationships that are represented in tables, graphs, and equations
- Construct tables, graphs, and symbolic equations that express linear relationships
- Interpret and translate information about linear relations given in a table, a graph, or an equation to one of the other forms
- Understand the connections between linear equations and the patterns in the tables and graphs of those equations: rate of change, slope, and y-intercept
- Solve linear equations
- Solve problems and make decisions about linear relationships using information given in tables, graphs, and symbolic expressions
- Use tables, graphs and equations of linear relations to answer questions
- Identify, calculate, describe, and apply the idea of a “unit rate” and relate that idea to the slope (rate of change) in a linear relationship.
- Apply the knowledge of proportional relationships to define what any point (x,y), on a graph means in terms of the situation
- Understand that writing an equivalent expression in a problem context can shed light on how quantities in the problem are related
- Solve inequality problems and make interpretations of the solution set in context of the problems
PA.CCSS.Math.Content.CC.2.2.7.B.1 Apply properties of operations to generate equivalent expressions. (7.EE.1)
PA.CCSS.Math.Content.CC.2.2.7.B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. (7.EE.2, 7.EE.3, 7.EE.4)
PA.CCSS.Math.Content.CC.2.1.7.D.1 Analyze proportional relationships and use them to model and solve real-world and mathematical problems. (7.RP.1, 7.RP.2)
PA.CCSS.Math.Content.CC.2.1.7.E.1 Ally and extend previous understandings of operations with fractions to operations with rational numbers. (7.NS.3)
Mathematical Practice Standards:
#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.
#3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
#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 middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. 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.
#7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. Structure can be found in tables, lists, area models and other diagrams.
#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. 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.
Concepts - Students will know:
- The concept of patterns of change between the independent and dependent variables for linear relationships
- How tables, graphs and equations represent linear patterns of change
- How to translate information about linear relations given in a table, graph or an equation to one of the other forms
- The y-intercept and slope of a line
- Negative rates of change and how they are represented in equations, tables, and graphs
- How to describe the information the variables and numbers in an equation represent
- How to find the solution to a problem using a table or graph
- The connection between solution in graphs and tables to solutions of equations
- How the y-intercept appears in tables and equations
- How to use the properties of equality to solve equations that are represented pictorially
- How to check solutions to equations
- How subtle shifts in equations effect the solution
- How to find the point of intersection of two lines
- The concept of slope as the ration of vertical change to horizontal change between two points on a line or ratio of rise over run
- How to use slope to sketch a graph of a line
- How to use the slope m and y-intercept b to write an equation in the form y=mx+b
- How to explore patterns among lines with the same slope
- How to explore patterns among lines whose slopes are negative reciprocals
- How to write an equation for an applied situation when the only information given is two data points
- How to compute unit rates associated with ratios of fractions, including quantities measured in like or different units
- Properties of operations
- How word problems lead to one- and two-step inequalities
Competencies -Students will be able to:
- Measure 10 meters
- Use a stop watch to time walking speed in seconds
- Find walking rate in meters per second
- Write an equation for distance using rate and time
- Use equations to predict variables given values from a situation
- Create tables given rates
- Graph points from a table
- Describe how change is represented in a table and graph
- Identify linear tables and equations
- Write an equation from a table and/or graph
- Describe the information that a point on a graph represents
- Compare two linear graphs
- Connect the concept of “starting amount” with y-intercept
- Make a table from a graph
- Describe strategies for finding points of intersection
- Describe the connection between steepness of a line and slope
- Set equations equal to each other to find points of intersection
- Make predictions about points between points on a line or between numbers in a table
- Describe how you can find the information for a variable in an equation
- Use “pouches and coins” to solve positive, whole number equations
- Write equations based on “pouch and coin” models
- Solve multi-step equations using equality
- Check answers to solved equations
- Measure the rise and run of school stairs in inches
- Find a stair’s ratio of rise to run
- Make a sketch of a set of stairs and label the rise to run measurements
- Identify the coefficient of a linear equation
- Find the slope and y-intercept represented in tables, graphs and equations
- Sketch and describe multiple graphs with parallel lines (same slope)
- Sketch and describe a pair of graphs with perpendicular lines (negative reciprocal slopes)
- Write linear equations from applied situations
- Write the coordinates of a point from a table
- Make predictions beyond the data set of a table
- Find the slope from two points on a line on a graph
- Find rate from a given situation
- Describe what expressions mean from a given situation
- Simplify expressions and give reasons for each step
- Evaluation expressions
- Use sales tax when evaluate expressions
- Compare “percent off” and “percent of” situations
- Find sale prices when given percent discount and sales tax situations
- Write inequalities
- Graph inequalities on a number line
- Solve inequalities and describe the solutions
- Explain why some values for a solution do not make sense
- Check-Up Quiz 1
- Partner Quiz
- Check-Up Quiz 2
- Common Core Additional Investigation 1 Check-Up Quiz- Graphing Proportions
- Common Core Additional Investigation 2 Check-Up Quiz- Equivalent Expressions
- Common Core Additional Investigation 3 Check-Up Quiz- Inequalities
- Common Core Unit Assessment- Moving Straight Ahead Part I
- Common Core Unit Assessment- Moving Straight Ahead Part II
- (optional) Unit Project- Conducting an Experiment
Elements of Instruction:
In Grade 6, instructional time focused on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. In this unit, the second area is emphasized and extended, with thorough analysis and application of algorithms and properties.
In two previous books, Variables and Patterns and Accentuate the Negative, students were asked to graph data in the coordinate plane and use symbols to represent relationships between variables. During Moving Straight Ahead, these skills are extended to identify, represent, and interpret patterns of change for linear relationships in tabular, graphical and symbolic forms (y=mx+b).
Also in the previous books, Variable and Patterns and Covering and Surrounding, students expressed relationships between variables in words, symbols, graphs, and tables. Now students are asked to write and interpret linear equations in y=mx+b.
In previous units, students learn about computing and interpreting ratios (Bits and Pieces II), finding rates of change in relationships between two variables (Variables and Patterns), and understanding positive and negative numbers (Accentuate the Negative). In this current unit, students learn about finding the slope of a line, interpreting slope as a ratio of vertical change to horizontal change between two points on a line, and interpreting slope as a rate of change of one variable to another.
In Bits and Pieces II, students found solutions to number sentences using fact families and in Variables and Patterns, students graphed relationships between two variables and found solutions using tables and graphs. Now in Moving Straight Ahead, they are to identify x- and y-intercepts from a graph, table or equation and to determine an equation of a line and find the solution to equations of the form y=mx+b and mx+b=nx+c (Moving Straight Ahead, pg. 9).
In the Common Core Additional Investigation 1 (CCAI1), students will extend their understanding of ratios and develop understanding of proportionality to solve problems using inverse variation relationships and converting rates to unit rates (pg.1). In MSA 1, students look at the change in the rate and its effects on various representations. They recognize that graphs of linear functions are straight lines and they begin to see that as the independent variable changes by a constant amount, there is a corresponding constant change in the dependent variable. At this point, some students will begin to recognize that the constant change is the coefficient of x in the equation y=mx+b.
As they proceed through the unit, constant rate of change and y-intercept are formalized. Students interpret the y-intercept as a special point on a line, as an entry in a table, or as the constant b in the equation y=mx+b. They also predict constant rate, decide whether relationships are decreasing or increasing, and begin to make connections among points on a line, a pair of data points on a table, and the solution to an equation.
Next, students find the ratio of vertical change to horizontal change between two points of a line. The connection between this ratio and constant rate of change is made explicit. They find the slope of a line given two points on a line and apply their knowledge of slope to explore lines that have the same slope (parallel lines) and lines that have slopes that are negative reciprocals (perpendicular lines). Graphing calculators are helpful during this unit to explore the slopes of many lines before students make their conjectures (Moving Straight Ahead, pg. 3).
In CCAI2, students must first understand that algebraic expressions can be written to represent problem situations. Have students explain what each value in their expressions represents. Before beginning the problems in this second Common Core Additional investigation, review the associative, commutative, and distributive properties, using different types of rational numbers (pg. 11).
Students use the properties of equality with equations in pictorial form and transition into solving equations symbolically by adding or subtracting the same number or variable or multiplying or dividing by the same nonzero number or variable on both sides of the equation. They also find the point of intersection of two lines (or the solution of a system of two linear equations) by setting the y values equal and then solving for x. The third Common Core Additional Investigation focuses on representing situations and solving problems using inequalities. There is a connection between solving equations and solving inequalities, but be sure to point out that there are differences as well. Review inequality symbols and the basic understanding of inequalities; why they are written and the meaning of open and closed circles in the graph (pg. 19).
Major misconceptions by and struggles for students in this unit include:
- Making the connection that the difference between numbers in a table and points on a graph is the rate of change
- Writing equations from a graph
- Writing equations from a table
- Confusing y-intercept with slope
- Confusing the independent variable with the dependent variable
- Forgetting to label the x- and y-axes in a table or graph
- Forgetting to provide a title for a table or graph
- Mismatching the y-intercept in a given table by finding the x-coordinate value of one instead of finding the x-coordinate value of zero.
- Creating an appropriate scale for a graph from a given situation or table of numbers
- Deciding whether or not to connect the points on a graph (discrete vs. continuous data)
- Describing patterns of change in words
- Coming up with a strategy to find a point of intersection for two separate linear situations
- Contextualizing and describing a point on a line for a situation
- Understanding the difference between a “head start” using distance and a “head start” using time
- Setting two expressions equal to each other and solving for a common unknown variable
- Keeping solution steps organized while solving two-step equations
- Transitioning from the “pouches and coins” equation model (all positive whole numbers) to abstract equations that include integers
- Knowing that points above or below a line are not solutions to a linear equation
- Inverting rise/run when finding slope
- Not making the connection that negative coefficients in linear equations mean “downward” slopes and that positive coefficients in linear equations mean “upward” slopes
- Distributing the factor outside of the parentheses to all integers inside of the parentheses during the Distributive Property
- Decontextualizing the properties of operations in given situations to write algebraic expressions
- Confusing open and closed circles when graphing inequalities
- Confusing greater than with less than symbols
Each lesson has differentiation options for each portion of the lesson. Additional differentiation options are listed with directions and student masters in the Teacher’s Guide.
- Special Needs Handbook
- Unit Projects
- Spanish Additional Practice and Skills guide
- Strategies for English Language Learners Guide
- “Extension” homework questions
- District-created notebooks
- Mathematical Reflections
- “Did You Know?” sections
- phschool.com and web codes
- “Connections” homework questions
- The real-world context embedded in lesson problems
Additional Resources / Games:
CMP2 and/or the Erie School District provide the following additional resources to aid students in achieving mathematical success.
- Additional Practice worksheets per investigation
- Skills Review worksheets to target key components of each investigation
- Parent letter to be sent home prior to beginning the unit to share with parents the skills, goals, and expectations of the coming unit.
- Assessment Resources workbook with extra test items (multiple choice, essay, open ended, question bank, etc)
- Investigation specific pre-generated notebooks that include tables, graphs, problem numbers, and all other items students may need to complete the investigation and all its parts. Students are provided with one per unit.
- Reflection questions at the end of each investigation to assess students’ comprehension of key concepts.
- phschool.com and web codes
- Transparencies of models, graphs, etc used within lesson(s)