# Up, Up and Away

Students will create graphs from a spreadsheet, discover linear relationships, and explain the real world meaning of slope. Students need concrete experiences in diverse and meaningful contexts to develop an understanding of the concepts of slope and function. Students measure circumference and flight time for deflating balloons. They create a spreadsheet of the data and create graphs using pairs of actual and projected measures from the experiment. They then examine the graphs to identify linear and non-linear relationships. Understanding slope as a rate of change is an important part of the study of functions, which in turn provides the basis for much of the study of algebra. They analyze the slope of the linear relationships and interpret the meaning of their graphs.

### Standards & Objectives

Academic standards
CCSS.Math.Content.4.NF.C.5
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective...
CCSS.Math.Content.5.G.A.1
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to...
CCSS.Math.Content.5.G.A.2
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of...
CCSS.Math.Content.5.OA.A.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
CCSS.Math.Content.6.EE.A.2
Write, read, and evaluate expressions in which letters stand for numbers.
CCSS.Math.Content.6.EE.B.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...
CCSS.Math.Content.6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can...
CCSS.Math.Content.6.EE.B.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...
CCSS.Math.Content.6.EE.B.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...
CCSS.Math.Content.6.EE.C.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...
CCSS.Math.Content.6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same...
CCSS.Math.Content.6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to...
CCSS.Math.Content.6.NS.C.7
Understand ordering and absolute value of rational numbers.
CCSS.Math.Content.6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and...
CCSS.Math.Content.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings...
CCSS.Math.Content.6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b != 0, and use rate language in the context of a ratio relationship....
CCSS.Math.Content.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams,...
CCSS.Math.Content.7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are...
CCSS.Math.Content.7.EE.B.4
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by...
CCSS.Math.Content.7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a...
CCSS.Math.Content.7.RP.A.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different...
CCSS.Math.Content.7.RP.A.2
Recognize and represent proportional relationships between quantities.
CCSS.Math.Content.7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and...
CCSS.Math.Content.8.EE.C.7
Solve linear equations in one variable.
CCSS.Math.Content.8.F.A.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal...
CCSS.Math.Content.8.F.A.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For...
CCSS.Math.Content.8.F.B.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a...
CCSS.Math.Content.8.F.B.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or...
CCSS.Math.Content.8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
GLE 0406.2.5
Add and subtract fractions with like and unlike denominators.
GLE 0506.3.1
Understand and use order of operations.
GLE 0506.3.2
Develop and apply the concept of variable.
GLE 0506.3.3
Understand and apply the substitution property.
GLE 0506.4.3
Describe length/distance relationships using the first quadrant of the coordinate system.
GLE 0606.1.1
Use mathematical language, symbols, and definitions while developing mathematical reasoning.
GLE 0606.2.3
Understand and use ratios, rates and percents.
GLE 0606.3.1
Write and solve two-step equations and inequalities.
GLE 0606.3.2
Interpret and represent algebraic relationships with variables in expressions, simple equations and inequalities.
GLE 0606.3.3
Extend order of operations to include grouping symbols and exponents.
GLE 0606.3.4
Use expressions, equations and formulas to solve problems.
GLE 0606.3.5
Use multiple representations including symbolic algebra to model and/or solve contextual problems that involve linear relationships.
GLE 0606.3.6
Understand and use the Cartesian coordinate system.
GLE 0706.1.5
Use mathematical ideas and processes in different settings to formulate patterns, analyze graphs, set up and solve problems and interpret solutions.
GLE 0706.2.1
Extend understandings of addition, subtraction, multiplication and division to integers.
GLE 0706.2.2
Understand and work with the properties of and operations on the system of rational numbers.
GLE 0706.2.3
Develop an understanding of and apply proportionality.
GLE 0706.2.4
Use ratios, rates and percents to solve single- and multi-step problems in various contexts.
GLE 0706.3.1
Recognize and generate equivalent forms for simple algebraic expressions.
GLE 0706.3.2
Understand and compare various representations of relations and functions.
GLE 0706.3.5
Understand and graph proportional relationships.
GLE 0706.3.7
Use mathematical models involving linear equations to analyze real-world phenomena.
GLE 0706.3.8
Use a variety of strategies to efficiently solve linear equations and inequalities.
GLE 0706.4.2
Apply proportionality to converting among different units of measurements to solve problems involving rates such as motion at a constant speed.
GLE 0706.4.4
Understand and use ratios, derived quantities, and indirect measurements.
GLE 0806.3.1
Recognize and generate equivalent forms for algebraic expressions.
GLE 0806.3.2
Represent, analyze, and solve problems involving linear equations and inequalities in one and two variables.
GLE 0806.3.4
Translate among verbal, tabular, graphical and algebraic representations of linear functions.
GLE 0806.3.5
Use slope to analyze situations and solve problems.
GLE 0806.3.6
Compare and contrast linear and nonlinear functions.
SPI 0406.2.5
Generate equivalent forms of common fractions and decimals and use them to compare size.
SPI 0406.3.1
Use letters and symbols to represent an unknown quantity and write a simple mathematical expression
SPI 0406.4.2
Graph and interpret points with whole number or letter coordinates on grids or in the first quadrant of the coordinate plane.
SPI 0406.4.3
Construct geometric figures with vertices at points on a coordinate grid.
SPI 0506.2.7
Recognize equivalent representations for the same number.
SPI 0506.2.9
Compare whole numbers, decimals and fractions using the symbols , and =.
SPI 0506.3.1
Evaluate algebraic expressions involving decimals and fractions using order of operations.
SPI 0506.3.2
Use variables appropriately to represent numbers whose values are not yet known.
SPI 0506.3.3
Find the unknown in single-step equations involving fractions and mixed numbers.
SPI 0506.4.5
Find the length of vertical or horizontal line segments in the first quadrant of the coordinate system, including problems that require the use of fractions and...
SPI 0606.2.6
Solve problems involving ratios, rates and percents.
SPI 0606.2.7
Locate positive rational numbers on the number line.
SPI 0606.2.8
Locate integers on the number line.
SPI 0606.3.1
Represent on a number line the solution of a linear inequality.
SPI 0606.3.2
Use order of operations and parentheses to simplify expressions and solve problems.
SPI 0606.3.3
Write equations that correspond to given situations or represent a given mathematical relationship.
SPI 0606.3.4
Rewrite expressions to represent quantities in different ways.
SPI 0606.3.5
Translate between verbal expressions/sentences and algebraic expressions/equations.
SPI 0606.3.8
Select the qualitative graph that models a contextual situation (e.g., water filling then draining from a bathtub).
SPI 0606.3.9
Graph ordered pairs of integers in all four quadrants of the Cartesian coordinate system.
SPI 0706.1.3
Recognize whether information given in a table, graph, or formula suggests a directly proportional, linear, inversely proportional, or other nonlinear relationship.
SPI 0706.2.1
Simplify numerical expressions involving rational numbers.
SPI 0706.2.5
Solve contextual problems that involve operations with integers.
SPI 0706.2.6
Express the ratio between two quantities as a percent, and a percent as a ratio or fraction.
SPI 0706.2.7
Use ratios and proportions to solve problems.
SPI 0706.3.1
Evaluate algebraic expressions involving rational values for coefficients and/or variables.
SPI 0706.3.3
Given a table of inputs x and outputs f(x), identify the function rule and continue the pattern.
SPI 0706.3.4
Interpret the slope of a line as a unit rate given the graph of a proportional relationship.
SPI 0706.3.5
Represent proportional relationships with equations, tables and graphs.
SPI 0706.3.6
Solve linear equations with rational coefficients symbolically or graphically.
SPI 0706.3.7
Translate between verbal and symbolic representations of real-world phenomena involving linear equations.
SPI 0706.3.8
Solve contextual problems involving two-step linear equations.
SPI 0706.3.9
Solve linear inequalities in one variable with rational coefficients symbolically or graphically.
SPI 0806.1.1
Solve problems involving rate/time/distance (i.e., d = rt).
SPI 0806.1.2
Interpret a qualitative graph representing a contextual situation.
SPI 0806.1.3
Calculates rates involving cost per unit to determine the best buy.
SPI 0806.3.2
Solve the linear equation f(x) = g(x).
SPI 0806.3.4
Translate between various representations of a linear function.
SPI 0806.3.5
Determine the slope of a line from an equation, two given points, a table or a graph.
SPI 0806.3.6
Analyze the graph of a linear function to find solutions and intercepts.
SPI 0806.3.7
Identify, compare and contrast functions as linear or nonlinear.
SPI 0806.4.2
Apply the Pythagorean theorem to find distances between points in the coordinate plane to measure lengths and analyze polygons and polyhedra.
SPI 0806.4.4
Convert between and within the U.S. Customary System and the metric system.
TSS.Math.4.NF.C.5
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective...
TSS.Math.5.G.A.1
Graph ordered pairs and label points using the first quadrant of the coordinate plane. Understand in the ordered pair that the first number indicates the...
TSS.Math.5.G.A.2
Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane and interpret coordinate values of points in...
TSS.Math.5.OA.A.1
Use parentheses and/or brackets in numerical expressions and evaluate expressions having these symbols using the conventional order (Order of Operations).
TSS.Math.6.EE.A.2
Write, read, and evaluate expressions in which variables stand for numbers.
TSS.Math.6.EE.B.5
Understand solving an equation or inequality is carried out by determining if any of the values from a given set make the equation or inequality true. Use...
TSS.Math.6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an...
TSS.Math.6.EE.B.7
Solve real-world and mathematical problems by writing and solving onestep equations of the form x + p = q and px = q for cases in which p, q, and x are all...
TSS.Math.6.EE.B.8
Interpret and write an inequality of the form x > c or x
TSS.Math.6.EE.C.9
Use variables to represent two quantities in a real-world problem that change in relationship to one another.
TSS.Math.6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side that joins two vertices (vertical or...
TSS.Math.6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent...
TSS.Math.6.NS.C.7
Understand ordering and absolute value of rational numbers.
TSS.Math.6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value...
TSS.Math.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, the ratio of wings to beaks...
TSS.Math.6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b != 0. Use rate language in the context of a ratio relationship.
TSS.Math.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems.
TSS.Math.7.EE.A.2
Understand that rewriting an expression in different forms in a contextual problem can provide multiple ways of interpreting the problem and how the...
TSS.Math.7.EE.B.4
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by...
TSS.Math.7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a...
TSS.Math.7.RP.A.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For...
TSS.Math.7.RP.A.2
Recognize and represent proportional relationships between quantities.
TSS.Math.7.RP.A.3
Use proportional relationships to solve multi-step ratio and percent problems.
TSS.Math.8.EE.C.7
Solve linear equations in one variable.
TSS.Math.8.F.A.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
TSS.Math.8.F.A.3
Know and interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
TSS.Math.8.F.B.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description...
TSS.Math.8.F.B.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear...
TSS.Math.8.G.B.6
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

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Learning objectives:

Lesson Objective:

Students will create graphs from a spreadsheet, discover linear relationships, and explain the real world meaning of slope.

### Lesson Variations

Blooms taxonomy level:
Analyzing

### Helpful Hints

Prior to the Lesson: Students should have some experience with graphing and interpreting graphs for sets of data. They should have some familiarity with functions and the concept of slope. It would be helpful if they had some prior experience with spreadsheets and graphing on computers.

Contributors: