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Graphing & charting

One variable & two variables

Questioning is the foundation of all learning.
The first step in rejecting not knowing is to ask, why?
Sweetland

Overview

Introduction

This page includes suggestions for teaching and developing graphing literacy across K-12 grade levels. Includes activities with pedagogical information.

Related graphing resources

Planning information

Background information

While this page reviews the use of charts as background information for developing graphing and the development of one and two variable graphing within data analysis . And graphing as one of four ways to solve two variable problems Algebra.

Background information necessary is the understanding of number values necessary for the numbers used to represent the values of the data along with problem solving. And data analysis

Intended learnings

Charting and graphing K-9 mapping

K - 3 average, bar graph, concrete real object graph, data, grid, picture graph, table, tally, trial.

  • One variable statistics - Introduction to a one variable chart. To displays data for a single variable and show its distribution - frequency (how often a value occurs) or range of values (highest to lwest) & measures of central tendencies (mean, median, mode) , or trend. Examples include bar charts, histograms, pie charts, and line graphs with only one dependent variable.
    • Sample problems for one variable statistics
      • Ping-Pong Ball Race 
      • Number of stars in a minute, 
      • How many M&M's of each color are in a package?
      • Plant growth unit with graphing plant height
      • M&M® Challenge - activity directions and worksheets for data analysis, mean, median, mode, & graphing of colors of M&M's. Can be used for probability.
      • Chart the number of letters in their first name and last name
      • Chart both the ages of learners in years and months
      • Bean toss - chart frequence of sums

4 - 6 broken line graph, descriptive statistics, equally likely events, extrapolate, frequency, generalization, histogram, inference, interpret, interpolate, line graph, line plot, mean, media, midway, mode, ordered pair, outcome, probability, quadrant, random, range, ratio, sample, scale drawin,g simple event, statistics.

    • Continue with one variable charts and graphs. A one variable line graph with only one dependent variable shows how a single variable changes over time or across a sequence. The x-axis usually represents time or categories, and the y-axis shows the variable's value, with points connected by a line to reveal trends or patterns.Graph rate of motel prices for different Monopoly places.
    • Use the rate of water flow from a shower head to graph the gallons of water used for a shower from 0-20 minutes. Find how much water is used when taking a bath in a regular and jacuzzi tub. Use the information to decide under what conditions it is better to take a bath or shower to conserve on water.
    • Pictures plots to practice plotting on a Cartesian Graph - Worksheet to plot a picture and create your own points to plot. Source Johnnie Ostermeyer
    • Find your class's hang time and vertical jump statistics - worksheet range, median, mean, mode, graph.
    • Graphs as pictures tennis ball and wadded paper toss and graph -
    • Pendulum swinging - Change or no change. Relationship or no relationship. Swing a pendulum by changing variablesof mass, graph, then repeat with different string lengths.

7 - 9 box plot, compound event, circle graph, error of measurement, extrapolate, mutually exclusive, outlier, quartile, scatter plot, skew, lines stem and leaf plot, whisker. What is the region bounded by y=sin(kx)+2, x=0, x=2π, and y=0, with k constant.

Social skills - Groups

Working in groups reduces stress and stereotyping: Groups provide a lower risk free environment to participate, make mistakes, and look for solutions.

Additionally learners sometimes view math more as a male subject or other racial preferences rather than gender and racial neutral.

Both can cause them to begin to doubt their competence in math and can believe they can’t do math.

Most learners, including females, tend to learn math better when taught using groups rather than the traditional competitive classroom methods of teacher demonstrating and lecturing.

Making this a truly cooperative lesson would further reduce the competition.

Activities that are a cooperative endeavor in finding solutions and justifying them rather competition against accuracy rather than time will also encourage more students.

Asking each learner group to prepare as an expert to demonstrate his or her method to his or her peers can provide motivation and focus.

Mathematical content - number value, geometry, measurement, algebra, data analysis, statistics, and probability

(What mathematics explains - enduring understanding, big ideas, generalizations)

Sets of related numbers can be visually represented in charts and graphs.

Concepts and facts - from algebra & patterns ...

  • Charts are a visual representation of a distribution of a set of data.
  • Graphs are a visual representation of a relationship between variables.
  • A system can have any number of equations and any number of variables. 
  • When there are only two variables it is possible to visualize the relationship with a graph on a coordinate plane.  
  • Each person in the world could pick a different three points on a line, because there are infinite ordered pairs as solutions for any line. All of which will graphed with the same line.
  • Simultaneous equations can be represented with graphs.
  • Systems of equations are dependent, consistent and independent.
  • Substitution and elimination methods can be used to solve simultaneous equations.
  • A system of equations is two equations that share variables. 
    x + y =10; -3x + y = 2
  • A system of equations can be represented graphically in different ways:
  • Variables are conditions that can be changed and that can affect outcomes.
  • Variables can represent, size, shape, temperature, amount, volume, rate, ...

Outcome

  • Solve and demonstrate the solutions to problems using graphing
  • Solve problems with graphing, substitution, addition, and determinants. Activity 6

Specific outcomes -

  1. Set up and label a coordinate system
  2. Understanding Cartesian coordinate systems (labeling axes, plotting points, ordered pairs, scaling axes, etc.
  3. Equations for straight line and forms of linear equations y=mx + b.
  4. Understanding of slope
  5. Parallel and perpendicular lines and their equations
  6. Provide appropriate scale 
  7. Represent given equation graphically, accurately and neatly
  8. Plot points accurately and neatly 
  9. Identify the solution set from the graph as an ordered pair 
  10. Accurately identify the variables.
  11. Translating the word problem into an equation. 
  12. Labeling the axes using the correct units.
  13. Provide a title that is a relationships.

Anticipated learner thinkings & misconceptions
(Source concepts & misconceptions)

Miconception - as concepts

  • Graphs only represent numbers. Not a relationship between a two sets of related values.
  • Slope is the rise or the run.
  • All equations produce straight lines, even quadratic, cubic, or exponential.

Miconception - as outcomes

  • Confusing x and y: Plotting points with coordinates reversed or mixing up which axis is x vs. y.
  • Use rise or run instead of slope as the relationship of rise over run.
  • Misinterpret negative slopes.
  • Think the x and y intercepts are the same or forget to check where the line crosses the axes.
  • Misplot points
  • Use a scale that creates distorted graphs.
  • Plotting only one point and guess the graph's shape.
  • Misread negative values .
  • Placing points in the wrong quadrants.

Mathematical Processes - Problem solving, representation, proof and reasoning, communications, and connections

(How mathematics inquires - process, skill, methodology)

Together people can use a general problem solving procedure to solve problems and verify their answers with different solutions to better convince each other of their validity.

Concepts and facts -

  • Solving problmes can be assissted with a problem solving heuristic.

Outcome

  • Solve a problem using a problem solving heuristic.
  • Use different representations to communicate, by demonstrating solutions.

Specific outcomes -

  1. Problem solving heuristic includes general steps for solving mathematical problems
  2. Use graphs to represent equations and mathematic relationships of variables.
  3. Solve two equations with two unknowns using four methods: graphing, substitution, addition, and determinants.
  4. Communicate with peers of the same and opposite sex mathematical ideas.

Habits of mind - values, attitudes

(Attitudes and values that contribute to mathematical success)

Curiosity, open-minded, skepticism, persistence, .

Concepts and facts -

  • Both sexes are equally capable of learning and teaching math.
  • Diverse populations can achieve similar results.
  • Cooperation brings together diverse ideas for the benefit or all.

Outcome - cooperation and planning

  • Develop attitudes and skills for cooperation.
  • Develop planning skills.

Specific outcomes -

  1. Cooperations looks, sounds, and feels like:

Scoring guides suggestions (rubric)

Graph creations (scoring guide)

Correctly graph a set of equations on the same Cartesian coordinate system

  • Set up and label a coordinate system
  • Understanding Cartesian coordinate systems (labeling axes, plotting points, ordered pairs, scaling axes, etc.
  • Equations for straight line and forms of linear equations
  • Understanding of slope
  • Parallel and perpendicular lines and their equations
  • Provide appropriate scale 
  • Represent given equation graphically, accurately and neatly
  • Plot points accurately and neatly 
  • Identify the solution set from the graph as an ordered pair 
  • Accurately identify the variables.
  • Translating the word problem into an equation. 
  • Labeling the axes using the correct units.
  • Provide a title that is a relationships.

Solve equations using graphing, substitution, addition, and determinants (scoring guide)

Graph linear equations, systems of equations 

  • Identify solution sets for systems of equations from graphs 
  • Identify systems of equations that are parallel/inconsistent, 
  • perpendicular/consistent, dependent and independent 
  • Identify solution sets 
  • Interpret algebraic, graphic and verbal representations of 
  • systems of equations 
  • Solve systems of equation using substitution 
  • Solve systems of equations using elimination 
  • Identify systems of equations that are consistent, inconsistent, dependent and independent 
  • 9. Graph solution sets after solving systems of equations algebraically
  • Students will use large chart paper to reproduce o the graphs for display

Communication of mathematical ideas.

 

Habits of mind for cooperation - listen, don't interrupt, respond to others ideas, use others ideas as example or as showing an exception

 

Pedagogical Overview & Strategies to achieve educational learnings

Based on learning cycle theory & method for instruction

Activities Sequence to provide sufficient opportunities for students to achieve the targeted outcomes.

Make sure learners have the prior knowledge identified in the background information.

  1. Activity 1 - Meat or cheese sandwich
  2. Activity 2 - Make it your way
  3. Activity 3 -
  4. Activity 4 -
  5. Activity 5 - Tee shirt decisions
  6. Activity 6 - Solving for two unknowns four ways. (8-12 Grade)

Focus question

Unit focus question:

How can we make s a visual representation for an equation or mathematical relationship of a variable or two variables.

Sub focus questions:

  1. What are the elements of a graph and how are they used. - Coordinate Graphing Notes

Materials

Lab notes

Resources

Lesson Plans

Activity 1 - Meat or cheese sandwich

Materials

Focus questions:

  1. How can I make a mathematical representation for sandwiches?

Learning outcomes:

  1. Make a table for 1o - meat - bologna or - American cheese on two sliced of bread and then represent it on a graph.

Suggested procedures overview:

  1. Put learners in groups, focus their attention, and assess their initial understanding of the focus questions.
  2. Activity - Make a table for 1o - meat - bologna or - American cheese on two sliced of bread and then represent it on a graph.

Exploration -

  1. Put learners in pairs.
  2. Ask. How can we make a mathematical representation for sandwiches?
  3. Say you need to make ten sandwiches how many slices of bread and how many pieces of cheese do you need?
  4. Fill in the table.

Invention -

  1. Explain and discuss with the learners

Discover

 

Activity 2 - Meat or cheese sandwich

Materials

Focus questions:

  1. How can I make a mathematical representation for sandwiches?

Learning outcomes:

  1. Make a table for 1o - sandwiches made your way and then represent it on a graph.

Suggested procedures overview:

  1. Put students in groups, focus their attention, and assess their initial understanding of the focus questions.
  2. Activity -

Exploration

  1. Put students in pairs.

Invention

  1. Recall and review

Discover


Activity 3 -

Materials:

Focus questions:

Learning outcomes:

  • T

Suggested procedures overview:

  1. Put students in groups, focus their attention, and assess their initial understanding of the focus questions.
  2. Activity -

Exploration

  1. Put learners in pairs.

Invention

  1. What

Discovery

  1. Discuss how


 

Activity 4 -

Materials

Focus questions:

  1. How does

Learning outcomes:

  1. Learners

Suggested procedures overview:

  1. Put students in groups, focus their attention, and assess their initial understanding of the focus questions.
  2. Activity -

Exploration

  1. Ask.

Invention

  1. Ask. How

Discover

Activity 5 - Tee shirt decisions

Materials:

Focus questions:

  1. How do we determine the best buy of T-shirts from companies with different pricing?

Learning outcomes:

  1. Calculate, graph, and explain similarities and differences between the cost for different amounts of t-shirts from two different companies as represented on a graph showing their relationships between quantity of shirts and their price.

Suggested procedures overview:

  1. Put students in groups, focus their attention, and assess their initial understanding of the focus questions.
  2. Activity - Use the pricing information to calculate, graph, and explain similarities and differences between the cost for different amounts of t-shirts from two different companies as represented on a graph showing their relationships between quantity of shirts and their price.

Exploration

  1. Organize learners into pairs or larger groups.
  2. Give them the pricing information and ask them to create a graph to use with a presentation to show the cost of different amounts of shirts to make a decision on which company to order from.
  3. Here are two companies selected for the best deals:
    (10, 150) ... I&C = 15n ; UTCo = 10n + 50

Ink & Cotton

T-shirts with your team name and logo!

$15.00 per shirt (S, M, L)

Urban Tee Co.

T-shirts with your team name and logo! (S, M, L)

One time set up charge of $50.00

$10.00 per shirt

Exploration

  1. Put learners into groups.
  2. Present them with the t-shirt challenge.
  3. Let them create their graph and present them.
  4. Display graphs.

Invention

  1. Each group explains their graph.
  2. Resolve any differences between slope and where the lines intersect and diverge.
  3. Together write a summary.
  4. Summary - If we order less than ____ we should order them from Ink & Cotton. If we order ________, we can order from either. If weorder more than ________ we should order from Urban Tee Co.

Discovery

  1. Write a similar problem to share with an other group.
  2. Exchange problems and creat a solution.
  3. Share solutions

Activity 6 - Solving for two unknowns four ways. (8-12 Grade)

Materials

Focus questions:

  1. What are the different ways to solve problems with two unknowns?

Learning outcomes:

  1. Solve a problem using graphing, substitution, addition, and determinants.
  2. Identify and explain attributes for graphing (x, y, y intercept, x intercept, slope, y=mx + b) as they apply to different problems.
  3. Develop a procedure for solving problems by graphing, substitution, addition, and determinants.
  4. Demonstrate method for solving problem by graphing, substitution, addition, and determinants; and generalize them to problems of a certain type.
  5. Identify variables and describe how they operate to effect other variables. (operational definition).
  6. Identiry systems of equations as dependent, consistent and independent.
  7. Use substitution and elimination methods to solve simultaneous equations.
  8. Know a system of equations is two equations that share variables. 
    x + y =10; -3x + y = 2
  9. Demonstrate that a system of equations can be represented graphically in different ways:

Pedagogical ideas

In this investigation learner explore equations by solving problems of two equations with two unknowns in four different ways: graphing, substitution, addition, and determinants and communication what an equation and graph represent.

In addition to graphing concepts algebra concepts include.

  • Systems of equations are dependent, consistent and independent.
  • Substitution and elimination methods can be used to solve simultaneous equations.
  • A system of equations is two equations that share variables. 
    x + y =10; -3x + y = 2
  • A system of equations can be represented graphically in different ways:

The jigsaw model asks learners to understand a method of solving equations, communicate their understanding to her or his peers, and to attempt to help the other group members to understand it and plan a demonstration for the method to the other peers. This enables every class member to have an area of expertise and empowers all learners to be valuable and capable among their peers, which helps improve their self-efficacy with math and social communications with peers.

Suggested procedures overview:

  1. Put students in groups, focus their attention, and assess their initial understanding of the focus questions.
  2. Activity - Have learners use the jig-saw method to learn and prepare to be an expert to demonstrate one of the four ways to solve problems with two unknowns.
  3. Repeat the process four time.
  4. Identify problems and select solution methods and discuss reasons for selecting each method.

Possible Activity Sequence

Exploration

  1. Organize learners into groups and pairs.
  2. Put learners into four work alike groups with equal numbers of girls and boys in each group. Consider separating girls and boys who usually associate with each other and students at the same skill levels and other diverse groups.
  3. Assign each group one of four methods for solving two equations with two unknowns: graphing, substitution, addition, and determinants.
  4. Tell learners to use their textbooks and supplementary books to learn how to use this method. They should work together to apply the method to various problems in the books and to make sure that everyone in the group can solve the equations using the given method.
  5. If necessary provide a problem for each group.
    1. Graphing - Ralph started a lawn service business and is going to pay him self two dollars an hour more than what he is paying Chris. What equation will describe both salaries? What would the wages be for different hour amounts? What would a graph of this relationship look like?
      • Graphing - Describe what each of the following are and the relationship to your problem: x, y, y intercept, x intercept, slope, y=mx + b, and create a procedure to demonstrate how to use graphing to solve real world problems and relate each of these ideas to a graphing method.

    Create other problems and describe ways to decide which of the four procedures might be most appropriate.

    1. Substitution -
    2. Addition -
    3. Determinants -
  6. Once they feel confident about using the method being learned by their group, the teacher should check their problems to be sure they understand it.
  7. When the teacher assesses the learners understand of the method, the group can begin to design its demonstration, which will demonstrate the method to others.
  8. The demonstration should have four part:
    1. Purpose,
    2. Materials,
    3. Procedure, and
    4. Proof and reasoning for their confidence of the solution with their demonstrated method.
    • Additionally - Each person in the group needs to make a contribution to the demonstration. The demonstration can be as long or short as the group feels is necessary to demonstrate the method and justify it's methodology. And each group member should be able to demonstrate their method.
  9. The teacher should assess each group’s demonstration to see how the method will be communicated and make copies of the demonstration for each student in the group.
  10. Reassign groups one of the four experts in each group.
  11. Each learner in the group should demonstrate his or her method for solving equations to the other group members.
  12. The learner should make sure that all of the members of the group can perform the four methods. When the groups feel they are confident about the methods, the class can regroup.

Invention

  1. Each learner in the group should choose a number from 1 to 4 (or up to the number of students in the group), which will serve to identify the student from each group who will be solving the problem.
  2. Randomly choose a number, and the student from each group with that number should come to the board.
  3. After sufficient time seated group members can provide hints and clues if necessary or give a thumbs up if they agree.

Discover

  1. Have learners create a problem of their own and pass it aroung to other group members who select a method for solving it.
  2. Select a couple of problems to solve with the suggested method and discuss the outcomes.

 

Lab Notes for activities

Lab notes 1 - Meat or cheese sandwich

Materials

Focus questions:

  • How can I make a mathematical representation for sandwiches?

Challenge

First fill in the table.

Sandwich Slice of Bologna or American cheese Slices of bread
1    
2    
3    
4    
5    
6    
7    
8    
9    
10    

 

Draw and label a graph for the relationship of your made your way sandwiches!

  • Label the axes using the correct units.
  • Provide a title that is a relationships.
  • What is the slope?
  • What is the x intercept?
  • What is the y intercept?
  • What is the equation for the line?

 

Lab notes 2 - Make it your way

Materials

Focus questions:

  • How can I make a mathematical representation for my favorite sandwich?

Challenge

Design yur favorite sandwich with all the fixens.

 

 

 

 

Then fill in the table.

Sandwich            
1            
2            
3            
4            
5            
6            
7            
8            
9            
10            

 

Draw and label a graph for the relationship of your made your way sandwiches!

  • Label the axes using the correct units.
  • Provide a title that is a relationships.
  • What is the slope?
  • What is the x intercept?
  • What is the y intercept?
  • What is the equation for the line?

Lab notes 3 -

Materials

Focus questions:

  • How d

 

Challenge

 

 

 

 


Lab notes 4 -

Materials

Focus questions:

  • How

Challenge

 

 

 

Lab notes 5 - Tee shirt decisions

Materials

Focus questions:

  • How can you best illustrate the cost of different amounts of goods that shows their range of pricing for different amounts of items?

Challenge

A research group of students researched prices for tee shirts and found two companies they thought would provide them for your group, but they aren't sure which would be best.

Here is the information for the two companies.

Make a graph to use to explain similarities and differences between the cost for different amounts of t-shirts from the two different companies that shows the relationships between quantity of shirts and their price.

Ink & Cotton

T-shirts with your team name and logo!

$15.00 per shirt (S, M, L)

Urban Tee Co.

T-shirts with your team name and logo! (S, M, L)

One time set up charge of $50.00

$10.00 per shirt

 

 

 

 

 

 

 

 

Lab notes 6 - Solving for two unknowns four ways

Materials

 

Challenge

Notes

Graphing - Ralph started a lawn service business and is going to pay him self two dollars an hour more than what he is paying Chris. What equation will describe both salaries? What would the wages be for different hour amounts? What would a graph of this relationship look like?

 

 

 

 

Substitution -

 

 

 

 

Addition -

 

 

 

 

Determinants

Support materials

Solutions for Solving for two unknowns four ways

Graphing Method
x + y = 4
x – y = 4
y = x + 4
y = x – 4

Solution: (4, 0)

 

x | y
____
0 | 4
1 | 3
2 | 2
3 | 1
4 | 0

Substitution Method

3x + 5y = 3
x + 4y = 8
3(8-4y) +5y = 3
24 – 12y + 5y = 3
-7y = -21
y = 3
x = -4
Solution: (-4, 3)

 

Determinants Method
3x – 2y = 7
3x + 2y = 9
D = 3 -2 = 6 – (-6) = 12
3 2

Dx = 7 -2 = 14 – (-18) = 32
9 2

Dy = 3 7 =27 – 21 = 6
3 9

x = Dx/D = 32/12 = 8/3

y = Dy/D = 6/12 = ½
Solution: (8/3, ½)

 

Addition Method

x + y = 2
3x – 2y = 0
2x + 2y = 4
3x – 2y = 0
5x = 4
X = 4/5
4/5 + y = 2
y = 10/5 – 4/5
y = 6/5
Solution: (4/5, 6/5)

 

 

 

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