6.2.1 Variables & Representations of Relationships
Overview
Patterns have a central component of change, repetition, or extension. At this level, students recognize and represent patterns with tables, graphs, and function rules, and are able to translate from one representation to another. Real-world situations provide meaningful opportunities and context to solve problems involving relationships between varying quantities. It is essential that students build an understanding of how contextual situations connect with patterns of change that allow them to connect a number in one set with a number in another set, resulting in the construction of a function. The eventual goal is to abstract from numbers and use algebraic representations to generalize numerical relationships and express mathematical ideas concisely.
All Standard Benchmarks
6.2.1.1 Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts.
6.2.1.2 Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations.
Benchmark Group A
- 6.2.1.1 Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts.
- 6.2.1.2 Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations.
What students should know and be able to do [at a mastery level] related to these benchmarks
- Use variables to represent quantities that can change;
- Recognize and use variables to represent additive (y = x + 2) and multiplicative (y = 2x) relationships between two varying quantities;
- Distinguish between independent and dependent variables in relationships;
- Represent relationships between two changing quantities with tables, graphs, or function rules;
- Translate between tables, graphs, and rules;
- Use tables, graphs, and function rules to solve real-world and mathematical problems.
Work from previous grades that supports this new learning includes:
- Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems;
- Use a rule or table to represent ordered pairs of positive integers and graph these ordered pairs on a coordinate system;
- Represent real-world situations using equations and inequalities involving variables. Create real-world situations corresponding to equations and inequalities;
- Use the associative, commutative and distributive properties and order of operations to generate equivalent expressions and to solve problems involving positive rational numbers.
NCTM Standards:
Understand patterns, relations, and functions
Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;
Relate and compare different forms of representation for a relationship;
Use mathematical models to represent and understand quantitative measurement
Model and solve contextualized problems using various representations, such as graphs, tables, and equations;
Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;
Use symbolic algebra to represent situations and to solve problems;
Analyze change in various contexts
Use graphs to analyze the nature of changes in linear relationships
Common Core State Standards (CCSS):
6. EE (Expressions and Equations) Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.6 Use variable to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.9 Use variables to represent two quantities in a real-world problem that can change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
Misconceptions
Student Misconceptions and Common Errors
- Students think that variables represent only one number.
- Students cannot distinguish between independent and dependent variables.
- Students misinterpret whether a relationship is additive or multiplicative given a table or graph.
- Students interpret y = 2x to mean that x is twice as large as y.
- Students misrepresent relationships involving subtraction or division (e.g., x - 3 or 3 - x; 3 ÷ x or x ÷ 3).
- Students incorrectly graph the ordered pair (x, y).
- Students believe that the axes have to be scaled in the same units when graphing.
Vignette
In the Classroom
This vignette tells how students use tables, graphs, and rules to explore how many meals they can pack for hungry children during their upcoming field trip.
Teacher: Next week our class will be taking a field trip to pack meals for starving children around the world. It is typical for volunteers to pack approximately 250 meals in two hours. That rate, 250 meals in 2 hours, is a relationship between two variables - time and the number of meals packed. Since the focus of our learning the past few weeks has been to represent relationships using graphs, tables, and rules, I think this situation provides a great opportunity for us to practice those skills.
Student: That's not fair. It seems like you find math problems in EVERY situation. Will there ever be a situation when you don't find a math problem?
Teacher: No, not really. Math is everywhere.
Student: I'm beginning to understand that.
Teacher: Back to the relationship between time and the number of meals packed. Take a few minutes now to represent this situation using a table. Be prepared to explain how you arrived at your answers.
Sample student work:
(Note: Some students reversed the columns in the table, leading to a discussion about dependent and independent variables.)
Teacher: How can this relationship be represented in a graph?
Student: We plot these points on a coordinate grid. I'd put time on the x-axis and meals packed on the y-axis.
Teacher: Does it matter which variable you put on the x-axis and which you put on the y-axis?
Student: Yes, the independent variable goes on the x-axis and the dependent variable goes on the y-axis.
Teacher: How did you decide which variable was which?
Student: The number of meals packed depends on the number of hours, so the number of hours is the independent variable.
Teacher: Yes, that's correct. Go ahead now and graph this relationship.
Student: How do I know if I should draw a line through the points after I plot them?
Teacher: Let's think about a point between 2 and 3 hours. Is it possible to pack meals for 2 ¼ hours?
Student: Well, I could pack meals for 2 hours and 15 minutes. I guess that would be the same as 2 ¼ hours.
Teacher: And how would you find the number of meals packed in 2 ¼ hours?
Student: I'd multiply 2 ¼ by 125. That's 281¼ meals.
Teacher: Is it possible to pack 281¼ meals?
Student: I suppose. That just means you completed 281 meals, and started on the next.
Student: I need another piece of graph paper. I can't fit my graph on my paper.
Teacher: Tell me how you set up your axes.
Student: I counted by ones, but I ran out of room on the y-axis.
Teacher: What's the largest y-value you want to graph?
Student: 1000.
Teacher: I don't think I have any graph paper that has 1000 units on it. How else could you organize the y-axis?
Student: I could count by 10s or maybe 50s, but I thought you had to use the same scales on both the x- and y- axes.
Student: No, then it would be impossible to fit some graphs on a piece of paper. It's only important to use equal intervals on each axis.
Teacher: Yes, that's true. Now go ahead and make your graphs.
Sample student work:
Teacher: Does anyone know what rule can be used to predict the number of meals that can be packed for t hours?
Student: That's easy. You must multiply the number of hours times 125.
Teacher: And how would you write that rule algebraically?
Student: M = 125t, where t is the time in hours and M represents the total number of meals packed.
Teacher: Which of these representations made it easier to find the rule?
Student: The table helped me more because it helped me see what the time needed to be multiplied by to find the number of meals packed.
Student: I liked the graph better, because I could see how much y increased for every unit that x increased
Teacher: So both representations could be used to find the rule. What is the advantage of knowing the rule?
Student: If you know the rule, you can predict the amount of meals made for any amount of hours.
Student: Or you could work backwards to predict how long it would take to pack a certain amount of meals.
Teacher: Tell me more about that.
Student: OK. Let's say you wanted to pack 10,000 meals. You could divide 10,000 by 125 and that tells you that it would take 80 hours to pack 10,000 meals.
Teacher: Good thinking. I have one more question for you. Do you think it's reasonable to expect that each of us will pack 250 meals during our 2 hour visit tomorrow?
Student: I don't know. That's about one meal every 30 seconds.
Teacher: Wow! I'm impressed. How did you figure that out so quickly>
Student: Easy. 125 meals per hour is the same as 125 meals in 60 minutes, since one hour has 60 minutes. To find the number of meals per minute, I need to divide 125 by 60. But I did an easier problem. I rounded 125 to 120 and divided 120 by 60. That gave me 2 meals every minute or 2 meals in 60 seconds. That's the same as 1 meal every 30 seconds.
Teacher: Sounds like we have our work cut out for us, but I know we're up to the challenge. What if we were able to work faster than typical volunteers? Let's say we could pack one meal every 20 seconds. How would that impact the number of meals we could pack?
Student: There you go again. That's a whole different math problem. I suppose we could go through the same process of making a table, graphing the points, and finding the rule.
Teacher: Yes, but this time I'd like to use the graphing calculators to represent the relationship between time and the number of meals packed. When we're finished, we can talk about the advantages of each representation.
Resources
Teacher Notes
- Students may have developed the misconception that variables represent only one number from solving equations such as x + 3 = 5, where x = 2 is the only value that makes the statement true. Using equations such as P = 7.00h, where P represents pay in dollars and h represents number of hours worked at $7.00 per hour, to determine pay for various numbers of hours will help students recognize that variables represent quantities that can change. As students begin to explore algebraic representation, they need to understand that a variable represents any value that makes the statement true. A variable, then, may represent one value, many values, or no value.
- Using equations that represent real-life situations, such as P = 7.00h, can also be used to help students distinguish between independent and dependent variables. Ask students, "Does the pay depend on the number of hours worked, or does number of hours worked depend on the pay?" Another way of saying this is, "Is the pay a function of the number of hours worked, or is the number of hours worked a function of the pay? Since the pay is a function of the number of hours worked, the equation P = 7.00h is sometimes written as f(h) = 7.00h.
- Since graphs of both additive and multiplicative relationships of two varying quantities can result in lines, it is easy for students to distinguish the relationships using a graph. It is helpful to have students translate the ordered pairs to a table for further examination. However, further misconceptions can occur when using tables. For example, students may misinterpret the multiplicative relationship shown in the table below to be additive by examining only the recursive nature of y (now + 2 = next), rather than considering the relationship between x and y.
This misconception can be addressed by extending student observations of this pattern to include the relationship between x and y as shown in the table below.
Students will also benefit from the opportunity to explore tables of additive and multiplicative relationships simultaneously. It is important to emphasize that when writing rules, you are expressing the relationship between two variables.
It is also important that students have multiple opportunities using a variety of representations to explore additive and multiplicative relationships. The chart below shows an example.
- When creating tables, the convention is to use the first column (row) for the independent variable, and the second column (row) for the dependent variable.
- When graphing, the convention is to use the x-coordinate to represent the independent variable and the y-coordinate to represent the dependent variable.
- It is helpful to refer to coordinates as ordered pairs to remind students that order matters. Remind students that the first coordinate represents the distance from 0 on the horizontal axis, while the second coordinate represents the distance from 0 on the vertical axis.
- It is likely that in previous graphing experiences, students have used the same scale for both axes. Asking students to graph an equation such as y = 365x, where x represents number of years and y represents total number of days will help students understand that it is not always practical to use the same scale for both axes. However, it is essential that students understand the necessity of using the same interval for each unit on an axis.
- When writing rules, the convention is to use x to represent the independent variable and y to represent the dependent variable.
- Students that interpret y = 2x to mean that x is twice as large as y will benefit from a discussion about equality. Remind students that since y = 2x, then y and 2x have the same value. If y = 4, then 2x must be 4. Therefore, x = 2.
- Remind students that order matters when writing function rules that involve subtraction and division, because those operations are not commutative.
- Function sense comes from looking for visual and number patterns and predicting outcomes from applying a rule. It helps students relate pictorial, symbolic, verbal, and concrete representations of a pattern and so develop multiple perspectives (Which is better - doubling $2 every year, or adding $50 every year?)
Fibonacci Trains
In this lesson, students use Cuisenaire Rods to build trains of different lengths and investigate patterns. Students make algebraic connections by writing rules and representing data in tables and graphs.
Representing Data
Students complete an organized chart, and create and interpret graphs to explore the relationship between a whale's length and its weight in these two units.
Bouncing Tennis Balls
In this lesson, students collect and record data using the real-world situation of bouncing a tennis ball. The data is represented in a table and graph, and used to explore the relationship between the dependent and independent variable in their experiment.
Counting Embedded Figures
Students look for patterns within given data and form generalizations for the problem, thereby sharpening algebraic skills.
Building Bridges
In this lesson, students transition from arithmetical to algebraic thinking by exploring problems that are not limited to single-solution responses. Values organized in tables and graphs are used to move towards symbolic representations.
Additional Instructional Resources
Grapher
This website provides a tool that allows you to type in a function rule and view its graph.
Lesson on Graphs
This lesson from NCTM is designed to strengthen understanding of the connections between graphs, tables, ordered pairs, and symbolic rules for linear patterns.
What's the Rule?
In this lesson, students display data in graphs and make generalizations to describe rules.
coordinate grid: a grid for locating points in a plane by using ordered pairs of numbers. It is formed by two number lines that intersect at right angles at their zero points. Example:
evaluate: to find the value. To evaluate algebraic expressions, particular numbers are substituted for variables before calculating. Example: To evaluate 7x for x = 5, x is replaced with 5, resulting in 35.
function: a special kind of relation in which every first value (x) in a set of ordered pairs (x, y) is paired with one and only one second value (y). Example: y = x + 3 is a function, since every value of x will be paired only with a number whose value is 3 more. Functions can be represented in various ways, including rules, graphs, and tables.
horizontal axis: positioned in a left-to-right position, parallel to the line of the horizon; referred to as the x-axis. Example:
rational number: any number that can be expressed in the form [math]\frac{a}{b}[/math], where a and b are integers and b ≠ 0. A rational number can always be represented by either a terminating or a repeating decimal. Examples: [math]\frac{2}{3}[/math]; 4 (which can be expressed as [math]\frac{4}{1}[/math]); 2.25 (which can be expressed as [math]\frac{225}{100}[/math]).
translate: change to a different form. Examples: 0.75 can be translated to [math]\frac{3}{4}[/math] or 75%; y = 2x can be translated to
variable: a quantity that changes or that can have different values; a letter is often used to represent a variable quantity. Example: in the expression 5n, n is a variable because it can have different values.
vertical axis: the vertical number line in the coordinate plane. Example:
x-axis: the horizontal number line in the coordinate plane. Example:
y-axis: the vertical axis. Example:
Reflection - Critical questions regarding the teaching and learning of these benchmarks
- What evidence shows that students recognize and use variables to represent relationships between varying quantities?
- What activities were used to connect real-life patterns of change expressed in words to tables, graphs, and function rules?
- What evidence shows that students can generalize patterns of change and represent them using rules?
- What evidence exists to show that students can distinguish additive and multiplicative relationships? Independent and dependent variables?
- What evidence shows that students can translate among tables, graphs, and rules?
- How did I use relevant real-world experiences to engage and motivate students?
Materials
- Representation as a Vehicle for Solving and Communicating
This NCTM article provides an activity that explores use of multiple representations in their classrooms, and builds understanding of representations as "tools that are vital for recording, analyzing, solving, and communicating mathematical data, problems, and ideas (p. 39)."
Keeley, P., & Rose, C. (2006). Mathematics curriculum topic study. Thousand Oaks, CA: Corwin Press.
Kilpatrick, J., Martin,W., & Schifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Minnesota's K-12 Mathematics Frameworks. (1998). St. Paul, MN: SciMathMN.
National Council of Teachers of Mathematics. (2010). Focus in grade 6 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Assessment
(DOK Level1)
- The cost C in dollars of a monthly phone plan can be found using the equation C = 19.99 + 0.15x, where x represents the number of text messages sent. Use the equation to find the cost of sending 78 text messages in one month.
Answer: $31.69
(DOK Level1)
2. Complete the table by using the rule y = x - 2.3.
Answers:
(DOK Level 2)
3. Which is the rule for the graph shown below?
a. y = 2x b. y = 2x - 1 c. y = x + 1 d. y = x + 2
Answer: d
(DOK Level 2)
4. Use the equation d = 50t, where d represents distance in miles and t represents time in hours, to make a table showing distance traveled in 1, 2, 3, or 4 hours.
Answer:
(DOK Level 3)
5. Write the rule for the table shown below.
Answer: y = 0.25 x or y = [math]\frac{1}{4}[/math]x
(DOK Level 4)
6. The perimeter P of a square can be found using the formula P = 4s, where s represents the length of one side. Describe how the perimeter of a square changes when the length of one side is decreased by 3.
Sample Answer: When the length of one side of a square is decreased by 3, the perimeter will decrease by 12. If the original perimeter is 4s, the perimeter of the smaller square will be 4(s - 3), or 4s - 12.
Differentiation
- Use manipulatives and pictorial representations to have students extend patterns.
- Provide graphic organizers, such as input/output charts, tables, and coordinate grids with labeled axes, to help students translate from one representation to another.
- Examine patterns of change that students encounter in everyday experiences to foster engagement and motivation.
- Patterns that Grow In this series of five lessons, students use logical thinking to create, identify, extend, and translate patterns.
- The word table has multiple meanings, and will require explicit instruction to clarify its meaning in mathematics.
- Use analogies to help students with the vocabulary. For example, in the same way a teacher substitute can replace a teacher, a number substitutes or replaces a variable in an algebraic expression so that it can be evaluated.
- It is essential that students can distinguish between dependent and independent variables in relationships. The analogy that children are dependent, or rely on their parents, can be used to help students understand that dependent variables rely on something else. Point out that dependent and independent have the same root word. However, independent begins with the prefix in, which means not and changes the meaning of independent to not dependent.
- Use graphic organizers such as the Frayer model shown below, for vocabulary development.
In this lesson of three units, students will represent data using tables, graphs, and rules to investigate a relationship between recursive functions and exponential functions.
In this lesson, students collect data using a rubber band bungee cord and a Barbie doll to create a scatter plot, determine a line of best fit, and make predictions.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
identifying and using variables to represent relationships between quantities that can change. | using real-world situations to explore relationships between quantities that change. |
representing relationships between varying quantities with tables, graphs, and rules. | posing problems that involve relationships between varying quantities that can be flexibly solved. |
translating patterns from one representation to another (tables, graphs, and rules). | asking students to consider the strengths and limitations of each representation. |
using patterns represented with input/output tables, charts, or graphs to determine rules. | providing students with multiple opportunities to explore a variety of patterns and make generalizations. |
extending patterns represented with tables, graphs, or rules. | asking students to make predictions and explain their reasoning. |
discussing and writing about patterns and relationships of varying quantities. | requiring students to communicate their thinking in oral and written forms. |
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