8.2.2 Sequences & Linear Functions
Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another.
Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.
Identify how coefficient changes in the equation f (x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects.
Represent arithmetic sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
For example: If a girl starts with $100 in savings and adds $10 at the end of each month, she will have 100 + 10x dollars after x months.
Represent geometric sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
For example: If a girl invests $100 at 10% annual interest, she will have 100(1.1x) dollars after x years.
Overview
Standard 8.2.2 Essential Understandings
In this standard, it is essential for students to be able to move fluidly between different representations of linear functions. Students move from their prior learning of linear relationships that are proportional to all linear functions. Tables, graphs and equations are used to find and interpret solutions to real-world linear situations. When students identify a situation as linear, it is essential that they are able to identify and make meaning of the slope and y-intercept. Students can understand more about linear when they compare linear function to other functions. As students explore other functions such as inverse and exponential functions, it is important that the tables, graphs, equations and situations are continuously compared to what they know about linear functions.
All Standard Benchmarks
8.2.2.1 Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another.
8.2.2.2 Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.
Example: Coordinates used for determining slope must contain integer values.
8.2.2.3 Identify how coefficient changes in the equation f (x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects.
8.2.2.4 Represent arithmetic sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
Example: If a girl starts with $100 in savings and adds $10 at the end of each month, she will have 100 + 10x dollars after x months.
8.2.2.5 Represent geometric sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
Example: If a girl invests $100 at 10% annual interest, she will have 100(1.1x) dollars after x years.
8.2.2 Sequences and Functions
8.2.2.1 Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another.
8.2.2.2 Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.
Example: Coordinates used for determining slope must contain integer values.
8.2.2.3 Identify how coefficient changes in the equation f (x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects.
8.2.2.4 Represent arithmetic sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
Example: If a girl starts with $100 in savings and adds $10 at the end of each month, she will have 100 + 10x dollars after x months.
8.2.2.5 Represent geometric sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.
Example: If a girl invests $100 at 10% annual interest, she will have 100(1.1x) dollars after x years.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Recognize linear relationships as they are expressed in a variety of formats;
- Given one form of a linear function, such as a table, words, equation, or graph, be able to transfer to any other form;
- Identify the slope and y -intercept of a linear function;
- Interpret the slope and y -intercept in the context of the given situation;
- Know how changing the coefficient changes the line on the graph;
- Know how to solve a proportional situation differently than a linear function that is not proportional;
- Students should be able to describe the pattern of a sequence by stating what is repeatedly being added or repeatedly being multiplied in the sequence;
- Students should be able to connect the sequence to the function that represents the sequence;
- Students will be able to translate between the equation, table, graph and verbal descriptions of the sequences;
- Students should make the connection between arithmetic sequences and linear functions as well as geometric sequences and exponential functions;
- Students will be able to use the different representations of arithmetic/geometric sequences to solve problems.
Work from previous grades that supports this new learning includes:
- Know how to represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs;
- Know how to calculate the constant of proportionality (unit rate or slope) given a proportional relationship;
- Use of exponents and exponential form;
- Understand linear functions (equations, tables, graphs).
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;
- Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.
- Represent and analyzemathematical situations and structures using algebraic symbols:
- 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, especially those that involve linear relationships.
- Use mathematical models to represent and understand quantitative relationships:
- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
- Analyze change in various contexts:
- Use graphs to analyze the nature of changes in quantities in linear relationships.
Common Core State Standards (CCSS)
- 8.F (Functions) Define, evaluate, and compare functions.
- 8.F.2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
- 8.F (Functions)Use functions to model relationships between quantities.
- 8.F.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 of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- 8.EE (Expressions and Equations)Understand the connections between proportional relationships, lines, and linear equations.
- 8.EE.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
- F-LE (Linear, Quadratic, & Exponential Models)Construct and compare linear, quadratic, and exponential models and solve problems.
- F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Misconceptions
- When given a table representation of a linear function with the first entry pair of the table not being "when x = 0," students sometimes give the first y value given as the y-intercept and not the y-value associated to x = 0.
- When calculating slope from a graph, students sometimes will not pay attention to the "direction" of the rise or the run which will sometimes cause the sign of the slope to be wrong.
- When given a table that doesn't have consecutive x-values, students sometimes will calculate the slope wrong. They will only pay attention to the change in y.
- Students get confused about which value is the y-intercept in the sequence.
- Students sometimes jump to conclusions and try to determine the pattern of the sequence by just looking at the first two terms. This leads students to sometimes mistake an arithmetic sequence for a geometric sequence (or vice versa) and then predict incorrect terms. For example, 1, 4 could be an arithmetic sequence (1, 4, 7, 10...) or a geometric sequence (1, 4, 16, 64...).
- In geometric sequences such as 16, 8, 4, 2..., students will want to say the pattern is divide by 2 instead of multiplying by 0.5.
Vignette
In the Classroom
In this vignette, students use a sequence of figures of toothpick triangles and a table of numbers as two ways to determine an algebraic expression.
Teacher: Today we are going to work with patterns and see what comes next.
Look at this pattern of figures. Who came come to the board and draw what they think the fourth figure would look like and explain the process.
Student: I can, it's easy. (Student draws the fourth figure). All we need to do is add one more triangle so that is three toothpicks, but one of the sides will overlap so I will subtract one.
Student: I thought of it a different way. I thought that each figure adds a triangle but because one line is already there I only need two to make the triangle.
Teacher: So who is right?
Student: I think they are both right since basically they are both adding two toothpicks each time to get the next figure.
Teacher: So let's say that bn denotes the nth box number. For example, b3 = 7 since figure three has 7 toothpicks. Let's see if we can come up with some sort of equation that will represent this pattern.
Let students work on their own for a bit.
Student: I think it's y = x + 2.
Teacher: I know you are probably used to using y = but here we are going to use different notation and use bn and n as our variables. And let's give your equation a try: bn = n + 2. So in Figure 3, n = 3. Substitute that into the equation and bn = 5, is that what we were supposed to get?
Student: No, we should have gotten 7. I see why he is using 2 in his equation because the pattern is adding two but I think that n should be multiplied by 2 instead of added to 2.
Student: So you think the equation is bn = 2n? But if I plug in 3 like the last time, I get 6 and we should have gotten 7. What if we say bn = 2n + 1? Would that work for all of them?
Teacher: Well, let's try one. What about Figure 5? How many toothpicks would be there?
Student: I just made it, and Figure 5 had 11 toothpicks. And it worked in the equation bn = 2n + 1.
Student: The equation I came up with works, but it's not the same as yours. I said that bn = 2(n - 1) + 3. I did this because I had three toothpicks in the first figure and then I kept adding on two for every figure after that. And it works. I plugged in 5 for n and bn = 11.
Teacher: Could it be possible that both work?
The teacher takes this time to revisit the distributive property if the students didn't catch on right away.
Student: I think they both work because if you simplify the second equation, you end up with exactly the same as the first.
Teacher: OK, now let's look at a different problem without the drawing. Here, the information is given in a table.
|
n |
1 |
2 |
3 |
4 |
5 |
|
an |
|
11 |
|
|
23 |
Student: So with this one, Figure 2 has 11 toothpicks and figure 5 has 23?
Teacher: Yes, that is what this table is saying. See if you can figure out the missing numbers in the table.
The teacher gives students time to work.
Student: Well, I noticed that the change was 4.
Teacher: How did you know that?
Student: Isn't it obvious?
Teacher: Well, not for some. Can you be a little more specific?
Student: Well, the bottom number went up 12 when the top number changed 3, so 12 / 3 equals +4 per change.
Teacher: This change is a sequence of numbers that has a constant rate of change is called the common difference. So I hear you say that the
$\text{common difference}=\frac{\text{increase in sequence values}}{\text{the number of steps it takes}}=\frac{23-11}{5-2}=4$.
Student: So last time with the picture drawing of the toothpicks, the equation was bn = 2n + 1, and the common difference was 2, here it's 4?
Teacher: Yes, so where did the 1 come from in the last equation and what would need to be in its place for this table?
Student: I noticed before that if we would have gone backwards and drawn a figure 0 there would have been 1 toothpick. Could that be what happens?
Teacher: Well, let's try it. If we went to where n = 0 in the table what value would be there?
|
n |
0 |
1 |
2 |
3 |
4 |
5 |
|
an |
|
7 |
11 |
15 |
19 |
23 |
Student: I think the number in the 0 spot should be 3. So then the equation would be an = 3 + 4n?
Student: I think she is right. It works for n = 5. 3 + 4*5 does equal 23. Essentially I think you start with 3 and take n steps forward at 4 toothpicks per step. This gives us an = 3 + 4n.
Note: Students have essentially just derived the slope formula and found the equation of a line given two points. All that is left to be done is transform the knowledge about sequences to correct language of linear equations.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- It is important for students to be able to translate between all representations of a function. Whenever students are working with a situation, they need to make connections between the context language, math language, table, graph and equation. When given one representation, students should be constantly creating the other representations of that same function.
- Most curricula will not address arithmetic and geometric sequences. They will address linear and exponential functions. The standard implies knowledge of exponential function, but never explicitly states "exponential." It is essential for students to learn how to write equations, make tables and graphs for exponential functions, and make the connection to geometric sequences.
- Students need to understand that rate of change/slope is what the dependent variable (output) changes when the independent variable (input) changes by one. Remind students to not only look at what the y-value changes by, but also what the x-value changes by in a table. Point out that sometimes to truly assess whether or not students understand the concept of linear functions, the question writers will make a test question where there is a linear relationship, but the x-value isn't changing by a constant amount so it's hard to distinguish linear vs. non-linear just by examining the y values. So it is important to expose students to tables that are not always in order and/or increase by a constant value. This makes students actually think about slope rather than just assume slope by looking at the patterns. For example:
|
x |
f(x) |
|
-2 |
3 |
|
2 |
5 |
|
4 |
6 |
|
10 |
9 |
- When converting a sequence to a function, it is common that the independent variable is the position of the number in the sequence. Therefore, the first value would be considered the first term. In order to find the y-intercept, students need to work backwards to find the previous term. For example, if the sequence is 3, 5, 7, 9..., the y-intercept would be 1.
- To convert from a sequence list to a table, have the students label the first column "term" and the second column "value."
- Some math books/resources show how to find the nth term in an arithmetic sequence by using the formula an = a0 + (n - 1)d. The term an is the term value we are searching for, a0 is the first term, n is the term being calculated and d is the common difference. Other math books will direct students to find the value of the zero term (y-intercept) and use y = mx + b. In this equation, y is the term value we are searching for, m is the common difference (slope), x is the term number being calculated and b is value of the "zero" term.
- The study of sequences lays the foundation for linear equations. Finding the patterns in arithmetic sequences and using them to find the next terms in the sequence prepares students for finding slope of a line.
- When students are finding the slope of a line, encourage them to go back to the graph and look to see if the line is increasing or decreasing. This will help them avoid making a sign error with the slope.
- This lesson compares linear patterns to exponential patterns:
Watch It Grow!
Suppose you are offered these two methods to be paid for 30 days of work.
- Plan 1-You receive $1000 the first day, and for each day following, you get $100 more than the day before. This means on day 2 you get $1100, on day 3 you get $1200 and so on. This continues for 30 days.
- Plan 2-You receive $1 the first day, and for each day following, your wage doubles. This means on day 2, you get $2, on day 3 you get $4 and so on. This continues for 30 days.
- Without doing any calculating, which plan sounds better? Do you think it will be a lot better or a little better?
- After you have made your preliminary choice, work with your group to calculate and compare the earnings under both plans. Decide how to display the information and report conclusions.
- You are then offered two additional plans.
- Plan 3-You receive $1000 the first day, and for each day following, you get $1000 more. How does this plan compare with Plans 1 & 2?
- Plan 4-You receive $1000 the first day, and an additional $10,000 for each day following. How does this plan compare with the other plans?
- How would you rank the four plans from best to worst? What variables might influence the order of your list?
- How does your ranking change if the number of days is changed? (Examine both fewer days and more days.)
About this Activity
- This activity will help students learn to appreciate the concept of geometric growth and how it differs from linear growth.
- The preliminary guesses can be kept private or discussed. Let students choose whether to share their guesses.
- Groups may want to use calculators or graphing calculators or the computer to help organize and display their comparisons of the two plans.
- Encourage groups to write equations to represent what they see happening.
- Ask questions such as:
- Which plan is better if the job suddenly ends after 5 days? 10 days? 15 days? 20 days?
- At what point does the "best plan" change from one plan to another? What happens to the graphs at that point?
- What factors - other than choosing the pay plan that yields the most money - affect your choice of plan?
Adapted from the 97 Frameworks Document.
Effects of m and b on a linear graph
A common problem when students learn about the slope-intercept equation y = mx + b is that they mechanically substitute for m and b without understanding their meaning. This lesson is intended to provide students with a method for understanding that m is a rate of change and b is the value when x = 0. This kinesthetic activity allows students to form a physical interpretation of slope and y-intercept by running across a football field. Students will be able to verbalize the meaning of the equation to reinforce understanding and discover that slope (or rate of movement) is the same for all sets of points given a set of data with a linear relationship.
Additional Instructional Resources
Linear Relationships: Tables, Equations, and Graphs
This website offers good ideas that model real world linear data using tables, graphs, rules and expressions.
Slope slider
This activity allows students or teachers to move a slider to examine what happens to the graph of a linear function if the slope is changed or if the y-intercept is changed.
Linear function resources
This website provides a list of resources to use focused on representing linear functions. It includes the connection to NCTM Focal Points in Grade 8.
Effects of changing slope or y-intercept
This TI-83/84 calculator and navigator activity explores the effects on the graph of changing the coefficient or constant in a linear equation.
linear function: a function whose rule may be written in the form f(x) = mx + b where m and b are real numbers. The term m represents the slope and b represents the y-intercept of the function. A linear function has a constant rate of change that results in a straight line graph.
arithmetic sequence: a sequence of numbers of the form: a, a + b, a + 2b, a + 3b, ... , a + (n - 1)b. There is a number that is constantly being added to each term to get the next term.
Example: 4, 7, 10, 13, 16, 19, 22, 25, ...
geometric sequence: a sequence of numbers of the form: a, ar, ar2, ar3,....., ar(n-1). Each term is being multiplied by a constant number to get the next term.
Example: 2, 4, 8, 16, 32, 64, ....
non-linear function: any function that does not follow a linear pattern of a constant rate of change, a straight line graph and an equation of the form f(x)=mx + b.
slope: the ratio of the vertical change to the horizontal change of a line on a graph. Slope represents the constant rate of change of a linear function. Given two points on a line slope is the ratio of the change in y to the change in x.
$\text{Slope}=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{\Delta y}{\Delta x}=\frac{\text{rise}}{\text{run}}=\frac{y2-y1}{x2-x1}$
y-intercept: the value on the y-axis where a graph crosses the y-axis.
domain: the set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs; the value that is the input in a function or relation.
range: the y-coordinates of the set of points on a graph; also, the y-coordinates of a given set of ordered pairs. The range is the output in a function or a relation.
independent variable: a variable whose value determines the value of other variables. Example: In the formula for the area of a circle, A = πr2, r is the independent variable, as its value determines the value of the area (A).
dependent variable: a variable whose value is determined by the value of an independent variable.
Example: In the formula for the area of a circle, A =πr2, A is the dependent variable, as its value depends on the value of the radius (r).
coefficient: the number multiplies times a product of variables or powers of variables in a term.
Example: 123 is the coefficient in the term 123x3y.
constant: a term or expression with no variables.
arithmetic sequence: a sequence of numbers in which the difference between any two consecutive terms is the same. In other words, an arithmetic sequence occurs when you add the same number each time as you move from one term to the next term in the sequence. This fixed number is called the common difference for the sequence.
geometric sequence: a sequence of numbers in which the ratio between any two consecutive terms is the same. In other words, you multiply by the same number each time to get the next term in the sequence. This fixed number is called the common ratio for the sequence.
exponential function: a function of the form y = a*bx where a represents the y-intercept and b represents the growth factor (the number multiplied by).
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- Why is it important for students to understand the different possible representations of a linear relationship: a verbal description, a numerical description (table or a set of ordered pairs), a geometrical description (graph) and an algebraic description (equation)? What are the strengths of each representation?
Taken from Focal Points: Focus In Grade 8, p.16. - What is the ultimate goal of having students do activities in which they explore equations, slopes and y-intercepts? What prior background do students bring to this discussion that can be built on?
Taken from Focal Points: Focus in Grade 8, p.32 - How did students demonstrate understanding of the materials presented?
- Did students make the connection between slope and rate of change?
- How did students communicate that they understand the meaning of the slope-intercept equation?
- What were some of the ways in which students illustrated that they were actively engaged in the learning process?
- Interactivate: Slope Slider. (n.d.). Shodor: A National Resource for Computational Science Education. Retrieved June 20, 2011, from this source.
- Algebra 1 Graphing Linear Equations. (n.d.). Glencoe Mathematics Online Study Tools. Retrieved June 20, 2011, from this source.
- Illuminations: Movement with Functions. (n.d.). Illuminations. Retrieved June 20, 2011, from this source.
- Integrated Algebra Practice A.A.34#1. (n.d.). Jefferson Math Project. Retrieved June 20, 2011, from this source.
- Interpreting the Effects of Changing Slope and Y-intercept. (n.d.). Welcome to Mr. Livingston's Algebra Assignments Page. Retrieved June 20, 2011, from this source.
- Linear Relationships: Tables, Equations, and Graphs. (n.d.). Utah Education Network. Retrieved June 20, 2011, from this source.
- Principles and standards for school mathematics. (2000). Reston, VA: National Council of Teachers of Mathematics.
- Schielack, J. F. (2010). Focus in grade 8: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
- Classroom Activities: TAKS: Effects of changing slope or y-intercept - Texas Instruments - US and Canada. (n.d.). Calculators and Education Technology by Texas Instruments - US and Canada. Retrieved June 20, 2011, from this source.
- iCost: Mathalicious. (n.d.). Mathalicious. Retrieved June 20, 2011, from this source.
Assessment
1.
DOK Level: 2
Answer: C
Taken from MCA III item sampler
2.
DOK Level: 2
Answer: B
Taken from MCA III item sampler
3.
DOK Level: 2
Answer: H
Taken from FCAT Math released test booklet
4.
DOK Level: 2
Answer: F
Taken from FCAT Math released test booklet
5.
DOK Level: 2
Answer: A
Taken from Massachusetts Comprehensive Assessment Released Exam
6.
DOK Level: 2
Answer: D
Taken from 2009 Texas TAKS test
7.
DOK Level: 2
Answer: F
Taken from 2009 Texas TAKS test
8.
DOK Level: 2
Answer: C
Taken from Minnesota MCA III item sampler
9.
DOK Level: 2
Answer: C
Taken from Minnesota MCA III item sampler
10.
DOK Level: 2
Answer: B
Taken from Massachusetts Comprehensive Assessment Released Exam
11.
DOK Level: 2
Answer: J
Taken from Massachusetts Comprehensive assessment
The following documents include assessment questions regarding these benchmarks.
Some questions regarding Interpreting the Effects of Changing Slope and y-intercept
Differentiation
- Give students multiple opportunities to experience the translation between graph, table, equation and situation. For example:
-
- Do a matching activity to put together all the representations of the same function. Use a table to organize. Then compare the different forms.
- Use a video to remind students of steps to translate from one representation to another.
- Use partners to describe their process.
- Look through magazines or catalogs to find real world examples of slope and linear functions. Compare the examples.
- Look at families of functions. For example, give students a group of three different functions that have something in common. Ask students to discuss how the functions compare to each other. Give students a chance to compare families of each representation.
- Ask students to define the word pattern. Where do they see patterns in their world? Continually go back to their understanding of pattern to help them represent linear patterns.
- Representing Patterns in Multiple Ways
This link provides some guided lesson plans to guide students through the process of understanding how to represent linear (arithmetic) patterns. Many organization maps are used including the Frayer model.
- Give students multiple opportunities to experience the translation between graph, table, equation and situation. For example:
- Use a table to show multiple linear functions in all different forms.
- Use a video to remind students of steps to translate from one representation to another.
- Use partners to describe their process.
- Look through magazines or catalogs to find real world examples of slope and linear functions. Compare the examples.
- Look at families of functions. For example, give students a group of three different functions that have something in common. Ask students to discuss how the functions compare to each other. Give students a chance to compare families of each representation.
- Ask students to define the word pattern. Where do they see patterns in their world? Continually go back to their understanding of pattern to help them represent linear patterns.
Representing Patterns in Multiple Ways
This link provides some guided lesson plans to guide students through the process of understanding how to represent linear (arithmetic) patterns. Many organization maps are used including the Frayer model.
- Introduce representations of non-linear models, and ask students to compare them to linear models.
- Give students more open-ended problems that involve linear representations.
- Describe the advantages and disadvantages of each representation.
Parents/Admin
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Students are: (descriptive list) |
Teachers are: (descriptive list) |
|
using graphing technologies to explore the effect of changing the m value (coefficient) in the equation. |
asking questions such as: |
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exploring and creating more than one representation of each linear function they explore. |
putting students in situations where they must use multiple representations of functions to solve problems. |
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using real world concrete situations to explore the meaning of linear functions. |
exposing students to a variety of tables that don't always have x values that increase by one or change by constant amount. |
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working with a variety of arithmetic and geometric sequences and connecting them to tables, graphs and equations. |
requiring students to not just solve problems involving functions, but to justify their solutions as well. |
Parent Resources
- Moving Straight Ahead: Linear Relationships
This review of the math involves the book Moving Straight Ahead, which includes information from this standard and a few others as well.
- Lines and slope
This site includes some basic examples and instructions for graphing linear functions.