Does the student appropriately identify independent and dependent quantities for the functional relationship described?

Can the student clearly explain why, in a particular situation, a quantity is independent or dependent? Does the student use appropriate language when describing the independent and dependent quantities for the functional relationship? Does the student describe a function as representing a dependence of one quantity on another? Does the student write an "if-then" or "input-output" sentence that accurately describes a functional relationship?

Teams of students are assigned a task that asks them to gather and record data sets of related quantities. Students gather data about the lengths of the first name and last name for each member of their class. Students gather data relating the height of a stack of identical cups to the number of cups in the stack.

Does the student clearly and accurately describe what it means for a relationship between two quantities to be a functional relationship? Can the student organize and record the collected data in an efficient and useful way (for example, length of name and month of birth of same person)?

Students are given a problem situation that can be described using a functional relationship. Develop a function rule or statement that shows how the charge depends on the length of the call.

What function rule can you use to describe the functional relationship between the cost and the length of the call? What inequality can be used to identify the lengths of time that Scott could talk for $3.00? Can the student verbalize the pattern of the functional relationship arising from the situation? Can the student write a function rule to describe the relationship between the quantities described in the situation?

Does the student write and solve equations or inequalities to answer questions arising from the situations? Can the student represent the function describing the relationship between the two quantities using a variety of ways, including a table, graph, description, picture, and function rule? Can the student use various representations of the functional relationship (table, graph, verbal description, diagram, function rule) to answer questions arising from a situation?

When given a series of buildings made of interlocking cubes that follow a pattern, students determine the relationship between the building number and the number of cubes needed to construct that building. Students construct buildings consisting of 2-cube bases and chimneys of increasing heights.

How did you represent the relationship between the building number and the number of cubes in the building? What equation can be used to determine the number of cubes in a building with a chimney that is exactly 12 cubes tall?

Can the student verbalize the pattern of the building number and the number of cubes in the building? Does the student recognize that the relationship between the quantities in the situation is a function? Does the student self-monitor and self-correct, noticing that the different representations should result in the same answers to questions? Can the student represent the function describing the relationship between the two quantities using a variety of ways, including concrete models, tables, graphs, verbal descriptions, diagrams, and function rules?

Can the student use various representations of the functional relationship (table, graph, verbal description, diagram, rule) to answer questions arising from the situation? Does the student appropriately use technology to graphically represent the function and answer questions arising from the situation?

Students determine the mailing weight of a box containing multiple copies of the same item after weighing the empty box and the one item.

Students interpret and make inferences and critical judgments using at least three representations of the relationship between the mailing weight of each box and the number of pamphlets in the box. What is the function rule relating the mailing weight of Box 1 containing pamphlets and the number of pamphlets in the box? How can you use the function rule to determine the weight of Box 1 with 25 pamphlets inside?

How could you use a graph to determine the number of pamphlets in Box 1 with a mailing weight of 1 kilogram?

Does the student use appropriate vocabulary (for example, relationship, trend, pattern, prediction, table, graph, plot, scatterplot)? Can the student create and use various representations of the functional relationship (table, graph, description, diagram, function rule) to make predictions and answer questions arising from the situation? Does the student use different representations to justify conclusions about the relationship between the data? Can the student write and solve equations or inequalities to answer questions arising from the situations? Can the student use the graphing calculator to create lists, tables, scatterplots, and so on? When given a variety of linear and quadratic functions represented in an assortment of ways (function rule, graph, concrete model, description, table), students identify and sketch the parent function for each.

Can the student identify the parent function for linear functions that are represented in a variety of ways (table, graph, verbal description, diagram, function rule)? Can the student identify the parent function for quadratic functions that are represented in a variety of ways (table, graph, verbal description, diagram, function rule)? Groups of students are given a set of cards that describe situations involving functional relationships.

The independent variable is the number of days Chris rents the roller blades, and the dependent variable is the total cost.

Can the student describe the mathematical domain and range for this functional relationship?

How can you use this information to help you describe a situation that the graph could represent? Can the student verbally describe the connections between the graph and the situation that they developed? Can the student study graphs and answer questions based on the information given in the graphs?

In groups, students investigate the relationship between two variables by collecting, organizing, representing, modeling, and interpreting two variable data sets in a problem situation. Students input the data into a graphing calculator to determine the function that describes the line of best fit. Students should realize that shoe sizes for men and women differ, as the following chart illustrates. What do you consider to determine a reasonable interval of values and a scale for each axis to construct a scatterplot of the data? Based on your scatterplot, would you say that shoe size and height are positively correlated, negatively correlated, or not correlated? Can the student collect and organize the collected data (relate shoe size and height to the same person)?

Can the student use a graphing calculator to create lists, tables, and scatterplots and to determine a line of best fit and so on? Can the student identify the correlation between the data sets as positive, negative, or no correlation (around 0) for data approximating linear situations?

Can the student model the relationship between shoe size and height using a linear function and use that model to make predictions?

When given a situation, students use symbols to represent the quantities in the situation and indicate what the symbols represent.

A plumbing company charges a flat fee of $35 plus $42 per hour for a residential service call. How did you use symbols to indicate how much the plumbing company will charge for a service call?

Students set up a table using the variables h and c to model the relationship between the hours worked and the total charge. What algebraic expression describes the relationship between the number of hours worked and the total charge of the plumbing company? Students solve problem situations by finding specific function values, simplifying polynomial expressions, transforming equations, factoring, and solving equations. The function that specifies the height of the bowling ball in terms of the time is h(t) = 144 - 16t2, where h(t) is feet above the ground and t is seconds. Can the student start with a function and determine an equation based on the function to solve a problem situation? Students use the commutative, associative, and distributive properties to determine equivalent expressions for the area of the following rectangle in terms of x.

How do you relate the concrete model for the area of the rectangle to each step in the multiplication process? Does the student use appropriate vocabulary when explaining how to simplify the algebraic expression?

Can the student apply the commutative, associative, and distributive properties to simplify algebraic expressions? Students are given a problem situation and a table of input-output values from the functional relationship modeled by the situation.

Determine an algebraic rule relating the total charge for a month to the number of days skated. Another way to express the relationship found in the situation is to use function notation: f(x) = 10 + 5x, where x is the number of days skated in one month and f(x) is the total cost to skate for that month.

Use the input-output table and the relationship expressed in function notation to determine how many days per month you could skate for a total cost of $40. How does the notation f(x) help show that cost depends on the number of days in this situation? Express the function in equation and function notation using variables other than y, x, and f(x). How effectively does the student describe the similarities and differences between equation and function notations? Does the student connect equation notation with function notation, such as y = 10 + 5x and f(x) = 10 + 5x? Can the student express the function in equation and function notation using variables other than y, x, and f(x)? Students work in pairs to determine whether or not the following data exhibit a constant rate of change and can be represented by a linear function.

Does the student clearly and accurately describe what it means for a relationship between two quantities to be a linear function?

Does the student express an understanding that linear functions can be represented in a variety of ways?

Does the student demonstrate an understanding that for linear functions the rate of change between data points is constant? Can the student use data sets to determine whether or not a relationship is a linear function? Students discuss problem situations for linear functions and determine reasonable domain and range values. What are reasonable input and output values for the function described in this problem situation?

Can the student identify the mathematical domains and ranges and determine reasonable domain and range values for given situations? How will you use words, symbols, a table, and a graph to represent the relationship between the amount of money Andrea earned and the number of hours she painted? Describe in words the dependency of the number of hours Andrea painted on the amount of money she earned. Is the rate of change constant in the amount earned with respect to the number of hours painted? How would each of your representations change if Andrea increased her hourly rate by $0.50?

Can the student verbalize the pattern between the number of hours Andrea painted and the amount of money she earned?

Can the student explain how various representations of the functional relationship (table, graph, description, rule) change when Andrea alters her hourly rate? Can the student represent the function describing the relationship between the two quantities using a variety of ways, including a table, graph, description, picture, and rule?

Students walk up a portable ramp with a side view and mark where they make their first step, second step, and so forth. They then measure and record the horizontal and vertical distances of each step and express the steepness of the ramp as a ratio of the measure of vertical change to the measure of horizontal change. Students record their measurements in a table, use the measurements to create a graph, write a function to describe the linear relationship of their measurements, and determine the slope using each representation (table, graph, algebraic representation). What function rule represents the linear relationship between the total horizontal and vertical distance traveled?

If the steepness of the ramp is a ratio of the measure of vertical change to the measure of horizontal change, how does the steepness of your ramp relate to the slope of your graph?

Does the student use appropriate language when describing the slope for the different representations? Does the student express an understanding that the rate of change (slope) for linear functions is constant? Does the student clearly explain that the determination of slope for different representations of the same linear function should yield the same result?

Did the student use an appropriate strategy to determine the slope for each representation? Can the student find the slope of a linear function from a table, graph, and algebraic representation?

Can the student interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs? Does the student use appropriate vocabulary, such as slope and y-intercept, when describing the effects of changing m and b on the graph of y = mx + b? What strategy does the student use to investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b?

Does the student use technology appropriately to investigate the effects of changes in m and b on the graph of y = mx + b? When given characteristics such as two points, a point and slope, or a slope and y-intercept, students find the equation of a line in y = mx + b form and graph the line. How can you use the points (2, 4) and (?1, ?5) to determine the equation of a line that contains these points?

Does the student use appropriate language when describing the slope, intercept, equation, and points on a line? Does the student use appropriate strategies to determine the equation of a line given some of the line?s characteristics? Can the student graph a line given two points, the slope and a point, and the slope and the y-intercept?

Can the student find the equation of a line given two points, the slope and a point, and the slope and the y-intercept? Which characteristics are easiest for the student to use to determine the equation of a line? Compare the y-intercept of the graph to the zeros of the function found in the table and the algebraic rule. Can the student clearly explain how to determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic rules? Can the student accurately describe the connection between the x-intercepts of a graph and the zeros of the function represented by the graph?

Can the student determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations? If the caterer originally charges a $25 set-up fee plus $4 per person, what is the functional relationship between the charges and the number of people? What is the graph of the linear function representing the catering charges and the number of people if she charges a $25 set-up fee plus $4 per person? If the caterer changes the price per person to $5 but leaves the set-up fee at $25, how does this affect the functional relationship between the charges and the number of people? Does the student use appropriate vocabulary when describing how changes in the situation affect the corresponding linear function?

Can the student connect changes in problem situations to changes in the graph, slope, and y-intercept?

The amount of chlorine needed in a pool increases proportionally (varies directly) as the amount of water increases.

If 5 units of chlorine are needed for 550 gallons of water, what function represents the amount of chlorine needed for x gallons of water?

What does it mean for the amount of chlorine needed to increase proportionally as the amount of water increases? How would you graph the function that describes the relationship between the amount of chlorine needed for a pool and the amount of water in the pool? Find a linear equation that describes the amount of chlorine needed for x gallons of water, and use this equation to determine how much chlorine is needed for 850 gallons of water. Is the student using an effective strategy to predict the amount of chlorine needed for a certain amount of water?

Can the student solve problems involving proportional change by using the linear function describing the situation? Can the student use a variety of representations and strategies to solve problems involving proportional relationships?

What function rule can you use to describe the relationship between the cost in dollars for the car rental and the number of miles driven? How can you use your knowledge about the slope and the y-intercept to write a rule for the linear relationship? Can the student identify the slope and y-intercept of the linear function from the problem situation? Does the student formulate equations or inequalities to answer questions arising from the situations? Can the student solve an equation or inequality to answer questions arising from the situation? Does the student correctly manipulate the equation or inequality to answer questions arising from the situations?

There are two types of confidence intervals we will talk about here: z-intervals and t-intervals.

This procedure is often used in textbooks as an introduction to the idea of confidence intervals, but is not really used in actual estimation in the real world. The sample size is greater than or equal to 30 and population standard deviation known OR Original population normal with the population standard deviation known. Suppose that in a sample of 50 college students in Illinois, the mean credit card debt was $346. Since we wish to estimate the mean, we immediately know we will be using either a t-interval or a z-interval. The indicates that we need to perform two different operations: a subtraction and an addition.

Another way to present this interval would be to calculate the margin of error and write .

The much more realistic scenario is using a t-interval to estimate an unknown population mean. Population standard deviation UNKNOWN and original population normal OR sample size greater than or equal to 30 and Population standard deviation UNKNOWN. Suppose that a sample of 38 employees at a large company were surveyed and asked how many hours a week they thought the company wasted on unnecessary meetings.

As before, since we are estimating a mean with a confidence interval, we know it will either be a t-interval or a z-interval. Now that we have that, we plug the values into the formula and do the calculations to get our two endpoints. Confidence intervals are most often calculated with tools like SAS, SPSS, R, (these are statistical calculations packages) Excel, or even a graphing calculator. All articles on Mathbootcamps are licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Interestingly, I found this older listing on 1st dibs for a brass coffee table very similar. Signed and numbered repousse brass and wood coffee table or side table featuring a removable panel of the sun and moon on the sides of the table. Please feel free to contact The English Room if you are interested in our interior design services in Charlotte or beyond.

Over at Richmond Thrifter, you can learn how to do just that with an old lamp base and picture frame.

The resources on this page have been updated and revised to align with the refined K-12 mathematics TEKS. A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.

They determine that the relationship is a function and represent the function using a variety of ways, including a function rule.

The table describes the building pattern by providing pictorial representations of the buildings for the first three building numbers.

What inequality can be used to determine the number of cubes in a building with a chimney that is up to 12 cubes tall? They may build tables, draw graphs, write algebraic formulas, or give verbal descriptions to help solve the problem. For each graph, the groups create and write a situation in which the value of one variable depends on the value of the other as represented by the graph.

Students predict the shoe size of a person whose height is 49 inches, 66 inches, and so on.

A simple way to include data for both genders is to convert to one gender's shoe size, as was done in the previous table. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. How can you determine algebraically how much the company will charge for a service call if you know how much time the plumber spent at the residence?

The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. Use x for the number of days skated in one month and y for the total cost to skate for the month. Compare the relationship expressed in function notation to the algebraic rule that you determined from the table. Use the input values from the table to determine the total cost for the month using the relationship expressed in function notation.

The student understands that linear functions can be represented in different ways and translates among their various representations. They represent the function in a variety of ways, using one representation to make another. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. Then, for each function given in the table below, students predict what they think the new graph will look like.

The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. They answer questions arising from the situation by formulating and solving equations and inequalities. They answer a question arising from the situation by formulating and solving an inequality using concrete models or graphs. Both are used to estimate an unknown population mean, but each is used in a different context. Well, in order to use a z-interval, we assume that (the population standard deviation) is known. Suppose that we also have reason to believe (from previous studies) that the population standard deviation of credit card debts for this group is $108. Looking a bit closer, we see that we have a large sample size (50) and we know the population standard deviation.

It is helpful to calculate them by hand once or twice to get a feel for the concept but you should also take the time to learn how to calculate them using one of these common tools.

Erin found her treasures on deep discount from the Goodwill, but you may have just the lamp and frame sitting in your very own closet! Students will continue to build on this foundation as they expand their understanding through other mathematical experiences. Students use functions to determine one quanity from another, to represent and model problem situations, and to analyze and interpret relationships. They use the data sets to determine whether or not there is a functional relationship between the quantities and explain why. They then determine that the relationship between the length of a student?s first name and last name is not a functional relationship. Data may reveal that two students studying the same amount of time earned different scores.

They then determine that the relationship between the height of a stack of cups and the number of cups in the stack is a functional relationship; that is, stack height depends on the number of cups in the stack.

The student is expected to describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations.

They answer questions arising from the situation by writing and solving equations or inequalities.

The student is expected to represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities. Students construct buildings, complete the table, write a function rule that expresses the relationship between the total number of interlocking cubes needed to construct the building and the building number, and graph the data from the relationship by hand or using a graphing calculator.

The student is expected to interpret and make decisions, predictions, and critical judgments from functional relationships. The student is expected to identify and sketch the general forms of linear (y = x) and quadratic (y = x2) parent functions. The student is expected to identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete.

Next they state the mathematical domain and range and describe a reasonable domain and range for the situation. The function rule for this situation is c = 12t, where c is the cost in dollars and t the number of days rented. The student is expected to interpret situations in terms of given graphs or creates situations that fit given graphs.

The student is expected to collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations. The student is expected to find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations.

The student is expected to use the commutative, associative, and distributive properties to simplify algebraic expressions.

The student is expected to connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

They discuss an alternate way to express the functional relationship using function notation. The student is expected to determine whether or not given situations can be represented by linear functions. The student is expected to determine the domain and range for linear functions in given situations.

The student is expected to use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.

Students create a table, graph, algebraic rule, and verbal description from the problem situation and make connections among these representations.

The student is expected to develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations. The student is expected to interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs. The student is expected to investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b. Next, students check their prediction with the graphing calculator, sketch the graph, and describe the change between y = x and the new graph.

The student is expected to graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.

The student is expected to determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations. The student is expected to interpret and predict the effects of changing slope and y-intercept in applied situations.

The students then connect changes in problem situations to changes in the graph, slope, and y-intercept.

The student is expected to relate direct variation to linear functions and solve problems involving proportional change.

The student is expected to analyze situations involving linear functions and formulate linear equations or inequalities to solve problems. The student is expected to investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities. As you can imagine, if we don’t know the population mean (that’s what we are trying to estimate), then how would we know the population standard deviation? Use this information to calculate a 95% confidence interval for the mean credit card debt of all college students in Illinois. Of course this is a very particular statement, so please make sure you study how to interpret confidence intervals in general and so you can understand exaclty what this means! All this means for us is that the formula will be very similar, but the critical value will no longer come from the normal distribution. We will use this to look up the value of in a table (a nice free version of that table can be found here. Calculate a 99% confidence interval to estimate the mean amount of time all employees at this company believe is wasted on unnecessary meetings each week. Finally, each team presents their task, their plan, and the data they collected to the class. Two students with a first name of equal length may have last names of a different length, so the length of a last name cannot be predicted using the length of the first name. Students also write the equations and inequalities used to answer questions arising from the situation. Because the cost increases $12 for each day Chris rents, the range of the problem situation is a subset of the natural numbers: $0, $12, $24, . Students, however, could convert men's and women's American sizes to the common metric sizes.

After investigating several functions, students write a general statement about the effects of changing . In the problem situation, the number of days he rents the roller blades can only be a whole number.

Determine the functional relationship that describes how the total charge depends on the length of a call in minutes.

This means that the domain of the problem situation is the set of whole numbers: 0, 1, 2, 3, 4, .

Students can relate function notation to a ?function machine.? They then use the relationship expressed using function notation to find values for the specific input values used in the table. How many of each box should Andrea use to mail the 5,000 pamphlets for as cheap as possible? Students determine that the output values are identical; they compare the output values to the output column of the table.

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