Objectives

Students will study scenarios that have solutions can be represented by linear equation. Students are going to:  - identify the linear equations that correspond to different real-world circumstances. - graph the linear solutions. - analyze linear solutions while considering the represented real-world scenario, and evaluate which solutions, if any, are valid given the real-world constraints of the problem.

Core Questions

- How do you construct, solve, graph, and analyze linear equations that represent relationships between quantities?

Vocabulary

- Constraint: A condition that a solution to an optimization problem must satisfy; a limiting condition of a variable. The set of solutions that satisfy all constraints is called the feasible set.  - Continuous Data: Data that can assume a range of numerical responses; frequently associated with physical measurements such as growth, decay, fluid motion, and others not subject to discrete enumeration.  - Coordinate Plane/Graph: A two-dimensional system in which the coordinates of a point are its distances from both a horizontal and a vertical line called the axes.  - Discrete Data: Data that takes on only disconnected values over any given interval.  - Linear Equation: An equation whose graph in a coordinate plane is a straight line.  - Ordered Pair: A pair of numbers (an x-coordinate and a y-coordinate) that designate the position of a point based on its distance from each axis.  - Slope: The ratio of the change in the vertical distance to the change in the horizontal distance of two points on a line. Slope measures the steepness of a line from left to right. The change in y divided by the change in x.$Δy \over Δx$.  - Slope-Intercept Form: y = mx + b.  - x-Coordinate: The first number in an ordered pair; it designates the distance along the horizontal axis.  - y-Coordinate: The second number in an ordered pair; it designates the distance along the vertical axis.

Materials

- graph paper for each group - Lesson 1 Exit Ticket handout (M-A1-4-1_Lesson 1 Exit Ticket)

Assignment

- Track each student's contribution to the group by keeping an eye on them during the three tasks. As each student contributes, have them explain how it improved the group's comprehension as well as their own contributions.  - Watch how well students describe the procedures required to resolve issues of this nature. Teachers can get a good idea of where gaps in a student's (and the class's) comprehension are by closely reviewing the responses that students provided during the self-evaluation and expression of their understanding of the lesson.

Supports

Scaffolding , Active Engagement  W: Students will have learned how to graph a linear solution to a problem and how a graph, which consists of a line or line segment, depicts all possible answers for a given scenario by the end of the session. They also understand how to use constraints in the original problem, as well as personal preferences, to help determine the optimal solution (if there is one). H: For students to actively participate in the lesson, they must be able to relate it to a specific example, ideally one that they have already experienced or may experience in the future. Using a sample problem and presenting it in a way that lets students explore the possible outcomes without thinking about the mathematical implications of the problem gets them interested and helps to persuade them that some simple problems have many (infinite) solutions, and it is important to be able to analyze those solutions.  E: In this lesson, students can visualize different problem solutions through graphing activities. The activities then progress to a more abstract level, with students working with the notion that because there are an endless number of solutions, they are best represented as points on a line (or line segment). In the final task, students work in pairs to create their own problems. By generating their own problems, students are better able to identify the key elements in a given problem and ensure that they understand the important elements of new problems that they come across.  R: A recap of the lesson's objectives and the students' starting points should be given to the class. Students can evaluate their progress and the new abilities they have gained by quickly going over the objectives and order of each activity.  E: Students must not only be able to solve new problem types when they are presented with them but also possess the ability to clearly understand how to approach problems of the new type. Students are encouraged to assess themselves by asking how they solved each challenge. They should review their own procedures and consider how to make their approach to solving problems better to ensure that they have incorporated all required details and actions.  T: This lesson can easily accommodate students with varying needs and learning styles. The class engages students of all learning styles by utilizing a range of tactics, allowing them to work in a variety of settings, and giving them chances to speak for their peers and apply their creativity to create problem situations.  O: This lesson's activities are set up to offer a lot of teacher-guided instruction. Students can more confidently investigate the connections between input and output, graphs and tables, and equations and roots once they have mastered sufficient teacher-directed abilities.

Procedures

Ask students the following question: "Suppose you have $20 and can buy chocolate candy for $5 a pound or fruit candy for $1 a pound. What do you buy and why?" Allow students to suggest what they can buy and why (for example, 4 pounds of chocolate candy because I don't like fruit candy.) After giving the students a few minutes to investigate the problem, point out that there are several chocolate and fruit candy combinations available that they can purchase. Remind students that they can buy candy in half-pounds, quarter-pounds, or any other fraction of a pound that they like. "Is there a certain number of different combinations of chocolate and fruit candy you can buy, or are the possibilities endless?" Pose the question to the class. After realizing that there are an infinite number of possible combinations (with a "perfect" scale capable of measuring minute fractions of a pound), explain: "We will try to figure out whether there's a pattern to the answers to these kinds of challenges. This type of situation, in which people have multiple options for how to spend their money, time, or energy, arises regularly, and it is critical to be able to evaluate all possible options to make the best decision possible. As you considered how many pounds of each type of candy to purchase, you found that your own tastes came into play. It's critical to think about these when examining your options in situations like these." Note: The activities in this lesson start with a lot of assistance and just examine the many solutions to a problem, observing how they look when graphed. The activities then progress to a more abstract and self-guided level, with students working with the notion that because there are infinitely many solutions, they are best represented as points on a line. Activity 1 (Small groups) This activity continues with more candy combinations: "What if you have $20 and can buy chocolate candy for $5 per pound or fruit candy for $1 per pound. What do you buy and why?" Draw the first quadrant of a Cartesian plane on the board or overhead, labeling the axes with ones.  Divide students into groups of three or four and ask them, "How many pounds can you get if you only buy chocolate candy?" (4 pounds)  Explain to students that the point you have drawn at (4, 0) on the Cartesian plane represents 4 pounds of chocolate candy and 0 pounds of fruit candy. "If you choose to purchase 1.5 pounds of chocolate candy, how many pounds of fruit candy will you receive?" (12.5 pounds) Guide the groups through the solution, demonstrating how to find the answer as needed. Then ask "What is the cost of a pound of chocolate candy?" ($5) "How do you find out how much for 1.5 pounds of candy?" (1.5 multiplied by $5 equals $7.50.) If students struggle with this concept, ask them how much it costs for 2 or 3 pounds of candies and how they got their answer. The realization that total cost is number of units (lbs) times unit cost ($/lb) is a key to setting up the problem; all students need to grasp this concept clearly to understand the rest of the lesson. Ask further, "How much money remains on your original $20 after purchasing 1.5 pounds of candy?" ($12.50) "How many pounds of fruit candy can you buy with $12.50?" (12.5) Again, ensure that students grasp the procedure for arriving at this answer: Divide the money you have by the unit cost. Once all students are able to come up with different answers for this scenario, have each group write down four different answers. Make sure to include a clear indication of how many pounds of each type of candy each group would buy, and represent their answers as ordered pairs of the form (x, y), where x stands for the pounds of chocolate candy and y for the pounds of fruit candy. Instruct each student group to ensure that students have at least one solution in which they buy only entire pounds of candy, as well as at least one solution in which they buy fractional amounts of candy. Allow students enough time to work on the various answers; while they are working, check in with each group to address any questions they may have and double-check that they are doing everything correctly. A Sample Table for Activity 1<figure class="image"><img style="aspect-ratio:574/409;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_86.png" width="574" height="409"></figure>Once all groups have finished, ask them to choose a spokesperson. Each group should ensure that their spokesperson has all of the arranged pairs that the group has agreed upon. Invite each representative to the board or overhead projector and select one of his or her group's solutions to plot on the Cartesian plane. Rotate through all of the spokespersons, each of whom will plot a point to represent one of the options proposed by his or her team.  The representative can sit down once his or her solutions have been expressed (either by graphing them or by another group doing so). After all of the group solutions have been plotted, ask the class, "Do the points we have graphed represent all of the solutions?" (no) Following this, ask: "What do you notice about all the solutions we graphed?"  Students should note that connecting all results can create a line segment. Ask students, "Do you get a line or a line segment?" Make sure students understand that the answers will not last forever because you only have $20. Connect the points with a line segment extending from (4, 0) to (0, 20). Ask the students,  "Does this line segment represent all possible solutions to the problem at hand?" (yes)  "How many solutions are there?" (an infinite number) Guide the class through a discussion on whether there are an endless number of solutions or if there are real-world constraints that limit the number of actual solutions in real life. Possible constraints include the accuracy level of the scale and the fact that the smallest unit of currency we have is the penny, both of which make certain calculations impossible (or require rounding). Graph of Money Spent on Chocolate and Fruit Candy, per pound<figure class="image"><img style="aspect-ratio:470/345;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_87.png" width="470" height="345"></figure>Activity 2 (same groups as Activity 1) Begin this activity by asking a new problem and giving each group a graph paper. Ask the class, "Suppose you have to read 50 pages for your English class this week, but you can only choose between two different books. Book A is a little easier, allowing you to read two pages per minute. Book B is harder, and you can only read half a page every minute. What are the possible book combinations that we can read?" Start by inviting students to think about Book A. "In one minute, how many pages from Book A can you read?" (2 Pages) "What about in 6 minutes?" (12 Pages) "What about in 20 minutes?" (40 Pages) "What about in x minutes?" (2x pages) If students struggle to understand the transition to the concept of abstract x here, ask them how they got their previous answers, such as "How did you arrive at 40 pages for 20 minutes?" Students should notice that they multiplied the number of minutes by two. [Use this realization to direct them to the unit "2x pages" for x minutes. Write 2x on the board and ask, "Assume you can only read $1 \over 2$ page of Book B in 1 minute. In 6 minutes, how many pages could you read?" (3 Pages) "How about in 20 minutes?" (10 Pages) "How about in y minutes?" ($1 \over 2$ * y) Activity 2, Book A and Book B Comparison<figure class="image"><img style="aspect-ratio:597/384;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_88.png" width="597" height="384"></figure>Ask learners: "How many total pages can we read if we read Book A for x minutes and Book B for y minutes?" Remind students that we already have phrases for the pages read in each book and that the word "total" implies adding. Write the equation 2x + $1 \over 2$y on the board, followed by an equal sign. If required, remind students of the original problem and then ask them what happens on the opposite side of the equal sign. (50) Number of Minutes to Read 50 Pages<figure class="image"><img style="aspect-ratio:549/321;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_89.png" width="549" height="321"></figure>Next, instruct the groups to graph 2x + $1 \over 2$y = 50  on their graph paper. Ask them to come up with as many ways to graph the line as possible. Tell them to be as precise as they can. While groups are working, walk around and answer questions, giving them hints; the two strategies they should use are creating an xy-chart and converting the equation to slope-intercept form. Ask, “What does the graph of  2x + $1 \over 2$y = 50 look like?” (a line)<figure class="image"><img style="aspect-ratio:489/389;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_90.png" width="489" height="389"></figure>"At what point does it begin? At what point does it end?" [(0, 100) and (25, 0)] "Concerning our initial problem, what does the point (0, 100) represent?" (read Book A for 0 minutes; read Book B for 100 minutes.) “The next version of this graph only displays the interval between (0, 100) and (25, 0).”<figure class="image"><img style="aspect-ratio:482/380;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_91.png" width="482" height="380"></figure>If students struggle to plot the figures on a graph, remind them of what the variables represent (the number of minutes spent reading Book A and Book B, respectively). Note: Since we aren't examining negative minutes spent reading either book, realistically, we are only interested in the portion from (0, 100) to (25, 0). For instance, our linear equation would indicate that we would need to read Book B for −4 minutes if we read Book A for 26 minutes. This does not make sense in the context of the problem. We are not constraining the minutes, however, negative minutes are not appropriate here. Ask them whatever point on the graph reflects the answer they might choose, and remind them that they do not have to select one of the endpoints. Have a conversation about the elements that may influence which solution is best for specific students—for example, how fascinating the book is, what the book is about, etc. Activity 3 (pairs) Tell students that in this activity, they will create their own problems, but first, the class will review similar circumstances to understand what kinds of variables are included in scenarios that they may be familiar with. If they're unfamiliar, it allows them to gain a new perspective. Give students time to consider the two problems from Activities 1 and 2 (the reading of Books A and B and the fruit/chocolate candy problem). "What do the two previous problems have in common?" Help students recognize that each problem has two variables (the number of pounds of each candy and the number of pages read from each book). Furthermore, the variables should be able to be divided into fractions. Point out that allowing the variables to reflect, say, a number of individuals alters the result because humans cannot be divided into fractional parts. Ideal variables include things like time and money. Take note of the changes between the two variables in the previous example. According to the glossary's definitions of discrete and continuous data, the number of pages is a discrete quantity. Minutes are likewise discrete numbers, despite the fact that time can be thought of and considered continuous.  Furthermore, students should understand that each problem has a constraint—something that limits how much of each of the two possibilities can be chosen. Ask students to identify the constraints in each problem (the amount of money in the first problem and the number of required pages in the second). Allow students to work in pairs and develop their own problems. As students work, move around the classroom and ask them about their two variables and constraints. Invite students to consider scenarios from their own lives. They can think about sports, or different jobs they have, different things they can be, different ways they can spend time, etc. Students in each group should write the group's word problem neatly on a piece of paper. Once each group has completed a word problem, students should pass it on to a nearby group, resulting in a new word problem for each. "Your group should identify the three key components of the problem: the constraint, the two variable quantities, and the problem setup before beginning to graph the solution." Each group should clearly identify each of the three main aspects of the problem and then develop an equation that represents all the possible solutions to the problem. Each group should then graph the answer to the problem assigned to them and specify the endpoints. It is necessary to remind students of their starting point and the knowledge they acquired during the lesson. Students can reflect on their progress and new skills after rapidly reviewing the width and sequencing of the activities. Ask students to recall the situation with chocolate and fruit sweets. "How many options did you think there were when we first started with that problem to choose the candy?" Ask students to determine the number of actual possibilities (assuming that we had a scale with infinite precision and could pay fractions of a penny): "What are the critical components of these types of problems?" (two variables and one constraint) Next, ask, "Why do the solutions end up being lines or line segments when we graph them?" Assist students in understanding how the problem's two variables relate to the idea that a two-variable equation can represent a line. Ask students if any variables were squared or cubed, or if they were simply multiplied by "regular" integers. Remind students that linear functions are defined as two variables, each to the first power, with specific exceptions for lines like x = a and y = b. This definition will help them recognize linear functions. Present the following two problems to the class: "Consider our original problem: chocolate candy is $5 per pound, fruit candy is $1 per pound, and we have $20 to spend. Consider our problem with reading books, but let's change it a little. Assume we can read 5 pages per minute in Book A and 1 page per minute in Book B and have just 20 minutes to read. In the first problem, we want to find the potential candy combinations; in the second problem, we want to identify the possible book combinations. In what ways are these two problems similar or different?" Students should take note of the evident differences but also recognize that the problems are fundamentally the same. Ask them to write what each variable symbolizes for each problem, as well as the limitation. Then, have students construct equations to reflect each case (the first one was completed earlier in class). Ask the question, "What do you notice about the equations (and hence the solutions) for the two problems?" Tell students that one of the most important concepts in mathematics is the ability to generalize various problems and break them down into their most fundamental components. This is what gives mathematics such power as a tool—the capacity to use the exact same tools to examine situations that appear to be very different. Students must not only be able to solve new problem types when they are presented with them but also have a well-defined strategy for approaching problems of this type. During this phase of the class, students are invited to reflect and analyze themselves, asking themselves how they solved each problem. They should not only review their own actions but also consider ways to improve their solution approach and ensure that all required information and stages are included. Assign a task to each pair of students: "Describe the steps you should take when faced with a problem that you think might have a solution that ir represented by a line." Ensure that student groups utilize the same scales for both the x and y axes. When each group has a clear, ordered set of steps, ask the class what the first step should be.; work with the class to develop a first step on which everyone can agree. Once the class has a step-by-step manual for solving issues whose solutions are linear equations, proceed through the remaining stages necessary to solve the problem. A suggested guide is: 1. Read the problem. 2. Identify the two variables and choose which letter will symbolize each quantity. 3. Identify the constraint. 4. Create an equation that links the two variables and the constraint. 5. Graph the equation in slope-intercept form or as an xy-chart. Once students have completed this portion of the course, distribute the course 1 Exit Ticket (M-A1-4-1_Lesson 1 Exit Ticket) for them to work on after class. Before leaving the classroom, have students share and discuss their answers, or go over the consequences. Extension: Students can start learning about the slope and the y-intercept, as well as how to use algebra to convert a standard-form equation to slope-intercept form. Students can work on a strategy to directly translate a problem scenario into a slope-intercept equation for graphing.

Layla Nguyen

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Graphing Linear Solutions (M-A1-4-1)

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Layla Nguyen

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Description

Students will study scenarios that have solutions can be represented by linear equation. Students are going to:
- identify the linear equations that correspond to different real-world circumstances.
- graph the linear solutions.
- analyze linear solutions while considering the represented real-world scenario, and evaluate which solutions, if any, are valid given the real-world constraints of the problem.

Grade

Grade 9 Grade 10