Writing and Graphing Linear Equations - DOC

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linear equations, graphing linear equations, the line, linear equation, slope-intercept form, ohio usa, point-slope form, two points, slope of the line, equation of a line, standard form, equation of the line, slope and y-intercept, graphing calculator, ordered pairs

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Overview of Sheltered Algebra Unit This Algebra I unit on writing and graphing linear equations is designed as a model for sheltered instruction for English language learners. It was developed for Algebra I class in a large urban high school. Classes have eighteen to twenty-five students and are scheduled in blocks of ninety minutes. The teacher has been trained by TEXTEAMS (Texas Teachers Empowered for Achievement in Mathematics and Science) Algebra I: 2000 and Beyond Institute. Part of the institute’s goal is for teachers to possess strategies that meet the needs of diverse student populations. Teachers use concrete learning experiences to build mathematical concepts and reasoning that are meaningful to the student. The Algebra I TEXTEAMS training inspires many of the activities in this unit, which is available through regional educational service centers. Technology has been integrated throughout the unit. In particular, students will be using a calculator based ranger (CBR) and graphing calculators. This technology is part of the TEKS and will be used in TAKS. Teachers are encouraged to receive training through the school district of the regional educational service centers on the use of the graphing calculator. A website has also been provided through Region IV ESC. The unit was designed as a model for sheltered instruction; however, it is good for all students due to the engaging activities and strong emphasis on vocabulary and accessing student’s prior knowledge. The presentation, materials, and pacing of this lesson have been further modified with second language acquisition strategies to accommodate a diverse class of native English speaking students and LEP students with a wide range of backgrounds and various levels of English proficiency (beginning, intermediate, and advanced). The sequence of lessons within the unit is only a suggestion. Teachers may need to make adjustments based on their students, resources, training and time. Lessons are not necessarily designed to be completed in one class period and may vary from the suggested length of time. The individual teacher determines the time needed to ensure that the students are learning.

Writing and Graphing Linear Equations Algebra One Lesson Slope Writing Linear Equations Given Slope and Y-Intercept TEKS c2A, c2B TAKS Objective 3

b1B, c2A, c2B, c2D, c2E, c2F

Objective 1 Objective 3

Writing Linear Equations Given a Table, a Graph, the Slope, the Y-Intercept, Two Points, and a Verbal Description

b1B, c2A, c2B, c2D, c2E

Objective 1 Objective 3

Graphing Lines

c2D

Objective 3

Evaluating Changes in Slope and Y-Intercepts of Graphs, Equations, and Problem Situations

c1C, c2C, c2E, c2F

Objective 3

Practicing Changes in Slope and Y-Intercept of Graphs, Equations, and Problem Situation

c1C, c2C, c2E, c2F

Objective 3

The above TEKS are specifically related to this unit; however, there are ongoing TEKS that are utilized in spiraling the instruction.

Inter-disciplinary Information for Algebra One Science Writing and Graphing Linear Equations with Science’s Motion Unit The use of the calculator based ranger (CBR) in Algebra translates perfectly to the speed lesson in Science. The students can take their knowledge from lesson 1 using the CBR and Ranger Program to apply it to the lesson associated with calculating speed.

Algebra One Unit Writing and Graphing Linear Equations Lesson Plan 1 Finding slope using graphs, tables, and two points Content Objectives The student will learn how to find slope given a graph, table, and two points. The student will understand and develop features of all four types of slopes. The student will use physical representations of the scenarios and work in pairs to work through the activity. SIOP Component(s): Preparation In order for the LEP student to best understand the content objectives, it is essential that it be clearly stated in simple language. The objectives should be given both orally and in written form. Language Objective The student will understand and define the following vocabulary: motion detector, rate, point, line, rise, run, graph, and slope. SIOP Component(s): Preparation It is important in the sheltered instructional setting to establish academic language objectives. LEP students not only learn content, but they acquire academic English skills in reading, writing, listening, and speaking. Special emphasis is placed on the development of math-specific vocabulary. Metacognitive Objectives The student will use a motion detector (CBR) to discover features of slope. The student will use a chart to organize information regarding slope. The student will work in cooperative groups for the Linear Motion activity.

SIOP Component(s): Preparation Metacognitive strategies that students acquire will continue to help them even when the teacher is not there to guide them. Strategies enable the student to plan, monitor, and evaluate learning independently. Materials  Motion detector (calculator based ranger, commonly referred to as CBR)  Assessment: Linear Motion – transparency (TEXTEAMS Algebra 1: 2000 and beyond pages 95 - 96)  Overhead graphing calculator  Student graphing calculators  Graph paper  Rulers  Slope Characteristic Chart SIOP Component(s): Preparation A variety of materials make the lesson clear, meaningful, and help provide concrete examples for the LEP student. Procedure Allow approximately two ninety-minute class periods (1) Prior knowledge should include plotting ordered pairs, simplifying fractions, independent and dependent variables, and finding patterns.

SIOP Component(s): Building Background It is important to access students’ prior knowledge. If there are knowledge gaps, steps should be taken to build the necessary background. This can be done through mini-lessons that precede the whole-group lesson. (2) The teacher needs to define the following terms as they come up in the lesson: motion detector, rate, point, line, rise, run, graph, and slope.

SIOP Component(s): Building Background, Comprehensive Input, Strategies, Interaction Vocabulary used in math is very specific and often difficult to define. The key terms to be defined and discussed in this lesson have multiple meanings. The teacher assists students in distinguishing the meaning that applies to the content, modeling correct usage. Students are encouraged to use the appropriate terms in class work and discussion. Have students create vocabulary cards using the verbal word association strategy. Teachers may reinforce vocabulary with posters or other visual representations in the classroom. (3) At this point, the class will go through the activity, ―Walk This Way.‖ The classroom should be set up with an aisle down the middle, wide enough for a student to easily walk down without distraction from other students. The motion detector should face the aisle from the front of the room and be connected to an overhead-graphing calculator. Make sure that all students can see both the student walking down the aisle and the data projected from the calculator on a screen in front of the room.

SIOP Component(s): Comprehensible Input, Strategies The classroom setting is designed to ensure that all students, including LEP students, are involved and encouraged to participate. Setting up the demonstration with the motion detector clarifies the content objective and makes instruction more meaningful through an authentic experience. (4) Explain to students that the motion detector works similarly to the way a police radar gun works by sending out an ultrasonic pulse that measures distance from the motion detector over a period of time. The data the motion detector retrieves will be displayed as a graph on the overhead projector.

SIOP Component(s): Comprehensible Input Providing real-life examples like the police radar gun helps the LEP student better understand the function of the motion detector. Students should be encouraged to supply other examples (the automatic fish finder that fisherman use to find fish under water, animals like bats that use sonar to get around, automatic doors and toilets that use a light beam). (5) To set up the motion detector for this activity, run the CBR Ranger program and follow the following steps: 1. Link the Ranger Program from the CBR to the graphing calculator. 2. Open the Ranger Program and press enter.

3. Go to main menu and enter number 1 setup/ sample.

4. Set up the next screen as follows: Realtime: Time: Display: Begin on: Smoothing: Units: yes 15 distance enter none feet

5. Put the arrow on start now at the top right hand corner. Face the motion detector toward the student walking and press enter. Have students volunteer to walk in front of the motion detector. Have each student walk differently—slow, fast, away from the motion detector, toward the motion detector, stand still, etc. After doing a few of these examples discuss how each style of walking affected the graph. For example, a slow walker has a very flat graph whereas a fast walker has a very steep graph. This is also a good time to review independent and dependent variables. SIOP Component(s): Lesson Delivery The teacher acts as a facilitator allowing students to discover the effect when rate, time, or direction are changed. With this activity, students are motivated and interested because they are actively involved and, through repetition, patterns develop and students can predict the results. (6) At this point, have a student come up with a scenario and then predict what the graph should look like. After the class has completed the discussion, have a student represent the scenario in front of the motion detector and determine if the class prediction is correct. For example, a student starts 8 feet from the motion detector and walks slowly toward the motion detector for 5 second. The student then stands still for 6 seconds and walks back quickly for 4 seconds. Distance in feet Time in seconds Do a few scenarios so that the students get a good understanding of what affects a line and how to draw it as a graph.

SIOP Component(s): Comprehensible Input, Strategies Having the students supply the scenario makes instruction more meaningful to them. Students take ownership in their own learning. (7) Pair students to answer the first three questions on the Assessment: Linear Motion transparency. Also assign one problem from problems 4-7 and one problem from problems 8-11 to each pair. As they finish working their problems, have each pair choose one problem to answer on the transparency. Discuss the answers as a class and review all that was discovered through the above activity.

SIOP Component(s): Interaction The teacher has lowered the anxiety level in the classroom for the LEP students by allowing them to work in a small group and to choose the problems they want to work on. The demonstration with the whole-class group provides a common, shared experience for the students. Changing to pair work not only allows the LEP student an opportunity to use and practice new knowledge but also get assistance from a classmate. Some suggestions for pairing students are: * pair students with same native language * allow students to choose their partner * pair a student who is knowledgeable in content with one who needs assistance In this case because of the reading that is necessary to ensure comprehension of the assigned tasks, the teacher may choose to group students by pairing a more proficient English reader with a student who is less proficient. The teacher may use large group instruction to work through one problem of each of the different types of questions in the handout. This provides a model of the expectation for completing the assignment. Through classroom observation, the teacher can determine if the students can understand and apply the content concepts. This time also allows for one-on-one teacherstudent time, which some LEP students may need.

By sharing their work with the whole group, students have the results of more than just working one problem. In addition, students gain confidence by sharing their own work. (8) Pass out a sheet of graph paper to each student. Have each student plot two ordered pairs on their graph paper and connect them using the ruler. Walk around the class as the students are doing this to make sure that all four types of slope (positive, negative, zero, and undefined) are being represented. If not, draw that specific slope on a sheet of graph paper to use later.

SIOP Component(s): Building Background, Review/Evaluation The teacher takes this opportunity to assess the students’ skills and prior knowledge as he/she monitors their work. (9) Have students compare their lines to their neighbor’s lines and talk about the differences and similarities. Have the students discuss some of the differences and similarities as a class and write them down on the board or on a transparency to use later.

SIOP Component(s): Strategies, Lesson Delivery In this activity, the student has an opportunity to share and learn from peers before being asked for input. Graphic organizers or T-charts are effective strategies for students to learn for recording their thoughts before the class discussions. The teacher may want to make directions more specific, requesting two similarities and two differences in the lines. When the teacher begins to record responses for similarities and differences, all students will be able to participate and contribute.

(10) Have each student take his or her ordered pairs and find the slope. The students do not know the concept of slope yet but lead them to discover how to find it. First have the students subtract their y values and their x values. Then have the students set up a fraction with y over x and simplify if necessary. Make the connection to match the graph activity completed earlier. Discuss with the students how this number represents the rate at which their line is moving up or down just like the graph was affected by how fast or slow someone walked.

SIOP Component(s): Building Background, Review/Evaluation It’s important to reinforce content-specific vocabulary. For example, when the student is asked to set up a fraction, use the terms numerator and denominator. Some students may have the concept but have not connected the word to the concept because of limited English skills. Using actions, visual aids, and gestures helps to define terms for LEP students. (11) Have students compare their answers and lines with their neighbor’s lines and talk about the similarities and differences. Discuss this as a class and come up with some generalizations. Write this down on the board or on a transparency to use later. SIOP Component(s): Comprehensible Input Some LEP students may not have the vocabulary to understand classroom instructions. For example, similarities and differences may be difficult for students with very limited English skills. It is important that the teacher explains the task in several ways. Graphic organizers can be especially helpful for students to organize their thoughts.

(12) Have the class organize their graphs by categories on the board or wall. As they are organizing the graphs, see what type of categories they are using. Hopefully, the students will be categorizing the lines by the types of slope (positive, negative, zero, and undefined). This is where the lines the teacher drew will be important to include so that all four types will be represented. The teacher may have to lead the students in that direction based on what they are doing. SIOP Component(s): Strategies Students can easily discriminate between graphs in this activity by using feature analysis. The graphs though created by different students and using different data, are excellent visuals. With this strategy, students will identify similarities and differences in the lines that are produced. (13) At this point pass out the characteristic chart to each student. Begin by talking about the four types of slope. Ask the students which group would represent positive slope, negative slope, zero slope, and undefined slope. Once all categories are labeled, have the students draw a sketch of each in the second row of the chart.

SIOP Component(s): Strategies Taking the feature analysis one step further, the teacher provides students with a graphic organizer to help categorize and label the different slopes. Graphic organizers provide an excellent tool to make the connection between graphic representations of the slope with the appropriate labels. (14) Review what the similarities and differences are of the four types of slope. Use the ones written on the board or on the transparency as a springboard to get the students started if necessary. As the features are discussed, have them fill in the third row of the chart.

Features: Rises to the right Goes up hill Steeper m > 1 Flatter 0< m <1 (reading from left to right)

Features: Features: Falls to the right Horizontal Goes down hill Straight Across Steeper m < -1 Flatter -1< m <0 (reading from left to right)

Features: Vertical Goes up and down

(15) Begin discussing the three ways to find slope: graphing, tables, and two points. Begin with graphing. Talk about how to look at two distinct points on the line and look at the rise (change in y) and the run (change in x). Write the slope as a fraction of y/x and simplify if necessary. Work through a couple of examples on the board or on a transparency.

2 3

Two points: (1, 5) and (-1, 2) The graph rises 3 and runs 2 so the slope is 3/2.

(16) Now discuss how to find slope given a table. Use an example problem of a table and talk about how to find the differences of the y values and the differences of the x values to find the patterns. Write the slope as a fraction of y/x and simplify if necessary. x -3 -1 1 3 y -1 2 5 8

-1 - -3 = 2 3–1=2

2 - -1 = 3 8–5=3

After finding the patterns of the table by looking at the differences, the slope can be identified by 3/2.

(17) Finally discuss how to find slope given two points. Put an example of two points on the board and have the students discuss and predict how to solve for slope. Work through the problem and write the slope as y/x. Two points: (1, 5) and (-1, 2) 5–2=3 1 - -1 2 or 2 – 5 = -3 = 3 -1 – 1 -2 2

It is key to remember that it does not matter which order pair gets subtracted first, just that both ordered pairs are subtracted in the same order. Keep all three examples on the board and/or transparency so the students can use them as a tool to do their own examples. (18) Put the students in groups of three. Each group will work out a type of problem (positive, negative, zero, or undefined) all three ways (graphing, table, and algebraically). Assign each group a type of slope to work with. Give them about 15 minutes to work the problems and have them put the examples on a blank transparency. Have each group explain its example to the class. Once all the groups have completed the activity, choose one of each type of slope to use as an example to fill in the rest of the chart. Give students time to finish up their charts. As they are completing the charts, walk around and answer any questions. (19) End the lesson with an overall discussion of the chart and slope. SIOP Component(s): Review/Evaluation In summarizing the lesson, the teacher stresses that students are able to use three different methods to find the slope. Students realize that through the use of different methods, they can arrive at the same slope. Technology Application The motion detector (CBR) is used to investigate slope. The teacher can show what a positive and negative slope look like by using a graphing calculator.

SIOP Component(s): Practice It is important that all students understand and are able to use a graphing calculator. LEP students may be unfamiliar with the technology and the teacher needs to direct teach these skills, explaining the use of the functions and allowing students time to practice using the calculator. This is an essential skill since students will be using the graphing calculator on the TAKS. Please see the information on the graphing calculator tutorial offered on the website by Region IV Education Service Center (www.esc4.net/math). Assessment An informal assessment will be given during the lesson through discussion and group work. A formal assessment will be given during the lesson through the completion of the chart and group presentations. SIOP Component(s): Review/Evaluation Immediate feedback to students and ongoing informal assessment by the teacher assures that students understand the content skill being taught. Teacher observations will help identify those students who may have gaps in their education and may need tutorials or reinforcement of key concepts. Mathematics is conceptual, and it is important for students to build a strong foundation of key concepts. Formal assessment evaluates the individual student’s comprehension and learning of the lesson’s objectives. In addition, the assessment measures the student’s procedural knowledge for problem solving. Extensions Find a graph or a table in a newspaper, magazine, or the Internet. Find the slope and explain the characteristics of that slope.

SIOP Component(s): Review/Evaluation Students respond and demonstrate understanding by using graphs and tables found in the media. This activity allows students to select graphics on topics that are of interest to them. The personal interest in the topic often motivates and leads the student into interpreting the results and a higher level of thinking.

Slope Characteristic Chart
POSITIVE SLOPE NEGATIVE SLOPE ZERO SLOPE UNDEFINED SLOPE

FEATURES:

EXAMPLES: GRAPH:

Table:

Slope: Two points:

Algebra One Unit Writing and Graphing Linear Equations Lesson Plan 2 Writing linear equations given slope and y-intercept Content Objective The student will use a motion detector to write an equation. SIOP Component(s): Preparation In order for the LEP student to best understand the content objective, it is essential that it be clearly stated in simple language. The objective should be given both orally and in written form. Language Objectives The student will understand and define the following vocabulary in his/her own words: slope, y-intercept, rate of change, and rate. The student will read, record, and interpret information on tables and graphs. SIOP Component(s): Preparation It is important in the sheltered instructional setting to establish academic language objectives. LEP students not only learn content, but also acquire academic English skills in reading, writing, listening, and speaking. Special emphasis is placed on development of math-specific vocabulary. Metacognitive Objectives The student will use technology to enhance learning. The student will create and use graphic organizers (tables, graphs). SIOP Component(s): Preparation Metacognitive strategies that students acquire will continue to help them even when the teacher is not there to guide them. Strategies enable the student to plan, monitor, and evaluate learning independently.

Materials  ―What’s My Trend?‖ student activity sheet - transparency and student copies (TEXTEAMS Algebra 1 2000 and Beyond page 167)  ―What’s My Trend?‖ graph page – transparency and student copies  ―What’s My Trend?‖ assessment – transparencies and student copies (TEXTEAMS Algebra 1 2000 and Beyond page 168 – 169)  Motion detector (CBR)  Overhead graphing calculator  Student graphing calculators  Measuring device SIOP Component(s): Preparation A variety of materials make the lesson clear, meaningful, and help provide concrete examples for the LEP student. Procedure Allow approximately two ninety-minute class periods (1) Prior knowledge should include slope (positive, negative, zero, undefined, steep, and flat), independent and dependent variables, domain and range, plotting and identifying ordered pairs, and patterns.

SIOP Component(s): Building Background In the sheltered math class, it is important to activate and assess students’ prior knowledge by reviewing vocabulary and key concepts. Teachers should not assume that students make connections automatically. It is necessary to explicitly create a link from prior knowledge or previous lessons to new lessons. In some cases, teachers think that LEP students lack comprehension or memory skills, but this may happen from a lack of or failure to activate prior knowledge.

SIOP Component(s): Comprehensible Input Remember that many math concepts are abstract; therefore it is important that LEP students are able to define terms in their own words rather than parrot the teacher or text. (2) At this point, the class will go through the activity, ―What’s My Trend?‖ The classroom should be set up with an aisle down the middle wide enough for a student to easily walk down without distraction from other students. The motion detector should face the aisle from the front of the room and be connected to an overhead graphing calculator. Make sure that all students can see both the student walking down the aisle and the data projected from the calculator on a screen in front of the room. This is the same setup as the previous lesson.

SIOP Component(s): Preparation, Lesson Delivery The classroom setting is designed to ensure that all students, including LEP students, are involved and encouraged to participate. Setting up the demonstration with the motion detector clarifies the content objective and makes instruction more meaningful through an authentic experience. (3) First pass out the student activity sheet (page 167) and graph page for students to write on throughout the lesson. To start the activity, the teacher begins with the first problem on the student activity sheet. Explain the scenario. ―Start 2 feet away from the motion detector and walk away at 1.5 ft/sec.‖ Have a student physically demonstrate the action by starting exactly 2 feet away from the motion detector and walking down the aisle from the motion detector approximately 1.5 ft/sec. Remind students that the motion detector works similar to the way a police radar gun works by sending out an ultrasonic pulse that measures distance from the motion detector over a period of time. After the students understand the scenario and how the motion detector works, have students complete the table.

t 0 1 2 3 t

d 2 3.5 5 6.5 2 + 1.5(t) = d

SIOP Component(s): Comprehensible Input Auditory information is sometimes difficult for LEP students to process. Through the use of visuals like the motion detector and graphing calculator, instruction becomes comprehensible. The step by step process in the demonstration guides LEP students from a real experience to articulating the same information in a table, in a graph, and orally. SIOP Component(s): Building Background, Comprehensible Input, Strategies Providing real-life examples like the police radar gun helps the LEP student better understand the function of the motion detector. Students should be encouraged to supply other examples (the automatic fish finder that fisherman use to find fish under water, animals like bats that use sonar to get around, automatic doors and toilets that use a light beam). SIOP Component(s): Strategies Using a graphic organizer such as a table to organize the data from the motion detector helps LEP students focus their attention and make connections between ordered pairs and the points on the graph. (4) Looking back to the previous lesson, have the students discuss or predict how they think the graph will look. Sketch a graph of the ordered pairs on the provided grid paper and draw the line that is represented by those ordered pairs. Have the students define the independent and dependent variables as well as the domain and range. The teacher should also graph it on large grid paper to help demonstrate the slope and y-intercept. Using the graph, determine the slope and the y-intercept and compare those values to the table.

Distance (feet) t 0 1 2 3 t

Time (seconds)

Point out that the differences of the y values over the differences of the x values is the slope/rate and the starting point is the y-intercept which is always at x or (t) = 0. d 2 3.5 5 6.5 2 + 1.5(t) = d

1–0=1 2–1=1 3–2=1

3.5 – 2 = 1.5 5 – 3.5 = 1.5 6.5 – 5 = 1.5

The above differences give us 1.5/1 which is the slope/rate and the y-intercept/starting point is 2. SIOP Component(s): Comprehensible Input Even though students have been taught to graph ordered pairs and lines in a previous lesson, it is important that the teacher model the skill. LEP students need the visual to reinforce and ensure understanding. LEP students sometimes experience difficulty in understanding instruction when teachers interchange terms with the same meaning. For example, y values over x values and y/x, both mean that the value of y is divided by the value of x. It is often necessary to clarify terminology by defining it in several different ways. (5) Have the students explain in words what happened. For example the total distance is 1.5 times the number of seconds plus the starting point of 2 feet. Then have the students write an equation that represents the written description (y = 2 + 1.5t or y= 1.5t + 2). The graphing calculator can be used to represent the table, graph, and

equation. To see how this works see technology applications for this lesson. SIOP Component(s): Practice The verbal expression provides LEP students with practice in using English. Math has a language of its own. Students learn to express a verbal statement in algebraic terms. (6) Now begin working on table 2. Have the students predict what the graph might look like given the scenario. Have a student volunteer to walk away from the motion detector at a constant rate. The data will be portrayed as a line on the graphing calculator. Use the trace function to get the distance at each second. Problem 2: Walk slowly away from the motion detector at a constant rate.

1–0=1 2–1=1 3–2=1

t 0 1 2 3 t

d 2.5 3.1 3.9 4.6 2.5 + .7(t) = d

3.1 – 2.5 = .6 3.9 – 3.1 = .8 4.6 – 3.9 = .7

Use the knowledge learned from the first table to find the slope and y-intercept. To find the slope on this table you will have to average the differences of the y values and divide them by the differences of the x values (.6 + .8 + .7 = .7 slope = .7/1). SIOP Component(s): Review/Evaluation Along with learning the identified key content terms, LEP students may need other words defined. For example, in this lesson it may be necessary to review and define the term average.

(7)

Have the students graph the table and write a description of the scenario in words and then write the same thing as an algebraic equation in slope-intercept form. Continue this same process for each table. When all the tables are complete, put the students into pairs and have them work through the assessment: ―What’s My Trend?‖ handout (page 168–169). Walk around to answer questions and to keep each pair on target. As a whole class, discuss the worksheet and go over the answers.

(8)

SIOP Component(s): Interaction The demonstration with the whole-class group provides a common, shared experience for the students. Changing to pair work not only allows the LEP student an opportunity to use and practice new knowledge but also to get assistance from a classmate. Some suggestions for pairing students are: * pair students with same native language * allow students to choose their partner * pair a student who is knowledgeable in content with one who needs assistance In this case because of the reading that is necessary to ensure comprehension of the assigned tasks, the teacher may choose to group students by pairing a more proficient English reader with a student who is less proficient. SIOP Component(s): Review/Evaluation The teacher may use large group instruction to work through one problem of each of the different types of questions in the handout. This provides a model of the expectation for completing the assignment. Through classroom observation, the teacher can determine if the students can understand and apply the content concepts. This time also allows for one-on-one teacherstudent time, which some LEP students may need. (9) Once the assessment is completed the teacher will discuss slopeintercept form (y = mx + b) and compare it to the equations the students wrote. Have them determine that ―m‖ is the slope/rate and ―b‖ is the y-intercept/starting point. After they understand the
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connection to slope-intercept form, introduce standard form (ax + by = c). Discuss each form as linear equations and give examples of how to transform an equation from standard form to slope-intercept form. SIOP Component(s): Practice It is important to note that with the demonstration and activities, the student not only develops an understanding of rates of change but can also express the information as a table, as a graph, in a verbal description, and finally as an algebraic equation. The skills have not been taught in isolation but combine to present the concept and show the relationship between the different skills. The gradual process of taking concrete examples to define abstract concepts scaffolds instruction so that the LEP student can exhibit increasing independence in learning. Technology Applications The motion detector (CBR) and graphing calculator will be used to learn and help write linear equations. This is accomplished using the Lists, Scatter Plots, y =, and linear regression. This can be done for all of the tables in this activity. The steps to accomplish this are as follows: 1. Push the STAT button and push enter on Edit.

2. The list screen should appear with L1, L2, etc. 3. Plug in the values for time into L1 and the values for distance into L2.

4. Push STAT PLOT by entering 2nd Y=.

5. Push enter on 1 and set up the screen to On, scatter plot: (first graph), Xlist: L1, Ylist: L2, Mark: any.

6. Push Window and set up your view window to include all the x values and all the y values. Give a few below the lowest numbers and few above the highest numbers.

7. Push Graph

8. Test your graph by plugging in your equation into y= and graph.

9. A linear regression can be found by going to Stat – Calc – 4:Linear Regression (ax + b). This gives you the most accurate equation for your data.

SIOP Component(s): Building Background It is important that all students understand and are able to use a graphing calculator. LEP students may be unfamiliar with the technology and the teacher needs to direct teach these skills, explaining the use of the functions and allowing students time to practice using the calculator. This is an essential skill since students will be using the graphing calculator on the TAKS. Please see the information on the graphing calculator tutorial offered on the website by Region IV Education Service Center (www.esc4.net/math). Assessment An informal assessment will be given during the lesson when the pairs are working on the assessment worksheet. The teacher should walk around and answer questions and informally rate the students’ understanding. A formal assessment (additional practice) could be given to assess individual understanding of the concept. The handout may also be used as additional practice or an alternative assignment for LEP students because it does not require extensive reading in English.

SIOP Component(s): Review/Evaluation Immediate feedback to students and ongoing informal assessment by the teacher assures that students understand the content skill being taught. Teacher observations will help identify those students who may have gaps in their education and may need tutorials or reinforcement of key concepts. Mathematics is conceptual, and it is important for students to build a strong foundation of key concepts. Formal assessment evaluates the individual student’s comprehension and learning of the lesson’s objectives. In addition, the assessment measures the student’s procedural knowledge for problem solving. Extensions The students will develop a graph for a given scenario by inputting their own points and scales. In the extension, each student will have to fill in the table, determine the rates, write a description, and write an equation in slopeintercept form and standard form.

What’s My Trend? Graph Page 1. 2.

Additional Practice Given the situation plot a graph; write a description, and an equation. 1. Susan started 3 feet from the motion detector and walked away at a constant rate of 2 ft/sec. t 0 1 2 3 t Description: Equation: 2. Use the given table to plot a graph, write a description, and an equation. t 0 1 2 3 t Description: Equation: Write an equation in slope-intercept form for each problem. Distance (feet) Distance (feet) Time (seconds) 3. 4. d 8 7.2 6.2 5.3 Distance (feet) Time (seconds) d Distance (feet)

Time (seconds)

5. slope: -3 y-intercept: 5

6. slope: ½ y-intercept: -2

7. slope: 4 y-intercept: 0

Transform each from standard form into slope-intercept form. Identify the slope and y-intercept in each. 8. 8x + y = 12 9. x – y = 9

Equation: Slope: Y-intercept: 10. 4x + 2y = -10

Equation: Slope: Y-intercept: 11. 3x – y = 6

Equation: Slope: Y-intercept:

Extension Fill in the blank grid below with a graph that demonstrates you riding in a car. Label the time and distance scales to help determine your graph. Using your scale and your graph fill in the table then write an equation and write a description.
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Distance (

)

5 Time (

10 )

2. Fill in the table. Time 0 5 10 20 t Distance

3. Write the equation for your graph and table.

4. Write a description represented by your graph, table, and equation.

Algebra One Unit Writing and Graphing Linear Equations Lesson Plan 3 Writing linear equations given a table, a graph, the slope, the y-intercept, two points, and a verbal description Content Objectives The student will discover how to take a written description to fill in a table, draw a graph, and write a symbolic equation. The student will also take a table and draw a graph, write a symbolic equation, and a verbal description. SIOP Component(s): Preparation In order for the LEP student to best understand the content objective, it is essential that it be clearly stated in simple language. The objective should be given both orally and in written form. Language Objectives The student will understand and define the following vocabulary: scales, rate of change, constant rate, motion detector (CBR), slope and y-intercept. The student will understand how to write a symbolic equation and verbal description. SIOP Component(s): Preparation It is important in the sheltered instructional setting to establish academic language objectives. LEP students not only learn content but also acquire academic English skills in reading, writing, listening, and speaking. Special emphasis is placed on development of math-specific vocabulary. Metacognitive Objectives The student will use physical representation of the scenarios and collaborate with a partner on the activity. The student will complete tables and graph data.

SIOP Component(s): Preparation Metacognitive strategies that students acquire will continue to help them even when the teacher is not there to guide them. Strategies enable the student to plan, monitor and evaluate learning independently. Materials  ―Wandering Around‖ Activity 1 sheets - transparencies and student copies (TEXTEAMS Algebra 1 2000 and Beyond pages 158 - 159)  ―Describe the Walk‖ Activity 2 sheets – transparencies and student copies (TEXTEAMS Algebra 1 2000 and Beyond pages 160 - 161)  Writing Equations Chart – transparency and student copies  Additional Practice  Quiz SIOP Component(s): Preparation A variety of materials make the lesson clear, meaningful, and help provide concrete examples for the LEP student. Procedure Allow approximately two ninety-minute class periods (1) Prior knowledge should include the use of the motion detector, independent and dependent variables, finding slope from a table or graph, plotting and identifying ordered pairs, patterns, slope-intercept form, standard form, and filling in a table given a rate and a starting point.

SIOP Component(s): Building Background Linking previous lessons to new learning can activate prior knowledge. Students do not automatically make connections, and the teacher may need to explicitly point out past learning. (2) The teacher needs to define the following terms: scale, rate of change, constant rate, motion detector, slope, and y-intercept.

SIOP Component(s): Building Background The teacher selects and focuses on critical key terms that will be used in the lesson. The vocabulary is defined as students work through the activity. Students will define the terms as the lesson progresses. Reinforcement will be oral, written, and visual. (3) First review the lesson on writing an equation given the slope and yintercept by leading the students in completing the first part of the writing equations chart. Example: slope = 3, y-intercept = -2; equation: y = 3x – 2 SIOP Component(s): Building Background Skills learned in a prior lesson are reviewed and linked to new concepts, building and reinforcing prior knowledge. (4) At this point the class will go through Activity 1: Wandering Around. Each student needs a copy of the handout, and the teacher needs to use the overhead copies to complete the activity. In this activity the student will look at a situation and fill in a table, a graph, a written symbolic equation, and a verbal description. The teacher will begin with a review of how the motion detector worked and the type of information it produces. If necessary, the teacher could do an example with the use of the motion detector to help remind the students how it works. The teacher will begin with the first problem by having a student read and walk through the scenario. ―Ryan was walking away from the motion detector at 2 feet per second. You missed where he started but you know that he was at the 9 foot mark when the timer called out the 3rd second.‖ Have a student demonstrate walking away from the motion detector and stop after 3 seconds and 9 feet away. This helps the students understand exactly what the scenario is explaining. After the students

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understand the scenario walk them through completing the table and graphing the ordered pairs.

t 0 1 2 3 4

d 3 5 7 9 11

Distance (feet)

Time (seconds)

The teacher will need to discuss the independent variable (time which is on the x-axis) and dependent variable (distance which is on the yaxis) along with the scales of the graph. SIOP Component(s): Comprehensible Input, Strategies, Lesson Delivery Students are actively engaged in problem solving. The teacher has the student reading the problem and acting out the problem. For LEP students that are struggling with English reading comprehension, this activity promotes understanding. The class participates in going through the stepby-step procedures for completing the table and graphing the data. (6) The teacher will need to lead the student to write the equation using the slope (rate) and the y-intercept (starting point). The teacher should show that the differences of y (distance) over the differences of x (time) is the slope and the starting point where x = 0 (time = 0) is the y-intercept. The equation should be written in slope-intercept form (d = 2t + 3 or y = 2x + 3). SIOP Component(s): Comprehensible Input Throughout this activity, the teacher models the reasoning process by verbalizing whatever comes to mind. This helps the student make connections and see the importance of the process rather than the product. (7) Now work on the second problem exactly as the first problem. In this problem the student is walking toward the motion detector, which

means the slope is going to be negative. Have the students fill in the table, graph, and write the equation. ―Madeline was walking toward the motion detector at 3 feet per second. You missed where she started, but you know that she was at the 9 foot mark at the 2nd second.‖
Distance (feet by 2’s) Time (seconds)

t 0 1 2 3 y = -3x + 15

d 15 12 9 6

SIOP Component(s): Comprehensible Input, Strategies Multiple examples are provided for students to develop a better understanding of the concepts. Each example has been selected to demonstrate a variety of situations that may occur. When students work independently they can refer to the models and notes from the class activity. (8) At this time the teacher can use the writing equations chart and begin making the connection on how to write an equation given the slope and a point. Algebraic example: slope is -3 and goes through the ordered pair (2,9) Using y = mx + b, plug in -3 for m, 2 for x, and 9 for y. Solve the equation for b. 9 = -3(2) + b 9 = -6 + b 15 = b So the equation of the line would be y = -3x + 15.

SIOP Component(s): Practice The teacher provides the student with ample guided practice. Problems used in practice are specifically chosen to link with the next concepts that will be introduced to the student. (9) Problems three and four are done exactly the same way as above except in these problems the scenarios provide two points. The students will have to determine the slope and the y-intercept from the table and write an equation. After completing these problems, the teacher can use the writing equations chart and begin making the connection on how to write an equation given two points. ―You looked up and Chet was walking. He was at the 6 foot mark at the 1st second and the 1 foot mark at the 2nd second.‖

Distance (feet) Time (seconds)

t 0 1 2 3 y = -5x + 11

d 11 6 1 -4

Algebraic Example: given the two points (1,6) and (2,1) First the student must find the slope. 1 – 6 = -5 2–1 1 Using y = mx + b, plug in -5 for m, and either 2 for x, and 1 for y or 1 for x and 6 for y. Solve the equation for b. 1 = -5(2) + b 6 = -5(1) + b 1 = -10 + b 6 = -5 + b 11 = b 11 = b So the equation of the line would be y = -5x + 11.

(10) In Activity Two the students will take the information from the table to graph the points, as well as write a symbolic equation and a verbal description. The symbolic equation will be written the same as the other activity using the slope and y-intercept, point and y-intercept, or two points. The written verbal description should be similar to the scenarios written in Activity One.
Distance (feet by 3’s) Time (seconds)

t 0 1 2

d 15 21 27

Symbolic equation: given the y-intercept and several points First the student must find the slope. 21 – 15 = 6 1–0 1 Using y = mx + b, plug in 6 for m and 15 for b. So the equation would be y = 6x + 15. Verbal description: Sam started 15 feet from the motion detector and walked away from it 6 feet per second.

SIOP Component(s): Comprehensible Input, Practice In this problem the teacher adds an additional skill, practice writing verbal descriptions for given data. This may seem difficult for some LEP students however, the teacher can refer to the description in prior problems to model writing. (11) Put the students in pairs to finish working the last three problems. As the students are working, the teacher should walk around and answer any questions. As the groups finish, have each group write in one part of the problem (graph, symbolic equation, and verbal description) on

the transparency to show the entire class. This will allow for discussion and summary of the entire lesson. SIOP Component(s): Interaction, Lesson Delivery The large group summary allows students to participate and share information. Even if a student does not understand all of the concepts or could not complete the entire task, students could still contribute. Technology Applications A graphing calculator can be used in several ways to check the students’ work. The students can check their equations and graphs by looking at the table and graphs on the graphing calculator. Inputting their equation into the y = screen and then pushing graph will check their graph and pushing 2nd graph will check their table. Remember to set up the window with the values on your table (remember to go below a few and above a few). The screens below will help demonstrate the above process. Additional help is available on the Region IV website graphing calculator tutorial.

SIOP Component(s): Building Background, Practice It is important that all students understand and are able to use a graphing calculator. LEP students may be unfamiliar with the technology and the teacher needs to direct teach these skills, explaining the use of the functions and allowing students time to practice using the calculator. This is an essential skill since students will be using the graphing calculator on the TAKS. Please see the information on the graphing calculator tutorial offered on the website by Region IV Education Service Center (www.esc4.net/math).

Assessment An informal assessment will be given during the lesson when the pairs are working on the activity sheet and writing up their answers on the transparencies. The teacher should walk around and answer questions and informally rate the students understanding. A formal assessment will be given to assess individual understanding of the concept in the form of additional practice. Another formal assessment will be given the following class period in the form of a quiz. SIOP Component(s): Review/Evaluation Immediate feedback to students and ongoing informal assessment by the teacher assures that students understand the content skill being taught. Teacher observations will help identify those students who may have gaps in their education and may need tutorials or reinforcement of key concepts. Mathematics is conceptual and it is important for students to build a strong foundation of key concepts. Formal assessment evaluates the individual student’s comprehension and learning of the lesson’s objectives. In addition, the assessment measures the student’s procedural knowledge for problem solving. Depending on the needs of the students, the teacher determines if and how to best use the additional practice assignment or if it is needed. The practice may be used as a tutorial for students who need the extra time and attention to better understand the concepts. Extensions Each student needs to find a table or a graph in a newspaper, magazine, or on the Internet. The student will find the slope (rate) and the y-intercept (starting point) and write a symbolic equation and a verbal description matching their data.

SIOP Component(s): Building Background, Comprehensible Input, Strategies, Practice, Lesson Delivery Students respond and demonstrate understanding by using graphs and tables found in the media. This activity allows students to select graphics on topics that are of interest to them. The personal interest in the topic often motivates and leads the student into interpreting the results and a higher level of thinking.

Writing Equations Chart

Type Given the slope and the y-intercept

Problem Given: slope = 3 y-intercept = -2 Plug in 3 for m and –2 for b

Solution Y = 3x –2 Or Y = 3x + -2

Given the slope = –3 ordered pair = (2,9) Y = -3x + 15 slope and an Ordered pair Plug in – 3 for m, 2 for x, and 9 for y 9 = -3(2) + b 9 = -6 + b 15 = b Given two two ordered pair (1,6) and (2,1) Y = -5x + 11 Ordered pairs Find slope 1 – 6 = -5
2–1 1 Plug in –5 for m, 2 for x and 1for y, or 1 for x and 6 for y 1 = -5(2) + b 6 = -5(1) + b 1 = -10 + b 6 = -5 + b 11 = b 11 = b
Distance (feet)

Given a Graph

Find the y-intercept and slope y-intercept is 3 slope is 2
Time (seconds)

Y = 2x + 3

Given a Table

Find the slope by difference of y over the difference of x Find the y-intercept where x = 0 x y 0 3 slope is 4 1 7 y-intercept is 3 2 11

Y = 4x + 3

Writing Equations Chart

Type Given the slope and the y-intercept Given the slope and an Ordered pair

Problem

Solution

Given two Ordered pairs

Given a Graph

Given a Table

Additional Practice Label the table and graph. Fill in the table, sketch the graph, and write a symbolic rule for the situation (write the equation for the table and graph).
1. Sean was walking away from the motion detector at 3 feet per second. You missed where he started, but you know that he was at the 15 foot mark when the timer called out the 4th second. TABLE GRAPH RULE

2. Stacey was walking toward the motion detector at 2 feet per second. You missed where she started, but you know that she was at the 4 foot mark at the 2nd second. TABLE GRAPH RULE

3. Jessica started 3 feet from the motion detector. You looked up and she was at 11 feet at the 2nd second. TABLE GRAPH RULE

4. You looked up and Aaron was walking. He was at the 4 foot mark at the 1st second and the 1 foot mark at the 2nd second. TABLE GRAPH RULE

Label the table and graph. Sketch the graph. Write a symbolic rule and a description of the walk.
5. 0 1 2 TABLE
TIME DISTANCE

GRAPH 1 3 5

RULE

6. 0 1 2

TABLE
TIME DISTANCE

GRAPH 1 5 9

RULE

7. 1 2 3

TABLE
TIME DISTANCE

GRAPH 6 4 2

RULE

8. 0 1 2

TABLE
TIME DISTANCE

GRAPH 0 3 6

RULE

9. 2 4 6

TABLE
TIME DISTANCE

GRAPH 4 2 0

RULE

10. 1 3 5

TABLE
TIME DISTANCE

GRAPH 2 4 6

RULE

QUIZ Label the table and graph. Fill in the table, sketch the graph, and write a symbolic rule for the situation (write the equation for the table and graph). 5. Sean was walking away from the motion detector at 2 feet per second. You missed where he started but you know that he was at the 10-foot mark when the timer called out the 4th second.
TABLE GRAPH RULE

6. Sue was walking toward the motion detector at 3 feet per second. You missed where she started but you know that she was at the 5-foot mark when the timer called out the 1st second.
TABLE GRAPH RULE

Label the table and graph. Sketch the graph. Write a symbolic rule and a description of the walk.
3. 0 1 2 TABLE
TIME DISTANCE

GRAPH 5 9 13

RULE

4. 2 3 4

TABLE
TIME DISTANCE

GRAPH 5 7 9

RULE

Algebra One Unit Writing and Graphing Linear Equations Lesson Plan 4 Graphing lines Content Objective The student will learn how to graph a line given two points, slope and point, and an equation. SIOP Component(s): Preparation In order for the LEP student to best understand the content objective, it is essential that it be clearly stated in simple language. The objective should be given both orally and in written form. Language Objective The student will understand and define the following vocabulary: graph, coordinate plane, x and y- axis, slope and y-intercept. SIOP Component(s): Preparation It is important in the sheltered instructional setting to establish academic language objectives. LEP students not only learn content but also acquire academic English skills in reading, writing, listening, and speaking. Special emphasis is placed on development of math-specific vocabulary. Metacognitive Objective The students will use a concrete model (transparency lines and coordinate plane) to graph and work in pairs to complete the activity. SIOP Component(s): Preparation Metacognitive strategies that students acquire will continue to help them even when the teacher is not there to guide them. Strategies enable the student to plan, monitor, and evaluate learning independently.

Materials  Large grid paper  Regular graph paper  Transparency lines or spaghetti (if using the transparency lines make a transparency of this handout and separate each line)  Graphing activity SIOP Component(s): Preparation A variety of materials make the lesson clear, meaningful, and help provide concrete examples for the LEP student. Procedure Allow approximately 1 ninety-minute period (1) Prior knowledge should include the understanding of slope, y-intercept, ordered pairs, and basic graphing on a coordinate plane.

SIOP Component(s): Review/Evaluation In this lesson, the teacher reviews terms and concepts previously taught in the unit. It is important to establish a strong foundation before proceeding and making links to new tasks. LEP students will benefit from having the important terms repeated and reinforced. (2) Pass out a sheet of graph paper to each student. Then give the students two points (such as (-2, 3) and (4, 2)) and ask them to graph the line. Label the line A. Then give the students a slope and a point (such as m = 2 and (-2, -5)) and have them graph that line. Label that line B. Have the students discuss and review the process they used to graph both lines A and B.

SIOP Component(s): Review/Evaluation, Interaction In reviewing the procedures for graphing lines, the teacher has the student making the graph, talking about the task in his/her own words. This reinforcement of the procedure demonstrates real understanding rather than mimicking memorized responses. If a student is having difficulty verbalizing the procedure, the teacher can paraphrase the student’s words using appropriate terminology and clarifying ideas. (3) Review slope-intercept form focusing the review on slope ―m‖ and yintercept ―b‖. Refer back to the previous lesson and discuss how the yintercept is the beginning point (the starting point in previous activity) and the slope is the rate (the rate in the previous activity). To help graph y = mx + b, stress that one starts with begin with ―b‖ and move with ―m‖. Take time to do some examples for them. (Example Problem Transparency)

SIOP Component(s): Strategies; Comprehensible Input The teacher provides students with an additional strategy, a mnemonic device that is used to help students recall information. Time should be spent in explaining the alternate representation of the letters m and b. This tool can be helpful for LEP students because the words are easily understood and represent the action on the graph. (4) The following activity will help reinforce and practice graphing. Pair up all students. Pass out a large coordinate plane and a transparency line or spaghetti to each student. Give them a problem (Example Problem Transparency) and have them use the transparency line or spaghetti to graph on the large coordinate plane. Have the pairs check their answer with each other to determine the correct answer. They can also use a graphing calculator to check their graphs. Continue doing practice problems until the class has a good grasp of graphing lines.

SIOP Component(s): Comprehensible Input, Practice LEP students build comprehension and understanding by practicing skills in a variety of activities. Even though students have been exposed to the skill and practiced graphing, changing the format of the practice keeps the student’s interest and actively engaged in learning. (5) At this time pass out the graphing game. Each pair needs a set of cards and a scorecard. Place the cards equation side up. Both students make sure the equation is in slope intercept form and then graph the equation using their large grid paper and transparency lines/spaghetti. When both students have completed the graph, check it by looking at the back of the card. If the student gets it correct, they give themselves two points. If it is incorrect, they lose a point. Continue play for 5 minutes. After 5 minutes, have the students total up their score and record it on the scorecard. Have the winners rotate around the room to another partner and continue play. This activity will allow the students to practice graphing and help peers teach each other.

SIOP Component(s): Interaction, Practice, and Strategies The game format provides yet another way for students to actively participate and interact with peers while getting additional practice and immediate feedback on their understanding of graphing. Technology Applications The graphing calculator can be used in this lesson to check their graphs. They could use them before the activity when they are in partners. They could also be allowed to use them in the game to check their answers before they look at the back of the card. If the calculator is used they might only get 1 point instead of 2 points. See lesson 3 for information on how to check equations and graphs.

SIOP Component(s): Practice It is important that all students understand and are able to use a graphing calculator. LEP students may be unfamiliar with the technology and the teacher needs to direct teach these skills, explaining the use of the functions and allowing students time to practice using the calculator. This is an essential skill since students will be using the graphing calculator on the TAKS. Please see the information on the graphing calculator tutorial offered on the website by Region IV Education Service Center (www.esc4.net/math). Assessment An informal assessment will be given as the pairs are working on the practice problems and during the game. A formal assessment could be given during the game by looking at the scorecards. SIOP Component(s): Review/Evaluation Immediate feedback to students and ongoing informal assessment by the teacher assures that students understand the content skill being taught. Teacher observations will help identify those students who may have gaps in their education and may need tutorials or reinforcement of key concepts. Mathematics is conceptual and it is important for students to build a strong foundation of key concepts. Formal assessment evaluates the individual student’s comprehension and learning of the lesson’s objectives. In addition, the assessment measures the student’s procedural knowledge for problem solving. Extensions Each student will write an equation and graph it for all four types of lines (positive, negative, horizontal, vertical).

Transparency Lines

Example Problems

(2,4) and (-1,0) (-2, -3) and (3, 6) slope = 3, (1, 3) slope = ½, (-2, 3) slope = -4, (-3, -4) slope = 1, (2, 5) Y = 2x + 2 Y=½x+3 Y = 4x - 1 Y = 1/4x - 2 Y = -2x + 2

Y = -1/2 x – 2 Y = -4x + 3 Y = -1x – 3 Y=x Y = -x 2x + y = 4 2x + 4y = -8 x + y = -2 x + 3y = 6 x – 2y = 10

Name: _____________________ Name: __________________________ Game 1: Your score Their score Game 1: Your score

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Name: ______________________ Game 1: Your score

Name: _____________________ Game 1: Your score

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y = 2x + 4

3x – y = -1

y=x–4

2x – y = 1

y=x+3

x – y = -4

y = 4x - 7

2x – y = 0

y = -2x + 4

3x + y = 1

y = -x – 4

2x + y = - 1

y = -x + 3

x+y=4

y = -x

2x + y = 0

y = 1/2x + 4

2x – 3y = -3

y = 2/3x – 4

x – 2y = 2

y = 3/4x + 3

x – 3y = 12

y = 3/4x

x – 2y = 0

y = -1/2x + 4

2x + 3y = 3

y = -2/3x – 4

x + 2y = -2

y = -3/4x + 3

x + 3y = -12

y = -3/4x

x + 2y = 0

(2, 4) and (-1, 2)

(-3, -4) and (0, 2)

(3, 0) and (-6, 0)

Slope = 3 Point = (-1, 0)

Slope = -2 Point = (3, 2)

Slope = 1/2 Point = (-2, -3)

Slope = - 2/3 Point = (3, 1)

Slope = - 1 Point = (2, 2)

Algebra One Unit Writing and Graphing Linear Equations Lesson Plan 5 Evaluating changes in slope and y-intercept of graphs, equations, and problem situations Content Objective The student will learn to evaluate how changes in slope and y-intercept affect a graph, an equation, and a problem situation. SIOP Component(s): Preparation In order for the LEP student to best understand the content objective, it is essential that it be clearly stated in simple language. The objective should be given both orally and in written form. Language Objectives The student will understand and define the following vocabulary: transform, graph, coordinate plane, x and y- axis, slope, and y-intercept. The student will write a verbal description explaining how changes in slope and yintercept affect the equation. SIOP Component(s): Preparation It is important in the sheltered instructional setting to establish academic language objectives. LEP students not only learn content but also acquire academic English skills in reading, writing, listening, and speaking. Special emphasis is placed on development of math-specific vocabulary. Metacognitive Objectives The students will use a graphing calculator to discover how transformations affect graphs, equations, and problem situations. Students will work in pairs to practice problems.

SIOP Component(s): Building Background In this lesson, the teacher reviews terms and concepts previously taught in the unit. It is important to establish a strong foundation before proceeding and making links to new tasks. LEP students will benefit from having the important terms repeated and reinforced.

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Review graphing of a linear equation from slope-intercept form focusing on slope as a rate and y-intercept as the starting point.

SIOP Component(s): Review/Evaluation It is important to continually reinforce vocabulary and concepts to ensure that all students are developing a clear understanding of the content objectives. Teachers may need to revise, reword, and/or redirect instruction to spiral content understanding. (8) Begin teaching transformations by beginning with the parent function y = x. Draw a picture of the graph and begin answering each of the discussion questions. The teacher should guide the students through the use of the graphing calculator to check each answer. What would happen to the graph and the equation if I moved it up two units? y=x+2 y-intercept would change to 2; slope would be the same What would happen to the graph and the equation if I moved it down 4 units? y=x–4 y-intercept would change to -4; slope would be the same What would happen to the graph and equation if I added 6 to the y-intercept? y=x+6 y-intercept would change to 6; slope would be the same What would happen to the graph and equation if I subtracted 10 from the yintercept? y = x – 10 y-intercept would change to -10; slope would be the same What would happen to the graph and equation if I multiplied the slope by 1? y = -x y-intercept stays the same; slope changes to -1 which changes the direction of the line

What would happen to the graph and equation if I multiplied the slope by 4? y = 4x
70

y-intercept stays the same; slope changes to 4 which makes the graph steeper What would happen to the graph and equation if I multiplied the slope by 1/2? y = -1/2x y-intercept stays the same; slope changes to –1/2 which changes the direction of the line and makes the graph flatter What would happen to the graph and equation if I multiplied the slope by -1 and added 3 to the y-intercept? y = -x + 3 y-intercept changes to 3; slope changes to -1 which changes the direction of the line All of these questions are examples of transformations of linear equations. At this point pair up the students and have them work through another example (such as y = 2x + 1) answering the same questions. Each pair should write the new equation and draw a graph for each question. When they finish this problem, have them use the graphing calculators to check their equations and graphs. SIOP Component(s): Strategies, Interaction, Review and Evaluation The questioning technique used by the teacher is designed to help students explore transformations. Answers are not linguistically demanding so that students with limited oral skills in English can participate. The teacher has walked students through the process, asking questions and thinking aloud. The LEP student benefits from following the thought process. This allows students an opportunity to ask their own questions as they develop. The activity also allows for self-evaluation and immediate feedback when students check the equations and graphs on the graphing calculator.

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Pass out the Problem Situation Activity to each student. Read the scenario aloud in class and ask a student to explain what the scenario is saying. Have the students fill in the table, write an equation, and graph a line for the scenario.

Time (weeks) 0 1 2 3 4 5 Equation: y = 20x + 1050 Amount Saved by $10’s Begin with 1050

Process $1050 1(20) + $1050 2(20) + $1050 3(20) + $1050 4(20) + $1050 5(20) + $1050

Amount Saved $1050 $1070 $1090 $1110 $1130 $1150

Weeks by 1’s SIOP Component(s): Comprehensive Input, Practice The application problem is clearly stated and the topic is relevant to high school students. This attracts the student’s attention and helps to keep him/her actively focused on completing the table and graphing the results.

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Begin answering the questions on the worksheet.

How would the equation and graph change if Tim made $200 during the summer? y = 20x + 200

The y-intercept (starting point) is lower and the slope is the same. How would the equation and graph change if Tim received $10 a week for chores? y = 10x + 1050 The y-intercept is the same but the slope is smaller (less steep). Compare the slopes and y-intercepts in the following two equations. In the first equation the slope is smaller (less steep) and the y-intercept is greater. What changes in the scenario resulted in a change in the steepness of a line? the slope What changes in the scenario resulted in a change in the starting point of a line? the y-intercept SIOP Component(s): Lesson Delivery The questions on the handout provide an opportunity for LEP students to practice writing in a very natural way. Responses do not require extensive academic writing skills but communicate essential information and ideas. (11) Close the lesson by discussing how transformations affect a graph, an equation, the slope, and the y-intercept.

SIOP Components(s): Strategies In summarizing the lesson, students are encouraged to reflect and consider the observations they made in both a real world and a hypothetical problem. The teacher monitors and guides responses and may need to paraphrase student responses to model the appropriate math vocabulary and English usage.

Technology Applications The graphing calculator can be used throughout this lesson to check their equations and graphs. To help the understanding of transformations, the graphing calculator can be used to investigate steepness, flatness, and varying starting points. SIOP Component(s): Building Background, Practice It is important that all students understand and are able to use a graphing calculator. LEP students may be unfamiliar with the technology and the teacher needs to direct teach these skills, explaining the use of the functions and allowing students time to practice using the calculator. This is an essential skill since students will be using the graphing calculator on the TAKS. Please see the information on the graphing calculator tutorial offered on the website by Region IV Education Service Center (www.esc4.net/math). Assessment An informal assessment will be through the discussion in class. A formal assessment can be assessed through the Problem Situation Activity sheet.

Formal assessment evaluates the individual student’s comprehension and learning of the lesson’s objectives. In addition, the assessment measures the student’s procedural knowledge for problem solving. Extensions Develop a scenario with a table, graph, and equation. Use that information to describe 4 changes in slope and y-intercept from the scenario. Write the new equations for each change.

Problem Situation Activity Tim worked all summer mowing lawns and made $1050. When school starts, Tim cannot work anymore but his parents pay him $20 a week for doing chores around the house. Instead of spending his money Tim has decided to save his money to purchase a car next year.

1. Make a table that represents the above scenario. Time (weeks) Process Amount Saved

2. Write an equation that represents how much money Tim will have saved after t weeks.

3. Sketch a graph.

4. How would the equation and graph change if Tim made $200 during the summer?

5. How would the equation and graph change if Tim received $10 a week for chores?

6. Compare the slopes and y-intercepts in the following two equations. y = 12x + 500 y = 20x + 250

7. What changes in the scenario resulted in a change in the steepness of a line?

8. What changes in the scenario resulted in a change in the starting point of a line?

Algebra One Unit Writing and Graphing Linear Equations Lesson Plan 6 Practicing transformations in slope and y-intercept of graphs, equations, and problem situations Content Objective The student will learn to evaluate how changes in slope and y-intercept affect a graph, an equation, and a problem situation. SIOP Component(s): Preparation In order for the LEP student to best understand the content objective, it is essential that it be clearly stated in simple language. The objective should be given both orally and in written form. Language Objectives The student will understand and define the following vocabulary: transform, graph, coordinate plane, x and y- axis, slope, and y-intercept. The student will write a verbal description explaining how changes in slope and yintercept affect the equation. SIOP Component(s): Preparation It is important in the sheltered instructional setting to establish academic language objectives. LEP students not only learn content but also acquire academic English skills in reading, writing, listening, and speaking. Special emphasis is placed on development of math-specific vocabulary. Metacognitive Objectives The student will use a graphing calculator to discover how transformations affect graphs, equations, and problem situations. The students will work in pairs to practice problem solving. The student will reflect and evaluate his/her participation in the different stations as to the ease and difficulty of the task.

SIOP Component(s): Preparation Metacognitive strategies that students acquire will continue to help them even when the teacher is not there to guide them. Strategies enable the student to plan, monitor and evaluate learning independently. Materials  Stations (Sheets for stations 1 – 6)  Station Worksheet  4 dice (2 with numbers 1- 6, 2 with positives and negatives)  Grid paper  Transparency lines  Student graphing calculators SIOP Component(s): Preparation A variety of materials make the lesson clear, meaningful and help provide concrete examples for the LEP student. Procedure Allow approximately 1 ninety-minute class period (1) Prior knowledge should include the understanding of transformations. SIOP Component(s): Building Background Transformation is an important concept in mathematics. The teacher presents the word’s meaning within the content area. The word is modeled by the teacher and defined by its use with familiar skills. (2) Review transformations by graphing, writing equations, and through problem situations.

SIOP Component(s): Building Background, Comprehensive Input Through the use of guided instruction of transformations, teachers activate, assess, and reteach content before beginning the station activities.

(3)

Put the students into pairs. Pass out the station worksheet to each student. Explain the six stations and how to rotate around them. SIOP Component(s): Comprehensive Input, Lesson Delivery Directions for the stations offer step-by-step instructions and clearly explain the process. It is best to simplify sentence structure in text used in directions. SIOP Component(s): Interaction Students work through the stations with a partner. This allows for collaboration and ample opportunities to communicate and learn from peers. Station 1: The students will use the grid paper and transparency lines. The student will graph the original problem with the transparency line and then translate line according to the instructions. The student will need to graph the original problem, graph the new problem and write the new equation on the station worksheet. (Similar to lesson 4) Station 2: The students will use a numbered die and a sign die. The original problem will be posted at the station. The student rolls the numbered die and the sign die twice. The first roll of each goes with the slope the second roll of each goes with the y-intercept. The number for the slope is what you multiply the slope by to find the new slope and the number for the y-intercept is what you add or subtract to find the new number. After the student gets the new equation they graph it and write a verbal description to explain the changes of the new equation on the station worksheet. (Similar to lesson 5)

Example:

y = 2x + 3 original problem First roll = 4 and + Second roll = 2 and –

y = 8x + 1 new problem ~ multiply positive 4 to the slope and subtract 2 from the y-intercept Graph the new equation. The new equation is steeper and remained positive and the y-intercept is smaller. Station 3: The students will use graphing calculators to explore the effects of changing slope and y-intercepts. This station will have 3 sets of 4 equations to graph and then a couple of questions to answer based on the graphs. The graphs and answers to questions should be written on the stations worksheet. Station 4: The students will look at a problem situation and fill in a table, a graph, write an equation, and answer some questions based on a scenario. Each answer should be placed on the station worksheet. (Similar to lesson 5) Station 5: The students will review graphing lines given in slope-intercept form and standard form. Each equation needs to be written and graphed on the station worksheet. (Similar to lesson 2) Station 6: The students will review writing an equation given a graph, table, or slope and y-intercept. Each equation needs to be written and graphed on the station worksheet. SIOP Component(s): Strategies, Interaction, Practice, Lesson Delivery Actively working at the different stations with a partner engages the LEP student in a variety of activities where they can express understanding of content objectives. This classroom setting also reduces the linguistic demands on students’ limited English skills. (4) After the stations are complete, have the students discuss the station that was the hardest and the station that was the easiest. This should allow for some good discussion of concepts.

SIOP Component(s): Planning Time has been built into the lesson for student self-evaluation. Students are asked to think about and give their opinion as to the ease and difficulty of each of the stations and the reasoning behind it. This also offers the teacher an opportunity to further assess the needs of students who may need more attention or tutorials. Technology Applications The graphing calculator is used on station 3 to discover more about transformations. SIOP Component(s): Building Background, Practice It is important that all students understand and are able to use a graphing calculator. LEP students may be unfamiliar with the technology and the teacher needs to direct teach these skills, explaining the use of the functions and allowing students time to practice using the calculator. This is an essential skill since students will be using the graphing calculator on the TAKS. Please see the information on the graphing calculator tutorial offered on the website by Region IV Education Service Center (www.esc4.net/math). Assessment An informal assessment will be done while observing the students as they work through the stations. The teacher may want to use a chart similar to the one on page X to identify students who need extra assistance. A simple check system identifies students who successfully completed the different stations. A formal assessment will be done through the station worksheet.

Formal assessment evaluates the individual student’s comprehension and learning of the lesson’s objectives. In addition, the assessment measures the student’s procedural knowledge for problem solving. Extensions Develop an example of a station that could be used to help practice transformations.

STATION 1
Use the grid paper and transparency line to graph the following equations and then to make transformations to those equations.

1. y = 2x + 4

Translate this graph down 2 units. Graph the new line. Write the new equation.

2. y = (1/2)x – 3

Translate this graph up 4 units. Graph the new line. Write the new equation.

3. 2x + y = 6

Translate this graph down 2 units. Graph the new line. Write the new equation.

STATION 2
84

Roll each die twice. The first roll of each goes with the slope the second roll of each goes with the y-intercept. The number for the slope is what you multiply the slope by to find the new slope, and the number for the y-intercept is what you add or subtract to find the new number. Write the new equation and a verbal description explaining the changes from the original equation. Example: y = 2x + 3 original problem First roll = 4 and + Second roll = 2 and – y = 8x + 1 new problem ~ multiply positive 4 to the slope and subtract 2 from the y-intercept Graph the new equation. The new equation is steeper and still positive and the y-intercept is smaller. Problems: 1. y = 3x + 1

2. y = 4x - 4

3. y = -6x + 8

4. y = -2x

STATION 3

Use a graphing calculator to graph the following equations on the same graph. Answer the following questions for each.
1. Graph: y=x y = 2x y = 4x y = 8x

Questions: a. How are these lines alike? b. How are these lines different? c. What happens to the graph as “m” gets bigger? (y = mx + b) 2. Graph: y=x y = 1/2x y = 1/4x y = 1/8x

Questions: a. How are these lines alike? b. How are these lines different? c. What happens to the graph as “m” gets smaller? (when m is between 0 and 1) 3. Graph: y=x y = 2x y = 2x + 3 y = -2x - 2

Questions: a. Explain how 2 and -2 in front of the “x” change the graph compared to y = x? b. Explain what the y-intercepts of 3 and -2 do to the graph?

STATION 4
86

Read the scenario below and answer all the questions.
Susie received a gift of $50. She also gets $10 a week for doing chores around the house. Instead of spending her money Susie has decided to save her money to purchase a new ring. 1. Make a table that represents the above scenario. Time (weeks) Process Amount Saved

2. Write an equation that represents how much money Susie will have saved after t weeks.

3. Sketch a graph.

4. How would the equation and graph change if Susie got $100?

5. How would the equation and graph change if Susie received $15 a week for chores? 6. Compare the slopes and y-intercepts in the following two equations. y = 15x + 50 y = 5x + 100

STATION 5

Sketch a graph of the following lines. Make sure they are in slopeintercept form ( y = mx + b). 1. y = -3x – 2

2. y = -1/3 x - 3 3. 4x – 2y = 6

4. 3x + 4y = -8

5. x + y = -3

STATION 6
88

Write an equation given a table, graph, two points, or slope and yintercept.

1. 0 1 2 3 4 2 4 6 8 10

2. slope = -2 y-intercept = 3

4. (2, 4) (4, 12)

5. -1 1 3 5 7
Station 1:

6. (0, 4) (1, 3) 0 3 6 9 12 Station Worksheet