Primary Lesson Designer(s):

Primary Lesson Designer(s):
21st Century Lessons Simultaneous Solutions Primary Lesson Designer(s): Stephanie Conklin

This project is funded by the American Federation of Teachers.

21st Century Lessons – Teacher Preparation
Please do the following as you prepare to deliver this lesson: Spend AT LEAST 30 minutes studying the Lesson Overview, Teacher Notes on each slide, and accompanying worksheets. Set up your projector and test this PowerPoint file to make sure all animations, media, etc. work properly. Feel free to customize this file to match the language and routines in your classroom. *1st Time Users of 21st Century Lesson: Click HERE for a detailed description of our project.

Lesson Overview (1 of 4) Lesson Objective Lesson Description
OBJECTIVE: Students will be able to determine if a point is the solution to a system of equations. Language Objective: Students will define and explain the meaning of simultaneous in real-world situations. Lesson Description This lesson provides a warm-up that reviews how students can solve a system of equations using a graph and table. Then, in a pair-share, students are challenged to solve a system of equations where the answer is not a whole number coordinate pair – thus, asking students to first make a table and then graph the system of equations to estimate the solution. Students will then see real-world examples of simultaneous using a hula hoop video and also a video of two airplanes which just avoided crashing. Students will then work to complete 2 questions which ask them to determine if a point is a solution to a system of equations. Then there will be time for practice problems in small groups, and lastly, the assessment will ask students to check a student’s work and explain mistakes through algebra and graphing.

Lesson Overview (2 of 4) Lesson Vocabulary Materials Common Core
Simultaneous: at the same time Simultaneous Equations: the equations share a common point that simultaneously proves both equations true. Materials There are two video links in this lesson: Airplane Video Hula Hoop Common Core State Standard 8.EE.8 – Analyze and solve pairs of simultaneous linear equations. 8.a – Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 8.b – Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. 8.c – Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Lesson Overview (3 of 4) Scaffolding
This lesson provides students with pop-up reminders and also checks using graphs to help connect students’ algebra with visuals. Students will connect prior knowledge to a new term, simultaneous, and will apply this word in both real-world situations and also in mathematical contexts. Students will also review how to solve an equation for y in both guided work and pair-work. Enrichment SWBAT graph to estimate solutions of systems of equations which are not whole number coordinate points. - The first example in notes will cover this topic, and teachers can push students to try other examples that are in standard form like: 2x + 3y = 12 and x + y = -7 to practice estimating. Online Resources for Absent Students This lesson from LearnZillion reviews graphing system of equations and specifically discusses how to write an equation in slope-intercept form, and then, how to graph lines once in this form. The teacher also discusses how to check a solution to see if it is true:

Lesson Overview (4 of 4) Before and After
Prior to this lesson, Lesson 1 called Intro to Systems of Equations establishes that the solution to a system of equations (with 1 solution) is the point (x, y) where the lines intersect. Students can create tables and graph to find the solution. Today’s lesson, will teach students how to use algebra to determine if a point is a solution. After this lesson students will then be introduced to two other kinds of solutions for systems of equations: infinitely many and no solution. Topic Background The airplane video showed in today’s class is an excellent real-world example of how and why systems of equations are important. This video from Discovery Channel discusses the job of an air traffic controller and how math, sonar and also aviation play into this career. The video is 50 minutes long but the first 1-2 minutes provides an overview of the career, where minutes 16:00 – 18:00 show the insider’s view of an air traffic controller’s station – and from minutes 18:15 – 21:00 explains a crash that took place because two planes crashed (some scenes of fire and explosion).

Warm Up OBJECTIVE: Students will be able to determine if a point is the solution to a system of equations. Language Objective: Students will define and explain the meaning of simultaneous in real-world situations. Determine the solution to the system of equations. Write your answer as a coordinate point. y = -x – 2 y = 4x + 3 Do we need to use a table and graph to solve systems of equations? Oh there are 2 ways we learned last class! What are they? Graphing or making tables both work! Help! How can we find the solution? (7 min) Time passed 7 In-Class Notes Students will review how to find the solution to a system of equations using both a table and a graph. This slide also prompts students (using the aviator bird) to compare and contrast which method is best – we will continue to explore this idea throughout the lesson and will also encourage students to talk through their thinking and solving process. The check button links to another slide which goes through the steps to check that the point (-1, -1) works in both equations. The solution to the system of equations is (-1, -1). Check Agenda

Agenda: 1) Warm Up 2) Launch – Turn and Talk
OBJECTIVE: Students will be able to determine if a point is the solution to a system of equations. Language Objective: Students will define and explain the meaning of simultaneous in real-world situations. 1) Warm Up 2) Launch – Turn and Talk 3) Explore – Hula Hoops and Planes -Watch, Turn and Talk 4) Summary 5) Practice – Pair Work (1 min) Time passed 8 In-Class Notes You may want to entice students into this lesson by telling them we will be watching not 1 but 2 fun videos! 6) Assessment

What other method can we use? Can you estimate the solution?
Launch – Turn and Talk Determine the solution to the system of equations by using the table below. x + y = y = 6x - 8 (2-3 min) Time passed 11 In-Class Notes Students are purposefully given the table in order to practice finding the solution to a system of equations using a table – BUT be aware!!! The students will not be able to find the solution in the table because the solution is a coordinate point that is a decimal. We can first let students estimate what the solution may be by looking at the tables. We can encourage students, using the bird’s prompt, to ask students to graph the system of equations. What other method can we use? Can you estimate the solution? Agenda

Launch – Turn and Talk x + y = 2 2y = 6x - 8
Determine the solution to the system of equations by using the table below. x + y = y = 6x - 8 __ = __ __ y = 3x - 4 -x x y = -x + 2 To write equations in slope-intercept form, always start by circling the y. Then look to see how you can cancel anything near y. (2-3 min) Time passed 14 In-Class Notes This slide will review how to write an equation in slope-intercept form. This is an area where students (at least mine!) always struggle and today’s review will just start to continue practice for students. More opportunities to review this will be in practice and homework. Before graphing, we need to write the equation in slope-intercept form How can we isolate y in the second equation? How can we isolate y in the first equation? This means, we need to isolate y! Agenda

Launch – Turn and Talk! x + y = 2 2y = 6x - 8
Determine the solution to the system of equations by graphing. x + y = y = 6x - 8 x + y = 2 -x x y = -x + 2 2y = 6x – y = 3x - 4 Solution: (1.5, 0.5) (4-5 min) Time passed 16 In-Class Notes This slide reviews how to solve for y before graphing an equation, and then reviews how to graph a systems of equations. The students will be able to tell that there is a point that is a solution – however the point is not two whole numbers. Students will see that their answer is not a whole number and can estimate on their graph, and use the visual above to show that the solution are two fractions. Before we graph, we need to write each equation in y = mx+ b form. We can see the solution is a decimal! Yippy! If there is a point of intersection, there is a solution! Let’s take a closer look!!! Agenda

Explore Launch Launch Launch Launch Launch
What does the word simultaneous look like? Launch Similar or same Launch Launch Launch What does the word simultaneous sound like? The root simul means at the same time in Latin. (1 min) Time passed 17 In-Class Notes This slide gives students and opportunity to make connections to the roots of words (simul) with other words like similar or same. You can also go over the root of the word in Latin with the second pop-up. Students may need to go over the pronunciation of the word simultaneous multiple times to practice – it’s hard to say! Agenda

Explore Launch At the same time Definition of Simultaneous:
(ex) In 2014, Guinness World Records reported that stage performer Marawa the Amazing spun the most hula hoops simultaneously,160 all by herself! What does spinning 160 hula hoops simultaneously look like? Click on my yellow hula hoop to watch Marawa! (2-3 min) Time passed 20 In-Class Notes After prompting students to connect simultaneous with similar – we can define the word as two events occurring at the same time. This fun video will show students 160 hula hoops being spun at same time – a great visual with which to connect the word simultaneous Preparation Notes Check to make sure that the video works. Agenda

Watch, Turn and Talk – Breaking News!!!
In 2014, the following tweet went viral: Dramatic near-miss as two planes almost collide on runway at Barcelona airport Launch Watch the video and answer the questions: Is this statement true? Be ready to defend your answer! The two planes were on the runway simultaneously. If the two planes were on the same place on the runway simultaneously, what would happened? No, the planes never were on the runway at the same time. (5 min) Time passed 25 In-Class Notes This slide will give students to connect simultaneous with system of equations by watching a video of 2 planes almost crashing into each other. This slide also allow students to actively discuss and apply the word simultaneous in real-world situations of high-interest. Preparation Notes - Make sure the video works The planes would have hit each other! Agenda

Just like in the warm-up!
Explore Simultaneous Equations are similar to a systems of equations. These equations have the _______ variables, usually x and y, and the ____ ______(x,y) which makes both equations true. Launch same same Launch point (1 min) Time passed 26 In-Class Notes Students will define the term simultaneous equations using a cloze statement (meaning one where they fill in terms) – and they are specifically filling in the words same to reinforce that equations must have the same variables and same coordinate point to prove both equations true. (-1, -1) is point of intersection and is true when plugged into both equations Just like in the warm-up! Agenda

Explore Launch Launch 1 = -4(1) + 5 1 = -4 + 5 1 = 1 ✔ 1 = 2(1) – 1
(ex 1) Is the point (1, 1) a solution to the system of equations? y = 2x y = -4x + 5 Launch Launch 1 = -4(1) + 5 1 = 1 = 1 ✔ 1 = 2(1) – 1 1 = 2 – 1 1 = 1 ✔ 1 = 2(1) – 1 1 = 2 – 1 1 = 2(1) – 1 Yes! (1, 1) is the solution since it is true for both equations. (4-5 min) Time passed 31 In-Class Notes You can help students work through the first equation (y = 2x – 1) and show that (1,1) does work – then you can ask students to work in pairs to plug(1,1) in for second equation y = -4x + 5. Students can then write a sentence or two showing that the point does work for both equations. Agenda

How can we check our work?
Explore Continued (ex 1) Is the point (1, 1) a solution to the system of equations? y = 2x - 1 y = -4x + 5 Yes! (1, 1) is the solution since it is true for both equations. (2 min) Time passed 33 In-Class Notes This check will reinforce for students that the solution to a system of equations will work both algebraically and graphically. Preparation Notes There is no graph provided for students – this slide is to provide a visual of students that solution is not graphically (-8, -1) You can extend this by asking students to graph by hand if time remains How can we check our work? Agenda

Explore – Pair-Share -1 = -3(-8) + 12 -1 = -3(-8) + 12 -1 = ¼(-8) + 1
(ex 2) Does the point (-8, -1) satisfy the system of equations? y = ¼x y = -3x + 12 -1 = -3(-8) + 12 -1 = -1 ≠ 36 -1 = -3(-8) + 12 -1 = -1 = 36 -1 = ¼(-8) + 1 -1 = -1 = -1 ✔ What does word satisfy mean in directions? Wait a second!!! Is (-8, -1) a solution to these systems of equations? (4 min) Time passed 37 In-Class Notes The first call out asks students to differentiate between directions in example 1 and example 2 – meaning that the wording of directions “is a point the solution to a system of equations” and “does the point satisfy the system of equations” basically are asking students to do the same thing. Students can work through plugging in the point for both equations, and should note that the second equation will not work. This slide reviews the not equal to sign, and also shows that No! (-8, -1) is not the solution since it is not true for both equations. Agenda

Is this the intersection point? How can we check our work?
Explore Continued y = ¼x + 1 y = -3x + 12 (ex 2) Does the point (-8, -1) satisfy the system of equations? No! (-8, -1) is not the solution since it is not true for both equations. (2 min) Time passed 39 In-Class Notes This check will reinforce for students that the solution to a system of equations will work both algebraically and graphically. Preparation Notes There is no graph provided for students on their hand out – this slide is to provide a visual of students that solution is not graphically (-8, -1) You can extend this by asking students to graph by hand if time remains Is this the intersection point? Where is (-8, -1)? How can we check our work? Agenda

Summary In order for a point to satisfy equations simultaneously, __________________ _______________________________. the point must prove true for both equations We can check our algebra work by ________. We then make sure that the point goes through _____ equations. graphing (1 min) Time passed 40 In-Class Notes This short summary will allow students to quickly review that a point must be true for both equations to be a solution. You can solicit answers from students prior to showing the filled in answer. both Agenda

Practice (8 min) Time passed 48 In-Class Notes
Students will work through practice problems in pairs or small groups to reinforce key concepts learned in today’s lesson. You can have students put answers on the board or use the solutions to have students check their work. Agenda

Practice (2-3 min) Time passed 51 In-Class Notes

Practice (2-3 min) Time passed 54 In-Class Notes

Assessment -5 = -2(-3) + 1 -2(-3) + -5 = -1 -5 = -6 + 1 6 + -5 = -1
Erika completes the following work and graphs both equations. Is she right, why or why not? (Her work is in purple) Does the point (-3, -5) satisfy the system of equations? y = -2x x + y = -1 -5 = -2(-3) + 1 -5 = -5 = -5 ✔ -2(-3) + -5 = -1 = -1 -1 = -1 ✔ (3 min) Time passed 57 In-Class Notes This assessment will help students to practice using both algebra and also graphing as a tool to determine the solution to a system of equations. Students may not quickly catch Erika’s mistake but when they look at the graph, they should be able to tell that (-3, -5) is not the solution since the point (0.5, 0) is shown on the graph. If you would like to scaffold this assessment, you could change the slide to say that Erika is wrong – please find her mistake. Based on student results from this formative assessment, you will have a sense if students have an understanding of solutions to systems of equations - you may want to plan on reviewing homework more if this question does not go well. Yes, (-3, -5) is solution since it makes both equations true. Agenda

Assessment -5 = -2(-3) + 1 -2(-3) + -5 = -1 -5 = 6 + 1 6 + -5 = -1
Erika is not right! The graph shows that the solution is not (-3, -5). We can also see that she made a mistake multiplying! Does the point (-3, -5) satisfy both equations simultaneously? y = -2x x + y = -1 -5 = -2(-3) + 1 -5 = 6 + 1 -5 ≠ 7 -2(-3) + -5 = -1 = -1 -1 = -1 ✔ (3 min) Time passed 60 In-Class Notes This slide shows the answer that Erika is not right – by highlighting that the solution is not the point (-3, -5). You can also prompt students to identify the solution or guess the solution which is (0.5, 0). Agenda

1st Time Users of 21st Century Lessons
Description of 21st Century Lessons: Welcome to 21st Century Lessons! We are a non-profit organization that is funded through an AFT (American Federation of Teachers) Innovation Grant. Our mission is to increase student achievement by providing teachers with free world-class lessons that can be taught via an LCD projector and a computer. 21st Century Lessons are extremely comprehensive; we include everything from warm–ups and assessments, to scaffolding for English language learners and special education students. The lessons are designed into coherent units that are completely aligned with the Common Core State Standards, and utilize research-based best practices to help you improve your students’ math abilities. Additionally, all of our lessons are completely modifiable so you can adapt them if you like. Next Slide Back to Overview

Standards for This Unit The lesson that you are currently looking at is part of a unit that teaches the following Common Core Standards: Next Slide Back to Overview

Requirements to teach 21st Century Lessons: In order to properly use 21st Century Lessons you will need to possess or arrange the following things: Required: PowerPoint for P.C. (any version should work) Note: Certain capabilities in the PowerPoint Lessons are not compatible with PowerPoint for Mac, leading to some loss of functionality for Mac PowerPoint users. An LCD projector Pre-arranged student groups of 2 – (Many lessons utilize student pairings. Pairs should be seated close by and be ready to work together at a moment’s notice. Scissors – at least 1 for every pair Next Slide Back to Overview

Strongly Suggested to teach 21st Century Lessons: Computer speakers that can amplify sound throughout the entire class “Calling Sticks” – a class set of popsicle sticks with a student’s name on each one A remote control or wireless presenter tool– to be able to advance the PowerPoint slides from anywhere in your classroom Personalize PowerPoints by substituting any names and pictures of children we included in the PowerPoint with names and pictures of your own students. Since many lessons utilize short, partner-processing activities, you will want a pre- established technique for efficiently getting your students’ attention. (“hands- up”, Count from “5” to “0” etc.) Project onto a whiteboard so you or your students can solve problems by hand. (Lessons often have a digital option for showing how to solve a problem, but you may feel it is more effective to show the work by hand on a whiteboard.) Internet connectivity – without the internet you may not have full functionality for some lessons. Next Slide Back to Overview

Lesson Preparation (Slide 1 of 2) We suggest spending minutes reviewing a lesson before teaching it. In order to review the lesson run the PowerPoint in “Slideshow “- Presenters View and advance to the “Lesson Overview” slide. By clicking on the various tabs this slide will provide you with a lot of valuable information. It is not necessary to read through each tab in order to teach the lesson, but we encourage you to figure out which tabs are most useful for you. Note: All of our lessons are designed to be taught during a minute class. If your class is shorter than this you will have to decide which sections to condense/remove. If your class is longer we suggest incorporating some of the “challenge” questions if available. Next Slide Back to Overview

Lesson Preparation (Slide 2 of 2) After reviewing the overview slide, click your way through the PowerPoint. As you go, make sure to read the presenter note section beneath each slide. The note section is divided into two sections: “In-Class Notes” and “Preparation Notes.” The In-Class Notes are designed to be concise, bulleted information that you can use “on the fly” as you teach the lesson. Included in In-Class Notes are: a) a suggested time frame for the lesson, so you can determine whether you want to speed up, slow down, or skip an activity, b) key questions and points that you may want to bring up with your students to get at the heart of the content, and c) answers to any questions being presented on the slide. The Preparation Notes use a narrative form to explain how we envision the activity shown on the slide to be delivered as well as the rationale for the activity and any insight that we may have. Next Slide Back to Overview

Features built into each PowerPoint lesson There are several features which have been incorporated into our PowerPoint lessons to help make lessons run more smoothly as well as to give you access to additional resources during the lesson should you want them. These features include: Agenda Shortcuts – On the agenda slide, click on any section title and you will advance to that section. Click the agenda button on any slide to return to the agenda. Action Buttons – On certain slides words will appear on the chalk or erasers at the bottom of the chalkboard. These action buttons give you access to optional resources while you teach. The most common action buttons are: Scaffolding – gives on-screen hints or help for that slide Answers – reveals answers to questions on that slide Challenge – brings up a challenge questions for students Agenda – will return you to the agenda at the beginning of the lesson Next Slide Back to Overview

21st Century Lessons The goal…
The goal of 21st Century Lessons is simple: We want to assist teachers, particularly in urban and turnaround schools, by bringing together teams of exemplary educators to develop units of high-quality, model lessons. These lessons are intended to: Support an increase in student achievement; Engage teachers and students; Align to the National Common Core Standards and the Massachusetts curriculum frameworks; Embed best teaching practices, such as differentiated instruction; Incorporate high-quality multi-media and design (e.g., PowerPoint); Be delivered by exemplary teachers for videotaping to be used for professional development and other teacher training activities; Be available, along with videos and supporting materials, to teachers free of charge via the Internet. Serve as the basis of high-quality, teacher-led professional development, including mentoring between experienced and novice teachers.

21st Century Lessons The people… Directors:
Kathy Aldred - Co-Chair of the Boston Teachers Union Professional Issues Committee Ted Chambers - Co-director of 21st Century Lessons Tracy Young - Staffing Director of 21st Century Lessons Leslie Ryan Miller - Director of the Boston Public Schools Office of Teacher Development and Advancement Wendy Welch - Curriculum Director (Social Studies and English) Carla Zils – Curriculum Director (Math) Shane Ulrich– Technology Director Marcy Ostberg – Technology Evaluator

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