Understanding By Design Unit Template
|Course Name |Algebra II Honors |Course Number |1200340 |
|Module Title |Module 01 – Linear Systems |Time Frame |9 days |
|Developed By |Algebra II Curriculum Revision Team |
|Identify Desired Results (Stage 1) |
|Content Standards |
|MA.912.A.3.10 |
|Write an equation of a line given any of the following information: two points on the line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, |
|or perpendicular to a given line, through a given point on the new line. |
|Moderate |
| |
|MA.912.A.3.3 |
|Solve literal equations for a specified variable. |
|Moderate |
| |
|MA.912.A.3.14 |
|Solve systems of linear equations and inequalities in two and three variables using graphical, substitution, and elimination methods. |
|Moderate |
| |
|MA.912.A.3.15 |
|Solve real-world problems involving systems of linear equations and inequalities in two and three variables. |
|High |
| |
|MA.912.A.3.6 |
|Solve and graph the solutions of absolute value equations and inequalities with one variable. |
|Moderate |
| |
|Understandings |Essential Questions |
|Enduring Understanding |Overarching |Topical |
|Solutions to linear systems can be no solution, one solution or infinitely many solutions. |How do you use mathematics to compare two or three| |
|We can use graphs to demonstrate a constant rate of change for linear equations. |linear situations? | |
|There are several approaches to solving various systems. |How do you use mathematics to represent situations| |
| |with a constant rate of change? | |
| |What are the benefits of knowing how to manipulate| |
| |formulas? | |
| |How do you use mathematics to represent situations| |
| |that require a range of solutions? | |
|Related Misconceptions | | |
|The method for solving the inequalities is the exact same as solving regular equalities. | | |
|Zero slope and no/undefined slope mean the same thing. | | |
|When solving absolute value equations or inequalities you can drop the absolute value symbols and solve | | |
|as before. | | |
|Knowledge |Skills |
|Students will know… |Students will be able to… |
|The formulas for the equation of a line including, slope-intercept, point-slope form and standard form. |Graph linear equations and identify key points such as x- and y-intercepts and slope. |
|The various methods to isolate a variable in a literal equation. | |
|The key components, slope and x- & y-intercepts, of a line. |Isolate a variable in a literal equation. |
|Recognize parallel and perpendicular lines. | |
|The methods of solving systems of equations using graphing, substitution, and elimination. |Find the equation in various forms (standard, slope-intercept, point-slope) of a line given key |
|The definition and concepts of absolute value and inequalities. |pieces of information such as slope, y-or x-intercept, one or two points. |
| | |
| |Write the equation of a parallel and perpendicular line given a line and a point. |
| | |
| |Recognize the equations of parallel and perpendicular lines. |
| | |
| |Solve a system of equations in 2 or 3 variables. |
| | |
| |Solve equations and inequalities containing absolute value. |
|Assessment Evidence (Stage 2) |
|Performance Task Description |
|Senior Tyler Smith budgets $1100 for a 4 day trip to Orlando. He expects to pay $600 on travel and lodging. The price per ticket for theme parks is $60 and he expects to pay an average of $20 per meal. |
|If T represents the number of tickets and M represents the number of meals, then give an inequality to represent the feasible options. |
|Using the inequality above, solve the inequality for M. |
|If he eats at least one meal per day and visits a minimum of 3 parks, write two inequalities one for the number of meals he can eat and one for the number of parks he can visit. |
|Using the three inequalities graph the feasible region (T is the independent variable and M is the dependent variable). |
|Are the maximum numbers of meals and tickets reasonable? |
|Is it possible/reasonable to visit 6 parks and eat 5 meals? |
|What is the best scenario (number of tickets, number of meals) for Tyler? Explain your suggestion. |
|Goal |To create travel suggestions for a high school senior to visit Orlando over 4 days on a budget of $1100. |
|Role |To create a travel package for the customer within their budget. |
|Audience |The customer visiting Orlando on a fixed budget. |
|Situation |The customer has $1100 to spend on a 4 day trip to Orlando and wants to visit multiple parks which cost $60 and has appropriate spending money for meals which cost |
| |$20 on average. |
|Product/Performance and Purpose |Travel package which would include the maximum number of parks and meals that are possible within the budget. |
|Standards |Solve real-world problems involving systems of linear equations and inequalities in two and three variables. |
| |Solve literal equations for a specified variable. |
| |Solve and graph the solutions of absolute value equations and inequalities with one variable. |
|Other Evidence |
|Systems of three equations and absolute value inequalities. |
|Learning Plan (Stage 3) |
|Where are your students headed? Where have they been? How will you make |Students should be able: |
|sure the students know where they are going? |Solve equations including absolute value and inequalities in one variable. |
| |Graph a linear equation and an inequality. |
| | |
| |In the future students will: |
| |Isolate a variable in a literal equation. |
| |Find the equation in various forms (standard, slope-intercept, point-slope) of a line given key pieces of information such as |
| |slope, y-or x-intercept, one or two points. |
| |Write the equation of a parallel and perpendicular line given a line and a point. |
| |Recognize the equations of parallel and perpendicular lines. |
| |Solve a system of equations in 2 or 3 variables. |
| |Solve equations and inequalities containing absolute value. |
|How will you hook students at the beginning of the unit? |Introduce the performance task at the beginning of the unit. |
|What events will help students experience and explore the big idea and |The Springboard MIU Choices and Otto’s Toy Factory CPM Unit 5 are the 2 exploration activities needed for success on the |
|questions in the unit? How will you equip them with needed skills and |performance task. In these activities the students will be exposed to a real world situation that require the development of |
|knowledge? |inequalities, graphing inequalities, finding a feasible region and making conclusions from the feasible region. |
|How will you cause students to reflect and rethink? How will you guide them|Ask questions that require students to write about the connections between graphical and algebraic solutions to systems of |
|in rehearsing, revising, and refining their work? |equations. |
| |How solving literal equations can be applied in other subjects such as science. |
| |Students will use self and peer assessment to revise their work. |
|How will you help students to exhibit and self-evaluate their growing |Students will share their work within a group or to the whole class. |
|skills, knowledge, and understanding throughout the unit? | |
|How will you tailor and otherwise personalize the learning plan to optimize |Differentiated activities including hands on and visual and oral presentations of content. |
|the engagement and effectiveness of ALL students, without compromising the | |
|goals of the unit? | |
|How will you organize and sequence the learning activities to optimize the |Present information in a logical, step-by-step fashion. (Teacher as a guide) |
|engagement and achievement of ALL students? |Move from basic skills to more advanced topics. |
|What FCAT benchmark(s) align most appropriately with the material in this |MA.A.3.4.2 – Selects and justifies alternative strategies, such as using properties of numbers, including inverse, identity, |
|unit? How can you connect, review, and assess this benchmark in a sensible |distributive, associative, and transitive, that allow operational shortcuts for computational procedures in real-world or |
|manner? |mathematical problems. |
| | |
| |MA.C.3.4.2 – Using a rectangular coordinate system (graph), applies and algebraically verifies properties of two- and |
| |three-dimensional figures including distance, midpoint, slope, parallelism, and perpendicularity. |
| | |
| |MA.D.1.4.1 – Describes, analyzes, and generalizes relationships, patterns, and functions using words, symbols, variables, tables,|
| |and graphs. |
| | |
| |MA.D.2.4.2 – Uses systems of equations and inequalities to solve real-world problems graphically, algebraically, and with |
| |matrices. |
| | |
| |Daily exemplars, warm ups and assessments can be used to review, assess and connect the benchmarks. |
|Module 01 Benchmarks |CPM Resources |SpringBoard Resources |Additional Resources |
|MA.912.A.3.10 Write an equation of a line | CPM Math 1 – Algebra I Skill Builders | Activity 1.2 | |
|given any of the following information: two | | | |
|points on the line, its slope and one point |Linear Equation EF-3, EF-13 | | |
|on the line, or its graph. Also, find an |Linear Inequalities LS-15 | | |
|equation of a new line parallel to a given |Slope –Intercept EF-113 | | |
|line, or perpendicular to a given line, |Standard EF-113 | | |
|through a given point on the new line. |Dependent variable EF-37, EF-50 | | |
| |Independent Variable EF-37, EF- 46 | | |
| |Domain EF-46 | | |
| |Range EF-50 | | |
| |Graphing EF-36 | | |
| |Sketch a graph EF-12 | | |
| | | | |
| | | | |
| |Intercepts EF-14 | | |
| |X-intercepts EF-30 | | |
| |Y-intercepts EF-29 | | |
| |Point if Intersection EF-18 | | |
| | | | |
|MA.912.A.3.3 Solve literal equations for a | Literal Equation EF-28, EF-104 | | |
|specified variable. | | | |
|MA.912.A.3.14 Solve systems of linear | Inequalities LS-9-13, 22,35,150 | Activities 1.1 and 1.2 |
|equations and inequalities in two and three |Graphing LS-16 | |rd%20Problems.pdf |
|variables using graphical, substitution, and |Linear LS-15 | | |
|elimination methods. |Systems LS-23 | | |
|MA.912.A.3.15 Solve real-world problems |Sharpening Pencils Lab EF-20 |Activities 1.1 and 1.2 | |
|involving systems of linear equations and |Systems of Equations BB-17 | | |
|inequalities in two and three variables. |Two variables | | |
| |Three variables LS-93, 106, 141,154 | | |
|MA.912.A.3.6 Solve and graph the solutions of|Absolute value LS-3, LS-13 |Activity 1.3 | |
|absolute value equations and inequalities | | | |
|with one variable. | | | |
|Day |Goals / Objectives / Outcomes |Activities |Assessment |
|1 |Diagnostic: |Student: |Questions: |
| | |CW: Assessment Game |How does the slope of the line relate to the graph of the line? |
| |Assess student on the following Algebra I | |What is the difference between the graph of an equation and the graph |
| |topics: |Introduction to Performance Task |of an inequality? |
| |-Solving | |What is the relationship between the slopes of parallel and |
| |-Graphing |HW: EF 3-9 |perpendicular lines? |
| |-Slopes & Intercepts | |What is the relationship between the slopes of horizontal and vertical |
| |-Factoring | |lines? |
| |-Inequalities | | |
| |-Order of Operations | | |
| |Current: |Teacher: |Exemplars: |
| |Review various Algebra I concepts. |Facilitate Assessment Game |Low: Graph the line y = 3x -5. |
| | |Review Outcome of Assessment Game |Moderate: Where does the line 2y+ 3x =12 |
| | | |cross the x-axis? Where does it cross the y-axis? |
| | | |High: Write equations for three different lines |
| | |Assessment Game Directions |with a slope of 4 and explain the relationship |
| | |This activity consists of a game where assessments are placed at each of 6|between the lines. Graph the lines and compare |
| | |stations (or more with a group larger than 24). Participants are expected |their graphs. |
| | |to answer the questions. Answers are placed on colored paper (keyed to | |
| | |each station) and turned in before moving to a new station. The color | |
| | |coding is used to identify which station the answers come from. This is a | |
| | |timed activity where each group has the same amount of time at each | |
| | |station to work on the questions. Suggested time is 7 minutes per station.| |
| | |The winner of the game is the group with the most correct answers. An | |
| | |important part of this activity is the debriefing of the game. The purpose| |
| | |of the game is to inform assess the students’ prior knowledge. Point out | |
| | |the scoring grid and how to keep track of each group’s score. It may be | |
| | |helpful to have each group write their names on a piece of paper and put | |
| | |that paper on the front table. As each timed session finishes the group | |
| | |puts their answer sheet (answers down) on their paper in the front. Then | |
| | |while the participants work at their new station, the teacher grades and | |
| | |records the results of the previous round. | |
| |Misconceptions: |Student Reflection: |
| | |What station was the most difficult for you? |
| | |What did you learn today? |
|2 |Diagnostic: |Student: |Questions: |
| |Write the equation of a line in |CW: Cornell notes/ toolkit and practice problems. |What is the slope formula, slope-intercept formula, standard formula, |
| |slope-intercept form given the graph of a |HW: Worksheet #1-16 |point-slope formula? |
| |line. | |
| | |f |Write the standard form of an equation? |
| |Current: |Teacher: |Exemplars: |
| |Write the equation of a line in standard |Direct instruction and questioning. |Low: Use the point-slope formula to write the equation of a line |
| |form, slope-intercept form or point-slope | |containing the point (-3, 1) and has a slope of 1/3. |
| |form given various information. |Cover increasing, decreasing , horizontal, and vertical description of a |Moderate: Determine the equation of the line that contains the points |
| | |line. |(4, -2) and (-1, 13). |
| | | |High: Suppose a manufacturer of graphing calculators has determined |
| | |Cornell notes/toolkit |that 10,000 calculators per week would be sold at a price of $95. At a|
| | |Demonstrate the proper use of Cornell notes |price of $90 it was estimated that 12,000 would be sold. Determine a |
| | | |linear function to predict a number of calculators to be sold at a |
| | | |given price. |
| |Misconceptions: |Student Reflection: |
| |Students may use point slope formula instead of the slope –intercept to graph linear equation. |What is the point-slope formula? |
| | |Describe the steps to find the equation of a line given two points. |
| |Zero slope and no/undefined slope mean the same thing. | |
| | | |
|3 |Diagnostic: |Student: |Questions: |
| |Write the equation of a line and identify |CW: Cornell notes/toolkit and worksheet #17-24 |How do you recognize parallel and perpendicular lines? |
| |the slope and y-intercept. | do you determine lines are parallel or perpendicular? |
| | |f | |
| |Calculate the slope and describe the line:|HW: Milepost #11 | |
| | | | |
| |(-2,13) and (-2, 7) | | |
| |(3,-7) and (0.,-5) | | |
| | | | |
| | | | |
| | | | |
| |Current: |Teacher: |Exemplars: |
| |Determine the equation of a line parallel |Direct instruction and questioning. |Low: Are the lines y=1/2x +3and y= 2x -7 parallel, perpendicular or |
| |or perpendicular to a given line through a| |neither? |
| |given point. |Allow students to use graph paper and scale the x- and y-axes. |Moderate: Are the lines 2x + 3y = 2 and 2x + 3y = -4 parallel, |
| | | |perpendicular or neither? |
| | |The teacher will also ensure that students are making accurate sketches |High: Are the vertices (0, 0), (4, 2) and (3, 4) the vertices of a |
| | |(labeling the axes, and plotting key points). |right triangle? Explain why. |
| | | | |
| | |The teacher will circulate to each group and ask questions during the | |
| | |class work time. | |
| | | | |
| | |The teacher will also ask questions as the groups present their solutions.| |
| | | | |
| |Misconceptions: |Student Reflection: |
| |Remember to change the sign for the perpendicular lines. |Explain how to determine if lines parallel and perpendicular. |
| | |What is the slope of any line parallel to the x-axis? |
|4 |Diagnostic: Given the equation 6x + 2y = |Student: |Questions: |
| |14, rewrite the equation in slope |CPM & |What is your first step in isolating a variable? |
| |intercept form. |CW: EF-28, 104, FCAT Math & Science Reference Sheet (Solve for different |How does the order of operations apply? |
| | |variables) |Have you thought of all possibilities? |
| |Are the following equations parallel, | | |
| |perpendicular or neither? | | |
| | | | |
| |4x +6y = 12 |HW: Milepost 7 | |
| |y - 4 = 2/3 (x +3) | | |
| |Current: |Teacher: |Exemplars: |
| |Solve literal equations. |Review vocabulary |Low: Solve for R: S = πs(R + r). |
| | |Explain the value of isolating a variable. |Moderate The surface area of a kite can be calculated by S= ½(wh1 |
| | | |–wh2). Solve the equation for w. List the steps you took. |
| | |Use all Algebra to demonstrate solving for variables other than x (i.e |High Sam says that the following equations are two ways to write the |
| | |physics, or chemistry equations) |SAME formula. Decide whether or not you agree with Sam. Explain how you|
| | | |made your decision. The two equation are: s = n/(n+1) and s/(s-1) = n.|
| | |Assign a samplings of formulas on the FCAT Math & Science Reference Sheet | |
| | |to solve for different variables (Supplies – copies of reference sheets) | |
| | | | |
| | | | |
| | |Literal Equations Lesson on Website | |
| | | | |
| |Misconceptions: |Student Reflection: |
| |Practical use across content area is not recognized. | |
|5 |Diagnostic: |Student: |Questions: |
| |Graph the following equations on the same |CW: BB- 17,27, 66, 95, Milepost #4 |Why is an equation written in the form |
| |axes. |HW |Ax +By=C called a linear equation? |
| | | |
| |y= -3/2 x + 1 |pdf |How many times do two linear equations intersect? |
| |y – ½ x – 3 |Link includes worksheets & answer key | |
| | | | |
| |Substitute the answer into the equations. | | |
| |What do you notice? | | |
| |Current: |Teacher: |Exemplars: |
| |Use substitution and elimination to solve | |Low Solve |
| |systems of equations. Graph systems of |Review substitution and elimination with student |7x –y =98 |
| |equations to determine the solution. | |3x -5y =10 |
| | |Assign BB-17 a & b during class. |Moderate Solve the following system: [pic] |
| | | |High |
| | |Supplies – graph paper & rulers |Gloves R' Us was having a great day at their warehouse. Before lunch |
| | | |they had sold a total of 82 pairs of gloves and sales were brisk. At |
| | | |the end of the day Dan, the owner, realized he just had the biggest day|
| | | |in Gloves R' Us history with a total of 127 pairs of gloves sold. As he|
| | | |was heading back to restock for the next day, he asked his new |
| | | |employee, Mittsy, how many of each (cloth and rubber) they had sold. |
| | | |Poor Mittsy, being new, did not know that she was supposed to keep them|
| | | |separate. All she knew was that the rubber gloves were $8.95 each, the |
| | | |cloth gloves were $6.95 each and the total sales for the day $1070.65. |
| | | |Help Mittsy keep her job by finding out how many of each type of glove |
| | | |was sold so Dan, the glove man, can restock. |
| |Misconceptions: |Student Reflection: |
| |The Solutions found algebraically are not related to the graphs |FX 11 |
|6 |How is a solution of two equations in two |Student: |Questions: |
| |variables represented? |Midterm Assessment (Mini Assessment – 30 minutes- Day 1-5) |Is there a systematic approach to solving systems of two equations? |
| | | |Three equations? |
| |What do you find when you are solving a |CW LS 93, 106,154, Milepost 21(selected problems) | |
| |system of equations? |HW |Identify the differences in solving two systems of equations versus |
| | |. |
| |Create a real-world situation can be |s%20Elimination.pdf | |
| |solved by using the systems of equations. | | |
| | | |
| |How do you verify that your solution is |s%20Substitution.pdf | |
| |correct? | | |
| | |Link includes worksheets & answer key | |
| |Current: |Teacher: |Exemplars: |
| |Solve systems of three equations | |Low Solve |
| | |The teacher will also ensure that students are following a systematic |[pic] |
| | |approach to solving three variable systems. | |
| | | |Moderate: Solve |
| | |The teacher will circulate and ask questions during the class work time. |8x -2y +6z =9 |
| | | |4x +5y =3 |
| | |The teacher will also ask questions as the groups present their solutions.|-4x +7y =21 |
| | | |High |
| | | |A group of students are busy studying for their Algebra final. They |
| | | |spent three evenings studying and of course they got hungry. Each day |
| | | |they sent out to the same place for food. Monday evening they |
| | | |purchased 5 hot dogs, 2 hamburgers and 4 orders of fries at a total |
| | | |cost of $23.00 before tax. Tuesday evening they purchased 2 hot dogs, |
| | | |4 hamburgers and 3 orders of fries at a total cost of $22.50 before |
| | | |tax. Wednesday evening they purchased 3 hot dogs, 3 hamburgers and 5 |
| | | |orders of fries at a total cost of $24.00 before tax. How much does |
| | | |each item cost before tax (hot dog, hamburger and fries)? A complete |
| | | |answer will include a definition of the variables, the system of |
| | | |equations to be solved, an organized algebraic approach to solving the |
| | | |problem and the answer written in sentence form. |
| |Misconceptions: |Student Reflection: |
| | |Milepost #21 Problem #8 |
|7 |Diagnostic: |Student: |Questions: |
| | | | |
| | |SpringBoard | |
| |Graph the following equations: |Choices | |
| | |Pages 5-9 #1-11 | |
| |y = 2x +1 | | |
| |y( 2x + 1 |Introduction to Performance Task (Rubics) | |
| | | | |
| |Which of the following points satisfies y(|HW : Systems of Inequalities | |
| |2x + 1? | |
| | | | |
| |( 3, -2) |Link includes worksheets & answer key | |
| |( 0, 9) | | |
| | | | |
| |Current: |Teacher: |Exemplars: |
| |Create a system of inequalities from a set|The teacher may want to create a warm-up to ensure understanding of |Low:Three cans of soda and two bags of chips cost $2.72 and two cans of|
| |of data, represent the system graphically,|graphing and solving linear equations and the use of Function notation. |soda and four bags of chips cost $3.92. What is the cost of each item? |
| |and determine its domain. | |A complete response will include an equation(s) and your work clearly |
| | |Separate the students in groups of four. Identify the four team members |shown |
| | |roles ( i.e. Team Leader, Reporter/Recorder, Time Keeper, Supply Person). |Moderate: Identify the inequalities that describe this region? |
| | | |[pic] |
| | |Administer ground rules for group work. |High: Gloves R' Us was having a great day at their warehouse. Before |
| | | |lunch they had sold a total of 82 pairs of gloves and sales were brisk.|
| | |Cover graphing equations that apply to feasible regions |At the end of the day Dan, the owner, realized he just had the biggest |
| | | |day in Gloves R' Us history with a total of 127 pairs of gloves sold. |
| | |After reading the introduction to Choices: |As he was heading back to restock for the next day, he asked his new |
| | | |employee, Mittsy, how many of each (cloth and rubber) they had sold. |
| | |Allow the students to summarize or paraphrase the process. Apply to real |Poor Mittsy, being new, did not know that she was supposed to keep them|
| | |life connection (i.e. shopping for school clothes). |separate. All she knew was that the rubber gloves were $8.95 each, the |
| | | |cloth gloves were $6.95 each and the total sales for the day $1070.65. |
| | |The teacher will circulate to each group and ask questions during the |Help Mittsy keep her job by finding out how many of each type of glove |
| | |class work time. |was sold so Dan, the glove man, can restock. |
| | | | |
| | |The teacher will question the groups as they present their solutions. | |
| | | | |
| |Misconceptions: |Student Reflection: |
| | |LS 105 |
|8 |Diagnostic: |Student: |Questions: |
| |Use substitution and elimination to solve |CW LS 3-5, 9-15, Compound Inequalities Worksheet | |
| |systems of equations. | | |
| | | | |
| |Graph systems of equations to determine |HW No Assignment | |
| |the solution. | | |
| |Current: |Teacher: |Exemplars: |
| |Compound Inequalities Absolute Value |The teacher will review math symbols ((, (, (, and (). |Low Solve and graph on a number line |
| |Inequalities, and Interval Notation | |1. |7r +3|< 11 |
| | | |2. |14-5x| ( 8 |
| | | |Moderate Solve and graph on a number line |
| | | |1. -|x+2| + 10 < 15 |
| | | |2. |x+2| +10 > 15 |
| | | |High : Explain the difference between the lower two graphs |
| | | |1. |7r +3|< 11 |
| | | |2. |14-5x| ( 8 |
| | | | |
| | | | |
| |Misconceptions: |Student Reflection: |
|9 |Diagnostic: |Student: |Questions: |
| | | | |
| |Current: Review & Assessment |Teacher: |Exemplars: |
| | | | |
| | |Revew | |
| | |Remediate | |
| | |Extend | |
| | | | |
| | |Performance Task assignment is due | |
| |Misconceptions: |Student Reflection: |
| |Current: |Teacher: |Exemplars: |
| |Misconceptions: |Student Reflection: |
From: Wiggins, Grant and J. Mc Tighe. (1998). Understanding by Design, Association for Supervision and Curriculum Development
ISBN # 0-87120-313-8 (ppk)
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