Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Unit 05: Algebraic Representations and Applications (13 days) Possible Lesson 01 (4 days) Possible Lesson 02 (9 days) POSSIBLE LESSON 02 (9 days) This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with district-approved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time frame to meet students needs. To better understand how your district is implementing CSCOPE lessons, please contact your child s teacher. (For your convenience, please find linked the TEA Commissioner s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.) Lesson Synopsis: Students solve linear equations with concrete models and make connections between concrete models, abstract, and symbolic representations. Students use tables, graphs, equations, and written descriptions to generate multiple representations of numerical relationships. TEKS: The Teas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Teas law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit. The TEKS are available on the Teas Education Agency website at http://www.tea.state.t.us/inde2.asp?id6148 8.2 Number, operation, and quantitative reasoning.. The student selects and uses appropriate operations to solve problems and justify solutions. The student is epected to: 8.2A Select appropriate operations to solve problems involving rational numbers and justify the selections. Supporting Standard 8.2B Use appropriate operations to solve problems involving rational numbers in problem situations. Readiness Standard 8.3 Patterns, relationships, and algebraic thinking.. The student identifies proportional or non-proportional linear relationships in problem situations and solves problems. The student is epected to: page 1 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days 8.3A Compare and contrast proportional and non-proportional linear relationships. Supporting Standard 8.4 Patterns, relationships, and algebraic thinking.. The student makes connections among various representations of a numerical relationship. The student is epected to: 8.4 Generate a different representation of data given another representation of data (such as a table, graph, equation, or verbal description). Readiness Standard 8.5 Patterns, relationships, and algebraic thinking.. The student uses graphs, tables, and algebraic representations to make predictions and solve problems. The student is epected to: 8.5A Predict, find, and justify solutions to application problems using appropriate tables, graphs, and algebraic equations. Readiness Standard 8.7 Geometry and spatial reasoning.. The student uses geometry to model and describe the physical world. The student is epected to: 8.7D Locate and name points on a coordinate plane using ordered pairs of rational numbers. Supporting Standard Underlying Processes and Mathematical Tools TEKS: 8.14 Underlying processes and mathematical tools.. The student applies mathematics to solve problems connected to everyday eperiences, investigations in other disciplines, and activities in and outside of school. The student is epected to: 8.14A Identify and apply mathematics to everyday eperiences, to activities in and outside of school, with other disciplines, and with other mathematical topics. 8.14B Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness. 8.14C Select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem. 8.14D Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, page 2 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days and number sense to solve problems. 8.15 Underlying processes and mathematical tools.. The student communicates about mathematics through informal and mathematical language, representations, and models. The student is epected to: 8.15A Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models. 8.15B Evaluate the effectiveness of different representations to communicate ideas. 8.16 Underlying processes and mathematical tools.. The student uses logical reasoning to make conjectures and verify conclusions. The student is epected to: 8.16A Make conjectures from patterns or sets of eamples and noneamples. 8.16B Validate his/her conclusions using mathematical properties and relationships. Performance Indicator(s): page 3 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Grade8 Unit05 PI01 Create a graphic organizer (e.g., four-corner model, concept map, etc.) that includes a table, graph, equation, and given verbal description of a real-life linear problem situation. Use a calculator to find the solution to the problem situation, and write an inequality statement using numbers and symbols when given a specific parameter within the problem. Justify the solution to the problem with each representation by locating and naming the solution on the graph, finding the solution in the table, and using appropriate operations to find the solution algebraically. Validate the solution process in a written justification, and describe the connections between the representations, detailing if the linear relationship is proportional or non-proportional. Sample Performance Indicator: The speed of a vehicle and a driver s reaction time determine the amount of time it will take a vehicle to stop. If driving on a dry street surface, a person takes 2.5 seconds to react to an emergency, and with reasonably good tires, the vehicle decelerates at a rate of 15 feet per second, then the formula: can calculate t, the time it takes a vehicle to stop in seconds, when traveling r, miles per hour. Create a four-corner model that includes the equation, verbal description, table, and graph for the problem situation. If a vehicle is traveling 65 miles per hour, use a calculator to calculate the time it will take for the vehicle to come to a complete stop. Write an inequality statement, using numbers and symbols, to identify the speeds in which a vehicle can stop in less than 10 seconds. Solve the problem by locating and naming each solution on the graph, finding each solution in the table, and using appropriate operations to find each solution algebraically. Validate each solution process in a written justification, and describe the connections between the representations, detailing if the linear relationship is proportional or non-proportional. Standard(s): 8.2A, 8.2B, 8.3A, 8.4, 8.5A, 8.7D, 8.14A, 8.14B, 8.14C, 8.14D, 8.15A, 8.15B, 8.16A, 8.16B ELPS ELPS.c.1C, ELPS.c.4B, ELPS.c.5G Key Understanding(s): Conjectures from everyday problem situations are helpful in validating patterns in tables, involving proportional and non-proportional linear relationships, and symbolically representing epressions or equations. Relationships among quantities may be epressed in a variety of forms. Different representations of data may be generated given one form of representation. The process of solving an equation involves using a plan or strategy to keep the values on both sides of the equation equally balanced and validating the solution for reasonableness. page 4 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days An ordered pair can be useful in everyday situations to communicate specific points on the coordinate plane. Underdeveloped Concept(s): Students may think equal means find the answer, rather than has the same value as. Some students may think variables are letters representing an object as opposed to representing a number or quantity of objects. Vocabulary of Instruction: equation equivalent equations equivalent epressions epression isolating the variable representations solution solving an equation variable Materials List: algebra tiles (12 1 tiles, 12 1 tiles, 5 tiles, 5 tiles) (1 set per student, 1 set per teacher) Bag of Algebra Tiles (12 1 tiles, 12 1 tiles, 5 tiles, 5 tiles) (1 set per student, 1 set per teacher) (previously created) calculator (1 per student) cardstock (2 sheets per 6 students, 3 sheets per 4 students) cardstock (24 sheets per teacher) cardstock (optional) (2 sheets of red, 2 sheets of green) (1 set per 4 students, 1 set per teacher) map pencil (1 red, 1 green) (1 set per student) math journal (1 per student) plastic zip bag (sandwich sized) (1 per 2 students, 1 per teacher) plastic zip bag (sandwich sized) (1 per 6 students, 1 per 4 students) plastic zip bag (sandwich sized) (1 per student, 1 per teacher) scissors (1 per teacher) square tiles (15 per 2 students, 15 per teacher) page 5 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days STAAR Reference Materials (1 per student) sticky notes (3 3 ) (30 per teacher) tape (masking) (1 roll per teacher) two-color counters (15 per 2 students, 15 per teacher) Attachments: All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website. Horses and Birds KEY Horses and Birds Variables and Epressions Notes KEY Variables and Epressions Notes I Have, Who Has Epress and Evaluate Student Recording Sheet KEY Epress and Evaluate Student Recording Sheet Epress and Evaluate Phrase Cards Epress and Evaluate Epression Cards Epress and Evaluate Solution Cards Variables and Epressions KEY page 6 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Variables and Epressions Variables and Epressions Practice KEY Variables and Epressions Practice Algebra Tiles Simplifying Algebraic Epressions KEY Simplifying Algebraic Epressions Which One Does Not Belong KEY Which One Does Not Belong Balance Scale Balance Scale Problems KEY Balance Scale Problems Parallel Representations on the Balance Scale KEY Parallel Representations on the Balance Scale Multiplication/Division Equations Addition/Subtraction Properties of Equality Algebra Tiles and One-Step Equations KEY Algebra Tiles and One-Step Equations Solving Equations Student Notes KEY page 7 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Solving Equations Student Notes Two-Step Equations and More KEY Two-Step Equations and More Mi-It Madness KEY Mi-It Madness Solve Equations Algebraically KEY Solve Equations Algebraically Formulating Equations KEY Formulating Equations Different Representations Practice KEY Different Representations Practice Different Representations KEY Different Representations Graphs and Data Representation KEY Graphs and Data Representation Graphs and Data Representations Etensions Equations and Data Representations KEY Equations and Data Representations page 8 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Quad Card Recording Sheet KEY Quad Card Recording Sheet Quad Card Activity GETTING READY FOR INSTRUCTION Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using the Content Creator in the Tools Tab. All originally authored lessons can be saved in the My CSCOPE Tab within the My Content area. Suggested Day Suggested Instructional Procedures Notes for Teacher 1 Topics: Introduction to variables and epressions Engage 1 Students use eperience and reasoning skills to review evaluating and simplifying epressions. Instructional Procedures: 1. Display teacher resource: Horses and Birds. 2. Place students in groups of 3 and instruct students to create a non-linguistic model to solve the problem in their math journal. Allow 5 minutes for student groups to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion about the different strategies students used to solve the problem. Ask: Spiraling Review ATTACHMENTS Teacher Resource: Horses and Birds KEY (1 per teacher) Teacher Resource: Horses and Birds (1 per teacher) Teacher Resource: Variables and Epressions Notes KEY (1 page 9 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher What strategy did you use to determine the number of horses and the number of birds if there is a total of 284 legs? Answers may vary. I drew a circle with 4 legs and another circle with 2 legs. I kept track of how many of each in a table until I reached 284; etc. How many legs would be on 5 horses? 5 birds? (number of horse legs: 5 4 20; number of bird legs: 5 2 10) How many total legs are there on 5 horses and 5 birds? (30) What numeric epression could you write to represent the total number of legs on 5 horses? 5 birds? (5 4 number of horse legs and 5 2 number of bird legs) What numeric epression would you write to represent the total number of legs on 5 horses and 5 birds? (5 4 5 2) What would be the correct order of operations to evaluate this epression? (multiply 5 4, then multiply 5 2, and then add the two products 20 10) How would you represent the number of legs on any number of horses? Birds? (4 h, where h represents the number of horses; 2 b, where b represents the number of birds) How would you represent the total number of legs on any number of horses and birds? (4 h 2 b, where h represents the number of horses and b represents the number of birds) What equation would you write to show you want this epression to have a value of 284? (4h 2b 284) Was your answer reasonable? Answers may vary. Yes, because ; etc. How many horses and how many birds would there be if there was a total of 70 legs? Answers may vary. 11 horses and 13 birds; 10 horses and 15 birds; etc. What equation would you write to show there is a total of 70 legs? (4h 2b 70) How could you use the description, Each horse has 4 legs and each bird has 2 legs to generate a table that would show the total number of legs for a given number of horses and birds? Answers may vary. per teacher) Handout: Variables and Epressions Notes (1 per student) MATERIALS math journal (1 per student) page 10 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher 3. Distribute handout: Variables and Epressions Notes to each student. Facilitate a class discussion about variables and epressions. Instruct students to complete the handout throughout the class discussion. Topics: ATTACHMENTS Multiple representations Eplore/Eplain 1 Students connect a verbal description to an algebraic epression. Instructional Procedures: Card Set: I Have, Who Has (1 set per 6 students) Teacher Resource: Epress and Evaluate Student Recording Sheet KEY (1 per teacher) page 11 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher 1. Prior to instruction, create a card set: I Have, Who Has for every 6 students by copying on cardstock, cutting apart, and placing in a plastic zip bag. Additionally, create card sets: Epress and Evaluate Phrase Cards; Epress and Evaluate Epression Cards; Epress and Evaluate Solution Cards for every 4 students by copying on cardstock, cutting apart, and placing in a plastic zip bag. 2. Place students in groups of 6 and distribute a card set: I Have, Who Has to each group. Instruct each student to select 4 cards from the set. Eplain to students that the student with the card with an * and the student statement, Who has four more than three times a number? will begin the game. The student reads their card aloud and the student with the matching card responds to by stating, I have 3 4, who has two less than one fifth of a number. This process continues until the students end on the card with an *. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion to debrief student solutions, as needed. 3. Place students in groups of 4. Distribute handout: Epress and Evaluate Student Recording Sheet to each student and card sets: Epress and Evaluate Phrase Cards; Epress and Evaluate Epression Cards; Epress and Evaluate Solution Cards to each group. 4. Instruct students to match each phrase card with an epression card and a solution card, and then record the matches on their handout: Epress and Evaluate Student Recording Sheet. Allow time for students to complete the matches. Monitor and assess students to check for understanding. Facilitate individual group discussions to clarify the understanding of how to evaluate an epression, as needed. Ask: Handout: Epress and Evaluate Student Recording Sheet (1 per student) Card Set: Epress and Evaluate Phrase Cards (1 set per 4 students) Card Set: Epress and Evaluate Epression Cards (1 set per 4 students) Card Set: Epress and Evaluate Solution Cards (1 set per 4 students) Teacher Resource: Variables and Epressions KEY (1 per teacher) Handout: Variables and Epressions (1 per student) Teacher Resource (optional): Variables and page 12 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher How do you write the epression, a number u times 5? (5u) What do you do with the u 8? (Substitute -8 for u in the epression 5u.) How do you solve the epression? (multiply 5 times -8) What is the solution for this epression? (-40) Why is the answer negative? (When you multiply a positive and a negative, the answer is negative.) Epressions Practice KEY (1 per teacher) Handout (optional): Variables and Epressions Practice (1 per student) 5. Distribute handout: Variables and Epressions to each student as independent practice and/or homework. MATERIALS cardstock (2 sheets per 6 students, 3 sheets per 4 students) scissors (1 per teacher) plastic zip bag (sandwich sized) (1 per 6 students, 1 per 4 students) ADDITIONAL PRACTICE The handout (optional): Variables and Epressions Practice may be used as additional practice if needed. page 13 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher 2 Topics: Simplifying algebraic epressions Eplore/Eplain 2 Students simplify algebraic epressions and translate written phrases to algebraic symbols. Instructional Procedures: 1. Prior to instruction, create a Bag of Algebra Tiles for each student and a Bag of Algebra Tiles for each teacher by placing 12 1 tiles, 12 1 tiles, 5 tiles, and 5 - tiles in a plastic zip bag. If algebra tiles are not available, use class resource: Algebra Tiles to create a Bag of Algebra Tiles for each student and a Bag of Algebra Tiles for each teacher by copying pages 1 and 3 on green cardstock and pages 2 and 4 on red cardstock, laminating, cutting apart, and placing 12 1 tiles, 12 1 tiles, 5 tiles, and 5 - tiles in a plastic zip bag. 2. Distribute a Bag of Algebra Tiles for each student and display a model of each shape of the algebra tiles. Facilitate a class discussion eplaining that algebra tiles are a concrete area model used to represent values and model algebraic symbols such as and 1. Spiraling Review ATTACHMENTS Class Resource (optional): Algebra Tiles (1 per 4 students, 1 per teacher) Teacher Resource: Simplifying Algebraic Epressions KEY (1 per teacher) Handout: Simplifying Algebraic Epressions (1 per student) MATERIALS 3. Display the small square algebra tile from a Bag of Algebra Tiles. Facilitate a class discussion to name the algebra tile. Ask: algebra tiles (12 1 tiles, 12 1 tiles, 5 tiles, 5 - tiles) (1 set per student, 1 set per teacher) plastic zip bag (sandwich sized) (1 per student, 1 per teacher) page 14 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures What is the name of this shape? (square) How do you find the area of a square? (s² or side times side) What is the width of this square algebra tile? (1) What is the length of the square algebra tile? (1) What is the area of this square tile? (1 times 1 1 unit 2 ) Notes for Teacher cardstock (optional) (2 sheets of red, 2 sheets of green) (1 set per 4 students, 1 set per teacher) scissors (optional) (1 per teacher) map pencil (1 red, 1 green) (1 set per student) Eplain to students that since the area of this tile is 1, these tiles are called unit tiles, and they represent 1. 4. Display the tile for net to the unit tile to show that the tile is 1 unit in width. Ask: What is the width of the larger tile? (It is unknown.) What variable can you use to represent an unknown? () What is the area of this tile if it is 1 unit in width and units in length? ( times 1 ) Eplain to students that since the area of this tile is, these tiles are called the tiles and will be used to represent most variables. 5. Display the following algebra tiles for the class to see. TEACHER NOTE Algebra tiles help students make sense of the language of algebra. They are concrete models of abstract thought. Middle school students (and most high school students) need a firm foundation in the use of algebra before moving to purely abstract algebraic manipulations. TEACHER NOTE The ² algebra tile will not be used page 15 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher until Algebra 1 for factoring and solving quadratic equations. TEACHER NOTE It is important to help students Ask: What do you think the red side represents? (negatives) Eplain to students that the color of positive values may be blue, green, yellow, or some other color but not red. 6. Facilitate a class discussion to clarify zero pairs. Ask: What does a red 1 unit and a green 1 unit equal? (zero, according to integer rules, negative 1 plus positive 1 equals zero.) What does a green unit and a red unit equal? (zero according to integer rules, negative plus positive equals zero.) What name represents this relationship? (zero pair) 7. Facilitate class discussion to practice using algebra tiles. Ask: How do you represent 3 using algebra tiles? (3 green 1 tiles) develop an understanding of the various ways variables are used. 1. Specific unknown Eample: 8 12 2. Variable- a pattern generalizer Eample: A L W 3. Variable- as quantities that vary in joint variation Eample: C 2 r Usiskin (1988) TEACHER NOTE Students have been eposed to variables throughout elementary school; however, studies indicate that most children have a vague understanding of the concept of a variable. Many students believe page 16 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures How do you represent (-5) using algebra tiles? (5 red 1 tiles) Notes for Teacher that 4 1 13 is different from 4y 1 13. How do you represent 2 1 using algebra tiles? (2 green tiles and 1 green 1 tile) How do you represent 3 4 using algebra tiles? (3 green tiles and 4 red 1 tiles) Remind students that a number and an unknown or variable written side-by-side, such as 3y, indicates multiplication of 3 and the unknown/variable: 3 y, or has the same value as y y y. TEACHER NOTE As a strategy for ELL students, line 8. Display the epression 4 2 2 3 for the class to see. Instruct students to use their Bag of Algebra Tiles to create a model of the epression. Allow time for students to complete the model. Monitor and assess students to check for understanding. Facilitate a class discussion about combining like terms and zero pairs. up the tiles and have students name the tiles as you write the epression under each name. Eample Ask: What can you do first to simplify this epression? (Look for zero pairs.) What zero pairs are displayed? (There are two sets of zero pairs for the tiles and two sets of zero pairs for the ones tiles.) What does the model look like once the zero pairs are removed? TEACHER NOTE As a strategy for ELL students, when removing zero pairs, model the removal of the tiles as follows. page 17 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher Use one hand. Place inde finger on the negative tile and the middle What algebraic epression can you use to represent this simplified model? (2 1) 9. Place students in pairs and distribute handout: Simplifying Algebraic Epression to each student. Instruct students to use their Bag of Algebra Tiles to model each epression and record the model, simplified model, and simplified epression. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion to debrief student solutions, as needed. finger on the positive tile and slide the two opposing colors off the screen at the same time. TEACHER NOTE The answer key will show the algebra tile models using the following symbolism. TEACHER NOTE Students may have some difficulty relating the epressions 2 10 and 1 9 as being equivalent. The students may have some difficulty relating the epressions (2 10) 2 and 5 as being equivalent. The students may have some difficulty relating the epressions page 18 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher 5 5 and as being equivalent. 3 Topics: Introduction to solving equations with a balance scale Engage 2 Students use eperience and reasoning skills to lay the foundation for connecting the parallel representation of models to the algebraic symbolism used to solve equations. Instructional Procedures: 1. Display set 1 from teacher resource: Which One Does Not Belong. Instruct students to analyze each diagram, determine and record the ratio of squares to circles for each diagram in their math journal, and for each set, determine which diagram does not have the same number of circles per square as the other diagrams in the set. Allow students 1 2 minutes to complete the activity. Monitor and assess students to check understanding. Facilitate a class discussion for students to justify how they determined which diagram was not correct. Ask: Spiraling Review ATTACHMENTS Teacher Resource: Which One Does Not Belong KEY (1 per teacher) Teacher Resource: Which One Does Not Belong (1 per teacher) MATERIALS math journal (1 per student) What does the balance scale imply? (The values on both sides of the balance scale are equal.) Which diagram does not belong? Eplain. (Diagram C: a square 1.5 circles. In the other diagrams, a square 2 circles.) How did you determine which diagram did not belong? Answers may vary. Diagram A: divided the objects on each side of the balance scale into 3 groups; Diagram B: divided the objects on each side of the balance scale into 3 groups; Diagram C: divided the objects on each side of the balance scale into 2 groups; Diagram D: 2 circles balance 1 square; Diagram E: removed 2 circles from each side of the balance scale and then divided TEACHER NOTE Solving application problems involves identifying the unknown and the given, translating the English phrase to algebraic symbols and checking answers page 19 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher the objects on each side of the balance scale into 3 groups; etc. How does Diagram E differ from the other diagrams? (It has circles on both sides of the balance scale.) What can you do so there are only squares on one side of the balance scale and the scale remains balanced? (Remove 2 circles from each side of the balance scale.) What operation does this action represent? (subtraction) with the conditions of the problem through the use of diagrams, tables, formulas, and graphs. 2. Display set 2 from the teacher resource: Which One Does Not Belong. Instruct students to analyze each diagram, determine and record the ratio for each diagram in their math journal, and determine which diagram does not have the same number of circles per square. Allow students 1 2 minutes to complete the activity. Monitor and assess students to check understanding. Facilitate a class discussion for students to justify how they determined which diagram was not correct. Ask: Which diagram does not belong? Eplain. (Diagram E: a square 2 circles. In the other diagrams, a square 1 circle.) How did you determine which diagram did not belong? Answers may vary. Diagram A: divided the objects on each side of the balance scale into 3 groups; Diagram B: removed 2 circles from each side of the balance scale and then divided the objects on each side of the balance scale into 3 groups; Diagram C: 1 circle balances 1 square; Diagram D: remove 1 circle from each side of the balance scale and then divide the objects on each side of the scale into 3 groups; Diagram E: divide the objects on each side of the balance scale into 2 groups; etc. Topics: ATTACHMENTS Solving one-step equations with a balance scale Teacher Resource: Balance page 20 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher Eplore/Eplain 3 Students apply previous knowledge to solve equations with pictorial representations. Instructional Procedures: 1. Prior to instruction create a Bag of Circles and Squares for every 2 students and a Bag of Circles and Squares for each teacher by placing 15 square tiles and 15 two-color counters in a plastic zip bag. 2. Place students in pairs. Distribute handouts: Balance Scale Problems and Balance Scale to each student and a Bag of Circles and Squares to each pair. 3. Display teacher resource: Balance Scale. Using a Bag of Circles and Squares model problem 1 from teacher resource: Balance Scale Problems. Instruct students to use their Bag of Circles and Squares and handout: Balance Scale to replicate the model and record the solution in their math journal. 4. Instruct student pairs to use their Bag of Circles and Squares and handout: Balance Scale to complete the remainder of the problems from handout: Balance Scale Problems and record their solutions in their math journal. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate small group discussions as needed. Ask: How could you use words to describe this diagram? Answers may vary. Three squares and 2 circles equal 8 circles; etc. What is the same on both sides of the balance scale? Answers may vary. Circles; etc. How do you decide what to remove (or add) from both sides of the scale? Answers may vary. Since there are circles on both sides, you need to get the squares by themselves, so you will remove 2 circles from both sides; etc. Scale (1 per teacher) Handout: Balance Scale (1 per student) Teacher Resource: Balance Scale Problems KEY (1 per teacher) Teacher Resource: Balance Scale Problems (1 per teacher) Handout: Balance Scale Problems (1 per student) Teacher Resource: Parallel Representations on the Balance Scale KEY (1 per teacher) Handout: Parallel Representations on the Balance Scale (1 per student) MATERIALS square tiles (15 per 2 students, 15 per teacher) two-color counters (15 per 2 page 21 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher Why must you perform the same action to each side of the balance scale? (So the scale will be in balance. If you change one side of the scale without doing the same thing to the other, the scale will not be balanced anymore.) 5. Distribute handout: Parallel Representations on the Balance Scale to each student as independent practice or homework. students, 15 per teacher) plastic zip bag (sandwich sized) (1 per 2 students, 1 per teacher) math journal (1 per student) TEACHER NOTE The purpose of the balance scale activity is to help students review their work with models for solving equations from Grade 7. 4 Topics: Solve one-step equations Eplore/Eplain 4 Students review mathematical properties and procedures to solve one-step equations algebraically and with concrete models. Instructional Procedures: 1. Prior to instruction cover each model on teacher resources: Addition/Subtraction Property of Equality and Multiplication/Division Equations with a sticky note. 2. Distribute a Bag of Algebra Tiles to each student. Spiraling Review ATTACHMENTS Teacher Resource: Addition/Subtraction Properties of Equality (1 per teacher) Teacher Resource: Multiplication/Division Equations (1 per teacher) Card Set: Algebra Tiles (1 page 22 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher 3. Display problem 1 from teacher resource: Addition/Subtraction Property of Equality. Model how to solve 2 5 using addition. Begin on the left side of the teacher resource uncovering each model representing the Addition Property of Equality while facilitating a class discussion about solving equations using addition. Ask: What does the left side represent? (2) What does the right side represent? (5) What does the line represent? () What part of the balance scale resembles the equal sign? (The fulcrum in the middle.) What was the rule for solving on the balance scale? (Whatever operation you perform to one side of the scale, you must perform the same operation to the other side of the scale.) Why? (To keep the scale balanced.) When working with the squares and circles, what was the goal? (To isolate circles on one side and squares on the other side.) What do you think will be the goal for this equation? (To isolate on one side and units on the other side.) This is called isolating the variable. We are going to eamine two ways of looking at solving equations containing a plus sign. What is happening in the second row of the model? (negative 2 is added to both sides) Why did I add negative 2 to both sides? (to get zero pairs on the left side) Why do I need zero pairs on the left side of the equal mark? (to get the by itself) What is the solution to the equation? ( 3) 4. Recover the left side of teacher resource: Addition/Subtraction Property of Equality. Model how to solve 2 5 using subtraction. Referencing the right side of teacher resource, uncover each model representing the per 4 students) Teacher Resource: Algebra Tiles and One-Step Equations KEY (1 per teacher) Handout: Algebra Tiles and One-Step Equations (1 per student) MATERIALS sticky notes (3 3 ) (30 per teacher) Bag of Algebra Tiles (12 1 tiles, 12 1 tiles, 5 tiles, 5 - tiles) (1 set per student, 1 set per teacher) (previously created) map pencil (1 red, 1 green) (1 set per student) TEACHER NOTE Equations are used to epress relationships between two page 23 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher Subtraction Property of Equality while facilitating a class discussion about solving equations using subtraction. Ask: What does the line represent? () What was the rule for solving on the balance scale? (Whatever operation you perform to one side of the scale, you must perform the same operation to the other side of the scale.) Why? (To keep the scale balanced.) What is the goal for this equation? (To isolate on one side and units on the other side.) What is happening in the second row? (2 is being removed from both sides) Why was 2 removed from both sides? (to get the by itself) What is the solution to the equation? ( 3) Why did both methods lead to the same answer for the value of "? (In both cases, balance is maintained. The variable is alone on one side of the equal sign, and I know what the variable represented in the problem.) How are the methods alike and how are they different? (In one problem I added zero pairs to isolate the variable. In the other, I subtracted.) How do you know the solution is correct? (When I substitute 3 for in the original equation and simplify both sides of the equation, I have a true statement: 2 5, 3 2 5.) 5. Display Problem 2 from teacher resource: Addition/Subtraction Property of Equality and facilitate a class discussion about each property while uncovering each row of the model. 6. Display Problem 1 from teacher resource: Multiplication/Division Equations to facilitate a class discussion, uncovering each row of the model. Ask: quantities. This functional relationship is important for algebra readiness. TEACHER NOTE It is very important to establish the understanding that is equivalent to. TEACHER NOTE It is important students view the as a symbol of the equality relationship between the left and right sides of an equation. TEACHER NOTE It is important for students to view solving an equation as performing the same operation on both sides of an equation to keep the equation balanced. TEACHER NOTE page 24 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher What does the line represent? () What part of the balance scale resembles the equal sign? (The fulcrum in the middle.) What was the rule for solving on the balance scale? (Whatever operation you perform on one side of the scale, you must perform on the other side.) Why? (To keep the scale balanced.) What operation is in the equation? (multiplication) What is the inverse operation of multiplication? (division) How can you isolate the variable? (Divide both sides of the equation by 4.) Why does this isolate the variable? (Because division undoes multiplication. The operation of division is the inverse of multiplication.) What does division mean? (Separating into equal groups.) How does that help you solve this equation? (I will separate the 8 units into 4 equal groups. This will give me two in each group.) What is the solution to the equation? ( 2) How do you know the solution is correct? (When I substitute 2 for in the original equation and simplify both sides of the equation, I have a true statement: 4 8, 4(2) 8.) Since there are some limitations to using algebra tiles to solve equations, students need to develop an understanding of the process with equations that are easily modeled using algebra tiles. For equations that are difficult to model using algebra tiles, it is important students are able to connect the process used with algebra tiles to the algebraic method for solving equations. The equation in Eample 1 is easily modeled using algebra tiles. page 25 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher 7. Display Problem 2 from teacher resource: Multiplication/Division Equations and facilitate a class discussion about each property while uncovering each row of the model. Eample 1 8. Display Problem 3 from teacher resource: Multiplication/Division Equations and facilitate a class discussion about each property while uncovering each row of the model. Ask: Why was this model difficult to create? (It is difficult to model dividing an algebra tile in half.) What could you do to solve this problem? Why? (I could double each side of the equal sign. Now, we could think of the tile representing only of 3. If we double both sides of the equation, we now have plus which is 1. The representation is now: 6. This keeps the balance and solves the problem with the fraction.) What operation is another way to double? (Multiply by 2.) What does multiplication mean? (Combining equal size groups.) How does that help you solve this equation? (I will combine 2 equal groups. This will give me 1 whole.) What is the solution to the equation? ( 6) How do you know the solution is correct? (When I substitute 6 for in the original equation and simplify both sides of the equation, I have a true statement:.) 9. Display Problem 4 from teacher resource: Multiplication/Division Equations and facilitate a class discussion using the questions above while uncovering each row of the model one at a time. The equation in Eample 2 may not be as easily modeled using algebra tiles, but students should connect the process used with algebra tiles to the algebraic method for solving equations. Eample 2 10. Place students in pairs. Distribute a red and green map pencil to each pair and handout: Algebra Tiles and One- Step Equations to each student. Instruct student pairs to complete the handout using the map pencils to represent page 26 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher the colors of the algebra tiles. Allow time for students to complete the activity. Monitor and assess students to check for understanding. 5 Topics: Solve multi-step equations Eplore/Eplain 5 Students review procedures to solve multi-step equations algebraically and with concrete models. Instructional Procedures: 1. Distribute handout: Solving Equations Student Notes to each student. 2. Display teacher resource: Solving Equations Student Notes and facilitate a class discussion to summarize the basic process for solving equations. 3. Instruct students to complete each eample on their handout: Solving Equations Student Notes. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion to debrief student solutions. 4. Distribute handout: Two-Step Equations and More and a Bag of Algebra Tiles to each student. 5. Display problem 1 from teacher resource: Two-Step Equations and More. Using algebra tiles, facilitate a class discussion to model solving two-step equations. Ask: How is this equation different from the other equations you modeled earlier? (This equation has two operations involved.) Spiraling Review ATTACHMENTS Teacher Resource: Solving Equations Student Notes KEY (1 per teacher) Teacher Resource: Solving Equations Student Notes (1 per teacher) Handout: Solving Equations Student Notes (1 per student) Teacher Resource: Two- Step Equations and More (1 per teacher) Teacher Resource: Two- Step Equations and More KEY (1 per teacher) Handout: Two-Step Equations and More (1 per student) page 27 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher What should you do first? (subtract 3) What should you do second? (divide by 2) How could you record your actions using algebraic symbols? (Two long bars with a positive sign can represent 2 and a square with a positive sign can represent the 3 and 7.) How can you verify your answer is reasonable? (Substitute the value for in the original equation and simplify both sides of the equation using the correct order of operations.) 6. Place students in pairs. Instruct student pairs to use their algebra tiles to complete problems 2 4 on their handout: Two-Step Equations and More. Allow time for students to complete the activity. Monitor and assess students to check for understanding. 7. Display problem 5 from teacher resource: Two-Step Equations and More. Using algebra tiles, facilitate a class discussion to model solving two-step equations. Ask: How is this equation different from the other equations you modeled earlier? (This equation has variables on both sides of the.) What should you do first? (Make a zero pair with the.) What should you do second? (Add 3 to isolate the variable.) How could you record your actions using algebra tiles? (The positive 2 will be represented by 2 long bars with positive signs, the negative 3 will be represented by 3 squares with negative signs, the will be represented with one long bar with a positive sign, and the negative 1 will be represented by one square with a negative sign.) How can you verify your answer is reasonable? (Substitute the value for in the original equation and simplify each side of the equation using the correct order of operations.) Teacher Resource: Mi-It Madness KEY (1 per teacher) Handout: Mi-It Madness (1 per student) Teacher Resource: Solve Equations Algebraically KEY (1 per teacher) Handout: Solve Equations Algebraically (1 per student) MATERIALS Bag of Algebra Tiles (12 1 tiles, 12 1 tiles, 5 tiles, 5 - tiles) (1 set per student, 1 set per teacher) (previously created) TEACHER NOTE While the focus of this portion of the lesson is to move students page 28 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher 8. Instruct student pairs to complete problems 6 9 from handout: Two-Step Equations and More. Allow time for students to complete the activity. Monitor and assess students to check for understanding. 9. Distribute handout: Mi-It Madness to each student. Instruct student pairs to solve each equation by selecting a response from Table B and recording it in the appropriate location in Table A. Remind students that not all the responses in Table B will be used. Allow time for student pairs to complete the activity. Monitor and assess students to check for understanding. 10. Distribute handout: Solve Equations Algebraically to each student as independent practice or homework. towards solving equations abstractly, students should be allowed to use the algebra tiles if they choose. Algebra tiles are shown on the high school assessments and it is appropriate to allow the use of tiles on assessments as well. TEACHER NOTE Students may have a conceptual understanding of solving equations and may need to show only equivalent equations after a step has been completed. They may not need to show the operation they are doing to both sides of the equation. This is not skipping a step." 6 Topics: Multiple representations Eplore/Eplain 6 Spiraling Review ATTACHMENTS page 29 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher Students generate a table, epression, or equation when given another algebraic representation. Instructional Procedures: 1. Display teacher resource: Formulating Equations. 2. Place students in pairs and distribute handout: Formulating Equations to each student. Instruct students to match the equations to the problem situations. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: How did you determine the situation that matched the equation? Answers may vary. I used process of elimination by determining which eample best fit the scenario; etc. What is the constant rate of change in each table? Eplain. Answers may vary. 60; etc. How did you determine the constant rate of change? ( ) Where is the constant rate of change in the equation? (The number being multiplied by the value.) Where is the constant rate of change in the table? (Look at the consecutive differences in the right hand column over the corresponding consecutive differences in the left hand column.) 3. Distribute handout: Different Representations Practice and a calculator to each student. Instruct student pairs to generate the missing representations for each problem situation, using the calculator to assist in the finding of solutions and verification of responses. Allow time for student to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion to debrief student solutions. Teacher Resource: Formulating Equations KEY (1 per teacher) Handout: Formulating Equations (1 per student) Teacher Resource: Formulating Equations (1 per teacher) Teacher Resource: Different Representations Practice KEY (1 per teacher) Handout: Different Representations Practice (1 per student) Teacher Resource (optional): Different Representations KEY (1 per teacher) Handout (optional): Different Representations (1 per student) MATERIALS page 30 of 144
Enhanced Instructional Transition Guide / Unit 05: Suggested Duration: 9 days Suggested Day Suggested Instructional Procedures Notes for Teacher calculator (1 per student) ADDITIONAL PRACTICE The handout (optional): Different Representations may be used as additional practice if needed. 7 Topics: Multiple representations Eplore/Eplain 7 Students generate a table, graph, epression, or equation when given another algebraic representation. Instructional Procedures: 1. Place students in pairs and distribute handout: Graphs and Data Representation and a STAAR Reference Materials to each student. Instruct students to generate tables and equations to match the given graphs and record a written description for each problem. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate individual group discussions about the activity, as needed. Ask: What are the two quantities represented in the graph? Answers may vary. Number of cups and number of pints; etc. Spiraling Review ATTACHMENTS Teacher Resource: Graphs and Data Representation KEY (1 per teacher) Handout: Graphs and Data Representation (1 per student) Teacher Resource: Graphs and Data Representation (1 per teacher) Teacher Resource: Graphs and Data Representations Etensions KEY (1 per page 31 of 144