Benchmark - Algebra Unit Plan
Peter Philips
Learning Facilitator, and Senior Editor at Tuitioncenter | CV/Resume/Cover Letter Writer | Helping Students and Professionals Succeed
Peter Philips
SPD -570
Methods of Teaching Math to Students with Mild to Moderate Disabilities
Prof. Peter Philips
June 28th 2023
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Benchmark - Algebra Unit Plan
Part 1: Strategies
The selected two-research based instructional approaches that I could utilize to help Fiona meet her identified objectives include explicitly teaching related vocabulary and the utilization of the concrete-representation-abstract (CRA) approach. The respective specific tips for implementing the strategies and rationalizing their selection are as follows.
Explicitly teaching related vocabulary
Explicitly teaching related vocabulary entails explicitly imparting and stressing the mathematical words/terms and language related to the ideas/concepts being taught (Paulsen, 2006; Paulsen, 2006b). This technique would be instrumental for Fiona as she has challenges understanding text while reading, which harms her comprehension of mathematics. Consequently, by concisely explaining and strengthening the vocabulary connected with algebraic equations and expressions, Fiona will have a firmer grounding in comprehending and resolving mathematics problems.
Implementation Pointers/Tips
?? Begin each lesson by unveiling and reviewing the main vocabulary words/terms associated with the specific concept/idea being taught (Paulsen, 2006; Paulsen, 2006b).
?? Provide thorough explanations and illustrations for each term while encouraging Fiona to utilize the terminology in exercises and discussions.
?? Utilize visual aids, such as flashcards or anchor charts, to help emphasize the significance/meaning of each word/term.
?? Encompass vocabulary-building exercises/activities, such as word puzzles or matching games, to make learning the words/terms more dynamic and engaging (Paulsen, 2006; Paulsen, 2006b).
Rationalization
Explicitly teaching related/associated vocabulary would benefit Fiona because it addresses her low reading understanding and reinforces her comprehension of math ideas/concepts. Consequently, by concentrating on the specific terminologies related to algebra (Paulsen, 2006; Paulsen, 2006b), Fiona will be better suited to understand and solve algebraic equations, thus improving her overall performance in mathematics.
The utilization of the concrete-representational-abstract (CRA) approach
The three-step concrete-representational-abstract approach aids learners in developing a solid comprehension of the concepts in mathematics (Paulsen, 2006; Paulsen, 2006b). It entails initially utilizing real-life items/objects manipulatives to introduce an idea or a concept, then representing the subject matter utilizing visual diagrams or frameworks, and eventually progressing to abstract depictions and symbolic writing/notation (Paulsen, 2006; Paulsen, 2006b). This approach is instrumental to Fiona, providing multiple entry points and graphic representations to improve her comprehension of algebraic concepts.
Implementation Pointers/Tips
?? Start by physically demonstrating and modeling algebraic expressions' subtraction, addition, division, and multiplication utilizing concrete manipulatives, such as algebra blocks or tiles (Paulsen, 2006; Paulsen, 2006b).
?? Progress to representational models/frameworks, such as pictures or diagrams, to showcase the association between operations, variables, and constants.
?? Finally, the transition to symbolic notation and abstract representations stresses the relationship between representational and concrete models (Paulsen, 2006; Paulsen, 2006b).
?? Encourage Fiona to utilize the CRA approach independently when solving problems and underline the progression from concrete to abstract thinking.
Rationalization
The CRA approach is ideal for Fiona because it provides a methodical and visual mechanism for learning algebraic concepts. Consequently, Fiona could develop a strong foundation of comprehension, improve her problem-solving abilities, and close the gap between her current mathematics performance and the higher-level ideas/concepts she needs to grasp by beginning with tangible manipulatives and eventually working towards abstract representations (Paulsen, 2006; Paulsen, 2006b).
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References
Paulsen, K. (2006). Algebra (Part 1): Applying Learning Strategies to Beginning Algebra. CASE STUDY UNIT. https://iris.peabody.vanderbilt.edu/wp-??????? content/uploads/pdf_case_studies/ics_alg1.pdf
Paulsen, K. (2006b). Algebra (Part 2): Applying Learning Strategies to ??????? Intermediate Algebra. CASE STUDY UNIT. https://iris.peabody.vanderbilt.edu/wp-??????? content/uploads/pdf_case_studies/ics_alg2.pdf
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Part 2: 3 Day Unit Plan
Section 1: Lesson Preparation
Teacher Candidate Name: Peter Philips
Grade Level:? 9th Grade
Unit/Subject: Math
Title of Unit: Exploring Algebraic Equations and Expressions
Brief Summary: This unit concentrates on developing skills/abilities in simplifying equations, solving expressions with variables, and solving and writing algebraic equations in real-life word problems. The unit will allow Fiona to strengthen her comprehension and proficiency in algebraic ideas/concepts.
Classroom and Student Factors/Grouping: Fiona is in a mainstream algebra class with learners at the ninth-grade level. She has an account of receiving instruction in the resource setting and is having challenges with reading comprehension. The class encompasses diverse students, including learners with IEPs, ELL students, and students with varying levels of mathematical proficiency.
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Day 1
Day 2
Day 3
National/State
Learning Standards
Indiana State Standard: PS.1: Simplify algebraic expressions using properties of numbers. (Indiana Department of Education, 2020).
Indiana State Standard: PS.1: Solve linear equations and inequalities with one variable (Indiana Department of Education, 2020).
Indiana State Standard: PS.1: Write and solve algebraic equations in real-life word problems (Indiana Department of Education, 2020).
Specific Learning
Target(s)/Objectives
v? Learners will simplify subtraction, addition, division, and multiplication equations by applying the distributive property and combining like values/terms (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Fiona will accurately simplify 80% of the given equations.
v? Learners will solve expressions with variables by isolating the variable utilizing inverse operations (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Fiona will solve 75% of the given expressions correctly.
v? Learners will write and solve algebraic equations based on real-life word challenges/problems (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Fiona will solve 70% of the word challenges/problems accurately.
Academic Language
Vocabulary:
?? Distributive property
?? Solution
?? Simplify
?? Combining like values/terms
Language Function(s):
?? Learners will detail the steps taken to simplify equations.
?? Learners will justify their solutions and approaches utilized to simplify equations.
?? Learners will communicate their comprehension of the distributive property and of combining like values/terms (Paulsen, 2006; Paulsen, 2006b).
Discourse and/or Syntax:
?? Learners will engage in collaborative discussions to improve their comprehension and learn from their peers.
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Vocabulary:
?? Variable
?? Solution
?? Isolate
?? Inverse operations
Language Function(s):
?? Learners will detail the steps taken to solve expressions with variables.
?? Learners will justify their solutions and approaches utilized to isolate the variable (Paulsen, 2006; Paulsen, 2006b).
?? Learners will communicate their comprehension of inverse operations.
Discourse and/or Syntax:
?? Learners will utilize appropriate mathematical vocabulary and sentence frameworks/structures when detailing their approaches and solutions.
Vocabulary:
?? Algebraic equations
?? Real-life word problems
?? Apply
?? Solution
Language Function(s):
?? Learners will detail/explain the steps taken to write algebraic equations based on real-life word challenges/problems (Paulsen, 2006; Paulsen, 2006b).
?? Learners will justify their solutions and approaches utilized to solve the equations.
?? Learners will communicate their comprehension of applying algebraic ideas/concepts to real-life instances.
Discourse and/or Syntax:
?? Learners will engage in whole-class discussions, group or partner discussions, and written explanations to express their comprehension and problem-solving approaches (Paulsen, 2006; Paulsen, 2006b).
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Unit Resources, Materials, Equipment, and Technology
v? Whiteboard or chart paper
v? Calculator (optional)
v? Markers or chalk
v? Simplifying equations worksheets
v? Resource materials detailing the distributive property and combining like values/terms (Paulsen, 2006; Paulsen, 2006b).
v? Markers or chalk
v? Solving expressions with variables worksheets
v? Whiteboard or chart paper
v? Calculator (optional)
v? Resource materials detailing inverse operations and approaches for isolating variables (Paulsen, 2006; Paulsen, 2006b).
v? Calculator (optional)
v? Whiteboard or chart paper
v? Real-life word problem worksheets
v? Markers or chalk
v? Resource materials providing instances of real-life word challenges/problems and their algebraic representations (Paulsen, 2006; Paulsen, 2006b).
v? Speech-to-text tool or assistive technology for Fiona's written expression support
Depth of Knowledge
Lesson Questions
Level 1: Recall
?? What is an equation?
?? Can you give an example of an addition equation?
?? What does it imply to simplify an equation (Paulsen, 2006; Paulsen, 2006b)?
Level 2: Skill/Concepts
?? How do you simplify addition equations? Can detail step by step?
?? What is the distributive property? How does it assist/help in simplifying equations?
?? Can you detail/explain what like values/terms are and how they are combined/enjoined in an equation?
Level 3: Strategic Thinking
?? Why is it significant to simplify equations?
?? How can you utilize the distributive property to simplify a multiplication equation?
?? Can you think of an instance/example where simplifying an equation is helpful in resolving a more complicated problem?
Level 4: Extended Thinking
?? Can you create/develop your addition equation and simplify it?
?? How can you utilize simplifying equations to check the accuracy of your calculations (Paulsen, 2006; Paulsen, 2006b)?
?? Can you detail how simplifying equations connects/relates to solving real-world challenges/problems?
Level 1: Recall
?? What is a variable?
?? Give one example of an expression with a variable.
?? What is the objective when resolving an expression with a variable?
Level 2: Skill/Concepts
?? What are inverse operations? How are they utilized to solve expressions with variables?
?? Can you detail the steps/process of isolating a variable in an equation?
?? How do you check the solution to an expression with a variable (Paulsen, 2006; Paulsen, 2006b)?
Level 3: Strategic Thinking
?? What approaches can you utilize to solve expressions with variables more efficiently?
?? How can you determine if your solution to an expression with a variable is appropriate/correct?
?? Can you think of a real-life instance where solving an expression with a variable is applicable?
Level 4: Extended Thinking
?? Can you create your expression with a variable and resolve it?
?? How can you detail the concept/notion of inverse operations to someone new at algebra concepts?
?? Can you identify a real-life challenge/problem and develop an expression with a variable to represent it (Paulsen, 2006; Paulsen, 2006b)?
Level 1: Recall
?? What is a real-life word challenge/problem?
?? Give an example of a real-life instance that can be represented by an algebraic equation (Paulsen, 2006; Paulsen, 2006b).
?? What does it imply to write and resolve an algebraic equation in a real-life word problem?
Level 2: Skill/Concepts
?? How do you find/identify the significant information in a real-life word challenge/problem to create/develop an algebraic equation?
?? Can you detail the steps to resolve an algebraic equation in a real-life word challenge/problem?
?? How do you know if your solution to a real-life word challenge/problem is reasonable?
Level 3: Strategic Thinking
?? What approaches can you utilize to approach complicated real-life word challenges/problems?
?? How can you verify the accuracy of your solution in a real-life context?
?? Can you think of a different mechanism to represent a given real-life word challenge/problem utilizing an algebraic equation?
Level 4: Extended Thinking
?? Can you create your real-life word challenge/problem and detail an algebraic equation to represent it?
?? How can you explain the process of writing and solving algebraic equations in real-life word challenges/problems to someone new at algebra concepts (Paulsen, 2006; Paulsen, 2006b)?
?? Can you evaluate a real-life instance and determine multiple equations that could represent the situation?
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Section 2: Instructional Planning
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Day 1
Day 2
Day 3
Anticipatory Set
Activity Description/Educator:
v? The educator will portray various math problems/challenges on the whiteboard (Paulsen, 2006; Paulsen, 2006b).
v? The educator will review the vocabulary terminologies related to simplifying equations.
v? The educator will engage in a whole-class discussion to review the responses and provide definitions for the vocabulary terms/words.
v? The educator will frame/model the process of simplifying a simple math challenge and observe the learners' comprehension before transitioning to small cluster/group work. The educator will also circulate and observe during small group assignments.
v? After the review, learners will get into small clusters/groups and work collaboratively to establish the sum on a Kahoot game displayed on the whiteboard.
Student Actions:
v? Learners will concentrate their attention on the whiteboard as the educator illustrates and models math challenges/problems.
v? Learners will discuss with a partner or in small clusters to define the vocabulary terms and solve the math challenges on their worksheets.
v? Learners will go over the responses/answers as a whole class (Paulsen, 2006; Paulsen, 2006b).
v? Learners will work in small clusters, collaborating to determine the sum on the Kahoot game displayed on the whiteboard.
Activity Description/Educator:
v? The educator will pose a real-life challenge/problem that involves variables and discuss its importance.
v? The educator will facilitate a whole-class engagement on the significance of solving expressions with variables.
v? The educator will explain and frame the steps to resolve expressions with variables utilizing inverse operations.
v? Learners will work in small clusters to practice solving expressions with variables, applying the approaches discussed.
v? The educator will circulate and provide support as needed (Paulsen, 2006; Paulsen, 2006b).
Student Actions:
v? Learners will engage in discussions and share their prior understanding related to real-life challenges and the need to resolve expressions with variables.
v? Learners will actively listen and participate in the whole-class engagement on the significance of solving expressions with variables.
v? Learners will observe and take notes as the educator frames the steps to solve expressions with variables (Paulsen, 2006; Paulsen, 2006b).
v? Learners will work collaboratively in small clusters, applying the approaches learned to solve expressions with variables.
Activity Description/Educator:
v? The educator will present a real-life word challenge/problem to the class and discuss its context.
v? The educator will guide the learners through identifying and translating the essential information into an algebraic equation.
v? Learners will work individually or in pairs to detail algebraic equations based on given real-life word challenges (Paulsen, 2006; Paulsen, 2006b).
v? Learners will solve the equations they wrote and cross-check their solutions.
v? The educator will facilitate a whole-class engagement to share and compare different mechanisms and solutions.
Student Actions:
v? Learners will actively listen and engage in the discussion about the real-life word challenge and its context.
v? Learners will engage in guided practice/activity as the educator showcases how to identify essential information/data and translate it into an algebraic equation.
v? Learners will work independently or in pairs to write algebraic equations based on provided real-life word challenges.
v? Learners will solve the equations they detailed and verify their solutions.
v? Learners will contribute to the whole-class engagement, sharing their mechanisms and solutions to the real-life word challenges (Paulsen, 2006; Paulsen, 2006b).
Presentation of Content
Multiple Means of
Representation
Simplifying Equations
Visual Representation:
?? Utilize a whiteboard or chart paper to visually illustrate the steps for simplifying equations.
?? Provide examples of equations and frame the process of simplification, underlining the distributive property and combining like values/terms.
?? Utilize different symbols or colors to distinguish various values/terms and operations in the equations (Paulsen, 2006; Paulsen, 2006b).
Example (Paulsen, 2006; Paulsen, 2006b):
Equation: 2x + 3 - (x - 4) + 2x
Visual representation:
2x + 3 - (x - 4) + 2x
2x + 3 - x + 4 + 2x
Manipulatives:
?? Provide manipulatives such as algebra tiles or counters to represent/serve as the values/terms in equations.
?? Learners can physically move and cluster the tiles to simplify the equations.
?? Encourage learners to utilize the manipulatives to visualize the process/steps for combining like values/terms.
Example (Paulsen, 2006; Paulsen, 2006b):
Equation: 2x + 3 - (x - 4) + 2x
Manipulative representation:
Group 2x tiles together
Combine the constant values/terms
Simplify the equation
Interactive Technology:
?? Use interactive technology instruments such as virtual manipulatives or online equation simplification activities from the Dreambox Learning, (2023) website.
?? These aides allow learners to interact with and manipulate values/terms and equations and receive prompt feedback on their simplification procedures (DREAMBOX LEARNING, 2023).
?? Provide links or access to online resource where learners can practice simplifying equations interactively (DREAMBOX LEARNING, 2023).
Example (Paulsen, 2006; Paulsen, 2006b):
Equation: 2x + 3 - (x - 4) + 2x
Interactive technology representation:
Drag and drop the terms/values to simplify the equation
Receive prompt feedback on the correct simplification procedure/steps
Solving Expressions with Variables
Visual Representation:
?? Utilize a whiteboard or chart paper to visually show the steps for solving expressions with variables (Paulsen, 2006; Paulsen, 2006b).
?? Provide examples of expressions and frame the procedure of isolating the variable utilizing inverse operations.
?? Underline the steps/process involved, such as applying the opposite operation and simplifying each side of the equation.
Example (Paulsen, 2006; Paulsen, 2006b):
Expression: 3x - 5 = 2x + 7
Visual representation:
3x - 5 = 2x + 7
3x - 2x = 7 + 5
x = 12
Concrete Examples:
?? Provide real-life instances or examples that involve expressions with variables.
?? Relate the examples to the learners' interests or daily lives to improve engagement.
?? Ask learners to identify the variables, set up the equation, and solve for the variable utilizing the appropriate steps/procedure.
Example:
Expression: The cost of a ticket to a concert is $50. Each additional ticket costs $15. If a group of colleagues bought a total of x tickets and spent $250 in total, how many additional tickets did they buy (Paulsen, 2006; Paulsen, 2006b)?
Solution: Identify the variables and develop the equation: 50 + 15x = 250
Solve for x utilizing inverse operations: x = 13
Online Illustrations and Simulations:
?? Provide access to online illustrations or simulations that visually indicate the procedure of solving expressions with variables.
?? These resources often provide step-by-step guidance and interactive components to engage learners in the solving process (Paulsen, 2006; Paulsen, 2006b).
?? Learners can follow along, practice, and receive prompt feedback on their solutions.
Example (Paulsen, 2006; Paulsen, 2006b):
Expression: 3x - 5 = 2x + 7
Online illustration/simulation:
Follow the procedure/guided steps to solve the expression
Check the solution and receive response on the correctness
Detailing and Solving Real-Life Word Challenges/Problems
Graphic Organizers:
?? Provide graphic templates or organizers that guide learners in organizing the significant information/data from real-life word challenges/problems.
?? Include sections/categories for identifying variables, translating the challenge/problem into an equation, and solving the equation (Paulsen, 2006; Paulsen, 2006b).
?? Learners can fill in the graphic organizers with the relevant data/information from different word challenges/problems.
Example:
Real-life word challenge/problem: A rectangular yard has a length of 8 meters and a width of x meters. The area of the yard is 40 square meters (Paulsen, 2006; Paulsen, 2006b). What is the value of x?
Graphic organizer:
Length: 8 meters
Width: x meters
Area: 40 square meters
Equation: 8 * x = 40
Solution: x = 5
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Visual Representations:
?? Utilize visual representations, such as models or diagrams, to showcase the real-life instances presented in word challenges/problems.
?? Link the visuals to the corresponding equations and solutions (Paulsen, 2006; Paulsen, 2006b).
?? Motivate learners to utilize the visuals to comprehend the problem context and derive the necessary equations.
Example (Paulsen, 2006; Paulsen, 2006b):
Real-life word challenge/problem: A rectangular yard has a length of 8 meters and a width of x meters. The area of the yard is 40 square meters. What is the value of x?
Visual representation:
Draw a rectangular yard with a length of 8 meters and an unknown width.
Label the dimensions and area on the diagram.
Write the equation: 8 * x = 40
Solve the equation to determine x.
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Real-Life Scenarios:
?? Provide various real-life word challenges/problems that relate to learners' interests, experiences or hobbies.
?? Present the challenges in different formats, such as detailed prompts, or images.
?? Motivate learners to identify the variables, develop the equations, and solve them to find solutions applicable to the given instance.
Example (Paulsen, 2006; Paulsen, 2006b):
Real-life word challenge: A bookstore is offering a discount of 20% on all books. Sarah bought a book for $25 after the discount. What was the original price of the book?
Real-life instance:
Present an instance explaining Sarah's book purchase and the discount offered.
Identify the variables and develop the equation:
0.8 * original price = $25
Solve the equation to determine the original price.
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Multiple Means of
Representation
Differentiation
English Language Learners (ELL):
v? Provide visual illustrations such as diagrams, charts, and guides to support comprehension.
v? Utilize simplified language and provide translation resources or bilingual dictionaries where necessary (Garnett, 1998; Paulsen, 2006; Paulsen, 2006b).
v? Pair ELL learners with English-speaking peers who can provide language support.
v? Provide sentence starters or sentence frames or to assist ELL learners express their perspectives and explanations.
Learners with Special Needs:
v? Provide additional guided practice and examples avenues.
v? Utilize concrete visual aids or manipulatives to represent equations (Garnett, 1998).
v? Provide step-by-step guide/procedures and provide additional support during independent exercises/practice.
v? Provide alternative mechanisms of illustrating comprehension, such as verbal explanations or drawing.
Learners with Gifted Abilities:
v? Provide more complicated equations or additional challenging problems.
v? Motivate learners to explore multiple approaches for simplifying equations.
v? Provide extension exercises or open-ended quizzes that require deeper thinking and evaluation (Paulsen, 2006; Paulsen, 2006b).
v? Allow learners to work independently or with minimal guidance if they demonstrate proficiency.
Early Finishers:
v? Provide extension exercises or challenge problems for early finishers.
v? Provide additional resources or online instruments such as DREAMBOX LEARNING (2023) for independent exploration and practice.
v? Encourage early finishers to assist their peers who may need extra support.
v? Provide avenues for early finishers to illustrate their comprehension through presentations or creative projects.
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English Language Learners (ELL):
v? Utilize visual representations such as graphs, diagrams, or models to support comprehension.
v? Provide avenues for peer collaboration and discussion to strengthen language abilities.
v? Provide translation resources or bilingual dictionaries where needed (Garnett, 1998).
v? Break down complicated challenges into smaller, more manageable steps/processes.
Learners with Special Needs:
v? Utilize concrete visual aids or manipulatives to portray expressions.
v? Provide step-by-step details/instructions and frame problem-solving approaches (Garnett, 1998).
v? Offer additional practice avenues and provide support during independent assignment.
v? Adjust the difficulty of challenges to match individual needs.
Learners with Gifted Abilities:
v? Provide additional problem-solving approaches or more challenging expressions to explore (Paulsen, 2006; Paulsen, 2006b).
v? Motivate learners to justify their responses/solutions and detail their problem-solving procedures.
v? Offer open-ended quizzes or real-world applications to strengthen comprehension.
v? Allow for independent exploration and provide avenues for learners to share their insights with the class.
Early Finishers:
v? Provide extension exercises or challenge problems for early finishers.
v? Provide additional resources or online instruments such as DREAMBOX LEARNING (2023) for independent exploration and practice.
v? Motivate early finishers to help their peers that require extra support.
v? Provide avenues for early finishers to present alternative solution mechanisms or create their problem-solving instances.
English Language Learners (ELL):
v? Provide visual illustrations and examples to support understanding (Garnett, 1998; Paulsen, 2006; Paulsen, 2006b).
v? Break down word challenges/problems into smaller segments/parts and scaffold the procedure of writing equations.
v? Encourage peer engagement and collaboration to develop language abilities.
v? Offer word banks or sentence frames to support ELL learners in expressing their ideas.
Learners with Special Needs:
v? Provide structured graphic organizers or templates to guide the process of writing equations.
v? Utilize real-life examples or instances relatable to learners' experiences (Garnett, 1998).
v? Provide additional practice avenues and provide support during independent assignment.
v? Adjust the difficulty of word challenges/problems to match individual needs.
Learners with Gifted Abilities:
v? Provide real-life and open-ended word problems that require critical thinking and application of algebraic ideas/concepts.
v? Encourage learners to create their word challenges/problems and solve them utilizing algebraic equations.
v? Offer avenues for learners to present their solutions and detail their problem-solving approaches.
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v? Provide resources for further extension or exploration, such as advanced word challenges or real-world applications.
Early Finishers:
v? Provide extension exercises or challenge problems for early finishers (Paulsen, 2006; Paulsen, 2006b).
v? Encourage early finishers to create their word challenges or real-world applications.
Application of Content
Multiple Means of
Engagement
Simplifying Equations
Visual Representation:
?? Utilize a whiteboard or chart paper to visually depict the steps for simplifying equations.
?? Provide examples of equations and frame the simplification process, underlining the distributive property and combining like values/terms (Paulsen, 2006; Paulsen, 2006b).
?? Utilize different signs or colors to distinguish different operations and terms in the equations.
Example (Paulsen, 2006; Paulsen, 2006b):
Equation: 2x + 3 - (x - 4) + 2x
Visual representation:
2x + 3 - (x - 4) + 2x
2x + 3 - x + 4 + 2x
Manipulatives:
?? Provide manipulatives such as algebra tiles or counters to indicate/depict the terms in equations.
?? Learners can physically move and group the tiles to simplify the equations.
?? Motivate learners to utilize the manipulatives to visualize the process of combining like values/terms.
Example (Paulsen, 2006; Paulsen, 2006b):
Equation: 2x + 3 - (x - 4) + 2x
Manipulative representation:
Group 2x tiles together
Combine the constant terms
Simplify the equation
Interactive Technology:
?? Use interactive technology instruments such as virtual manipulatives or online equation simplification exercises (DREAMBOX LEARNING, 2023).
?? These tools/instruments allow learners to interact with the equations, manipulate values/terms, and receive prompt feedback on their simplification process.
?? Provide links or access to online resources such as DREAMBOX LEARNING (2023) where learners can practice simplifying equations interactively.
Example:
Equation: 2x + 3 - (x - 4) + 2x
Interactive technology representation:
Drag and drop the terms (values) to simplify the equation
Receive prompt feedback on the correct simplification procedure/steps
Solving Expressions with Variables
Visual Representation:
?? Utilize a whiteboard or chart paper to visually depict the steps for solving expressions with variables (Paulsen, 2006; Paulsen, 2006b).
?? Provide examples of expressions and frame the process of isolating the variable utilizing inverse operations.
?? Highlight the procedure/steps involved, such as applying the opposite operation and simplifying each side of the equation.
Example (Paulsen, 2006; Paulsen, 2006b):
Expression: 3x - 5 = 2x + 7
Visual representation:
3x - 5 = 2x + 7
3x - 2x = 7 + 5
x = 12
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Concrete Examples:
?? Provide real-life instances or examples that involve expressions with variables.
?? Relate the examples to the learners' interests or daily lives to improve engagement.
?? Ask learners to find/identify the variables, develop the equation, and solve for the variable utilizing the appropriate steps.
Example (Paulsen, 2006; Paulsen, 2006b):
Expression: The cost of a ticket to a concert is $50. Each additional ticket costs $15. If a group of friends bought a total of x tickets and spent $250 in total, how many additional tickets did they buy?
Concrete example:
Find/Identify the variables and develop the equation: 50 + 15x = 250
Solve for x utilizing inverse operations:
x = 13
Online Illustrations and Simulations:
?? Provide access to online illustrations or simulations that visually depict the process of solving expressions with variables.
?? These resources often provide step-by-step guidance and interactive components to engage learners in the solving process.
?? Learners can follow along, practice, and receive prompt responses on their solutions (Paulsen, 2006; Paulsen, 2006b).
Example:
Expression: 3x - 5 = 2x + 7
Online demonstration/simulation:
Follow the guided procedure/steps to solve the expression
Check the solution and receive response on the correctness
Detailing and Solving Real-Life Word Challenges/Problems
Graphic Organizers
?? Provide graphic organizers such as problem-solving storyboards or templates to help learners arrange data and find key elements in word problems.
?? These organizers can help learners in translating real-life instances into math equations or expressions (Paulsen, 2006; Paulsen, 2006b).
?? Motivate learners to utilize the graphic organizers to break down the challenge/problem, find variables, and develop an equation or inequality that depicts the situation.
Example:
Real-life Word challenge/Problem: Mia has $40 and wants to buy a new video game that costs $25. She also wants to save at least $15 for the upcoming movie. How much money should Mia earn to meet her goals?
Graphic Organizer:
?? Identify/underline the given information: Mia has $40, the video game costs $25, and she wants to save $15.
?? Determine/find the unknown: The amount of money Mia needs to earn.
?? Develop an equation or inequality: Let x stand for the amount of money Mia needs to earn.
?? The equation is x = 25 + 15.
Peer Discussions and Collaboration
?? Facilitate peer discussions and collaboration to encourage learners to evaluate and solve real-life word challenges/problems together.
?? Assign small clusters or pairs of learners to work on word problems, share their approaches, and detail their reasoning to each other.
?? Encourage learners to seek clarity by asking questions and discuss different mechanisms of solving the problems (Paulsen, 2006; Paulsen, 2006b).
Example:
Real-life Word Challenge/Problem: John has a rectangular yard with a length of 12 meters and a width of 8 meters. He wants to increase the length by 3 meters and the width by 2 meters. What will be the area of the enlarged yard?
Peer Discussions and Collaboration:
?? Assign manageable groups and provide the word challenge/problem.
?? Have learners discuss and share their mechanisms of solving the problem.
?? Motivate learners to detail their views and listen to different perspectives.
?? Facilitate a class discussion where groups can present their solutions and discuss the different strategies used.
Authentic Applications
?? Provide real-life instances or applications where learners can apply their math skills to solve challenges in meaningful contexts.
?? Utilize examples from daily life, including shopping, budgeting, or sports, to engage learners and make associations between math and their experiences.
?? Motivate learners to evaluate the scenario, establish relevant data/information, and translate it into math equations or expressions.
Example:
Real-life Word Challenge/Problem: Sarah wants to buy a new bicycle that costs $350. She has saved $150 and plans to save $30 every week. How many weeks will it take for Sarah to save enough funds to buy the bicycle?
Authentic Application:
?? Discuss the instance with learners, stressing the real-life context.
?? Ask learners to find/identify the relevant data/information and variables (Paulsen, 2006; Paulsen, 2006b).
?? Guide learners in creating an equation or inequality to depict the scenario and solve for the unknown variable.
Multiple Means of
Engagement
Differentiation
English Language Learners (ELL):
v? Provide visual illustrations, such as diagrams or charts to depict the steps/procedure for simplifying equations.
v? Provide language support resources such as bilingual dictionaries to help ELL learners comprehend key vocabulary.
v? Pair ELL learners with English-speaking colleagues who can provide additional support and explanations (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Learners with special needs:
v? Provide simplified equations with fewer values/terms or smaller numbers for learners who could be having challenges with complicated equations.
v? Utilize manipulatives, such as number lines or counters, to help learners visualize and comprehend the process of simplification.
v? Offer additional time and support for exercises/practice and mastery of the idea/concept.
Learners with gifted abilities:
v? Provide additional challenges or advanced equations for learners who grasp the idea/concept quickly.
v? Motivate them to explore different approaches or extensions of simplifying equations, such as applying the distributive property in unique mechanisms (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Allow them to engage and discuss their mechanisms with other learners who could also be advanced in mathematics.
Early finishers:
v? Prepare extension exercises for early finishers to work on independently after completing the assigned tasks.
v? Provide additional practice challenges or puzzles linked with simplifying equations.
v? Offer access to online math games from DREAMBOX LEARNING (2023) or interactive resources that reinforce the concept of simplifying equations.
English Language Learners (ELL):
v? Utilize visual illustrations, such as diagrams or charts, to depict the steps/procedure for solving expressions with variables (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Provide sentence frames or sentence stems to facilitate ELL learners in detailing their problem-solving approaches.
v? Offer language support resources such as bilingual dictionaries to help ELL learners comprehend key vocabulary (Garnett, 1998).
Learners with special needs:
v? Provide simplified expressions with smaller numbers or fewer variables for learners who could find complex expressions challenging.
v? Utilize concrete examples or real-life instances to assist learners link the concept of solving expressions with variables to practical scenarios.
v? Provide additional support and time for practice and mastery of the concept/ideas, including one-on-one or small group instruction.
Learners with gifted abilities:
v? Provide additional challenges or advanced expressions for learners who grasp the concept quickly.
v? Motivate them to explore different problem-solving approaches or alternative mechanisms of solving expressions with variables.
v? Allow them to engage and discuss their strategies with other learners that are also advanced in mathematics (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Early finishers:
v? Prepare extension exercises for early finishers to work on independently after completing the assigned tasks.
v? Offer additional more challenging expressions or word problems that require multiple steps to resolve.
v? Provide access to online math games or interactive resources from DREAMBOX LEARNING (2023) that underline the concept of solving expressions with variables.
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English Language Learners (ELL):
v? Provide real-life word challenges/problems with simpler language and shorter sentences for ELL learners (Garnett, 1998).
v? Provide sentence frames or sentence stems to facilitate ELL learners in detailing and explaining their algebraic equations.
v? Motivate peer engagement and discussion to improve understanding and language development (Garnett, 1998; Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Learners with special needs:
v? Provide structured graphic organizers or templates to assist learners organize the data/information from real-life word challenges/problems and create algebraic equations.
v? Utilize concrete examples or visual aids to facilitate learners in comprehending the context of the word challenges/problems.
v? Provide additional support and time for practice and mastery of writing and resolving algebraic equations in real-life word challenges/problems.
Learners with gifted abilities:
v? Provide more challenging or multi-step real-life word problems that require higher-order thinking and problem-solving abilities.
v? Motivate learners to explore alternative strategies to solving the word problems or to create their unique real-life examples/instances.
v? Allow them to share and discuss their answers/responses with other learners that are also advanced in mathematics (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Early finishers:
v? Prepare extension exercises that further challenge early finishers in applying their knowledge of detailing and solving algebraic equations in real-life word challenges/problems.
v? Provide open-ended tasks where early finishers can develop their own word problems and challenge their classmates to resolve them.
v? Provide access to online math resources or platforms such as DREAMBOX LEARNING (2023) that offer a variety of real-life word challenges for additional practice and enrichment.
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Assessment of Content
Multiple Means of
Expression
Simplifying Equations
Visual Representation:
?? Utilize a whiteboard or chart paper to visually depict the steps/procedure for simplifying equations.
?? Provide examples of equations and frame the simplification procedure, underlining the distributive property and enjoining like terms.
?? Utilize different signs or colors to distinguish different terms/values and operations in the equations (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Example:
Equation: 2x + 3 - (x - 4) + 2x
Visual representation:
2x + 3 - (x - 4) + 2x
2x + 3 - x + 4 + 2x
Manipulatives:
?? Provide manipulatives such as algebra tiles or counters to depict the values/terms in equations.
?? Learners can physically move and cluster the tiles to simplify the equations.
?? Motivate learners to utilize the manipulatives to visualize the process of combining like values/terms.
Example:
Equation: 2x + 3 - (x - 4) + 2x
Manipulative depiction:
Group 2x tiles together
Combine the constant values/terms
Simplify the equation
Interactive Technology:
?? Use interactive technology instruments such as virtual manipulatives or online equation simplification exercises from DREAMBOX LEARNING (2023).
?? These instruments allow learners to interact with the equations, manipulate values/terms, and receive prompt response on their simplification procedure.
?? Provide links or access to online resources such as DREAMBOX LEARNING (2023) where learners can practice simplifying equations interactively.
Example:
Equation: 2x + 3 - (x - 4) + 2x
Interactive technology depiction:
Drag and drop the values/terms to simplify the equation
Receive prompt response on the correct simplification procedure/steps
Using the illustrations above, learners can break down the process of simplifying equations into clear steps. They are essential as they help them identify like terms, apply distributive property where needed, and combine terms to simplify the equation.
Formative and Summative Assessments:
Formative Assessment:
?? Provide various equations for learners to simplify.
?? Ask them to utilize the graphic organizer to illustrate their step-by-step process.
?? Observe their mechanism and provide responses/feedback on their comprehension of combining like terms and simplifying equations.
?? Utilize their responses to address any misconceptions or challenges (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Summative Assessment:
?? Create various equations that require simplification.
?? Ask learners to apply the steps from the provided illustrations and graphic organizer to simplify each equation.
?? Evaluate their ability to correctly combine like values/terms and simplify the equations.
?? Provide feedback/responses on their performance and utilize it to guide future instruction (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
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Solving Expressions with Variables
Visual Representation:
?? Utilize a whiteboard or chart paper to visually depict the procedure/steps for solving expressions with variables.
?? Provide examples of expressions and frame the procedure of isolating the variable utilizing inverse operations (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
?? Underline the procedure/steps involved, such as applying the opposite operation and simplifying each side of the equation.
Example:
Expression: 3x - 5 = 2x + 7
Visual representation:
3x - 5 = 2x + 7
3x - 2x = 7 + 5
x = 12
Using whiteboard or chart paper:
?? Provide real-life instances or examples that involve expressions with variables.
?? Associate the examples with the learners' interests or daily lives to improve engagement.
?? Ask learners to identify/find the variables, develop the equation, and solve for the variable utilizing the appropriate procedure/steps.
Example:
Expression: The cost of a ticket to a concert is $50. Each additional ticket costs $15. If a group of friends bought a total of x tickets and spent $250 in total, how many additional tickets did they buy?
Solution for the example:
Identify the variables and set up the equation: 50 + 15x = 250
Solve for x using inverse operations: x = 13
Online Demonstrations and Simulations:
?? Provide access to online illustrations or simulations that visually depict the process of solving expressions with variables (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
?? These resources often provide step-by-step guidance and interactive aspects to engage learners in the solving process.
?? Learners can follow along, practice, and receive prompt responses on their solutions.
Example:
Expression: 3x - 5 = 2x + 7
Online illustrations/simulation:
Follow the guided steps/procedure to solve the expression
Check the solution and receive response/feedback on the correctness
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Utilizing the above-depicted illustrations, learners can methodically resolve expressions with variables. Consequently, they aid in variable isolation, equation simplification, and solution determination (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Formative and Summative Assessments:
Formative Assessment:
?? Provide various expressions with variables for learners to solve.
?? Ask them to utilize the whiteboard or chart paper to illustrate their step-by-step process.
?? Observe their mechanism and provide response/feedback on their comprehension of isolating the variable and solving the expression.
?? Utilizing their responses to address any misconceptions or challenges.
Summative Assessment:
?? Develop a set of expressions with variables that need to be solved.
?? Ask learners to apply the steps from the whiteboard or chart paper to solve each expression.
?? Evaluate their ability to correctly isolate the variable and determine the solution (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
?? Provide response/feedback on their performance and utilize it to guide future instruction.
Writing/Detailing and Solving Real-Life Word Challenges/Problems
Graphic Organizers:
Provide templates or graphic organizers that guide learners in organizing the significant information from real-life word challenges/problems. Include sections for identifying variables, translating the challenge/problem, and solving the equation.
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Example Graphic Organizer:
Title: Real-Life Word Challenge/Problem: The Gas Station Dilemma
1. Problem Description:
?? Briefly detail the instance presented in the word challenge/problem.
?? Include relevant information and data (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
2. Identifying Variables:
?? Identify the variables involved in the challenge/problem and what they imply/meanings.
?? List the variables and their corresponding quantities or values.
3. Translating the Challenge/Problem:
?? Write/Detail an equation or expression that represents the problem mathematically.
?? Utilize the identified variables to develop an equation that models the scenario.
4. Solving the Equation:
?? Illustrate the step-by-step procedure of solving the equation to determine the solution.
?? Include any necessary calculations, simplifications or inverse operations (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
5. Solution:
?? State/outline the solution to the word challenge/problem in a concise manner.
?? Provide any relevant units of measurement or context.
Utilizing the above depicted graphic designer/illustrations, learners can systematically evaluate and resolve real-life word challenges. Consequently, it is essential for them in identifying the variables, translate the challenge/problem into an equation, and applying math operations to determine the solution.
Formative and Summative Assessments:
Formative Assessment:
?? Provide various real-life word challenges/problems associated with the topic (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
?? Ask learners to identify the variables, translate the challenges/problems into equations, and resolve them.
?? Observe learners' problem-solving approaches and provide feedback.
?? Utilize their responses to address any misconceptions or challenges.
Summative Assessment:
?? Develop a comprehensive real-life word challenge/problem that encompass various concepts covered.
?? Ask learners to identify variables, translate the challenge/problem, and solve the equation to determine the solution (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
?? Evaluate their ability to apply the learned skills and concepts in a real-world context.
?? Provide response/feedback on their performance and utilize it to guide future instruction.
Multiple Means of
Expression
Differentiation
English Language Learners (ELL):
v? Provide visual illustrations such as diagrams, charts, and labeled examples to facilitate comprehension.
v? Pair ELL learners with fluent English speakers or provide bilingual support if available.
v? Utilize sentence starters or sentence frames to assist ELL learners in detailing their steps and solutions (Garnett, 1998).
v? Offer additional practice tasks and avenues for ELL learners to strengthen their comprehension of simplifying equations.
Learners with Special Needs:
v? Provide a simplified category of equations with fewer variables or terms for learners struggling with complex equations (Garnett, 1998).
v? Break down the procedure of simplifying equations into smaller steps, providing visual prompts and illustrations.
v? Provide hands-on manipulatives, such as algebra tiles or counters, for learners who benefit from tactile learning.
v? Provide additional support and time from a special education educator or instructional assistant to facilitate understanding.
Learners with Gifted Abilities:
v? Provide challenging extension exercises or problems that require applying the idea of simplifying equations in more complex contexts.
v? Encourage independent exploration and discovery of alternative approaches/mechanisms of simplifying equations.
v? Provide avenues for learners to present their responses/solutions and explanations to the class, promoting deeper comprehension and collaboration.
Early Finishers:
v? Prepare additional online or worksheet exercises that offer more advanced equations or variations on the idea/concept of simplification.
v? Motivate early finishers to help their peers or engage with others who could need additional support (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Provide access to enrichment resources, such as math games or interactive online platforms available at DREAMBOX LEARNING (2023), to further engage and challenge early finishers.
English Language Learners (ELL):
v? Provide visual aids, such as diagrams or models, to facilitate comprehension of the steps/procedure for solving expressions with variables (Garnett, 1998).
v? Provide translation tools such as bilingual dictionaries for bilingual support to assist ELL learners in comprehending math terms and instructions.
v? Utilize graphic organizers or graphic representations to visually plan the steps involved in solving expressions with variables.
v? Provide sentence starters or sentence frames to guide ELL learners in detailing their approaches and solutions.
Learners with Special Needs:
v? Break down the procedure for solving expressions with variables into smaller steps and provide visual illustrations or prompts for each step (Garnett, 1998).
v? Provide concrete examples or real-life instances to assist learners comprehend the practical application of solving expressions with variables (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Provide additional practice challenges with varying levels of difficulty to underline the concept and develop confidence.
v? Allow the utilization of manipulatives or calculators as necessary to facilitate problem-solving and computation.
Learners with Gifted Abilities:
v? Provide challenging problems or instances that require critical thinking and applying the idea/concept of solving expressions with variables in novel approaches.
v? Encourage learners to explore multiple approaches for solving expressions with variables and justify their mechanisms.
v? Offer avenues for independent study or research into advanced topics linked to solving equations with variables.
v? Provide open-ended assignments that allow learners to create their expressions with variables and challenge their classmates to solve them.
Early Finishers:
v? Prepare additional online or worksheet tasks that offer more complex expressions with variables or avenues for extension.
v? Motivate early finishers to develop their expressions with variables and challenge their colleagues to solve them (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Provide access to online resources including math games that offer interactive exercises and avenues for further exploration.
v? Motivate early finishers to help their colleagues or engage with others who could need additional support or guidance.
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English Language Learners (ELL):
v? Provide visual illustrations, such as pictures or diagrams, to assist ELL learners comprehend the context of the real-life word challenges/problems (Garnett, 1998; Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
v? Utilize realia or physical items/objects associated with the problem's context to improve comprehension.
v? Break down complex word challenges/problems into simpler phrases or sentences to make them more accessible for ELL learners.
v? Offer translation tools and bilingual support such as bilingual dictionaries to help ELL learners in comprehending unfamiliar vocabulary or phrases in the word challenges/problems.
Learners with Special Needs:
v? Simplify the language utilized in real-life word challenges/problems, removing unnecessary information or complex sentence frameworks (Garnett, 1998).
v? Provide visual supports, such as graphic organizers or visual illustrations, to assist learners understand and outline the information in the word problems.
v? To enhance relevance and engagement, offer concrete examples or real-life instances that link with the learners' interests or experiences.
v? Provide step-by-step guidance on addressing and solving real-life word challenges/problems, utilizing visual prompts or checklists.
Learners with Gifted Abilities:
v? Provide open-ended and more challenging real-life word problems that require critical thinking and problem-solving abilities beyond the basic application.
v? Motivate learners to explore multiple solution approaches, considering different mechanisms and justifying their choices.
v? Provide avenues for learners to develop real-life word problems, challenging their classmates to resolve them.
v? Offer extension assignments that evaluate and analyze real-life word challenges/problems in different contexts or propose alternative resolutions.
Early Finishers:
v? Provide extra sets of real-life word challenges/problems at varying difficulty levels to challenge early finishers.
v? Motivate early finishers to develop their real-life word challenges and share them with classmates.
v? Provide access to online resources or math problem-solving applications offering various real-life word challenges/problems.
v? Offer avenues for early finishers to engage with their classmates or help other learners who could need extra support in resolving the word challenges (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b).
Extension Activity and/or Homework
Extension Activity and/or Homework
Extension Activity:
Assign learners additional equations to simplify after practicing simplifying equations utilizing the distributive property and combining like terms (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b). These equations can involve more complicated expressions or multiple variables. Consequently, this extension activity will challenge learners to apply their comprehension of simplification to more advanced equations.
Example:
Equation: 3(2x + 4) - 2(3x - 5)
Extension Activity:
Simplify the equation:
Notably, I will support and guide Fiona during the extension exercises, ensuring she has access to additional resources and individualized instruction as needed (Harvard Family Research Project, 2013).
Extension Activity:
Once learners have mastered and practiced solving expressions with variables utilizing inverse operations, provide them with more challenging expressions to resolve. These expressions can involve multiple variables or require additional procedures/steps to isolate the variable (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b). Consequently, this extension exercise/activity will allow learners to improve their problem-solving abilities and solidify their understanding of equation solving.
Example:
Expression: 2(3x - 5) + 4 = 3(x + 2) – 2
Extension Activity:
Solve the expression for x:
Notably, I will support and guide Fiona during the extension exercises, ensuring she has access to additional resources and individualized instruction as needed (Harvard Family Research Project, 2013).
Extension Activity:
Present more complex real-life scenarios for learners after they have mastered creating and resolving algebraic equations based on real-world word challenges or problems. These scenarios should require critical thinking and problem-solving abilities to find/identify the important information, translate it into an equation, and solve for the unknown (Indiana Department of Education, 2020; Paulsen, 2006; Paulsen, 2006b). Consequently, this extension exercise will challenge learners to apply their algebraic understanding to more challenging and realistic situations.
Example:
Real-Life Example/Scenario: A rectangular yard has a length that is 4 meters longer than twice its width. The perimeter of the yard is 38 meters. Find the dimensions of the yard.
Extension Activity:
Detail/write an algebraic equation to depict/represent the situation and solve it to find the dimensions of the yard:
Notably, I will support and guide Fiona during the extension exercises, ensuring she has access to additional resources and individualized instruction as needed (Harvard Family Research Project, 2013).
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References
DREAMBOX LEARNING. (2023). Teacher tools. DreamBox Learning - Online Math & Reading Solutions ?????? for Students K-12. https://www.dreambox.com/teachertools
Garnett, K. (1998). Math learning disabilities. Journal of CEC.
Harvard Family Research Project. (2013). Tips for administrators, teachers, and families: How to share data effectively. Tips for Administrators, Teachers, and Families: How to Share Data Effectively / ?????? Browse Our Publications / Publications & Resources / HFRP - Harvard Family Research Project. ??????? https://archive.globalfrp.org/publications-resources/browse-our-publications/tips-for-??? administrators-teachers-and-families-how-to-share-data-effectively
Indiana Department of Education. (2020). Indiana Academic Standards Mathematics: Algebra I. Working ???? Together for Student Success. https://www.in.gov/sboe/files/Math-Standards-2020.pdf
Paulsen, K. (2006). Algebra (Part 1): Applying Learning Strategies to Beginning Algebra. CASE STUDY UNIT. https://iris.peabody.vanderbilt.edu/wp-content/uploads/pdf_case_studies/ics_alg1.pdf
Paulsen, K. (2006b). Algebra (Part 2): Applying Learning Strategies to Intermediate Algebra. CASE ?? STUDY UNIT. https://iris.peabody.vanderbilt.edu/wp-??????? content/uploads/pdf_case_studies/ics_alg2.pdf