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FREE K-12 standards-aligned STEM

curriculum for educators everywhere!

Find more at TeachEngineering.org .

• TeachEngineering
• Problem Solving

Lesson Problem Solving

Grade Level: 8 (6-8)

(two 40-minute class periods)

Lesson Dependency: The Energy Problem

Subject Areas: Physical Science, Science and Technology

• Print lesson and its associated curriculum

Curriculum in this Unit Units serve as guides to a particular content or subject area. Nested under units are lessons (in purple) and hands-on activities (in blue). Note that not all lessons and activities will exist under a unit, and instead may exist as "standalone" curriculum.

• Energy Forms and States Demonstrations
• Energy Conversions
• Watt Meters to Measure Energy Consumption
• Household Energy Audit
• Light vs. Heat Bulbs
• Efficiency of an Electromechanical System
• Efficiency of a Water Heating System
• Solving Energy Problems
• Energy Projects

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Engineering connection, learning objectives, worksheets and attachments, more curriculum like this, introduction/motivation, associated activities, user comments & tips.

Scientists, engineers and ordinary people use problem solving each day to work out solutions to various problems. Using a systematic and iterative procedure to solve a problem is efficient and provides a logical flow of knowledge and progress.

• Students demonstrate an understanding of the Technological Method of Problem Solving.
• Students are able to apply the Technological Method of Problem Solving to a real-life problem.

Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN) , a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g. , by state; within source by type; e.g. , science or mathematics; within type by subtype, then by grade, etc .

Ngss: next generation science standards - science.

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International Technology and Engineering Educators Association - Technology

State standards, national science education standards - science.

Scientists, engineers, and ordinary people use problem solving each day to work out solutions to various problems. Using a systematic and iterative procedure to solve a problem is efficient and provides a logical flow of knowledge and progress.

In this unit, we use what is called "The Technological Method of Problem Solving." This is a seven-step procedure that is highly iterative—you may go back and forth among the listed steps, and may not always follow them in order. Remember that in most engineering projects, more than one good answer exists. The goal is to get to the best solution for a given problem. Following the lesson conduct the associated activities Egg Drop and Solving Energy Problems for students to employ problem solving methods and techniques. 

Lesson Background and Concepts for Teachers

The overall concept that is important in this lesson is: Using a standard method or procedure to solve problems makes the process easier and more effective.

The specific process of problem solving used in this unit was adapted from an eighth-grade technology textbook written for New York State standard technology curriculum. The process is shown in Figure 1, with details included below. The spiral shape shows that this is an iterative, not linear, process. The process can skip ahead (for example, build a model early in the process to test a proof of concept) and go backwards (learn more about the problem or potential solutions if early ideas do not work well).

This process provides a reference that can be reiterated throughout the unit as students learn new material or ideas that are relevant to the completion of their unit projects.

Brainstorming about what we know about a problem or project and what we need to find out to move forward in a project is often a good starting point when faced with a new problem. This type of questioning provides a basis and relevance that is useful in other energy science and technology units. In this unit, the general problem that is addressed is the fact that Americans use a lot of energy, with the consequences that we have a dwindling supply of fossil fuels, and we are emitting a lot of carbon dioxide and other air pollutants. The specific project that students are assigned to address is an aspect of this problem that requires them to identify an action they can take in their own live to reduce their overall energy (or fossil fuel) consumption.

The Seven Steps of Problem Solving

1.  Identify the problem

Clearly state the problem. (Short, sweet and to the point. This is the "big picture" problem, not the specific project you have been assigned.)

2.  Establish what you want to achieve

• Completion of a specific project that will help to solve the overall problem.
• In one sentence answer the following question: How will I know I've completed this project?
• List criteria and constraints: Criteria are things you want the solution to have. Constraints are limitations, sometimes called specifications, or restrictions that should be part of the solution. They could be the type of materials, the size or weight the solution must meet, the specific tools or machines you have available, time you have to complete the task and cost of construction or materials.

3.  Gather information and research

• Research is sometimes needed both to better understand the problem itself as well as possible solutions.
• Don't reinvent the wheel – looking at other solutions can lead to better solutions.
• Use past experiences.

4.  Brainstorm possible solutions

List and/or sketch (as appropriate) as many solutions as you can think of.

5.  Choose the best solution

Evaluate solution by: 1) Comparing possible solution against constraints and criteria 2) Making trade-offs to identify "best."

6.  Implement the solution

• Develop plans that include (as required): drawings with measurements, details of construction, construction procedure.
• Define tasks and resources necessary for implementation.
• Implement actual plan as appropriate for your particular project.

7.  Test and evaluate the solution

• Compare the solution against the criteria and constraints.
• Define how you might modify the solution for different or better results.
• Egg Drop - Use this demonstration or activity to introduce and use the problem solving method. Encourages creative design.
• Solving Energy Problems - Unit project is assigned and students begin with problem solving techniques to begin to address project. Mostly they learn that they do not know enough yet to solve the problem.
• Energy Projects - Students use what they learned about energy systems to create a project related to identifying and carrying out a personal change to reduce energy consumption.

The results of the problem solving activity provide a basis for the entire semester project. Collect and review the worksheets to make sure that students are started on the right track.

Learn the basics of the analysis of forces engineers perform at the truss joints to calculate the strength of a truss bridge known as the “method of joints.” Find the tensions and compressions to solve systems of linear equations where the size depends on the number of elements and nodes in the trus...

Through role playing and problem solving, this lesson sets the stage for a friendly competition between groups to design and build a shielding device to protect humans traveling in space. The instructor asks students—how might we design radiation shielding for space travel?

A process for technical problem solving is introduced and applied to a fun demonstration. Given the success with the demo, the iterative nature of the process can be illustrated.

The culminating energy project is introduced and the technical problem solving process is applied to get students started on the project. By the end of the class, students should have a good perspective on what they have already learned and what they still need to learn to complete the project.

Hacker, M, Barden B., Living with Technology , 2nd edition. Albany NY: Delmar Publishers, 1993.

Other Related Information

This lesson was originally published by the Clarkson University K-12 Project Based Learning Partnership Program and may be accessed at http://internal.clarkson.edu/highschool/k12/project/energysystems.html.

Contributors

Supporting program, acknowledgements.

This lesson was developed under National Science Foundation grants no. DUE 0428127 and DGE 0338216. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: August 16, 2023

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Problem solving

Problem solving lesson plan

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Problem Solving: Lesson plan

Problem solving: Presentation slides

Demonstrating your skills quick fire activity

Problem solving in practice: Interactive worksheet

Our problem solving content focuses on one of these skills and develops understanding of the six stages of problem solving, as well as identifying different types of situations in which young people might already be using these skills. Furthermore, it encourages them to use an adaptive approach, explaining that different types of problems can be approached in different ways.

The activities on this page support your teaching of these skills through an independent activity, quick activities or a full length, curriculum-linked lesson plan. 

Teaching resources:

• Problem solving: Lesson plan and presentation slides – full lesson plan including icebreaker for use with a group of students in the classroom
• Demonstrating your skills: Quick-fire activity  – 10 minute activity for a group of students in the classroom, can be used as an icebreaker for the lesson plan
• Problem solving in practice: Interactive worksheet – activity for independent learning whether remote or in class

Lesson plan

(60 -75 minutes)

This lesson is designed to equip young people with an adaptable approach to solving problems, large or small. It includes a short film and scenarios that encourage development of practical problem solving skills which can be useful for learning, day to day life, and when in employment.

By the end of the lesson, students will be able to:

• Identify problems of different scales and what is needed to solve them
• Illustrate the use of an adaptable approach to solving problems
• Understand that problem solving is a core transferable skill and identify its usefulness in a work setting
• Work on a problem solving activity in a team

The lesson aims to reinforce students’ understanding of the potential future applications of this skill as they move into the world of work, particularly in an activity differentiated for an older or more able group on creating new opportunities.

Quick-fire activity

(5 - 10 minutes)

The demonstrating your skills quick-fire activity focuses on helping young people understand the key skills that are needed in the workplace, including the importance of problem solving.

Students will be asked to name the skills being demonstrated in a variety of scenarios, and identify ways they’re already using those skills in this short activity.

You might find it useful as a starter or icebreaker activity to begin a lesson, or at the end to allow students to put what they have just learnt in the Problem solving lesson into practice.

Interactive worksheet

(20 - 25 minutes)

Please note that students below the age of 14 cannot sign up for their own LifeSkills account. Any independent tasks must be printed or downloaded and provided digitally for them to complete as they are currently hosted on educator pages.

The Problem solving in practice interactive worksheet introduces some of the themes from the full lesson plan and gives students some practical strategies for problem solving, including introducing the six stages of problem solving. The worksheet can be printed or completed digitally, so can be used flexibly to give students practise putting their problem solving skills into action. You might choose to assign it:

• As homework following the Problem solving lesson
• For independent study
• For remote learning

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Other lessons that may prove useful for students to build on these activities include the  Adaptability  and  Innovation and idea generation  lessons. Alternatively, consider encouraging them to apply their skills through  Steps to starting a business  or the  Social action toolkit .

Why not build problem solving in as a focus in your students’ wider curriculum? Refer to our  Content guide to find out how this resources can be used as part of your teaching.

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Social action toolkit

Build a comprehensive social action programme and support young people to access enriching experiences that build transferable skills for work.

Think Like an Engineer: Solving Problems from Start to Finish

What you will learn.

Welcome to our lesson on planning a fun summer vacation! By the end of this lesson, you will be able to explain why it’s important to work together to solve problems, just like engineers do. You will also learn how to gather ideas from friends, design a project, and test your creations to see how well they work. Finally, you’ll understand how to improve your designs based on what you find out during testing. Get ready to think creatively and have fun while learning!

Let's Think

• What problem did Squeaks and his friends need to solve for Juniper?
• How did the friends work together to come up with a solution for the missing boat?
• What materials did they use to build the raft, and why were they chosen?

Lesson Article

Planning a fun summer vacation.

Hey there! We’re getting ready for a big summer vacation. All our friends are going to meet up and swim at the lake. Everyone will be there, including Juniper. It’s going to be so much fun!

The Missing Toy Boat

Uh-oh! Squeaks went to find his toy boat because Juniper was going to ride on it at the lake. But it seems like the boat is missing. Squeaks thinks he might have lent it to someone who didn’t return it. This is a big problem because Juniper can’t go out on the lake without a boat.

Thinking Like Engineers

We don’t want Juniper to miss out on the fun, so we decided to think like engineers to solve this problem. Engineers are great at coming up with solutions. We thought about building a simple boat, like a raft, for Juniper.

Gathering Information

Before building, engineers gather information. We talked to our friends to get ideas. Dino suggested making Juniper fly across the lake, but we decided to focus on building a raft instead. Bill and Webb helped us realize that the raft needs to be big enough and have a handle for safety.

Designing the Raft

We found some materials around the fort: a kitchen sponge, craft sticks, cardboard, paper, modeling clay, aluminum foil, corks, and a big flat rock. We tested these materials to see which ones float. The craft sticks, corks, cardboard, and aluminum foil floated well, so we decided to use them for our raft designs.

Creating Our Designs

Each of us came up with a design for the raft. Juniper designed a raft with aluminum foil, Squeaks used craft sticks with a handle, and Sam used cardboard with corks and a sail. We needed to test these designs to see which one worked best.

Testing the Prototypes

We built prototypes of our designs and tested them. Juniper’s raft floated but wasn’t stable. Squeaks’ raft floated well but was too small. Sam’s raft floated at first but got soggy over time. None of the rafts were perfect, so we needed to improve them.

Improving Our Design

We combined the best ideas from each design. We used craft sticks for sturdiness, made the raft wide and flat for space, and added sides for safety. This new design was ready for testing.

The Final Raft

Our final raft design floated, was big enough for Juniper, and kept her safe even if there were waves. We worked together like engineers to solve the problem, and now we’re ready for our exciting vacation!

Unexpected Surprise

Just as we finished, Dr. Turtleman showed up with Squeaks’ missing boat! He had borrowed it for some experiments. Even though we didn’t need the raft anymore, we had a lot of fun thinking like engineers and solving the problem together.

Thanks for joining us on this adventure! If you want to have more fun with us, you can subscribe to our channel. See you next time at the fort!

Discussion Questions

• What do you think it means to “think like an engineer”? Can you think of a time when you had to solve a problem by coming up with a creative solution?
• Have you ever lost something important, like Squeaks did with his toy boat? How did you feel, and what did you do to find it or solve the problem?
• If you were to design your own raft, what materials would you use and why? Can you think of any other objects around your home that might float?

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Spark discussion, perseverance, and enjoyment of mathematics..

IM 6–8 Math is a problem-based core curriculum rooted in content and practice standards to foster learning and achievement for all. Students learn by doing math, solving problems in mathematical and real-world contexts, and constructing arguments using precise language. Teachers can shift their instruction and facilitate student learning with high-leverage routines that guide them in understanding and making connections between concepts and procedures.

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This lesson supports students in identifying problems and creating a guide to solve problems on individual, local, and global levels.

Step 1 - Inquire: Students play the Knot Game and reflect on their collective ability to solve the problem at hand.

Step 2 - Investigate: Students learn about the ozone layer, identify the solutions, and brainstorm general strategies that are effective for solving problems.

Step 3 - Inspire: Students create a problem-solving guide to address a climate change-related problem.

Learning Outcomes

Students will be able to:

• Analyze and reflect on problems and solutions.
• Take action to address individual, local, or global problems.

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Lesson Plan

Problem-solving steps, view aligned standards, learning objectives.

Students will be able to identify a problem and engage in problem-solving steps to come up with a solution to strengthen their responsible decision-making competency.

Introduction

• Bring students together in a circle, either seated or standing.
• Bring blocks with you to the circle.
• Show the student the blocks and ask them to watch you build a tall castle.
• After you build it, bring out two figurines that you would like to play with in the castle.
• Say out loud, "Hmm....there seems to be a problem. This castle is too small for my toys. How can I make this bigger?"
• Take a few big breaths into your belly and ask the class to breathe with you.
• Explain that deep breathing can help you stay calm and focused when solving a problem.
• Ask the class, "How can I make this castle bigger?"
• Ask, "Would someone like to come up and think of a new way to solve this problem?"
• Invite one student to come up to solve the problem (i.e., make a larger castle).
• Ask your student volunteer, "What would you do here? How would you test a solution?"
• Give the student time to come up with a solution to make a larger castle.
• Identify the problem.
• Look at what solutions have already been tried.
• Think of new ways to solve the problem.
• Try it out!
• Thank your student volunteer for showing you how to solve the problem and send them back to their seats.
• Say, "All of us sometimes come across problems we have to solve, and sometimes friends can help us, too!"

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William: The Problem Solver (KS3 Roman to Norman Britain 6/7)

Subject: History

Age range: 11-14

Resource type: Lesson (complete)

Last updated

19 August 2024

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This lessons explores the tactics used by William the Conqueror to secure control following the Battle of Hastings in 1066 and encourages students to consider the wider dynamics of problem solving in general.

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Roman to Norman Britain (7 lessons with end-of-topic assessment)

7 lessons exploring Roman, Anglo Saxon and Norman Britain aimed at KS3, with end of unit assessment and mark scheme .

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enVision Mathematics - Middle School Math Curriculum

enVision® Mathematics offers a unique combination of problem-based learning and visual learning to develop conceptual understanding. This instructional model has been consistently successful and effective for middle school students across the nation.

• Motivate students through ownership of their learning with Let’s Investigate!
• Harness storytelling for instruction with engaging 3-Act Math problems
• Leverage student interests with Pick a Project and enVision STEM Project’s varied contexts, modalities, and final products
• Easy accessibility to meaningful digital content on the award-winning Savvas Realize® LMS

Grades 6-8 Math Curriculum Built for Success

Set students up for success in your class and beyond with a math curriculum that meets today’s challenges.

Student-Centered Mathematics

enVision ’s 3-Act Math, Let’s Investigate!, and Pick a Project components connect mathematical thinking to familiar real world scenarios so students stay engaged.

Personalized and Adaptive Learning

Formative and summative assessments plus tools like MathXL® for School practice and enrichment and Savvy™ Adaptive Practice tailor assignments and content to each student’s interests and learning level.

Monitor and Support Student Understanding

Assess students’ progress, customize content, and reach or exceed state standard proficiency through the Savvas Realize® platform.

Comprehensive and Flexible Planning Materials

Editable lesson presentation slides allow teachers to present content and engage each student with customized content relevant to the students’ world around them.

Middle School Math Curriculum Built for Students, Teacher, and Families

Unique and innovative lessons, motivating student projects, enlightening interactivities powered by desmos™, supportive professional development, family engagement resources.

• Problem-Based Learning Real-world math problems foster collaboration skills. By evaluating options and presenting their own solutions, students stay engaged throughout the lesson.
• Productive Struggle Let’s Investigate! and 3 Act Math Modeling lessons allow students to experience productive struggle through inviting problem solving.

• Real-World Application Flexible student activities such as enVision STEM and Pick a Project provide opportunities for students to explore math concepts with real world application.

• Concept Visualization Embedded interactivities throughout lessons help students visualize concepts

• Instructional Support and Insight Professional development videos, such as Using Manipultives videos, provide valuable instructional support and insight into student learning.

• Family-Friendly Support Every topic and lesson comes with family-friendly support that offers compatibility with Google Translate so that families can access resources in 300+ languages.

Award-Winning Digital Lesson Support

Savvas Realize® provides access to all the enVision Mathematics Grades 6-8 program’s digital resources and downloadable, editable print materials to meet every educational standard.

Offline Accessibility

Learning does not stop when students have no internet access. enVision ensures access to resources offline, automatically updating their work when reconnected!

Further Enhancements to the enVision 6-8 Math Program

• Bilingual Support
• Meet Your Students Where They Are
• Personalized Programs

Embedded Spanish-Language Materials

Matemáticas is built with comprehensive program resources that support students who learn in Spanish. Fully integrated within the 6-8 courseware, resources include Spanish text, audio, and video. Easy-to-navigate content is fully customizable. All English and Spanish assets are provided in one course, so teachers and students do not have to toggle between multiple locations. Empower students with the ability to think and communicate mathematically in Spanish.

SuccessMaker® Math helps learners at every level

This adaptive intervention program continuously personalizes math instruction for student growth or differentiation.

Momentum Screener & Diagnostic + Growth - Math

An easy and reliable way to identify student needs, assign the right content, and measure growth, delivered on Savvas Realize®.

School Stories

In these inspirational stories, you'll learn about what schools and districts from across the country are doing to help students succeed and shape the future of education.

Frequently Asked Questions

enVision® Mathematics © 2024 for grades 6-8 is the only middle grades math program that combines problem-based learning and visual learning to deepen students’ conceptual understanding. enVision is used by classrooms across the country and around the world. The latest enVision is even better with new digital Let’s Investigate! lessons which provide students with opportunities to take ownership of deeper exploration into problem-based learning. Ensure successful implementation with the comprehensive teacher support based on the 5 Practices.

enVision packs a unique one-two punch. Lessons start with Problem-Based Learning (PBL), where students must think critically about a real-world math problem, evaluate options, collaborate, and present solutions. This is followed by Visual Learning to solidify the underlying math concepts. It’s the best way to help kids better understand math ideas.

The program is made up of the following program components:

• Teacher’s Edition - Available in digital or print, the Teacher’s Edition includes wrap-around pages that provide direct instruction and teaching suggestions to engage students. The Interactive Teacher’s Edition online features annotation models and downloadable lesson resources.
• Student Edition - Interactive Student Edition—available in digital or print write-in format.
• enVision Digital - enVision digital courseware on Savvas Realize® includes robust digital tools that give teachers flexibility to use a digital, print, or blended format in their classrooms. Teachers can customize the program to rearrange content, upload their own content, add links to online media, and edit resources and assessments. All program resources, including personalized practice, remediation, and assessments are available in one location for easy lesson planning and presentation Students will use technology to interact with text and activities, and they can write directly in their digital Student Edition to make interaction with text more meaningful. Students will engage in activities that will inspire conceptual understanding, classroom discourse, and build their mathematical thinking skills, while learning to formulate and defend their own opinions.

The learning model in the enVision program—problem-based learning, visual learning, and data-driven differentiated instruction—has been researched and verified as effective. Core instruction used for every lesson has been shown to be effective for developing conceptual understanding.

enVision Mathematics features comprehensive differentiated instruction and intervention support to allow access for all students. The program’s balanced instructional model provides appropriate scaffolding, differentiation, intervention, and support for a broad range of learners, and is designed to facilitate conceptual understanding of mathematics for students at a range of learning levels.

Comprehensive, built-in differentiation resources support all levels of learners, including those with learning disabilities and ELLs, through personalized, adaptive learning. The program meets a variety of student needs and provides Response to Intervention (RtI) during each lesson, at the end of each lesson, at the end of each Topic, and any time as indicated in the Teacher’s Edition. A description of RtI tiered instructional resources for the program is included in the Teacher’s Program Overview for each grade. The following are examples of tiered instructional support found online for each lesson.

Tier 1 ongoing Intervention includes the following resources that can be used during the lesson:

• Prevent Misconceptions. During the Visual Learning Example, a remediation strategy is included to address a common misconception about the lesson concept.
• Error Intervention (If... Then...). During Practice & Problem Solving, error intervention identifies a common error and provides remediation strategy
• Reteaching Set. This set is provided before independent practice to develop understanding prior to practice.
• MathXL for School: Practice & Problem Solving, during the lesson, includes personalized practice for the Practice & Problem Solving portion of the lesson, along with Additional Practice or Enrichment; auto‐scored with on‐screen help, including Help Me Solve This and View an Example tools, tutorial videos, Math Tools, and one‐click animated glossary access.

Tier 2 strategic intervention includes the following resources that can be used at the end of the Lesson:

• Intervention Activity. This supports teachers working with small groups of struggling students.
• Reteach to Build Understanding. This provides guided reteaching as a follow‐up to the intervention activity.

Tier 3 intensive intervention instruction is delivered daily outside of the core math instruction, often in a one‐to‐one situation. The Math Diagnosis and Intervention System can be used for this purpose, for example.

• Variety of Instructional Strategies
• Multisensory instruction is provided in online Solve & Discuss It!/Explore It!/Explain It! activities that include audio, Visual Learning
• Animation Plus, Virtual Nerd videos, interactive MathXL for School: Practice & Problem Solving, Additional Practice, and Enrichment, online digital math tools, and online math games.

The authorship team is made up of respected educational experts and researchers whose experiences working with students and study of instructional best practices have positively influenced education. Contributing to enVision with a mind to the evolving role of the teacher and with insights on how students learn in a digital age, these authors bring new ideas, innovations, and strategies that transform teaching and learning in today’s competitive and interconnected world.

• Dr. Robert Q. Berry, III is an Associate Professor at the University of Virginia in the Curry School of Education with an appointment in Curriculum Instruction and Special Education. A former mathematics teacher, he teaches elementary and special education mathematics methods courses in the teacher education program at the University of Virginia. Additionally, he teaches a graduate level mathematics education course and courses for in-service teachers seeking a mathematics specialist endorsement.
• Zachary Champagne taught elementary school students in Jacksonville, Florida for 13 years. Currently he is working as an Assistant in Research at the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCRSTEM) at Florida State University.
• Dr. Randall Charles is Professor Emeritus in the Department of Mathematics at San Jose State University, San Jose, California. His research interests have focused on problem solving with several NCTM publications including Teaching and Assessing Problem Solving, How to Evaluate Progress in Problem Solving, and Teaching Mathematics Through Problem Solving. In recent years Dr. Charles has written and talked extensively on Big Ideas and Essential Understandings related to curriculum, teaching, and assessment.
• Francis (Skip) Fennell, PhD, is emeritus as the L. Stanley Bowlsbey professor of education and Graduate and Professional Studies at McDaniel College in Maryland, where he continues to direct the Brookhill Institute of Mathematics supported Elementary Mathematics Specialists and Teacher Leaders Project. A mathematics educator who has experience as a classroom teacher, principal, and supervisor of instruction, he is a past president of the Association of Mathematics Teacher Educators (AMTE), the Research Council for Mathematics Learning (RCML), and the National Council of Teachers of Mathematics (NCTM).
• Eric Milou is a Professor in the Department of Mathematics at Rowan University in Glassboro, NJ. He is an author of Teaching Mathematics to Middle School Students. Recently, his focus has been on approaches to mathematical content and the use of technology in middle grades classrooms.
• Dr. Jane Schielack is Professor Emerita in the Department of Mathematics and a former Associate Dean of Assessment and PreK-12 Education in the College of Science at Texas A&M University. A former elementary teacher, Dr. Schielack has pursued her interests in working with teachers and students to enhance mathematics learning in the elementary and middle grades. She has focused her activities for improving mathematics education in two main areas: teacher education and professional development and curriculum development.
• Jonathan Wray has involvement and leadership in a number of organizations and projects. His interests include the leadership roles of mathematics coaches/specialists, access and equity in mathematics classrooms, the use of engaging and effective instructional models to deepen student understanding, and the strategic use of technology in mathematics to improve teaching and learning.
• How do I sign up for an enVision digital demo? enVision digital courseware on Savvas Realize® includes robust digital tools that give teachers flexibility to use a digital, print, or blended format in their classrooms. Teachers can customize the program to rearrange content, upload their own content, add links to online media, and edit resources and assessments. Program resources, personalized practice, remediation, and assessments are available in one location for easy lesson planning and presentation.

enVision Mathematics is designed to achieve a coherent progression of mathematical content within each course and across the program, building lesson to lesson. Every lesson includes online practice instructional examples as the progression of topics builds, allowing students additional practice with these skills and to develop a deeper conceptual understanding.

At the beginning of every topic, teachers are provided with support for the focus of the topic, how the topic fits into an overall coherence of the grade and across grades, the balance of rigor in the topic, and how the practices enrich the mathematics in the topic. Carefully designed learning progressions achieve coherence across grades:

Coherence is supported by common elements across grades, such as Thinking Habits questions for math practices and diagrams for representing quantities in a problem. Coherence across topics, clusters, and domains within a grade is the result of developing mathematics as a body of interconnected concepts and skills. Across lessons and standards, coherence is achieved when new content is taught as an extension of prior learning—developmentally and mathematically. (For example, Solve & Share at the start of lessons engages students in a problem-based learning experience that connects prior knowledge to new ideas.)

Look Back! and Look Ahead! connections are highlighted in the Coherence part of Topic Overview pages in the Teacher’s Edition.

The Topic Background: Rigor page shows teachers how the areas of rigor will be addressed in the topic, and details how conceptual understanding, procedural skill and fluency, and application builds within each topic to provide the rigor required.

On the first page of every lesson, the Lesson Overview includes sections titled Focus, Coherence, and Rigor. The Rigor section highlights the element or elements of rigor emphasized in the lesson, which may be one, two, or all three. Features in every lesson support each element, but the emphasis will vary depending on the standard being developed in the lesson. The core instructional model features support for conceptual understanding, procedural fluency, and application during both instruction and practice, as described below.

• Problem-Based Learning Step 1 Problem-Based Learning supports coherence by helping students connect what they already know to a problem in which new math ideas are embedded. When students make these connections, conceptual understanding emerges. Students are given time to struggle to make connections to the mathematical ideas and conceptual understandings. They can choose to represent their thinking and learning in a variety of ways. Physical and online manipulatives are available.
• Visual Learning Step 2 Visual Learning further develops understanding of the lesson ideas through classroom conversations. The Visual Learning Example features visual models to help give meaning to math language. Instruction is stepped out to help students visually organize important ideas. Students perform better on procedural skills when the procedures make sense to them. Procedural skills are developed through careful learning progressions in the Visual Learning Example.
• Assess and Differentiate Step 3 Assess and Differentiate features a Lesson Quiz and a comprehensive array of intervention, on-level, and advanced resources for all learners, with the goal that all students have the opportunity for extensive work in the state standards. Leveled practice with scaffolding is included at times. Varied problems are provided and math practices are identified as appropriate. Higher Order Thinking problems offer more challenge. Students have ample opportunity to focus on conceptual understanding and procedural skills and to apply the mathematics they just learned to solve a range of problems.
• How does the program identify performance gaps? At the start of the school year, schools have the opportunity to implement norm-referenced and validated assessments to identify students’ strengths and areas for growth. The new award-winning Momentum Screener & Diagnostic + Growth - Math work directly with the enVision Mathematics course on Savvas Realize to inform instruction and provide robust student data. As a result of the Diagnostic assessment, teachers are armed with flexible instructional recommendations personalized to every student.

enVision Mathematics portrays diverse individuals and groups in a variety of settings and backgrounds. The program has been reviewed and approved for unbiased and fair representation. The selections in enVision Mathematics include a wide variety of contemporary, classic, and multicultural authors.

Our educational materials feature a fair and balanced representation of members of various cultural groups, including racial, ethnic, and religious groups; males and females; older people; and people with disabilities. The program integrates social diversity throughout all of its lessons, and includes a balanced representation of cultures and groups in multiple settings, occupations, careers, and lifestyles. We strive to accurately portray diverse groups within our society as well as diversity within groups. Our programs use language that is appropriate to and respectful of our cultural diversity. We involve members of diverse ethnic and cultural groups in the concept development of our products as well as in the writing, editing, illustration, and design.

• What is Pick a Project? Pick a Project is one of the motivating activities in enVision Mathematics , giving students a choice by letting them pick from a selection of math projects. Pick a Project launches each enVision topic and engages students in a real-world math project that accommodates different learning styles and interests. Students work independently, with a partner, or in small groups. The math problem activates prior knowledge and is a great way to deepen understanding during the entire topic.
• How does the relationship between enVision Mathematics and Desmos benefit students? Exclusive integration of Desmos into Savvas Realize® offers a groundbreaking interactive experience designed to foster conceptual understanding through highly visual interactives that bring mathematical concepts to life. Embedded interactives powered by Desmos and animated examples engage students and deepen conceptual understanding. Allowing students to manipulate data and see an immediate effect on graphs, number lines, etc. clarifies concepts as students are learning new content. Unique to enVision , the Desmos best-in-class graphing calculator and brand new geometry tools are available to middle and high school enVision students anytime, anywhere, both online and offline through Savvas Realize.

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Curriculum  /  Math  /  8th Grade  /  Unit 2: Solving One-Variable Equations  /  Lesson 6

Solving One-Variable Equations

Lesson 6 of 12

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Solve equations with variables on both sides of the equal sign.

Common Core Standards

Core standards.

The core standards covered in this lesson

Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.

Foundational Standards

The foundational standards covered in this lesson

7.EE.B.4 — Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Understand why solving an equation with variables on both sides of the equal sign follows the same rules as solving an equation with variables on one side.
• Use inverse operations to solve equations with variables on both sides of the equal sign.
• Analyze and fix incorrect solutions to equations. 

Suggestions for teachers to help them teach this lesson

• Solving equations with variables on both sides of the equal sign is new for 8th graders. Though the approach to solving these equations (simplifying, using inverse operations) is the same as what students are used to, there are now more ways that an equation can be solved and additional places where errors may be made (such as combining terms across the equal sign).
• Continue to emphasize the importance of maintaining balance in the equation through each “move” students make. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

25-30 minutes

The equation  $$2x+3=7$$  can be shown using a balance, like below, where each bag contains an unknown number of tiles,  $$x$$ .

Solving this equation can be shown using the picture by keeping the balance level. The solution below shows  $$x=2$$ .

a.   Solve $$5x+1=2x+7$$  in two ways: symbolically, the way you usually do with equations, and also with pictures of a balance. Show how each step you take symbolically is shown in the pictures.

b.   Solve the equation $$4x=x+1$$ using pictures and symbols. Discuss any issues that arise.

Guiding Questions

Solving Equations , accessed on Aug. 31, 2017, 1:47 p.m., is licensed by Illustrative Mathematics under either the  CC BY 4.0  or  CC BY-NC-SA 4.0 . For further information, contact Illustrative Mathematics .

The equation below is solved incorrectly three different ways. 

For each solution, find and explain the error(s) that occurred. Then find the correct solution for $$y$$ .

$${12-4y=7y+10}$$

What value for $$x$$ would make the following equation true?

$$20-(3x-9)-2=-(-11x+1)$$

Grade 8 Mathematics > Module 4 > Topic A > Lesson 6 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

A set of suggested resources or problem types that teachers can turn into a problem set

15-20 minutes

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

Three expressions are shown below.

Using the expressions in the chart, find the value of  $$x$$  in each situation described.

a.   A = C

b.   A + B = C

Student Response

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• EngageNY Mathematics Grade 8 Mathematics > Module 4 > Topic A > Lesson 6 — Exercises and Problem Set; do not include problems with "no solution"; spread these problems out over Lesson 6 and Lesson 7
• Open Middle Create an Equation with a Given Solution
• Open Middle Multi-Step Equations - Positive (or Negative) Solution
• Open Middle Multi-Step Equations - Smallest (or Largest) Solution — Challenge
• Kuta Software Free Algebra 1 Worksheets Equations — Multi-Step Equations
• EngageNY Mathematics Grade 8 Mathematics > Module 4 > Topic A > Lesson 4 — Exercises and Problem Set

Topic A: Simplifying Expressions and Verifying Solutions

Write equivalent expressions using properties of operations and verify equivalence using substitution.

Define a solution to an equation. Solve and check solutions to 1 and 2 step equations.

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Topic B: Analyzing and Solving Equations in One Variable

Justify each step in solving a multi-step equation with variables on one side of the equation.

8.EE.C.7.A 8.EE.C.7.B

Write and solve multi-step equations to represent situations, with variables on one side of the equation.

Model with equations using a three-act task.

Write and solve multi-step equations to represent situations, including variables on both sides of the equation.

Understand that equations can have no solutions, infinite solutions, or a unique solution; classify equations by their solution.

Solve and reason with equations with three types of solutions.

Use equations to model a business plan and determine the break-even point.

Topic C: Analyzing and Solving Inequalities in One Variable

Solve and graph inequalities with variables on one side of the inequality (optional).

Solve and graph inequalities with variables on both sides of the inequality (optional).

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