lesson 7 problem solving

Curriculum  /  Math  /  7th Grade  /  Unit 6: Geometry  /  Lesson 7

Lesson 7 of 21

Criteria for Success

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

Solve real-world and mathematical problems using the relationship between the circumference of a circle and its diameter. 

Common Core Standards

Core standards.

The core standards covered in this lesson

7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Foundational Standards

The foundational standards covered in this lesson

Measurement and Data

4.MD.A.3 — Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

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

• Know the formula that relates the circumference and diameter of a circle:  $${{C=\pi d}.}$$
• Use the formula  $${C=\pi d}$$ to solve problems.
• Understand that the distance around a closed semi-circle is half of the circumference added to the diameter of the circle.

Suggestions for teachers to help them teach this lesson

Lessons 6 and 7 focus on the relationship between a circle’s circumference and its diameter. In Lesson 7, students solve real-world and mathematical problems involving circumference, including problems involving semi-circles.

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

Basil saw a strange old bicycle at the museum. It had one very big wheel and one very small one. 

a.   Using the measurements that Basil found, what is the circumference of the big wheel?

b.   How far would you travel in one turn of the big wheel? Give your answer in feet and inches.

c.   How many times must the cyclist turn the big wheel to travel 1 mile? A mile is 1,760 yards. Give your answer to the nearest 10 turns.

d.   How many times does the small wheel turn when the cycle travels 1 mile?

Guiding Questions

Historic Bicycle  from the Summative Assessment Tasks for Middle School is made available through the Mathematics Assessment Project  under the  CC BY-NC-ND 3.0  license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed March 10, 2018, 11:11 a.m..

Two figures are shown below. Figure A is a semi-circle, and Figure B is composed of a square and two semi-circles. 

Find the distance around each figure. 

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

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

New plants and bushes are to be planted along a path that goes around a park. The shape of the park is shown below. What is the distance around the park?

Student Response

An example response to the Target Task at the level of detail expected of the students.

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.

• Inside Mathematics Performance Assessment Tasks Grades 3-High School Which is Bigger — Grade 7
• EngageNY Mathematics Grade 7 Mathematics > Module 3 > Topic C > Lesson 16 — Problem Set #4–7
• Open Up Resources Grade 7 Unit 3 Practice Problems — Lesson 4
• Smarter Balanced Assessment Consortium: Sample Items Item MAT.07.ER.2.0000G.A.295

Topic A: Angle Relationships

Identify and determine values of angles in complementary and supplementary relationships.

Use vertical, complementary, and supplementary angle relationships to find missing angles. 

Use equations to solve for unknown angles. (Part 1)

Use equations to solve for unknown angles. (Part 2)

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Topic B: Circles

Define circle and identify the measurements radius, diameter, and circumference. 

Determine the relationship between the circumference and diameter of a circle and use it to solve problems. 

Determine the relationship between the area and radius of a circle and use it to solve problems.

Solve real-world and mathematical problems using the relationship between the area of a circle and its radius.

Solve problems involving area and circumference of two-dimensional figures (Part 1).

7.G.B.4 7.G.B.6

Solve problems involving area and circumference of two-dimensional figures (Part 2).

Topic C: Building Polygons and Triangles

Draw two-dimensional geometric shapes using rulers, protractors, and compasses. 

7.G.A.2 7.G.B.5

Determine if three side lengths will create a unique triangle or no triangle. 

Identify unique and identical triangles. 

Determine if conditions describe a unique triangle, no triangle, or more than one triangle.

Topic D: Solid Figures

Identify and describe two-dimensional figures that result from slicing three-dimensional figures.

Find the surface area of right prisms.

 Find the surface area of right pyramids.

Find the volume of right prisms and pyramids.

Solve real-world and mathematical problems involving volume.

Distinguish between and solve real-world problems involving volume and surface area.

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Adding and Subtracting to Solve Problems

7.1: Positive or Negative? (5 minutes)

CCSS Standards

Building On

Building Towards

Routines and Materials

Instructional Routines

• Think Pair Share

The purpose of this warm-up is to have students reason about an equation involving positive and negative rational numbers using what they have learned about operations with rational numbers.

Arrange students in groups of 2. Give students 30 seconds of quiet think time and ask them to give a signal when they have an answer and a strategy for the first question. Then have them discuss their reasoning with a partner. Ask for an explanation, and then ask if everyone agrees with that reasoning.

Then give students 30 seconds of quiet think time and ask them to give a signal when they have an answer for the second question. Then have them discuss their reasoning with a partner.

Student Facing

Without computing:

• Is the solution to \(\text-2.7 + x = \text- 3.5\) positive or negative?

Select  all  the expressions that are solutions to \(\text-2.7 + x = \text- 3.5\) .

• \(\text-3.5 + 2.7\)
• \(3.5 - 2.7\)
• \(\text-3.5 - (\text-2.7)\)
• \(\text-3.5 - 2.7\)

Student Response

Teachers with a valid work email address can  click here to register or sign in for free access to Student Response.

Activity Synthesis

Ask several student to share which expressions they chose for the second question. Discuss until everyone is in agreement about the answer to the second question.

7.2: Phone Inventory (10 minutes)

• MLR8: Discussion Supports

Positive and negative numbers are often used to represent changes in a quantity. An increase in the quantity is positive, and a decrease in the quantity is negative. In this activity, students see an example of this convention and are asked to make sense of it in the given context.

Arrange students in groups of 2. Give students 30 seconds of quiet work time followed by 1 minute of partner discussion for the first two problems. Briefly, ensure everyone agrees on the interpretation of positive and negative numbers in this context, and then invite students to finish the rest of the questions individually. Follow with whole-class discussion. 

A store tracks the number of cell phones it has in stock and how many phones it sells.

The table shows the inventory for one phone model at the beginning of each day last week. The inventory changes when they sell phones or get shipments of phones into the store.

• What do you think it means when the change is positive? Negative?
• What do you think it means when the inventory is positive? Negative?
• Based on the information in the table, what do you think the inventory will be at on Saturday morning? Explain your reasoning.
• What is the difference between the greatest inventory and the least inventory?

Tell students that we often use positive and negative to represent changes in a quantity. Typically, an increase in the quantity is positive, and a decrease in the quantity is negative.

Ask students what they answered for the second question and record their responses. Highlight one or two that describe the situation clearly. 

Ask a few students to share their answer to the third question, and discuss any differences. Then discuss the answer to the last question.

7.3: Solar Power (15 minutes)

• Anticipate, Monitor, Select, Sequence, Connect
• MLR6: Three Reads

It is common to use positive numbers to represent credit and negative numbers to represent debts on a bill. This task introduces students to this convention and asks them to solve addition and subtraction questions in that context. Note that whether a number should be positive or negative is often a choice, which means one must be very clear about explaining the interpretation of a signed number in a particular context (MP6).

For the second question, monitor for students who find the amount each week and sum those, and students who sum the value of the electricity used and the value of the electricity generated separately, and then find the sum of those.

Arrange students in groups of 2. Give students 4 minutes of quiet work time followed by partner discussion. Follow with a whole-class discussion.  

Han's family got a solar panel. Each month they get a credit to their account for the electricity that is generated by the solar panel. The credit they receive varies based on how sunny it is.

Attribution: Solar panels on a roof, by Pujanak (Own work). Public Domain. Wikimedia Commons. Source .

Here is their electricity bill from January.

In January they used $ 83.56 worth of electricity and generated $ 6.75 worth of electricity.

• In July they were traveling away from home and only used $ 19.24 worth of electricity. Their solar panel generated $ 22.75 worth of electricity. What was their amount due in July?

The table shows the value of the electricity they used and the value of the electricity they generated each week for a month. What amount is due for this month?

• What is the difference between the value of the electricity generated in week 1 and week 2? Between week 2 and week 3?

Are you ready for more?

While most rooms in any building are all at the same level of air pressure, hospitals make use of "positive pressure rooms" and "negative pressure rooms."  What do you think it means to have negative pressure in this setting?  What could be some uses of these rooms?

Teachers with a valid work email address can  click here to register or sign in for free access to Extension Student Response.

Ask one or more students to share their answer to the first question and resolve any discrepancies.

Ask selected students to share their reasoning for the second questions. Discuss the relative merits of different approaches to solving the problem.

Finish by going over the solution to the third question. Point out that the bill will reflect a negative number in the amount due section, but we can interpret this to mean that the family receives a credit, and it will be applied to their next bill.

7.4: Differences and Distances (15 minutes)

Optional activity.

In grade 6, students practiced finding the horizontal or vertical distance between points on a coordinate plane. In this activity, students see that this can be done by subtracting the \(x\) or \(y\) -coordinates for the points (MP7). Students continue to work with the distinction between  distance  (which is unsigned) and  difference  (which is signed) (MP6). This prepares them finding the slope of a line and the diagonal distance between points in grade 8.

Arrange students in groups of 2. Give students 3 minutes of quiet work time followed by partner discussion. Follow with a whole-class discussion.   

Plot these points on the coordinate grid: \(A= (5, 4), B= (5, \text-2), C= (\text-3, \text-2), D= (\text-3, 4)\)

• What shape is made if you connect the dots in order?
• What are the side lengths of figure \(ABCD\) ?
• What is the difference between the \(x\) -coordinates of \(B\) and \(C\) ?
• What is the difference between the \(x\) -coordinates of \(C\) and \(B\) ?

How do the differences of the coordinates relate to the distances between the points?

Plot these points on the coordinate grid: \(A= (5, 4), B= (5, \text-2), C= (\text-3, \text-2), D= (\text-3, 4)\)

Main learning points:

When two points in the coordinate plane lie on a horizontal line, you can find the distance between them by subtracting their \(x\) -coordinates.

When two points in the coordinate plane lie on a vertical line, you can find the distance between them by subtracting their \(y\) -coordinates.

• The distance between two numbers is independent of the order, but the difference depends on the order.

Discussion questions:

• Explain what makes the distance between two points and the difference between two points distinct.
• Explain how you would find the vertical or horizontal distance between two points.
• Explain how you would find the vertical or horizontal difference between two points.

Lesson Synthesis

What are some situations where adding and subtracting rational numbers can help us solve problems?

7.5: Cool-down - Coffee Shop Cups (5 minutes)

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Student Lesson Summary

Sometimes we use positive and negative numbers to represent quantities in context. Here are some contexts we have studied that can be represented with positive and negative numbers:

• temperature
• an account balance
• electricity flowing in and flowing out

In these situations, using positive and negative numbers, and operations on positive and negative numbers, helps us understand and analyze them. To solve problems in these situations, we just have to understand what it means when the quantity is positive, when it is negative, and what it means to add and subtract them.

Description: <p>Five line segments on a coordinate plane with the origin labeled “O”. The numbers negative 5 through 5, are indicated on the horizontal axis and the numbers negative 4 through 5 are indicated on the vertical axis. Each line segment is either vertical or horizontal. The first line segment is vertical that begins at the point negative 5 comma 3 and ends at the point negative 5 comma negative 4; the line segment is labeled with the expression 3 minus negative 4. The second line segment is horizontal and begins at the point negative 5 comma negative 4 and ends at the point negative 2 comma negative 4; the line segment is labeled with the expression negative 2 minus negative 5. The third line segment is vertical and begins at the point negative 2 comma negative 4 and ends at the point negative 2 comma 2; the line segment is labeled with the expression 2 minus negative 4. The fourth line segment is horizontal and begins at the point negative 2 comma 2 and ends at the point three comma 2; the line segment is labeled with the expression 3 minus negative 2. The fifth line segment is vertical and begins at the point 3 comma 2 and ends at the point 3 comma five; the line segment is labeled with the expression 5 minus 2.</p>

Remember: the distance between two numbers is independent of the order, but the difference depends on the order.

Video Summary

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

TE Newsletter

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.

View aligned curriculum

Do you agree with this alignment? Thanks for your feedback!

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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Lesson Title: An Interactive Review of Intermediate Math Concepts

Wondering how to make your high school math classes more engaging and effective?

In this math lesson plan, math students in grades 7-10 will have fun while they review key concepts learned throughout the semester. Students will engage in a dynamic scavenger hunt that combines technology with collaborative learning. This interactive activity will challenge students to apply their mathematical knowledge in a variety of contexts, while working individually or in teams. Watch as your students deepen their understanding of math concepts through problem-solving and critical thinking! 

Grade: 7 - 10

Subject: Math, Algebra, Physics

• Students will review and demonstrate their understanding of key math concepts taught throughout the semester using an interactive Goosechase Scavenger Hunt.
• Smartphones or tablets with Goosechase app installed
• Access to the internet
• Writing notebooks or digital devices for problem solving
• Pens or pencils

Preparation:

• Create an Experience on the Goosechase app with a series of Math Missions.
• Find example Missions in the Goosechase Templates below.  
• Run your Experience during one class period or for the entire week.
• Photo Mission: “Find an object in your surroundings that represents a geometric shape (e.g., a cylinder, sphere, or rectangle). Take a photo of the object and label the shape.”
• Video Mission: "Record a short video explaining the probability of flipping a coin and getting heads. Include an example in your explanation.”
• Text Mission: " Research a famous mathematician and write a short paragraph about their contributions to math.”
• Adapt the difficulty of the Missions based on the proficiency levels of your students to keep the activity inclusive and engaging for everyone.
• Prepare hints or resources that might help students solve the Missions if needed.
• Divide students into teams and assign each team a mobile device with the app, or have students complete the scavenger hunt individually.
• Dedicate time to ensuring all students have the Goosechase app downloaded and joined to the Experience to avoid any tech issues.

Lesson Plan Outline:

• Begin by explaining the objectives of the lesson
• Review some of the skills you have taught throughout the writing unit
• Briefly demonstrate how to use the Goosechase app.
• Go over the rules and expectations for the scavenger hunt, including safety guidelines and respect for school property.
• Read through Missions with students, emphasizing the importance of evidence and analysis in their responses.
• Monitor team progress and provide assistance as necessary.
• If working in teams, encourage them to strategize and collaborate to solve the Missions.
• Gather students to discuss their experiences.
• Review some of the key challenges and correct any misunderstandings.
• Announce the winning team (optional) and celebrate all participants.
• Allow each team to share one challenge they found particularly interesting or difficult.

Optional Post-Experience Activities:

• Assign further review questions for students to complete 
• Assign a reflection essay where students describe what they learned and how they felt about using the scavenger hunt for review.
• Observe student participation and engagement during the scavenger hunt.
• Review the correctness of the answers submitted through Goosechase.

Ready Made Experience Templates to Try

Physics on the Playground

Use this game to complete missions on the playground by using the physics concepts we learned!

What's your Angle?

Bringing math to life! How many of these angles and concepts can you find in your real life?

Algebra Parabolas

Complete these math Missions to review using and solving for parabolas in this algebra Experience.

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Review articles, unit 1: proportional relationships, unit 2: rates and percentages, unit 3: integers: addition and subtraction, unit 4: rational numbers: addition and subtraction, unit 5: negative numbers: multiplication and division, unit 6: expressions, equations, & inequalities, unit 7: statistics and probability, unit 8: scale copies, unit 9: geometry.