Putting Math to Work

Type of Unit: Problem Solving

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Write ratios and rates.
Write and solve proportions.
Solve problems involving scale.
Write and solve equations to represent problem situations.
Create and interpret maps, graphs, and diagrams.
Use multiple representations (i.e., tables, graphs, and equations) to represent problem situations.
Calculate area and volume.
Solve problems involving linear measurement.

Lesson Flow

Students apply and integrate math concepts they have previously learned to solve mathematical and real-world problems using a variety of strategies. Students have opportunities to explore four real-world situations involving problem solving in a variety of contexts, complete a project of their choice, and work through a series of Gallery problems.

First, students utilize their spatial reasoning and visualization skills to find the least number of cubes needed to construct a structure when given the front and side views. Then, students select a project to complete as they work through this unit to refine their problem-solving skills. Students explore the relationship between flapping frequency, amplitude, and cruising speed to calculate the Strouhal number of a variety of flying and swimming animals. After that, students explore the volume of the Great Lakes, applying strategies for solving volume problems and analyzing diagrams. Next, students graphically represent a virtual journey through the locks of the Welland Canal, estimating the amount of drop through each lock and the distance traveled. Students have a day in class to work on their projects with their group.

Then, students have two days to explore Gallery problems of their choosing. Finally, students present their projects to the class.

Students explore the relationship between the flapping frequency, the amplitude, and the cruising speeds of a variety of animals to calculate their Strouhal numbers.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesAnalyze the relationship between the variables in an equation.Write formulas to show how variables relate.Communicate findings using multiple representations including tables, charts, graphs, and equations.

Students create a bar graph showing the Strouhal numbers for a variety of birds and bats and use their graph and other data to compare the Strouhal numbers of the different animals to analyze variation and to make predictions.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesAnalyze the relationship among the variables in an equation.Write formulas to show how variables relate.Calculate ranges of Strouhal numbers and use these ranges to make predictions.Communicate findings using multiple representations including tables, charts, graphs, and equations.Create bar graphs.

Allow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery ProblemsThe SS Edmund FitzgeraldStudents solve mathematical problems about the sinking of the ship Edmund Fitzgerald.SpiralsStudents learn about the mathematics of spirals. They see spirals in nature and connect spirals to the Fibonacci.Ship It!Students learn about shipping containers and use a unit of measure that is only used in the shipping industry the twenty-foot equivalent unit (TEU).Rideau Canal WaterwayStudents compare information about the Rideau Canal and compare it with the Welland Canal.A Rule of ThumbStudents learn about a “rule of thumb” that people use to estimate the speed of a train they are riding on. They investigate the mathematics of this rule.IntersectionStudents use information on a map to calculate where two streets will intersect.Tolstoy's ProblemStudents learn about Leo Tolstoy, a Russian writer who wrote two of the greatest novels of all time. They solve a problem that Tolstoy found very interesting.The Dog RunStudents imagine having 22 meters of wire fencing for a dog run. They investigate how the area of the dog run changes as the length varies.Bodies of WaterStudents investigate a claim on the Runner's World website about the amount of water in the body of a 160-pound man.

How much water is in the Great Lakes? Students read and interpret a diagram that shows physical features of the Great Lakes and answer questions based on the diagram. They find the volume of each of the Great Lakes, as well as all five lakes combined, and make a bar graph to represent the volumes.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills.Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesRead and interpret graphs and diagrams.Solve problems involving volume.

Students first create a diagram that represents the distance a ship drops in each of a series of locks. Students create their diagrams based on a video of an actual ship traveling through the locks. Students need to use contextual clues in order to determine the relative drops in each of the locks.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills.Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesRead and interpret maps, graphs, and diagrams.Solve problems that involve linear measurement.Estimate length.Critique a diagram.

Student groups make their presentations, provide feedback to other students' presentations, and get evaluated on their listening skills.Key ConceptsIn this culminating event, students must present their project plan and solution to the class. The presentation allows students to explain their problem-solving plan, to communicate their reasoning, and to construct a viable argument about a mathematical problem.Students also listen to other project presentations and provide feedback to the presenters. Listeners have the opportunity to critique the mathematical reasoning of others.Goals and Learning ObjectivesPresent project to the class.Give feedback on other project presentations.Exhibit good listening skills.

Student groups continue to make their presentations, provide feedback to other students' presentations, and get evaluated on their listening skills.Key ConceptsIn this culminating event, students must present their project plan and solution to the class. The presentation allows students to explain their problem-solving plan, to communicate their reasoning, and to construct a viable argument about a mathematical problem.Students also listen to other project presentations and provide feedback to the presenters. Listeners have the opportunity to critique the mathematical reasoning of others.Goals and Learning ObjectivesPresent project to the class.Give feedback on other project presentations.Exhibit good listening skills.Reflect on the problem-solving process.

Student groups continue to make their presentations, provide feedback to other students' presentations, and get evaluated on their listening skills.Key ConceptsIn this culminating event, students must present their project plan and solution to the class. The presentation allows students to explain their problem-solving plan, to communicate their reasoning, and to construct a viable argument about a mathematical problem.Students also listen to other project presentations and provide feedback to the presenters. Listeners have the opportunity to critique the mathematical reasoning of others.Goals and Learning ObjectivesPresent project to the class.Give feedback on other project presentations.Exhibit good listening skills.Reflect on the problem-solving process.

Students choose a project idea and a partner or group. They write a proposal for their project.Key ConceptsProjects engage students in the applications of mathematics. It is important for students to apply mathematical ways of thinking to solve rich problems. Students are more motivated to understand mathematical concepts if they are engaged in solving a problem of their own choosing. In this lesson, students are challenged to identify an interesting mathematical problem and to choose a partner or a group to work collaboratively on solving that problem. Students gain valuable skills in problem solving, reasoning, and communicating mathematical ideas with others.Goals and Learning ObjectivesIdentify a project idea.Identify a partner or group to work collaboratively on a math project.

Students design and work on their projects in class. They review the project rubric and, as a class, add criteria relevant to their specific projects.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills.Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Use multiple representations—including tables, graphs, and equations—to organize and communicate data.Articulate strategies, thought processes, and approaches to solving a problem and defend why the solution is reasonable.

Students critique the diagrams of other students from the previous lesson and receive feedback about their own diagrams. Students revise their diagrams from the first part of the lesson based on the feedback they receive.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesRead and interpret maps, graphs, and diagrams.Solve problems that involve linear measurement.Estimate length.Critique a diagram.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features and/or concepts and will help them to pay close attention to salient information. Students need to know their goal is to develop and refine their problem solving skills.

Students are presented with a front view and a side view of a cube structure. They use spatial reasoning to picture what the entire structure looks like and to determine the least number of cubes they would need to build the structure.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesVisualize three-dimensional spaces.Solve problems that require spatial reasoning.Design and implement a problem-solving plan.Articulate strategies, thought processes, and approaches to solving a problem and defend why the solution is reasonable.

Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the concept of a ratio.Write ratios as percents.Describe data using measures of center.Display and interpret data in dot plots, histograms, and box plots.Lesson FlowStudents begin to think about probability by considering the relative likelihood of familiar events on the continuum between impossible and certain. Students begin to formalize this understanding of probability. They are introduced to the concept of probability as a measure of likelihood, and how to calculate probability of equally likely events using a ratio. The terms (impossible, certain, etc.) are given numerical values. Next, students compare expected results to actual results by calculating the probability of an event and conducting an experiment. Students explore the probability of outcomes that are not equally likely. They collect data to estimate the experimental probabilities. They use ratio and proportion to predict results for a large number of trials. Students learn about compound events. They use tree diagrams, tables, and systematic lists as tools to find the sample space. They determine the theoretical probability of first independent, and then dependent events. In Lesson 10 students identify a question to investigate for a unit project and submit a proposal. They then complete a Self Check. In Lesson 11, students review the results of the Self Check, solve a related problem, and take a Quiz.Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think about methods for randomly sampling a population to ensure that it is representative. In Lesson 13, students collect and analyze data for their unit project. Students begin to apply their knowledge of statistics learned in sixth grade. They determine the typical class score from a sample of the population, and reason about the representativeness of the sample. Then, students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes, and decide whether increasing the sample size improves the results. In Lesson 16 and Lesson 17, students compare two data sets using any tools they wish. Students will be reminded of Mean Average Deviation (MAD), which will be a useful tool in this situation. Students complete another Self Check, review the results of their Self Check, and solve additional problems. The unit ends with three days for students to work on Gallery problems, possibly using one of the days to complete their project or get help on their project if needed, two days for students to present their unit projects to the class, and one day for the End of Unit Assessment.

Students begin to formalize their understanding of probability. They are introduced to the concept of probability as a measure of likelihood and how to calculate probability as a ratio. The terms discussed (impossible, certain, etc.) in Lesson 1 are given numerical values.Key ConceptsStudents will think of probability as a ratio; it can be written as a fraction, decimal, or a percent ranging from 0 to 1.Students will think about ratio and proportion to predict results.Goals and Learning ObjectivesDefine probability as a measure of likelihood and the ratio of favorable outcomes to the total number of outcomes for an event.Predict results based on theoretical probability using ratio and proportion.

Students collect and analyze data for their unit project.Students are given class time to work on their project. Some students may choose to use the time to collect data (if their project is an experiment based on experimental probability), while others will use the tools (spinners, coin toss, number cube, etc.) to collect their data. Students should use the time to analyze their data, finding the theoretical (if possible) probability and comparing it to the experimental results.Key ConceptsStudents will apply what they have learned about probability to work on their project, including likelihood of events, determining theoretical and experimental probability, comparing results to calculations, and using simulations to establish probability.Students may also use data analysis tools to discuss their results.Goals and Learning ObjectivesComplete the project, or progress far enough to complete it outside of class.Review concepts of probability (simple probability, compound events, experimental vs. theoretical probability, simulations).

Students estimate the length of 20 seconds by starting an unseen timer and stopping it when they think 20 seconds has elapsed. They are shown the results and repeat the process two more times. The first and third times are recorded and compiled, producing two data sets to be compared. Students analyze the data to conclude whether or not their ability to estimate 20 seconds improves with practice.Key ConceptsMeasures of center and spreadLine plots, box plots, and histogramsMean absolute deviation (MAD)Goals and Learning ObjectivesApply knowledge of statistics to compare sets of data.Use measures of center and spread to analyze data.Decide which graph is appropriate for a given situation.

Students extend their understanding of compound events. They will compare experimental results to predicted results by calculating the probability of an event, then conducting an experiment.Key ConceptsStudents apply their understanding of compound events to actual experiments.Students will see there is variability in actual results.Goals and Learning ObjectivesContinue to explore compound independent events.Compare theoretical probability to experimental probability.

Students begin learning about compound events by considering independent events. They will consider everyday objects with known probabilities. Students will represent sample spaces using lists, tables, and tree diagrams in order to calculate the probability of certain events.Key ConceptsCompound events are introduced in this lesson, building upon what students have learned about determining sample space and probabilities of single events.Terms introduced are:multistage experiment: an experiment in which more than one action is performedcompound events: the combined results of multistage experimentsindependent events: compound events in which the outcome of one does not affect the outcome of the otherGoals and Learning ObjectivesLearn about compound events and sample spaces.Use different tools to find the sample space (tree diagrams, tables, lists) of a compound event.Use ratio and proportion to solve problems.SWD: Go over the mathematical language used throughout the module. Make sure students use that language when discussing problems in this lesson.

Students will apply their knowledge of statistics learned in sixth grade. They will determine the typical class score from a sample of the population, and reason about the representativeness of the sample.Students analyze test score data from a fictitious seventh grade class and make generalizations about district-wide results. They then compare the data to a second seventh grade class and reason about whether these are random samples. Students will review measures of center and spread as they find evidence to draw conclusions about the data.Key ConceptsSample size will be considered as it affects the conclusions of an analysis of a population.Students will review tools that they used in sixth grade to analyze data, such as measures of center and spread, and different types of graphs.Goals and Learning ObjectivesExplore sample size.Look at the effects of using a nonrandom sample.Review tools used to analyze data.