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.

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.

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 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 will begin to think about probability by considering how likely it is that their house will be struck by lightning. They will consider the relative likelihood of familiar events (e.g., outdoor temperature, test scores) on the continuum between impossible and certain. Students will discuss where on the continuum "likely," "unlikely," and "equally likely as unlikely" are.Key ConceptsAs students begin their study of probability, they look at the likelihood of events. Students have an intuitive sense of likelihood, even if no numbers or ratios are attached to the events. For example, there is clearly a better chance that a specific student will be chosen at random from a class than from the entire school.Goals and Learning ObjectivesThink about the concept of likelihood.Understand that probability is a measure of likelihood.Informally estimate the likelihood of certain events.Begin to think about why one event is more likely than another.SWD: Students with disabilities may need additional support seeing the relationships among problems and strategies. Throughout this unit, keep anchor charts available and visible to assist them in making connections and working toward mastery. Provide explicit think alouds comparing strategies and making connections. In addition, ask probing questions to get students to articulate how a peer solved the problem or how one strategy or visual representation is connected or related to another.

Students need many concrete experiences with fractions to develop a deep understanding of the three models of fractions: area, linear and set models. Teachers need to address all three models in well-designed instructional activities so that students develop a rich concept of fractions that they can use to make sense of numbers, operations, measurement and probability. The Math Tours include: activities, problem solving, games, writing to learn, templates, math-literature connections, and web links. Each page has a left navigation bar to easily take you through the tour and back to the homepage or the math topics page.

The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common misconception related to multiplication.

This book seeks to provide students with a deep understanding of the definitions, examples, theorems, and proofs related to measure, integration, and real analysis. The content and level of this book fit well with the first-year graduate course on these topics at most American universities. This textbook features a reader-friendly style and format that will appeal to today's students.

In this task students work with partners to measure themselves by laying multiple copies of a shorter object that represents the length unit end to end. It gives students the opportunity to discuss the need to be careful when measuring.

This lesson will require students to practice measuring real-world objects in different units and evaluate the most effective units to use for solving problems.

This short video and interactive assessment activity is designed to teach fifth graders about single object length ii (english units).

Students learn about the statistical analysis of measurements and error propagation, reviewing concepts of precision, accuracy and error types. This is done through calculations related to the concept of density. Students work in teams to each measure the dimensions and mass of five identical cubes, compile the measurements into small data sets, calculate statistics including the mean and standard deviation of these measurements, and use the mean values of the measurements to calculate density of the cubes. Then they use this calculated density to determine the mass of a new object made of the same material. This is done by measuring the appropriate dimensions of the new object, calculating its volume, and then calculating its mass using the density value. Next, the mass of the new object is measured by each student group and the standard deviation of the measurements is calculated. Finally, students determine the accuracy of the calculated mass by comparing it to the measured mass, determining whether the difference in the measurements is more or less than the standard deviation.

In this Cyberchase video segment, the CyberSquad helps resolve a dispute about the squares on the Gollywood Walk of Fame.

This unit describes the concepts of quality measurement and performance improvement. The unit begins by setting the context of known quality problems in healthcare and then describes how quality is measured and efforts to improve it. The unit also discusses the role of information technology, incentives for quality improvement, and quality measurement under meaningful use.

This lesson helps to build procedural skill with identifying unit fractions on a number line (and ruler). The partitioning of number lines and rulers is used here to introduce the idea that fractional lengths are invariant. This work develops students' understanding that a length does not change if its position on a number line or ruler changes.

A very common aim of epidemiological investigation is to estimate the frequency of disease in a population. This is of particular importance in the case of surveillance and disease monitoring systems, and is commonly the central aim of many descriptive studies. There are two main measures of disease frequency used by epidemiologists - the prevalence and the incidence of disease, which each measure different aspects of disease. The survival time, which is closely associated with the incidence, is another measure commonly used. Counts of disease are not commonly used in epidemiological studies, although they can be useful when deciding upon resource requirements when implementing disease control strategies.

Measures of effect and impact can be calculated using the same contingency table used to calculated measures of strength of association.