Proportional Relationships

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Understand what a rate and ratio are.
Make a ratio table.
Make a graph using values from a ratio table.

Lesson Flow

Students start the unit by predicting what will happen in certain situations. They intuitively discover they can predict the situations that are proportional and might have a hard time predicting the ones that are not. In Lessons 2–4, students use the same three situations to explore proportional relationships. Two of the relationships are proportional and one is not. They look at these situations in tables, equations, and graphs. After Lesson 4, students realize a proportional relationship is represented on a graph as a straight line that passes through the origin. In Lesson 5, they look at straight lines that do not represent a proportional relationship. Lesson 6 focuses on the idea of how a proportion that they solved in sixth grade relates to a proportional relationship. They follow that by looking at rates expressed as fractions, finding the unit rate (the constant of proportionality), and then using the constant of proportionality to solve a problem. In Lesson 8, students fine-tune their definition of proportional relationship by looking at situations and determining if they represent proportional relationships and justifying their reasoning. They then apply what they have learned to a situation about flags and stars and extend that thinking to comparing two prices—examining the equations and the graphs. The Putting It Together lesson has them solve two problems and then critique other student work.

Gallery 1 provides students with additional proportional relationship problems.

The second part of the unit works with percents. First, percents are tied to proportional relationships, and then students examine percent situations as formulas, graphs, and tables. They then move to a new context—salary increase—and see the similarities with sales taxes. Next, students explore percent decrease, and then they analyze inaccurate statements involving percents, explaining why the statements are incorrect. Students end this sequence of lessons with a formative assessment that focuses on percent increase and percent decrease and ties it to decimals.

Students have ample opportunities to check, deepen, and apply their understanding of proportional relationships, including percents, with the selection of problems in Gallery 2.

Students determine whether a relationship between two quantities that vary is a proportional relationship in three different situations: the relationship between the dimensions of the actual Empire State Building and a miniature model of the building; the relationship between the distance and time to travel to an amusement park; and the relationship between time and temperature at an amusement park.Key ConceptsWhen the ratio between two varying quantities remains constant, the relationship between the two quantities is called a proportional relationship. For a ratio A:B, the proportional relationship can be described as the collection of ratios equivalent to A:B, or cA:cB, where c is positive.Goals and Learning ObjectivesIdentify proportional relationships.Explain why a situation represents a proportional relationship or why it does not.Determine missing values in a table of quantities based on a proportional relationship.

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 estimate the length of 50 seconds by starting an unseen timer and stopping it when they think 50 seconds has elapsed. The third attempt is recorded and compiled into a data set, which students then compare to the third attempt from the previous lesson when they estimated the length of 20 seconds. Students analyze the data to make conclusions about how well seventh grade students can estimate lengths of time.Students repeat the timing activity for 50 seconds, but only the third trial is recorded. The task today is to compare this set of data with the third trial for 20 seconds. Students will need to deal with the difference in the spread of data, as well as how to compare the data sets. Students will be reminded of Mean Absolute Deviation (MAD), which will be a useful tool in this situation.Key ConceptsStudents apply the tools learned in Unit 6.8:Measures of center and spreadMean absolute deviation (MAD)Goals and Learning ObjectivesApply knowledge of statistics to compare different sets of data.Use measures of center and spread to analyze data.

This is three examples of "Type III" documents that I have prepared for students to write about math. Our school uses the Collins system of writing, but these could be adapted to a variety of classroom settings. The topics could be middle school or early high school topics, and they are set up to allow students to indepedently express their knowledge of the given topics in short essay form.

In addition to the writing prompts, I have included a rubric that students use to peer-edit each others' rough drafts. Students then use this feedback form to make the necessary corrections and submit a subsequent final draft.

This short video and interactive assessment activity is designed to teach fourth graders about distance between items.

This short video and interactive assessment activity is designed to teach fifth graders about distance between items.

This short video and interactive assessment activity is designed to teach fifth graders about solving multi-step distance problems.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En el módulo 8, el módulo final del año, los estudiantes amplían su comprensión de las relaciones completas a través de la lente de la geometría. A medida que los estudiantes componen y descomponen formas, comienzan a desarrollar una comprensión de las fracciones unitarias como partes iguales de un todo.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In Module 8, the final module of the year, students extend their understanding of part–whole relationships through the lens of geometry.  As students compose and decompose shapes, they begin to develop an understanding of unit fractions as equal parts of a whole.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This short video and interactive assessment activity is designed to teach second graders an overview of telling time to 1-minute intervals.

This short video and interactive assessment activity is designed to teach first graders an overview of telling time to quarter and hour.

8.01L is an introductory mechanics course, which covers all the topics covered in 8.01T. The class meets throughout the fall, and continues throughout the Independent Activities Period (IAP).

Students investigate the accuracy of sundials and the discrepancy that lies between "real time" and "clock time." They track the position of the sun during the course of a relatively short period of time as they make a shadow plot, a horizontal sundial, and a diptych sundial. (The activity may be abridged to include only one or two of the different sundials, instead of all three.)

Most research in the realm of relationships has examined that which can go wrong in relationships (e.g., conflict, infidelity, intimate partner violence). I summarize much of what has been examined about what goes right in a relationship and call these positive relationship deposits. Some research indicates that relationships need five positive interactions for every negative interaction. Active-constructive responding, gratitude, forgiveness, and time spent together are some sources of positive deposits in one’s relational bank account. These kinds of deposits can reduce the negative effects of conflict on marriage and strengthen relationships.

This is a first course in project management, one in which students will learn the knowledge, skills, and abilities necessary to be an effective project manager. They will learn how to plan, execute, and monitor a project. The course will cover the latest theories and concepts on scoping, stakeholder management, team leadership, budgeting and contracting, scheduling, quality control and assurance, and risk management. Students will have the option to apply their learning to a real-world project. 

This learning module (Lesson 1 of Unit 5) is part of a course called Project Management Fundamentals and may either be completed individually as a stand-alone topic, or part of a trio of learning modules on time management, or as part of the course.

This learning module (Lesson 2 of Unit 5) is part of a course called Project Management Fundamentals and may either be completed individually as a stand-alone topic, or part of a trio of learning modules on  time management, or as part of the course.