Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.

Students work in pairs to critique and improve their work on the Self Check. Students complete a task similar to the Self Check with a partner.Key ConceptsTo critique and improve the task from the Self Check and to complete a similar task with a partner, students use what they know about solving inequalities, graphing their solutions, and relating the inequalities to a real-world situation.Goals and Learning ObjectivesSolve algebraic inequalities.Graph the solutions of inequalities using number lines.Write word problems that match algebraic inequalities.Interpret the solution of an inequality in terms of a word problem.

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 analyze the graph of a proportional relationship in order to find the approximate constant of proportionality, to write the related formula, and to create a table of values that lie on the graph.Key ConceptsThe constant of proportionality determines the steepness of the straight-line graph that represents a proportional relationship. The steeper the line is, the greater the constant of proportionality.On the graph of a proportional relationship, the constant of proportionality is the constant ratio of y to x, or the slope of the line.A proportional relationship can be represented in different ways: a ratio table, a graph of a straight line through the origin, or an equation of the form y = kx, where k is the constant of proportionality.Goals and Learning ObjectivesIdentify the constant of proportionality from a graph that represents a proportional relationship.Write a formula for a graph that represents a proportional relationship.Make a table for a graph that represents a proportional relationship.Relate the constant of proportionality to the steepness of a graph that represents a proportional relationship (i.e., the steeper the line is, the greater the constant of proportionality).

Lesson OverviewStudents calculate the constant of proportionality for a proportional relationship based on a table of values and use it to write a formula that represents the proportional relationship.Key ConceptsIf two quantities are proportional to one another, the relationship between them can be defined by a formula of the form y = kx, where k is the constant ratio of y-values to corresponding x-values. The same relationship can also be defined by the formula x=(1k)y , where 1k is now the constant ratio of x-values to y-values.Goals and Learning ObjectivesDefine the constant of proportionality.Calculate the constant of proportionality from a table of values.Write a formula using the constant of proportionality.

Students explore the idea that not all straight lines are proportional by comparing a graph representing a stack of books with a graph representing a stack of cups. They recognize that all proportional relationships are represented as a straight line that passes through the origin.Key ConceptsNot all graphs of straight lines represent proportional relationships.There are three ways to tell whether a relationship between two varying quantities is proportional:The graph of the relationship between the quantities is a straight line that passes through the point (0, 0).You can express one quantity in terms of the other using a formula of the form y = kx.The ratios between the varying quantities are constant.Goals and Learning ObjectivesUnderstand when a graph of a straight line is and when it is not a proportional relationship.Recognize that a proportional relationship is shown on a graph as a straight line that passes through the origin (0, 0).Make a table of values to represent two quantities that vary.Graph a table of values representing two quantities that vary.Describe what each variable and number in a formula represents.

Students continue to explore the three relationships from the previous lessons: Comparing Dimensions, Driving to the Amusement Park, and Temperatures at the Amusement Park. They graph the three situations and realize that the two proportional relationships form a straight line, but the time and temperature relationship does not.Key ConceptsA table of values that represent equivalent ratios can be graphed in the coordinate plane. The graph represents a proportional relationship in the form of a straight line that passes through the origin (0, 0). The unit rate is the slope of the line.Goals and Learning ObjectivesRepresent relationships shown in a table of values as a graph.Recognize that a proportional relationship is shown on a graph as a straight line that passes through the origin (0, 0).

CK-12 Foundation's Middle School Math € Grade 6 Flexbook covers the fundamentals of fractions, decimals, and geometry. Also explored are units of measurement, graphing concepts, and strategies for utilizing the book's content in practical situations.

In this Assessment Routine, students use a Mind Pie chart to express how comfortable and confident they feel about certain topics and activities they will encounter during the field experience. The chart provides the instructor with some information about the group, which they can use to inform their instruction. It also gives students an idea of what to expect from the field experience. This activity does not explicitly illuminate student misconceptions, rather, it serves as an opportunity for students to access and reflect on their prior knowledge and experience.

Methylmercury contamination within fish populations is an important toxin that affect human, animal, and environmental health, serving as a carcinogen (cancer-causing agent) and endocrine-disruptor (compounds that in some way alter the signaling of the hormone system. The impacts of exceeding safe dietary methylmercury levels were tragically made clear in Ontario, Canada, where a First Nations community in Grassy Narrows are living with the consequences of methylmercury poisoning in the fish supply. The fish were contaminated due to the dumping of mercury in the traditional waterways of the First Nation community. In 2016, there were highly publicized protests in Muskrat Falls, Labrador, Canada, where the Inuit people raised direct concerns about the potential for a proposed Nalcor Energy hydroelectric dam, to increase mercury levels in fish in those waters, which are an integral part of their traditional diet. Despite significant protests, the project was completed in 2019 and 41 km were flooded. This module uses these real-world examples as a jumping-off point for exercises that will guide case-study driven discussion on mathematical, biological and ethical concerns.

Methylmercury contamination within fish populations is an important toxin that affect human, animal, and environmental health, serving as a carcinogen (cancer-causing agent) and endocrine-disruptor (compounds that in some way alter the signaling of the hormone system. The impacts of exceeding safe dietary methylmercury levels were tragically made clear in Ontario, Canada, where a First Nations community in Grassy Narrows are living with the consequences of methylmercury poisoning in the fish supply. The fish were contaminated due to the dumping of mercury in the traditional waterways of the First Nation community. In 2016, there were highly publicized protests in Muskrat Falls, Labrador, Canada, where the Inuit people raised direct concerns about the potential for a proposed Nalcor Energy hydroelectric dam, to increase mercury levels in fish in those waters, which are an integral part of their traditional diet. Despite significant protests, the project was completed in 2019 and 41 km were flooded. This module uses these real-world examples as a jumping-off point for exercises that will guide case-study driven discussion on mathematical, biological and ethical concerns.

Learn about position, velocity, and acceleration graphs. Move the little man back and forth with the mouse and plot his motion. Set the position, velocity, or acceleration and let the simulation move the man for you.

This video explains how to use graphs, including how to set up a graph, the use of variables, and how to interpret the data. Disclaimer: Host website is responsible for accessibility compliance. Educator is responsible for accommodations.

The purpose of this lesson is to teach students about the three dimensional Cartesian coordinate system. It is important for structural engineers to be confident graphing in 3D in order to be able to describe locations in space to fellow engineers.

Open Resources for Community College Algebra (ORCCA) is an open-source, openly-licensed textbook package (eBook, print, and online homework) for basic and intermediate algebra. At Portland Community College, Part 1 is used in MTH 60, Part 2 is used in MTH 65, and Part 3 is used in MTH 95.

In this video lesson, students will learn about linear programming (LP) and will solve an LP problem using the graphical method. Its focus is on the famous "Stigler's diet" problem posed by the 1982 Nobel Laureate in economics, George Stigler. Based on his problem, students will formulate their own diet problem and solve it using the graphical method. The prerequisites to this lesson are basic algebra and geometry. The materials needed for the in-class activities include graphing paper and pencil. This lesson can be completed in one class of approximately one hour. If the teacher would like to cover the simplex algorithm by George Dantzig as an alternative solution method, an additional whole class period is suggested.

Student groups use a "real" 3D coordinate system to plot points in space. Made from balsa wood or wooden dowels, the system has three axes at right angles and a plane (the XY plane) that can slide up and down the Z axis. Students are given several coordinates and asked to find these points in space. Then they find the coordinates of the eight corners of a box/cube with given dimensions.

Using Avida-ED freeware, students control a few factors in an environment populated with digital organisms, and then compare how changing these factors affects population growth. They experiment by altering the environment size (similar to what is called carrying capacity, the maximum population size that an environment can normally sustain), the initial organism gestation rate, and the availability of resources. How systems function often depends on many different factors. By altering these factors one at a time, and observing the results, students are able to clearly see the effect of each one.

Students observe four different classroom setups with objects in motion (using toy cars, a ball on an incline, and a dynamics cart). At the first observation of each scenario, students sketch predicted position vs. time and velocity vs. time graphs. Then the classroom scenarios are conducted again with a motion detector and accompanying tools to produce position vs. time and velocity vs. time graphs for each scenario. Students compare their predictions with the graphs generated by technology and discuss their findings. This lesson requires assorted classroom supplies, as well as motion detector technology.

SYNOPSIS: In this lesson, students use New Jersey precipitation data to create graphs and discuss climate change.

SCIENTIST NOTES: This lesson has students working on their data analysis skills through the use of graphs which help students to interpret New Jersey’s precipitation data and how it relates to climate change. A class discussion encourages students to think critically about the raw data. Students then work independently to graph the precipitation over time, finding a line of best fit and the equation for the line. This is followed by a discussion of the relationship between time and precipitation. Data forecasting is touched upon when students are asked to think about what data they would need next and what is predictable about the data. Students then choose one of two choices that allow them to compare and contrast visually represented data. This is a well-rounded lesson that relays the information of climate change through graphing and data analysis and is recommended for teaching.

POSITIVES:
-This lesson can be used independently to practice application of math and reasoning skills or as ang point for longer research into data displays.
-Students can use graph paper or any digital platform schools and teachers are already familiar with.
-Students are given voice and choice in this lesson.
-Students learn to apply math skills to current situations to explore and explain relationships in nature.
-Students defend their chosen quantities and levels of accuracy in displaying data.

ADDITIONAL PREREQUISITES:
-Students should have some basic understanding of graphing, plotting points, and the relationship between x & y-axes.
-Students should have a basic idea of an equation of a line, line of best fit, and slope.
-Students should have a basic understanding of other types and purposes of graphs and charts.

DIFFERENTIATION:
-Teachers can adjust the degree of difficulty based on the math level of each class.
-If using a digital graphing platform, teachers and students can manipulate data to explore related questions.
-Teachers can explore deeper the purpose of different kinds of graphs in highlighting different parts of the same data set.
-Teachers can bring in a variety of graphs from scientific journals or magazines, such as National Geographic, as instructional tools.
-Teachers can extend this project to have students or classes graph the relationship between precipitation and time for all 50 states. Students can then display their graphs and conclusions. Teachers can moderate discussions comparing and contrasting various states and regions or make a conclusion as a whole.
-Using the same website resources, students can explore the average maximum and minimum temperature table. They can explore the relationship between temperature and precipitation using various graphs. Teachers can then use this to discuss causation and/or correlation.
-Teachers can use the lesson to introduce causation and correlation, asking students if there is a correlation between precipitation and climate change.