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.

Students interpret verbal descriptions of situations and determine whether the situations represent proportional relationships.Key ConceptsIn a proportional relationship, there has to be some value that is constant.There are some relationships in some situations that can never be proportional.Goals and Learning ObjectivesIdentify verbal descriptions of situations as being proportional relationships or notUnderstand that some relationships can never be proportionalUnderstand that for two variable quantities to be proportional to one another, something in the situation has to be constant

Students represent and solve percent decrease problems.Key ConceptsWhen there is a percent decrease between a starting amount and a final amount, the relationship can be represented by an equation of the form y = kx where y is the final amount, x is the starting amount, and k is the constant of proportionality, which is equal to 1 minus the percent change, p, represented as a decimal: k = 1 – p, so y = (1 – p)x.The constant of proportionality k has the value it does—a number less than 1—because of the way the distributive property can be used to simplify the expression for the starting amount decreased by a percent of the starting amount: x – x(p) = x(1 – p).Goals and Learning ObjectivesDetermine the unknown amount—either the starting amount, the percent change, or the final amount—in a percent decrease situation when given the other two amounts.Make a table to represent a percent decrease problem.Write and solve an equation to represent a percent decrease problem.

Students represent and solve percent increase problems.Key ConceptsWhen there is a percent increase between a starting amount and a final amount, the relationship can be represented by an equation of the form y = kx where y is the final amount, x is the starting amount, and k is the constant of proportionality, which is equal to 1 plus the percent change, p, represented as a decimal: k = 1 + p, so y = (1 + p)x.The constant of proportionality k has the value it does—a number greater than 1—because of the way the distributive property can be used to simplify the expression for the starting amount increased by a percent of the starting amount: x + x(p) = x(1 + p).Goals and Learning ObjectivesDetermine the unknown amount—either the starting amount, the percent change, or the final amount—in a percent increase situation when given the other two amounts.Make a table to represent a percent increase problem.Write and solve an equation to represent a percent increase problem.

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).

Students have an opportunity to review their own work on the Self Check in the previous lesson, consider feedback that addresses specific aspects of their work, examine a different approach to the problem from the Self Check, and then use what they learned to solve a closely related problem.Key ConceptsStudents reflect on their work, review and critique student work on the same problem, and then apply their learning to solve a similar problem.Goals and Learning ObjectivesUse teacher comments to refine their solution strategies for a proportional relationship problemDeepen their understanding of proportional relationships.Synthesize and connect strategies for representing and investigating proportional relationships.Critique given student work involving proportional relationships.Apply deepened understanding of proportional relationships to a new problem situation.

In Part 2 of this two-part lesson, students review and revise their work on the Self Check task based on feedback from you and their peers and use what they’ve learned to solve similar problems.Key ConceptsStudents apply their knowledge, review their work, and make revisions based on feedback from you and their peers. This process creates a deeper understanding of the concepts.Goals and Learning ObjectivesUse feedback to refine solution strategies on the Self Check task.Deepen understanding of percent change.Apply deepened understanding to solve similar problems.

Students watch a video showing three different ways to solve a problem involving a proportional relationship, and then they use each method to solve a similar problem. Students describe each approach, including the mathematical terms associated with each.Key ConceptsThree methods for solving problems involving proportional relationships include:Setting up a proportion and solving for the missing valueFinding the unit rate and multiplyingWriting and solving a formula using the constant of proportionalityGoals and Learning ObjectivesSolve a problem involving a proportional relationship in three different ways: set up a proportion and solve for a missing value, use a unit rate, and use the constant of proportionality to write and solve a formula.

In Part l of this two-part lesson, students use an interactive to place percent increase and percent decrease signs between monetary amounts to indicate the correct increase or decrease between the amounts of money. They must also place the correct decimal multiplier  between the two amounts to show what decimal to multiply the original amount by to get the final amount.Key ConceptsStudents apply understanding of percent change situations to systematize and generalize patterns in relating two amounts by multiplication.Goals and Learning ObjectivesIdentify the percent increase or percent decrease between two amounts.Identify the decimal multiplier that when multiplied by the original amount results in the final amount.Reason abstractly and quantitatively.

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.

This lesson teaches about Rates, Ratios, and Unit Rates, using visually appealing art and real-life examples that will engage teenagers.

 This Remote Learning Plan was created by Brandee Drahota in collaboration with Rick Meyer as part of the 2020 ESU-NDE Remote Learning Plan Project. Educators worked with coaches to create Remote Learning Plans as a result of the COVID-19 pandemic.The attached Remote Learning Plan is designed for preschool students. This counting lesson uses the story The Very Hungry Caterpillar by Eric Carle, as an introduction to practice counting skills and one to one correspondence. The expected lesson length is 30 minutes and includes online and offline options. 

Every month we will give you a math/statistics brain teaser that lets you test your knowledge with a fun problem.Many of the ideas in these Teasers come from thoughts formed by some of the great mathematical/statistical geniuses in history.

All the activities in this lesson are addition and subtraction based. It is not designed to introduce addition and subtraction, rather, to supplement and enrich lessons already being taught. This lesson is not designed to be completed in one sitting. It may be done throughout an entire addition and subtraction unit. These activities may be used as starter activities when introducing new math concepts, particularly those that relate to addition and subtraction.

This paper makes recommendations for developing mathematics instruction for English Language Learners (ELLs) aligned with the Common Core State Standards. The recommendations can guide teachers, curriculum developers, and teacher educators as they develop their own ways of supporting mathematical reasoning and sense-making for ELLs.Some instructional recommendations discussed in the paper include: Focus on ELL students' mathematical reasoning, not the correctness of their mathematical language use. Shift to a focus on mathematical discourse practices; move away from simplified views of language. Support ELL students as they engage in complex mathematical language. Use ELL students' language and experiences as resources. Provide professional development to enhance teachers' awareness of ways to support ELs as they develop both language and mathematical knowledge.

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 goal of this task is to give students familiarity using the formula for the area of a circle while also addressing measurement error.