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Home > INSTRUCTION > State Standards and Frameworks > Mathematics > gr7_RP_Analyze_Proportional_Relationships_Unit_Overview

 Gr. 7 Unit: 7.RP.A.1-3: Analyze Proportional Relationships and Use Them to Solve Real-World and Mathematical Problems

Unit Overview

Essential Questions: Question

  • What are the types/varieties of situations in life that depend on or require the application of ratios and proportional reasoning?

  • Lesson Plans and Seeds

    Lesson Plan A.2: Proportional Relationships and Similarity

    Lesson Seed A.2: Creating a Graph from a Table

    Lesson Seed A.2: Calculator and Direct Variation

    Download Seeds, Plans, and Resources (zip)

    Unit Overview

    Content Emphasis By Clusters in Grade 7

    Progressions from Common Core State Standards in Mathematics

    Send Feedback to MSDE’s Mathematics Team

    Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery..

    Unit Overview Question

    This unit extends knowledge of fractions and ratios from previous grades, and it develops understanding of proportionality to solve single-step and multi-step problems, and to distinguish proportional relationships from other relationships (see Key Advances from Previous Grades). Knowledge and understanding of ratios and proportionality can be used to solve a variety of problems, including percent, scale drawings, and unit rate. Graphs of proportional relationships extend unit rate and connect it to slope of a line.

    Teacher Notes: Question

    As explained in the draft Progressions for the Common Core State Standards in Mathematics (10 September 2011), two instructional perspectives provide insight for the instruction of ratios in relation to proportional reasoning: (1) ratio as a composed unit or batch, and (2) ratio as a fixed numbers of parts.

    1. Ratio as a unit or batch, for example, there are 3 cups of apple juice for every 2 cups of grape juice in the mixture. This way uses a composed unit: 3 cups apple juice and 2 cups grape juice. Any mixture that is made from some number of the composed unit is in the ratio 3 to 2. In the table, each of the mixtures of apple juice and grape juice are combined in a ratio of 3 to 2:

      F1

    2. Ratio as a combined number of parts, for example a mixture is made from 3 parts apple juice and 2 parts grape juice, where all parts are the same size, but can be any unit.

    apple juice:

    grape juice:

    F2 Each part represents the same amount, but can be any amount, such as 2 cups or 5 liters
    If 1 part is: 1 cup 2 cups 5 liters 3 quarts
    Amount of apple juice: 3 cups 6 cups 15 liters 9 quarts
    Amount of grape juice: 2 cups 4 cups 10 liters 6 quarts

    In the table, each mixture of apple juice to grape juice is in a proportional relationship of 3 to 2, regardless of the unit.

    Enduring Understandings:Question

    At the completion of this unit on the use of properties of operations to generate equivalent expressions, the student will understand that:

    1. A ratio is a multiplicative comparison of two quantities.
    2. Proportional reasoning involves relationships among ratios.
    3. Proportional situations involve multiplicative relationships.
    4. Situations based on proportional reasoning can involve multiple representations.
    5. Ratios and proportional reasoning interconnects with topics in geometry, probability, and rational numbers.

    Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document): Question

    1. 7.RP.A.2: Students in grade 7 need ample opportunity to refine their ability to recognize, represent, and analyze proportional relationships in various ways, including through the use of tables, graphs, and equations.
    2. 7.EE.B.3: Applying properties of operations to calculate with rational numbers in any form while using tools strategically, converting between forms as appropriate, and assessing the reasonableness of answers using mental computation and estimation strategies is a major capstone standard for arithmetic and its applications.

    Possible Student Outcomes:Question

    The student will:

    1. Determine, represent, and calculate proportional relationships between quantities in tables, graphs, equations, diagrams, and verbal descriptions.
    2. Compute unit rates and distinguish them from other ratios.
    3. Use proportional relationships to solve problems involving simple interest, tax, markups and markdowns, gratuities, commissions, fees, percent increase and decrease, percent error, scale drawings, and probability, among others.
    4. Solve problems involving unit rates associated with ratios of fractions (e.g., if a person walks ½ mile in each ¼ hour, the unit rate is the complex fraction ½/¼ miles per hour or 2 miles per hour)
    5. Analyze proportional relationships in geometric figures.
    6. Approximate the probability of simple and complex events.

    Evidence of Student Learning:

    The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

    Fluency Expectations and Examples of Culminating Standards:Question

  • Students solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically (7.EE.3). This work is the culmination of many progressions of learning in arithmetic, problem solving, and mathematical practices.
  • Common Misconceptions:Question

    Students may

    1. Confuse the significance of the numerator compared to the denominator.
    2. Believe that a denominator with a greater digit automatically has a greater value than a fraction with a lesser denominator, e.g., F3
    3. Rely on one configuration for setting up proportions without realizing that other configurations may also be correct (see discussion of within ratios and between ratios on page 2).
    4. Have difficulty calculating unit rate, recognizing unit rate when it is graphed on a coordinate plane, and realizing that unit rate is also the slope of a line.
    5. Misinterpret or not have mastery of the precise meanings and appropriate use of ratio and proportion vocabulary.
    6. Miscomprehend the difference between additive reasoning versus multiplicative reasoning.

    Interdisciplinary Connections:

    Interdisciplinary connections fall into a number of related categories:

  • Literacy standards within the Maryland Common Core State Curriculum
  • Science, Technology, Engineering, and Mathematics standards
  • Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.
  • Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:

  • Items purchased from vendors
  • PARCC prototype items
  • PARCC public release items
  • Maryland Public release items
  • Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)

    Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.

    Part I – Focus Cluster
    Ratios and Proportional Reasoning

    unit rate: Given a specific quantity in dollars, inches, centimeters, pounds, hours, etc., the unit rate is the amount for one standard of measurement (i.e., one dollar, one inch, one centimeter, one pound, one hour, etc.) relative to the given quantity.

    ratio: A ratio is the quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities). The three ways of writing a ratio include: 9 to 10, and 9:10, and F4

    complex fraction: A complex fraction has a fraction for the numerator or denominator or both.

    proportional relationship: A proportional relationship is a collection of pairs of
       numbers that are in equivalent ratios (Example: F5 ). A proportional relationship also can be described by an equation of the form y = kx, where k is a positive constant (often called a “constant of proportionality”)

    within ratios: A ratio of two measures in the same setting is within ratios.

    F6

    between ratios: A ratio of two corresponding measures in different situations is considered between ratios. Example: If a person walks 4 miles in 2 hours, how many miles will she walk in 3 hours? Let m = miles.

    F7

    additive reasoning versus multiplicative reasoning: Proportional situations are based on
      multiplicative relationships. Equal ratios result from multiplication or division, not addition
      or subtraction.
      Consider any proportion, for example F8 What operation was used to convert F9? Was additive reasoning used: F10 ? Or was multiplicative reasoning used:F11

    The example below shows an application of multiplicative reasoning:

    Last month, two bean plants were measured at 9 inches tall and 15 inches tall. Today they are 12 inches tall and 18 inches tall, respectively. Which bean plant grew more during the month, the 9-inch bean plant or the 15-inch bean plant?

    Using additive reasoning, we know that each plant added 3 inches in a month. To determine
       plant grew more, however, we use multiplicative reasoning. In other words,
      we want to know what proportion of the original plant height is represented by 3 inches.
       The 3-inch increase representsF12 of the first plant’s growth, whereas the 3-inch increase representsF13 of the second plant’s growth. SinceF14 the 9-inch plant grew more.

    Proportion: A proportion is a statement of equality of two ratios; an equation whose members are ratios.

    identify that a proportional relationship intersects (0, 0): As shown on the graphs below, when the value of x is 0, the value of y is also 0, or (0, 0). For example, (a) when zero electricity is being generated, zero pounds of carbon are emitted; (b) when zero ounces of oregano are weighed, the cost is zero dollars; (c) when a cone has a height of zero, the cone flattens into a 2-dimensional circle, and thus, has volume of zero.

    a. F15    b. F16

    c. F17

    determine other points using (1, r), where r is the unit rate: In the ordered pair (1, r),
       x-value “1” represents one standard of measurement. The y-value “r” represents
       a given specific quantity.

    For example in the table “A Moving Automobile,” the x-value is time (hours). The y-value “r” is distance traveled (miles). The unit rate is r miles per hour, or F18
    According to the graph, unit rate is (1, 50) or F19
    To determine other points: 1. multiply by F20 ,F21 ,F22.
    2. identify (2, 100) orF24 ,(3, 150) orF25 ;and (6, 300) or F28 , respectively.
    F27

    percent error: We often assume that each measurement we make in mathematics or science is true and accurate.  However, sources of error often prevent us from being as accurate as we would like.  Percent error calculations are used to determine how close to the true values our experimental values actually are. The value that we derive from measuring is called the experimental, or observed, value. A true, ortheoretical, value can be found in reference tables.


    The percent error can be determined when the theoretical value is compared to the experimental value according to the equation below:

    F29

    Example:  A student measured the volume of a 2.50 liter container to be 2.38 liters. What is the percent error in the student's measurement?
       percent error = F30 F31

    cross-product algorithm: This algorithm is a strategy for determining a missing value in a proportion. The proportion can be set up using either as a within ratios proportion or as a between ratios proportion, as shown in the diagrams below.

    F32

    factor of change algorithm: This algorithm is a strategy for determining a missing value
    in a proportion. A unit-rate approach is used, whereby the factor of change (rate of a single unit) is established first from the given values in one ratio. Then, the missing value from the other ratio is computed by multiplying the known value in the other ratio by the factor of change. Example using the within ratios format: Three candies cost a total of $2.40. At that same price, how much would 10 candies cost? The within ratios proportion is: F33 . To determine the factor of change (unit cost) for one candy, divide $2.40 by 3. The factor of change is F34 (or $0.80 for one candy), meaning that as the number of candies increases by one, the total cost increases by $0.80. So, 10 candies ($0.80 x 10) would cost $8.00.

    Part II – Instructional Connections outside the Focus Cluster

    rational numbers: Numbers that can be expressed as an integer, as a quotient of integers
      (such as F35 , F36 , 7), or as a decimal where the decimal part is either finite or repeats infinitely (such as 2.75 and 33.3333…) are considered rational numbers.

    probability: Probability is used to describe the likeliness that an event will happen.

    relative frequency: When a collection of data is separated into several categories, the number of items in a given category is the absolute frequency. The absolute frequency divided by the total number of items is the relative frequency. Out of 50 middle school students, 18 are sixth graders; 18 is the absolute frequency. The relative frequency is

    F37

    outcome: A possible end result of a probability experiment is referred to as an outcome.

    event: An event is a collection of possible outcomes.

    simple event: An event is a collection of possible outcomes. A simple event is a single outcome of an experiment. For example, tossing two number cubes and getting a 7 is an event. Getting a 7 specifically with a 2 and a 5 is considered a simple event.

    compound event: An event is a collection of possible outcomes. An event that consists of two or more events is a compound event. The probability of a compound event can be determined by multiplying the probability of one event by the probability of a second event. Some compound events (independent events) do not affect each other's outcomes, such as rolling a number cube and tossing a coin. Other compound events do affect each other's outputs (dependent events). For example, if you take two cards from a deck of playing cards, the likelihood of second card having a certain quality is altered by the fact that the first card has already been removed from the deck.

    sample space: Sample space is the collection of all possible outcomes in a probability experiment.

    simulation: Simulation is a technique used for answering real-world questions or making decision in complex situations where an element of chance is involved.

    Additional Resources: Question

    1. http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g
      _k6_2012_06_27.pdf
    2. www.bbc.co.uk/skillswise/topic/ratio-and-proportion
    3. Litwiller, B., Bright, G. (2002). Making Sense of Fractions, Ratios, and Proportions, Reston, VA: National Council of Teachers of Mathematics
    4. Dase, P.H. (2007). Texas Instrument Graphing Calculator Strategies: Algebra (pp. 49-56).Huntington Beach, CA: Shell Education
    5. http://www.azed.gov/standards-practices/mathematics-standards/

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      Last Updated 3/9/2020 2:16 PM