Essential Questions:
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
Unit Overview
Content Emphasis By Clusters in Grade 7
Progressions from Common Core State Standards in Mathematics
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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..
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.
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.
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:
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:
In the table, each mixture of apple juice to grape juice is in a proportional relationship of 3 to 2, regardless of the unit.
At the completion of this unit on the use of properties of operations to generate equivalent expressions, the student will understand that:
The student will:
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.
Students may
Interdisciplinary Connections:
Interdisciplinary connections fall into a number of related categories:
Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:
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
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: ). 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.
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.
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 What operation was used to convert ? Was additive reasoning used: ? Or was multiplicative reasoning used:
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 represents of the first plant’s growth, whereas the 3-inch increase represents of the second plant’s growth. Since 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. b. c.
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.
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:
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 =
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.
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: . To determine the factor of change (unit cost) for one candy, divide $2.40 by 3. The factor of change is (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 , , 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
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: