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
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..
For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:
The Common Core Standards Writing Team (10 September 2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at: http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf
Vertical Alignment: Vertical curriculum alignment provides two pieces of information: (1) a description of prior learning that should support the learning of the concepts in this unit, and (2) a description of how the concepts studied in this unit will support the learning of additional mathematics.
Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the over-arching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.
Overarching Unit Standards
Supporting Standards within the Domain
Instructional Connections outside the Cluster
7.RP.A.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like and/or different units.
N/A
7.G.A.1:
7.SP.C.6:
7.SP.C.8:
7.RP.A.2: Recognize and represent proportional relationships between quantities.
7.RP.A.2a:
7.RP.A.2b:
7.RP.A.2c:
7.RP.A.2d:
7.EE.A.3:
Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.
In this unit, educators should consider implementing learning experiences which provide opportunities for students to:
Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.>
Standard
Essential Skills and Knowledge
Clarification
7.RP.A.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like and or different units. (i.e., if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction miles per hour or 2 miles per hour).
Within Ratios:
Between Ratios:
Ratio as a complex fraction:
Ratio as a fraction: Ratio as a unit rate:
All three of these ratios are equivalent; thus, in proportion to one another.
“within” ratios vs. “between” ratios If a person walks 4 miles in 2 hours, how many miles will she walk in 3 hours? Let m = miles
Set up of the proportion using “within ratios” format:
Set up of the proportion using “between ratios” format:
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 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 which 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.
proportional relationship intersects (0, 0):
When the value of x is 0, the value of y is 0, or (0, 0).
For example:
determine other points using (1, r), where r is the unit rate: In the ordered pair (1, r), the x-value “1” represents one standard of measurement. The y-value “r” represents a given specific quantity.
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.
According to the graph, unit rate is (1, 50) or . To determine other points, multiply by , , , and identify (2, 100) or ; (3, 150) or ; and (6, 300) or, respectively.
7.RP.A.3: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
cross-product 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
factor of change 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.