## Unit Overview

**Essential Questions: **

**Lesson Plans and Seeds**

Lesson Plan A.3: Cross Sections Polyhedral

Lesson Seed A.1: Scale Drawings

**Download Seeds, Plans, and Resources (zip)**

**Content Emphasis By Clusters in Grade 7**

**Progressions from Common Core State Standards in Mathematics**

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

This unit builds on student understanding of ratio concepts and reasoning, as well as on proportional relationships to create scale drawings. The unit also focuses on the origin of two-dimensional figures that result from slicing three-dimensional figures.

**Teacher Notes:**

- Students should be well-grounded in their ability to identify two-dimensional figures based on identifying characteristics, and in the classifications of two- and three-dimensional figures.
- Students should have prior knowledge of ratio concepts and language.

**Enduring Understandings:**

At the completion of this unit on drawing, constructing, and describing geometrical figures and their relationships, the student will understand that:

- Representation of geometric ideas and relationships allows multiple approaches to solving geometric problems, and connects geometric interpretations to other contexts in life.
- Changing the dimensions of an object effects linear measure, area, and volume.
- Proportional relationships express how dimensions of a geometric figure change in relationship to each other.
- A constant ratio exists between corresponding sides of similar figures.
- Geometric figures can be represented on the coordinate plane.

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

- PARCC has not provided examples of opportunities for in-depth focus related to the cluster “Draw, Construct, and Describe Geometrical Figures and Describe the Relationships between Them.”

**Possible Student Outcomes:**

Students will be able to:

- Solve problems involving scale drawings.
- Compute actual lengths and areas from scale drawing.
- Reproduce a scale drawing at a different scale.
- Draw geometric shapes with given conditions.
- Describe two-dimensional figures that result from slicing a right rectangular prism and right rectangular pyramid.

**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:**

**Common Misconceptions:**

Students may

- have misconceptions about correctly setting up proportions.
- not realize that when the scale for two objects (i.e., the actual geographic distance between Baltimore and New York versus distance on a map) changes, the dimensions of the static object (actual distance between Baltimore and New York) cannot change.
- be confused about how to read a ruler, mixing up standard and metric measures, or not placing the ruler correctly on the dimension to be measured.
- not be sure which protractor scale to use when measuring angles, particularly when the orientation of the angle is not horizontal.

**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
Use Properties of Operations to Generate Equivalent Expressions**

*right rectangular prism:*

a.
b. c. |
A right rectangular prism is a 6-sided polyhedron in which all faces are quadrilaterals that meet to form right angles at the vertices (see prism a. and prism b.). Prism c is an oblique rectangular prism because not all faces meet to form right angles at the vertices. |

*right rectangular pyramid:*

a.
b. |
A right rectangular pyramid has a rectangle as the base and four isosceles triangles as the faces (see pyramid a.). An oblique rectangular pyramid has a rectangle as the base, with two scalene triangles and two isosceles triangles as faces (see pyramid b.). |

**Part II – Instructional Connections outside the Focus Cluster**

** unit rate: ** Unit rate is the ratio of two different measurements in which the second term

is 1. Example: 6:1 6 miles/one gallon, , (In fraction form, the denominator is 1.)

** ratio: ** A ratio is a comparison of 2 quantities. It is written as the quotient of two numbers

(part to part and part to whole). 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”)

** proportion: **A proportion is a statement of equality of two ratios; that is written as an

equation. Example: 3:9 = 4:12 or =

** 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.

** within ratios: **A ratio of two measures in the same setting is within ratios. Example: If a person walks 4 miles in 2 hours, how many miles will she walk in 3 hours? Let

*m*= miles.

** 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 to ?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 > (or > , the 9-inch plant grew more.

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.

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 According to the graph, unit rate is (1, 50) or To determine other points: 1. multiply by 2. identify (2, 100) or, (3, 150) or ; and (6, 300) or , respectively. |

** 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:

percent error =

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 =

*
*