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The content that follows is what the teacher sees on the Lesson Overview page for each lesson in the teacher course.
Unit 5, Lesson 2: Distinguishing Between Proportional and Non-Proportional Relationships
Lesson Overview
Focusing Question: How can I distinguish between proportional and non-proportional situations?
Learning Outcomes
- I can distinguish between proportional and non-proportional situations in tables, graphs, and equations.
- I can identify examples of proportional and non-proportional situations in mathematical or real-world contexts.
| Texas Essential Knowledge and Skills (TEKS) |
| 8.5F Proportionality. The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. The student is expected to distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0. |
| 8.5H Proportionality. The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. The student is expected to identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems. |
| 8.1F Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to analyze mathematical relationships to connect and communicate mathematical ideas. |
| English Language Proficiency Standards (ELPS) |
| 5.B Cross-curricular second language acquisition/writing. The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to write using newly acquired basic vocabulary and content-based grade-level vocabulary. |
Overview of Process Standard(s)
In this lesson, students analyze proportional and non-proportional relationships to connect attributes of proportional or non-proportional relationships from a variety of representations and communicate how those relationships are either proportional or non-proportional. Students may:
- Use a graph to locate the y-intercept of the relationship in order to distinguish between proportional or non-proportional relationships.
- Use a table of values to determine the y/x ratios for each number pair in order to distinguish between proportional or non-proportional relationships.
- Use an equation to determine if there is a non-zero constant term (i.e., b ≠ 0) in order to distinguish between proportional or non-proportional relationships.
Prior Learning Supports
Students began learning about proportional relationships in 6th grade.
- 6.4A The student is expected to compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.
- 6.6C The student is expected to represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
In 7th grade, students formalized proportional and linear relationships.
- 7.4A The student is expected to represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
- 7.4C The student is expected to determine the constant of proportionality (k = y/x) within mathematical and real-world problems.
- 7.7A The student is expected to represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
In 8th grade, students make direct comparisons between proportional and non-proportional linear relationships as they develop the concept of a linear function. Linear functions all have a constant rate of change. Different subsets of linear functions have different characteristics within the linear function family. For example, proportional relationships are linear functions in that they have a constant rate of change and they have the additional common characteristic that the starting point or y-intercept is 0.
Lesson Plan
See the subsections below for lesson components and guidance for instruction.
- Exploration
- Play the instructional video to launch instruction.
- Guide students through the activity.
- Provide students with the Blackline Master of the Student Pages or assign the Google Slides as appropriate.
- Use the Answer Key as necessary.
- Explanation
- Assign students the ePub to provide direct instruction on the content along with guided practice through examples and a set of practice questions.
- Performance Task
- As a formative assessment, use the performance task to determine what students know about the topic.
- Performance tasks have four versions: on-level, simplified, enriched, and scaffolded.
- Allow students to work in pairs or small groups if desired.
- If there are multiple performance tasks, select one or encourage self differentiation by allowing students to select a task based on their interests and comfort levels.
Lesson Components
Pacing Guide
- Exploration (20-25 minutes)
- Explanation and Practice (20-25 minutes)
- Performance Task (30-35 minutes)
Teaching Hints
- Provide students the opportunity to compare and contrast proportional linear relationships with non-proportional linear relationships in a variety of representations. Use graphics organizers such as a Venn diagram or T-chart to compare and contrast the attributes of the two types of linear relationships.
- Encourage students to make connections among characteristics of proportional and non-proportional linear relationships in multiple representations (e.g., graphs, tables, equations).
Teacher Materials
- Graphing technology or scientific calculators (1 per student)
Student Materials
- Pencil
- Paper
Vocabulary
One way to support students’ acquisition of new vocabulary terms is to pre-teach the vocabulary terms using a strategy such as a Frayer model. For these key terms, we provide partially completed Frayer models to which students should add as they move through the lesson and develop a deeper understanding of the vocabulary term. Click on the term to see the Frayer model.
Use this template for additional vocabulary words.
