MA.8.F.1.3

Analyze a real-world written description or graphical representation of a functional relationship between two quantities and identify where the function is increasing, decreasing or constant.

Clarifications

Clarification 1: Problem types are limited to continuous functions.

Clarification 2: Analysis includes writing a description of a graphical representation or sketching a graph from a written description.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Functions
Status: State Board Approved

Benchmark Instructional Guide

• Function

Vertical Alignment

Previous Benchmarks

Next Benchmarks

Purpose and Instructional Strategies

In grade 7, students determined one of the key features of a proportional relationship, its constant of proportionality, from its graph. In grade 8, students determine where a function is increasing or decreasing from its graph. In Algebra 1, students will compare key features of linear and nonlinear functions represented algebraically, graphically, in tables or written descriptions.
• Graphs can be described many ways. Using knowledge of functions, equations, and graphs, students should be able to describe a graph in words and should be able to draw a graph if given a qualitative description.
• When working with graphs, instruction includes students reasoning through asking questions such as:
• Does the graph represent a function?
• Does the graph show increase, decrease, both, or neither?
• Are there intervals of the domain in which the graph shows that the function increases, decreases, or stays constant?
• Does the graph represent a linear function?
• What is the (approximate) slope of a given interval within the graph?
• Students should be given opportunities to analyze graphs individually and with others (MTR.4.1).
• When sketching a graph based on a written description, students may use curved or straight lines to represent portions of increase or decrease based on the description.
• Problems where students are creating a graph are note expected to differentiate between linear and nonlinear functions to represent the written description.
• For example, if the description has a rapid increase, a student can sketch a curve that increases rapidly or a straight line with a steep slope.
• For example, in grade 8 students are not expected to recognize curves that represent exponential growth or decay.

Common Misconceptions or Errors

• Students may invert domain and range.
• Students may incorrectly describe increasing, decreasing or constant intervals using elements outside of the domain.

Strategies to Support Tiered Instruction

• Teacher reviews vocabulary and the difference between the terms. Once students understand that domain represents the input variable (independent variable), they can make sense of real-world problems to accurately identify domain and range.
• Teacher poses questions to encourage discourse to gain information about graphs. Students have the opportunity to discuss what they see and know from the graph and have the ability to make inferences about its description.
• Instruction includes providing real-world situations and having the students identify domain and range for the situation, giving justification as to their reasoning. Students can create their own situation and identify the domain and range for their situation.
• Teacher provides examples of different characteristics of graphs. Once students can identify basic attributes of the graphs, they can begin to reason through more specific questions about each graph.
• For example, describing if the graph is increasing or decreasing, if the graph is a linear function, if the graph is a function, what is the slope of the graph (if linear), etc.
• Instruction includes creating an anchor chart with students describing increasing, decreasing, or constant graph.

The graph describes the number of bacteria in a culture over time.

Describe in detail the relationship between the number of bacteria in the culture and time. Include where it is increasing, decreasing or remaining constant.

Instructional Items

Instructional Item 1
Sketch a graph of the representation described below.
Madison is studying the growth of bacteria in food and learned it has four phases. Label the axes and show a graph of the four stages, assuming an initial bacterium count of 50.
Phase 1: No growth in the number of cells for the first hour.
Phase 2: Rapid growth in the number of bacteria for the next two hours.
Phase 3: Growth stops for one hour as nutrients are used up and waste accumulates.
Phase 4: All bacteria gradually die off during the final four-hour phase.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.F.1.AP.3: Given a functional relationship displayed as a graph, identify where the function is increasing, decreasing or constant.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Formative Assessments

Population Trend:

Students are asked to describe the relationship between two quantities in a nonlinear function.

Type: Formative Assessment

Jet Fuel:

Students are asked to analyze and describe the relationship between two linearly related quantities.

Type: Formative Assessment

Graph the Ride:

Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship.

Type: Formative Assessment

Bacterial Growth Graph:

Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship.

Type: Formative Assessment

Taxi Ride:

Students are asked to sketch a graph from a verbal description.

Type: Formative Assessment

Bike Race:

Students are asked to evaluate three verbal descriptions and to state why each does or does not match a given graph.

Type: Formative Assessment

Lesson Plan

Constructing and Calibrating a Hydrometer:

Students construct and calibrate a simple hydrometer using different salt solutions. They then graph their data and determine the density and salinity of an unknown solution using their hydrometer and graphical analysis.

Type: Lesson Plan

Original Student Tutorials

Math Models and Social Distancing:

Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.

Type: Original Student Tutorial

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

Perspectives Video: Expert

Using Mathematics to Optimize Wing Design:

Nick Moore discusses his research behind optimizing wing design using inspiration from animals and how they swim and fly.

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiasts

Asymptotic Behavior in Shark Growth Research:

Fishery Scientist from Florida State University discusses his new research in deep sea sharks and the unusual behavior that is found when the data is graphed.

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Kites, Wind, and Speed:

Lofty ideas about kites helped power a kayak from California to Hawaii.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

MFAS Formative Assessments

Bacterial Growth Graph:

Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship.

Bike Race:

Students are asked to evaluate three verbal descriptions and to state why each does or does not match a given graph.

Graph the Ride:

Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship.

Jet Fuel:

Students are asked to analyze and describe the relationship between two linearly related quantities.

Population Trend:

Students are asked to describe the relationship between two quantities in a nonlinear function.

Taxi Ride:

Students are asked to sketch a graph from a verbal description.

Original Student Tutorials Mathematics - Grades 6-8

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Math Models and Social Distancing:

Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Original Student Tutorials

Math Models and Social Distancing:

Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.

Type: Original Student Tutorial

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.