Standard #: MA.8.F.1.3


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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.)
Grade: 8
Strand: Functions
Standard: Define, evaluate and compare functions.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • 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.

 

Instructional Tasks

Instructional Task 1 (MTR.6.1)
The graph describes the number of bacteria in a culture over time.
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

Course Number1111 Course Title222
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

Access Point Number Access Point Title
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

Formative Assessments

Name Description
Population Trend

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

Jet Fuel

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

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.

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.

Taxi Ride

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

Bike Race

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

Lesson Plan

Name Description
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.

Original Student Tutorials

Name Description
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.

Cruising Through Functions

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

Perspectives Video: Expert

Name Description
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.

Download the CPALMS Perspectives video student note taking guide.

Perspectives Video: Professional/Enthusiasts

Name Description
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.

Download the CPALMS Perspectives video student note taking guide.

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]

Download the CPALMS Perspectives video student note taking guide.

Problem-Solving Tasks

Name Description
Battery Charging

This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them.

Bike Race

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

Modeling with a Linear Function

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Tides

This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

Riding by the Library

In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

Velocity vs. Distance

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.

Student Resources

Original Student Tutorials

Name Description
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.

Cruising Through Functions:

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

Problem-Solving Tasks

Name Description
Bike Race:

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

Modeling with a Linear Function:

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Tides:

This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

Riding by the Library:

In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

Velocity vs. Distance:

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.



Parent Resources

Problem-Solving Tasks

Name Description
Battery Charging:

This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them.

Bike Race:

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

Modeling with a Linear Function:

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Tides:

This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

Riding by the Library:

In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

Velocity vs. Distance:

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.



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