Use functions to model relationships between quantities.


8.F.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Describing, Analyzing, and Sketching graphs of functions

At this stage, children are taking the concept of graphs to a whole new level.  The interpretation of the graph must include the relationships which exist between the two quantities being graphed.  Students are to both interpret and create scenarios given related to graphs.


Students are expected to describe the trend of the graph, i.e. increasing, decreasing or no change:

Taken from PARCC EOY Test





Different representations could be an equation, table, graph or verbal description

Given a graph of a situation, students provide a verbal description of the situation.





-7< x < -3




3 < x < 5



No Change

-1< x < 1

5 < x < 7

In addition, given a situation, students are supposed to describe what is happening by looking at a graph of the situation.  Below, you will see a graph of a scenario, students are asked to describe in words (using the story line) to interpret each part of the graph.


The graph below shows a student‘s trip to school. This student walks to Bobby walk to his friend, Michael’s house and together, they ride a bus to school. The bus stops once before arriving at school.


Describe how each part A-E of the graph relates to the story.

Part A: Bobby is walking at a constant rate of speed to Michael’s house.


Part B: Bobby and Michael are waiting for the bus.


Part C: Both boys are on the bus as it travels the speed limit through town.


Part D: This is the bus stop on the way to school


Part E: The bus continues through town at the speed limit.


The last piece of this standard requires students to write a narrative of a given graph.  The scenario can begin by being set up (as in the following example) or it can be a graph given with not axes labeled.  Of course the latter is much more difficult and therefore scaffolding by using these other experiences is a must.  Diagram A below shows the scaffold for writing a narrative from scratch.  Diagram B is the writing of a narrative after much practice.

Writing a story for a graph is difficult for most students. Scaffold the experience with multiple experiences with a wide variety of scenarios.

Diagram A


Sunflower Seed Graphs

Ian and his friends were sitting on a deck and eating sunflower seeds. Each person had a bowl with the same amount of seeds. The graphs below all show the amount of sunflower seeds remaining in the person’s bowl over a period of time.


Write sentences that describe what may have happened for each person.

Diagram B


Using the axes below, create a story about the relationship between two quantities.   


a. Write a story about the relationship between two quantities.  Any quantities can be used (e.g., distance and time, money and hours, age and growth).  Be creative!  Include keywords in your story like increase and decrease to describe the relationship.


b. Label each axis with the quantities of your choice and sketch a graph of the function that models the relationship described in the story.

From Engage NY 8.6 lesson 4


Explanations and Examples

Coherence and Connections: Need to Know

Grade Below

Grade Level

Grade Above









Illinois Assessment of Readiness Mathematics Evidence Tables



Statement Key

Evidence Statement Text





Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear).

i) Tasks may or may not have a context






Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

i) Tasks may or may not have a context



Classroom Resources

8.F.5 Daily Discourse


Graphing Stories - Teachers need to preview the videos that are on the site.  Each video shows an action (going up stairs, swinging on a rope, etc) then students graph the action that they are viewing.  The video is time stamped by seconds to assist students with the independent variable (x-axis).  Then they follow it up with the correct graph.            



A complete lesson with teacher guide on matching graphs to the situation. Card Sort can be done in groups or individually.

Math Assessment Project




Desmos – Free Online graphing



HOT Questions

1.  Below the following graphs are three stories about walking from your locker to your class.


     Two of the stories correspond to the graphs. Match the graphs and the stories. Write stories for the other two graphs.          Draw a graph that matches the third story.

a.  I started to walk to class, but I realized I had forgotten my notebook, so I went back to my locker and then I went quickly at a constant rate to class.


b.  I was rushing to get to class when I realized I wasn’t really late, so I slowed down a bit.


c.  I started walking at a steady, slow, constant rate to my class, and then, realizing I was late,  I ran the rest of the way at a steady, faster rate.



Graph C and Story A, Graph B and story B, Graph D and Story C  Graph A is constant rate.Story could be: I walked to class and didn’t stop or rush just walked at a steady rate.



2.  Indicate which graph matches the statement. Give reasons for your answer.

     A bicycle valve’s distance from the ground as a boy rides at a constant speed.


The graph is b because the valve will go up then down as it moves in a circular motion.



3.  A child swings on a swing, as a parent watches from the front of the swing.


Graph a because the the longer the child swings the higher he/she goes which will cause the parent to be farther away with each swing.