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.
Official wording from the Common Core State Standards for Mathematics (Β© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Not every graph comes with numbers. A qualitative graph shows the shape of a relationship: when a quantity rises, falls, stays level, speeds up or slows down. Students read such graphs as stories. A graph of a skateboarder's height might climb steadily up a ramp, flatten at the top, then drop sharply; each section of the line describes a part of the ride.
The standard works in both directions. Given a graph, students describe where the function is increasing or decreasing, where it is constant, and whether each piece is linear (a steady rate) or nonlinear (changing speed). Given a description, such as 'the bath fills quickly, someone gets in, and later the water drains slowly', students sketch a graph with the right features in the right order. Accuracy here is about shape and relative steepness rather than exact coordinates, which makes it a powerful check on whether students really understand what a graph shows.
A distance-time graph shaped like a hill does not mean someone walked up a hill. It shows moving away from a point and then returning.
A flat section of a speed-time graph means constant speed, not standing still. What 'flat' means depends on the quantity on the vertical axis.
Students may sketch walking and running as the same slope. A faster change must be shown by a steeper section.
Some sketches have the graph moving left. Time on the horizontal axis only moves forward, so the graph must always progress to the right.
Ava walks from home to a friend's house for 10 minutes, stays there for 20 minutes, then jogs home in 5 minutes. Sketch distance from home against time and describe each section.
Answer: A rising line, a flat section, then a steeper falling line ending at distance 0 after 35 minutes.
Ask students to act out a graph: one walks in front of the class while others sketch distance from the board, then swap roles with a graph to perform. Motion detectors, if available, make the feedback immediate. Matching cards of stories and graphs, including a few distractors, gets students arguing about steepness and order.
Test items typically show a graph with labelled intervals and ask which describes the situation, or give a description and four sketches. Teach students to check the order of events, the direction of each section and the relative steepness before choosing.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: B) From 6 a.m. to 2 p.m.
The function is increasing where the graph rises from left to right, which is from 6 a.m. to 2 p.m.
Answer: C) Not moving, staying at the same distance
Distance from home is not changing, so the person is staying in one place.
Answer: D) Increasing and getting less steep
Volume increases throughout, but the rate slows, so the graph rises less and less steeply.
Answer: 5 (also accepted: 5 minutes)
The flat section runs from 4 to 9 minutes, which is 9 - 4 = 5 minutes.
Answer: B) The steepest part
Steepness shows the rate of change. The steepest section is where the quantity changes fastest.
A full lesson with slides, activities and an exit ticket on describing graphs qualitatively, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson βA printable, differentiated worksheet on 8.F.B.5 with an answer key, ready in about a minute.
Make a worksheet βTurn describing graphs qualitatively into a quiz students answer online that marks itself, with a class summary for you.
Build a test βIt means describing the shape and behavior of a graph (increasing, decreasing, steady, curving) without needing exact numbers.
Yes. Axis labels with the quantities, and key events marked along the time axis, make a sketch readable even without a numeric scale.