Functions

How can we determine what kind of function we are looking at without being told its name?

Display several graphs without labels. Learners individually record their observations before sharing.

• Notice: What do you notice? What patterns do you see? Which graphs seem related?
• Wonder: Why are some graphs straight? Why does one graph have a sharp corner? Why do some graphs have an endpoint? Why does one graph have an S-like shape?

Give groups a collection of graphs. Learners sort them according to their own criteria. They may initially create categories such as: straight; curved; V-shaped; starts at a point; S-shaped.

• "What rule did your group use to create your categories?"
• "Could another group classify them differently?"
• "What relationship do you notice between the equation and the shape of the graph?"

This encourages learners to recognize that mathematical classification requires defensible criteria, not simply visual guessing.

Groups present their classification. Other groups are encouraged to challenge respectfully:

• "What evidence supports your answer?"
• "Would your rule work for this equation?"
• "Can you find an example that contradicts your rule?"

This explicitly develops critical thinking and mathematical argumentation. Groups construct: Equation → Pattern → Graph → Characteristics → Possible Real-Life Situation.

Only after learners have constructed their own categories does the teacher introduce the formal terminology. The teacher connects the students' discovered descriptions to the formal mathematical concepts. Say: "You discovered five different patterns. Mathematicians have names for these patterns..."

• Linear
• Absolute Value
• Quadratic
• Square Root
• Cube Root

Given an unfamiliar equation, learners must:

• classify it;
• identify evidence from the equation;
• predict the shape of the graph; and
• explain their reasoning.

Critical Thinking Focus: Analyze → Justify → Predict

Reinforce classification of linear, absolute value, quadratic, square root, and cube root functions.

• Look around your home, community, textbook, or a teacher-provided online resource for 5 examples of graphs, equations, or situations that represent mathematical relationships.
• For each example, identify the possible function type.
• If you are unsure, use Desmos to graph the equation and examine its shape.
• Complete the table below.
• Choose one example and explain why you classified it that way.

Can we predict the shape of a graph just by examining its equation?

Learners receive the following equations. Without graphing, they draw what they think each graph looks like.

Present the function. Ask the student to generate the table of values by substituting the assigned value of x. Then, identify the ordered pair and let the student plot it to see the graph. Learners discover that the graph forms a straight line.

Give the questions to the groups in advance. Present the function. Learners investigate rather than simply being told about symmetry.

• What happens when x changes from negative to positive?
• Why do the two sides of the graph look symmetrical?
• What feature of the equation might explain this?

Ask the learner to find a partner. Each student will compare their work and how to graph their function to each other and answer the following questions. To verify their answers, ask the learners to double check it with the graphing technology (Desmos/GeoGebra).

• "What changed?"
• "Where did the graph move?"
• "How can you prove that?"

Learners receive a new equation and must:

• predict its graph;
• sketch it;
• explain their prediction;
• verify with technology; and
• explain any discrepancy.

Develop the ability to sketch graphs without immediately relying on technology. For each function:

• Look at the equation.
• Predict what the graph will look like.
• Identify at least 3 points.
• Sketch the graph on graphing paper.
• Use Desmos to check your answer.
• Circle any differences between your prediction and the digital graph.
• Explain what you learned from the comparison.

What does an equation tell us about where a graph can exist?

Learners investigate the following functions by constructing a table and graph.

• Why doesn't the graph continue to the left?
• Does the same restriction occur?
• What evidence do you have?
• How can you distinguish a square-root function from a cube-root function?

Group the learners to compare the two graphs and complete the table.

Give learners an equation without its graph. Ask: "Predict where the graph begins." Learners sketch first, then verify digitally.

Each learner explains to a partner: "How can you distinguish a square-root function from a cube-root function?" The partner asks for evidence.

Present contradictory student explanations. Ask: Who is correct? Prove your answer.

• Student A: "The square-root function can have any real number as its input."
• Student B: "The square-root function has restrictions on its input."
• Student C: "The cube-root function has a restricted domain like the square-root function."

Strengthen understanding of square-root and cube-root functions, particularly domain, range, and graph behavior. Graph the functions using Desmos/GeoGebra.

• Before graphing, predict the possible values of x and y.
• Record your observations.
• Compare the square-root and cube-root graphs.
• Explain why the two graphs behave differently.

How do we decide which function best represents a real-life situation?

Present: A delivery company charges a fixed fee plus a charge for every kilometer travelled. Ask: "How could we represent this situation mathematically?" Do not give the equation. Learners discuss:

• What changes?
• What stays constant?
• What is the dependent variable?
• What is the independent variable?

Groups receive different situations. Groups must determine: Situation → Variables → Pattern → Function → Equation → Graph.

• Situation A: Taxi fare based on distance.
• Situation B: Distance from a target value.
• Situation C: Height of a ball over time.
• Situation D: Side length of a square based on its area.
• Situation E: Side length of a cube based on its volume.

Groups must answer: "Why is your chosen function appropriate?" They must provide at least three pieces of evidence. For example:

• behavior of the variables;
• shape of the graph;
• equation structure;
• behavior of the situation.

Learners modify their models and predict the consequences. This promotes transfer and deeper conceptual understanding.

• What if the fixed fee increases?
• What if the rate changes?
• What if the object is launched from a different height?
• What if the dimensions change?

Learners submit:

• situation;
• variables;
• selected function;
• equation;
• table;
• graph;
• solution;
• interpretation; and
• justification of the model.

Apply functions to authentic situations and develop mathematical modeling skills. Find one real-life situation at home or in your community involving a relationship between two quantities (e.g., transportation fare and distance; working hours and earnings; product quantity and total cost; distance and time; area and side length; volume and side length).

• Identify the two variables.
• Determine which function type best represents the situation.
• Write an equation.
• Create a table and draw the graph manually or create it using Desmos/GeoGebra.
• Explain whether your model is reasonable.

"How can functions help us explain, predict, and solve problems in the real world?"

Ask the learners to reconstruct a Function Concept Map from memory: Five Functions → Equation → Graph → Characteristics → Real-Life Application.

Learners choose an authentic situation involving mathematics (business; transportation; sports; science; engineering; construction; environmental issues; personal finance; school/community situations). They must answer:

• What is the problem?
• What quantities are involved?
• What function could model the situation?
• Why did you choose that function?
• What equation represents it?
• What does the graph tell us?
• What solution does the model provide?
• What assumptions did you make?
• What are the limitations of your model?

Groups present their models. The audience becomes a panel of mathematical critics using the rubric. Learners must defend, evaluate, and possibly revise their mathematical models. The panel may ask:

• Why did you select this function?
• Could another function work?
• What evidence supports your model?
• What happens if one variable changes?
• Is your model realistic?
• What are the limitations?
• How would you improve the model?

Based on the feedback given by their classmates and the teacher, learners will also do a self-assessment using the rubric.

For learners who have not yet achieved mastery: Concrete/Visual → Guided → Collaborative → Independent. Use:

• graph cards;
• equation cards;
• partially completed tables;
• guided questioning;
• peer teaching;
• technology-assisted exploration.

For advanced learners: "Can one real-life situation be modeled by more than one type of function?" Learners compare competing models and determine which is more appropriate. They must defend their choice using mathematical evidence.