Parabolic Characteristics

Syllabus Indicators:

• Graph simple non-linear relations, with and without the use of digital technologies (ACMNA296)
  • Complete tables of values to graph simple non-linear relationships and compare these with graphs drawn using digital technologies
• Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technologies as appropriate (ACMNA239)

Learning goals:

• Identify on a graph the x and y intercepts and maximum or minimum value of a parabola of the form y = ax² + b
• Recognise that quadratics of the form y = -ax² have a negative orientation

Working Mathematically Skills:

• Connect shape of a non-linear graph with distinguishing features of equation (Communicating, Reasoning)

Resources:

• Guess Who? Suggestion Equations
• Desmos
• Graph paper
• iPads for student use (at least 1 per pair)

Description of Lesson:

Introduction: (15 min)

• Activity One: What's up?
  • Students sketch these functions using tables of values, and check using technology to explore relationships of positive and negative orientations.

• Discussion: What did students notice about the graphs?
  • Hopefully they noticed that some of the graphs were facing different directions to those they've seen before. Discuss why this is, and generalise the role of the negative sign in front of equations.
• Introduce key terms:
  • X and y intercepts
  • Orientation
  • Symmetrical
  • Minimum/maximum value

Body: (35 min)

• Activity Two: Guess Who? [LIT and TECH]
  • To play this game, students will create their own quadratic and exponential equations and try to guess each others. Pair students up so they are roughly matched for ability, which will help to ensure the equations they choose are suitable for each other's stage of learning. Students can use the document in Resources for a ready made equation if they would prefer.
  • Guide students in creating their own equations of the form y = ax² + b or y = ae^x + b. They can make them more interesting and difficult with these steps:
    • Change the value of a to an integer other than one.
    • Add a value for b.
    • Make one or both of the values negative.
    • Choose bigger values for a and b!
  • Using an iPad, students enter the equation into Desmos to explore the effects of changing values. Sitting back to back, they must take turns asking questions about the other's curve using the terms:
    • x and y intercepts
    • orientation
    • symmetrical
    • minimum/maximum value
  • Some example questions:
    • Is your minimum or maximum value greater than 10?
    • Is your curve positively or negatively oriented?
    • Is your curve symmetrical?
  • The students use the information from their dialogue to try to accurately sketch the other person's curve on paper.
  • Note: For this activity, students only need to refer to the coordinates of the vertex, they do not need to calculate them themselves.

Conclusion: (5 mins)

Reiterate the key learning ideas for the lesson.

Lesson Evaluation Questions:

• Did all of your students understand and meet the learning goals?
• Were all students engaged and participating?
• Were the activities appropriate and relevant for the stage of my learners?
• Did the use of technology enrich the quality of the lesson?
• What would you change about the activities next time?

Homework/Enrichment Activities:

Students are to respond to this prompt:

"What is a possible equation for this curve?"

Variations:

• If iPads are not available, graphs can be printed off with the coordinates of key features labelled.
• Advanced students can select equations from the Guess Who document's 5.2/5.3 Level list for a further challenge. Alternatively, they can substitute their own values into the full quadratic equation y = ax² + bx + c. They should be paired up for this.
• The homework activity is already low floor/high ceiling, but it can still be adapted further to best suit your students. Use your own intuition in this.