Circles

ACARA Syllabus Indicators:

• Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technologies as appropriate (ACMNA239)
  • Use digital technologies to graph simple quadratics, exponentials and circles.

Learning goals:

• Recognise the formula for a circle as x² + y² = r²
• Use the formula for a circle to derive the radius
• Sketch circles centred at the origin

Working Mathematically Skills:

• Connect the shape of a non-linear graph with the distinguishing features of its equation (Communicating, Reasoning)
• Derive the non-linear relationship of a circle to represent possible locations of enemy ships in Battleships activity (Problem Solving)

Resources:

• Desmos Battleships activity [Activity idea sourced from here, I just made it into a Desmos Classroom activity since the original platform for the activity no longer exists]

Description of Lesson:

Introduction: (20 mins)

• Activity One: Battleships [TECH]
  • Pose the situation in the image to students; allow them to plot guesses of where the enemy ship might be through Desmos Student, entering their guesses using the (x,y) function. Let discussion lead to a rationale for a circular relationship.

• Discussion:
  • We can see a relationship between these points on our graphs; they are all the same distance away from our enemy ship. How could we represent this circular relationship algebraically?

Body:

• Activity Two: Battleships (30 mins)
  • Now students can apply their knowledge to a game of Battleships! Provide students with graph paper like the one in the resources (one graph per student), and pair them up.
  • Students can place five army submarines (shown as dots) on their graphs. The coordinates of their submarines must be integer values. Students need to keep the locations of their submarines secret from their partners, who are the enemy.
  • To play, students take turns guessing coordinate locations of their opponents submarines. If they do not guess correctly, their opponent must tell them how many kilometres their guess was from their nearest submarine. One square = 1 km. For example, "your guess was 3km away from one of my submarines". The player can then narrow down the location of that enemy submarine by testing other integer coordinates 3km away from their incorrect guess.
  • The first player to find all their enemies submarines wins!

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:

On Desmos, students are to create a circle with equation x² + y² = r² , with a slider for r. Students explore what effect varying r has on the circle, and sketch their findings in their books.