Mathematics · Year 9

Active learning ideas

Investigating Vertical Translations of y=x^2

Active learning works for this topic because visual and hands-on approaches help students distinguish between shifts and stretches. Students see the exact numerical change to every y-value, which clarifies that the parabola moves without changing shape or width.

• ACARA Content Descriptions
• AC9M9A06

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Stations Rotation
• 30 min · Pairs

Pairs Plotting: Table and Graph Comparison

Pairs create tables of values for y = x² and y = x² + 3 using x from -3 to 3. They plot both graphs on grid paper and mark the vertices. Pairs then repeat for y = x² - 2 and note patterns in a shared chart.

How does adding a constant 'c' to y=x^2 change the y-values in the table?

Facilitation Tip

During Pairs Plotting, circulate and ask each pair to mark the vertex on both graphs in the same color to spotlight the vertical shift.

What to look for

Provide students with the equation y=x^2+3. Ask them to: 1. State the coordinates of the vertex. 2. Describe how the graph compares to y=x^2. 3. Calculate the y-value when x=2.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 02

• Stations Rotation
• 35 min · Small Groups

Small Groups: Digital Slider Investigation

In small groups, students use Desmos or GeoGebra to graph y = x² + c and vary c from -5 to 5. They record vertex positions and sketch three examples. Groups share one key observation with the class.

What is the relationship between the constant 'c' and the vertical position of the parabola's vertex?

Facilitation Tip

In Small Groups: Digital Slider Investigation, remind students to record the vertex coordinates each time they adjust c before moving on.

What to look for

Display two graphs on the board: y=x^2 and y=x^2-2. Ask students to identify the value of 'c' for the second graph and explain, in one sentence, how the vertex has moved.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 03

• Stations Rotation
• 25 min · Whole Class

Whole Class: Predict and Verify Demo

Display y = x² on a projector. Students predict vertex and two points for y = x² + 4, then reveal the graph. Repeat for c = -3, with class voting on predictions before discussion.

Compare the graphs of y=x^2 and y=x^2+c by plotting key points.

Facilitation Tip

For Whole Class: Predict and Verify Demo, pause after predictions and ask three volunteers to share their reasoning before revealing the answer.

What to look for

Pose the question: 'If you were given the graph of y=x^2 and asked to shift it upwards by 5 units, what would be the new equation? How would you explain your reasoning to someone who has never seen this before?'

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 04

• Stations Rotation
• 20 min · Individual

Individual: Multiple Shifts Challenge

Each student plots y = x² + c for c = 1, 2, -1 on one set of axes from provided tables. They label vertices and write one sentence on the shift rule.

How does adding a constant 'c' to y=x^2 change the y-values in the table?

Facilitation Tip

During Individual: Multiple Shifts Challenge, provide graph paper with labeled axes to prevent scaling errors and keep focus on the shift.

What to look for

Provide students with the equation y=x^2+3. Ask them to: 1. State the coordinates of the vertex. 2. Describe how the graph compares to y=x^2. 3. Calculate the y-value when x=2.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Teachers approach this topic by starting concrete, moving to visual, then abstract. Students first calculate and plot points to see the constant shift in tables, then overlay graphs to observe unchanged shape and width. Avoid rushing to the rule; let students articulate the pattern after they have evidence. Research shows linking the constant c to the y-shift of every point reduces later confusion with horizontal shifts or stretches.

When students finish, they will accurately identify the vertex of any vertical translation of y=x² and describe how the graph has moved compared to the original. They will connect the constant c in the equation to the vertical shift of every point on the curve.

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Watch Out for These Misconceptions

• During Pairs Plotting: Table and Graph Comparison, watch for students who describe the parabola as getting taller or wider when c is positive.

  Have the pair list the y-values for y=x² and y=x²+5 side by side and calculate the difference. They will see the y-values increase by exactly 5, confirming the shape and width remain the same.

• During Small Groups: Digital Slider Investigation, listen for students who say the vertex moves left or right when they increase c.

  Ask each group to pause the slider at three values of c and record the vertex coordinates. They will observe the x-coordinate stays at 0 while the y-coordinate changes to c.

• During Whole Class: Predict and Verify Demo, watch for students who believe a negative c flips the parabola upside down.

  Have students sketch y=x² and y=x²-3 on the same axes. They will see the curve still opens upward but sits lower, confirming the orientation does not change.

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Methods used in this brief

• Stations Rotation