Mathematics - Senior Secondary 1 - Quadratic equations (III)

Quadratic equations (III)

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TERM: 2ND TERM

WEEK: 3

Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Quadratic Equations (III)
Focus:
i. Plotting graphs involving quadratic and linear functions
ii. Using plotted graphs to solve equations
iii. Finding gradients, maximum and minimum values of a curve
iv. Solving quadratic and linear equations graphically
v. Word problems leading to quadratic equations

 

SPECIFIC OBJECTIVES

By the end of the lesson, students should be able to:

  1. Construct tables of values for quadratic and linear functions.
  2. Plot graphs for quadratic and linear functions accurately.
  3. Solve quadratic equations using graphs.
  4. Identify maximum and minimum values of a quadratic graph.
  5. Solve real-life problems leading to quadratic equations using graphs.

 

INSTRUCTIONAL TECHNIQUES

  • Guided demonstration
  • Graphical plotting
  • Discussion
  • Group work
  • Practical exercises

 

INSTRUCTIONAL MATERIALS

  • Graph boards
  • Graph books
  • Rulers and pencils
  • Mathematical sets
  • Pre-drawn graphs on manila paper

 

PERIOD 1 & 2: Plotting a Graph of a Quadratic and a Linear Function

PRESENTATION

Step

Teacher’s Activity

Student’s Activity

Step 1 – Introduction

Recaps previous lessons on quadratic expressions and equations. Introduces the idea of using graphs to solve them.

Students listen and ask questions.

Step 2 – Construct Table of Values

Demonstrates how to choose values of x and compute corresponding y values for both quadratic and linear functions (e.g., y = x² - 4x + 3 and y = x + 1).

Students calculate values and fill in tables.

Step 3 – Plotting

Guides students to draw the axes, choose suitable scales, label the axes, and plot the points. Draws the curve and straight line.

Students follow along and plot in their graph books.

Step 4 – Interpretation

Leads students to observe the points of intersection and interpret them as solutions to the equation.

Students identify and record solutions.

NOTE ON BOARD
To plot quadratic y = ax² + bx + c and linear y = mx + c:

  1. Make a table of values.
  2. Draw axes with chosen scale.
  3. Plot points and connect.
  4. Points of intersection = solution(s) to equation.

 

EVALUATION (5 exercises)

  1. Complete the table of values for y = x² - 4x + 3 for x = 0 to 5.
  2. Plot the graph of y = x² - 4x + 3.
  3. Plot the graph of y = x + 1.
  4. Find the points of intersection of the two graphs.
  5. What are the solutions to x² - 4x + 3 = x + 1?

 

CLASSWORK (5 questions)

  1. Plot y = x² - 2x - 3 for x = -1 to 4.
  2. Plot y = 2x - 1.
  3. Find the intersection points.
  4. What equation do the intersection points solve?
  5. Use the graph to find the roots of x² - 2x - 3 = 2x - 1.

 

ASSIGNMENT (5 tasks)

  1. Create a table for y = x² - 6x + 8.
  2. Create a table for y = 2x + 1.
  3. Plot both graphs and find their intersection points.
  4. What equation is solved at the points of intersection?
  5. Explain in your own words how graphs can be used to solve equations.

 

PERIOD 3 & 4: Maximum/Minimum Values and Gradient of Curves

PRESENTATION

Step

Teacher’s Activity

Student’s Activity

Step 1 – Introduction

Reviews plotted quadratic graphs and highlights the turning point. Explains concept of maximum and minimum.

Students listen and ask questions.

Step 2 – Identifying Max/Min

Demonstrates how to find the highest/lowest point on the graph and the corresponding x and y values.

Students identify turning points on their graphs.

Step 3 – Gradient Concept

Explains the concept of gradient as rate of change. Shows that the gradient of a curve varies and can be estimated at points using tangents.

Students draw tangent lines and estimate gradients.

Step 4 – Guided Practice

Guides students to find min/max values and estimate gradients at different points on the curve.

Students work in pairs to complete the task.

NOTE ON BOARD

  • Maximum value: highest point on curve
  • Minimum value: lowest point on curve
  • Gradient = steepness; slope of the tangent at a point

 

EVALUATION (5 exercises)

  1. From your graph, what is the minimum value of y?
  2. At what value of x does y reach a maximum/minimum?
  3. Estimate the gradient at x = 1 using a tangent.
  4. Estimate the gradient at x = 3.
  5. Explain the meaning of a negative gradient.

 

CLASSWORK (5 questions)

  1. Draw a tangent to the curve y = x² - 4x + 3 at x = 2.
  2. Estimate the gradient at that point.
  3. Identify the minimum point on the graph.
  4. What does the minimum point represent?
  5. Is the graph increasing or decreasing at x = 1?

 

ASSIGNMENT (5 tasks)

  1. Plot y = -x² + 4x - 2.
  2. What is the maximum value of y?
  3. At what value of x does this occur?
  4. Draw a tangent at x = 1 and estimate its slope.
  5. What is the difference between a maximum and minimum point?

PERIOD 5: Solving Word Problems Using Graphs

PRESENTATION

Step

Teacher’s Activity

Student’s Activity

Step 1 – Introduction

Presents real-life problems that can be modeled with quadratic equations.

Students listen and try to relate.

Step 2 – Modeling

Shows how to translate word problems into quadratic equations. Example: "The height of a ball t seconds after being thrown is given by h = -5t² + 20t."

Students note steps of modeling.

Step 3 – Graphing

Leads students to plot the height-time graph. Guides them to find the time the ball reaches the maximum height, and when it hits the ground.

Students plot and interpret graph.

Step 4 – Solution

Students use the graph to answer real-life questions: maximum height, when it hits ground, etc.

Students respond using the graph.

NOTE ON BOARD

  • Translate word problem → equation
  • Make table of values
  • Plot and interpret graph
  • Use graph to answer questions

 

EVALUATION (5 exercises)

  1. A ball’s height is given by h = -5t² + 20t. Plot this for t = 0 to 5.
  2. What is the maximum height?
  3. At what time does the ball hit the ground?
  4. At what time does it reach the maximum height?
  5. What does the curve represent?

 

CLASSWORK (5 questions)

  1. A rectangular field’s area is 100 m². Its length is x and width is (x - 5). Find x.
  2. Form a quadratic equation from the problem.
  3. Plot the graph and find dimensions.
  4. Identify the value of x that gives maximum area.
  5. What is the maximum area?

 

ASSIGNMENT (5 tasks)

  1. A toy is thrown upwards and its height is modeled by h = -2t² + 12t.
  2. Create a table of values for t = 0 to 6.
  3. Plot the graph of h against t.
  4. Use your graph to find maximum height.

When will the toy return to the ground?