Lesson Plan for Secondary Two Mathematics Topic: Graphs for Linear Equations in Two Unknowns

Lesson ObjectivesAt the end of the lesson, students would have:

Drawn graphs of linear equations with two unknowns given equations, paired-coordinates, gradient or y-intercept. Manipulated and form the equation y=mx + c given the graph Understand how the m (gradient) and c (y-intercept) values affect the graph Solved various problems involving linear equations and its graphs

Next lesson: Review graphs of linear equations. Solve simultaneous equations using graphical method

Time Activities Teacher Students Comments15min Introduce the

topic using a verbal problem (testing prior knowledge and showing relevance of topic to life)

Introduce topic using amusement park cost problem. Giving the scenario and organizing the information in a table on the board.

Scenario:There is an amusement park that’s recently opened. You and a bunch of friends are checking out the prices so that you can estimate how much money to request from your parents. There is an admission fee of $10 for students. Each ride and game is $5 per go. If you are interested in 3 rides and 2 games, how much money would you spend? Your friend, May is going on all 12 rides and not interested in games, how much would she need? If Bob is not interested in rides or games and just wants to be there to socialize, does he have to pay? Giving more random nos. and ask students to come out with a formula for this problem. Link using the values on the board to draw

a line graph. Test for prior knowledge by asking students

to link the formula to the graph. Ask if students are able to identify m and c. Divide students into groups according to

their strengths if necessary. Give out worksheet to all students. Explain

how the worksheets work for both groups.

Listen carefully to scenario and give answers when prompted.

Write down the questions and copy example from the board.

In the first lesson, I will explain the purpose of grouping.

45 mins Activity One – Learning Track

L Go through the key

points with

P Explain to students

how to proceed

L Work through

the worksheets,

P Go through the

key points and

To cater to different learning level, so that

students, emphasizing the important points.

Using examples and work through the worksheets with students.

Ask students to complete steps along the way.

with the worksheet.

Be clear that students are welcome to listen in with the rest of the class at any time without telling me.

highlighting important tips and fill in the blanks.

Answer questions when prompted and ask questions when in doubt.

do a ‘self-check’ by ticking the boxes.

Highlight important tips along the way.

Proceed to answering questions if ALL boxes are ticked.

At any point, if student feels that he/she needs help, they should listen to the explanation with the learners.

weaker students get proper explanation and capable students are not wasting time listening to what they already know. They should proceed to practice questions.

5 mins RBT- Refresh Brain Time(toilet time)

Give a couple of math puzzles on the board.

Go for toilet break, drink. Try solving the puzzles while

waiting for others to return.30 mins Activity Two

– Role reversal

Explain the activity to the students.

Explain the requirement of question 4 using examples.

Assign each student with a question to start off with ‘randomly’.

Students must work through that particular question.

Using that question, student must now pretend to be ‘teacher’ and explain that question to me, the ‘student’.

Students are allowed to switch question if they genuinely have trouble with the one given.

Give everyone 10 mins to start on their questions and then proceed to individual students.

Students must start with the question given and prepare to verbally explain their working to me.

While waiting for their turn, students will proceed with the rest of the questions.

‘Randomly’ assign means that I can match the questions to the ability of the students so that everyone can be involved in the activity and students can build on their confidence. By students explaining to me, I can check for understanding.

20 mins Question and Answer time

Help students with questions from school work.

Hand out solutions to the worksheet

Students can ask questions from homework.

If students have no

given today. Give out practice worksheet

(homework) Conclude the lesson by asking to

students to think of 3 key points to share with their neighbour.

Release students

questions, they should proceed to the challenge questions from the worksheet.

Students should mark their own work from the worksheets and carefully do corrections at home.

Share 3 key points that they have learnt today with their neighbour.

Self-reflections:

Name: _______________________

Graphs of Linear Equations- LEARNING TRACK

Introduction:

Formula:

Linear Equation: let x be _____________and y be _____________

:

No. of rides and games

Cost

1. All graphs of linear equations must be straight lines.

2. All linear equations takes the form of y = mx + c (m and c will be covered later)

*Can you identify which of the following equations are linear?

y=6x – 4 x+2=y s =4t +10 x=6 g=3h

3x – y +5=0 4x=y – 9 a2=3b – 5 2x+y=10 1 – 6u = v

REMEMBER:

3. m stands for gradient. Gradient is the steepness of the line. It is like climbing a hill or ladder, for every step you walk across, you are also going up at the same time.

/

The larger the gradient, the _____________the line. If the gradient is positive, the line slopes upwards to the ___________.If the gradient is negative, the line slopes upwards to the ___________.If two lines have the same gradient, then they must be _______________You can find the gradient of a line using any 2 points on the graph.

m = y

2

– y1

(rise) Example:

x2 –x

1 (run)

Follow the teacher’s instruction and fill

in the blanks.

Highlight all key points!

Go through each key

point carefully and tick

each point when you

can understand it. Proceed

to the questions only when you can

confidently tick ALL points!

Gradient is

‘Rise’Run

run

rise

y = - x

y = x y = 2x

y = 6x y

x

?

4. c stands for the y-intercept. It is the point at which the line cuts the y-axis. *Identify the y-intercept of each graph below.

Value of c Line s:

Line t:Line u:Line v:

In the linear equation, it is the constant. (eg. like the entrance fee of the amusement park)

It is the y value when x = ____

5. Every straight line has a specific formula, y = mx + c. Every point that sits on the line must follow this formula or rule. Therefore, when the paired-coordinate values are substituted into the equation and the answer is valid, the point must be on the line.

Example: y = - x + 3Prove that (2,1) sits on the line and (1,6) does not using substitution.

`\\\\

6. To draw a line graph given the equation, you only need 2 points to link up, but use 3 to be sure.

1) Construct a table of x and y values. 2) Choose 3 consecutive x values. *Choose wisely! Keep it low.3) Substitute the x values into the equation to get the corresponding y values.4) Choose your scale carefully. Look at the range of your values.5) Plot the 3 points on the graph and link up to form the line. 6) Be sure to LABEL your graph with the equation!

Example: Plot the graph of y=2x -7 on the graph above (in section 5)

v

s

t u

(2,1)

(1,6)

Practice time!

1. Complete the table. Then plot the coordinates and draw the graph of each equation on the axes provided.

a) y = x + 2 b) y = 2x

x

y

c) y = – x – 3 d) y = 5 – x

x

y

e ) 2x + 5y = 10 f) 2y – 3x + 6 = 0

x

y

WAIT! CHECK!

m = 1

(+ve slopes up to the right)

c =

x

y

g) 1.5 – y = 0 h) x + y – 3 = 0

x

y

2. Sketch the graph of y = mx + c where:

x

y

x

y

3. Match the equation with its graph.

m > 0, c > 0 y

x0

m < 0, c < 0 y

x0

m = 0, c > 0 y

x0

m < 0, c = 0 y

x0

3x + y = 21 x + 7y = 48 x + 2y = 105x + 3y = 30 x + 5y = 51 6x + 7y = 77x + 2y = 1 2x - 7y = 49 6x - 7y = 7

1.   2.   3.  

4.   5.   6.  

7.   8.   9.  

Now can you do the reverse?

4. Given 2 coordinates, plot the graph and write down the equations of the straight lines. Use the dice given to generate the x and y values of the coordinates of the 2 points or simply use your lucky numbers. For those of you who like to challenge yourself, be sure to include lots of negative numbers. Use the graph paper of the following page for all your equations and make sure you label each line carefully!

*Hint: All linear equations are y=mx+cTherefore you just need to find m and c from your graph.

a) Point 1 ( ____, ____) Point 2 ( ____, ____) b) Point 1 ( ____, ____) Point 2 ( ____, ____)c) Point 1 ( ____, ____) Point 2 ( ____, ____)d) Point 1 ( ____, ____) Point 2 ( ____, ____)e) Point 1 ( ____, ____) Point 2 ( ____, ____)

5. Using the graph paper below, draw the graphs of the lines y = 0, x = 6 and y = – x + 1. Calculate the numerical value of the area of the triangle bounded by the lines y = 0, x = 6 and y = – x + 1.

6. State if the ordered pair lies on the line. Show working to prove.

a) (– 1, – 4) x – 2y = 7 b) (10,3) 3x +5y =15

c) (–, 2) 4x +y = 0 d) (3,3) 7x – 8y + 3 = 0

7. a) Write down the equation of the vertical line that passes through point (3, - 2).

b) Write down the equation of the horizontal line that passes through point (11, 8).

8. The points (d, 0), (0, e) and (4, f) lie on the line 5x – 4y = 8. a) Find the values of d, e and f. b) State whether ( – 3, 4) lies on the line.

9. Find the values of p, q and r if the following points lie on the line 2x + 3y = 9.a) ( – 3, p) b) (2q – 3, – 1) c) (4r, 5 – 3r)

10. For each of the tables shown, determine whether it is a linear function.

a) b)

c) d)

Challenge begins…

11. a) The line x = – 3 meets the x-axis at P. Write down the coordinates of P.b) The line y = 4 meets the y-axis at Q. Write down the coordinates of Q.c) The line x = – 3 and y = 4 meet at R. Write down the coordinates of R.d) Calculate (i) the area of ∆PQR and (ii) perimeter of ∆PQR.

12. a) Given the equation 3x – 4y – 12 = 0, complete the table below.

x 0 4 8

y

b) Draw the graph of 3x – 4y – 12 = 0.c) Given that the point (h, 1.8) is a solution to the equation 3x – 4y – 12 = 0, find the value of h. d) The graph 3x – 4y – 12 = 0 intersects the x-axis at the point A and the y-axis at the point B.

Find the area of triangle AOB where O is the origin.

13. It is given that the points (2, – 4) and (– 8, 1) lie on the line hx + ky + 6 = 0.a) Find the values of h and k.

x 0 1 2 3 4 5

y 5 10 20 40 80 160

x 1 2 3 4 5 6

y 21 17 13 9 5 1

b) Draw the above graph for – 2 ≤ x ≤ 6.c) Write down the y-intercept of the graph.d) If (r, – 5) lie on the graph, find the value of r.

14.a) Draw the graph of each of the following equations on the same axis.(i) x = 1 (ii) x = – 2 (ii) y = 2x (iv) y = 2x + 8

b) Name the figure formed by these four lines.c) Find the area of the figure formed.

15.The point A is the x-intercept of the line x + 3y = 10.Find the coordinates of A. Another point B is the y-intercept of the line 3x – 5y = 15. Find the coordinates of B. Hence, find the numerical value of the area of triangle AOB, where O is the origin.