Student activity 1
Assume all balances in these equations are balanced unless told otherwise.
1.
What can we tell about the weights of the shapes used in this balance?
2.
What can we tell about the weight of a cylinder and the weight of a box in the
following balance?
3.
What can we tell about the weight of a cylinder and the weight of a
following balance?
4.
What can we tell about the weight of a
in the
in the following balance?
9
X
5.
What can you tell about X in the following balance?
6.
What can you tell about Y in the following balance?
7.
What would you need to add to the left hand side to make this balance balanced?
8.
How would you achieve balance in this situation?
9
Y
7
9
4
846
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
9.
What can we tell about the weight of a
in the following balance?
10. What can you tell about X in the following balance? Explain how you got your
answer.
8
2X
11. What can you tell about Y in the following balance? Explain how you got your
answer.
18
2Y
19
X+1
12. What can you tell about X in the following balance? Explain how you got your
answer.
20
X+2
13. What can you tell about X in the following balance? Explain how you got your
answer.
X
9
14. If x=8, what will you do to achieve balance?
X
9
15. If we know this balance is not balanced, what number can x not be?
16. When a balance is balanced how do the weights on the right hand side relate to
those on the left hand side? What do a balance and an equation have in common?
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
Student Activity 2
1. Complete the table of rules for backtracking.
RULES FOR
BACKTRACKING
ORIGINAL
BACK (REVERSE )
ACTION
ACTION
+
X
2.
John thinks of a number, multiplies it by 3 and adds 2 to his answer. The result is
17.
a. Using Backtracking, what number did he think of?
b. Write an equation to represent this problem.
c. Solve the equation.
d. How did your answers for part a. and c. relate?
3.
Sarah thinks of a number, multiplies it by 4 and adds 5 to her answer. The result is
25.
a. Using Backtracking, what number did she think of?
b. Write an equation to represent this problem.
c. Solve the equation.
d. How did your answers for part a. and c. relate?
4.
Dillon thinks of a number, multiplies it by 3 and subtracts 5 from his answer. The
result is 7.
a. Using Backtracking, what number did he think of?
b. Write an equation to represent this problem.
c. Solve the equation.
d. How did your answers for part a. and c. relate?
5.
Saoirse thinks of a number and divides it by 2 and adds 5 to her answer. The result
is 9.
a. Using Backtracking, what number did she think of?
b. Write an equation to represent this.
c. Solve the equation.
d. How did your answers for part a. and c. relate?
6.
Susan thinks of a number and divides it by 3 and subtracts 5 from her answer. The
result is 14.
a. Using Backtracking, what number did she think of?
b. Write an equation to represent this.
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
c.
d.
Solve the equation.
How did your answers for part a. and c. relate?
Solve the following equations:
a. 2x=4
b. 3x+1=9
7.
4x-4=44
d.
e.
11x-5=39
c.
5x+4=21
f.
11=3x-4
Student activity 3
N.B. When asked to solve equations, always check answers.
1. Solve the following equations:
2x=8
2x+1=9
2x-1=7
3x-7=2x
4x+6=2x
3x-4=2x+9
2x+3x+4=19
1.
Brendan thinks of a number, adds 3 and the answer is 15. How can this be
represented as an equation? Solve the equation and check your answer.
2.
Joanne thinks of a number then subtracts 5 and the answer is 10. How can this be
represented as an equation? Solve the equation and check your answer.
3.
Martha has a certain number of sweets in a bag and she gives half to Mary and Mary
gets 20. How can this be represented as an equation? Solve the equation and check
your answer.
4.
A farmer has a number of cows and he plans to double that number next year,
when he will have 24. How can this be represented as an equation. Solve the
equation and check your answer.
5.
A new student enters a class and the class now has 25 students. How can this be
represented as an equation. Solve the equation and check your answer.
6.
The temperature increases by 18 degrees and the temperature is now 15. How can
this be represented as an equation? Solve the equation and check your answer.
7.
A farmer doubles the amount of cows he has and then buys 3, he now has 29.
Represent this as an equation. How many did he originally have?
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
8.
Emma and her twin brother will have a total age of 42 in 5 year’s time. Represent
this as an equation. How old are they at the moment?
Solve the following equations and show what backtracking actions you did for each
equation and check your answers:
a. 2x=8
b. 2x+1=x+7
10. Peter cut a length of ribbon into five equal parts. Each part was 30 cm long. How
can this be represented as an equation? Solve the equation to discover how long
the ribbon was before it was cut.
9.
11. Write a story that the equation 2x=10 could represent.
12. Write a story that the equation 2x+5=11 could represent.
13. Write a story that the equation 3x-5=13 could represent.
14. Write a story that the equation 3x-5=2x+13 could represent.
Student Activity 4
N.B. When asked to solve equations, always check answers.
1. Can you solve the equation 2x=2x+1? Why or why not?
2.
Think of a number, double it and subtract three and the answer is my original
number. Represent this as an equation and solve it to find your original number.
3.
A table’s length is 6 metre longer than its width and the perimeter of the table is
24 metres. Allow x to represent the width of the table write an
equation to represent this problem and solve the equation.
4.
A woman is twice the age of her daughter, who is now 23. Find an
equation to represent this situation and solve the equation.
5.
A woman is twice the age of her daughter, who in 3 years time will be 33. Find an
equation to represent this situation and solve the equation.
6.
Mark had some cookies He gave ½ of them to his friend John. They divided the
remaining cookies evenly between his other three friends and each got 4. How
many cookies did Mark originally have? Design an equation to represent this
situation and solve the equation.
7.
Chris has €400 in his bank account and he deposits €5 per week. His brother Ben
has €582 in his account and withdraws €8 per week. If this pattern continues, how
many weeks will it be before as they have the same amounts in their bank
accounts?
The sum of three consecutive numbers is 51. What are the numbers?
Draft 01
©Project Maths Development Team
www.projectmaths.ie
Page 1 of 13
8.
9.
a.
b.
Make a list of 4 points on this line.
What is added to each x to give the y
value?
c.
So is it true to say the line is y=x+3?
d.
Can we read from the graph the point
where y=0 or x+3=0?
e.
Solve the equation x+3=0 by algebra.
f.
Do you get the same answer when you
graph the line y= x+3 and find where it cuts the x axis as you get when you
solve the equation x+3=0 by algebra?
10. Complete the following table and draw
the corresponding graph and use the
graph to find where 2x+2=0 cuts the x
axis.
x
-2
-1
0
1
2
3
Y=2x+2
a.
Solve the equation2x+2=0 by algebra.
b.
Do you get the same answer when you graph the line y=2x+2 and find where
it cuts the x axis as you do when you solve the equation 2x+2=0 by algebra?
11. Solve the equation 2x-6=0 graphically.
12. Solve the equation 2x-3=0 graphically.
Student Activity 5
Note it is always good practice to check solutions where possible.
Draft 01
©Project Maths Development Team
www.projectmaths.ie
Page 1 of 13
1.
List a set of possible solutions for x+ y=12.
2.
How does the equation of the form x + y=12 differ from an equation of the form 2x3=5?
3.
Two numbers add up to 11. The difference in these numbers is 1. Write 2 equations
to represent this problem. Draw graphs of these equations. Find where the 2 lines
meet. What were the 2 numbers?
4.
1000 tickets were sold to a concert. x adults attended and y children attended.
Write an equation for the number of people who attended the concert in terms of x
and y. Draw a graph of this equation. Adult tickets cost €10, a child's ticket costs
€5, and a total of €500 was collected. Write the amount of money collected in
terms of x and y. Draw a graph of this equation on the same axis as the previous
line. Where do the two lines meet? How many tickets of each kind were sold?
Simultaneous equations are a set of equations with 2 or more
unknowns.
Solve the following simultaneous equations:
2x+y=10
x+2y=10
X=3
y=3
3x+y=10
x-y=2
2x+3y=13
x-2y=3
2x+3y=13
x-2y=3
5.
3x+y=13
x-3y=1
6.
A certain book costs €4 and 3 of these books plus 5 CDs cost €27. Write an equation
in terms of x to represent this information. Solve this equation to find the cost of a
CD.
7.
In the school canteen 1 roll and 2 pieces of fruit cost €4.20 and 3 rolls and 1 piece
of fruit cost €9.60. Write two equations in terms of x and y to represent this
information. Solve these equations to find the cost of a roll and the cost of a piece
of fruit.
8.
John is the owner of a shop. If he hires 4 sales assistants and 1 security guard, his
daily payroll is €480, while 2 sales assistants and 1 security guard require a daily
payroll of €300. Write two equations in terms of x and y to represent this
information. Solve these equations. What are the daily wage of a sales assistant
and the daily wage of a security guard?
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
9.
If the sum of two numbers a and b is 45 and their difference is 3. Write two
equations in terms of a and b to represent this information. Solve these equations
to find the two numbers.
10. A car park charges €a to enter and €b per hour after that. John pays €14 for 4 hour
parking and Sara pays €20 for 6 hours parking. Write two equations in terms of a
and b to represent this information. Solve these equations to find the cost to enter
the car park and the cost per hour of parking.
11. 5 oranges and 3 apples cost €2.10 and 3 oranges and 1 apple cost €1.10. Write two
equations in terms of x and y to represent this information. Solve these equations
to find the cost of an orange and the cost of an apple.
12. There are a number of rabbits and budgies in a cage. Altogether there are 29 heads
and 98 legs. Represent this problem as an equation and solve the equation. How
many of each animal was in the cage?
13. Twice the first number plus 3 times the second number is 7. 5 times the first
number minus twice the second number is 9. Write two equations in terms of x and
y to represent this information. Solve these equations to find the two numbers.
14. Solve the simultaneous equations 3x+2y=11 and 2x+3y=9.
15. Write a story that the following set of simultaneous equation could represent:
2x+y=11
x-2y=3
Student activity 6
Note it is always good practice to check solutions.
1. What is meant by the factors of a number and find the factors of 12?
2.
If xy=0, what value must either x or y or both have?
Note: The roots of a quadratic equation are another name for its solution set.
3. Solving an equation means finding a value for the unknown(s) that make the
equation true. Solve the equation (x-2) (x-3) = 0.
4.
Solve the equation (x-4) (x-5) = 0.
5.
Solve the equation (x-4) (x+5) = 0.
6.
What values of x make the equation (x-2) (x+4) = 0 true?
7.
Find the roots of (x-4)(x+5)=0
8.
Solve the equation (x-3) (x+2) = 0. State what the roots of (x-3) (x+2) = 0 are.
9.
Solve the equation x(x-1) = 0.
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
10. Solve the equation x(x-2) = 0.
11. Find a positive value for x that makes the equation(x-4) (x+2) = 0 true.
12. Solve the equation x(x+4) = 0.
13. If xy=0 can x=6 and y=6? Explain.
2
14. Find 2 values for x that make the equation x - 25 = 0 true.
15. Think of a number square it. Subtract 25 from it and the answer is 0. Represent this
as an algebraic equation and solve the equation.
2
16. Factorise x - 25 = 0 and solve the equation.
17. What are the roots of the equation represent in the graph below?
18. What are the roots of the equation
represent in the graph below?
19. You know the solutions to a quadratic equation are 2 and 3. Write the equation?
20. You know the solutions to a quadratic equation are 2 and -3. Write the equation?
21. You know the solutions to a quadratic equation are -2 and -3. Write the equation?
22. You know the solutions to a quadratic equation are 2 and ¾. Write the equation?
23. You know the roots of an equation are 2 and 3. Write the equation?
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
Student Activity 7
Note it is always good practice to check solutions.
2
1. Find the factors of x +5x+4. Using the factors you got for the first part of this
2
equation, solve x +5x +4=0.
2.
Find the factors of x2+6x+8. Using the factors you got for the first part of this
equation, solve x2 +6x +8=0.
3.
Find the factors of x2-9x+8. Using the factors you got for the first part of this
equation, solve x2 -9x +8=0.
4.
Find the factors of x2-6x+8. Using the factors you got for the first part of this
equation, solve x2 -6x +8=0.
5.
Find the factors of 2x2 - 5x +3. Using the factors you got for the first part of this
equation, solve 2x2 - 5x +3=0. (Higher level.)
6.
Find the factors of 3x2-10x+8. Using the factors you got for the first part of this
equation, solve 3x2-10x+8 =0. (Higher level.)
7.
Solve the equation x(x-1) = 6.
8.
Solve the equation x2 +2x=3.
9.
When a number is added to its square the result is 12. Represent this problem as an
equation and solve the equation to find the number.
10. One number is 3 greater than the other and their produce is 28. Write an algebraic
equation to represent this and solve this equation.
2
11. The area of a garden is 50 cms and you know the width of the garden is 5 cms less
than the breadth. Represent this as an algebraic equation and then solve the
equation. Use this information to find the dimensions of the garden.
2
12. A garden with an area of 99 m has length x m and its width is 2m longer than its
length. Write its area in term of x. Solve the equation to find the length and width
of the garden.
13. The product of 2 consecutive numbers is 110. Represent this as an algebraic
equation and solve the equation
14. Represent this problem as an equation and solve the equation to find a value for x.
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie
15.
Draw the graph of x2+2x +1 for values of x between -2 and 2. Where does
this graph cut the x axis?
2
b. Factorise and solve x +2x +1=0.
c. What do you notice about the values you got for part a) and part b)?
a.
16.
Draw the graph of x2+3x+2 for values of x between -3 and 3. Where does
this graph cut the x axis?
2
b. Factorise and solve x +3x +2=0.
c. What do you notice about the values you got for part a) and part b)?
a.
17. What is the relationship between the roots of a quadratic and where the graph of
the same quadratic cuts the x axis?
2
18. The area of the triangle opposite is 8 cms . Find the
length of the base and the height of the triangle.
19. One number is 2 greater than another number. When
these two numbers are multiplied by each other the
result is 15. Represent this problem as an equation
and solve the equation.
2
20. Solve x -36=0.
Higher level
21.
Draw the graph of 3x2-10x+8=0 for values between -3 and 3. Where does this
graph cut the x axis?
2
b. Factorise and solve 3x -10x+8=0.
c. What do you notice about the values you got for part a) and part b)?
a.
Draft 01
©Project Maths Development Team
Page 1 of 13
www.projectmaths.ie