Integer games provide some fun while building and strengthening your Integer skills.
Search the internet for some of the many interactive Integer games and applets.
What is the difference between a Java applet and a Flash applet?
Technology
+/− or (−) is normally on the bottom row of your calculator.
Can you complete a WebQuest from start to finish?
Chapter Review 1
Exercise 2.16
1 Change the following words to an integer:
a) A drop of 2°C. b) A increase of 15%.
c) A profit of $25. d) Shortened by 12 cm.
2 Copy each pair of numbers and place the correct < or > between them.
a) 3 ˉ2 b) ˉ4 1 c) ˉ2 ˉ5
3 Arrange the integers in ascending order (smallest to largest):
a) 3, 0, ˉ1, ˉ5 b) 2, ˉ1, 0, ˉ2 c) ˉ4, ˉ3, 1, ˉ2 4 Calculate the following:
a) 6 − 4 b) ˉ2 + 1 c) ˉ5 – 4
5 Calculate the following:
a) 7 − 5 b) ˉ4 + 2 c) ˉ6 – 1 6 Joseph owes each of six friends $40.
What is Joseph's balance.
7 The temperature at midnight was 4 °C. If the temperature dropped a further 9 °C, what is then the temperature?
8 Ashley borrowed $150, repaid $70, and borrowed another $20.
What is Ashley's balance?
9 Abhaya is driving at 110 km/h. Abhaya slows down by 30 km/h and then a further 30 km/h. What is Abhaya's speed?
10 The Roman civilisation began around 500 BC and finished around 475 AD.
How long did the Roman civilisation last?
−ˉ3 = 3 −ˉ1 = 1
Exercise 2.17
1 Change the following words to an integer:
a) A loss of $250 million. b) A rise of 15 m.
c) $500 under value. d) No change.
2 Copy each pair of numbers and place the correct < or > between them.
a) 5 ˉ1 b) ˉ3 3 c) ˉ4 ˉ3
3 Arrange the integers in ascending order (smallest to largest):
a) 2, ˉ4, ˉ2, 3 b) 4, ˉ2, ˉ4, 0 c) ˉ3, 1, ˉ2, ˉ1 4 Calculate the following:
a) 2 − 3 b) ˉ7 + 2 c) ˉ1 – 3
5 Calculate the following:
a) 4 − 1 b) ˉ2 + 2 c) ˉ9 – 2 6 Andre owes each of four friends $70.
What is Andre's balance.
7 The temperature at midnight was −4°C. If the temperature dropped a further 3°C, what is then the temperature?
8 Ming borrowed $350, repaid $220, and borrowed another $60.
What is Ming's balance?
9 A submarine at 120 m below sea level rose 50 m. What is the position of the submarine relative to sea level?
10 Mercury has a melting point of −40°C and a boiling point of 357°C. What is Mercury's temperature range from melting point to boiling point?
−ˉ3 = 3 −ˉ1 = 1
A LITTLE BIT OF HISTORY
Karl Friederick Gauss (1777-1855) was one of the greatest mathematicians of all time.
Before Gauss entered the Göttingen University in 1795, he had independently discovered Bode's law, the binomial theorem, the arithmetic and geometric mean, the law of quadratic reciprocity, and the prime number theorem.
In his early twenties, Gauss constructed a
heptakaidecagon, a regular polygon with 17 sides, using only a straight edge and a compass. A construction that had puzzled mathematicians for hundreds of years.
Gauss was so proud that he wanted a heptakaidecagon on his tombstone. The stonemason said it would be too difficult and would look like a circe.
1+2+3+ ... +98+99+100 = ???
Extend and apply the distributive law to the expansion of algebraic expressions.
Factorise algebraic expressions by identifying (highest common factor) of numeric and algebraic expressions.
Simplify algebraic expressions involving the four operations.
A TASK
Supposedly, Gauss's teacher asked the class to sum the integers from 1 to 100 while the teacher talked to a parent.
As the teacher walked out of the room to see the parent, Gauss called out the answer.
y What is the shortcut that Gauss probably used?
y Can the shortcut be used for other problems?
y Brainstorm a number of real life applications.
y Report your findings (poster, oral report, etc.).
Sum 1 to 10 is 55.
Exercise 3.1
Let x represent the total number of students in a maths class. Write an algebraic expression for the number of students in the classroom if:
a) 4 students were absent. x − 4
b) 2 new students entered the room. x + 2 c) y students from another class entered the room. x + y d) the number of students in the room doubled. 2x e) one-third of the students left the room. x − x3
1 Let x represent the total number of students in a maths class. Write an algebraic expression for the number of students in the classroom if:
a) 3 students were absent.
b) 5 new students entered the room.
c) b students from another class entered the room.
d) the number of students in the room tripled.
e) half of the students left the room.
2 Let a represent the amount of money ($) in the cookie jar. Write an algebraic expression for the amount of money in the cookie jar for each of the following:
a) $1.50 is added to the jar.
b) $5.25 is taken out of the jar.
c) $b is added to the jar.
d) $t is taken out of the jar.
e) the amount of money in the jar is doubled.
3 The cupboard contains x dinner plates, y soup plates, and z cups.
Write algebraic expressions for each following situation:
a) the total number of dinner plates, soup plates, and cups.
b) the total number of plates (no cups).
c) the number of cups in the cupboard after 3 cups are taken.
d) the number of cups in the cupboard after 5 cups are added.
4 Write algebraic expressions for each of the following:
a) the sum of x and y.
b) add a and 7.
c) twice b plus 10.
d) the difference between m and n.
e) the product of a and g.
f) b cubed.
g) triple x and add five times the value of p.
h) square x and subtract double x then add 7.
i) the product of m and the square of c.
Warm Up
Algebra is an efficient way of solving millions and millions of problems.
2x and 2×x are the same thing.
x÷3 and x3 are the same thing.
"I was x years old in the year x2."
Augustus De Morgan (when asked about his age).
Substitution
Exercise 3.2
Find the value of 2x − 1 if x = 3 2x − 1 = 2×3 − 1
= 6 − 1
= 5
Find the value of 4a − b2 if a = 2, b = 3 4a - b2 = 4×2 − 32
= 8 − 9
= ˉ1
1 Find the value of each of the following algebraic expressions given that x = 4 and y = 2.
a) 3x b) 2y c) x + 5
d) y − 5 e) x ÷ 6 f) y3
g) x + y h) x − y i) y − x
j) 3x2 − 5 k) 5x − 2y l) 5y − 2y2 + 10 2 If x = 4, what is the value of 2x−3x2?
3 y = 8 − 3x, what is the value of y when x = 1.25?
4 B = 3h2 What is the value of B when h = 25?
5 What is the value of 2x2 − 5x + 7 when x = ˉ2?
The area A, of a rectangle of length l, and width w, is given by the algebraic formula: A = lw.
Find the area of the rectangle if length = 12.2 m and width = 4.6 m A = lw
= 12.2 m × 4.6 m
= 56.12 m2
6 The area, A, of a rectangle of length, l, and width, w, is given by the algebraic formula: A = lw. Find the area of each of the following rectangles:
a) length = 6 cm and width = 5 cm.
b) length = 12.2 m and width = 4.6 m.
7 The perimeter, P, of a rectangle of length, l, and breadth, b, can be calculated using either the algebraic formula: P = 2(l + b) or the formula: P = 2l + 2b.
Calculate the perimeter of each of the following rectangles using both formulas:
a) length 5 cm and breadth 2 cm.
b) length 16.5 cm and breadth 9.7 cm.
8 The area of a triangle, A, with base b, and height h, is given by the algebraic formula: A = 0.5bh. Calculate the area of each of the following triangles:
a) base of 6 mm and height of 2 mm.
b) base of 8.2 cm and height of 5.8 cm.
When using substitution in algebra, a variable such as x or y is replaced with its value.