Vocabulary variable algebraic expression term polynomial
Polynomials
As science becomes more complex, it becomes more important to communicate mathematical ideas clearly. Imagine if Einstein always had to write out his famous equation E = mc2 as “energy equals mass times the
square of the speed of light.” Using letters or symbols to represent unknown amounts is called algebra. Algebra is a basic building block of advanced mathematical and scientifi c thinking. How can you use algebra to communicate your mathematical ideas?
CHAPTER
3
Number Sense and Algebra
2 Substitute into and evaluate algebraic expressions involving exponents.
2 Describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three.
2 Derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents.
2 Relate understanding of inverse operations to squaring and taking the square root.
2 Extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents.
2 Add and subtract polynomials with up to two variables.
2 Multiply a polynomial by a monomial involving the same variable.
2 Expand and simplify polynomial expressions involving one variable.
degree of a term degree of a polynomial like terms
Chapter Problem
Alysia has selected the letter E to design the logo for her school team, the Eagles.
The design will be used to make different-sized crests for clothing such as jackets, sweaters, and baseball caps. How can Alysia make sure that, when the crest is made
1. Add or subtract. a) 7 5 b) 10 3 c) 5 (9) d) 5 (4) e) (4) 6 f) 7 9 g) (3) (11) h) (4) (8) 2. Evaluate. a) (2) (2) 1 b) 10 (3) c) 5 (7) 7 d) 4 (3) (2) e) (9) 6 f) 1 (1) g) (5) 8 h) (8) 9 (2) 3. Multiply. a) 3 (8) b) (4) (6) c) (8) 4 d) (5)(6) e) 12(5) f) 2(20) 4. Divide. a) (8) 4 b) 9 (3) c) d) e) 25 (5) f) 36 (4) 6 6 16 8
Multiply and Divide Integers
The product or quotient of two integers of the same sign gives a positive result. The product or quotient of two integers of opposite signs gives a negative result.
5 3 15 14 (7) 2 3 (4) 12 1 8 8 5 6 30 15 (5) 3 2 is the opposite of 2. 1 is the opposite of 1.
Add and Subtract Integers
To add integers, you can use a number line. • Start at the first integer.
• Add the second integer by drawing an arrow.
• The arrow points to the right if the second integer is positive, and to the left if it is negative.
• The answer is at the tip of the arrow. (1) (2) 3 2 (3) 1 ⫺2 ⫺3 ⫺1 0 1 2 2 3 2 1 0
Opposite integers add to zero. For example, (4) (4) 0.
Subtracting an integer is the same as adding the opposite.
5 (2) 5 (2) 7 (3) (1) (3) (1) 4
Multiply Rational Numbers
To multiply fractions, multiply the numerators together and multiply the denominators together.
When a numerator and a denominator share a common factor, you can divide it out before multiplying.
When negative fractions or decimals occur, apply the same rules as for products of integers.
Check:
0.25 0.5 0.125 Using a scientific calculator:
ç0.25\*0.5 =
Using a graphing calculator:
–0.25 * 0.5 e
Using a scientific calculator:
ç3 \∫4 *1 ∫5 =
Using a graphing calculator:
–3 ÷4 *1 ÷5 k1 e 3 20 a34b 15 a3 4b 1 5 1 2 21 3 1 3 1 4 2 2 3 3 4 ––3 4 2 –– 3 5. Multiply. a) b) c) d) e) f) 6. Multiply. a) b) c) 0.6 (0.95) d) (0.3)(0.4) e) 2.5 (3.2) f) 8(3.8) a103 b a56b a25b 14 1 2 a1 1 2b 38 45 a34b a15b 3 5 a 2 9b 13 14 4 5 1 3
My scientific calculator may need different keystrokes. I’ll check the manual.
When I multiply a negative by a positive, I get a negative result.
3.1
Petra likes to run at the track to keep in shape. This year, to motivate herself, she
will record her training progress visually. What are some ways that she can do this?
Investigate
How can you model length, area, and volume using concrete materials?
Part A: Model Length
Petra’s running record for the first 2 weeks of the year is shown below. Petra used algebra tiles to model each distance. She used the side length of a unit tile to represent 1 km.
Algebra tiles are tools that can be used to model measured quantities.
A unit tile is a square tile that measures 1 unit by 1 unit. It can be used as a counter. On July 14, Petra ran 4 km.
A tile model is a good way of tracking Petra’s progress visually.
䊏 algebra tiles 䊏 linking cubes
Tools
unit tile
Build Algebraic Models
Using Concrete Materials
Date Distance (km) Progress
January 1 1 January 3 2 January 6 2 January 8 3 January 10 3 January 14 4
What other concrete materials could I use to represent one unit?
In this book, positive algebra tiles are green and negative algebra tiles are white.
One weekend, Petra cross-country skiied around a lake three times. She did not know the distance around the lake, so she used a
, x, to represent it.
An x-tile is a rectangular tile that is used to represent the variable x. It has the same width as the side length of a unit tile. You can use an
x-tile to describe any unknown value.
The total distance can be modelled using three x-tiles:
This means that on the weekend, Petra skiied a total of 3x kilometres, where x is the distance around the lake.
1. Use tiles to represent each length. a) 6 km
b) an unknown distance
c) an unknown distance and back d) 3 km plus an unknown distance
2. Use tiles to model each .
a) 7 b) 4x c) x 3 d) 3x 2 3. Write an algebraic expression for each model.
a) b)
c) d)
4. Create an algebraic expression of your own and build a tile model to represent it. Record the expression and the model. Part B: Model Area
You can use an x2-tile to represent an unknown area, measured in
square units, if you let the side length of each tile represent x units. Note that the side length of an x2-tile is equal to the length of an x-tile.
algebraic expression x-tile
variable
䊏 a quantity whose value can change (or vary)
䊏 usually represented by a letter
variable
䊏 a mathematical phrase made up of numbers and variables, connected by addition or subtraction operators
䊏 can be used to model real-life situations
䊏 2x 3 is an algebraic expression
algebraic expression
Expressions are sometimes confused with equations. For example, 2x 3 is an expression, but 2x 3 1 is an equation. Equations contain an equal sign, expressions do not.
onnections
Literac
1 4 9
(square units) x2-tile
x2-tile
x-tile x-tile
This model shows that Petra mowed one square lawn 6 times over a summer. The total area mowed, in square metres, is 6x2, where x is the side length, in metres, of the square lawn.
5. Use algebra tiles to represent each area. a) 16 square units
b) 25 square units c) 3x2
d) 5x2
6. Use algebra tiles to represent each expression. a) x2 2
b) 2x2 x c) x2 3x 2 d) 3x2 5x 1 Part C: Model Volume
7. A cube is a rectangular prism with length, width, and height all equal.
a) Use linking cubes to build a model of a cube that has a side length of 3 cm. Sketch your model.
b) What is the volume, in cubic centimetres, of this cube? c) Express the volume of the cube as a power.
8. Repeat step 7 for a cube with a side length of 5 cm.
9. Describe other concrete materials you could use to build an algebraic model of volume.
10. Suppose you do not know the side length of a cube. You can use the variable x to represent the side length.
a) Sketch the cube and label its length, width, and height. b) Write an algebraic expression for the area of one face of
the cube.
c) Write an algebraic expression for the volume of the cube.
11. Reflect Describe how concrete materials can be used to build
algebraic models of length, area, and volume. Use words, expressions, and diagrams to support your explanation.
Key Concepts
䊏 Concrete materials, such as algebra tiles and linking cubes, can be used to build algebraic models.
䊏 You can build length models with algebra tiles.
䊏 You can build area models with algebra tiles.
䊏 You can build volume models with linking cubes.
Communicate Your Understanding
State the length and width of each. a) a unit tileb) an x-tile c) an x2-tile
a) How are the length and width of a unit tile and an x-tile related? b) How are the length and width of an x-tile and an x2-tile related? a) Suggest two other objects that could be used to model length. b) Suggest two other objects that could be used to model area. c) For your answers to parts a) and b), identify any advantages or
disadvantages of each object.
Explain how concrete materials can be used to model each type of measurement. Include a diagram to support each explanation. a) length b) area c) volume
Practise
1. Which expression is represented by the algebra tile model? A 4x2 2x 5 B 4x2 2x 5 C 4x2 2x 5 D 4x2 2x 5 C4 C4 C3 C3 C2 C2 C1 C1 x-tile ⫺x-tile x2-tile ⫺x2-tile unit tile ⫹1 ⫺1 negative unit tile
2. Use tiles to model each algebraic expression. a) x2 3x
b) 2x2 5 c) 3x2 x 2 d) x2 2x 4
3. Write the algebraic expression represented by each model. a)
b)
c)
d)
Connect and Apply
4. Each unit tile represents 1 km that Miko rode her bicycle. Find each distance.
a) b) c) d)
5. Create an algebraic expression of your own, using x2-tiles, x-tiles, and unit tiles, and build a tile model to represent it. Record the expression and the model.
6. a) Build a volume model to represent a cube with length, width, and height all equal to 4 cm. Sketch the model and label the length, width, and height.
b) What is the volume? Write this as a power.
c) Write an expression for the area of one face as a power. Evaluate the area of one face.
The Greek mathematician Diophantus (about 250-275) was the first person to use a letter to represent an unknown.
7. A cube has a volume of 216 cm3.
a) What side length of the cube would give this volume? b) Determine the area of one face of the cube.
8. The area of one face of a cube is 49 m2.
a) What side length of the cube would give this area? b) Determine the volume of the cube.
Extend
9. Build an area model using tiles that have length and width as indicated.
a) length x 3, width x b) length x 4, width x 1
10. A cube has a volume of 8 cm3. Find the total surface area of all six faces.
11. Math Contest Mersenne numbers are numbers of the form 2n 1.
Father Marin Mersenne (15881648) was especially interested in prime numbers of this form. One conjecture about Mersenne numbers is that numbers of the form 2p 1 are prime if p is prime.
Investigate this conjecture and write a brief report of your findings. 12. Math Contest Find the smallest possible value of ab cd efif a, b,
c, d, e, and f are all different and are chosen from the values 1, 2, 3,
4, 5, and 6.
13. Math Contest When 3040 4030is written in expanded form, the number of zeros at the end of the number is
A 30 B 40 C 70 D 120 E 1200
14. Math Contest Fermat numbers are numbers of the form 22n 1. Pierre de Fermat (16011665) conjectured that all numbers of this form are primes. Investigate this conjecture. Write a brief report of your findings.
3.2
Work With Exponents
Suppose the trend in the cartoon continues: every day each new customer tells two new friends at school about the Barney Burger. How many new customers will Barney get each day? Sammy friend Day 1 Day 2 friend friend friend friend friend
Investigate
How can you use exponential models to describe growth patterns?
1. Copy and complete the table. In the last column, write the number of new customers as a power of 2.
2. Barney’s is open 7 days a week. Use this model to determine how many new customers Barney should expect on Day 7. Explain how you found your answer.
3. Use this model to determine how many new customers Barney should expect on Day 14. Is this answer realistic? Why or why not? 4. Estimate the number of students at your school. How long would
it take for everyone at your school to find out about Barney’s? Describe how you found your answer, and identify any assumptions you made.
5. Suppose that each new customer told three friends about Barney’s, instead of two, and that this trend continued. Use exponents to help explain your answers to the following. a) How many new customers should Barney expect after 2 days? b) How many new customers should Barney expect after 4 days? c) How much more quickly would word reach all the students
at your school? Explain.
6. Reflect Explain how exponents are useful in describing growth
patterns.
A power is a product of identical factors and consists of two parts: a base and an exponent.
24 is a power
base exponent
The base is the identical factor, and the exponent tells how many factors there are.
242 2 2 2
exponential form expanded form
Day New Customers Expanded Form Power
1 2 2 21
2 4 22 22
3 4
Example 1
Evaluate Powers
Write in expanded form, and then evaluate. a) 25 b) (3)3 c) (3)4 d) 34 e) 3.53 f) Solution a) 25 2 2 2 2 2 32 b) (3)3 (3) (3) (3) 27 c) (3)4 (3) (3) (3) (3) 81 d) 34 (3 3 3 3) 81 e) 3.53 3.5 3.5 3.5 42.875 ç3.5 Y3 = f)
Example 2
Apply Exponents to Solve Problems
Mega-Box Jumbo Drum
Which container holds more popcorn? How much more? Assume that each container is filled just to the top. Round your answer to the nearest cubic centimetre.
8 27 2 2 2 3 3 3 a23ba23ba23b a23b3 a2 3b 3
There is an odd number of negative factors. The answer is negative.
There is an even number of negative factors. The answer is positive.
The base of this power is 3, not 3. The negative sign in front makes the result negative.
To multiply fractions, I multiply numerators together and multiply denominators together.
9.2 cm 9.2 cm
Popcorn
9.2 cm 9.0 cm 10.4 cmSolution Mega-Box
The Mega-Box is in the shape of a cube. Apply the formula for the volume of a cube.
V s3 s is the side length of the cube.
9.23 ç9.2 Y3 =
778.688
The Mega-Box holds about 779 cm3of popcorn. Jumbo Drum
The Jumbo Drum is in the shape of a cylinder. Apply the formula for the volume of a cylinder.
V r2h r is the radius of the base and h is the height of the cylinder.
(5.2)2(9.0) The radius is half the diameter: 10.4 2 5.2 764.54 çπ*5.2 x*9 =
The Jumbo Drum holds about 765 cm3of popcorn.
The Mega-Box holds 14 cm3more popcorn than the Jumbo Drum.
Key Concepts
䊏 Powers are a useful way to express repeated multiplication. For example,
4 4 4 43
䊏 A power consists of a base and an exponent, e.g., 43.
• The baseis the identical factor.
• The exponent tells how many factors there are.
䊏 Powers sometimes appear in formulas. When evaluating expressions involving powers, follow the correct order of operations.
Communicate Your Understanding
Identify the base and the exponent of each power. a) 34 b) c) (2)6 d) 26 e) 1.22 a) Evaluate each power in question 1.
b) Explain why the answers to parts c) and d) are different.
C2 C2 a12b4 C1 C1 779 765 14 Selecting Tools Representing
Reasoning and Proving
Communicating
Connecting Reflecting Problem Solving
The first step in evaluating the volume of a cylinder is to substitute the known values for
r and h into the formula for the volume of a
cylinder: V r2h. Describe the next step. Which expressions would you evaluate using a calculator? Explain.
a) 23 b) (4)2 c) (1.25)4 d) 82 e) 76 f) (0.1)3
Practise
For help with questions 1 to 5, see Example 1.
1. Which is 6 6 6 6 written as a power?
A 64 B 64
C 46 D 1296
2. Which is 35written in expanded form? A 3 5 B 5 5 5 C 3 3 3 3 3 D 243 3. Write each expression as a power.
a) (5) (5) (5)
b) 1.05 1.05 1.05 1.05 1.05 1.05 c)
4. Write each power in expanded form. Then, evaluate the expression.
a) (4)3 b) 0.82 c)
5. Evaluate.
a) 93 b) (7)2 c) 24
d) e) f) 1.22
g) 18 h) (1)55 i) 0.53 6. Evaluate. Remember to use the correct order of operations.
a) 25 42 b) 53 52 c) 13 16 12 d) (32 42) (34 43) e) a2 f) 500(1.08)5 3b 3 a34b2 a23b4 a56b3 a34b4 a35b a35b a35b C4 C4 C3 C3 r h
7. Substitute the given values into each expression. Then, evaluate the expression. Round your answers to one decimal place where necessary. a) 6s2 s 5 b) r2 r 2.5 c) a2 b2 a 3, b 4 d) r2h r 2.3, h 5.2 e) r3 r 1.5 f) x2 2x 24 x 6
Connect and Apply
8. a) Evaluate each power.
(2)2 (2)3 (2)4 (2)5
b) Examine the signs of your answers. What pattern do you notice? c) Explain how you can tell the sign of the answer when a power
has a negative base. Create and use examples of your own to illustrate your explanation.
9. Listeria is a type of bacteria that can cause dangerous health problems. It doubles every hour. The initial population of a sample of Listeria is 800.
a) Copy and complete this table, which shows the population of Listeria over time.
b) Construct a graph of population versus time. Use a smooth curve to connect the points. Describe the shape of the graph.
c) What will the population be after • 1 day? • 2 days?
d) The symptoms of food poisoning can start as quickly as 4 h after eating contaminated food or as long as 24 h later. Discuss why some types of food poisoning begin quickly and others much more slowly.
4 3
Time (min) Population of Listeria
0 800 60 1600 120 180 240 Technology Tip If your calculator does not have a πkey, use 3.14 as an approximate value for .
10. E. coli is a type of bacteria that lives in our intestines and is necessary for digestion. It doubles in population every 20 min. The initial population is 10.
a) Copy and complete the table. Refer to your table from question 9 to complete the second column.
b) When will the population of E. coli overtake the population of Listeria?
c) What population will the two cultures have when they are equal?
11. The durations (lengths of time) of musical notes are related by powers of , beginning with a whole note. Copy and complete the table.
12. Refer to question 11. Look at the pattern in the last column. Extend this pattern backward to write the power form for a whole note. Does this answer make sense? Use a calculator to evaluate this power. Describe what you observe.
1 2
Food and water contaminated with E. coli (escherichia coli) can be very dangerous, but infections are easily treatable with certain antibiotics.
Did You Know?
Time (min) Population of Listeria Population of E. Coli
0 20 40 60 80 100 120
Note Symbol Duration (in beats) Power Form
whole 1 half —1 2
(
1 — 2)
1 quarter —1 4(
1 — 2)
2 eighth sixteenth thirty-second Selecting Tools RepresentingReasoning and Proving
Communicating
Connecting Reflecting Problem Solving
13. Chapter Problem Alysia has selected the letter E to design the logo for her school team, the Eagles.
The design will be used to make different-sized crests for clothing such as jackets, sweaters, and baseball caps. The height of the crest is twice the width. How can Alysia make sure that, when the crest is made larger or smaller, the proportions will not change?
a) Find an expression for the area of the crest in terms of the width. b) Determine the area of a crest with a width of 8 cm.
c) Determine the height of a crest with an area of 72 cm2.
14. Uranium is a radioactive material that emits energy when it changes into another substance. Uranium comes in different forms, called isotopes. One isotope, U-235, has a half-life of 23 min, which means that it takes 23 min for a sample to decay to half its original amount. a) Suppose you started with a 100-mg sample of U-235. Copy and
complete the table.
b) Construct a graph of the amount, in milligrams, of U-235 remaining versus time, in minutes. Describe the shape of the graph.
c) Approximately how much U-235 will remain after 2 h?
d) How long will it take until only 1 mg of U-235 remains?
e) Use the pattern in the table to write an expression, using powers of , for the original amount of U-235. Does this make sense?
1 2
Number of
Half-Life Periods Time (min)
Amount of U-235 Remaining (mg) Expression 0 0 100 1 23 50 100
(
—1 2)
1 2 46 100(
—1 2)
2 3 4Uranium is used as a fuel source in nuclear fission reactors, which provide 48% of Ontario’s electrical power. CANDU (Canada Deuterium
Uranium) reactors are among
the safest nuclear power generators in the world.
Extend
15. Uranium-233 is another isotope that is used in nuclear power generation. 1 kg of U-233 can provide about the same amount of electrical power as 3 000 000 kg of coal. This number can be written in scientific notation as 3 106.
a) Another isotope of uranium, U-238, has a half-life of
4 500 000 000 years. Write this number in scientific notation. b) What is the half-life of U-238, in seconds? Write your answer
in scientific notation.
c) The number 6.022 1023is a very important number in chemistry. It is called “one mole.” One mole is the amount of a substance that contains as many atoms, molecules, ions, or other elementary units as the number of atoms in 12 g of carbon-12. Carbon-12 is the basic building block of living things. Write one mole in standard notation.
d) Describe any advantages you see to using scientific notation. 16. Refer to the cartoon at the beginning of the section. Suppose that
every new customer returns to Barney’s every day for lunch, in addition to recruiting two new customers.
a) How many customers in total will Barney have • 2 days after Sammy’s first visit?
• 5 days after Sammy’s first visit?
b) On which day will Barney’s reach 500 new customers for lunch? c) Write an expression that gives the total number of new lunch
customers n days after Sammy’s first visit.
d) Describe any assumptions you must make in finding your answers.
17. Math Contest Determine the last digit of the number 31234when written in expanded form. Justify your answer.
18. Math Contest If 3x 729, the value of x is
A 3 B 5 C 6 D 7 E 8
19. Math Contest Numbers are called perfect powers if they can be written in the form xyfor positive integer values of x and y. Find
all perfect powers less than 1000.
20. Math Contest xxis always greater than yyas long as x > y. For what
whole-number values of x and y is xy > yx? Justify your answer.
Scientific notation is a
convenient way to write very large or very small numbers. In scientific notation, the value is expressed as the product of a number between 1 and 10 and a power of 10. For example,
56 000 000 000 5.6 1010
0.000 342 3.42 104
Scientific calculators express numbers in scientific notation when there are too many digits for the display. Multiply 1234 by 1000. Repeat until the output appears in scientific notation. You can enter a number in scientific notation into a scientific calculator. For example, to enter 5.6 1010, press
ç5.6 ÍEE 10 =
To enter 3.42 104, press
ç3.42 ÍEE 4 \=
Not all calculators show scientific notation, or let you enter such numbers, in exactly the same way. Experiment or refer to the user’s manual for your calculator.
onnections
3.3
The 100-m dash is one of the most exciting events in track and field. If you ran this race, how many centimetres would you run? How many millimetres is this?
Investigate A
How can you simplify expressions involving products and quotients of powers?
Part A: Patterns Involving Powers of 10
In the metric system, length measures are related by powers of 10. For example, there are 10 mm in 1 cm. This makes it easy to convert one unit of length to another. Note that hectometres and decametres are uncommon units.
1. How many metres are in 1 km? Write this as a power of 10.
2. Copy and complete the table.
Unit Number of these in 1 m Power of 10
decimetre 10 101
centimetre 100
millimetre
Discover the
Exponent Laws
A Canadian named Donovan Bailey set the men’s world record for the 100-m dash at the 1996 Olympics. He ran the race in 9.84 s, and was considered at that time to be the world’s fastest human.
Did You Know?
kilometre (km) 1000 m hectometre (hm) 100 m decametre (dam) 10 m decimetre (dm) 0.1 m centimetre (cm) 0.01 m millimetre (mm) 0.001 m metre (m)
To convert to a smaller unit, move the decimal point to the right.
To convert to a larger unit, move the decimal point to the left.
kilometre (km) 1000 m hectometre (hm) 100 m decametre (dam) 10 m decimetre (dm) 0.1 m centimetre (cm) 0.01 m millimetre (mm) 0.001 m metre (m)
I can use the metric ladder to help. I count the steps on the ladder to find the exponent.
3. a) Multiply the number of centimetres in 1 m by the number of metres in 1 km. What does this answer give you? b) Write the product in part a) using powers of 10. Write the
answer as a power of 10.
4. Repeat step 3 for millimetres instead of centimetres.
5. Reflect Look at the exponents in the powers of 10 in your answers
to steps 3b) and 4b). Describe how these numbers are related. Part B: Products of Powers
How can you simplify expressions containing products of powers with the same base?
6. Copy and complete the table, including an example of your own.
7. What do you notice about the bases of the powers in each product in the first column?
8. Look at the exponents in the first column for each product. How does the sum of the exponents compare to the exponent in the last column?
9. Reflect Explain how you can write a product of powers using a
single power. Use your example to illustrate your explanation. 10. Write a rule for finding the product of powers by copying and
completing the equation xa xb 䊏.
Part C: Quotients of Powers
How can you simplify expressions containing quotients of powers with the same base?
11. Copy and complete the table, including an example of your own.
Quotient Expanded Form Single Power
55 53 5 5 5 5 5 5 5 5 5 2 74 71 106 104 27 26 p8 p5
Create your own example I can reduce
common factors.
Quotient Expanded Form Single Power
3234 (33)(3333) 36
4343
6461
242223 k3k5
12. What do you notice about the bases of the powers in each quotient in the first column?
13. Look at the exponents in the first column for each quotient. How do they relate to the exponent of the single power in the last column?
14. Reflect Explain how you can write a quotient of powers using a
single power. Use your example to illustrate your explanation. 15. Write a rule for finding the quotient of powers by copying and
completing the equation xa xb 䊏 .
The patterns in the activity above illustrate two exponent laws. The exponent laws are a set of rules that allow you to simplify expressions involving powers with the same base.
Product Rule
When multiplying powers with the same base, add the exponents to write the product as a single power:
xa xb xa b Quotient Rule
When dividing powers with the same base, subtract the exponents to write the quotient as a single power:
xa xb xa b
Example 1
Apply the Product Rule
Write each product as a single power. Then, evaluate the power. a) 32 33 b)52 5 52 c) (2)4 (2)3 d) e) 0.14 0.12 Solution a) 32 33 32 3 35 243 ç 3 Y 5 = a12b3a12b2
The bases are the same, so I can add the exponents.
b) 52 5 52 52 51 52 55 3125 c) (2)4 (2)3 (2)4 3 (2)7 128 d) e) 0.14 0.12 0.14 2 0.16 0.000 001
Example 2
Apply the Quotient Rule
Write each product as a single power. Then, evaluate the power. a) 87 85 b) 47 4 43 c) d) Solution a) 87 85 87 5 82 64 a34b3 a34b2 a34b5 (0.5)6 (0.5)3 1 32 1 1 1 1 1 2 2 2 2 2 a12b a12b a12b a12b a12b a12b5 a12b32 a12b3 a12b2
When no exponent appears, I know that it is 1.
52 5 52 52 51 52
Now, I can add the exponents: 2 1 2 5
To find the sixth power of 0.1, I need to multiply 0.1 by itself six times. In the product, there will be six digits after the decimal point.
The bases are the same, so I can subtract the exponents.
b) 47 4 43 47 41 43 47 1 43 46 43 46 3 43 64 c) (0.5)6 3 (0.5)3 0.125 d) 1
Investigate B
How can you simplify expressions involving powers of powers?
1. Copy and complete the table, including an example of your own. a34b5 a34b5 a34b3 2 a34b5 a34b3a34b2 a34b5 (0.5)6 (0.5)3 Divide in order from left to right.
Apply the product rule first to simplify the numerator.
Anything divided by itself equals 1.
What if I use the quotient rule?
(
3 — 4)
5 —————(
3 — 4)
5(
3 — 4)
5 — 5(
3— 4)
0I know the answer is 1. I wonder if an exponent of 0 always gives an answer of 1?
Power of a Power Expanded Form Single Power
(22)3 (22) (22) (22)
(2 2) (2 2) (2 2) 26 (53)4
(104)2
2. Look at the exponents in the first column for each case. How do they relate to the exponent of the single power in the last column?
3. Reflect Explain how you can write a power of a power using
a single power. Use your example to illustrate your explanation. 4. Write a rule for finding the power of a power by copying and
completing the equation (xa)b 䊏.
The patterns in Investigate B illustrate another exponent law. Power of a Power Rule
A power of a power can be written as a single power by multiplying the exponents.
(xa)b xa b
Example 3
Apply the Power of a Power Rule
Write each as a single power. Then, evaluate the power. a) (32)4 b) [(2)3]4 c) d) (0.23)2 Solution c a23b2d2 a) (32)432 4 38 6561 c) b)[(2)3]4(2)3 4 (2)12 4096 d)(0.23)2 0.23 2 0.26 0.000 064 16 81 2 3 2 3 2 3 2 3 a23b4 a23b2 : 2 c a23b2d2
Example 4
Simplify Algebraic Expressions
Simplify each algebraic expression by applying the exponent laws. a) y3 y5 b) 6p7 3p3
c) a2b3 a6b4 d)
Solution
a) y3 y5 y3 + 5 Apply the product rule. y8 b) 6p7 3p3 2p7 3 2p4 c) a2b3 a6b1 a8b4 d) Divide. u3 2v5 4 uv
Key Concepts
䊏 The exponent laws are a way to simplify expressions involving powers with the same base.
䊏 When multiplying powers with the same base, add the exponents:
xa xb xa b
䊏 When dividing powers with the same base, subtract the exponents:
xa xb xa b
䊏 When finding the power of a power, multiply the exponents: (xa)b xa b 16u3v ˛˛ 5 16u2v˛˛ 4 16 u1 2 v ˛˛ 3 2 16u2v ˛˛˛˛ 2 2 (2) 8 u u2 v ˛˛ 3 v ˛˛˛ 2 42u2(v˛˛˛ 2)2 2uv˛˛˛ 3 8u2v ˛˛˛ 2 (4uv˛˛˛ 2)2 2uv˛˛˛ 3 8u2v ˛˛˛ 2 (4uv˛˛˛ 2)2
Exponent laws only apply to powers with the same base. • First, I add exponents of a:
a2 6 a8
• Then, I add exponents of b :
b3 1 b4 Divide the numeric factors, 6 3 = 2.
Simplify numerator and denominator first. The exponent in the denominator applies to all the factors inside the brackets.
Communicate Your Understanding
Identify which exponent law you can apply to simplify each expression. If no exponent law can be used, explain why not. a) 63 62 b) (m2)3 c) 34 43 d) a2b a3b4 e) f)
Create an example involving powers where you can a) add exponents
b) multiply exponents c) subtract exponents
Look at part d) of Example 4. Suppose that u 3 and v 2: Original expression Simplified expression
uv
a) Which expression would you rather substitute into to evaluate the expression, and why?
b) What is the value of the expression after substituting the given values?
Practise
For help with questions 1 and 2, see Example 1.
1. Which is 73 72expressed as a single power?
A 76 B 75
C 79 D 496
2. Apply the product rule to write each as a single power. Then, evaluate the expression.
a) 34 37 b) 24 2 23 c) (1)5 (1)6 d) a2 5b 3 a25b3 2uv˛˛ 3 8u2v ˛˛˛ 2 (4uv˛˛˛ 2)2 C3 C3 C2 C2 u4v ˛˛ 5 w˛˛˛ 2x ˛˛ 3 p3q2 pq C1 C1
For help with questions 3 and 4, see Example 2.
3. Which is 117 115expressed as a single power?
A 1112 B 111.4 C 12 D 112
4. Apply the quotient rule to write each as a single power. Then, evaluate the expression.
a) 128 122 b) (6)5 (6)2 (6)2
c) d)
For help with questions 5 and 6, see Example 3.
5. Which is (54)2expressed as a single power?
A 58 B 56
C 256 D 258
6. Apply the power of a power rule to write each as a single power. Then, evaluate the expression.
a) (42)2 b) [(3)3]2 c) [(0.1)4]2 d)
7. Simplify using the exponent laws. Then, evaluate. a) 52 53 54 b) 37 35 3
c) d) (2)4 (2)5 [(2)3]3
For help with questions 8 and 9, see Example 4.
8. Simplify.
a) y4 y2 b) m8 m5 c) k2 k3 k5 d) (c3)4 e) a2b2 a3b f) (2uv2)3 g) m2n mn2 h) h2k3 hk i) (a3b)2
Connect and Apply
9. Simplify. a) 12k2m8 4km5 b) 8a5 (2a3)2 c) (x2)3 (3x2)2 d) e) f) (3a2b)2 (ab)2 g) h) i) 30g ˛˛˛ 2h (2gh)2 5gh˛˛˛˛˛ 2 6gh (3xy2)3 (4x2y) (2x˛˛˛˛˛ 2y ˛ 2)2 5c˛˛˛ 3d 4c2d2 (2c˛˛˛˛˛ 2d ˛ )2 3f˛˛˛ 4g3 8fg4 (6f˛˛˛˛˛ 2g ˛ 3)2 4d4w3 6dw4 3d3w 8dw2 (0.53)4 0.56 0.54 c a32b3d2 0.16 0.14 0.12 a34b4 a34b
The notation that we use for powers, with a raised number for the exponent, was invented by Réné Descartes (1596-1650). Descartes used this notation in his text Géométrie, published in 1637. In this famous text, Descartes connected algebra and geometry, starting the branch of mathematics called Cartesian geometry.
10. Consider the expression .
a) Substitute x 3 and y 1 into the expression. Then, evaluate the expression.
b) Simplify the original expression using the exponent laws. Then, substitute the given values and evaluate the expression. c) Describe the advantages and disadvantages of each method. 11. A crawlspace in a space station has the shape of a rectangular prism.
It is about 100 cm high, 10 m wide, and 1 km long. What is the volume enclosed by the crawlspace?
12. The probability of tossing heads with a standard coin is , because it is one of two possible outcomes. The probability of tossing three heads in a row is or .
a) What is the probability of tossing • six heads in a row?
• 12 heads in a row?
b) Write each answer in part a) as a power of a power. 13. a) What is the probability of rolling a 6 with a standard
number cube?
b) What is the probability of rolling four 6s in a row? c) What is the probability of rolling a perfect square with a
number cube?
d) What is the probability of rolling eight perfect squares in a row? e) Write each answer in parts b) and d) as a power of a power. Achievement Check
14. Consider the expression .
a) Substitute m 4 and n 3 into the expression and evaluate it. b) Simplify the original expression using the exponent laws.
c) Substitute m 4 and n 3 into the simplified expression and evaluate it.
d) What did you notice? What are the advantages and disadvantages of the two methods?
e) Josie made two errors in copying the above expression. She wrote , but she still got the correct answer.
Explain how this is possible. 3m2n 4mn2 (2mn)2 3mn 3m2n 4m3n2 (2m2n)2 3mn 1 8 a12b3 1 2 5xy˛˛˛ 2 2x ˛˛ 2y (2xy)2
3 6
2
Rectangular prism is the mathematical name for a box.onnections
Extend
15. You can multiply and divide numbers in scientific notation by applying the exponent laws. For example,
(3 105) (2 104) (9 108) (3 105) 3 2 105 104
6 109
3 103
Evaluate each of the following. Express each answer in scientific notation and then in standard notation.
a) 4 102 2 103 b) 1.5 104 6 106 c) (8 107) (2 105) d) (3.9 1012) (3 108)
16. a) Predict the screen output of your scientific or graphing calculator when you enter the following calculation: (3 105) (2 106). b) Is the answer what you predicted? Explain the answer that the
calculator has provided.
17. a) Predict the screen output of your scientific or graphing calculator when you enter the following calculation: (3 1018) (6 102). b) Is the answer what you predicted? Explain the answer that the
calculator has provided.
18. a) Evaluate (2 105)3. Express your answer in both scientific and standard notation.
b) Explain how you can evaluate a power of a number expressed in scientific notation. Create an example of your own to help illustrate your explanation.
19. Math Contest Copy and complete the table to make the square a multiplicative magic square (the product of every row, column, and diagonal is equal).
20. Math Contest If x , place the following values in order from least to greatest: x, x2, x3, , 1 x 2x 1 9 a19b16 a15b12 a9b6 a13b10 a21b18 a7b4 93 108 3 1 10 5 9 108 3 105
3.4
Communicate With Algebra
You have seen how algebra tiles can be used to model algebraic expressions.
The model below shows that Petra mowed a square lawn of unknown area once a week for 4 weeks.
The algebraic representation is 4x2. Look at each part of this expression. What does the 4 represent? What does the x2represent?
The expression 4x2is called a . A term consists of two parts:
4 x2
coefficient variable
When you represent an algebraic expression using algebra tiles, the variable in the expression tells you which type of tile to use. To represent x2, use an x2-tile.
The number of tiles corresponds to the coefficient. Since the expression is 4x2, there are four x2-tiles.
term 䊏 an expression formed by
the product of numbers and/or variables
term
Polynomial Number of Terms Type of Polynomial
3x2 2x 2 binomial
2m 1 monomial
4x2 3xy y2 3 trinomial
a 2b c 3 4 four-term polynomial
Expression Coefficient Variable Comments
7x 7 x
4.9t2 4.9 t2 The negative sign is included with the
coefficient. 1 — 2bh 1 — 2 bh
The variable can consist of more than one letter or symbol.
k2 1 k2 When the coefficient is not shown, it is 1.
6 6 none A term with no variable is called a
constant term, or simply a constant.
The coefficient is also called the numerical coefficient. It is a number only.
The variable is also called the
literal coefficient. It consists
of one or more variables and their exponents, if they exist. Exponents on the variables belong to the literal coefficient, because they represent a product of variables: x2 x x.
onnections
Literac
䊏 an algebraic expression consisting of one or more terms connected by addition or subtraction operators
polynomial
The prefixes of the polynomial names have the following meanings: • mono means 1 • bi means 2 • tri means 3 onnections Literac
Example 1
Identify Coefficients and Variables
Identify the coefficient and the variable of each term.
a) Jim earns $7 per hour at his part-time job. If he works for x hours, his earnings, in dollars, are 7x.
b) The depth, in metres, of a falling stone in a well after t seconds is 4.9t2.
c) The area of a triangle with base b and height h is bh.
d) The area of a square with side length k is k2. e) Amir walks 6 blocks to school.
Solution a) b) c) d) e)
A can be classified by the number of terms it has.
Example 2
Classify Polynomials by Name
Classify each polynomial by the number of terms it has.
a) 3x2 2x b) 2m c) 4x2 3xy y2 d) a 2b c 3 Solution a) b) c) d)
Type of Polynomial Number of Terms Examples
monomial 1 x, 3y, 4a2, 5
binomial 2 2x 3, ab 2a, 0.4x2 x trinomial 3 2x2 3x 1, a 2b c polynomial 1 2 3x2 2x has two terms, 3x2and 2x.
I can find the number of terms by looking for the addition and subtraction operators that separate the terms: 4x2 3xy y2
Two operators separate the three terms in this trinomial.
Polynomial Term With Highest Degree Degree of Term in Column 2 Degree of Polynomial x 3 x 1 first 5x2 2x 5x2 2 second 3y3 0.2y 1 3y3 3 third 7x2y4 x6y x6y 7 seventh
Term Sum of Exponents Degree
x2 2 2 3y4 4 4 0.7u1 1 1 2a2b1 2 1 3 3 2 — 3x 1y1 1 1 2 2 5 0 0
䊏 the degree of the highest-degree term
degree of a polynomial
The degree of the first term is 1. The degree of the second term is 0. The highest degree is 1.
The degree of the first term is 2 4. The degree of the second term is 6 1.
䊏 the sum of the exponents on the variables in a term
degree of a term
Example 3
Classify Terms by Degree
Find the .
a) x2 b)3y4 c) 0.7u
d) 2a2b e) f) 5
Solution
Look at the exponents of the variables. Add them if there is more than one. a) b) c) d) e) f)
Example 4
Classify Polynomials by Degree
Find the . a) x 3 b) 5x2 2x c) 3y3 0.2y 1 d) 7x2y4 x6y Solution a) b) c) d)
degree of each polynomial 2 3xy degree of each term
Remember, when no exponent appears on a variable, the value of the exponent is 1. For example, 4u 4u1.
If the term has no variable at all, then the degree is 0.
Example 5
Use an Algebraic Model to Solve a Problem
Cheryl works part-time as a ski instructor. She earns $125 for the season, plus $20 for each children’s lesson and $30 for each adult lesson that she gives.
a) Write an expression that describes Cheryl’s total earnings for the season. Identify the variable and the coefficient of each term and explain what they mean. b) One winter, Cheryl gave eight children’s lessons and
six adult lessons. What were her total earnings? Solution
a) Cheryl’s total earnings can be described by the polynomial expression 20c 30a 125.
b) Substitute c 8 and a 6, and evaluate the expression. 20c 30a 125
20(8) 30(6) 125 160 180 125 465
Cheryl’s total earnings for this season were $465.
Key Concepts
䊏 Algebraic expressions can be used to communicate mathematical ideas.
䊏 A term is the product of a coefficient and variable part. 䊏 A polynomial can be a single term or a combination of terms
using addition or subtraction operators. 䊏 A polynomials can be classified
• by the number of terms it has
• by its degree
Term Variable Meaning of Coefficient
20c c represents the number of children’s lessons.
20 represents the earnings per children’s lesson. 30a a represents the number of
adult lessons.
30 represents the earnings per adult lesson.
125 There is no variable. Cheryl has fixed earnings of $125 per season.
Communicate Your Understanding
Create two examples of each.a) monomial b) binomial
c) trinomial d) four-term polynomial
Julio says that the term x2has a coefficient of 2 and a variable x. Is Julio correct? Explain.
a) Are these expressions equivalent? Explain. 2w 1t 2w t
b) Are these expressions equivalent? Explain. 3x 1 3x
c) Explain when you must write the number 1, and when you do not need to.
Practise
For help with question 1, see Example 1.
1. Identify the coefficient and the variable part of each term.
a) 2y b) 3x c) mn
d) x2 e) w2 f) 0.4gh3
For help with questions 2 and 3, see Example 2.
2. 7x2 3xy 4y2is a
A monomial B binomial C trinomial D term
3. Classify each polynomial by the number of terms. a) 2x b) 6y2 2y 1
c) a b d) 3u2 uv 2v2 e) 3k2 k f) m 0.2n 0.3 mn
For help with questions 4 and 5, see Example 3.
4. The degree of 4u 5u2 9 is
A 1 B 2 C 3 D 0
5. State the degree of each term.
a) 5x2 b) 6y c) 3 d) u2v4 e) 1x2y3 f) 0.2a2b 3 1 2 1 2 1 2 C3 C3 C2 C2 C1 C1
For help with question 6, see Example 4.
6. State the degree of each polynomial.
a) 3x 4 b) y2 3y 1 c) m 2m3
d) a3b2 8a2b5 e) 2x2y4 xy3
For help with questions 7 and 8, see Example 4.
7. In a TV trivia show, a contestant receives 500 points for a correct answer and loses 200 points for an incorrect answer. Let c represent the number of correct answers and i represent the number of incorrect answers. Which expression describes a contestant’s total points?
A 500c 200i B 500c 200i C 500i 200c D 500i 200c
8. A hockey team earns 2 points for a win and 1 point for a tie. Let
w represent the number of wins and t represent the number of ties.
a) Which expression can be used to describe the total points? A 2w 1 B w t C 2w 1t D 2w t b) Is there more than one correct answer? Justify your answer. 9. Substitute the given values and evaluate each expression.
a) 3x 5 x 2
b) 4y 4 y 2
c) a2 2b 7 a 4, b 1 d) 2m2 3n 8 m 2, n 5
Connect and Apply
10. The students at Prince Albert Public School sell magazine subscriptions to raise money. The school receives 37% of the money paid for the subscriptions.
a) Choose a variable to represent the money paid for the subscriptions.
b) Using your variable from part a), write an expression for the amount of money the school will receive.
c) Tyler sold one magazine subscription to his aunt for $25.99. Calculate the amount the school receives on this sale.
d) The sum of all the subscription orders was $4257.49. Use your formula to calculate how much the school will receive for this fundraiser.
2 5
11. Meredith has a summer job at a fitness club. She earns a $5 bonus for each student membership and a $7 bonus for each adult membership she sells.
a) Write a polynomial expression that describes Meredith’s total bonus.
b) Identify the variable and the coefficient of each term and explain what they mean.
c) How much will Meredith’s bonus be if she sells 12 student memberships and 10 adult memberships?
12. An arena charges $25 for gold seats, $18 for red seats, and $15 for blue seats.
a) Write an expression that describes the total earnings from seat sales.
b) Identify the variable and the coefficient of each term and explain what they mean.
c) How much will the arena earn if it sells 100 gold seats, 200 blue seats, and 250 red seats?
13. On a multiple-choice test, you earn 2 points for
each correct answer and lose 1 point for each incorrect answer. a) Write an expression for a student’s total score.
b) Maria answered 15 questions correctly and 3 incorrectly. Find Maria’s total score.
14. a) Describe a situation that can be modelled by an algebraic expression.
b) Select variables and write the expression.
c) Illustrate your expression using algebra tiles or a diagram. 15. Write a response to this e-mail from a classmate.
FROM: ManuelS
TO: JillP
SUBJECT: changing a decimal to a percent Hi Jill,
Too bad I missed today’s class. I copied your notes and I get most of it, I guess. I’m just a little confused about the difference between a term and a polynomial. Can you please help me out?
Thanks, Manuel
16. Chapter Problem Alysia is designing a logo for her school team, the Eagles. The design will be used to make different-sized crests for clothing such as jackets, sweaters, and baseball caps. How can Alysia make sure that, when the crest is made larger or smaller, the shape will not change? The height will always be double the width.
a) If w represents the width, what expression represents the height? b) How high will a crest that is 5 cm wide be?
c) How wide will a crest that is 25 cm high be? Achievement Check
17. In a soccer league, teams receive 3 points for a win, 2 points for a loss, and 1 point for a tie.
a) Write an algebraic expression to represent a team’s total points. b) What variables did you choose? Identify what each variable
represents.
c) The Falcons’ record for the season was 5 wins, 2 losses, and 3 ties. Use your expression to find the Falcons’ total points. d) The 10-game season ended with the Falcons tied for second place
with the same number of points as the Eagles. The Eagles had a different record than the Falcons. How is this possible?
Extend
18. Alberto is training for a triathlon, where athletes swim, cycle, and run. During his training program, he has found that he can swim at 1.2 km/h, cycle at 25 km/h, and run at 10 km/h. To estimate his time for an upcoming race, Alberto rearranges the formula
distance speed time to find that: time .
a) Choose a variable to represent the distance travelled for each part of the race. For example, choose s for the swim.
b) Copy and complete the table. The first row is done for you.
c) Write a trinomial to model Alberto’s total time.
d) A triathlon is advertised in Kingston. Participants have to swim 1.5 km, cycle 40 km, and run 10 km. Using your expression from part c), calculate how long it will take Alberto to finish the race. e) Is your answer a reasonable estimate of Alberto’s triathlon time?
Explain.
Part of the Race Speed (km/h) Distance (km) Time (h)
swim 1.2 s ———s 1.2 cycle run distance speed
19. Ashleigh can walk 2 m/s and swim 1 m/s. What is the quickest way for Ashleigh to get from one corner of her pool to the opposite corner?
a) Predict whether it is faster for Ashleigh to walk or swim.
b) Ashleigh can walk at a speed of 2 m/s. The time, in seconds, for Ashleigh to walk is , where w is the distance, in metres, she walks. Use this relationship to find the travel time if Ashleigh walks around the pool.
Path 1: Walk the entire distance.
c) Write a similar expression to represent the time taken for
Ashleigh to swim a distance s. Her swimming speed is 1 m/s. Use this relationship to find the travel time if Ashleigh swims straight across.
Path 2: Swim the entire distance.
d) Which route is faster, and by how much? 20. Refer to question 19.
a) Do you think it will be faster for Ashleigh to walk half the length and then swim? Explain your reasoning.
Path 3: Walk half the length, then swim.
b) Find the travel time for this path. Compare this with your answers to question 19.
c) Do you think this is the fastest possible path? Find the fastest path and the minimum time required to cross the pool, corner to opposite corner. Describe how you solved this.
w 2 10 m 25 m 10 m 25 m I can use the formula
time ——dis—————tance speed to calculate Ashleigh’s travel time.
Sixteen year-old Marilyn Bell of Toronto was the first person to swim across Lake Ontario. In September 1954 Marilyn swam from Youngstown, NY, to the CNE grounds in Toronto. The 51.5 km distance took her 21 h.
21. Some algebraic expressions involve more than one variable. You can model these using an expanded set of algebra tiles.
Fly By Night Aero Insurance company charges $500 for liability, plus 10% of the value of the plane, plus $300 per seat. Let v represent the value of the plane and s represent the number of seats. The cost of the insurance is modelled by C 500 0.1v 300s. A four-seat Piper Cherokee valued at $30 000 would cost
500 0.1(30 000) 300(4) or $4700 per year to insure. a) Explain how you would use tiles to
represent 0.1v.
b) Explain how you would use tiles to represent 300s.
c) Build an algebraic model to describe the cost for airplane insurance for the Piper Cherokee, using algebra tiles, diagrams, or virtual algebra tiles. d) Find the cost of insurance for a
50-seat plane, valued at $500 000.
22. Math Contest If 3m 5 23 and 2n 7 21, the value of 3m 2n is
A 20 B 44
C 46 D 54 E 56
23. Math Contest If ax by cz 18 144 and a, b, and c are all prime
numbers, the value of x y z is
A 7 B 10 C 11
D 12 E 20
24. Math Contest Find value(s) of m for which . Is there more than one possible value of m? Explain.
a14bm a12b2m
x-tile y-tile y2-tile
Virtual Algebra Tiles With
The Geometer’s Sketchpad
®
You can create and manipulate virtual algebra tiles using computer software such as The Geometer’s Sketchpad®. You can build algebraic models, plus you can change the length of the variable tiles.
Investigate
How can you build algebraic models using virtual algebra tiles?
1. a) Start The Geometer’s Sketchpad® and open the sketch Algebra Tiles.gsp.
b) Read the instructions and click on the Algebra Tiles page button. 2. Explore the pre-made tiles.
a) Click on the Show example tiles button.
b) Click and drag the top of the x-slider. Which tiles are affected by the x-slider?
c) Repeat part b) for the y-slider.
d) The unit tile seems to have a slider next to the x-slider. Try to change its length, and describe what happens. Why can you not change the dimensions of the unit tile?
Use Technology
䊏 computer equipped with
The Geometer’s Sketchpad®
䊏 Algebra Tiles.gsp
For the rest of this activity, you will work only with • unit tiles
• x-tiles (horizontal and vertical)
• x2-tiles
3. Clear the workspace and bring out only the algebra tiles you need: • Click on Hide example tiles.
• Click and hold the Custom Tool icon at the left side of the screen.
• Select 1 (the unit tile) and place a unit tile somewhere on the workspace.
• Click and hold the Custom Tool again and select x (horizontal) and place a tile on the workspace.
• Repeat for the x (vertical) and x^2 (x2) tiles.
Technology Tip If you do not see the list of tools when you click on the
Custom Tool icon, try
holding the left mouse button for a couple of seconds, and they will appear.
Technology Tip You can add more tiles by using the Custom Tool or by using the Copy and Paste commands in the Edit menu. To copy an entire tile, use the Selection Arrow Tool to click and drag a dashed box around the tile you want to copy. Then, paste it and move it wherever you like.
Once you have all four tiles, choose the Selection Arrow Tool and click somewhere in the white space to deselect the last object you created.
4. Explore the relationships between the tiles. Arrange the tiles as shown, by clicking and dragging them one at a time.
a) Move the x-slider and describe what happens to the length and width of each tile.
b) Describe how the length of the x-tile is related to the length and width of the x2-tile. Why is this so?
c) Describe how the width of the x-tile is related to the length and width of the unit tile. Why is this so?
Tile Length (changes/does
not change) Width (changes/does not change) unit x (horizontal) x (vertical) x2
