Download Try Before You Buy

SAMPLE CHAPTER
7229_FM_pii-xiv.qxp
4/1/10
4:25 PM
Page iii
Table of Contents
Introduction
Issues in Teaching Mathematics
The Power of Being Comfortable With the Math We Teach
What Teaching to Big Ideas Means
How This Resource is Organized
Who This Resource is For
Chapter 1 Algebra
BIG IDEAS FOR ALGEBRA
Using Variables to Generalize
Different Ways We Use Variables
Representing Generalizations in Different Ways
Functions and Relations
Domain and Range of Relations
Functions as Special Relations
Operating with Functions
Composition of Functions
Inverse Functions
Manipulating Algebraic Expressions
Simplifying and Evaluating Algebraic Expressions
Polynomials
Introducing Polynomials
Representing Polynomials
Simplifying Polynomials
Operations with Polynomials
Factoring
Linear Relations
Representing a Linear Relation
Characteristics of Linear Relations
Real-world Linear Situations
Direct and Partial Variation
Slopes of Graphs of Linear Relations
Equations of Linear Relations
Solving Linear Equations
Solving Systems of Linear Equations
Solving Linear Inequalities
NEL
Table of Contents
iii
7229_FM_pii-xiv.qxp
4/1/10
4:25 PM
Page iv
Quadratic Relations
Representations of Quadratic Functions
Real-world Quadratic Situations
Characteristics of Quadratic Functions
Transforming y = x 2
Solving Quadratic Equations
Inverse of Quadratic Relations
Polynomial Relations
Representing Polynomial Relations
Graphing Polynomials
Solving Polynomial Equations
Rational Functions
Graphing Rational Functions
Characteristics of Rational Functions
Exponential Relations
Representing Exponential Relations
Graphing Exponential Relations
Real-world Exponential Situations
Solving Exponential Equations
Inverse Exponentials: Logarithms
Trigonometric Functions
Periodic Functions
Describing Trigonometric Functions
Real-world Trigonometric Situations
Characteristics of Trigonometric Graphs
Graphing a Trigonometric Function
Sequences and Series
Pattern Rules
Arithmetic Sequences and Series
Geometric Sequences and Series
Fibonacci Sequences
Chapter 2 Number
BIG IDEAS FOR NUMBER
Classification of Numbers
Rational Numbers
Representing and Comparing Rational Numbers
Operations with Rational Numbers
Irrational Numbers
Non-repeating, Non-terminating Decimals
Irrational Square Roots
Irrational Numbers as a Subset of Real Numbers
Comparing Irrational Numbers
Matrices
Operations with Matrices
iv
Table of Contents
NEL
7229_FM_pii-xiv.qxp
4/1/10
4:25 PM
Page v
Representing Numbers
Percents
Absolute Value
Proportional Reasoning
Ratio and Rate
Solving Ratio, Rate, and Percent Problems
Powers and Roots
Logarithms
Chapter 3 Geometry
BIG IDEAS FOR GEOMETRY
Points, Lines, and planes
Parallelism and Perpendicularity
Symmetry
Properties of Polygons
Triangles
Quadrilaterals
Regular Polygons
Circle Properties
Angle Relationships in a Circle
Chords in a Circle
Tangents to a Circle
Describing Position
Cartesian Coordinates in 3-space
Polar Coordinates
Bearings
Transformations
Representations of 3-D Shapes
Isometric Drawings
Base Plans
Orthographic Drawings
Vector Geometry
Operations with Vectors
Uses of Vectors
Chapter 4 Measurement
BIG IDEAS FOR MEASUREMENT
Measurement Units
Imperial/Metric Conversions
Unit Analysis
Radian Measure
Precision and Measurement Error
Area and Perimeter
Optimizing Area and Perimeter Measures
NEL
Table of Contents
v
7229_FM_pii-xiv.qxp
4/1/10
4:25 PM
Page vi
The Pythagorean Theorem
Extending the Pythagorean Theorem
Using the Pythagorean Theorem
Surface Area and Volume
Volume and Surface Area of Prisms
Volume and Surface Area of Cylinders
Volume and Surface Area of Pyramids and Cones
Volume and Surface Area of Spheres
Optimizing Volume and Surface Area Measurements
Trigonometry
Similar Triangles
Primary Trigonometric Ratios
Determining Unknown Right Triangle Measures
Inverse Ratios
Angle of Elevation and Angle of Depression
Unknown Measures in Complex Situations
Using Trig to Develop New Formulas
Other Trigonometric Ratios
Properties of Trigonometric Ratios
Trigonometric Identities
Trigonometry in Acute Triangles
Solving Trigonometric Equations
Using Trigonometry with Vectors
Chapter 5 Data and Probability
Data
BIG IDEAS FOR DATA
Data Collection
Designing Experiments to Collect Data
Data Collection Methods
Summarizing, Displaying, and Analyzing Data
Univariate versus Bivariate Data
Univariate Data Summary Statistics
Displaying Univariate Data
Displaying Bivariate Data
Cause and Effect With Bivariate Data
Controlling Bivariate Data Experiments
Probability
BIG IDEAS FOR PROBABILITY
Introducing Probability
Theoretical Probability
Tree Diagrams
Area Models
Combinations and Permutations
vi
Table of Contents
NEL
7229_FM_pii-xiv.qxp
4/1/10
4:25 PM
Page vii
Pascal’s Triangle and Combinations
Odds
Calculating Theoretical Probability
Experimental Probability
Law of Large Numbers
Simulations
Probability Distributions
Expected Values E(x)
Types of Probability Distribution
References
Glossary
Index
NEL
Table of Contents
vii
7229_Ch02_p079-106.qxd
4/7/10
3:31 PM
Page 79
Chapter 2
Number
Big Ideas for Number
Although number is a more fundamental part of the curriculum in the elementary grades than in Grades 9 to 12, secondary students must still deal with
important number ideas. Because the numbers they are working with now are
more abstract, it is even more important that students think of their number
work in terms of the big ideas described below and throughout the chapter.
All number work in Grades 9 to 12 can be filtered through the following
six big ideas.
BIG IDEAS FOR NUMBER
1. Numbers tell how many or how much.
2. Classifying numbers provides information about the characteristics of the numbers.
A knowledge of big ideas can
help teachers interpret their
curriculum; choose, shape,
and create tasks; and use
questioning to help students
make powerful connections.
3. There are many equivalent representations for a number or
numerical relationship. Each representation may emphasize
something different about that number or relationship.
4. Numbers are compared in many ways. Sometimes they are
compared to each other. Other times, they are compared to
benchmark numbers.
5. The operations of addition, subtraction, multiplication, and
division hold the same fundamental meanings no matter the
domain in which they are applied.
6. There are many algorithms for performing a given operation.
In each description below, an informal version of the big idea has been provided (in italics). Some teachers might prefer this version of the big idea as
it enables the teacher to discuss and speak about the big ideas using language that is more accessible to students.
BIN 1. Numbers tell how many or how much.
Most often, we use numbers to describe how many (e.g., how many people at the
show) or how much (e.g., what percent of the room is full).
For example, whether a number is a rational number such as - 45 , an irrational number such as p, a complex number such as 3 + i, or what we call
NEL
Each teaching and planning
idea in this chapter will indicate
one or more Big Ideas for
Number (BIN) as the focus.
Some teachers might prefer a
more informal version of each
big idea that uses more accessible language for students.
7229_Ch02_p079-106.qxd
4/7/10
3:31 PM
Page 80
infinity, or whether it is represented as a power, a radical, a continued fraction, or a logarithm, it usually tells how many or how much. The exception is
when numbers are used as labels, for example, a PIN or a bar code.
BIN 2. Classifying numbers provides information about the characteristics of the numbers.
Sometimes knowing one thing about a number tells you a lot more about it than
you realized.
Part of the essence of
mathematics is that we deduce
new information from known
information.
Part of the essence of mathematics is that we deduce new information
from known information. For example, knowing that a number is irrational tells us there is no way to represent it as the quotient of two integers. Knowing that a number is complex tells us that the number may have
an imaginary number component involving i, the square root of -1.
BIN 3. There are many equivalent representations for a number or
numerical relationship. Each representation may emphasize something different about that number or relationship.
Different representations of numbers show different things about them, and
which representation is more useful depends on the situation.
To be fluent with numerical
calculations, students benefit
from being able to express a
number in different ways.
To be fluent with numerical calculations, students benefit from being able to
express a number in different ways. For example, it is helpful to use relationships between powers, often called exponent laws, to determine 2 5 * 55 by
rewriting the expression as (2 * 5)5 or 105.
BIN 4. Numbers are compared in many ways. Sometimes they are
compared to each other. Other times, they are compared to benchmark numbers.
Numbers can be compared in different ways; sometimes we compare them to
each other, and sometimes we compare them to familiar numbers.
Humans internalize the
meaning of a limited number
of benchmarks and anchor
other numbers to these.
Realizing that 12 is about 1.5, that e is about 3, and that 2.9 * 108 is about
300 million is how we best understand the size of these numbers. Humans
internalize the meaning of a limited number of benchmarks and anchor
other numbers to these.
A second way we compare numbers is in terms of which is greater. Once
students are in the world of irrational and complex numbers, this is sometimes more difficult for them than it was with counting numbers, and so
attention to this is important.
Another way we compare numbers is using ratios (or rates or percents).
It is important for success at the secondary level that students start thinking
of numbers like 81 multiplicatively in terms of other numbers (e.g.,
81 = 27 * 3) rather than only additively (e.g., 81 = 80 + 1).
BIN 5. The operations of addition, subtraction, multiplication, and
division hold the same fundamental meanings no matter the
domain in which they are applied.
Each operation (addition, subtraction, multiplication, and division) means
more or less the same thing no matter what objects are being used.
Students in Grades 9 to 12 will use familiar operations in new situations, for
example, multiplying radicals and subtracting matrices. They need to know
that the meanings they already know for these operations still hold.
Students will extend the operations they use to taking powers and roots,
which are extensions of multiplication and division.
80
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:31 PM
Page 81
BIN 6. There are many algorithms for performing a given operation.
There are many ways to perform an operation with numbers.
Today’s approach to mathematics education strongly values students’ using
meaningful approaches, not memorized procedures, to perform calculations.
This big idea underlies all work with operations in any number system.
BIN 6 underlies all operation
work in any number system.
Classification of Numbers
Students through Grade 8 have had opportunities to work with whole numbers, integers, decimals, and fractions. They have met irrational numbers,
such as p, but do not really have a sense of what makes them different from
rational numbers. Although they know that putting such numbers in a calculator results in a very long decimal display, they still do not really know the
difference between repeating decimals, which represent rationals, and irrational numbers, where the decimal does not repeat.
In Grades 9 to 12, students should have the opportunity to distinguish
between classes of numbers and see how the classes interrelate. Ultimately,
the Venn diagram below should be meaningful to them.
Complex numbers
2 + 3i, √–5
Real numbers
2,
3
0.45, –0.6, –5 1
3
–2, –306
Integers
0
3, 107
Whole numbers
Natural numbers
2π, √2
Rational numbers
Irrational numbers
Teaching Idea
2.1
Have students think about
the numbers used to describe
different lengths. You might
prompt them to think about
lengths related to rectangles
and circles.
To bring out BIN 1, ask students to tell when a length
might be an irrational
number and when it might
be rational. [e.g., The circumference of a circle with radius
2 is irrational, but the radius
itself is rational.]
(See Big Ideas for Number, p. 79.)
Students should recognize that
• complex numbers include real numbers
• real numbers include rational numbers (positive or negative
terminating or repeating decimals) and irrational numbers (positive
or negative non-terminating, non-repeating decimals)
• rational numbers include integers, positive fractions and their opposites,
and positive terminating and repeating decimals and their opposites
• integers include whole numbers (i.e., 0, 1, 2, 3, ...) and their opposites
• whole numbers include natural numbers (i.e., the counting numbers
1, 2, 3, 4, ...) and zero
Rational Numbers
Students are generally introduced to the term rational numbers in Grade 9,
but, in fact, they have worked with rational numbers, specifically integers
and positive fractions, in earlier grades. At this level, they learn that
NEL
Chapter 2: Number
81
7229_Ch02_p079-106.qxd
4/7/10
3:31 PM
2.2
Teaching Idea
Ask students for a situation
where - 34 might tell how
much something is. [e.g.,
how much the temperature
changes if it goes down a
part of a degree]
To bring out BIN 1, ask:
Could you describe the same
situation using any rational
number? Explain. [No; e.g.,
temperature can only get so
high or so low.]
(See Big Ideas for Number, p. 79.)
Page 82
• rational numbers include zero, all positive integers and their opposites, and all positive fractions and their opposites
• rationals can be expressed in either decimal form (e.g., 4.302) or
fraction form (e.g., - 35 )
• any number is rational if it can be represented as the quotient of two
integers (e.g., -32 is ( - 2) , 3, - 1.2 is ( - 12) , 10, and 4.111... is 37 , 9)
• rational numbers tell how many or how much (BIN 1)
Although students are not likely to meet non-integer negative rational
numbers on a regular basis in their everyday life, some applications, such as
temperatures or decreases in quantity, can be introduced to help students
better understand their meaning.
Representing and Comparing Rational Numbers
When representing and comparing rational numbers, students must consider both their signs and their values.
IMPORTANT POINTS ABOUT REPRESENTING AND COMPARING RATIONALS
• Every negative rational number can be written in any of these forms:
a
-a a
b , - b , or - b , where a and b are positive integers.
• Comparisons of rational numbers follow the rules for comparing integers
and comparing fractions and decimals.
• Between any two rational numbers, there is an infinite number of other
rational numbers.
• Every negative rational number can be written in any of these forms:
a
a
b , b ,
or - ba , where a and b are positive integers.
A number such as - 34 can be written as
To explain why
x
4x
-4x
x
-3
4
=
3
- 4 , we
-3
4
and x =
3
–3
4
0
3
- 4.
Multiply both sides by 4.
Multiply both sides by -1.
Divide both sides by -4.
3
- 4,
then
To explain why - 34 =
If we divide the distance from 0 to -3 on a number
line into 4 equal pieces, we can represent this as
(-3) , 4 (recall that dividing by a number can mean
creating that many equal groups). The measure of
the piece from 0 to the first division mark can therefore be represented by -43 .
or as
can use equations as shown below:
= -43
= -3
= 3
= -34
If x =
-3
4
-3
4
-3
4 , we
=
3
- 4.
can use a number line as shown below:
From the number line we can see that the distance -43
is the opposite of the distance 34 (the measure of
the piece from 0 to the first division mark when we
divide the distance from 0 to 3 into 4 equal parts,
3 , 4). Since the opposite of 43 is - 34 , that means -43
must be equal to - 34 .
3
3
–3
4
0
3
4
3
34
• Comparisons of rational numbers follow the rules for comparing integers
and comparing fractions and decimals.
This is an example of BIN 2—
knowing that a rational is negative tells you that it has to be
less than any positive.
Any negative rational is less than any positive rational or 0:
• A negative rational farther from 0 is less than one closer to 0
(e.g., -0.8 6 -0.6).
• A positive rational farther from 0 is greater than one closer to 0
(e.g., 0.8 7 0.6).
82
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:31 PM
Page 83
Two positive fractions can be compared in a variety of ways (BIN 4), as
described below:
Using Common Denominators
Using Common Numerators
Using Benchmarks
3
8
3
8
3
3
9
8 6 10 , since 8 is less
9
1
10 is greater than 2 .
5
8,
6 since 3 pieces of the same
size or unit (18 ) are less than
5 pieces of that size or unit.
3
10 ,
since 3 larger pieces
7
(eighths are larger than tenths)
are greater than 3 smaller pieces.
Notice that although we tend to emphasize comparisons involving common
denominators, probably since we use common denominators to add and
subtract fractions, using a common numerator can make a lot of sense in
6
3
particular situations. For example, it is much easier to compare 15
to 29
by
6
3
6
6
87
96
renaming 15 as 30 (29 > 30 ) than by renaming them as 435 and 435 .
Note that all of these methods presuppose that the same whole is being
9
9
compared. For example, if a student is comparing 23 of 10
to 34 of 12 , the 32 of 10
2
3
is greater, even though 3 is less than 4 . Students need to learn that the size of
the whole must be considered in comparing fractions and that if no size is
mentioned, we assume a whole of 1.
If one positive fraction is greater than another, it is farther to the right on the
number line:
This means that its opposite is farther to the left, so the opposite is less than
the opposite of the other fraction. That is, if ba > dc and ba and dc are positive,
then - ba < - dc .
PLANNING IDEA
than 12 and
2.3
Teaching Idea
Ask students to describe the
rational 2.345 345 345... so
that another student would
understand its size.
To focus on BIN 1 and 4, ask
how they could use benchmark numbers to help them.
[e.g., I would think of
2.345 345 345... as a bit
closer to 2.5 than 2
because 2.25 is exactly halfway between 2 and 2.5.
–2.5
–2
–2.345 345 345… ]
(See Big Ideas for Number, p. 79.)
2.1
Curriculum
A curriculum expectation/outcome expects students to demonstrate
an understanding of rational numbers by comparing and ordering them.
Big Idea Filter
BIN 4. Numbers are compared in many ways. Sometimes they are compared
to each other. Other times, they are compared to benchmark numbers.
Lesson Focus and Consolidation
You might focus the lesson on the different approaches to take to compare rationals and then use consolidating questions such as: How do
you know that - 23 7 - 34 ? [e.g., - 23 is 31 to the right of 1 and - 34 is only 14 to
the right of -1. Since
compare
- 23
1
4
6 13 , - 23 is closer to 0 and is greater.] Would you
3
4
and in the same way? Explain. [e.g., I would still use a benchmark
to help me but I would use 0, not 1, this time, and I would use it differently:
- 23 6 0 and 34 7 0, so - 23 is less than 34 .]
• Between any two rational numbers, there is an infinite number of other
rational numbers.
Students at this level should understand the density of rationals; that is,
between any two rationals there is always another rational. This becomes
relevant when students are asked, for example, to create a graph for a quadratic function that is wider than another but narrower than yet another.
There are a variety of ways to show this. For example,
7
10
• Using Common Denominators: 51 = 35
and 74 = 20
35 , so 35 is between
1
4
5 and 7 .
4
• Using Common Numerators: 51 = 20
, so 104 is between 51 and 47 .
• Adding Numerators and Denominators: 1 + 4 = 5 and 5 + 7 = 12,
5
so 12
is between 51 and 47 .
NEL
Chapter 2: Number
83
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
2.4
Teaching Idea
Have students compare the
following:
4
13
4
- 13
and
and
6
19
6
- 19
1
13
1
- 13
8
19
8
- 19
and
and
To bring out BIN 3 and 4, ask:
What strategy did you use to
6
4
compare 13
and 19
? Did you
use the same strategy for
1
8
13 and 19 ? Explain. [e.g.,
4
common numerators: 13
= 12
39
6
=
and 19
12
38
and 12
39 6
12
38 ,
so
Page 84
It might be interesting for students to see a graphical representation of
the notion that, if you add the numerators and add the denominators of two
given fractions, you end up with a fraction that is in-between:
If students draw a graph like the one shown here, they will see that the size
of a fraction is related to the steepness of the line joining the point (denominator, numerator) to (0, 0)—the steeper the line, the greater the fraction. It
is clear that the fraction resulting from adding the numerators and denominators is between the two given fractions because its slope is between the
slopes of the other two lines.
9
8
8
5
7
6
6 19
; no, I used bench1
marks: 13
is really close to 0,
8
1
8
6 19
]
but 19 is almost 12, so 13
4
13
(See Big Ideas for Number, p. 79.)
Numerator
How did these comparisons
affect how you compared the
negative rationals? [e.g., I
used “opposite thinking.”
6
4
I knew that if 13
6 19
, then
4
6
- 13 7 - 19 because if a positive fraction is less than
another positive fraction, its
opposite is greater than the
other number’s opposite; I did
the same for the other pair.]
6
5
4
5
4
3
1
3
2
1
0
0
1
2 3 4 5
Denominator
6
One by-product of looking for rationals between two others is an introduction to compound fractions, that is, fractions where the numerator or
denominator is, itself, a fraction and not a whole number. For example, students should recognize that
3 12
5
is a fraction between 35 and 45. They can then use
what they know about equivalent representations (BIN 3) to figure out that
3 12
5
7
a
is 10
3 12
5
=
3 12 * 2
5 * 2
=
7
10 b
in standard form.
Operations with Rational Numbers
Teaching Idea
2.5
Have students think about a
situation when they might
want to find a rational
4
number between - 11
and
5
- 11. [e.g., to find a line with
a slope between the slopes
4
of the lines y = - 11
x + 2
5
and y = - 11
x + 8]
To bring out BIN 3, ask: How
4
would you rename - 11
and
5
to find a rational
- 11
number between them?
[e.g., Multiply the numerator
and denominator of both by
8
9
2 to get - 22
and - 10
22 ; - 22 is
in-between them.]
(See Big Ideas for Number, p. 79.)
84
When adding, subtracting, multiplying, and dividing rationals, students need
to consider the signs of the numbers involved as well as how to combine
them. Many students are fairly comfortable with combining decimals, but
less so combining rational numbers in fraction form.
Adding and Subtracting Signed Numbers
It is important for students in Grades 9 to 12 to understand the following
about adding and subtracting rational numbers:
IMPORTANT POINTS ABOUT ADDING AND SUBTRACTING SIGNED NUMBERS
•
•
•
•
•
The sum of a number and its opposite is 0 (the zero principle).
The sum of any two negatives is negative.
The sum of any two positives is positive.
The sum of a positive and a negative can be positive, negative, or 0.
Adding a positive and a negative number is simplified by using the zero
principle.
• Adding a negative is the same as subtracting its positive opposite.
• To subtract a signed number, you can use either the take-away meaning
of subtraction or the missing-addend meaning.
• To subtract a positive or negative number, you can add its opposite.
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 85
• The sum of a number and its opposite is 0 (the zero principle).
The zero principle can be stated as either (-1) + 1 = 0 or (-n) + n = 0.
Mathematicians define -1 as the number to add to 1 to get 0, which explains
the first statement of the zero principle. The second version of the zero principle is an application of the distributive principle; i.e., if (-1) + 1 = 0, then
n[(-1) + 1] = n * 0, so (-n) + n = 0.
• The sum of any two negatives is negative.
This is because if you start to the left of 0 at a negative number and then
move farther to the left (because you are adding a negative number), you
are still to the left of 0.
• The sum of any two positives is positive.
This is because if you start to the right of 0 at a positive number and then
move farther to the right (because you are adding a positive number), you
are still to the right of 0.
• The sum of a positive and a negative can be positive, negative, or 0.
• If the positive is farther from 0 than the negative, the sum is positive.
This is because adding the negative does not bring the value all the
way back to 0 from the initial positive position.
• If the positive is closer to 0 than the negative, the sum is negative.
This is because adding the positive does not bring the value all the
way back to 0 from the initial negative position.
• If the positive and negative are opposites, the sum is 0.
• Adding a positive and a negative number is simplified by using the zero principle.
For example, - 23 + 43 = [(- 23) + 23] + 23 = 0 + 23, or 23 . The trick is to break
up the number farther from 0 into two parts, one of which is the opposite of
the number nearer 0.
• Adding a negative is the same as subtracting its positive opposite.
This can be explained in different ways. For example,
Using a Number Line Model
Using the Zero Principle
Adding a negative value is modelled on a number line by using an
arrow of the appropriate magnitude pointing to the left. The same
model is used for subtracting its
opposite. For example, both
2
4
2
4
3 + (- 3 ) and 3 - 3 are represented
below using the same model:
Another way to look at this is based
on the zero principle. For example,
to add 34 + (- 23), we break up 34 into
a 23 part and another part, so that
we can apply the zero principle. The
other part is 43 - 23.
4
3
1
3
0
4
3
2
+ (- 23) = (43 - 23) +
( 3 )
=
4
3
=
2
3
+ (- 23) =
4
3
2
3
-
4
3
+
2
3
2
3
+ (- 23)
(- 23)
Have students simplify each:
5.2 + (-4.7)
5.2 + 4.7
-5.2 + (-4.7)
To focus on BIN 3, ask: Why
do you use the zero principle
to add a positive and a negative rational number, but not
to add two positive rational
numbers or two negative
rational numbers? [When
you add two positives or two
negatives, there are no
opposites to combine to
make zero.]
(See Big Ideas for Number, p. 79.)
Teaching Idea
2.7
-5 - (-4)
=
4
3
-
2
3.
2
3
• To subtract a signed number, you can use either the take-away meaning of
subtraction or the missing-addend meaning (BIN 5).
For example,
- 43 - (-1) = - 13, because if you take away -1 from - 43 , or -113, - 13 is left.
- 43 - (-1) = - 13, because you have to add - 13 to -1 to get to - 43.
NEL
2.6
Have students simplify each:
+ 0
4
3
1
4
3
Therefore,
4
3
4
3
2
3
Teaching Idea
- 35 - (- 15)
-312 - (-4)
To focus on BIN 5, ask students how they interpreted
the subtraction each time.
[e.g., For -5 - (-4),
I compared -5 to -4; for
- 35 - (- 15), I used take-away;
for -312 - (-4), I added to -4
to get to -312.]
(See Big Ideas for Number, p. 79.)
Chapter 2: Number
85
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 86
• To subtract a positive or negative number, you can add its opposite.
A number line model can be used to explain why subtracting a positive is
the same as adding its opposite. Equations and the missing-addend meaning
of subtraction can be used to explain why subtracting a negative is the same
as adding its opposite. For example,
Subtracting a Negative
Subtracting a Positive
Write a missing-addend equation:
4
1 ( 3 )
1 - (- 43) = . S - 43 + . = 1
4
Add 34 to both sides and simplify:
13
- 43 +
4
3
+ . = 1 +
0 + . = 1 +
0
1
1
3
. = 1 +
4
3
4
3
4
3
So if 1 - (- 43) = . and . = 1 + 43,
1 + (- 43) = 1 -
4
3
then 1 - (- 43) = 1 + 43.
Multiplying and Dividing Signed Numbers
As with adding and subtracting signed numbers, students must understand
the following important points about multiplying and dividing signed
numbers.
IMPORTANT POINTS ABOUT MULTIPLYING AND DIVIDING SIGNED NUMBERS
• The product or quotient of two numbers with different signs is negative.
• The product or quotient of two numbers with the same sign is positive.
• The product or quotient of two numbers with different signs is negative.
• The product of a positive and a negative is negative, since the positive
factor can be thought of as the number or fraction of an amount
whose size is the negative factor. For example, 23 * (-4) = 23 of -4,
which can be thought of as 23 of the distance that is 4 units to the left
of 0, so it has to be negative.
• The quotient of a positive and negative is negative, since division is
the opposite of multiplication. For example, -4 , 23 is negative, since
if 23 of an amount is -4, which is negative, the amount must be negative.
• The product or quotient of two numbers with the same sign is positive.
• The product of two negatives is positive since it makes sense that the
result is the opposite of the product of a positive and a negative. For
example, -3 * - 23 should be the opposite of 3 * - 23, which is negative, and the opposite of a negative is positive.
• The quotient of two negatives is positive since division is the opposite
of multiplication. For example, if -5 , (- 34) = n, then - 34n = -5, so
n must be positive.
• The product of two positives is positive because a * b means
a groups of b, and if b is positive, the product must be a positive
amount. For example, 7 * 8 = 56 because 7 groups with 8 items
in each group is 56 items altogether.
Adding and Subtracting Numbers in Fraction Form
Students must understand the following important points about adding and
subtracting numbers in fraction form.
86
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 87
IMPORTANT POINTS ABOUT ADDING AND SUBTRACTING FRACTIONS
• To add or subtract two proper fractions with different denominators,
c
a
b and d , a grid of dimensions b by d can be used.
• Numerators and denominators cannot be separately added to create a sum.
• It is sometimes more efficient to leave mixed numbers in that form to add
or subtract them rather than forming improper fractions.
• Although it is acceptable to change fractions to decimals to operate with
them, this sometimes makes the computation more difficult.
• To add or subtract two proper fractions with different denominators, ba and dc ,
a grid of dimensions b by d can be used.
Using a grid of dimensions b by d makes it easy to model ba + dc or ba - dc . The
grid automatically expresses the fractions as equivalent fractions with a
common denominator. For example, to subtract 34 - 23 ,
Step 1
Start with a 3 by 4 grid.
Model 34 by filling 3 of
the 4 columns with
blue counters.
Step 2
Step 3
2
3,
To subtract you want to take away all the
counters in 2 rows (since 2 rows is 23 of the grid).
This is done in two steps.
Move counters to fill
2 rows.
3
4
9
or 12
of the grid has
blue counters.
Now 23 of the grid
can be taken away.
Remove the counters
from the 2 rows.
1
12 of the grid is
2
9
8
3
4 - 3 = 12 - 12
left.
=
1
12
• Numerators and denominators cannot be separately added to create a sum.
The process of adding numerators and denominators results in a fraction
between the two being added. For example, with 32 and 14 , 37 (23 ++ 14) is between
2
1
4 and 3 and is not their sum (which would be greater than both of the original fractions).
• It is sometimes more efficient to leave mixed numbers in that form to add or
subtract them rather than forming improper fractions.
Instead of automatically changing mixed numbers to their improper fraction equivalents, students should first analyze the numbers involved to see if
it might be easier to work with the numbers in their mixed-number form.
For example,
The subtraction 613 - 356 can be thought of as the
amount that must be added to 356 to reach 613 (the
missing-addend meaning of subtraction). Determining
the missing addend in this situation is fairly simple.
Working with this representation is much less likely
to lead to computational errors than working with
23
the improper fractions 19
3 - 6 , a process that requires
multiplication of fairly large whole numbers (BIN 3).
NEL
1
6
5
36
4
2
1
3
6
1
22
1
63
613 - 356 = ___ S 356 + ___ = 613
356 + 212 = 6 13 S 613 - 3 56 = 212
Chapter 2: Number
87
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 88
• Although it is acceptable to change fractions to decimals to operate with
them, this sometimes makes the calculation more difficult.
Many students avoid fractions, sometimes even at the teacher’s suggestion,
and convert all fractions to decimals. Although this sometimes simplifies calculations because the student can then use the decimal functions on a calculator, sometimes the calculation is actually harder than it needs to be.
Consider, for example, the situation of asking students which sum is
greater: 23 + 58 or 12 + 49. It is clearly preferable for a student to realize that
the second sum must be less because each of the addends is less than to
bother converting all the fractions to decimals and then adding.
PLANNING IDEA
2.2
Curriculum
A curriculum expectation/outcome expects students to demonstrate an
understanding of rational numbers by solving problems that involve arithmetic operations on rational numbers.
Big Idea Filter
BIN 6. There are many algorithms for performing a given operation.
Lesson Focus and Consolidation
You might focus the lesson on alternative approaches for adding and subtracting rational numbers in fraction form to solve problems. A consolidating
question might be as follows: The temperature in the morning was -312°.
It dropped to -1513° later in the day. How much did the temperature drop?
Solve the problem using two or more different strategies. Which strategy
did you prefer? Why? [-1156°; e.g., one way to solve is -1513 - (-312) S
-312 + __ = -1513, so I added - 12 to get to -1512 and then went backward 61
to get to -1513, which is adding -1156 altogether; another way is -1513 - (-312) S
-1513 + (+312) S -1513 + 313 + 16 = -12 + 16 = -1156; I prefer the first way
because I was able to sketch a number line to help me.]
Multiplying and Dividing Numbers in Fraction Form
Students must understand the following important points about multiplying
and dividing numbers in fraction form.
IMPORTANT POINTS ABOUT MULTIPLYING AND DIVIDING FRACTIONS
• The product of two fractions can be thought of as the area of a rectangle
with those dimensions.
• Multiplying a positive number by a positive proper fraction results in a
lesser number. Multiplying by an improper positive fraction results in a
greater number.
• The quotient of two fractions can be thought of as the number of groups
of the divisor that fit in the dividend.
• Another way to think about ba , dc is to decide what to multiply by dc to get ba .
• Although it is acceptable to change fractions to decimals to operate with
them, this sometimes makes the computation more difficult.
• The product of two fractions can be thought of as the area of a rectangle
with those dimensions.
This approach assumes that the dimensions of the grid are 1 unit by 1 unit
and that each factor describes a fraction of one of the dimensions.
The grid is created using the denominators of the factors. This approach also
88
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 89
shows students why the numerators and denominators of the factors are
multiplied to determine the part and the whole of the product. For example,
A 5 by 3 grid is used to model 35 * 23.
The diagram shows why the numerators and denominators are multiplied to determine the part and the
whole in the product.
3
5
• The shaded part shows an array based on multiplying the numerators.
• The whole is an array based on multiplying the
denominators.
2
3
10
2
3
3
2
3
5 * 3 is 5 of 3 or 5 of 15 .
10
of 15 , we could think
The diagram also shows that
3
5
2
3
Since of is the same as
3
5
*
2
3
=
6
15
1
5
3
5
3
5
is the area of a rectangle with
dimensions 53 of 1 and 23 of 1.
3
5
of 10 is 2, so
of 10 is 6, so
6
of 10 fifteenths is 6 fifteenths, or 15
.
• Multiplying a positive number by a positive proper fraction results in a lesser
number. Multiplying by an improper positive fraction results in a greater
number.
These examples of BIN 2 are explained below:
c
a
a
c
a
c
• Since b * d is b of d , then if b is less than 1, the part of d must be less
than dc . For example, 32 of 98 is part of 89 , so 23 *
c
a
a
c
8
9
6 89 .
a
c
• Since b * d is b of d , then if b is greater than 1, the part of d must be
greater than
c
d. For
example, 34
of is all of and a bit more, so 43 *
8
9
8
9
8
9
7 89 .
• The quotient of two fractions can be thought of as the number of groups of
the divisor that fit in the dividend.
This is an instance of BIN 5:
3
1
3
1
• 5 , 5 = 3, since there are 3 one-fifths in 3 fifths.
15
3
15
1
8
15
8
• 8 , 5 = 8 , since 8 = 40 and 5 = 40, and 40 , 40 tells how many
8 fortieths are in 15 fortieths, or 15 , 8, which is 15
8.
Teaching Idea
2.8
Have students simplify each:
• Another way to think about ba , dc is to decide what to multiply by dc to get ba .
Equivalent fractions can be used to explain this. For example, 38 , 15 =
a , c
30
2
2
c
30
15
a
c
c
80 , 10 . If 10 * d = 80 , then d = 8 . Notice that b , d is actually b , d .
c
Another way to look at the idea of what to multiply by d to get ba is to
think ba , dc = ba * dc because (ba * dc) * dc = ba .
- 34 , (- 14)
- 34 , 13
- 45 , 2
To focus on BIN 6, ask: For
which calculation would
you think of the division as
sharing? Why? [only for
- 45 , 2, since it does
not make sense to share
among 13 of a person
• Although it is acceptable to change fractions to decimals to operate with
(- 34 , 13) or - 14 of a person
This explains why ba , dc 7 1 if ba 7 dc , but ba , dc 6 1 if ba 6 dc (i.e., if ba 7 dc ,
then at least 1 group of dc fits in ba . If ba 6 dc , then not even 1 group of dc fits in ba ).
them, this sometimes makes the computation more difficult.
Consider, for example, the situation of multiplying 32 * 37. It is clearly preferable for a student to realize that 23 of 3 sevenths is 2 sevenths (72 ) (because
1
1
2
3
2
7 = 3 * 7 and 3 of 3 * 7 is 7 ) than to bother converting all the fractions to
decimals and multiplying. Working with decimals in some cases also results
in an approximated answer, for example, 23 * 37 0.67 * 0.43 0.29, and
0.29 Z 27.
(- 34 , (- 14), but it does make
sense to share among
2 people (- 45 , 2)] What
meaning would you use for
the other divisions? [e.g.,
for - 34 , (- 14), how many
groups of - 14 fit into - 34 , and
for - 34 , 13 , reverse multiplication (missing factor)]
(See Big Ideas for Number, p. 79.)
NEL
Chapter 2: Number
89
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 90
PLANNING IDEA
2.3
Curriculum
A curriculum expectation/outcome might expect students to demonstrate
an understanding of rational numbers by solving problems that involve
arithmetic operations on rational numbers.
Big Idea Filter
BIN 5. The operations of addition, subtraction, multiplication, and division
hold the same fundamental meanings no matter the domain in which they
are applied.
Lesson Focus and Consolidation
You might focus the lesson on bringing out the similarity in meanings of
operations with rational numbers to familiar meanings when operating with
whole numbers and then use a consolidating question such as: How does
what you know about multiplying integers and fractions explain why - 23 * 23
is greater than - 23? [e.g., The product is negative because the product
of a negative and a positive is negative; multiplying by a positive fraction
means taking that part of the other number, so 23 * 23 , or 32 of 32 , is less than 32 .
If 32 7 23 * 23 , then ( - 23 ) 6 23 * ( - 23 ).]
Teaching Idea
2.9
Ask students to provide several examples of rational
decimals and irrational decimals and ask them how they
know which are rational.
[e.g., rational: 0.12, 0.111...,
0.234 234 234...; irrational:
0.313 113 111..., 0.102 030 405...;
rational numbers are
repeating or terminating
decimals that can be written
as fractions, but irrationals
can’t]
To focus on BIN 2, tell students that a number can be
written as the decimal
3.121 231 231 231 23..., and
then ask: How do you know
this is a rational number,
not an irrational number?
[Even though the 12 at the
start does not repeat, the
rest of the decimal is
repeating, so it is a fraction.
The fraction is the sum
12
123
+ 99900
.]
of 3 + 100
(See Big Ideas for Number, p. 79.)
Some students think that all
square roots are irrational.
Provide students with opportunities to see that this is not
always the case; for example,
125 or 10.09 are not irrational
since they are equal to 5 and
0.3, respectively.
90
Irrational Numbers
It might be useful to revisit the diagram on page 81 that shows how irrational numbers and rational numbers are separate sets that, together, make
-3
up the real numbers. Examples of irrational numbers are p, 17, and 12
.
Non-repeating, Non-terminating Decimals
Once students understand that every repeating decimal is a fraction, they can
deal with the definition of an irrational number as a non-repeating, nonterminating decimal. Initially, although understanding the meaning of nonterminating is not difficult, students struggle with what non-repeating means.
An example might be necessary to clarify this idea. For example, students
might learn that 0.1010010001... (where an additional 0 is used in each subsequent grouping before the 1 is written again) is an example of a non-repeating
decimal. It is fairly obvious that the 10, the 100, the 1000, and so on do not
repeat. What is less clear is that there is never a group of digits that repeats.
This demands reasonably sophisticated thinking and is difficult for many students. Most simply take it on faith that if the teacher says it does not repeat, it
does not. Technically, students must realize that because the number of zeroes
keeps growing, there is no way that any group of digits can repeat.
Irrational Square Roots
The rationale for why a square root like 12 is irrational is fairly complex. Most
students can understand why it cannot be 1.4, 1.41, or 1.414 (since 1.42 = 1.96,
1.412 = 1.9881, and 1.4142 = 1.999 396, none of which is 2), but they cannot
really be sure that if they go out far enough into the sequence, the decimal will
not terminate and the value of the square will not be 2. In fact, students need to
follow some pretty sophisticated logic—a proof by contradiction. The proof
that 12 cannot be rational and, as a result, must be irrational, follows:
Let us assume 12 is rational. If 12 is rational, you can write it as the
quotient of two integers in lowest terms (with no factors in common), so
let ba represent that quotient.
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
If 12 = ba , then (12)2 =
Page 91
a2
b2 ,
2 =
a2
b2 ,
and 2b2 = a 2.
If a2 is a multiple of 2, then a2 must be even.
Since even2 = even, and odd2 = odd, then a must be even.
If a is even, then a is the product of 2 and another integer, which might be
called k. Then a = 2k and a2 = 4k2.
Since 2b2 = a2 (from above), then 2b2 = 4k2 and b2 = 2k2.
Since even2 = even and odd2 = odd, then b must also be even.
If both a and b are even, they must have a factor in common (2), which
violates the assumption that was made that ba is in lowest terms.
That means that there is no way 12 can be the quotient of two integers in
lowest terms, contradicting the assumption. Thus, 12 is irrational.
Many students will find this complicated, and will simply accept the
teacher’s word. Some students will be able to handle such thinking.
Irrational Numbers as a Subset of Real Numbers
Teaching Idea
Students may eventually stop focusing on whether numbers are rational or
irrational and simply think of the entire set of real numbers, including both
subsets. Ultimately, both rational and irrational numbers are used for the same
purpose, to tell how many or how much (BIN 1). Students will need to recognize that each point on a number line is associated with a single real number,
and vice versa. What is challenging for some students is that there is room for
irrational numbers on the number line, knowing that rational numbers are
dense; that is, there is an infinite number of rationals between any two given
rationals. Again, this requires a leap of faith. Using more sophisticated mathematics, it can be proven that the infinity of the real numbers is actually greater
than the infinity of the rational numbers.
Comparing Irrational Numbers
Students might compare irrational numbers by using a rational approximation or using other information about the irrational, for example, about its
square (BIN 4). For example,
• 12 6 1.5, since 12 is about 1.4 and 1.4 6 1.5.
• 12 6 13, since a square of area 2 is smaller than a square of area 3,
so the side length of the first square, 12, is less than the side length
of the second square, 13.
PLANNING IDEA
2.4
2.10
Ask students to determine
the circumference and area
of circles with radii of 1, 2, 3,
4, and 5 units in terms of p.
[2p, p; 4p, 4p; 6p, 9p;
8p, 16p; 10p, 25p]
To focus on BIN 1 and 4,
ask students whether the
number of units associated
with the circumference is
usually more or less than the
number of square units associated with the area. Then
ask them if they think this is
true no matter what the
radius. [When the radius is
greater than 2, the area
number is greater than the
circumference; when the
radius is 2, the area number
is equal to the circumference
number; and when the
radius is less than 2, the
area number is less than
the circumference.]
(See Big Ideas for Number, p. 79.)
Curriculum
A curriculum expectation/outcome might expect students to compare irrational numbers.
Big Idea Filter
BIN 4. Numbers are compared in many ways. Sometimes they are compared
to each other. Other times, they are compared to benchmark numbers.
Lesson Focus and Consolidation
You might focus the lesson on the fact that irrational numbers might be
compared using more familiar, but close, rational values and then use a consolidating question such as: How can you use what you know about rational
numbers to decide whether 4p is greater or less than 1139? [e.g., Since
p 7 3, 4p 7 12. Since 139 6 144, 1139 6 1144, or 1139 6 12. That
means 4p 7 1139.]
NEL
Chapter 2: Number
91
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 92
Matrices
Matrices are often used to show
connectivity in networks of
points called vertices joined by
paths called edges.
Matrices (singular matrix) are rectangular arrays of data. Matrices are used
to efficiently describe a lot of data, and, in appropriate situations, can be
added, subtracted, multiplied, or divided in meaningful ways.
The size of a matrix is indicated by counting the number of rows first and
the number of columns second. For example, a matrix with 4 rows and
6 columns is called a 4 by 6 (or 4 * 6) matrix. Matrices that have only 1 row
or 1 column are called vectors.
Here is an example of a 4 by 3 matrix that represents the number of books
in a library borrowed by people of different ages in different categories.
Ages 6 to 12
Ages 13 to 18
Adults younger than 65
Seniors
Picture books
320
42
134
16
Fiction
82
729
1040
411
Nonfiction
142
511
793
345
Usually a matrix is represented by a capital letter. For example, L.
Because the decision about which data are in rows versus which data are
in columns is arbitrary, the rows and columns can be reversed to form a
transposed matrix for example, LT.
Library Matrix
Transposed Library Matrix
The rows are the 4 age ranges and
the columns are the number of
books in 3 categories.
The rows are the number of books
in 3 categories and the columns
are the 4 age ranges.
320
42
L = D
134
16
320
LT = C 82
142
82
729
1040
411
142
511
T
793
345
42
729
511
134
1040
793
16
411 S
345
Operations with Matrices
Addition and Subtraction
If the library matrix above represents the numbers of transactions in one month
and a similar matrix represents the information for another month, it makes
sense to add corresponding terms to create a matrix to show the total transactions over two months, and it makes sense to subtract corresponding terms to
create a matrix to show how many more transactions were in month 1 than in
month 2. Adding matrices can be thought of as joining them and subtracting as
comparing them or seeing what is left, just as with whole numbers (BIN 5).
Two simple 2 by 2 matrices are used to demonstrate addition and subtraction of matrices:
Some calculators allow students
to add, subtract, multiply, and
divide matrices.
92
c
2
-4
3
1
d + c
6
3
-4
2 + 1
d = c
6
-4 + 3
3 - 4
3
d = c
6 + 6
-1
c
2
-4
3
1
d - c
6
3
-4
2 - 1
d = c
6
-4 - 3
3 - (-4)
1
d = c
6 - 6
-7
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
-1
d
12
7
d
0
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 93
Multiplication
Teaching Idea
You can multiply a matrix by a number, often called a scalar, by multiplying
each term by that number. For example, if the number of library books in
each category is tripled, each entry is multiplied by 3.
A simple 2 by 2 matrix is used to demonstrate scalar multiplication:
3 * c
7
3 * 1
d = c
0
3 * (-7)
1
-7
3 * 7
3
d = c
3 * 0
-21
Have students think about
how operations with
matrices compare to operations with real numbers.
To focus on BIN 5, ask: How
are adding and subtracting
matrices like adding and subtracting real numbers? How is
multiplying matrices like multiplying real numbers? How is
it different? [You add and
subtract matrices in the same
way as you add or subtract
numbers except you have to
do a lot of calculations, one
for each pair of corresponding terms. Adding still means
putting together and subtracting still means how much
more one thing is than
another. Multiplying is different since you’re not really
making equal groups or areas
of rectangles like you would
with real numbers.]
21
d
0
You can also multiply two matrices, although not by multiplying corresponding terms. For example, suppose matrix A below shows the marks
of 4 students (the 4 rows) in a course on projects, tests, and quizzes (the
3 columns), and matrix B tells the weightings for projects, tests, and quizzes:
95%
87%
A = D
73%
66%
92%
81%
82%
71%
83%
79%
T
88%
74%
0.4
B = C 0.4 S
0.2
Notice that the number of columns in the first matrix matches the number
of rows in the second one (a requirement for multiplying matrices).
It makes sense to multiply the 3 values in each row by the 3 weightings and
add them to get a final score for each student. That is, in fact, how matrix
multiplication is handled:
95%
87%
A * B = D
73%
66%
*
*
*
*
0.4
0.4
0.4
0.4
+
+
+
+
92%
81%
82%
71%
*
*
*
*
0.4
0.4
0.4
0.4
+
+
+
+
83%
79%
88%
74%
*
*
*
*
2.11
(See Big Ideas for Number, p. 79.)
0.2
0.2
T
0.2
0.2
91.4%
83.0%
= D
T
79.6%
69.6%
Notice that the resulting product matrix has the number of rows
of the first matrix and the number of columns of the second one.
If there are 2 different weighting combinations for the teacher to consider,
matrix B will have 2 columns.The same sort of process, where each element in
a row in matrix A is multiplied term by term by each element in a column in
matrix B and the values added applies to the second column as well.
The process is described below in general terms:
If A = E
n11
n21
n12
n22
n1b
n2b
Á
Á
Á
Á
na1
na2
NEL
11
a2
21
ab
b1
m12
m22
m1c
m2c
Á
Á
Á
Á
mb1
nab
Á
A * B =
n11m11 + n12m21 + Á + n1bmb1
n21m11 + n22m21 + Á + n2bmb1
Á
E
Á
n m + n m + Á + n m
a1
U and B = E
m11
m21
mb2
mbc
Á
n11m12 + n12m22 + Á + n1bmb2
n21m12 + n22m22 + Á + n2bmb2
Á
Á
n m + n m + Á + n m
a1
12
a2
22
ab
U , then
b2
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
n11m1c + n12m2c + Á + n1bmbc
n21m1c + n22m2c + Á + n2bmbc
Á
U
Á
n m + n m + Á + n m
a1
1c
a2
2c
ab
Chapter 2: Number
bc
93
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 94
If an a * b matrix is multiplied by a b * c matrix, the result is an a * c
matrix. So if a 3 by 5 matrix is multiplied by a 5 by 6 matrix, the product
matrix will be 3 by 6.
Because of the way matrices are multiplied, with each value in a row of the
first matrix being multiplied by a corresponding value in the column of the
second one, it is essential that the number of rows in the right-hand matrix
match the number of columns in the left-hand matrix. For example, a 1 by 3
matrix can be multiplied by a 3 by 2 matrix, but only if the 1 by 3 matrix is on the
left. If it were on the right, the number of columns in the left matrix would be 2
and would not match the number of rows in the right one (which would be 1).
One of the applications of matrix multiplication is to describe connections
in networks. Another application involves working with simultaneous equations.The motivation for multiplying matrices in the manner described above
is, in fact, to be able to represent a system of equations such as 2x + 3y = 12
and -3x - y = 7 as shown below, in order to solve it:
c
x
12
3
d * c d = c d
7
-1
y
2
-3
Division
As with multiplication, matrices can be divided by scalars by dividing each
value by the scalar. For example,
c
60
75
90
60 , 3
d , 3 = c
96
75 , 3
90 , 3
20
d = c
96 , 3
25
30
d
32
A situation where this might happen is when the values in the original
matrix represent the areas of 4 rectangles with widths of 3, and the resulting
matrix describes their lengths.
Although matrices cannot be formally divided, inverse multiplication can
be used (BIN 6). At a level beyond high school, students learn how to do this.
Representing Numbers
We represent numbers in different ways (BIN 3), sometimes to make a computation simpler, sometimes to better compare one number to another, and
sometimes to highlight some property of that number. This is true whether
we are representing whole numbers as products of factors, representing
fractions as decimals or percents, representing lengths using absolute values,
representing ratios as equivalent ratios, representing repeated multiplication using powers, representing fractional powers as roots, or representing
exponentiation logarithmically.
Percents
Students are usually comfortable with whole-number percents between 0%
and 100%. Students have learned to interpret x% as x out of 100 squares.
For example,
A model for 25%
94
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
A model for 12.5%
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 95
The percents that cause difficulties are greater than 100% and less than
1%, for example, 0.2%, which is often confused with 0.2, 20%, or 2%.
Teaching Idea
2.12
Ask students to write a
variety of decimal percents,
such as 4.2%, 0.8%, and
1.5%, as decimals.
A model for 2%, or 0.02
A model for 20%, or 0.2 A model for 0.2%, or 0.002
It is important that students have opportunities to work with these more
difficult percents. If students always think of a percent as a fraction with a
denominator of 100, it makes sense that, for example,
• 210% =
• 0.5% =
210
100 ,
which is a bit more than 2.
0.5
100 , which
is 21 of 1% (not 12 , which is 0.5).
To focus on BIN 3, ask: How
could you have predicted
that the decimal representation involved thousandths?
[4.2% = 0.042; 0.8% = 0.008,
and 1.5% = 0.015; since
whole-number decimals are
hundredths, to get rid of the
decimal in the fraction, I’d
multiply by 10 or 100 or
1000, so the denominator
would be thousandths or ten
thousandths.]
(See Big Ideas for Number, p. 79.)
Percent Growth
Another area of difficulty for many students is percent growth, yet this will
be important for secondary students working with, for example, compound
and simple interest and population growth. For example, when we say that a
value has grown 150%, we actually mean that the final value is 250% of
what it previously was, a different way to compare numbers (BIN 4). It is
very easy for students to get confused about this. You might use visual or
pictorial representations like that below to help make sense of this (BIN 3).
20
20
10
Original = 20
Growth = 20 + 10 (100% + 50% of original)
Final value = 20 + 30, or 50 (2.5 times, or 250% of original)
It is critical that students think about what the percent growth is a percent
of. For example, if we say that a population grew 10% and is now 2 000 000,
that 10% growth is 10% of the original population, not 10% of 2 000 000.
Many students determine 10% of 2 000 000 and then subtract to end up with
1 800 000 (but a 10% growth on 1 800 000 is 1 980 000). If a 10% growth
results in 2 000 000, then 2 000 000 is 110% of the original value, and we
divide 2 000 000 by 1.1 to determine the original population of 1 818 182.
Again, a visual might help (see below), but what is most important is that
students think about which number to take the percent of.
10% of P
Original population (P)
2 000 000
Absolute Value
As students work in higher grades, there are a number of instances where
the magnitude of a quantity, and not its direction (i.e., sign), is of interest.
The notation for the absolute value, or the magnitude regardless of direction, is ƒ ƒ . So ƒ -3 ƒ = ƒ 3 ƒ = 3. The absolute value represents the distance
from 0 and is always positive. The negative of an absolute value is negative.
NEL
Considering quantity and not
direction is not difficult for most
students, but they do need to
become accustomed to the
absolute value notation.
Chapter 2: Number
95
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 96
Algebraically, we write ƒ x ƒ for the distance of x from 0.We can write ƒ x - 3 ƒ
to represent the distance of x from 3. When solving equations involving
absolute values, we have to consider two possibilities. So, for example, the
solutions to the equation ƒ x - 3 ƒ = 2 are 1 and 5 because the distance could
be from x to 3 (from 1 to 3 is 2) or from 3 to x (from 3 to 5 is 2).
Students can use absolute value to solve problems. For example, suppose a
student is looking for all values on the real number line that are within
2 units of 45 in either direction. In this case, they are actually solving the
inequality ƒ x - 45 ƒ 6 2 because the direction on the number line is irrelevant.
PLANNING IDEA
2.5
Curriculum
A curriculum expectation/outcome expects students to demonstrate an
understanding of the absolute value of real numbers.
Big Idea Filter
BIN 1. Numbers tell how many or how much.
Lesson Focus and Consolidation
You might focus a lesson on how numbers are sometimes used to describe
how much something is in terms of distance or length. To consolidate the
lesson, you might ask: When does it make sense to use an absolute value to
describe a distance? When does it not? [e.g., It makes sense if you want to
know how far apart two numbers are, but it does not make sense if you
have to know which is greater; for example, (-3) - (-4) and (-4) - (-3)
have the same absolute value, but if you’re subtracting, you’d want to know
the direction you’re going in so you need to pay attention to the signs.]
Proportional Reasoning
Proportional reasoning
underpins a significant
amount of the number work
from Grades 5 to 9.
Proportional reasoning underpins a significant amount of the number work
from Grades 5 to 9. Students use proportional reasoning when solving
problems involving ratio, rate, and percent; when writing one number as a
factor of another; and when creating scale diagrams, but also in higher
grades when factoring expressions or studying similarity or trigonometry.
Proportionality is a way of describing a multiplicative relationship. One
variable is proportional to another if one variable is always a constant multiple
of the other. For example, the numbers in the 2 times table (2, 4, 6, 8, ...) and the
corresponding numbers in the 3 times table (3, 6, 9, ...) are proportional; when
the sequences are compared (BIN 4), each number in the second sequence
is 1.5 times the number it matches in the first sequence. In contrast, the sets
of numbers 1, 2, 3, ... and 1, 4, 9, ... , the natural numbers and their squares, are
not proportional; there is no constant multiplier that relates terms in the
first sequence to corresponding terms in the second sequence.
Direct and Inverse Proportion
Two quantities, A and B, are directly proportional when A increases by a
certain factor if B increases by the same factor. For example, the variable a
representing the number of arms for n people and the variable n representing the number of people are directly proportional quantities; if the
number of arms, a, doubles, so does the number of people, n, and vice versa.
Often this proportionality factor is called a scale or a proportionality constant.
If y and x are proportional, we can always write y = kx or x = k1 y. In the
case of the arms and people, if y represents the number of people and x represents the number of arms, k has a value of 2.
96
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 97
Here are some other examples of directly proportional quantities:
• The perimeter of a square and its side length are proportional
quantities, since the perimeter is always 4 times the side length.
• The value of a number of dimes is proportional to the value of the
same number of quarters, since the value of the dimes is always 25 of
the value of the quarters.
• The distance on a map drawn to scale is always proportional to the
real distance it represents, based on the scale of the map.
Other times, two quantities, A and B, are inversely proportional, that is,
A = kB1 . This is the case when A increases by a certain factor, while B
decreases by the same factor. For example, the time required to complete a
task is inversely proportional to the number of people working on the task
(assuming everyone works at an equal rate). For example, if there are 3 times
as many people working, the task should be completed in 13 of the time.
Much of the work in secondary mathematics and science is about determining whether two variables are proportional or not and, if so, what the
proportionality constant, k, is. In order for students to be successful at this, it
is essential that they recognize multiples when they see them. For some students, review work on factoring numbers may be crucial.
PLANNING IDEA
Much of the work in secondary
mathematics and science is
about determining whether
two variables are proportional.
2.6
Curriculum
A curriculum expectation/outcome expects students to solve proportions.
Big Idea Filter
BIN 3. There are many equivalent representations for a number or numerical
relationship. Each representation may emphasize something different about
that number or relationship.
Lesson Focus and Consolidation
You might focus a lesson on helping students see that solving a proportion
is really about writing an equivalent ratio or rate in a more useful form.
A consolidating question might be: Jeff wants to know how far he can go
in 1.3 h if he can go 87 km in 1 h. Explain why he is really trying to find an
87
87
equivalent ratio for 1.0
with a denominator of 1.3. [1.0
is the number of
kilometres per hour, so if you multiply the numerator and denominator by
1.3, you’ll get an equivalent ratio for the number of kilometres in 1.3 h.]
Ratio and Rate
Ratios are comparisons of numbers where the comparison is multiplicative.
For example, if a boy’s height at age 9 is 75% of his adult height, the ratio
3 : 4 compares the 9-year-old’s height to the adult’s height. Since the ratio is
multiplicative, if one 9-year-old’s height is 6 cm more than another’s, the difference in the adults’ heights is not 6 cm, but 43 * 6 cm.
The focus in higher grades is on multiplicative relationships rather than
additive ones, since multiplicative relationships underlie linear relations, a
topic that is significant at these higher levels.
Sometimes, ratios are part-to-whole ratios as in the height ratio above
(with the child’s height being the part and the adult’s height being the
whole), and sometimes they are part-to-part (BIN 4). For example, the ratio
1 : 2 : 3 for concrete (the ratio of cement to sand to gravel) is a part-to-part
ratio because it compares the number of units needed for each of the 3 parts
that make up the concrete recipe. A part-to-whole ratio that describes the
same situation could be 1 : 6 (1 unit of cement to make 6 units of concrete)
or 2 : 6 (2 units of sand to make 6 units of concrete).
NEL
Chapter 2: Number
97
7229_Ch02_p079-106.qxd
4/7/10
Teaching Idea
3:32 PM
2.13
Ask students to model the
part-to-part ratio 4 : 5 using
10 counters. [e.g., 4 red counters and 5 blue counters]
To focus on BIN 3 and 4, ask:
Why does it make sense that
the ratios 4 : 5 and 8 : 10
describe the same situation
even though 10 and 8 are
farther apart than 4 and 5?
Use an example to help you
explain. [e.g., If there are 5
boys for every 4 girls, then if
there are 2 groups of 9 children, there would be
10 boys for every 8 girls. It
doesn’t matter how far apart
they are.]
(See Big Ideas for Number, p. 79.)
Page 98
Percents are special ratios where the second term is 100. Calculations
involving percents are not essentially different from calculations involving
other ratios and rates.
Rates compare values where the measurement units involved are different.
For example, a rate might describe kilometres per hour, dollars per item, area
required in a classroom per child, and so on. Some people consider rates special ratios and others make a distinction between ratios and rates. The differences are sometimes subtle. For example, if a map uses a distance of 1 cm to
represent a distance of 1 km, we need to use the ratio 100 000 : 1 to describe the
scale ratio (so the units are the same, namely centimetres), but we can also use
the rate of 1 cm to 1 km to describe the same comparison (BIN 3 and 4).
Rates are important everyday calculations. We often compare prices
based on rates, for example, how much per millilitre for each of 2 different
brands of laundry detergent. There is often an emphasis on using unit rates
(e.g., comparing how much per 1 mL as opposed to how much per 100 mL)
to make these comparisons easier, but it is important for students to recognize that rates could be compared differently depending on the numbers
involved and the context. For example, suppose Cookie Brand A is $3 per
dozen and Brand B is $2.25 for 8 cookies. It is hard to tell at first glance
which price is better. Units rates could be used: Brand A costs 25¢ per
(1) cookie, and Brand 2 costs a bit more than 28¢ per (1) cookie; however, it
might be easier to compare them using the price per 4 cookies: Brand 1 costs
$1 for 4 cookies, but Brand 2 costs a bit more than $1.12 for 4 cookies. In
essence, a 4-cookie unit is being used instead of a 1-cookie unit.
PLANNING IDEA
2.7
Curriculum
A curriculum expectation/outcome expects students to compare the properties of direction and partial variation.
Big Idea Filter
BIN 2. Classifying numbers provides information about the characteristics of
the numbers.
Lesson Focus and Consolidation
You might focus a lesson on what we know when two variables are related
using partial variation as opposed to direct variation. Consolidating
questions might be: You know that y = 3x + 2 describes the relationship
between x and y. How do you know that the corresponding values for x and
y are not proportional? [e.g., There is no one number you can multiply x by
to get y every time, since if x = 1, you’d multiply y by 5, but if x = 2, you’d
multiply y by 8.] What two sets of corresponding values are proportional?
[x and y - 2] Why are they proportional? [e.g., If you double x, then
y = 6x + 2, so y - 2 = 6x instead of 3x—it’s been doubled.] Is there always a
set of proportional values with partial variation? [e.g., Yes, for y = mx + b,
it would be x and y - b, and the multiplier is m.]
Solving Ratio, Rate, and Percent Problems
Much of the number work in Grades 7 to 10 involves working with ratios,
rates, and percents. Often, a ratio or rate is provided in one form, and an
equivalent ratio or rate is needed to solve a problem.
Using Ratio Tables
Students can use ratio tables to solve problems such as the following:
98
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 99
A lead content greater than 400 parts per million in soil is dangerous to
crops planted in that soil. A soil test is taken of 480 cm3 of soil. How much
lead is allowable?
To solve this problem, the student realizes that the ratio 400 : 1 000 000
must be changed to an equivalent ratio with a second term of 480 (BIN 3). A
table can be created where the values in each column represent equivalent
ratios. The first column is the given ratio. In order to determine the equivalent ratio needed, subsequent columns (equivalent ratios) are created by
multiplying or dividing existing columns or by adding or subtracting combinations of existing columns.
0.16 + 0.032 = 0.192
400 80 480
Lead
Soil
400
1 000 000
0.04
100
0.16
400
10 000
4
0.032
80
0.192
480
Since 0.192 : 480 is equivalent to 400 : 1 000 000,
the problem is solved; there can only be 0.192 cm3
of lead.
5
Columns can be multiplied or divided by any number to get equivalent
ratios, since it is just a matter of thinking of the ratio as a fraction and then
multiplying or dividing numerator and denominator by the same amount.
We can use the distributive property to explain the thinking behind
adding or subtracting combinations of columns:
Column 1 is x : y, column 2 is mx : my, and column 3 is nx : ny.
mx + nx = x(m + n)
[add the values in columns 2 and 3 of the top row]
my + ny = y(m + n)
[add the values in columns 2 and 3 of the bottom row]
x(m + n)
y(m + n)
[create a ratio using the sums]
x(m + n)
y(m + n)
=
x
y
[simplify to observe that the new ratio is equivalent to the others]
Tables can also be used to solve rate problems. For example,
A car travels 171 km in 1.8 h. If it continues at that speed,
how far will it go in 3 h?
Distance (km) 171
Time (h)
1.8
3
57
0.6
570
6
10
285
3
The car will go 285 km in 3 h.
2
Notice that to use ratio or rate tables effectively, students need to use
their number sense to look for relationships that relate one of the original
values to the desired new value. Although students who lack this number
sense will find it more difficult to use these tables, the result will likely be an
improvement in that number sense.
Using Proportions
Sometimes we describe the equivalence of two ratios using an equation;
when we do this, we create what is called a proportion. Proportions can be
used to solve the same problems as ratio tables. For example, for the soil
400
l
problem above, the proportion is 1 000
000 = 480 , with l being the unknown
amount of lead. Solving the proportion for l solves the problem.
NEL
Chapter 2: Number
99
7229_Ch02_p079-106.qxd
4/7/10
3:32 PM
Page 100
400
1 000 000
400
1 000 000
=
l
480
* 480 =
l
480
192 000
1 000 000
* 480
= l
0.192 = l
When students cross-multiply,
they are often following a rote
procedure that skips important
steps; it should be discouraged,
especially for struggling students.
When students cross-multiply, they are often following a rote procedure
that skips important steps. Although this method does work, it should be discouraged, especially for struggling students, until they have a better conceptual
understanding of what solving proportions is all about, as it does not promote
the same level of number sense that other methods promote. For example,
x
contrast the number sense exhibited in solving 68 = 28
by cross-multiplying to
get 8x = 6 * 28 and dividing by 8 with recognizing that 86 is really a ratio of
3 : 4, so it makes sense that x must be seven 3s since 28 is seven 4s.
Note that the difficult part for many students when solving problems
using a proportion is deciding how to set up the two ratios. For example, for
the problem above, they might wonder whether the 480 belongs in the
numerator or the denominator. They need to think about what the various
parts of each ratio represents. For example, they might think, “Since
400
400 : 1 000 000 or 1 000
000 represents the ratio of lead to soil, then the numerator should be the lead, l, and the denominator should be the 480.” Many
teachers encourage students to write the names of the items being com400
l
pared beside the ratio; for example, lead
soil = 1 000 000 = 480 .
You or a student might initially wonder about the value of using a ratio table
400
l
compared to simply solving the proportion 1 000
000 = 480. The advantage to the
student is that he or she can think proportionally without the additional
abstraction of solving an algebraic equation, but, even more than that, the student can take as many steps as is helpful to get to the desired result. This alternative algorithm (BIN 6) is valuable for students who need that extra flexibility.
PLANNING IDEA
2.8
Curriculum
A curriculum expectation/outcome expects students to solve problems
involving ratios, rates, and directly proportional relationships in various
contexts using a variety of methods.
Big Idea Filter
BIN 3. There are many equivalent representations for a number or numerical
relationship. Each representation may emphasize something different about
that number or relationship.
Lesson Focus and Consolidation
You might focus a lesson on how using more traditional fraction equivalence
methods to solve proportions is essentially the same as using a ratio table.
x
A consolidating question might be: Jeff solved 68 = 28
by first writing 68 as
the equivalent fraction 34 and then multiplying the 3 by 7. How might he
have shown the same thing with a ratio table? [e.g., He could have a
column with 6 and 8, then divide both numbers by 2 to make a column
with 3 and 4, and then multiply both numbers by 7 to get a column with
21 and 28.] Which method would you use and why? [e.g., the ratio table
because it keeps me organized]
Powers and Roots
Just as students in early grades learn to use multiplication to more efficiently record repeated addition, for example, using 7 * 3 instead of
100
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:33 PM
Page 101
3 + 3 + 3 + 3 + 3 + 3 + 3, students in higher grades learn to use powers
to more efficiently record repeated multiplication, for example, using 37
instead of 3 * 3 * 3 * 3 * 3 * 3 * 3. They learn that in the expression
ab, a is the base and b is the exponent: a is the value that is repeatedly multiplied and b is the number of times it appears as a factor.
Some students will wonder whether there are shortcut ways to record
repeated subtraction and repeated division. There are, in fact. Division is a
shortcut for repeated subtraction; for example, we write 54 , 9 to determine how many 9s can be subtracted from 54 before reaching 0 (54 - 9 1
9 - 9 - ...). We use roots to describe repeated division; for example, 32 5 or
5
132 is used to determine what number must be used as a repeated factor
5 times to result in 32.
Powers
To determine powers with a calculator, students must become familiar with
the necessary buttons to press. Usually, to determine, for example, 1.243,
a student presses 1.24, then the key yx, and then the key 3.
Geometric Interpretations of Powers
Having been introduced to square and cubic units of measurement in earlier
grades, students should explicitly make the connection between the powers
of 2 and 3 and the areas and volumes of squares and cubes. They should
relate, for example, 42 to the area of a square face of a cube with an edge
length of 4 and 43 to the volume of the cube. They will generalize this to variables later: x2 as the area of a square with side length x and x3 as the volume
of a cube with edge length x.
Teaching Idea
2.14
Once a student knows that
one number can be written
as a power of another, he or
she knows even more things
about that number. Have students determine the values
of several powers of 4. [e.g.,
16, 64, 256]
To focus on BIN 2, ask:
Suppose a number is 4 x
where x is an even counting
number. What else do you
know about 4 x? [e.g., It’s a
power ofx 2 (22 x), it’s a power
of 16 (162 ), it’s more than 15,
and it’s even.]
(See Big Ideas for Number, p. 79.)
4 cm
4 cm
4 cm
Asquare face 42; Vcube 43
Place Value Connections to Powers
Students’ first exposure to powers tends to be in the context of place value.
They learn that the place value columns represent decreasing powers of 10 as
they move right. Even if 0 and negative exponents are introduced this way, it
is only later that students generalize the meanings of these exponents.
Thousands
Hundreds
Tens
Ones
Tenths
103
102
101
100
10ⴚ1
Hundredths Thousandths
10ⴚ2
10ⴚ3
Exponent Laws
In order to calculate with and simplify powers, students need to be familiar
with the exponent laws. These are sometimes called exponent rules.
EXPONENT LAWS
1. The product of two powers with the same base can be expressed as a single
power with the same base by adding the exponents; i.e., am * an = am + n.
2. The quotient of two powers with the same base can be expressed as a single
m
power with the same base by subtracting the exponents; i.e., aan = am - n.
3. The product of two powers with different bases but the same exponent
can be expressed as a single power with the same exponent by multiplying the bases; i.e., am * bm = (ab)m.
4. The quotient of two powers with different bases but the same exponent
can be expressed as a single quotient with the same exponent by dividing
m
the bases; i.e., bam = (ba )m.
5. A power raised to a power can be represented as a single power with the
same base by multiplying the exponents; i.e., (am)n = amn.
NEL
Chapter 2: Number
101
7229_Ch02_p079-106.qxd
4/7/10
3:33 PM
Page 102
LAW 1 The product of two powers with the same base can be expressed as a
single power with the same base by adding the exponents.
For example, 34 * 35 = 34 + 5 = 39. This is based on the definition of a
power. Since 34 means four 3s multiplied together and 35 means five 3s multiplied together, altogether nine (4 + 5 = 9) 3s are multiplied together.
34 * 35 = (3 * 3 * 3 * 3) * (3 * 3 * 3 * 3 * 3) = 39
So 34 * 35 = 34 + 5 = 39.
Teaching Idea
2.15
Exponent laws can be used
to simplify calculations since
they offer alternative, but
simpler, representations of
quantities. Have students use
a calculator to write 22 * 52
in standard form. [100]
To focus on BIN 3, ask students how they could use the
exponent laws to represent
45 * 254 in an alternative
form to determine its value
without a calculator and what
that form might be. [e.g.,
45 * 254
= 44 * 254 * 4
= (4 * 25)4 * 4
= 1004 * 4
= 100 000 000 * 4
= 400 000 000]
(See Big Ideas for Number, p. 79.)
LAW 2 The quotient of two powers with the same base can be expressed as a
single power with the same base by subtracting the exponents.
For example, 35 , 33 = 35 - 3 = 32. This is based on the definition of a
power. Since 35 means five 3s multiplied together and 33 means three 3s
multiplied together, there are two (5 - 3 = 2) 3s left to be multiplied after
the fraction is simplified.
35 , 33 =
3 * 3 * 3 * 3 * 3
3 * 3 * 3
=
3 * 3
1
= 32
So 35 , 33 = 35 - 3 = 32.
LAW 3 The product of two powers with different bases but the same exponent
can be expressed as a single power with the same exponent by multiplying the
bases.
For example, 2 4 * 34 = (2 * 3)4 = 64.
2 4 * 34 is the product of four 2s and four 3s:
(2 * 2 * 2 * 2) * (3 * 3 * 3 * 3)
2 * 2 * 2 * 2 * 3 * 3 * 3 * 3 = 2 * 3 * 2 * 3 * 2 * 3 * 2 * 3
= (2 * 3) * (2 * 3) * (2 * 3) * (2 * 3)
= (2 * 3)4 = 64
So 2 4 * 34 = (2 * 3)4 = 64.
LAW 4 The quotient of two powers with different bases but the same exponent can be expressed as a single quotient with the same exponent by dividing
the bases.
For example, 12 4 , 34 = (12 , 3)4 = 4 4.
12 4 , 34 =
12 * 12 * 12 * 12
3 * 3 * 3 * 3
12
12
12
= (12
3) * (3) * (3) * (3)
4
4
4
= (12
3 ) = (12 , 3) = 4
4
4
4
4
So 12 , 3 = (12 , 3) = 4 .
LAW 5 A power raised to a power can be represented as a single power with
the same base by multiplying the exponents.
For example, (34)2 = 34 * 2 = 38.
(34)2 is the product of two powers: 34 * 34.
34 * 34 = (3 * 3 * 3 * 3) * (3 * 3 * 3 * 3) = 38
So (34)2 = 34 * 2 = 38.
Explaining Why a0 1 and a -n a1n
Students can use what they know about the exponent laws to give meaning
to expressions like a0 and an. In effect, they are using alternative representations of each of these expressions to make sense of them (BIN 3).
Using Exponent Laws to Deduce Why a0 1
Using Patterns to See Why a0 1
ax
ax
ax
ax
[using exponent law 2]
34 = 3 * 3 * 3 * 3 = 81
[because ax = ax (a Z 0)]
33 = 3 * 3 * 3 = 27
= ax - x = a0
= 1
If a0 =
ax
ax
and 1 =
ax
ax
if a Z 0, then a0 = 1 (a Z 0).
For example, 10 = 1, ( - 5)0 = 1, and 130 = 1.
102
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
32
= 3 * 3 = 9
31 = 3 f , 3
30 = 1
f , 3
r , 3
r , 3
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:33 PM
Page 103
Using a0 1 and the Exponent Laws to Deduce Why an 1 an
0
Since a0 = 1, then aan =
1
an
[divide both sides by an (a Z 0)].
Based on the exponent laws, aan = a0n = an (a Z 0).
0
0
If aan =
1
an
and aan = an, then a1n = a-n (a Z 0).
0
For example, 4-3 =
1
.
43
Some students will wonder why a cannot be 0 for a0 = 1. It might be
useful to help them see that, although a0 = 1 for every non-zero value of a,
0x = 0 for every non-zero value of x. These two facts make it impossible to
define 00. From one perspective, it should be 0, but from the other perspective, it should be 1. It is for this reason that we do not define 00.
PLANNING IDEA
2.9
Curriculum
A curriculum expectation/outcome expects students to apply the exponent
laws in expressions.
Big Idea Filter
BIN 3. There are many equivalent representations for a number or numerical
relationship. Each representation may emphasize something different about
that number or relationship.
Lesson Focus and Consolidation
You might focus a lesson on recognizing how the exponent laws can be used
to simplify computations. A consolidating question might be: Why might it
be useful to use the exponent laws to rename 25 * 55 as 105? [e.g., You can
easily write 105 in standard form without a calculator, but not 55.]
Roots
Just as division undoes multiplication and subtraction undoes addition,
taking roots undoes taking powers; for example,
116 = 4 and 42 = 16 (the square root is the inverse of squaring)
3
43
= 64 (the cube root is the inverse of cubing)
4
34
= 81 (the 4th root is the inverse of raising to the 4th power)
164 = 4 and
181 = 3 and
The type of root is indicated
in the corner of the root
symbol; if no value is indicated, it is a square root.
Geometric Interpretations of Square Root and Cube Root
Students have learned in earlier grades that the square root of a number is
the side length of a square with that area, and the cube root of a number is
the edge length of a cube with that volume. For example,
• Since the area of a 1.4 * 1.4 square is 1.96, the square root of 1.96 is
1.4 (11.96 = 1.4).
• Since the volume of a 1.2 * 1.2 * 1.2 cube is 1.728, the cube root of
3
1.728 is 1.2 (11.728
= 1.2).
PLANNING IDEA
2.10
Curriculum
A curriculum expectation/outcome expects students to determine the
meaning of a negative exponent.
Big Idea Filter
BIN 5. The operations of addition, subtraction, multiplication, and division
hold the same fundamental meanings no matter the domain in which they
(continued)
are applied.
NEL
Chapter 2: Number
103
7229_Ch02_p079-106.qxd
4/7/10
3:33 PM
Page 104
2.10
PLANNING IDEA
(continued)
Lesson Focus and Consolidation
You might focus a lesson on helping students see how they can use what
they already know about division to make sense of the rule an = n1a . A consolidating question might be: How does what you know about the division
of whole numbers help you understand why 72 is the same as 73 , 75?
[e.g., I know from the exponent laws that
73 , 75 = 7 - 2. I also know that
73
3
5
7 , 7 can be written as the fraction 75 , since fractions are another way to
describe division. I also know that you get an equivalent fraction by dividing
the denominator and 3numerator by the same thing. If I divide the numerator
and denominator of 775 by 73, I’m left with 72 in the denominator and 1 in the
numerator, which is what 7 - 2 must mean.]
Rational Exponents
Just as the exponent laws help students understand what a0 and a–n mean,
1
m
they also help students understand what a n and a n mean.
1
m
Deducing What an Means
1
1
1
1
1
1
1
1
1
Deducing What a n Means
1
1
1
a = a1 = a2 + 2 = a2 * a2 = (a2)2
a = 1a * 1a = (1a)2
1
1
Then (1a)2 = (a2)2. So 1a = a2.
1
1
Suppose an is raised to the power m:
1
1
m
(an)m = an * m = a n .
Using the exponent laws,
1
1
m
a = a1 = a3 + 3 + 3 = a3 * a3 * a3 = (a3)3
3
3
3
3 3
a = 1a
* 1a
* 1a
= ( 1a)
1
1
3
3 3
3
= a3.
Then ( 1a) = (a3) . So 1a
1
1
1
1
1
1
a = a1 = a4 + 4 + 4 + 4 = a4 * a4 * a4 * a4 = (a4)4
4
4
4
4
4 4
a = 1a
* 1a
* 1a
* 1a
= ( 1a)
1
1
4 4
4
4
Then ( 1a) = (a4) . So 1a = a4, and so on.
1
1
1
a n is also equal to (am * n) or (am)n .
1
1
n
n m.
Using an = 1a
, we know that (am)n = 1a
m
1
1
n
If a n = (am)n and (am)n = 1am,
m
m
n m
n m
then a n = 1a
and a n = (1a)
.
n
Therefore, an = 1a.
Students can use these alternatives to simplify computations. For
4 8
example, to determine the 4th root of 38 (23
), students can think
1
1
8
4 8
8
8
*
2
23 = (3 )4 = 3 4 = 34 = 3 , or 9.
Logarithms
Just as students learn that square roots undo squaring and cube roots undo
cubing, they learn how to use logarithms to undo exponentiation. In other
words, if ax = b, then loga b = x (read “the log of b to the base a
is x”). Students think about the logarithm as the exponent in the exponential
situation, and the logarithm base as the base of the power. For example,
since 35 = 243, then log3 243 = 5.
Logarithm Laws
In order to work with logarithms, students need to be familiar with the following laws as well as the exponent laws.
LOGARITHM LAWS
1. The logarithm of a product is the sum of the logarithms of the factors;
i.e., loga bc = loga b + loga c.
2. The logarithm of a quotient is the difference of the logarithms of the
dividend and divisor; i.e., loga bc = loga b - loga c.
3. The logarithm of a power of a number is a multiple of the logarithm of
that number; i.e., loga bc = c loga b.
104
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL
7229_Ch02_p079-106.qxd
4/7/10
3:33 PM
Page 105
LAW 1 The logarithm of a product is the sum of the logarithms of the factors.
For example, log5 (53 * 54) = log5 53 + log5 54. This is based on the definition of a logarithm and assumes the exponent law for multiplying powers
with the same base. Two different proofs are shown:
log5 53 = 3
log5 54 = 4
53 * 54 = 57
log5 57 = 7
log5 (53 * 54) = 7 = 3 + 4
log5 (53 * 54) = log5 53 + log5 54
Let y = log5 (53 * 54).
5y = 53 * 54
= 53 + 4
y = 3 + 4
log5 53 = 3
log5 54 = 4
log5 53 + log5 54 = 3 + 4
= y
= log5 (53 * 54)
LAW 2 The logarithm of a quotient is the difference of the logarithms of the
dividend and divisor.
4
example, log5 553
54
53. This, too, is
For
= log 5
- log 5
based on the definition
of a logarithm and assumes the exponent law for dividing powers with the
same base. Two different proofs are shown below:
4
log5 53 = 3
Let y = log5 553.
log5 54 = 4
5y =
54 , 53 = 51
log5 51 = 1
= 54 - 3
y = 4 - 3
log5 54 = 4
4
log5 553 = 1 = 4 - 3
log5 53 = 3
4
log5 553 = log5 54 - log5 53
54
53
log5
54
- log5 53 = 4 - 3
= y
Two different proofs are shown
because one might make more
sense than another to a student.
Teaching Idea
2.16
Have students represent the
equation (43)3 = 262 144
using logarithms. [e.g.,
log4 262 144 = 9]
To focus on BIN 3, ask them
to think of a number of
other ways, using exponents
and/or logarithms and/or
multiplication, to represent
the equation (43)3 = 262 144.
[e.g., 49 = 262 144;
218 = 262 144;
log2 262 144 = 18]
(See Big Ideas for Number, p. 79.)
4
= log5 553
LAW 3 The logarithm of a power of a number is a multiple of the logarithm of
that number.
For example, log5 (53)4 = 4 * log 5 53. This logarithm law corresponds to
the exponent law relating to powers of powers being written as a single
power. Two different proofs are shown below:
(53)4 = 512
log5 512 = 12
log5 53 = 3
log5 (53)4 = 4 * 3
log5 (53)4 = 4 * log5 53
Let y = log5 (53)4.
5y = (53)4
= 53 * 4
y = 3 * 4
log5 53 = 3
4 * log5 53 = 4 * 3
= y = log5 (53)4
Working with logarithms allows the user to work with small numbers
rather than large ones and to turn calculating powers into calculating multiples in order to solve problems. A common example is to use logarithms to
determine how long it will take the principal in a savings account to reach a
specified amount at a given compound interest rate.
Suppose someone deposits $2000 at 4% interest and wants to know how
long it will take to double its value. A possible solution follows:
NEL
Chapter 2: Number
105
7229_Ch02_p079-106.qxd
4/7/10
3:33 PM
Notice that an equation that
initially involved exponents is
solved as a linear situation
because logarithms are used.
Page 106
• The value of a deposit at an r% annual interest rate compounded
n times per year is A = P(1 + nr )nt, where A is the amount, P is the
original deposit, r is the annual rate, and t is the number of years.
• If the compounding is continuous (i.e., the limit when compounding
is done in shorter and shorter periods of time), A = P(1 + nr )nt can be
written as A = Pe rt, because the special number e, with a value of
2.718..., can be written as e = lim (1 + 1x)x.
xS q
• If 4000 = 2000ert, then 2 = ert. That means rt = ln 2, which is the natural logarithm of 2. (Note that natural logarithms, those reported to
the base e, are written as ln x rather than loge x.)
ln 2
= 0.693
• Since r = 0.04 (4% interest), then t = 0.04
0.04 = 17.3 years.
As a consequence of the logarithm laws, students should also discover the following (when a Z 0 and b Z 0):
• loga 1 0, which reflects that a0 = 1 for any non-zero value of a.
• loga 1x loga x, which reflects that 1x = x -1, means
loga 1x = loga x–1 = - loga x (based on the third logarithm law).
log x
• logb x logaa b , which describes how to handle a change of base. This is
useful since calculators only report logarithms to, base 10 and base e.
For example, to solve x - 2 = log3 157, the right-hand side might be
rewritten as
log 10 157
log 10 3 . The
reason this relationship holds is as follows:
Suppose x is written as a power with two different bases, a and b:
at = x and bq = x.
If at = x and bq = x, then bq = at.
Since at = x and bq = x, then an = b.
If an = b, then bq = (an)q = anq.
But bq = at, so anq = at. So nq = t or q = nt .
Since q = logb x, t = loga x, and n = loga b, then logb x =
log a x
log a b .
A special case is the relationship between logarithms of base 10
and natural logarithms: ln x =
log 10 x
log 10 e
x
= 2.303 log10 x.
= log10 0.4329
Since 10 is more than e2, it makes sense that the power of e used to
make a number is more than twice the power.
PLANNING IDEA
2.11
Curriculum
A curriculum expectation/outcome expects students to solve problems involving
exponential equations algebraically using common bases and logarithms.
Big Idea Filter
BIN 3. There are many equivalent representations for a number or numerical
relationship. Each representation may emphasize something different about
that number or relationship.
Lesson Focus and Consolidation
You might focus a lesson on how an expression like ax = b can be read another
way, as loga b = x. You might consolidate a lesson with a question such as:
Consider the list of exponent laws and logarithm laws I have circulated. (You
should circulate a list where either the exponent laws or the logarithm laws are
mixed up.) Which law goes with which and why? [Exponent Law 1 matches
Logarithm Law 1 since they are both about the product of powers with the
same base. Exponent Law 2 matches Logarithm Law 2 since they are both
about the quotient of powers with the same base. Exponent Law 5 matches
Logarithm Law 3 since they are both about raising a power to a power.] How
does looking at the matched laws help you see why some people call taking
powers and taking logarithms inverse, or opposite, operations? [Each exponent
law is just the reverse of the corresponding matching logarithm law.]
106
BIG IDEAS for Teaching Mathematics, Grades 9 to 12
NEL