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The following sample lesson, covering Standard AI2.0, was
taken from pages 171-175 of the Teacher Guidebook for
Instructional Strategies for Student Achievement: California
Mathematics. Student worksheets accompanying this lesson can
be found in the Student Workbook, pages 49-52, and the answer
keys for the worksheets appear in the Teacher Guidebook on
pages 176-177.
Copyright © 2004 by Educational Testing Service. All rights reserved. ETS, the ETS logo,
and Pathwise are registered trademarks of Educational Testing Service. Pathwise is a
trademark of Educational Testing Service.
Algebra 1
Standard 1A2.0
Students understand and use such operations as taking the opposite, finding the reciprocal,
taking a root, and raising to a fractional power. They understand and use the rules of
exponents.
Purpose
Using inverse operations and
finding reciprocals are
necessary skills for solving
equations and inequalities. The
laws of exponents make
simplifying easier. These
concepts are the building blocks
for more advanced mathematics
and applications.
Real-Life Examples
✔ A contractor building a square deck with an area of
䊐
256 square feet takes the square root of the area to find
the side length of 16 feet.
✔ To divide 1 of a pizza evenly among her 5 guests, Paula
䊐
2
uses the reciprocal of 5, which is 15 , and multiplies
by 12 . Using the multiplicative inverse, she is able to
determine that each guest should receive 101 of the whole
pizza.
✔
䊐 A clockmaker uses the formula t = 2π 9l.8 to determine
that the 30-meter pendulum in a bell tower clock that he
designed will take approximately 11 seconds to complete
a full swing.
Building Blocks
Students should have prior knowledge of the following topics:
■ Integers
■ Exponents, roots, and powers
■
Fractions
■
Solving equations
Stepping Stones
The following standards are related to Standard 1A2.0:
● 7NS1.2 – Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating
decimals) and take positive rational numbers to whole-number powers.
●
7NS2.4 – Use the inverse relationship between raising to a power and extracting the root of a perfect
square integer; for an integer that is not square, determine without a calculator the two integers between
which its square root lies and explain why.
●
7AF2.1 – Interpret positive whole-number powers as repeated multiplication and negative whole-number
powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate
expressions that include exponents.
●
1A5.0 – Students solve multi-step problems, including word problems, involving linear equations
and linear inequalities in one variable and provide justification for each step.
䊊
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1A2.0
|
Algebra 1
Progress Monitor
Pr
oblems to PPose
ose
Problems
Questions to Ask
What numbers would you use to
represent “an elevation of 25 feet”
and “a depth of 15 feet?”
•
What are the signs of the numbers you chose?
•
What is a positive integer? What is a negative integer?
•
What phrases could represent the opposite of the numbers
you chose?
What is the value of 4 + (−4)?
•
What result do you get when you add an integer and its
opposite?
•
Explain why the opposite of a number is referred to as its
additive inverse.
•
What is a reciprocal?
•
What value will you get every time you multiply a number
by its reciprocal?
•
How is the reciprocal different from the additive inverse?
Evaluate 64 and − 64 .
•
How are the results similar? How are they different?
Write 5 i 5 i 5 i 5 using an exponent.
•
Which number is the base?
•
Which number is the exponent?
•
What does 54 mean?
•
Are these expressions equal?
•
What is the relationship between the values of the two
expressions?
Multiply
1
4
by its reciprocal.
Algebra 1
Evaluate − 32 and (−3)2 .
䊊
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Algebra 1
|
Possible Student Responses to Questions
An elevation of 25 feet can be
represented by the number 25. A
depth of 15 feet can be represented by
the number −15.
•
The elevation is represented by a positive number, while the
depth is represented by a negative number.
•
A positive integer is a whole number that is greater than 0.
A negative integer is a whole number that is less than 0.
•
The opposite of 25 is − 25. A phrase that could
represent − 25 is “a depth of 25 feet.” The opposite
of −15 is 15. A phrase that could represent 15 is “an
elevation of 15 feet.”
•
When a number is added to its opposite, the result is 0.
•
The opposite of a number is called its additive inverse
because when the two numbers are added, they “cancel”
each other out, i.e., their sum is always 0.
•
The reciprocal of a number is its multiplicative inverse.
•
The product of any number and its reciprocal is 1.
•
A reciprocal is a multiplicative inverse rather than an
additive inverse.
•
The results are similar in that their absolute value is 8. The
results are different in that the first is positive and the
second is negative.
•
The base is 5.
•
The exponent is 4.
•
54 means “the number 5 multiplied by itself 4 times.”
•
No, the expressions have different values.
•
The values 9 and − 9 are additive inverses.
4 + (−4) = 0
1
4
×4 = 1
64 = 8 and − 64 = −8
5 i 5 i 5 i 5 = 54
−32 = −9 and (−3)2 = 9
Algebra 1
Ans
wers to Pr
oblems
Answ
Problems
1A2.0
䊊
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1A2.0
Algebra 1
Basic Instruction
Concrete Model:
To help students comprehend the concept of
additive inverses, use a model of negative and
positive tiles or the idea of charged particles.
Step 1: Represent the equation with algebra tiles
and x’s to represent the variable.
(Unshaded squares are negative units,
and shaded squares are positive.)
4x − 6
Algebra Tile Model
Negative 3:
3x + 2
× × × ×
Tiles of different colors represent positive and
negative numbers. Combinations represent
addition and subtraction problems, and
zero (0) is represented by an equal number of
tiles of both colors:
Positive 3:
=
× × ×
Step 2: To undo − 6, add + 6 to both sides. On the
left, the six tiles of each color “cancel out,”
leaving 4 x = 3 x + 8.
Zero:
4x − 6 + 6
Charged Particles Model
× × × ×
=
3x + 2 + 6
× × ×
Positive numbers are represented with
plus (+) signs.
Positive 3: + + +
Algebra 1
Negative numbers are represented with
minus (−) signs.
Negative 3: −−−
A combination represents addition and
subtraction problems, and opposite signs
cancel each other out:
4 + (−2) = + + + + − −
Step 3: Add the opposite of 3x, which is − 3x, to
both sides. Since x represents the variable,
use a circled x to represent each negative
occurrence of the variable.
4x + (−3x)
=
3x + (−3x) + 8
× × × ×
× × ×
× × ×
× × ×
Balance Model:
Use the idea of balance to help students apply the
additive inverse property when solving equations.
To get rid of an amount on one side, students add
the opposite of that amount. Whatever is added to
one side of the balance must be added to the other
side to keep the balance level. To solve
4 x − 6 = 3 x + 2, students can use a balance with
algebra tiles:
䊊
Step 4: The result is x = 8.
x
=
8
×
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Algebra 1
|
1A2.0
Match Game:
To help students understand the laws of exponents, play the following matching game. Use one color to create
a deck of exponent problem cards, and use a different color to make a second deck of answer cards. Students
should match the cards from the two decks. (You can also play this game with a two-column transparency,
with problems in one column and the answers in another.)
For instance, you might include (x2y3)4, x 2 i x 3 , 37 ÷ 33 , and x a i x b in the problem deck and x8y12, x5, 34,
and x a+b in the answer deck.
Data Analysis,
Stats, and Probs
Make It One:
Have students use the idea of “making 1” when solving equations with fractional coefficients. Remind
students that the goal is to isolate the variable; that is, to have just 1 of that variable by itself. Using the
reciprocal will eventually lead to just 1 of the variable remaining.
For example, to solve 35 x = 6, ask students what 35 must be multiplied by to result in a product of 1. Because
any number multiplied by its reciprocal yields 1, multiply by 35 .
3 5 15
× = =1
5 3 15
5 3
5
15
30
× x = ×6 ➔
x=
➔ x = 10
3 5
3
15
3
Break the Fraction:
Instead of trying to distribute a fraction, students can use the multiplicative inverse to make an equation easier
to solve. Instruct students to multiply both sides of the equation by the reciprocal of the fractional factor:
3
8 3
8
(9 x + 7) = 6 ➔ × (9 x + 7) = × 6 ➔ 9 x + 7 = 16 ➔ x = 1
8
3 8
3
Algebra 1
This strategy may pose problems for students when a fractional coefficient occurs that has not been factored
from all terms. In this case, instruct students to factor out the fraction from both terms before multiplying by
the reciprocal. This can be done using the following “trick” of knowing that the product of a number and its
reciprocal is equal to 1:
2
2
2  5
2
2
2
x + 10 = 8 ➔ x +   (10) = 8 ➔ x + (25) = 8 ➔ ( x + 25) = 8



5
5
5 2
5
5
5
This equation can now be solved by multiplying both sides by
5
2
to “undo” the fraction 25 .
䊊
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The following sample lesson, covering Standards 7AF4.1,
7AF4.2, AI4.0 and AI5.0, was taken from pages 163-167 of
the Teacher Guidebook for Instructional Strategies for Student
Achievement: California Mathematics. Student worksheets
accompanying this lesson can be found in the Student
Workbook, pages 47-48, and the answer keys for the
worksheets appear in the Teacher Guidebook on pages 168-170.
Copyright © 2004 by Educational Testing Service. All rights reserved. ETS, the ETS logo,
and Pathwise are registered trademarks of Educational Testing Service. Pathwise is a
trademark of Educational Testing Service.
Algebra and Functions
|
7AF4.1
7AF4.2
1A4.0
1A5.0
Standards 7AF4.1, 7AF4.2, 1A4.0, 1A5.0
7AF4.1
7AF4.1–Solve two-step linear equations and inequalities in one variable over the
rational numbers, interpret the solution or solutions in the context from which they
arose, and verify the reasonableness of the results.
7AF4.2
7AF4.2– Solve multi-step problems involving rate, average speed, distance, and time or a
direct variation.
1A4.0
1A4.0–Students simplify expressions before solving linear equations and inequalities in
one variable, such as 3(2x – 5 ) + 4(x – 2 ) = 12.
1A5.0
1A5.0–Students solve multi-step problems, including word problems, involving linear
equations and linear inequalities in one variable and provide justification for each step.
Many real-life problems can be
modeled using single-variable
equations and inequalities, and
the ability to manipulate these
equations enables students to
solve the problems. In addition,
solving single-variable
equations is a precursor to the
more advanced study of systems
of linear equations.
Algebra and
Functions
Purpose
Real-Life Examples
✔ A page in the high school newspaper is 8 12 inches wide,
䊐
and the margins account for 2 81 inches. The page has
three columns, and the margins between columns
are 163 inches each. The editor could use the
 
equation 2 81 + 2 163  + 3 x = 8 12 to find that x = 2; that is,
that each column should be 2 inches wide.
✔ A fishing shop spends 85¢ to buy enough materials to
䊐
make three flies. The shop then sells the flies for 68¢
each. To determine the number of flies that must be sold
to make $200 in profit, the shop owner could use the
inequality 0.68 x − 0.85( 13 x ) > 200. Solving that equation,
the owner would know that he must sell 505 flies.
Building Blocks
Students should have prior knowledge of the following topics:
■ Order of operations
■ Rational numbers
■ Distributive property
■ Translating word problems
Stepping Stones
The following standard is related to Standards 7AF4.1, 7AF4.2, 1A4.0, and 1A5.0:
●
7AF1.1 – Use variables and appropriate operations to write an expression, an equation, an inequality,
or a system of equations or inequalities that represents a verbal description (e.g., three less than a
number, half as large as area A).
䊊
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7AF4.1
7AF4.2
1A4.0
1A5.0
|
Algebra and Functions
Progress Monitor
Pr
oblems to PPose
ose
Problems
Solve for x:
x
3
= 7.
Solve for x: x − 6 = −14.
Solve for x: 6 − 3 x = 5 x.
Algebra and
Functions
Solve for x: x − 2 > 10.
The length of the side of a square is x.
If each side’s length is decreased
by 3, what is the perimeter of the
new square?
Questions to Ask
•
What is the first step in solving this problem?
•
What operation must be used?
•
How do you know that your answer is reasonable?
•
What is the first step in solving this problem?
•
How do you know that your answer is reasonable?
•
What is the first step in solving this problem?
•
What operations must be used to solve this problem?
•
Is your answer reasonable?
•
How would this problem be different if there were an equal
sign (=) instead of a greater than symbol (>)?
•
Is x >12 equivalent to this problem?
•
Which of these expressions is equivalent to your
expression?
•
4 x − 12
4( x − 3)
•
•
•
When a 5-pound weight is placed
on a spring scale, the spring stretches
to a length of 15 centimeters. When
a 10-pound weight is placed on
the scale, the spring stretches to
30 centimeters. How long will
the spring stretch if a weight of
15 pounds is placed on the spring?
䊊
x −3+ x −3+ x −3+ x −3
(2 x − 6 ) + (2 x − 6 )
•
Create a table that shows the relationship between the
weight and the length of the spring.
•
Write an equation that shows how the weight w (in pounds)
is related to the length of the spring s (in centimeters).
•
Why is this a direct variation?
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Algebra and Functions
|
Possible Student Responses to Questions
x = 21
•
Multiply both sides by 3.
•
Multiplication.
•
One way to ensure reasonableness is to replace x by 21
in the original equation. Another way is to think: “Is 21
divided by 3 equal to 7? Yes, so the answer is correct.”
•
Add 6 to both sides.
•
By replacing x by −8, a true equation is
formed: − 8 − 6 = −14.
•
Add 3x to both sides.
•
Addition and division.
•
Yes. Replacing x by
•
For solving the problem, the steps would be the same.
However, the answer would be x = 12 instead of x >12.
•
Yes.
4( x − 3), or equivalent
•
All four of the expressions are equivalent to the expression
shown to the right. (Students may use various methods to
arrive at any of these expressions.)
45 centimeters
•
x = −8
x = 34
x >12
3
4
Algebra and
Functions
Ans
wers to Pr
oblems
Answ
Problems
7AF4.1
7AF4.2
1A4.0
1A5.0
results in a true equation.
Weight (lbs)
Spring (cm)
5
15
10
30
15
?
•
s = 3w, or w = s ÷ 3
•
The length of the spring increases proportionally with the
weight. In addition, the information seems to indicate that
when w = 0, s = 0, and all direct variations must pass
through the origin.
䊊
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|
7AF4.1
7AF4.2
1A4.0
1A5.0
Algebra and Functions
Basic Instruction
What is x?
When solving problems for which an algebraic equation is required, students must understand how the
variable is being used. Consequently, have students write a let statement every time they use an algebraic
equation. A let statement identifies the variable and what it represents, such as “Let x = Alyssa’s age.”
Such a statement is important when solving problems like, “Alyssa is 2 years older than twice Kyle’s age.
What is Alyssa’s age?” The expression 2 x + 2 could be used when solving this problem, but the variable x
represents Kyle’s age, not Alyssa’s. By identifying the variable at the beginning, students will have more
success interpreting the results when they solve the problem.
Draw Pictures:
Drawing pictures allows students to make a connection between abstract equations and a concrete, real-life
situation. Ask students to draw pictures that represent various situations. For instance, if the rectangle
below represents today’s high temperature, have students draw pictures that represent yesterday’s high
temperature (8 degrees colder) and tomorrow’s projected high temperature (5 degrees warmer). In addition,
have students label the rectangles with algebraic expressions (x −8 and x + 5, respectively).
Algebra and
Functions
x+5
x
x−8
Yesterday’s
High
Temperature
䊊
Today’s
High
Temperature
Tomorrow’s
High
Temperature
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Algebra and Functions
|
7AF4.1
7AF4.2
1A4.0
1A5.0
Students might then use their pictures to answer several questions, including:
• What is the difference between yesterday’s high temperature and tomorrow’s projected high
temperature? (13 degrees)
• What is the average high temperature for these 3 days? (x −1 degrees)
• If yesterday’s high temperature was 65 degrees, what will tomorrow’s high temperature be? (78 degrees)
• Ursula noticed that 15 times today’s high temperature is equal to 9 times the sum of yesterday’s and
tomorrow’s high temperatures. What is today’s high temperature? (9 degrees)
Require Students to Justify:
When solving algebraic equations, ask students to explain their work at each step and give a good reason for
why it is correct. At first, you may not want to require that they use the proper terminology as long as their
justifications are valid; eventually, however, you will want students to use such terms as variable, exponent,
operation, distributive property, and additive inverse.
Scaffold the Learning:
Algebra and
Functions
When students become competent solving one-step linear equations with integer coefficients, have them
move on to equations involving more than one operation. At first, these multi-step equations should contain
only integers, and they probably should have integer solutions. However, when students demonstrate
proficiency, begin using equations with any rational number. The following set of equations demonstrates
the progression from a one-step equation to a multi-step equation to an equation involving non-integers.
3 x = 24
3 x + 6 = 24
1
x + 6 = 24
3
Classroom Competitions:
Multi-step equations lend themselves to team relay competitions, which can lead to increased student learning.
Using four-person teams, have the first student perform the first step in solving an equation. The first student
should then pass the paper to a second student, who should perform the second step. Continue until the fourth
student finishes the solution.
Classroom Practice:
Although drill practice is not necessarily the best way to learn, it can be effective for demonstrating and
practicing skills such as order of operations. Give students problems involving the order of operations one
at a time. As each problem appears on the chalkboard or overhead projector, students should write their
answers either on a small dry-erase board (if available) or on a sheet of paper. When they’re done, students
should hold their papers above their heads. This will enable you to make a quick assessment of student
competency, help you decide which topics to re-teach, and help you form groups for cooperative work, since
you’ll want to group students who understand with students who are having difficulty.
䊊
Instructional Strategies for Student Achievement 167
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The following sample lesson, covering Standard AI6.0, was
taken from pages 179-182 of the Teacher Guidebook for
Instructional Strategies for Student Achievement: California
Mathematics. Student worksheets accompanying this lesson can
be found in the Student Workbook, pages 53-56, and the answer
keys for the worksheets appear in the Teacher Guidebook on
pages 183-184.
Copyright © 2004 by Educational Testing Service. All rights reserved. ETS, the ETS logo,
and Pathwise are registered trademarks of Educational Testing Service. Pathwise is a
trademark of Educational Testing Service.
Algebra 1
|
1A6.0
Standard 1A6.0
Students graph a linear equation and compute the x- and y-intercepts (e.g.,
graph 2 x + 6 y = 4 ).
Purpose
Real-Life Examples
✔ A laser printer requires 30 seconds to warm up and then
䊐
takes 5 seconds to print each page. The total time in
seconds, t, to complete a print job consisting of p pages
is expressed by the linear equation t = 5 p + 30. Duante
used the graph below to determine that it would take
2 minutes to print an 18-page document.
Time to Print (seconds)
Graphing linear equations is an
essential component of many
algebraic applications. Students
should be able to identify the
equation of a line by examining
its graph. Along with slope, a
line’s x- and y-intercepts
provide the bases for graphing
and analyzing the graphs of
linear equations.
125
100
75
t = 5p +
50
30
25
0
2
4
6
Number of Pages
8
Algebra 1
Building Blocks
Students should have prior knowledge of the following topics:
■ Arithmetic computations—addition,
■ Coordinate geometry—reading coordinate grids
subtraction, multiplication, division
and plotting points
■
Basic algebra—substitution,
solving linear equations in one variable
■
Tables and graphs
Stepping Stones
The following standards are related to Standard 1A6.0:
● 7AF1.5 – Represent quantitative relationships graphically, and interpret the meaning of a specific
part of a graph in the situation represented by the graph.
●
7AF3.3 – Graph linear functions, noting that the vertical change (change in y-value) per unit of
horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”)
is called the slope of a graph.
●
1A7.0 – Students verify that a point lies on a line, given the equation of the line. Students are able
to derive linear equations by using the point-slope formula.
●
1A9.0 – Students solve a system of two linear equations in two variables algebraically and are able
to interpret the answer graphically. Students are able to solve a system of two linear inequalities in
two variables and to sketch the solution sets.
䊊
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1A6.0
|
Algebra 1
Progress Monitor
Pr
oblems to PPose
ose
Problems
Questions to Ask
Write the equation 3 x + 4 y = 12 in
slope-intercept form.
•
What is the slope-intercept form of a linear equation?
•
What is the first step in converting this equation into slopeintercept form?
•
Why is this form of the equation called the slope-intercept
form?
•
Which intercept can you identify simply by looking at the
equation? What is its value?
•
What is the difference between an x-intercept and a
y-intercept?
•
If the x-intercept is represented by an ordered pair, what is
its y-coordinate?
•
How can you use this information to find the x-intercept of
the line?
•
What are the x- and y-intercepts of this line?
•
How can you use this information to graph the line?
•
Write this equation in slope-intercept form. What is the
slope of the line? Does this match the slope of the line in
your graph?
•
Why wouldn’t it be possible to graph the
line − 5 x + 4 y = 0 just by plotting the x- and y-intercepts?
What are the x- and y-intercepts of the
line y = 7 x − 14?
Graph the line 4 x − 6 y = −12.
Algebra 1
䊊
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Algebra 1
|
Possible Student Responses to Questions
y = − 34 x + 3
•
An equation that is in the form y = mx + b, where m and b
are constants, is said to be in slope-intercept form.
•
The first step is to isolate the y term on the left side of the
equation.
•
The form y = mx + b, is called slope-intercept form
because m is the slope of the line and b is its y-intercept.
•
When an equation is written in the slope-intercept
form y = mx + b, its y-intercept is b. Therefore, the
y-intercept of this line is −14.
•
An x-intercept indicates the point at which the line crosses
the x-axis. A y-intercept indicates the point at which the
line crosses the y-axis.
•
The y-coordinate of an x-intercept is always 0, because it is
a point on the x-axis.
•
Substitute 0 for y in the original equation, and solve for x.
•
The x-intercept is (−3, 0), and the y-intercept is (0, 2).
•
Both of these points lie on the line. Therefore, to graph the
line, plot these two points and connect them.
•
The slope-intercept form of the equation is y = 23 x + 2. The
slope of the line is 23 , which matches the graph.
•
The line − 5 x + 4 y = 0 passes through the origin.
Therefore, its x- and y-intercept are both (0, 0). Plotting one
point is insufficient for drawing a line; a second point is
needed.
The x-intercept is (2, 0), and the
y-intercept is (0, − 14).
y
12
4x
y
–6
=–
x
O
Algebra 1
Ans
wers to Pr
oblems
Answ
Problems
1A6.0
䊊
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|
1A6.0
Algebra 1
Basic Instruction
Graphing Games:
Make a Table:
Have your students practice graphing linear
equations by playing a great classroom game: Hit the
Points. On an overhead projector, display several
points on a coordinate graph. Ask students to find
the equation of a line that hits as many points on the
graph as possible. Students receive a score of 1 for a
line passing through one point, a score of 3 for a line
passing through two points, and a score of 7 for a
line passing through three points.
The simplest way for students to graph a line is
to make a table of values and plot points. Have
students choose values for x and solve for the
corresponding y-values. Encourage students to
use values that result in points that are easy to
plot, i.e., points whose coordinates do not
involve fractions or large numbers. For instance,
in making a table for the equation y = 13 x + 7,
students should choose values for x that are
multiples of 3; because the coefficient of x is
1
× 4 = 1, the resulting y-values will be integers:
4
The graph below can be used the first time you play
this game. It contains points with integer coordinates
only.
y
O
x
x
y
0
7
3
8
9
10
Since all of the points should lie on a straight
line, plotting two points will be enough to
determine the line. However, always have
students plot at least three points. This provides
a built-in check. If all three points do not lie on a
straight line, students should realize that they
made an error.
Algebra 1
Slope-Intercept Form:
An equation in slope-intercept
form y = mx + b, is easy to graph because it
clearly indicates the slope and y-intercept of the
line. Teach students to convert to this form by
isolating the variable y using basic algebra. The
example below shows how to convert
5 x + 2 y = 10 to slope-intercept form:
After students understand the game, you may wish to
increase the difficulty by including points with
fractional or decimal coordinates. You may also
want to alter the rules by requiring students to hit all
of the points with the smallest number of lines
possible.
䊊
5 x + 2 y = 10
2 y = −5 x + 10
5
y =− x +5
2
To graph the line, students should note that the
constant (5) represents the y-intercept, so the line
passes through (0, 5). Likewise, the coefficient
of x represents the slope, meaning that the line
rises − 5 while it runs 2; from (0, 5), this makes
the line pass through (2, 0).
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The following sample lesson, covering Standard AI15.0, was
taken from pages 207-212 of the Teacher Guidebook for
Instructional Strategies for Student Achievement: California
Mathematics. Student worksheets accompanying this lesson can
be found in the Student Workbook, pages 63-64, and the answer
keys for the worksheets appear in the Teacher Guidebook on
pages 213-214.
Copyright © 2004 by Educational Testing Service. All rights reserved. ETS, the ETS logo,
and Pathwise are registered trademarks of Educational Testing Service. Pathwise is a
trademark of Educational Testing Service.
Algebra 1
|
1A15.0
Standard 1A15.0
Students apply algebraic techniques to solve rate problems, work problems, and percent
mixture problems.
Purpose
The key assessment of student
learning in any subject is
whether students can apply their
knowledge and skills to a new
problem situation. This standard
focuses on applying knowledge
to rate, work, and percent
mixture problems that we
commonly encounter in
real life.
Real-Life Examples
Algebra 1
✔ Law firms hire typists to do the large volumes of word
䊐
processing that legal cases require. In order to meet the
deadlines in a case, the supervisor of word processing
must know the rate at which each staff member types.
If Martha types 60 words per minute and Arthur types
50 words per minute, it will take them, working together,
about 8 hours to type a 120-page report.
✔ When purchasing a car, smart consumers consider more
䊐
than just the sticker price. One important consideration is
fuel economy, i.e., the number of miles that a car gets for
each gallon of fuel. If a Speedster gets 22 miles per
gallon and a Weekender gets 28 miles per gallon, a
person who drives 12,000 miles will need about
117 fewer gallons of gas with the Weekender.
✔
䊐 For a barbecue, Ali combines 2 pounds of hamburger
meat that is 15% fat with 3 pounds of meat that is 4% fat.
He determined that the total mixture is approximately
8.4% fat, which fits within the restrictions of his wife’s diet.
Building Blocks
Students should have prior knowledge of the following topics:
■ Percentage, rate, base, discount, and interest
■ Distance, rate, and time
■
Percent problems
■
Ratio, proportion
■
Formulas for area, volume, and other quantities
Stepping Stones
The following standards are related to Standard 1A15.0:
● 7NS1.6 – Calculate the percentage of increases and decreases of a quantity.
●
7AF1.1 – Use variables and appropriate operations to write an expression, an equation, an
inequality, or a system of equations or inequalities that represents a verbal description (e.g., three
less than a number, half as large as area A).
●
7AF4.2 – Solve multi-step problems involving rate, average speed, distance, and time or a direct
variation.
䊊
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1A15.0
|
Algebra 1
Progress Monitor
Pr
oblems to PPose
ose
Problems
Look for a pattern in the
x- and y-values. Complete
the table, and write an
equation that shows the
relationship between
x and y.
Questions to Ask
x
1
2
3
4
y
3
5
7
Jardy types 300 words in 10 minutes,
and Claire types 600 words in
15 minutes. How long will it take
them to type a 5250-word research
paper?
What value of y corresponds to an x-value of 4?
•
By what number could you multiply each value of x to get a
value close to y?
•
What additional amount must be added to or subtracted
from this product to get each y-value?
•
How could a table be used to solve this problem?
•
How many words does Jardy type each minute?
•
How many words will Jardy type in t minutes?
•
How many words does Claire type each minute?
•
How many words will Claire type in t minutes?
•
Combined, how many words will they type in t minutes?
•
How could an equation be used to solve this problem?
•
Draw a picture to represent the amount of equipment that
Melissa carries. Then, draw a picture to represent the
amount that Hasina carries.
•
If the weight of Melissa’s equipment is m pounds, what is
the weight of Hasina’s equipment?
•
What equation shows that the difference between the two
packages is 10 pounds?
Algebra 1
•
Melissa and Hasina have 16 pounds
of equipment to take on a
backpacking trip. Since Hasina is
stronger, they divide the equipment so
that Hasina’s pack weighs 10 pounds
more than Melissa’s pack. How many
pounds of equipment does each of
them carry?
䊊
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Algebra 1
Ans
wers to Pr
oblems
Answ
Problems
x
1
2
3
4
y
3
5
7
9
y = 2x +1
It will take them 75 minutes, or
1 hour and 15 minutes, to type the
paper.
1A15.0
Possible Student Responses to Questions
•
Each time that 1 is added to an x-value, 2 is added to the
corresponding y-value. Therefore, a y-value of 9
corresponds to an x-value of 4.
•
Multiplying each x-value by 2 gives a number close to the
corresponding y-value.
•
Adding 1 to twice the x-value gives the corresponding
y-value.
•
The table below could be helpful. When 5250 appears in
the fourth column, the corresponding value in the first
column is the number of minutes required.
Time
(minutes)
Claire
Jardy
Total
Words
350
5
200
150
10
400
300
700
15
600
450
1050
:
:
:
:
75
3000
2250
5250
•
Jardy types 300 ÷ 10 = 30 words each minute.
•
Jardy types 30t words in t minutes.
•
Claire types 600 ÷ 15 = 40 words each minute.
•
Claire types 40t words in t minutes.
•
In t minutes, they type a total of 70t words.
•
Solve 70t = 5250 for t.
•
The following picture could be used to represent the
situation visually:
Algebra 1
Melissa carries 3 pounds, and Hasina
carries 13 pounds.
|
10 pounds
m
m
Melissa’s Pack
Hasina’s Pack
•
The weight of Hasina’s equipment is m + 10 pounds.
•
m + (m + 10) = 16
䊊
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1A15.0
Algebra 1
Basic Instruction
Tables:
Tables help students organize given information and
identify missing pieces. When students see that there
is a blank in a table, they will realize that a variable
must be used. With repeated use of tables, students
will start to categorize problems by the type of
formula needed and the type of solution for which
they are asked.
You can illustrate the usefulness of tables by using
one to solve the following question.
Miguel is 4 years younger than Alicia. If the
total of their ages in 7 years will be 36, how old
are they now?
The unknown is Alicia’s age, which can be
represented by a. The rest of the table can be filled
in by using the relationship of each piece to Alicia’s
age.
Current Age
Age in 7 Years
Alicia
a
a+7
Miguel
a–4
a+3
Not Needed
2a + 10
Total
Algebra 1
The table yields the equation 2 a + 10 = 36. Solving
this gives a = 13 and a − 4 = 9. Thus Alicia is
13 years old and Miguel is 9 years old.
Questioning Techniques for
Understanding Rate:
Algebra has been described as the “generalization of
arithmetic.” As such, a series of questions that make
the generalization explicit can be helpful to students.
For every algebraic situation, you may first want to
ask several questions using numbers. These
questions can lead from concrete numeric examples
to the abstract use of variables.
䊊
For instance, help students to understand rates
by asking the following series of questions:
1. If it took you 3 hours to do your
homework last night, how much
homework did you complete in 1 hour?
That is, what is your rate per hour?
(Three hours for all of the homework
means that 13 of the homework was
completed each hour.)
2. If it took 5 hours to do your homework,
how much homework did you complete
per hour? That is, what is your rate per
hour? (Five hours for all of the
homework means that 15 of the
homework was completed each hour.)
3. If it took x hours to complete your
homework, how much did you complete
per hour? That is, what is your rate per
hour? (x hours for all of the homework
means that 1x of the homework was
completed each hour.)
4. If it took 5 hours to complete your
homework, how much would you
complete in 2 hours? In 3 hours? In
t hours? (Respectively, you would
complete 25 , 35 , and 5t .)
This strategy will help with work questions, such
as the following:
Reshawn can mow the lawn in 5 hours, and
Juan can mow the lawn in 3 hours. How
long will it take them to mow the entire lawn
if they work together?
Reshawn will mow 5t of the lawn in t hours and
Juan will mow 3t in t hours, so combined it will
take them 3t + 5t to complete the entire job,
which is 1. Solving the equation 3t + 5t = 1 yields
t = 158 hours, meaning that it will take them just
under 2 hours to mow the lawn when working
together.
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Algebra 1
|
1A15.0
Diagrams:
Diagrams help students to visualize and make sense of a problem. The most basic pictures can often help
students break the problem into easier pieces and realize which operations they need to use.
For the mowing problem on the previous page, a rectangle could be used to represent the entire lawn. Since
Reshawn mows 13 per hour and Juan mows 15 per hour, the following visual solution would be obtained:
Entire Lawn
1
5
1
3
1
5
Amount Mowed
by Reshawn
During 1st Hour
Amount
Mowed
by Juan
During
1st Hour
Amount Mowed
by Reshawn
During 2nd Hour
Amount
Mowed
by Juan
During
2nd Hour
Amount Mowed
During 2nd Hour
Algebra 1
Amount Mowed
During 1st Hour
Data Analysis,
Stats, and Probs
1
3
This diagram will help students to see that, in 2 hours, Reshawn and Juan will mow the entire lawn and a little
more. Although the answer is not as precise as the algebraic solution, it does give students a clear
representation of the situation.
Formula Sheet:
To master this standard, students must be proficient at manipulating formulas. Review the basic formulas that
they will encounter, and require students to make their own study sheets. Formulas for area and volume,
interest, distance, and rates (miles per hour, miles per gallon) should be included.
䊊
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1A15.0
Algebra 1
Basic Instruction
Use Fractions:
Rates should be expressed as ratios with a unit of measure in both the numerator and the denominator.
Consider the following problem:
A car gets 20 miles per gallon of gas. How many gallons are needed to travel 100 miles?
miles
To solve this problem, it is first necessary to express 20 miles per gallon as 20gallon
. Students should also
miles
realize that the rate “gallons per mile” can be obtained by flipping the fraction, i.e., that 20gallon
corresponds
1 gallon
0.05 gallons
to 20 miles = mile .
The solution can then be found in either of 2 ways:
100 miles 20 miles
=
x
gallon
20 x = 100
x = 5 gallons
100 miles ×
0.05 gallons
= 5 gallons
mile
The Beginning, Middle, and End:
The beginning is the problem, and the end is the answer. In the middle, students should set up and solve an
equation. However, students often skip steps when the answer is readily apparent. Unfortunately, they then
have trouble solving more difficult problems.
To help students overcome this, present a series of problems of increasing difficulty that cannot all be solved
mentally. Then change the values in the problems. Show students how variables allow one to solve equations
for changing situations with ease.
Algebra 1
Label Analysis:
For work, rate, and mixture problems, the unit of measure is an important component of the correct answer.
In addition, it provides an easy way for students to determine the reasonableness of their answer. Encourage
students to do a “label analysis” to check the units.
For example, if you want to determine the distance when the speed and time are known, a unit analysis shows:
miles minutes
×
≠ miles
hour
1
but
miles hours
×
= miles
hour
1
Final Check:
Require students to check their work by substituting their answer into the original equation. Be sure they
understand that the check is an important part of the solution and that it is a required step for every problem.
The check is good practice with substitution and will help students to find their own errors. If necessary, both
the label analysis and the check can be included as part of the grading rubric.
䊊
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