The following sample lesson, covering Standard 7AF1.2, was
taken from pages 123-128 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 31-32, and the answer
keys for the worksheets appear in the Teacher Guidebook on
pages 129-130.
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
|
7AF1.2
Standard 7AF1.2
Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2.
Purpose
Real-Life Examples
✔ An adult admission ticket to the zoo costs $5, a teen
䊐
ticket costs $4, and a child ticket costs $2. The total cost
of admission for two adults, two teenagers, and three
children will be (2 × 5) + (2 × 4) + (3× 2) = $24.
✔ At the hardware store, Larry purchased two patio lights
䊐
for $25.75 each and one window shade for $16.99. The
state charges 7 12 % sales tax. His total bill was
(2 × 25.75 + 16.99) ×1.075 = $73.63.
✔ Shari’s debate club sold 3 cakes at $8 each, 4 dozen
䊐
cookies at $3 per dozen, and 5 pies at $7 each. The club
paid $5 to advertise the sale in the school newspaper. If
four members of the club divide the proceeds from the
sale evenly, then Shari’s share of the money
is ( 3×8+4×34+5×7 )−5 = $16.50.
Algebra and
Functions
Order matters! To be sure that
mathematical expressions
always compute to the same
value, a set of rules must be
used that establishes a specific
order for evaluating complex
expressions and solving multistep problems. This order
ensures that the value of the
expression is always the same.
This set of rules is referred to as
the order of operations.
Building Blocks
Students should have prior knowledge of the following concepts:
■ Variables
■ Exponents
■ Distributive property
■ Grouping symbols ( ), [ ], { } , −,
■
Evaluating arithmetic expressions
Stepping Stones
The following standards are related to Standard 7AF1.2:
● 7NS1.2 – Add, subtract, multiply, and divide rational numbers (integers, fractions,
and terminating decimals) and take positive rational numbers to whole-number powers.
● 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).
● 7MR1.1 – Analyze problems by identifying relationships, distinguishing relevant from irrelevant
information, identifying missing information, sequencing and prioritizing information, and
observing patterns.
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7AF1.2
|
Algebra and Functions
Progress Monitor
Pr
oblems to PPose
ose
Problems
Questions to Ask
8 + 16 ÷ 4
•
Which operation should be performed first?
•
Which operation should be done second?
•
Switch the order in which you perform the two operations.
Does the value of the expression change?
•
Does it matter which operation is performed first when
simplifying an expression?
•
Which operation should be performed first?
•
What would be the result if the problem was 12 × (2 ÷ 8)?
•
If the results are the same, then why does it matter in which
order you perform the operations?
•
Both of the operations involved are subtraction. How do
you decide which to perform first?
•
Is the value of this expression the same as the value
of (20 − 10) − 5 ? How about 20 − (10 − 5)?
•
What are the grouping symbols in this expression?
•
What operation should be done first? Why?
•
What other grouping symbols may be used to group?
•
What operations are involved in this expression?
•
Evaluate this expression for n = −2 and h = 5.
12 × 2 ÷ 8
Algebra and
Functions
20 − 10 − 5
(6 × (3 + 5)) + 6 ÷ 3
Evaluate
4n
h
if n = 6 and h = 8.
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Algebra and Functions
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Possible Student Responses to Questions
12
•
Division; 16 ÷ 4 = 4
•
Addition; 8 + 4 = 12
•
If the addition is done first and the division second, the
result is 6.
•
Yes, as this example shows, the order of operations can
change the value of an expression.
•
The multiplication is done first because it occurs to the left
of the division.
•
12 × 0.25 = 3
In this example, the answer is the same whether the
multiplication or the division is done first. However, this is
not always the case, especially in more complex problems
involving several operations. Even when not strictly
necessary, the order of operations should be applied
because it always results in the correct answer.
3
•
•
The subtractions should be done left to right.
•
The value of (20 − 10) − 5 is also 5, because the
parentheses indicate the same first operation as the order of
operations. However, the expression 20 − (10 − 5) is 15,
since now the second subtraction is performed first.
(6 × (3 + 5)) + 6 ÷ 3
•
The grouping symbols are parentheses ().
(6 ×8) + 6 ÷ 3
48 + 6 ÷ 3
48 + 2
50
•
The operation in the innermost grouping symbols, (3 + 5),
should be done first.
•
Other grouping symbols include brackets [], braces {}, the
radical sign
, and the fraction bar −.
•
Multiplication is used in the numerator, and division is
implied by the fraction bar.
− 85
5
3
•
Algebra and
Functions
Ans
wers to Pr
oblems
Answ
Problems
7AF1.2
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7AF1.2
Algebra and Functions
Basic Instruction
A Matter of Order:
Ask students to simplify the following mathematical expression: 5 + 4 × 3 − 6 ÷ 3. If order of operations is
neglected and all operations are performed in order from left to right, then the result is 7. However, if order of
operations is used, then the expression simplifies to 15.
Order of operations is also important when solving application problems. To find the surface area of a cube
with 10-cm sides, students must first find the area of one face and then multiply by 6:
(10 cm)2 × 6 = 600 cm 2 . Multiplying 10 by 6 and then squaring would result in an incorrect answer.
Students are frequently taught the mnemonic “Please Excuse My Dear Aunt Sally” (PEMDAS) to help
remember the order of operations: parentheses, exponents, multiplication or division, and addition or
subtraction. The “or” situations for multiplication or division (MD) and addition or subtraction (AS) are
frequently overlooked so that students’ use of order of operations is not successful. There are several teaching
strategies that will help students see and understand this process more clearly.
Steps to Simplification:
Algebra and
Functions
Order of operations is a set of steps that group operations together and proceed from left to right within each
step. The organizer below visually reinforces both the steps and the left-to-right progression.
Simplification
Grouping
Symbols
Powers and
Roots
Multiplication or
Division
Addition or
Subtraction
• Grouping Symbols: These include parentheses (), braces {}, brackets [], the fraction bar or −, and
the radical sign
. Grouping symbols can contain other grouping symbols, as in the expression
(4 + 3) 2 + 7  × 5. Students should always start with the innermost set of grouping symbols and


progress outward.
• Powers or Roots: The word exponent is often used in place of these words, the result being that
students tend not to recognize the need to evaluate the radical.
• Multiplication or Division: Using PEMDAS, students often forget that either multiplication or
division is done as they occur, left to right. Because M comes before D in PEMDAS, some students
believe that multiplication must be done before division. The “or” needs to be emphasized. A good
example is 4 + 6 ÷ 3 ÷ 2 × 5 ÷ 10 × 2 = 5. This expression shows that neither multiplication nor division
takes precedence over the other; students should perform each of the operations from left to right as
they encounter them.
• Addition or Subtraction: As with multiplication and division, emphasize the “or” since addition or
subtraction should be done as they are encountered, left to right. A good example is
10 + 6 − 8 + 7 − 9 = 6, which shows that neither addition nor subtraction take precedence.
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Algebra and Functions
|
7AF1.2
Another Mnemonic:
PEMDAS is the most frequently taught mnemonic, but there might be others that make more sense
to students. The following mnemonic uses GS instead of P, which teaches students to do the work in all
grouping symbols, not just parentheses. It also uses PR for powers and roots, instead of just E for exponents.
Gertrude Salamander
(Grouping Symbols)
Plays Really
(Powers or Roots)
Mean Dominoes
(Mult or Div)
After Supper.
(Add or Sub)
Emphasize that the words go in pairs with a left-to-right “or” order.
Vertical Format:
This process involves having students work down the page, lining up numbers with the previous step. Students
use the steps for order of operations to move from one line to the next. This strategy will help students do one
step at a time and facilitate reviewing what has been done. This is a valuable tool for students as they move on
to higher levels of mathematics.
Algebra and
Functions
(2 × 25.75 + 16.99) ×1.075
(51.50 + 16.99) ×1.075
(67.49) ×1.075
73.63
Round ‘Em Up:
In this strategy, students circle all operations that occur between addition or subtraction signs, creating
groups of operations to compute first.
7 + 2 • 5 – 3(8 – 4)
1. Addition and subtraction are the last operations completed, so use
these signs to separate the rest of the elements of the expression to
see what needs to be done first.
7 + 2 • 5 – 3(8 – 4)
2. Circle these terms to make them a group.
Parentheses remind students, “Do me first!”
7 + 10 – 3(4)
3. Simplify each term.
7 + 10 – 12
4. Add and subtract from left to right.
5
5. Simplification is complete.
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7AF1.2
Algebra and Functions
Basic Instruction
Vocabulary Notes:
A mathematical expression is like a phrase that contains ideas, but no complete thoughts.
In English class, this could be: the blue sky
In math class, this is: 6 × 7 + 4
On the other hand, a mathematical equation conveys a complete statement.
In English class, this could be: The blue sky is filled with clouds.
In math class, this is: 6 × 7 + 4 = 46.
An easy way to distinguish between the two is to note that an equation contains an equal sign (=), while
an expression does not (just as in English class, a complete sentence must contain both a noun and a verb,
while a fragment or expression might not).
Algebra and
Functions
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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.
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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.
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The following sample lesson, covering Standards 7MG2.1 and
7MG2.2, was taken from pages 235-239 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 71-72, and
the answer keys for the worksheets appear in the Teacher
Guidebook on pages 240-241.
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.
Measurement and Geometry
|
7MG2.1
7MG2.2
Standards 7MG2.1, 7MG2.2
7MG2.1
7MG2.1–– Use formulas routinely for finding the perimeter and area of basic
two-dimensional figures and the surface area and volume of basic three-dimensional figures,
including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and
cylinders.
7MG2.2–
7MG2.2–Estimate and compute the [surface] area of more complex or irregular two- and
three-dimensional figures by breaking the figures down into more basic geometric objects.
Purpose
From home remodeling to
surveying, geometric formulas
for perimeter, area, and volume
have myriad applications.
Breaking complex figures into
more basic parts, or in general
breaking a complex problem
down into simpler parts, is one
of the most fundamental
problem-solving strategies, and
geometry gives a good visual
illustration of the technique.
Real-Life Examples
✔ The border of a sandbox at the park is to be made from
䊐
wooden planks. The sandbox measures 20 feet by 30 feet.
The city planners need to budget for 100 feet of planks to
surround the sandbox.
✔ A construction crew will need to buy 1 cubic yard of
䊐
1
1
concrete to pour a pillar that’s 3 yd by 3 yd by 9 yd tall,
since 13 yd × 13 yd × 9 yd =1 yd3.
✔ To cover the three sides of a shower enclosure 8 feet tall,
䊐
3 feet wide, and 6 feet long, a homeowner will need to
buy (8 × 3) + (8 × 6) + (8 × 3) = 96 square feet of tiles.
Building Blocks
Measurement
and Geometry
Students should have prior knowledge of the following topics:
■ Identifying geometric figures
■ Unit conversions and dimensional analysis
Stepping Stones
The following standards are related to Standards 7MG2.1 and 7MG2.2:
●
●
●
●
7MG1.1 – Compare weights, capacities, geometric measures, times, and temperatures within and
between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic
centimeters).
7MG1.2 – Construct and read drawings and models made to scale.
7MR2.1 – Use estimation to verify the reasonableness of calculated results.
7MR3.1 – Evaluate the reasonableness of the solution in the context of the original situation.
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7MG2.1
7MG2.2
|
Measurement and Geometry
Progress Monitor
Pr
oblems to PPose
ose
Problems
Questions to Ask
What is the area of a 4 foot by 6 foot
rectangle? What is its perimeter?
•
What are the units of the answers?
•
Can you justify your answers with a picture?
A cardboard box is 16 inches high,
12 inches wide, and 18 inches deep.
How many cubic inches of packing
material do you need to fill it?
•
How can you tell if this problem is asking for surface area
or volume?
What is the area of the shaded part of
the square below?
•
What is the area of the large square?
•
What is the area of each of the four unshaded triangles?
•
Why is the shaded area half the area of the whole square?
•
Is 100 meters related to the area or the circumference?
•
How does the circumference of a semicircle relate to a
circle?
•
What practical application would this diameter have for
someone designing a track?
2
2
Measurement
and Geometry
2
2
2
2
2
2
A 100-meter-long piece of track is
curved into a semicircle. What is the
diameter of the semicircle?
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Measurement and Geometry
|
Ans
wers to Pr
oblems
Answ
Problems
Possible Student Responses to Questions
Area: 24 square feet;
•
The units for area are square feet, or ft2; the units for
perimeter are just feet, or ft.
•
The following picture could be used to represent this
situation. The 24 individual squares represent the area, and
each small square is 1 ft2. The perimeter is the distance
around the rectangle.
Perimeter: 20 feet
7MG2.1
7MG2.2
6 ft
4 ft
4 ft
6 ft
3456 cubic inches
•
The question is asking for volume because it asks for the
amount of space to be filled.
8 square units
•
16 square units
•
Each triangle is (2 × 2) = 2 square units.
•
Visually, the area covered by the four unshaded triangles is
equal to the area covered by the shaded square, which
implies that each has the same area.
•
Circumference
•
The circumference of a semicircle is equal to half the
circumference of a circle, just as a semicircle is half of a
circle.
•
Knowing the diameter—or the radius—makes it easier to
construct a semicircle of specific dimensions. It would be
more difficult to construct based on only the circumference.
Measurement
and Geometry
A 100-meter-long piece of track is
curved into a semicircle. What is the
diameter of the semicircle?
1
2
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7MG2.1
7MG2.2
|
Measurement and Geometry
Basic Instruction
Arranging formulas:
Many students have trouble remembering the many different formulas for different shapes. It is helpful to give
them a framework for organizing the formulas, such as:
Figures with Two Bases
Figures with One Base and a Point
Two-dimensional
Triangle: one-half base times height
Rectangle: base times height
Parallelogram: base times height
Trapezoid: average of the bases,
times height
Circle: πr 2
Three-dimensional
Prism: base times height
Cylinder: base times height
Pyramid: one-third base times height
Cone: one-third base times height
Point out to your students that a “pointy” object in two dimensions has half the area of the corresponding
“boxy” thing. Show them the following drawing to justify that a triangle’s area is half that of the
corresponding parallelogram:
b
h
h
b
Then reason, by analogy, that just as the area of a pointy object is half the area of the boxy thing in two
dimensions, the volumes of pointy objects are one-third as large as that of the corresponding boxy thing in
three dimensions.
Dimensional Analysis:
Measurement
and Geometry
The sphere formulas don’t fit into the classification above. They’re also harder to derive (requiring calculus,
or Cavalieri’s principle). Many students have trouble remembering which formula, 4πr 2 or 43 πr 3, gives the
surface area of the sphere and which gives the volume. If you emphasize that quantities for area must have
square units, and quantities for volume must have cubic units, then the students can figure out which is
which. It will also help them understand why a formula like 12 (b 1+ b 2)h is possible for the area of a
trapezoid, while 12 b1b2 h would not be. Similarly, if the problem is asking for an area, then the students’ work
should involve multiplying two lengths, or squaring one length.
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238 Instructional Strategies for Student Achievement
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Measurement and Geometry
|
7MG2.1
7MG2.2
Cutting and pasting:
Show your students that cutting and pasting explains why parallelograms and rectangles have the same area
formula.
h
b
Use this picture to remind your students why they have to use the perpendicular height (and not the side length)
in the formula for the area of a parallelogram, A = bh.
Lattice polygons:
When a polygon has its vertices on a grid, as shown, students are often unsure how to begin. Encourage them to
draw a rectangle around the outside, and then subtract the extra triangles:
(5× 4) − 1 (3× 3) − 1 (2 × 2) − 1 (3× 2) − 1 (2 ×1) = 9.5 square units
2
2
2
2
Of course, it’s also often possible to cut the figure up inside. For smaller polygons, it may be practical to simply
count the squares, pairing up fractional squares that add up to 1.
Measurement
and Geometry
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The following sample lesson, covering Standard 7NS1.2, was
taken from pages 17-21 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 3-4, and the answer
keys for the worksheets appear in the Teacher Guidebook on
pages 22-24.
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.
Number Sense
|
7NS1.2
Standard 7NS1.2
Number Sense
7NS1.2– Add, subtract, multiply, and divide rational numbers (integers, fractions, and
terminating decimals) and take positive rational numbers to whole-number powers.
Purpose
All students need to understand
the basic arithmetic functions
involving positive and negative
rational numbers, especially
since common fractions like
1 2
, , and 14 occur frequently.
2 3
Real-Life Examples
✔ A weather forecaster reports that the actual temperature
䊐
is 8ºF but that with the wind chill, it feels like − 13°F.
The wind chill makes it feel 21ºF colder, since
8 − (−13) = 21.
✔
䊐 The value of a company’s stock gained $1.48 per share in
2001 but lost $2.43 per share in 2002. If the value per
share was $5.56 at the beginning of 2001, then the value
at the end of 2002 was 5.56 + 1.48 − 2.43, or $4.61.
✔ The isotope of ruthenium known as Ru-106 has a
䊐
half-life of 1 year, meaning that only 12 of it will remain
after 1 year, only 14 will remain after 2 years, and so on.
Consequently,
the amount remaining after n years
 n
1 

is  2  of the original amount.
Building Blocks
Students should have prior knowledge of the following topics:
■ Less than, greater than
■ Binary operations (+, −, ×, ÷)
■ Comparison of fractions and decimals
■ Signed numbers (positive, negative)
■ Exponents
■ Rational numbers
■
Absolute value
Stepping Stones
The following standards are related to Standard 7NS1.2:
● 7NS2.2 – Add and subtract fractions by using factoring to find common denominators.
● 7NS2.5 – Understand the meaning of the absolute value of a number; interpret the absolute value as
the distance of the number from zero on a number line; and determine the absolute value of real
numbers.
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17
7NS1.2
|
Number Sense
Progress Monitor
Number Sense
Pr
oblems to PPose
ose
Problems
Questions to Ask
What is the value of (−42) + (32)?
•
Is the value positive or negative? Why?
•
What is the absolute value of − 42? What is the absolute
value of 32?
•
Is the value positive or negative? Why?
•
If the signs weren’t there, which number would be
larger: 87 or 65 ?
•
How do you add or subtract fractions with different
denominators?
•
Is the value positive or negative? Why?
•
Would the answer differ if the problem asked for the value
of (24)(−12)?
•
Is the value positive or negative? Why?
•
Would the answer differ if the problem asked for the value
 
of (−54) ÷  12  ?
•
Is the value positive or negative? Why?
•
How would the answer be different if the problem asked for
the value of (−2)5 ?
•
How would the answer be different if the problem asked for
 5
the value of − 12  ?
What is the value of
− 7 
 
8
 
+  65  ?
What is the value of (−24)(12)?


What is the value of (54) ÷ − 12  ?
What is the value of (−2)4 ?
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18
Instructional Strategies for Student Achievement
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Number Sense
Possible Student Responses to Questions
−10
•
•
−1
24
•
•
−288
−108
16
7NS1.2
Negative, because the number with the larger absolute
value (−42) is negative.
−42 = 42, and 32 = 32
Negative, because the number with the larger absolute
 
value − 87  is negative.
7
8
•
Convert both fractions to equivalent fractions with common
denominators—in this case, the common denominator is 24.
Then, you can add or subtract the numerators, as required.
•
Negative. When a positive number is multiplied by a
negative number, the result is negative.
•
The answers would be the same. The absolute values of the
numbers are the same, so the absolute value of the answer
(288) would be the same. In addition, one factor is negative,
so the answer must be negative.
•
Negative. When a positive number is divided by a negative
number (or vice versa), the result is negative.
•
The answer would be the same—the same absolute values
are involved, and one of the numbers is negative.
•
Positive. Any number raised to an even power is positive.
•
The answer would be − 32. The exponent of 5 indicates
that (−2)5 = −2 × (−2)4 . Consequently, it must
be − 2 ×16 = −32.
•
The numerator would be 15 , and the denominator would
be (−2)5 , so the answer would be − 321 .
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Number Sense
Ans
wers to Pr
oblems
Answ
Problems
|
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19
7NS1.2
|
Number Sense
Basic Instruction
Number Line:
Number Sense
When dealing with positive and negative numbers, you can use a number line to provide a concrete
representation of abstract ideas. Positive numbers can be represented by arrows to the right, negative
numbers by arrows to the left. Addition doesn’t affect an arrow’s direction; subtraction, on the other hand,
turns it around.
Students sometimes have difficulty using a number line when several signs are involved, such as when they
need to add a negative number or, worse, subtract a negative number. Arrows can be helpful. Which way
does the arrow point when a negative number is subtracted, as in 3 − (−2)? This problem can be represented
as follows, since (−2) would be represented by an arrow to the left, but subtraction turns it around:
3
(2)
0
5
A similar diagram could be used to represent the addition problem 3 + 2.
Fact Families:
A fact family is a collection of related mathematical statements. To explore addition and subtraction of
signed numbers, use the same numbers with positive and negative signs, and then show what happens when
the numbers are added or subtracted. Have students describe the relationships between the results.
4+2=6
4−2 = 2
4 + (−2) = 2
4 − (−2) = 6
−4 + 2 = − 2
−4 − 2 = − 6
−4 + (−2) = −6
−4 − (−2) = − 2
After examining the fact family above, students can see that 4 + 2 is the same as 4 − (−2) and is the opposite
of both − 4 + (−2) and − 4 − 2.
Fact families can be used with multiplication and division, too:
䊊
20
(3) × (5) =
(3) × (−5) =
(−3) × (5) =
(−3) × (−5) =
(24) ÷ (3) =
(24) ÷ (−3) =
(−24) ÷ (3) =
(−24) ÷ (−3) =
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Number Sense
|
7NS1.2
Pattern Recognition:
Number Sense
When signed numbers are multiplied, patterns result, as shown below. The pattern of answers in the first
column begins 8, 4, 0... but then what? Students should see that the pattern continues − 4, − 8. A similar
pattern can be used to show what happens when negative numbers are multiplied, as in the second column.
2× 4 = 8
−4 × 2 = −8
1× 4 = 4
−4 ×1 = −4
0× 4 = 0
−4 × 0 = 0
−1× 4 =
−4 × (−1) =
−2 × 4 =
−4 × (−2) =
Have students complete the following patterns to understand powers of negative numbers. You may allow
students to use a calculator to fill in the answers, but have them describe the pattern in words to make sure
they understand what is happening.
21 =
22 =
23 =
24 =
(−2)1 =
(−2)2 =
(−2)3 =
(−2)4 =
Factoring:
Negative numbers with exponents can be taught using factoring.
Demonstrate that (−2)5 = (−2) × (−2) × (−2) × (−2) × (−2) and that each pair of (−2)s that are multiplied
together “cancel out” the negative signs. Consequently, if the exponent is even, all the negatives cancel out,
and the result is positive. If the exponent is odd, there is a negative sign left over, and the result is negative.
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