Download Math 7 Pacing Guide 2015-2016 - Fredericksburg City Public Schools

Math 7 Pacing Guide 2015-2016
Walker-Grant Middle School, Fredericksburg City School
ALL CONCEPTS IN THIS COURSE SHOULD BE APPROACHED USING THE PROCESS GOALS, STRATEGIES, AND THE REOCCURRING THEMES
Process Goals:
Problem Solving
Communication
Logic and Reasoning
Connections
Representations
Quarter 1
Probability (7.9) (7.10)
-
-
Theoretical and Experimental
Law of Large Numbers
Tree Diagrams and the Counting Principle
Compound Events
Statistics (7.11)
-
Review Measures of Center
Frequency Distributions
Histograms
Stem and Leaf Plots
Compare and Contrast
Strategies:
Graphical – math picture
Numerical – table of values or numbers only
Symbolic – abstract formula or equation
Verbal – oral or written words
Concrete – modeled situation
Quarter 2
Fractions, Decimals, and Percents (7.1)
-
-
Conversions
Compare and Order with Scientific
Notation
Practical Problems
Order of Operations (7.13) (7.16)
-
Algebraic Vocabulary
Distributive Property
Write Verbal Expressions
Evaluate Expressions
Themes
Fractions and Basic Skills
Comparing and Ordering
Problem Solving
Writing
Properties
Quarter 3
Quarter 4
Proportions (7.4)
-
Ratios
-
Practical Problems
SOL Review
Consumer Math (7.4)
-
Percent of a Number
Tax, Tip, Total Cost
-
Discount and Sale Price
Volume and Surface Area (7.5)
-
-
Integers and Absolute Value (7.1)(7.3)(7.16)
-
-
Model and Calculate Integer Operations
Identity Property
Inverse Property
Zero Property
Review Perimeter and Area
Review Rectangular Prisms
Cylinders
Practical Problems
Equations and Inequalities (7.14) (7.15)
-
Review Graphing Inequalities
Inverse Operations
Model and Solve
Properties
Practical Problems
Similar Figures and Quadrilaterals (7.6) (7.7)
-
-
Corresponding Sides
Corresponding and Congruent Angles
Similarity Statements
Properties (7.16)
-
-
Commutative Property
Associative Property
Perfect Squares and Negative Exponents (7.1)
Patterns and Sequences (7.2)
-
Scientific Notation (7.1)
-
Convert between Standard Form and Scientific
Notation
Compare and Order
-
Arithmetic and Geometric
Tables
Functions (7.12)
-
Transformations (7.8)
-
-
Translations
Reflections
Rotations
Dilations
Review Coordinate Plane
Function Vocabulary
Tables and Graphs
1
Math 7 Pacing Guide 2015-2016
SOL 7.1
Quarter: 1, 2
Walker-Grant Middle School, Fredericksburg City School
Unit(s): Integers and Absolute Value, Perfect Squares and Negative Exponents, and Scientific Notation
Language Forms and Conventions
Linguistic Complexity
Vocabulary
6.2 a) frac/dec/% ‐ b) ID from representation; d) compare/order 8.1 b) compare/order fract/dec/%, and scientific notation
Vertical Articulation
6.5 investigate/describe positive exponents, perfect squares
8.5 a) determine if a number is a perfect square; b) find two consecutive whole numbers between which a square root lies
7.1 The student will
a) investigate and describe the concept of negative exponents for powers of ten;
Reporting Category:
b) determine scientific notation for numbers greater than zero; (without calculator)
Number and Number
c)
compare and order fractions, decimals, percents and numbers written in scientific notation; (without calculator)
Sense (16 CAT items)
d) determine square roots; (without calculator) and
e) identify and describe absolute value for rational numbers.
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills
Vocabulary/WIDA Standards
a) Recognize powers of 10 with negative

When should scientific notation
Perfect square, square root, absolute value, fraction,

Negative exponents for powers of 10 are used to
exponents by examining patterns.
be used?
decimal, percent, exponent, base, power, negative
represent numbers between 0 and 1.
b) Write a power of 10 with a negative
Scientific notation should be
exponent, scientific notation, standard form,
3 1
exponent in fraction and decimal form.
equivalent, greater than, less than
used whenever the situation
(e.g., 10 = 3 = 0.001).
Write a number greater than 0 in scientific
calls for use of very large or very c)
10
Levels 1-2
notation.
small numbers.
What does it mean to find the absolute value of a

Negative exponents for powers of 10 can be
d) Recognize a number greater than 0 in
number? To find the number’s distance from zero.
investigated through patterns such as:
scientific notation.

How are fractions, decimals and
e)
Compare
and
determine
equivalent
percents related?
2
List the perfect squares from 1 to 100.
10 =100
relationships between numbers larger than
Any rational number can be
1,4,9,16,25,36,49,64,81,100
1
0
written
in
scientific
notation.
represented
in
fraction,
decimal
Levels 3-5
10 = 10
f)
Represent a number in fraction, decimal,
and percent form.
When should scientific notation be used?
0
and percent forms.
10 = 1
Scientific notation should be used whenever the
g) Compare, order, and determine equivalent
situation calls for use of very large or very small

What
does
a
negative
exponent
1 1
1 = 0.1
10 =
numbers.
relationships among fractions, decimals,

mean when the base is 10?
1
10 10
and percents. Decimals are limited to the
A base of 10 raised to a negative
Write a power of 10 with a negative exponent in
thousandths place, and percents are limited
exponent represents a number
fraction and decimal form.

A number followed by a percent symbol (%) is
to the tenths place. Ordering is limited to
between 0 and 1.
Levels 1-2
equivalent to that number with a denominator of 100
no more than 4 numbers.
Absolute value
3 60
Equivalent
h) Order no more than 3 numbers greater than
(e.g., 5 = 100 = 0.60 = 60%).

How is taking a square root
Exponent
0 written in scientific notation.
different from squaring a
i)
Determine the square root of a perfect

Scientific notation is used to represent very large or very
number?
square less than or equal to 400.
small numbers.
Squaring a number and taking a
j)
Demonstrate absolute value using a number
Levels 3-5
square root are inverse

A number written in scientific notation is the product of
line.
Scientific Notation
operations.
two factors — a decimal greater than or equal to 1 but
Perfect Square
k) Determine the absolute value of a rational
less than 10, and a power of 10
number.

Why is the absolute value of a
4
l)
Show that the distance between two
number positive?
(e.g., 3.1  105= 310,000 and 2.85 x 10 = 0.000285).
rational numbers on the number line is the
The absolute value of a number
absolute value of their difference, and apply
represents distance from zero

Equivalent relationships among fractions, decimals, and
this principle to solve practical problems.†
on a number line regardless of
percents can be determined by using manipulatives
direction. Distance is positive.
(e.g., fraction bars, Base-10 blocks, fraction circles,
graph paper, number lines and calculators).

A square root of a number is a number which, when multiplied by itself, produces the given number (e.g.,

The square root of a number can be represented geometrically as the length of a side of the square.
The absolute value of a number is the distance from 0 on the number line regardless of direction.
1 1

(e.g., 2 2 )
121 is 11 since 11 x 11 = 121).
Notes:
2
Math 7 Pacing Guide 2015-2016
SOL 7.2
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Quarter: 2
Unit(s): Patterns and Sequences
6.17 ID/extend geometric/arithmetic sequences
Reporting Category:
7.2 The student will describe and represent arithmetic and geometric sequences using variable expressions.
Number and Number
Sense (16 CAT items)
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills

In geometric sequences, students must determine what
each number is multiplied by in order to obtain the next
number in the geometric sequence. This multiplier is
called the common ratio. Sample geometric sequences
include
2, 4, 8, 16, 32, …; 1, 5, 25, 125, 625, …;
and 80, 20, 5, 1.25, ….


When are variable expressions
used?
a)
Analyze arithmetic and geometric
sequences to discover a variety of patterns.
Variable expressions can
express the relationship
between two consecutive
terms in a sequence.
b)
Identify the common difference in an
arithmetic sequence.
c)
Identify the common ratio in a geometric
sequence.
d)
Given an arithmetic or geometric sequence,
write a variable expression to describe the
relationship between two consecutive
terms in the sequence.
Vocabulary
In the numeric pattern of an arithmetic sequence,
students must determine the difference, called the
common difference, between each succeeding number
in order to determine what is added to each previous
number to obtain the next number.
A variable expression can be written to express the
relationship between two consecutive terms of a
sequence
Language Forms and Conventions
If n represents a number in the sequence 3, 6, 9, 12…,
the next term in the sequence can be determined using
the variable expression n + 3.
If n represents a number in the sequence 1, 5, 25, 125…,
the next term in the sequence can be determined by
using the variable expression 5n.
Sequence, term, arithmetic sequence, common
difference, geometric sequence, common ratio,
consecutive terms, variable expression
Levels 1-2
A ____________is an ordered list of numbers.
(A sequence is an ordered list of numbers.)
Linguistic Complexity

Vocabulary/WIDA Standards
In an ______________ ____________, each term is
found by adding the common difference to the
previous term. (In an arithmetic sequence, each term is
found by adding the common difference to the
previous term.)
Levels 3-5
_________ ____________is the difference between
terms in an arithmetic sequence.
(Common difference is the difference between terms
in an arithmetic sequence.)
What is the common ratio in the sequence below:
5, 10, 15, 20, 25… (The common ratio is two.)
Levels 1-2
Sequence
Term
Arithmetic Sequence
Geometric Sequence
Levels 3-5
Common Difference
Common Ratio
Notes:
3
Math 7 Pacing Guide 2015-2016
SOL 7.3
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Quarter: 1
Unit(s): Integer Operations
6.3 a) ID/represent integers; b)order/compare integers; c) ID/describe absolute value of integers
6.6 a) mult/div fractions
Reporting Category:
7.3 The student will
Computation and
a) model addition, subtraction, multiplication and division of integers; and
Estimation
b) add, subtract, multiply, and divide integers. (without calculator)
(16 CAT items)
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills
Integers are used in practical situations, such as
temperature changes (above/below zero), balance in a
checking account (deposits/withdrawals), and changes
in altitude (above/below sea level).


Concrete experiences in formulating rules for adding
and subtracting integers should be explored by
examining patterns using calculators, along a number
line and using manipulatives, such as two-color
counters, or by using algebra tiles.
Concrete experiences in formulating rules for
multiplying and dividing integers should be explored by
examining patterns with calculators, along a number
line and using manipulatives, such as two-color
counters, or by using algebra tiles.
The sums, differences, products
and quotients of integers are
either positive, zero, or
negative. How can this be
demonstrated?
This can be demonstrated
through the use of patterns and
models.
a)
Model addition, subtraction, multiplication
and division of integers using pictorial
representations of concrete manipulatives.
b)
Add, subtract, multiply, and divide integers.
c)
Simplify numerical expressions involving
addition, subtraction, multiplication and
division of integers using order of
operations.
d)
Solve practical problems involving addition,
subtraction, multiplication, and division
with integers.
Vocabulary


Integer, whole number, opposites, sum, difference,
product, quotient, number line, zero pair
Levels 1-2
Linguistic Complexity
The set of integers is the set of whole numbers and their
opposites
(e.g., … –3, –2, –1, 0, 1, 2, 3, …).
Draw a number line to represent the integers from
negative three to positive three.
Identify all of integers in a given list of numbers.
Levels 3-5
In the problem 4 + (-9), how many zero pairs would I
use? In this problem, I would use four zero pairs.
Levels 1-2
Language Forms and Conventions

Vocabulary/WIDA Standards
Integer
Whole number
Opposites
Levels 3-5
Zero Pair
Sum
Product
Notes:
4
Math 7 Pacing Guide 2015-2016

A proportion is a statement of equality between two
ratios.

a c
A proportion can be written as b = d , a:b = c:d, or a is to
b as c is to d.


Essential Teacher Questions

A proportion can be solved by finding the product of the
means and the product of the extremes. For example, in
the proportion a:b = c:d, a and d are the extremes and b
and c are the means. If values are substituted for a, b, c,
and d such as 5:12 = 10:24, then the product of
extremes (5  24) is equal to the product of the means
(12  10).
In a proportional situation, both quantities increase or
decrease together.

In a proportional situation, two quantities increase
multiplicatively. Both are multiplied by the same factor.

A proportion can be solved by finding equivalent
fractions.
What makes two quantities
proportional?
Two quantities are proportional
when one quantity is a
constant multiple of the other.
Essential Knowledge and Skills
a)
Write proportions that represent equivalent
relationships between two sets.
b)
Solve a proportion to find a missing term.
c)
Apply proportions to convert units of
measurement between the U.S. Customary
System and the metric system. Calculators
may be used.
d)
Apply proportions to solve practical
problems, including scale drawings. Scale
factors shall have denominators no greater
than 12 and decimals no less than tenths.
Calculators may be used.
e)
Using 10% as a benchmark, mentally
compute 5%, 10%, 15%, or 20% in a
practical situation such as tips, tax and
discounts.
f)
Solve problems involving tips, tax, and
discounts. Limit problems to only one
percent computation per problem.

A rate is a ratio that compares two quantities measured
in different units. A unit rate is a rate with a
denominator of 1. Examples of rates include miles/hour
and revolutions/minute.

Proportions are used in everyday contexts, such as
speed, recipe conversions, scale drawings, map reading,
reducing and enlarging, comparison shopping, and
monetary conversions.

Proportions can be used to convert between measurement systems. For example: if 2 inches is about 5 cm, how many inches are in 16 cm?
–
Ratio, proportion, equivalent fractions, cross multiply,
rate, unit rate, scale drawing, units, fraction, percent,
scale factor, increase, decrease
Levels 1-2
Is this an example of a proportion?
Levels 3-5
Give a real-life example of when you would use a scale
drawing.
Levels 1-2
Fraction
Percent
Equivalent Fractions
Levels 3-5
Proportion
2inches 5cm

x
16cm

A percent is a special ratio in which the denominator is 100.

Proportions can be used to represent percent problems as follows:
–
Vocabulary/WIDA Standards
Vocabulary
Reporting Category:
Computation and
Estimation
(16 CAT items)
Understanding the Standard
Linguistic Complexity
Vertical Articulation
Quarter: 3
Unit(s): Proportions, Consumer Math
6.7 solve practical problems involving add/sub/mult/div decimals 8.3 a) solve practical problems involving rational numbers, percent, ratios, and prop; b) determine percent inc/dec
6.1 describe/compare data using ratios
8.3 a) solve practical problems involving rational numbers, percent, ratios, and prop
7.4 The student will solve single-step and multistep practical problems, using proportional reasoning.
Language Forms and Conventions
SOL 7.4
Walker-Grant Middle School, Fredericksburg City School
percent
part

100
whole
Notes:
5
Math 7 Pacing Guide 2015-2016
Walker-Grant Middle School, Fredericksburg City School
SOL 7.5

The area of a circle is computed by squaring the radius
and multiplying that product by  (A = r2 , where  
22
3.14 or 7 ).


A rectangular prism can be represented on a flat surface
as a net that contains six rectangles — two that have
measures of the length and width of the base, two
others that have measures of the length and height, and
two others that have measures of the width and height.
The surface area of a rectangular prism is the sum of the
areas of all six faces ( SA  2lw  2lh  2 wh ).
A cylinder can be represented on a flat surface as a net
that contains two circles (bases for the cylinder) and one
rectangular region whose length is the circumference of
the circular base and whose width is the height of the
cylinder. The surface area of the cylinder is the area of
the two circles and the rectangle (SA = 2r2 + 2rh).

The volume of a rectangular prism is computed by
multiplying the area of the base, B, (length times width)
by the height of the prism (V = lwh = Bh).

The volume of a cylinder is computed by multiplying the
area of the base, B, (r2) by the height of the cylinder (V
= r2h = Bh).

There is a direct relationship between changing one
measured attribute of a rectangular prism by a scale
factor and its volume. For example, doubling the length
of a prism will double its volume. This direct relationship
does not hold true for surface area.

How are volume and surface
area related?
Volume is a measure of the
amount a container holds while
surface area is the sum of the
areas of the surfaces on the
container.

How does the volume of a
rectangular prism change when
one of the attributes is
increased?
There is a direct relationship
between the volume of a
rectangular prism increasing
when the length of one of the
attributes of the prism is
changed by a scale factor.
a)
Determine if a practical problem involving a
rectangular prism or cylinder represents the
application of volume or surface area.
b)
Find the surface area of a rectangular prism.
c)
Solve practical problems that require
finding the surface area of a rectangular
prism.
d)
Find the surface area of a cylinder.
e)
Solve practical problems that require
finding the surface area of a cylinder.
f)
Find the volume of a rectangular prism.
g)
Solve practical problems that require
finding the volume of a rectangular prism.
h)
Find the volume of a cylinder.
i)
Solve practical problems that require
finding the volume of a cylinder.
j)
Describe how the volume of a rectangular
prism is affected when one measured
attribute is multiplied by a scale factor.
Problems will be limited to changing
attributes by scale factors only.
k)
Describe how the surface area of a
rectangular prism is affected when one
measured attribute is multiplied by a scale
factor. Problems will be limited to changing
attributes by scale factors only.
Area, rectangle, adjacent sides, circumference, circle,
squaring, radius, diameter, pi, product, rectangular
prism, cylinder, base, face, net, sum, surface area,
volume, length, width, height
Levels 1-2
Linguistic Complexity
The area of a rectangle is computed by multiplying the
lengths of two adjacent sides.
Determine if a practical problem involving a
rectangular prism or cylinder represents the
application of volume of surface area.
Levels 3-5
How does the volume of a rectangular prism change
when one of the attributes is increased? (The volume
of a rectangular prism will increase by the scale
factor.)
Levels 1-2
Language Forms and Conventions

Vocabulary
Quarter: 2
Unit(s): Volume and Surface Area
6.9 make ballpark comparisons between U.S. Cust/metric system
8.7 a) investigate/solve practical problems involving volume/surface area of prisms, cylinders, cones, pyramids;
6.10 a) define π; b) solve practical problems w/circumference/area
b) describe how changes in measured attribute affects volume/surface area
Vertical Articulation
of circle; c) solve practical problems involving area and perimeter
8.11 solve practical area/perimeter problems involving composite plane figures
given radius/diameter; d) describe/determine volume/surface area
of rectangular prism
7.5 The student will
Reporting Category:
a) describe volume and surface area of cylinders;
Measurement
b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and
(13 CAT items)
c)
describe how changing one measured attribute of a rectangular prism affects its volume and surface area.
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills
Vocabulary/WIDA Standards
Length
Width
Height
Radius
Diameter
Levels 3-5
Circumference
Volume
Surface Area
Notes:
6
Math 7 Pacing Guide 2015-2016
Understanding the Standard

Two polygons are similar if corresponding (matching)
angles are congruent and the lengths of corresponding
sides are proportional.

Congruent polygons have the same size and shape.

Congruent polygons are similar polygons for which the
ratio of the corresponding sides is 1:1.

Similarity statements can be used to determine
corresponding parts of similar figures such as:
ABC ~ DEF
 A corresponds to  D
AB corresponds to DE


The traditional notation for marking congruent angles is
to use a curve on each angle. Denote which angles are
congruent with the same number of curved lines. For
example, if  A congruent to  B, then both angles will
be marked with the same number of curved lines.
Congruent sides are denoted with the same number of
hatch marks on each congruent side. For example, a side
on a polygon with 2 hatch marks is congruent to the side
with 2 hatch marks on a congruent polygon.
Essential Teacher Questions

How do polygons that are
similar compare to polygons
that are congruent?
Congruent polygons have the
same size and shape. Similar
polygons have the same shape,
and corresponding angles
between the similar figures are
congruent. However, the
lengths of the corresponding
sides are proportional. All
congruent polygons are
considered similar with the
ratio of the corresponding sides
being 1:1.
Essential Knowledge and Skills
a)
Identify corresponding sides and
corresponding and congruent angles of
similar figures using the traditional notation
of curved lines for the angles.
b)
Write proportions to express the
relationships between the lengths of
corresponding sides of similar figures.
c)
Determine if quadrilaterals or triangles are
similar by examining congruence of
corresponding angles and proportionality of
corresponding sides.
d)
Given two similar figures, write similarity
statements using symbols such as
ABC ~ DEF ,  A corresponds to 
D, and
AB corresponds to DE .
Vocabulary/WIDA Standards
Vocabulary
Reporting Category:
Measurement
(13 CAT items)
Polygon, quadrilateral, triangle, proportional, similar
figures, corresponding, ratio, similarity statements,
congruent, hatch marks, angles
Levels 1-2
Linguistic Complexity
Vertical Articulation
Quarter: 3
Unit(s): Similar Figures and Quadrilaterals
6.2 frac/dec/% ‐ a) describe as ratios; b) ID from representation; c) equiv relationships; 8.10 a) verify the Pythagorean Theorem; b) apply the Pythagorean Theorem
6.12 determine congruence of segments/angles/polygons
7.6 The student will determine whether plane figures – quadrilaterals and triangles – are similar and write proportions to express the relationships between
corresponding sides of similar figures.
Are these two figures similar or not?
Levels 3-5
Write a similarity statement to represent the
corresponding sides of two similar figures.
Levels 1-2
Language Forms and Conventions
SOL 7.6
Walker-Grant Middle School, Fredericksburg City School
Triangle
Congruent
Similar Figures
Levels 3-5
Corresponding Sides
Corresponding Angles
Notes:
7
Math 7 Pacing Guide 2015-2016
SOL 7.7
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Quarter: 3
Unit(s): Similar Figures and Quadrilaterals
6.13 ID/describe properties of quadrilaterals 8.6 a) verify/describe relationships among vertical/adjacent/supplementary/complementary angles; b) measure angles < 360°

A parallelogram is a quadrilateral whose opposite sides
are parallel and opposite angles are congruent.

A rectangle is a parallelogram with four right angles. The
diagonals of a rectangle are the same length and bisect
each other.

A square is a rectangle with four congruent sides whose
diagonals are perpendicular. A square is a rhombus with
four right angles.

A rhombus is a parallelogram with four congruent sides
whose diagonals bisect each other and intersect at right
angles.

A trapezoid is a quadrilateral with exactly one pair of
parallel sides.

A trapezoid with congruent, nonparallel sides is called
an isosceles trapezoid.

Quadrilaterals can be sorted according to common
attributes, using a variety of materials.

A chart, graphic organizer, or Venn diagram can be
made to organize quadrilaterals according to attributes
such as sides and/or angles.

Why can some quadrilaterals
be classified in more than one
category?
Every quadrilateral in a subset
has all of the defining attributes
of the subset. For example, if a
quadrilateral is a rhombus, it has
all the attributes of a rhombus.
However, if that rhombus also
has the additional property of 4
right angles, then that rhombus
is also a square.
a)
Compare and contrast attributes of the
following quadrilaterals: parallelogram,
rectangle, square, rhombus, and trapezoid.
b)
Identify the classification(s) to which a
quadrilateral belongs, using deductive
reasoning and inference.
Quadrilateral, parallelogram, rectangle, square,
rhombus, trapezoid, isosceles trapezoid, congruent,
parallel, right angle, Venn Diagram
Levels 1-2
Linguistic Complexity
A quadrilateral is a closed plane (two-dimensional)
figure with four sides that are line segments.
Draw a picture of a parallelogram, rectangle, square,
rhombus, and trapezoid.
Levels 3-5
Complete a Venn Diagram to represent the similarities
and differences between a rectangle and a trapezoid.
Levels 1-2
Language Forms and Conventions

Vocabulary
Reporting Category:
7.7 The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.
Geometry
(13 CAT items)
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills
Vocabulary/WIDA Standards
Right Angle
Square
Rectangle
Levels 3-5
Quadrilateral
Parallelogram
Parallel Sides
Notes:
8
Math 7 Pacing Guide 2015-2016
SOL 7.8
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Quarter: 3
Unit(s): Transformations
6.11 a) ID coordinates of a point in a coordinate plane; b) graph ordered pairs in coordinate plane
8.8 a) apply transformations to plane figures; b) ID applications of transformations
f)
Sketch the image of a right triangle or
rectangle that has been rotated 90° or 180°
about the origin.
g)
Sketch the image of a right triangle or
rectangle that has been reflected over the
x- or y-axis.
h)
Sketch the image of a dilation of a right
triangle or rectangle limited to a scale factor
of
1 1
, , 2, 3 or 4.
4 2
Language Forms and Conventions
Linguistic Complexity
Vocabulary
Reporting Category:
7.8 The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane.
Geometry
(13 CAT items)
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills
Vocabulary/WIDA Standards

How does the transformation of
Coordinate plane, ordered pair, transformation,
a) Identify the coordinates of the image of a
 A rotation of a geometric figure is a turn of the figure
a figure affect the size, shape
rotation, translation, reflection, dilation, polygon,
right triangle or rectangle that has been
around a fixed point. The point may or may not be on the
and position of that figure?
image, preimage, point, prime, center of rotation, scale
translated either vertically, horizontally, or
figure. The fixed point is called the center of rotation.
factor, horizontal, vertical
a combination of a vertical and horizontal
Translations, rotations and
 A translation of a geometric figure is a slide of the figure
Levels 1-2
translation.
reflections do not change the
in which all the points on the figure move the same
What is it called when you turn a figure around a fixed
size
or
shape
of
a
figure.
A
b) Identify the coordinates of the image of a
distance in the same direction.
point? (It is a called a rotation when you turn a figure.)
dilation of a figure and the
right triangle or rectangle that has been
 A reflection is a transformation that reflects a figure
original figure are similar.
Are the images on the coordinate plane the same or
rotated 90° or 180° about the origin.
different? (The images are different.)
across a line in the plane.
Reflections, translations and
c)
Identify the coordinates of the image of a
rotations usually change the
 A dilation of a geometric figure is a transformation that
right triangle or a rectangle that has been
Levels 3-5
position of the figure.
changes the size of a figure by scale factor to create a
reflected over the x- or y-axis.
A mirror is an example of what type of transformation?
similar figure.
(A mirror is a reflection.)
d) Identify the coordinates of a right triangle
 The image of a polygon is the resulting polygon after the
or rectangle that has been dilated. The
A translation, rotation, reflection, and dilation are all
transformation. The preimage is the polygon before the
examples of what? (They are all examples of
center of the dilation will be the origin.
transformations.)
transformation.
e) Sketch the image of a right triangle or
 A transformation of preimage point A can be denoted as
rectangle translated vertically or
Levels 1-2
the image A (read as “A prime”).
horizontally.
Image
Rotation
Levels 3-5
Coordinate Plane
Ordered Pair
Reflection
Notes:
9
Math 7 Pacing Guide 2015-2016
SOL 7.9
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Quarter: 1
Unit(s): Probability
6.16 a) compare/contrast dep/indep events; b) determine probabilities for dep/indep events
8.12 determine probability of indep/dep events with and without replacement
Theoretical probability of an event is the expected
probability and can be found with a formula.

Theoretical probability of an event =
number of possible favorable outcomes
total number of possible outcomes


The experimental probability of an event is determined
by carrying out a simulation or an experiment.
The experimental probability =
number of times desired outcomes occur
number of trials in the experiment
In experimental probability, as the number of trials
increases, the experimental probability gets closer to the
theoretical probability (Law of Large Numbers).
What is the difference between
the theoretical and
experimental probability of an
event?
Theoretical probability of an
event is the expected
probability and can be found
with a formula. The
experimental probability of an
event is determined by carrying
out a simulation or an
experiment. In experimental
probability, as the number of
trials increases, the
experimental probability gets
closer to the theoretical
probability.
a)
Determine the theoretical probability of an
event.
b)
Determine the experimental probability of
an event.
c)
Describe changes in the experimental
probability as the number of trials
increases.
d)
Investigate and describe the difference
between the probability of an event found
through experiment or simulation versus
the theoretical probability of that same
event.
Vocabulary/WIDA Standards
Probability, favorable outcomes, possible outcomes,
theoretical probability, experimental probability,
event, trials, Law of Large Numbers
Levels 1-2
How many possible outcomes are on a coin?
(There are 2 possible outcomes on a coin.)
Levels 3-5
What is the probability of rolling an even number on a
die? (The probability of rolling an even number is 3/6
or 1/2)
If we are looking to find the probability of picking a day
of the week that starts with “T”, how many favorable
outcomes are there? (Since Tuesday and Thursday
both start with a “T”, there would be two favorable
outcomes.)
Levels 1-2
Language Forms and Conventions


Linguistic Complexity

Vocabulary
Reporting Category:
7.9 The student will investigate and describe the difference between the experimental probability and theoretical probability of an event.
Probability and
Statistics (21 CAT items)
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills
Possible Outcomes
Trials
Levels 3-5
Theoretical Probability
Experimental Probability
Notes:
10
Math 7 Pacing Guide 2015-2016
SOL 7.10
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Quarter: 1
Unit(s): Probability
6.16 a) compare/contrast dep/indep events; b) determine probabilities for dep/indep events
8.12 determine probability of indep/dep events with and without replacement
Reporting Category:
7.10 The student will determine the probability of compound events, using the Fundamental (Basic) Counting Principle.
Probability and
Statistics (21 CAT items)
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills

Tree diagrams are used to illustrate possible outcomes
of events. They can be used to support the Fundamental
(Basic) Counting Principle.
A compound event combines two or more simple
events. For example, a bag contains 4 red, 3 green and 2
blue marbles. What is the probability of selecting a
green and then a blue marble?
The Fundamental (Basic)
Counting Principle is a
computational procedure used to
determine the number of
possible outcomes of several
events.
 What is the role of the
Fundamental (Basic) Counting
Principle in determining the
probability of compound events?
a)
Compute the number of possible outcomes
by using the Fundamental (Basic) Counting
Principle.
b)
Determine the probability of a compound
event containing no more than 2 events.
Vocabulary
 What is the Fundamental (Basic)
Counting Principle?
Fundamental Counting Principle, product, probability,
possible outcomes, events, compound events,
independent events, dependent events
Levels 1-2
Linguistic Complexity

The Fundamental (Basic) Counting Principle is a
computational procedure to determine the number of
possible outcomes of several events. It is the product of
the number of outcomes for each event that can be
chosen individually (e.g., the possible outcomes or
outfits of four shirts, two pants, and three shoes is 4 · 2 ·
3 or 24).
How many possible outcomes are on a single die?
(There are 6 possible outcomes on a single die.)
Levels 3-5
How can you use the Fundamental Counting Principle?
(You can use it to multiply the number of outcomes for
each event.)
Levels 1-2
Product
The Fundamental (Basic)
Counting Principle is used to
determine the number of
outcomes of several events. It is
the product of the number of
outcomes for each event that can
be chosen individually.
Language Forms and Conventions

Vocabulary/WIDA Standards
Levels 3-5
Fundamental Counting Principle
Compound Events
Notes:
11
Math 7 Pacing Guide 2015-2016
7.11
The student, given data in a practical situation, will
a) construct and analyze histograms; and
b) compare and contrast histograms with other types of graphs presenting information from the same data set.
Understanding the Standard

All graphs tell a story and include a title and labels that
describe the data.

A histogram is a form of bar graph in which the
categories are consecutive and equal intervals. The
length or height of each bar is determined by the
number of data elements frequency falling into a
particular interval.
Essential Teacher Questions

What types of data are most
appropriate to display in a
histogram?
Numerical data that can be
characterized using consecutive
intervals are best displayed in a
histogram.
Essential Knowledge and Skills
a)
Collect, analyze, display, and interpret a
data set using histograms. For collection
and display of raw data, limit the data to 20
items.
b)
Determine patterns and relationships within
data sets (e.g., trends).
c)
Make inferences, conjectures, and
predictions based on analysis of a set of
data.
d)
Compare and contrast histograms with line
plots, circle graphs, and stem-and-leaf plots
presenting information from the same data
set.
Vocabulary/WIDA Standards
Vocabulary
Reporting Category:
Probability and
Statistics (21 CAT items)
8.13 a) make comparisons/predictions/inferences, using information
displayed in graphs; b) construct/analyze scatterplots
Graph, title, label, data, histogram, bar graph,
elements, interval, consecutive, frequency distribution,
item, construct, comparison, prediction, inference
Levels 1-2
Linguistic Complexity
Vertical Articulation
Quarter: 1
Unit: Statistics
6.14 a) construct circle graphs; b) draw conclusions/make predictions, using circle graphs; c) compare/contrast graphs
Should a bar graph or circle graph be used to display
parts of a whole? (A circle graph should be used to
display parts of a whole.)
Levels 3-5
What is the difference between a bar graph and a
histogram? (A histogram has intervals.)
Levels 1-2
Title
Prediction
Comparison
Language Forms and Conventions
SOL 7.11
Walker-Grant Middle School, Fredericksburg City School
Levels 3-5
Bar Graph
Interval
Histogram

A frequency distribution shows how often an item, a
number, or range of numbers occurs. It can be used to
construct a histogram

Comparisons, predictions and inferences are made by examining characteristics of a data set displayed in a variety of graphical representations to draw conclusions.

The information displayed in different graphs may be examined to determine how data are or are not related, ascertaining differences between characteristics (comparisons), trends that suggest what new data
might be like (predictions), and/or “what could happen if” (inference).
Notes:
12
Math 7 Pacing Guide 2015-2016
Quarter: 2
Unit(s): Functions
8.14 make connections between any two representations (tables, graphs, words, rules)

Rules that relate elements in two sets can be
represented by word sentences, equations, tables of
values, graphs, or illustrated pictorially.

A relation is any set of ordered pairs. For each first
member, there may be many second members.

A function is a relation in which there is one and only
one second member for each first member.

As a table of values, a function has a unique value
assigned to the second variable for each value of the
first variable.

Rules that relate elements in two
sets can be represented by word
sentences, equations, tables of
values, graphs or illustrated
pictorially.
a)
Describe and represent relations and
functions, using tables, graphs, rules, and
words. Given one representation, students
will be able to represent the relation in
another form.
Vocabulary/WIDA Standards
Complete a function table.
Input, x
Output, y
1
3
2
4
3
5
4
Levels 3-5
As a graph, a function is any curve (including straight
lines) such that any vertical line would pass through the
curve only once.
Some relations are functions; all functions are relations.
Relations, functions, table, graph, rule, elements,
members, vertical line, illustrate, ordered pairs
Levels 1-2
Use a table of values to graph a linear function.
Levels 1-2
Language Forms and Conventions

 What are the different ways to
represent the relationship
between two sets of numbers?
Essential Knowledge and Skills
Vocabulary
Reporting Category:
7.12 The student will represent relationships with tables, graphs, rules, and words.
Patterns, Functions, and
Algebra (21 CAT items)
Understanding the Standard
Essential Teacher Questions
8.17 ID domain, range, indep/dep variable
Linguistic Complexity
SOL 7.12
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Ordered Pair
Table
Graph
Levels 3-5
Vertical Line
Illustrate
Notes:
13
Math 7 Pacing Guide 2015-2016
Quarter: 2
Unit(s): Order of Operations
6.8 evaluate whole number numerical expressions using order of operations
Vertical Articulation
The student will
a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; and
b) evaluate algebraic expressions for given replacement values of the variables.
Understanding the Standard

An expression is a name for a number.

An expression that contains a variable is a variable
expression.

An expression that contains only numbers is a numerical
expression.

A verbal expression is a word phrase (e.g., “the sum of
two consecutive integers”).

A verbal sentence is a complete word statement (e.g.,
“The sum of two consecutive integers is five.”).



An algebraic expression is a variable expression that
contains at least one variable (e.g., 2x – 5).
An algebraic equation is a mathematical statement that
says that two expressions are equal
(e.g., 2x + 1 = 5).
To evaluate an algebraic expression, substitute a given
replacement value for a variable and apply the order of
operations. For example, if a = 3 and b = -2 then 5a + b
can be evaluated as:
5(3) + (-2) = 15 + (-2) = 13.
Essential Teacher Questions

Essential Knowledge and Skills
How can algebraic expressions
and equations be written?
a)
Write verbal expressions as algebraic
expressions. Expressions will be limited to
no more than 2 operations.
Word phrases and sentences
can be used to represent
algebraic expressions and
equations.
b)
Write verbal sentences as algebraic
equations. Equations will contain no more
than 1 variable term.
c)
d)
e)
Translate algebraic expressions and
equations to verbal expressions and
sentences. Expressions will be limited to no
more than 2 operations.
Identify examples of expressions and
equations.
Apply the order of operations to evaluate
expressions for given replacement values of
the variables. Limit the number of
replacements to no more than 3 per
expression.
Vocabulary/WIDA Standards
Vocabulary
7.13
Expression, variable expression, numerical expression,
verbal expression, verbal sentence, algebraic
expression, algebraic equation, evaluate, substitute,
translate, constant, term
Levels 1-2
Linguistic Complexity
Reporting Category:
Patterns, Functions, and
Algebra (21 CAT items)
8.4 evaluate algebraic expressions using order of operations
8.1 a) simplify numerical expressions involving positive exponents, using rational numbers, order of
operations, properties
What acronym can help you remember the order of
operations? (GEMDAS can help you remember the
order of operations.)
Levels 3-5
What do the letters in GEMDAS stand for? (Grouping,
Exponents, Multiply, Divide, Add, Subtract)
What is the difference between a numerical and an
algebraic expression? (A numerical expression only has
numbers and an algebraic expressions contains at least
one variable.)
Levels 1-2
Language Forms and Conventions
SOL 7.13
Walker-Grant Middle School, Fredericksburg City School
Evaluate
Substitute
Levels 3-5
Algebraic Expression
Numerical Expression
Notes:
14
Math 7 Pacing Guide 2015-2016
Reporting Category:
Patterns, Functions, and
Algebra (21 CAT items)
Understanding the Standard

An equation is a mathematical sentence that states that
two expressions are equal.

A one-step equation is defined as an equation that
requires the use of one operation to solve
(e.g., x + 3 = – 4 ).


The inverse operation for addition is subtraction, and
the inverse operation for multiplication is division.
A two-step equation is defined as an equation that
requires the use of two operations to solve
 When solving an equation, why is
it important to perform identical
operations on each side of the
equal sign?
An operation that is performed
on one side of an equation must
be performed on the other side
to maintain equality.
Essential Knowledge and Skills
a)
Represent and demonstrate steps for solving
one- and two-step equations in one variable
using concrete materials, pictorial
representations and algebraic sentences.
b)
Solve one- and two-step linear equations in
one variable.
c)
Solve practical problems that require the
solution of a one- or two-step linear
equation.
Vocabulary/WIDA Standards
Equation, equal, equivalent, operations, inverse
operations, solution, variable
Levels 1-2
x7
 4 ).
3
An equation must always have an _____ sign.
(An equation must always have an equal sign.)
Levels 3-5
What is the inverse operation you must use to solve
x + 3 = 10? (You have to subtract 3 form both sides.)
Levels 1-2
Language Forms and Conventions
(e.g., 2x + 1 = -5; -5 = 2x + 1;
Essential Teacher Questions
Vocabulary
Vertical Articulation
Quarter: 2
Unit(2): Equations and Inequalities
6.18 solve one‐step linear equations in one variable 8.15 a) solve multistep linear equations in one variable (variable on one and two sides of equations); b) solve two‐step linear
inequalities and graph results on number line; c) ID properties of operations used to solve
7.14 The student will
a) solve one- and two-step linear equations in one variable; and
b) solve practical problems requiring the solution of one- and two-step linear equations.
Linguistic Complexity
SOL 7.14
Walker-Grant Middle School, Fredericksburg City School
Equivalent
Variable
Operations
Levels 3-5
Inverse Operations
Solution
Notes:
15
Math 7 Pacing Guide 2015-2016
The student will
a) solve one-step inequalities in one variable; and
b) graph solutions to inequalities on the number line.
Understanding the Standard

A one-step inequality is defined as an inequality that
requires the use of one operation to solve
(e.g., x – 4 > 9).

The inverse operation for addition is subtraction, and
the inverse operation for multiplication is division.

When both expressions of an inequality are multiplied
or divided by a negative number, the inequality symbol
reverses (e.g., –3x < 15 is equivalent to x > –5).

Solutions to inequalities can be represented using a
number line.
Essential Teacher Questions


Essential Knowledge and Skills
How are the procedures for
solving equations and
inequalities the same?
a)
Represent and demonstrate steps in solving
inequalities in one variable, using concrete
materials, pictorial representations, and
algebraic sentences.
The procedures are the same
except for the case when an
inequality is multiplied or
divided on both sides by a
negative number. Then the
inequality sign is changed from
less than to greater than, or
greater than to less than.
b)
Graph solutions to inequalities on the
number line.
c)
Identify a numerical value that satisfies the
inequality.
Vocabulary/WIDA Standards
Vocabulary
7.15
Inequality, operations, inverse operations, solution,
variable, greater than, greater than or equal to, less
than, less than or equal to
Levels 1-2
Linguistic Complexity
Reporting Category:
Patterns, Functions, and
Algebra (21 CAT items)
Quarter: 2
Unit(s): Equations and Inequalities
6.20 graph inequalities on number line 8.16 graph linear equation in two variables
Would the inequality x > 5 require you to use an open
or a closed circle on a graph? (It would require you to
use an open circle.)
Levels 3-5
What do you have to do when you multiply or divide
both sides by a negative number? (You must reverse
the inequality sign.)
Levels 1-2
How is the solution to an
inequality different from that of
a linear equation?
In an inequality, there can be
more than one value for the
variable that makes the
inequality true.
Language Forms and Conventions
SOL 7.15
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Greater than
Less than
Levels 3-5
Inequality
Notes:
16
Math 7 Pacing Guide 2015-2016
SOL 7.16
Vertical Articulation
Walker-Grant Middle School, Fredericksburg City School
Quarter: 1, 2
Unit(s): Integers and Absolute Value, Properties, Order of Operations, Equations and Inequalities
6.19 a) investigate/recognize identity properties for add/mult; b) multiplicative property of zero; c) inverse preperty for mult
8.15 c) ID properties of operations used to
solve equations
7.16
The student will apply the following properties of operations with real numbers:
a)
the commutative and associative properties for addition and multiplication;
Reporting Category:
b)
the distributive property;
Patterns, Functions, and
c)
the additive and multiplicative identity properties;
Algebra (21 CAT items)
d)
the additive and multiplicative inverse properties; and
e)
the multiplicative property of zero
Understanding the Standard
Essential Teacher Questions
Essential Knowledge and Skills
The commutative property for multiplication states that
changing the order of the factors does not change the
product (e.g., 5 · 4 = 4 · 5).

The associative property of addition states that
regrouping the addends does not change the sum
[e.g., 5 + (4 + 3) = (5 + 4) + 3].

Why is it important to apply
properties of operations when
simplifying expressions?
Using the properties of
operations with real numbers
helps with understanding
mathematical relationships.
a)
Identify properties of operations used in
simplifying expressions.
b)
Apply the properties of operations to
simplify expressions.
The associative property of multiplication states that
regrouping the factors does not change the product
[e.g., 5 · (4 · 3) = (5 · 4) · 3].

Subtraction and division are neither commutative nor
associative.

The distributive property states that the product of a
number and the sum (or difference) of two other
numbers equals the sum (or difference) of the products
of the number and each other number
[e.g., 5 · (3 + 7) = (5 · 3) + (5 · 7), or
Vocabulary


Levels 1-2
Linguistic Complexity
The commutative property for addition states that
changing the order of the addends does not change the
sum (e.g., 5 + 4 = 4 + 5).
14 x 0 = 0 is an example of the _____ property.
(14 x 0 = 0 is an example of the zero property.)
Language Forms and Conventions

Vocabulary/WIDA Standards
Commutative property, Associative property,
Distributive property, Identity property, Inverse
property, Zero property, additive, multiplicative.
opposites, reciprocal
Zero Property
Additive
Multiplicative
Levels 3-5
What does the word “inverse” mean? (It means use the
opposite or reciprocal of a number.)
Levels 1-2
Levels 3-5
Inverse
Reciprocal
Commutative Property
5 · (3 – 7) = (5 · 3) – (5 · 7)].

Identity elements are numbers that combine with other numbers without changing the other numbers. The additive identity is zero (0). The multiplicative identity is one (1). There are no identity elements for
subtraction and division.

The additive identity property states that the sum of any real number and zero is equal to the given real number (e.g., 5 + 0 = 5).

The multiplicative identity property states that the product of any real number and one is equal to the given real number (e.g., 8 · 1 = 8).

1
Inverses are numbers that combine with other numbers and result in identity elements [e.g., 5 + (–5) = 0; 5 · 5 = 1].

The additive inverse property states that the sum of a number and its additive inverse always equals zero [e.g., 5 + (–5) = 0].

1
The multiplicative inverse property states that the product of a number and its multiplicative inverse (or reciprocal) always equals one (e.g., 4 · 4 = 1).

Zero has no multiplicative inverse.

The multiplicative property of zero states that the product of any real number and zero is zero.

Division by zero is not a possible arithmetic operation. Division by zero is undefined.
Notes:
17
Math 7 Pacing Guide 2015-2016
Walker-Grant Middle School, Fredericksburg City School
18