Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
6th Grade
Standard 1 – Number Sense
Students compare and order positive and negative integers*, decimals,
fractions, and mixed numbers. They
find multiples* and factors*.
6.1.1 Understand and apply the basic concept of negative numbers (e.g., on a
number line, in counting, in
temperature, in “owing”).
Example: The temperature this morning was -6° and now it is 3°. How much has
the temperature
risen? Explain your answer.
6.1.2 Interpret the absolute value of a number are the distance from zero on
a number line, and find the
absolute value of real numbers.
Example: Use a number line to explain the absolute values of -3 and of 7.
6.1.3 Compare and represent on a number line positive and negative integers,
fractions, decimals (to
hundredths), and mixed numbers.
Example: Find the positions on a number line of 3.56, -2.5, 1 6
5 , and -4.
6.1.4 Convert between any two representations of numbers (fractions,
decimals, and percents) without the
use of a calculator.
Example: Write 8
5 as a decimal and as a percent.
6.1.5 Recognize decimal equivalents for commonly used fractions without the
use of a calculator.
Example: Know that 3
1 = 0.333 …, ½ = 0.5, 5
2 = 0.4, etc.
6.1.6 Use models to represent ratios.
Example: Divide 27 pencils to represent the ratio 4:5.
6.1.7 Find the least common multiple* and the greatest common factor* of
whole numbers. Use them to
solve problems with fractions (e.g., to find a common denominator to add two
fractions or to find the
reduced form for a fraction).
Example: Find the smallest number that both 12 and 18 divide into. How does
this help you add the
fractions 12
5 and 18
7 ?
*positive and negative integers: …, -3, -2, -1, 0, 1, 2, 3, …
*multiples: e.g., multiples of 7 are 7, 14, 21, 28, etc.
*factors: e.g., factors of 12 are 1, 2, 3, 4, 6, 12
*least common multiple: e.g., least common multiple of 4 and 6 is 12
*greatest common factor: e.g., greatest common factor of 18 and 42 is 6
Sixth Grade
Mathematics
Page 1 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 2 – Computation
Students solve problems involving addition, subtraction, multiplication, and
division of integers. They solve
problems involving fractions, decimals, ratios, proportions, and percentages.
6.2.1 Add and subtract positive and negative integers.
Example: 17 ÷ -4 = ?, -8 - 5 = ?
6.2.2 Multiply and divide positive and negative integers.
Example: Continue the pattern: 3 x 2 = ?, 2 x 2 = ?, 1 x 2 = ?, 0 x 2 = ?, -
1 x 2 = ?, -2 x 2 = ?, etc.
6.2.3 Multiply and divide decimals.
Example: 3.265 x 0.96 = ?, 56.79 ÷ 2.4 = ?
6.2.4 Explain how to multiply and divide positive fractions and perform the
calculations.
Example: Explain why 8
5 ÷ 16
15 = 8
5 x 15
16 = 3
2 .
6.2.5 Solve problems involving addition, subtraction, multiplication, and
division of positive fractions and
explain why a particular operation was used for a given situation.
Example: you want to place a towel bar 9 ¾ inches long in the center of a
door 27 ½ inches wide. How
far from each edge should you place the bar? Explain your method.
6.2.6 Interpret and use ratios to show the relative sizes of two quantities.
Use the notations: a/b, a to b, a:b.
Example: A car moving at a constant speed travels 130 miles in 2 hours.
Write the ratio of distance to
time and use it to find how far the car will travel in 5 hours.
6.2.7 Understand proportions and use them to solve problems.
Example: Sam made 8 out of 24 free throws. Use a proportion to show how many
free throws Sam
would probably make out of 60 attempts.
6.2.8 Calculate given percentages of quantities and solve problems involving
discounts at sales, interest
earned, and tips.
Example: In a sale, everything is reduced by 20%. Find the sale price of a
shirt whose pre-sale price
was $30.
6.2.9 Use estimation to decide whether answers are reasonable to decimal
problems.
Example: your friends says that 56.79 ÷ 2.4 = 2.36625. Without solving,
explain why you thing the
answer is wrong.
6.2.10 Use mental arithmetic to add or subtract simple fractions and
decimals.
Example: Subtract 6
1 from ½ without using pencil and paper.
Sixth Grade
Mathematics
Page 2 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 3 – Algebra and Functions
Students write verbal expressions and sentences as algebraic expressions and
equations. They evaluate
algebraic expressions, solve simple linear equations, and graph and
interpret their results. They investigate
geometric relationships and describe them algebraically.
6.3.1 Write and solve one-step linear equations and inequalities in one
variable and check the answers.
Example: The area of a rectangle is 143 cm² and the length is 11 cm. Write
an equation to find the
width of the rectangle and use it to solve the problem. Describe how you
will check to be sure that your
answer is correct.
6.3.2 Write and use formulas with up to three variables to solve problems.
Example: You have P dollars in a bank that gives r% simple interest per
year. Write a formula for the
amount of interest you will receive in one year. Use the formula to find the
amount of interest on $80 at
6% per year.
6.3.3 Interpret and evaluate mathematical expressions that use grouping
symbols such as parentheses.
Example: Find the values of 10 – (7 – 3) and of (10 – 7) – 3.
6.3.4 Use parentheses to indicate which operation to perform first when
writing expressions containing more
than two terms and different operations.
Example: Write in symbols: add 19 and 34 and double the result.
6.3.5 Use variables in expressions describing geometric quantities.
Example: Let l, w, and P be the length, width, and perimeter of a rectangle.
Write a formula for the
perimeter in terms of the length and width.
6.3.6 Apply the correct order of operations and the properties of real
numbers (e.g., identity, inverse,
commutative*, associative*, and distributive* properties) to evaluate
numerical expressions. Justify
each step in the process.
Example: Simplify 3(4 - 1) + 2. Explain your method.
6.3.7 Identify and graph ordered pairs in the four quadrants of the
coordinate plane.
Example: Plot the points (3, -1), (-6, 2) and (9, -3). What do you notice?
6.3.8 Solve the problems involving linear functions with integer* values.
Write the equation and graph the
resulting ordered pairs of integers on a grid.
Example: A plant is 3 cm high the first time you measure it (on Day 0). Each
day after that the plant
grows by 2 cm. Write and equation connecting the height and the number of
the day and draw its
graph.
Sixth Grade
Mathematics
Page 3 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
6.3.9 Investigate how a change in one variable relates to a change in a
second variable.
Example: In the last example, what do you notice about the shape of the
graph?
*commutative: the order when adding or multiplying numbers makes no
difference (e.g., 5 + 3 = 3 + 5), but
note that this is not true for subtraction or division
*associative: the grouping when adding or multiplying numbers make s no
difference (e.g., in 5 + 3 + 2, adding
5 and 3 and then adding 2 is the same as 5 added to 3 + 2), but note that
this is not true fro subtraction and
division
*distributive: e.g., 3(5 + 2) = 3 X 5 + 3 x 2
*integers: …, -3, -2, -1, 0, 1, 2, 3 …
Sixth Grade
Mathematics
Page 4 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 4 – Geometry
Students identify, describe, and classify the properties of plane and solid
geometric shapes and the
relationships between them.
6.4.1 Identify and draw vertical*, adjacent*, complementary, and
supplementary* angles and describe these
angle relationships.
Example: Draw two parallel lines with another line across them. Identify all
pairs of supplementary
angles.
6.4.2 Use the properties of complementary, supplementary, and vertical
angles to solve problems involving
an unknown angle. Justify solutions.
Example: Find the size of the supplement to an angle that measures 122°.
Explain how you obtain
your answer.
6.4.3 Draw quadrilaterals* and triangles from given information about them.
Example: Draw a quadrilateral with equal sides but no right angles.
6.4.4 Understand that the sum of the interior angles of any triangle is 180°
and that the sum of the interior
angles of any quadrilateral is 360°. Use this information to solve problems.
Example: find the size of the third angle of a triangle with the angles of
73° and 49°.
6.4.5 Identify and draw two-dimensional shapes that are similar*.
Example: Draw a rectangle similar to a given rectangle, but twice the size.
6.4.6 Draw the translation (slide) and reflection (flip) of shapes.
Example: Draw a square and then slide it 3 inches horizontally across your
page. Draw the new
square in a different color.
6.4.7 Visualize and draw two-dimensional views of three-dimensional objects
made from rectangular solids.
Example: Draw a picture of an arrangement of rectangular blocks from the
top, front, and right-hand
side.
*vertical angle: angles 1 and 3, or 2 and 4
*adjacent angles: angles 1 and 2 or 2 and 3, etc.
*complementary angles: two angles whose sum is 90°
*supplementary angles: two angles who sum is 180° (angels 1 and 2)
*quadrilateral: a two-dimensional figure with four sides
1
4
3
2
*similar: figures that have the same shape but may not have the same size
Sixth Grade
Mathematics
Page 5 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 5 – Measurement
Students deepen their understanding of the measurement of plane and solid
shapes and use this
understanding to solve problems. They calculate with temperature and money,
and choose appropriate units
of measure in other areas.
6.5.1 Select and apply appropriate standard units and tools to measure
length, area, volume, weight, time,
temperature, and the size of angles.
Example: A triangular sheet of metal is about 1 foot across. Describe the
units and tools you would
use to measure its weight, its angles, and the length of its sides.
6.5.2 Understand and use larger units for measuring length by comparing
miles to yards and kilometers to
meters.
Example: How many meters are in a kilometer?
6.5.3 Understand and use larger units for measuring area by comparing acres
and square miles to square
yards and square kilometers to square meters.
Example: How many square meters are in a square kilometer?
6.5.4 Understand the concept of the constant π as the ratio of the
circumference to the diameter of a circle.
Develop and use the formulas for the circumference and area of a circle.
Example: Measure the diameter and circumference of several circular objects.
(Use string to find the
circumference.) With a calculator, divide each circumference by its
diameter. What do you notice
about the results?
6.5.5 Know common estimates of π (3.14, 7
22 ) and use these values to estimate and calculate the
circumference and the area of circles. Compare with actual measurements.
Example: Find the area of a circle of radius 15 cm.
6.5.6 Understand the concept of significant figures and round answers to an
appropriate number of
significant figures.
Example: You measure the diameter of a circle as 2.47 m and use the
approximation 3.14 for π to
calculate the circumference. Is it reasonable to give 7.7558 m as your
answer? Why or why not?
6.5.7 Construct a cube and rectangular box from two-dimensional patterns and
use these patterns to
compute the surface area of these objects.
Example: Find the total surface area of a shoe box with length 30 cm, width
15 cm, and height 10 cm.
6.5.8 Use strategies to find the surface area and volume of right prisms*
and cylinders using appropriate
units.
Example: Find the volume of a cylindrical can 15 cm high and with a diameter
of 8 cm.
6.5.9 Use a formula to convert temperatures between Celsius and Fahrenheit.
Example: What is the Celsius equivalent of 100°F? Explain your method.
Sixth Grade
Mathematics
Page 6 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
6.5.10 Add, subtract, multiply, and divide with money in decimal notation.
Example: Share $7.25 among five people.
*right prism: a three-dimensional shape with two congruent ends
that are polygons and all other faces are rectangles
Sixth Grade
Mathematics
Page 7 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 6 – Data Analysis and Probability
Students compute and analyze statistical measures for data sets. They
determine theoretical and
experimental probabilities and use them to make predictions about events.
6.6.1 Organize and display single-variable data in appropriate graphs and
stem-and-leaf plots*, and explain
which types of graphs are appropriate for various data sets.
Example: This stem-and-leaf diagram show a set of test scores for your class:
Stem Leaf
6 2 3 7
7 1 5 5 6 8 9
8 0 1 1 2 3 3 5 7 8 8
9 1 2 2 3 3 4
Find your score of 85 in this diagram. Are you close to the top or the
bottom of the class on this test?
6.6.2 Make frequency tables for numerical data, grouping the data in
different ways to investigate how
different groupings describe the data. Understand and find relative and
cumulative frequency for a data
set. Use histograms of the data and of the relative frequency distribution,
and a broken line graph for
cumulative frequency, to interpret the data.
Example: A bag contains pens in three colors. Nine students each draw a pen
from the bag without
looking, then record the results in the frequency table shown. Complete the
column showing relative
frequency.
Relative
Color Frequency Frequency
Red 2 2/9
Green 4
Purple 3
6.6.3 Compare the mean*, median*, and mode* for a set of data and explain
which measure is most
appropriate in a given context.
Example: Twenty students were given a science test and the mean, median and
mode were as
follows:
mean = 8.5, median = 9, mode = 10.
What does the difference between the mean and the mode suggest about the
twenty quiz scores?
Sixth Grade
Mathematics
Page 8 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
6.6.4 Show all possible outcomes for compound events in an organized way and
find the theoretical
probability of each outcome.
Example: A box contains four cards with the numbers 1 through 4 written on
them. Show a list of all
the possible outcomes if you draw two cards from the box without looking.
What is the theoretical
probability that you will draw the numbers one and two? Explain your answer.
6.6.5 Use data to estimate the probability of future events.
Example: Teams A and B have played each other 3 times this season and Team A
has won twice.
When they play again, what is the probability of Team B winning? How
accurate do you think this
estimate is?
6.6.6 Understand and represent probabilities as ratios, measures of relative
frequency, decimals between 00
and 1, and percentages between 0 and 100 and verify that the probabilities
computed are reasonable.
Example: the weather forecast says that the chance of rain today is 30%.
Should you carry an
umbrella? Explain your answer.
*stem-and-leaf plot: see diagram in the first example
*mean: the average obtained by adding the values and dividing by the number
of values
*median: the value that divides a set of data (written in order of size)
into two equal parts
*mode: the most common value in a set of data
Sixth Grade
Mathematics
Page 9 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 7 – Problem Solving
Students make decisions about how to approach problems and communicate their
ideas.
6.7.1 Analyze problems by identifying relationships, telling relevant from
irrelevant information, identifying
missing information, sequencing and prioritizing information, and observing
patterns.
Example: Solve the problem: “Develop a method for finding all the prime
numbers up to 100.” Notice
that any numbers that 4, 6, 8, … divide into also divide exactly by 2, and
so you do not need to test 4,
6, 8, …
6.7.2 Make and justify mathematical conjectures based on a general
description of a mathematical question
or problem.
Example: In the first example, decide that you need to test only the prime
numbers as divisors, and
explain it in the same way as for 4, 6, 8, ….
6.7.3 Decide when and how to break a problem into simpler parts.
Example: In the first example, decide to find first those numbers not
divisible by 2.
Students use strategies, skills, and concepts in finding and communicating
solutions to problems.
6.7.4 Apply strategies and results from simpler problems to solve more
complex problems.
Example: In the first example, begin by finding all the prime numbers up to
10.
6.7.5 Express solutions clearly and logically by using the appropriate
mathematical terms and notation.
Support solutions with evidence in both verbal and symbolic work.
Example: In the first example, use a hundreds chart to cross off all
multiples of 2 (except 2), then all
multiples of 3 (except 3), then all multiples of 5 (except 5), etc. Explain
why you are doing this.
6.7.6 Recognize the relative advantages of exact and approximate solutions
to problems and give answers to
a specified degree of accuracy.
Example: Calculate the perimeter of a rectangular field that needs to be
fenced. How accurate should
you be: to the nearest kilometer, meter, centimeter, or millimeter? Explain
your answer.
6.7.7 Select and apply appropriate methods for estimating results of
rational-number computations.
Example: Measure the length and height of the walls of a room to find the
total area. Estimate an
answer by imagining meter squares covering the walls.
6.7.8 Use graphing to estimate solutions and check the estimates with
analytic approaches.
Example: use a graphing calculator to estimate the coordinates of the point
where the straight line y =
8x - 3 crosses the x-axis. Confirm your answer by checking it in the
equation.
6.7.9 Make precise calculations and check the validity of the results in the
context of the problem.
Example: In the first example, check whether some of the numbers not crossed
out are in fact primes.
Students determine when a solution is complete and reasonable and move
beyond a particular problem by
generalizing to other situations.
6.7.10 Decide whether a solution is reasonable in the context of the
original situation.
Example: In the first example, decide whether your method was a good one –
did it find all the prime
numbers efficiently?
Sixth Grade
Mathematics
Page 10 of 11
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Sixth Grade
Mathematics
Page 11 of 11
6.7.11 Note the method of finding the solution and show a conceptual
understanding of the method by solving
similar problems
Example: Use a hundreds chart to find all the numbers that are multiples of
both 2 and 3.
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
7th Grade
Standard 1 – Number Sense
Students understand and use scientific notation* and square roots. They
convert between fractions and
decimals.
7.1.1 Read, write, compare and solve problems using whole numbers in
scientific notation.
Example: Write 300,000 in scientific notation.
7.1.2 Compare and order rational* and common irrational* numbers and place
them on a number line.
Example: Place in order: -2, 8
5 , -2.45, 0.9, π, -1 ¾.
7.1.3 Identify rational and common irrational numbers from a list.
Example: Name all the irrational numbers in the list: =2, 8
5 , -2.45, 0.9, π, -1 ¾.
7.1.4 Understand and compute whole number power of whole numbers.
Example: 35 = 3 X 3 X 3 X 3 X 3 = ?
7.1.5 Find the prime factorization* of whole numbers and write the results
using exponents.
Example: 24 = 2 x 2 x 2 x 3 = 2³ X 3.
7.1.6 Understand and apply the concept of square root.
Example: Explain how you can find the length of the hypotenuse of a right
triangle with legs that
measure 5 cm and 12 cm.
7.1.7 Convert terminating decimals* into reduced fractions.
Example: Write 0.95 as a fraction.
*scientific notation: a shorthand way of writing numbers using power of ten
(e.g., 300,000 = 3 x 105)
*rational number: any number that can be written as a ratio of two integers*
(e.g., ½, 8
5 , 9
23 )
*integers: …, -3, -2, -1, 0, 1, 2, 3 …
*irrational number: any number that cannot be written as a ratio of two
integers (e.g., π, 3 , 7π)
*prime factors: e.g., prime factors of 12 are 2 and 3, the two prime numbers
that divide 12
*terminating decimals: decimals that do not continue indefinitely (e.g.,
0.362, 34.1857)
Seventh Grade
Mathematics
Page 1 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 2 – Computation
Students solve problems involving integers*, fractions, decimals, ratios,
and percentages.
7.2.1 Solve addition, subtraction, multiplication, and division problem that
use integers, fractions, decimals,
and combinations of the four operations.
Example: the temperature one day is 5°. It then falls by 3° each day for 4
days and, after that, rises by
2° each day for 3 days. What is the temperature on the last day? Explain
your method.
7.2.2 Calculate the percentage increase and decrease of a quantity.
Example: The population of a country was 36 million in 1990 and it rose to
41.4 million during the
1990s. What was the percentage increase in the population?
7.2.3 Solve problems that involve discounts, markups, and commissions.
Example: A merchant buys CDs for $11 wholesale and marks up the price by
35%. What is the retail
price?
7.2.4 Use estimation to decide whether answers are reasonable in problems
involving fractions and
decimals.
Example: Your friend says that 3 8
3 x 2 9
2 = 10. Without solving, explain why you think the answer is
wrong.
7.2.5 Use mental arithmetic to compute with simple fractions, decimals, and
powers.
Example: Find 34 without using pencil and paper.
*integers: …, -3, -2, -1, 0, 1, 2, 3, …
Seventh Grade
Mathematics
Page 2 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 3 – Algebra and Functions
Students express quantitative relationships using algebraic terminology,
expressions, equations, inequalities,
and graphs.
7.3.1 Use variables and appropriate operations to write an expression, a
formula, an equation, or an
inequality that represents a verbal description.
Example: Write in symbols the inequality: 5 less than twice the number is
greater than 42.
7.3.2 Write and solve two-step linear equations and inequalities in one
variable and check the answers.
Example: Solve the equation 4x – 7 = 12 and check your answer in the
original equation.
7.3.3 Use correct algebraic terminology such as variable, equation, term,
coefficient*, inequality, expression,
and constant.
Example: Name the variable, terms, and coefficient in the equation: 7x + 4 =
67.
7.3.4 Evaluate numerical expressions and simplify algebraic expressions by
applying the correct order of
operations and the properties of rational numbers* (e.g., identify, inverse,
commutative*, associative*,
distributive*). Justify each step in the process.
Example: Simplify 3(4x + 5x – 1) + 2(x+3) by removing the parentheses and
rearranging. Explain each
step you take.
7.3.5 Solve an equation or formula with two variables for a particular
variable.
Example: Solve the formula C = 2πr for r.
7.3.6 Define slope as vertical change per unit of horizontal change and
recognize that a straight line has
constant slope or rate of change.
Example: Examine a table of values and make a conjecture about whether the
table represents a
linear function.
7.3.7 Find the slope of a lien from its graph.
Example: Draw the graph of y = 2x = 1. Choose two points on the graph and
divide the change in yvalue
by the change in x-value. Repeat this for other pairs of points on the
graph. What do you notice?
7.3.8 Draw the graph of a line given the slope and one point on the line, or
two points on the line.
Example: Draw the graph of the equation with slope of 3 and passing through
the point with
coordinates (0, -2).
7.3.9 Identify functions as linear or nonlinear and examine their
characteristics in tables, graphs, and
equations.
Example: A plant is growing taller according to the formula H = 2d + 3,
where H is the height after d
days. Draw the graph of this function and explain what the point where it
meets the vertical axis
represents. Is this graph linear or nonlinear?
Seventh Grade
Mathematics
Page 3 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
7.3.10 Identify and describe situations with constant or varying rates of
change and know that a constant rate
of change describes a linear function.
Example: In the last example, how will the graph be different if the plant’s
speed of growth changes?
*coefficient: e.g., 7 is the coefficient in 7x
*rational number: any number that can be written as a ratio of two integers*
(e.g., 2
1 , 6
5 , 9
23 )
*integers: …, -3, -2, -1, 0, 1, 2, 3, …
*commutative property: the order when adding or multiplying numbers make no
difference (e.g., 5 + 3 = 3 + 5),
but note that this is not true for subtraction and division
*associative: the grouping when adding or multiplying numbers makes no
difference (e.g., in 5 + 3 +2, adding 5
and 3 and then adding 2 is the same as 5 added to 3 + 2), but note that this
is not true for subtraction or
division
*distributive: e.g., 3(5 + 2) = 3(5) + 3(2)
Seventh Grade
Mathematics
Page 4 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 4 – Geometry
Students deepen their understanding of plane and solid geometric shapes by
constructing shapes that meet
given conditions and by identifying attributes of shapes.
7.4.1 Understand coordinate graphs and use them to plot simple shapes, find
lengths and areas related to
the shapes and find images under translations (slides), rotations (turns),
and reflections (flips).
Example: Draw the triangle with vertices (0,0), (3,0), and (0,4). Find the
lengths of the sides and the
area of the triangle. Translate (slide) the triangle 2 units to the right.
What are the coordinates of the
triangle?
7.4.2 Understand that transformations such as slides, turns, and flips
preserve the length of segments, and
that figures resulting from slides, turns, and flips are congruent* to the
original figures.
Example: In the last example, find the lengths of the sides and the area of
the new triangle.
7.4.3 Know and understand the Pythagorean Theorem and use it to find the
length of the missing side of a
right triangle and the lengths of other line segments. Use direct
measurement to test conjectures about
triangles.
Example: Use the length and width of your classroom to calculate the
distance across the room
diagonally. Check by measuring.
7.4.4 Construct two-dimensional patterns (nets) for three-dimensional
objects, such as right prisms*,
pyramids, cylinders, and cones.
Example: Draw a rectangle and two circles that will fit together to make a
cylinder.
*congruent: same shape and size
*right prism: a three dimensional shape with two congruent
Ends that are polygons and all other sides are rectangles
Seventh Grade
Mathematics
Page 5 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 5 – Measurement
Students compare units of measure and use similarity* to solve problems.
They compute the perimeter, area,
and volume of common geometric objects and use the results to find measures
of less regular objects.
7.5.1 Compare lengths, areas, volumes, weights, capacities, times, and
temperatures within measurement
systems.
Example: The area of the school field is 3 acres. How many square yards is
that? Explain your
method.
7.5.2 Use experimentation and modeling to visualize similarity problems.
Solve problems using similarity.
Example: At a certain time, the shadow of your school building is 36 feet
long. At the same time, the
shadow of a yardstick held vertically is 4 feet long. How high is the school
building?
7.5.3 Read and create drawings made to scale, construct scale models, and
solve problems related to scale.
Example: On a plan of your school, your classroom is 5 cm long and 3 cm
wide. The actual classroom
is 10 m long. How wide is it? Explain your answer.
7.5.4 Use formulas for finding the perimeter and area of basic two-
dimensional shapes and the surface area
and volume of basic three-dimensional shapes, including rectangles,
parallelograms*, trapezoids*,
triangles, circles, right prisms*, and cylinders.
Example: Find the surface area of a cylindrical can 15 cm high and with a
diameter of 8 cm.
7.5.5 Estimate and compute the area of more complex irregular two-
dimensional shapes by dividing them
into more basic shapes.
Example: A room to be carpeted is a rectangle 5 m by 4 m. A semicircular
fireplace of diameter 1.5 m
takes up some of the floor space. Find the area to be carpeted.
7.5.6 Use objects and geometry modeling tools to compute the surface area of
the faces and the volume of a
three-dimensional object built from rectangular solids.
Example: Build a model of an apartment building with blocks. Find its volume
and total surface area.
*similarity: figures that have the same shape but may not have the same size
*parallelogram: a four-sided figure with both pairs of opposite sides
parallel
*trapezoid: a four-sided figure with one pair of opposite sides parallel
*right prism: a three-dimensional shape with two congruent ends that
are polygons and all other sides are rectangles.
Seventh Grade
Mathematics
Page 6 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 6 – Data Analysis and Probability
Students collect, organize, and represent data sets and identify
relationships among variables within a data
set. They determine probabilities and use them to make predictions about
events.
7.6.1 Analyze, interpret, and display data in appropriate bar, line, and
circle graphs and stem-and-leaf plots*,
and justify the choice of display.
Example: You survey the students in your school to find which of three
designs for a magazine cover
they prefer. To display the results, which would be more appropriate: a bar
chart or a circle graph?
Explain your answer.
7.6.2 Make predictions from statistical data.
Example: Record the temperature and weather conditions (sunny, cloudy, or
rainy) at 1 p.m. each day
for two weeks. In the third week, use your results to predict the
temperature from the weather
conditions.
7.6.3 Describe how additional data, particularly outliers, added to a data
set may affect the mean*, median*,
and mode*.
Example: You measure the heights of the students in your grade on a day when
the basketball team is
playing an away game. Later you measure the players on the team and include
them in your data.
What kind of effect will including the team have on the mean, median, and
mode? Explain your
answer.
7.6.4 Analyze data displays, including ways that they can be misleading.
Analyze ways in which the wording
of questions can influence survey results.
Example: On a bar graph of a company’s sales, it appears that sales have
more than doubled since
last year. Then you notice that the vertical axis starts at $5 million and
can see that sales have in fact
increased from $5.5 million to $6.2 million.
7.6.5 Know that if P is the probability of an event occurring, then 1 – P is
the probability of that event not
occurring.
Example: The weather forecast says that the probability of rain today is
0.3. What is the probability
that it won’t rain?
7.6.6 Understand that the probability of either one or the other of two
disjoint events* occurring is the sum of
the two individual probabilities.
Example: Find the probability of rolling 9 with two number cubes. Also find
the probability of rolling 10.
What is the probability of rolling 9 or 10?
7.6.7 Find the number of possible arrangements of several objects using a
tree diagram.
Example: A state’s license plates contain 6 digits and one letter. How many
different plates can be
made if the letter must always be in the third position and the first digit
cannot be zero?
Seventh Grade
Mathematics
Page 7 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
*stem-and-leaf plot: e.g., this one shows 62, 63, 67, 71, 75, 75, 76, etc.
Stem Leaf
6 2 3 7
7 1 5 5 6 8 9
8 0 1 1 2 3 5 5 7 8 8
9 1 2 2 3 3 4
*mean: the average obtained by adding the values and dividing by the number
of values
*median: the value that divides a set of data written in order of size into
two equal parts
*mode: the most common value
*disjoint events: events that cannot happen at the same time
Seventh Grade
Mathematics
Page 8 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 7 – Problem Solving
Students make decisions about how to approach problems and communicate their
ideas.
7.7.1 Analyze problems by identifying relationships, telling relevant from
irrelevant information, identifying
missing information, sequencing and prioritizing information, and observing
patterns.
Example: Solve the problem: “The first three triangular numbers are shown in
the diagram below.
Find an expression to calculate the nth triangular number.”
● ● ●
● ● ● ●
● ● ●
1 3 6
Decide to look for patterns.
7.7.2 Make and justify mathematical conjectures based on a general
description of a mathematical question
or problem.
Example: In the first example, notice that three dots make and equilateral
triangle for the number 3 and
six dots make the next equilateral triangle.
7.7.3 Decide when and how to divide a problem into simpler parts.
Example: In the first example, decide to make a diagram for the fourth and
fifth triangular numbers.
Students use strategies, skills, and concepts in finding and communicating
solutions to problems.
7.7.4 Apply strategies and results from simpler problems to solve more
complex problems.
Example: In the first example, list the differences between any two
triangular numbers.
7.7.5 Make and test conjectures by using inductive reasoning.
Example: In the first example, predict the difference between the fifth and
sixth numbers and use this
to predict the sixth triangular number. Make a diagram to test your
conjecture.
7.7.6 Express solutions clearly and logically by using the appropriate
mathematical terms and notation.
Support solutions with evidence in both verbal and symbolic work.
Example: In the first example, use words, numbers, and tables to summarize
your work with triangular
numbers.
7.7.7 Recognize the relative advantages of exact and approximate solutions
to problems and give answers to
a specified degree of accuracy.
Example: Calculate the amount of aluminum needed to make a can with diameter
10 cm that is 15 cm
high and 1 mm thick. Take π as 3.14 and give your answer to appropriate
accuracy.
7.7.8 Select and apply appropriate methods for estimating results of
rational-number computations.
Example: Measure the dimensions of a swimming pool to find its volume.
Estimate an answer by
working with an average depth.
7.7.9 Use graphing to estimate solutions and check the estimates with
analytic approaches.
Example: Use a graphing calculator to find the crossing point of the
straight lines y = 2x + 3 and
Seventh Grade
Mathematics
Page 9 of 10
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Seventh Grade
Mathematics
Page 10 of 10
x + y = 10. Confirm your answer by checking it in the equations.
7.7.10 Make precise calculations and check the validity of the results in
the context of the problem.
Example: In the first example, check that your later results fit with your
earlier ones. If they do not,
repeat the calculations to make sure.
Students determine when a solution is complete and reasonable and move
beyond a particular problem by
generalizing to other situations.
7.7.11 Decide whether a solution is reasonable in the context of the
original situation.
Example: In the first example, calculate the 10th triangular number and draw
the triangle of dots that
goes with it.
7.7.12 Note the method of finding the solution and show a conceptual
understanding of the method by solving
similar problems.
Example: Use your method from the first example to investigate pentagonal
numbers.
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
7th Grade
Standard 1 – Number Sense
Students understand and use scientific notation* and square roots. They
convert between fractions and
decimals.
7.1.1 Read, write, compare and solve problems using whole numbers in
scientific notation.
Example: Write 300,000 in scientific notation.
7.1.2 Compare and order rational* and common irrational* numbers and place
them on a number line.
Example: Place in order: -2, 8
5 , -2.45, 0.9, π, -1 ¾.
7.1.3 Identify rational and common irrational numbers from a list.
Example: Name all the irrational numbers in the list: =2, 8
5 , -2.45, 0.9, π, -1 ¾.
7.1.4 Understand and compute whole number power of whole numbers.
Example: 35 = 3 X 3 X 3 X 3 X 3 = ?
7.1.5 Find the prime factorization* of whole numbers and write the results
using exponents.
Example: 24 = 2 x 2 x 2 x 3 = 2³ X 3.
7.1.6 Understand and apply the concept of square root.
Example: Explain how you can find the length of the hypotenuse of a right
triangle with legs that
measure 5 cm and 12 cm.
7.1.7 Convert terminating decimals* into reduced fractions.
Example: Write 0.95 as a fraction.
*scientific notation: a shorthand way of writing numbers using power of ten
(e.g., 300,000 = 3 x 105)
*rational number: any number that can be written as a ratio of two integers*
(e.g., ½, 8
5 , 9
23 )
*integers: …, -3, -2, -1, 0, 1, 2, 3 …
*irrational number: any number that cannot be written as a ratio of two
integers (e.g., π, 3 , 7π)
*prime factors: e.g., prime factors of 12 are 2 and 3, the two prime numbers
that divide 12
*terminating decimals: decimals that do not continue indefinitely (e.g.,
0.362, 34.1857)
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
ALGEBRA 1
Standard 1 – Operations with Real Numbers
Students simplify and compare expressions. They use rational exponents, and
simplify square roots.
A1.1.1 Compare real number expressions.
Example: Compare the sizes of: 36 + 64 and 64 36 + .
A1.1.2 Simplify square roots using factors.
Example: Explain why 48 = 3 4
A1.1.3 Understand and use the distributive, associative, and commutative
properties.
Example: Simplify (6x² - 5x + 1) – 2(x² + 3x – 4) by removing the
parentheses and rearranging.
Explain why you can carry out each step.
A1.1.4 Use the laws of exponents for rational numbers.
Example: Simplify 253/2.
A1.1.5 Use dimensional (unit) analysis to organize conversions and
computations.
Example: Convert 5 miles per hour to feet per second.
Standard 2 – Linear Equations and Inequalities
Students solve linear equations and inequalities in one variable. They solve
word problems that involve linear
equations, inequalities, or formulas.
A1.2.1 Solve linear equations.
Example: Solve the equation 7a + 2 = 5a – 3a + 8.
A1.2.2 Solve equations and formulas for a specified variable.
Example: Solve the equation q = 4p – 11 for p.
A1.2.3 Find solution sets of linear inequalities when possible numbers are
given for a
variable.
Example: Solve the inequality 6x – 3 > 10 for x in the set {0, 1, 2, 3, 4}.
A1.2.4 Solve linear inequalities using properties of order.
Example: Solve the inequality 8x – 7 ≤ 2x + 5, explaining each step in your
solution.
A1.2.5 Solve combined linear inequalities.
Example: Solve the inequalities -7 > 3x + 5 > 11.
A1.2.6 Solve word problems that involve linear equations, formulas, and
inequalities.
Example: You are selling tickets for a play that cost $3 each. You want to
sell at least $50
worth. Write and solve an inequality for the number of tickets you must sell.
Algebra
Mathematics
Page 1 of 8
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 3 – Relations and Functions
Students sketch and interpret graphs representing given situations. They
understand the concept of a function
and analyze the graphs of functions.
A1.3.1 Sketch a reasonable graph for a given relationship.
Example: Sketch a reasonable graph for a person’s height from age 0 to 25.
A1.3.2 Interpret a graph representing a given situation.
Example: Jessica is riding a bicycle. The graph below shows her speed as it
relates to the time
she has spent riding. Describe what might have happened to account for such
a graph.
A1.3.3 Understand the concept of a function, decide if a given relation is a
function, and link equations to
functions.
Example: Use either paper or a spreadsheet to generate a list of values for
x and y in y = x².
Based on your data, make a conjecture about whether or not this relation is
a function. Explain
your reasoning.
A1.3.4 Find the domain and range of a relation.
Example: Based on the list of values from the last example, what do the
domain and range of y
= x² appear to be? How can you decide whether you are correct?
Algebra
Mathematics
Page 2 of 8
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 4 – Graphing Linear Equations and Inequalities
Students graph linear equations and inequalities in two variables. They
write equations of lines and find and
use the slope and y-intercept of lines. They use linear equations to model
real data.
A1.4.1 Graph a linear equation.
Example: Draw the graph of the line with slope 3 and y-intercept -2.
A1.4.2 Find the slope, x-intercept and y-intercept of a line given its
graph, its equation, or two points on the
line.
Example: Find the slop and y-intercept of the line 4x + 6y = 12.
A1.4.3 Write the equation of a line in slope-intercept form. Understand how
the slope and y-intercept of the
graph are related to the equation.
Example: Write the equation of the line 4x + 6y = 12 in slope-intercept
form. What is the slope
of this line? Explain your answer.
A1.4.4 Write the equation of a line given appropriate information.
Example: Find an equation of the line through the points (1,4) and (3,10),
then find an equation
of the line through the point (1,4) perpendicular to the first line.
A1.4.5 Write the equation of a line that models a data set and use the
equation (or the graph of the
equation) to make predictions. Describe the slope of the line in terms of
the data, recognizing that
the slope is the rate of change.
Example: As your family is traveling along an interstate, you note the
distance traveled every 5
minutes. A graph of time and distance shows that the relation is
approximately linear. Write the
equation of the line that fits your data. Predict the time for a journey of
50 miles. What does the
slope represent?
A1.4.6 Graph a linear inequality in two variables.
Example: Draw the graph of the inequality 6x + 8y ≥ 24.
Algebra
Mathematics
Page 3 of 8
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 5 – Pairs of Linear Equations and Inequalities
Students solve pairs of linear equations using graphs and using algebra.
They solve pairs of linear inequalities
using graphs. They solve word problems involving pairs of linear equations.
A1.5.1 Use a graph to estimate the solution of a pair of linear equations in
two variables.
Example: Graph the equations 3y – x = 0 and 2x + 4y = 15 to find where the
lines intersect.
A1.5.2 Use a graph to find the solution set of a pair of linear inequalities
in two variables.
Example: Graph the inequalities y ≤ 4 and x + y ≤ 5. Shade the region where
both inequalities
are true.
A1.5.3 Understand and use the substitution method to solve a pair of linear
equations in two variables.
Example: Solve the equations y = 2x and 2x + 3y = 12 by substitution.
A1.5.4 Understand and use the addition or subtraction method to solve a pair
of linear equations in two
variables.
Example: Use subtraction to solve the equations: 3x + 4y = 11,
3x + 2y = 7.
A1.5.5 Understand and use multiplication with the addition or subtraction
method to solve a pair of linear
equations in two variables.
Example: Use multiplication with the subtraction method to solve the
equations: x + 4y = 16, 3x
+ 2y = -3.
A1.5.6 Use pairs of linear equations to solve word problems.
Example: The income a company makes from a certain product can be
represented by the
equation y = 10.5x and the expenses for that product can be represented by
the equation y =
5.25x + 10,000, where x is the amount of the product sold and y is the
number of dollars. How
much of the product must be sold for the company to reach the break-even
point?
Algebra
Mathematics
Page 4 of 8
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 6 – Polynomials
Students add, subtract, multiply, and divide polynomials. They factor
quadratics.
A1.6.1 Add and subtract polynomials.
Example: Simplify (4x² - 7x + 2) = (x² + 4x – 5).
A1.6.2 Multiply and divide monomials.
Example: Simplify a²b5 ÷ ab².
A1.6.3 Find powers and roots of monomials (only when the answer has an
integer exponent).
Example: Find the square root of a²b6.
A1.6.4 Multiply polynomials.
Example: Multiply (n + 2)(4n - 5).
A1.6.5 Divide polynomials by monomials.
Example: Divide 4x³y² + 8xy4 - 6x2y5 by 2xy2.
A1.6.6 Find a common monomial factor in a polynomial.
Example: Factor 36xy2 + 18xy4 – 12x2y2.
A1.6.7 Factor the difference of two squares and other quadratics.
Example: Factor 4x2 - 25 and 2x2 - 7x + 3.
A1.6.8 Understand and describe the relationships among the solutions of an
equation, the zeros of a
function, the x-intercepts of a graph, and the factors of a polynomial
expression.
Example: A graphing calculator can be used to solve 3x2 - 5x - 1 = 0 to the
nearest tenth. Justify
using the x-intercepts of y = 3x2 - 5x - 1 as the solutions of the equation.
Algebra
Mathematics
Page 5 of 8
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Algebra
Mathematics
Page 6 of 8
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Standard 7 - Algebraic Fractions
Students simplify algebraic ratios and solve algebraic proportions.
A1.7.1 Simplify algebraic ratios.
x2 – 16
x2 + 4x
Example: Simplify
A1.7.2 Solve algebraic proportions.
Example: Create a tutorial to be posted to the school's web site to instruct
beginning students in
the steps involved in solving an algebraic proportion. Use
4
5 + x = 7
5 3 + x as an example.
Standard 8 - Quadratic, Cubic, and Radical Equations
Students graph and solve quadratic and radical equations. They graph cubic
equations.
A1.8.1 Graph quadratic, cubic, and radical equations.
Example: Draw the graph of y = x2 - 3x + 2. Using a graphing calculator or a
spreadsheet
(generate a data set), display the graph to check your work.
A1.8.2 Solve quadratic equations by factoring.
Example: Solve the equation x2 - 3x + 2 = 0 by factoring.
A1.8.3 Solve quadratic equations in which a perfect square equals a constant.
Example: Solve the equation (x - 7)2 = 64.
A1.8.4 Complete the square to solve quadratic equations.
Example: Solve the equation x2 - 7x + 9 = 0 by completing the square.
A1.8.5 Derive the quadratic formula by completing the square.
Example: Prove that the equation ax2 + bx + c = 0 has solutions x = a
ac b b
2
4 2 − ± − .
A1.8.6 Solve quadratic equations by using the quadratic formula.
Example: Solve the equation x2 - 7x + 9 = 0.
A1.8.7 Use quadratic equations to solve word problems.
Example: A ball falls so that its distance above the ground can be modeled
by the equation s =
100 – 16t2, where s is the distance above the ground in feet and t is the
time in seconds.
According to this model, at what time does the ball hit the ground?
A1.8.8 Solve equations that contain radical expressions.
Example: Solve the equation 6 + x = x.
A1.8.9 Use graphing technology to find approximate solutions of quadratic
and cubic equations.
Example: Use a graphing calculator to solve 3x2 - 5x - 1 = 0 to the nearest
tenth.
Algebra
Mathematics
Page 7 of 8
Archdiocese of Washington Catholic Schools
Academic Standards
Mathematics
Algebra
Mathematics
Page 8 of 8
Standard 9 - Mathematical Reasoning and Problem Solving
Students use a variety of strategies to solve problems.
A1.9.1 Use a variety of problem solving strategies, such as drawing a
diagram, making a chart, guess-and-
check, solving a simpler problem, writing an equation, and working backwards.
Example: Fran has scored 16, 23, and 30 points in her last three games. How
many points must
she score in the next game so that her four game average does not fall below
20 points?
A1.9.2 Decide whether a solution is reasonable in the context of the
original situation.
Example: John says the answer to the problem in the first example is 10
points. Is his answer
reasonable? Why or why not?
Students develop and evaluate mathematical arguments and proofs.
A1.9.3 Use the properties of the real number system and the order of
operations to justify the steps of
simplifying functions and solving equations.
Example: Given an argument (such as 3x + 7 > 5x + 1, and therefore -2x > -6,
and therefore x >
3), provide a visual presentation of a step-by-step check, highlighting any
errors in the
argument.
A1.9.4 Understand that the logic of equation solving begins with the
assumption that the variable is a
number that satisfies the equation, and that the steps taken when solving
equations create new
equations that have, in most cases, the same solution set as the original.
Understand that similar
logic applies to solving systems of equations simultaneously.
Example: Try "solving" the equations x + 3y = 5 and 5x + 15y = 25
simultaneously, and explain
what went wrong.
A1.9.5 Decide whether a given algebraic statement is true always, sometimes,
or never (statements
involving linear or quadratic expressions, equations, or inequalities).
Example: Is the statement x2 - 5x + 2 = x2 + 5x + 2 true always, sometimes,
or never? Explain
your answer.
A1.9.6 Distinguish between inductive and deductive reasoning, identifying
and providing examples of each.
Example: What type of reasoning are you using when you look for a pattern?
Al.9.7 Identify the hypothesis and conclusion in a logical deduction.
Example: What is the hypothesis and conclusion in this argument: If there is
a numbers such
that 2x+ 1 = 7, then x = 3.
A1.9.8 Use counter
examples to show that statements are false, recognizing that a single
counterexample is sufficient to prove a
general statement false.
Example: Use the demonstration-graphing calculator on an overhead projector
to produce an
example showing that this statement is false: all quadratic equations have
two different
solutions.