ACHIEVING & SURPASSING MATH K-12 AMO THIS YEAR:
EVERY TARGETED MINUTE COUNTS!
“Our energies to redirect the conversations away from test data and more
toward children will not happen overnight. But it will happen. The stars
are aligned for reasonable discussion about public education in Virginia.”
~ Dr. Ben Kiser, the Executive Director of VASS
PREPARED FOR THE COLLABORATIVE LEARNING NETWORK OF THE
VIRGINIA ASSOCIATION OF SCHOOL SUPERINTENDENTS
BY DAN MULLIGAN, FLEXIBLECREATIVITY.COM
OCTOBER 2014
TABLE OF CONTENTS
PREPARING FOR COMPUTER ADAPTIVE ASSESSMENT
3
AUTHENTIC ASSESSMENT: SAMPLE SOL SCRIMMAGE
18
ALIGNMENT OF DOK AND RIGOR/RELEVANCE FRAMEWORK
22
RIGOR: PRODUCTS, ROLES, AND STRUCTURES
28
DOK/QUADRANTS: VERBS AND PRODUCTS
36
DOK/QUADRANTS: QUESTIONS
37
PERFORMANCE ASSESSMENTS: INTRODUCTION
38
PUTTING IT TOGETHER: TYPE OF THINKING + DEPTH OF THINKING
40
PERFORMANCE ASSESSMENT: A DEEPER UNDERSTANDING
45
STUDENT STANDARDS-BASED MATH PRACTICES
49
2
A CHILD CENTERED APPROACH TO THE SOL: PREPARING FOR COMPUTER ADAPTIVE TEST (CAT)
KINDERGARTEN
Name: ______________________________
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – At a Glance
Target for Understanding: Whole Number Concepts and Introduction to Fractions
K.1
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a. Match each member of one set with each member of
another set, using the concept of one-to-one
correspondence to compare the number of members
between sets, where each set contains 10 or fewer
objects
b. Compare and describe two sets of 10 or fewer objects,
using the terms more, fewer, and the same
c. Given a set of objects, construct a second set which has
more, fewer or the same number of objects
K.2
a. Count orally the number of objects in a set containing 15
or fewer concrete objects, using one-to-one
correspondence, and identify the corresponding numeral
b. Identify written numerals from 0 through 15 represented
in random order
c. Select the numeral from a given set of numerals that
corresponds to a set of 15 or fewer concrete objects
d. Write the numerals from 0 through 15
e. Write a numeral that corresponds to a set of 15 or fewer
concrete objects
f. Construct a set of objects that corresponds to a given
numeral, including an empty set
K.3
a. Identify the ordinal positions first through tenth using
ordered sets of ten concrete objects and/or pictures of
such sets presented from
1. left-to-right
2. right-to-left
3. top-to-bottom
4. bottom-to-top
continued
3
K.4
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a. Count forward from 0 to 100
b. Count backward from 10 to 0
c. Recognize the relationships of one more than and one
less than a number using objects (i.e., five and one more
is six; and one less than ten is nine)
d. Group 100 or fewer objects together into sets of fives or
tens and then count them by fives or by tens
e. Investigate and recognize the pattern of counting by fives
to 100, using a variety of tools
f. Investigate and recognize the pattern of counting by tens
to 100, using a variety of tools
K.5
a. Recognize fractions as representing parts of equal
size of a whole
b. Given a region, identify half and/or a fourth of the
region
c. Given a set, identify half and/or a fourth of the set
4
GRADE 1 Target for Understanding
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – At a Glance
Target for Understanding: Place Value and Fraction Concepts
1.1
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a. Count by rote from 0 to 100, using the correct name for
each numeral
b. Use the correct oral counting sequence to tell how many
objects are in a set
c. Write numerals correctly
d. Write each numeral from 0 to 100
e. Read two-digit numbers when shown a numeral, a
Base-10 model of the number, or a pictorial
representation of the number
f. Identify the place value (ones, tens) of each digit in a
two-digit numeral (e.g., The place value of the 2 in the
number 23 is tens. The value of the 2 in the number 23
is 20)
g. Group a collection of objects into sets of tens and
ones. Write the numeral that corresponds to the
total number of objects in a given collection of
objects that have been grouped into sets of tens and
ones
1.2
a. Count by ones, twos, fives, and tens to 100, using
concrete objects, such as counters, connecting cubes,
pennies, nickels, and dimes
b. Demonstrate a one-to-one correspondence when
counting by ones with concrete objects or
representations
c. Skip count orally by twos, fives and tens to 100 starting
at various multiples of 2, 5, or 10
d. Count backward by ones from 30
continued on next page
5
GRADE 1
Name: _________________________________
page 2 of 2
1.3
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a. Represent a whole to show it having two equal parts
1
2
and identify one half ( 2 ), and two halves ( 2 )
b. Represent a whole to show it having three equal parts
1
2
and identify one third ( 3 ), two thirds ( 3 ) and three
3
thirds ( 3 )
c. Represent a whole to show it having four equal parts
1
2
and identify one fourth ( 4 ), two fourths ( 4 ), three
3
4
fourths ( 4 ) and four fourths ( 4 )
d. Identify and model halves, thirds, and fourths of a
whole, using the set model (e.g., connecting cubes and
counters), and region/area models (e.g., pie pieces,
pattern blocks, geoboards, paper folding, and drawings)
e. Name and write fractions represented by drawings or
concrete materials for halves, thirds, and fourths
f. Represent a given fraction using concrete materials,
pictures, and symbols for halves, thirds, and fourths.
For example, write the symbol for one-fourth, and
represent it with concrete materials and pictures
6
GRADE 2
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – At a Glance
Target for Understanding: Place Value, Number Patterns, and Fraction Concepts
2.1
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a. Demonstrate the understanding of the ten-to-one
relationships among ones, tens, and hundreds, using
manipulatives (e.g., beans and cups, Base-10 blocks,
bundles of 10 sticks)
b. Determine the place value of each digit in a three-digit
numeral presented as a pictorial representation (e.g., a
picture of Base-10 blocks) or as a physical
representation (e.g., actual Base-10 blocks)
c. Write numerals, using a Base-10 model or picture
d. Read three-digit numbers when shown a numeral, a
Base-10 model of the number, or a pictorial
representation of the number
e. Identify the place value (ones, tens, hundreds) of each
digit in a three-digit numeral
f. Determine the value of each digit in a three-digit
numeral (e.g., in 352, the 5 represents 5 tens and its
value is 50)
g. Round two-digit numbers to the nearest ten
h. Compare two numbers between 0 and 999
represented pictorially or with concrete objects
(e.g., Base-10 blocks), using the words greater than,
less than or equal to
2.2
a. Count an ordered set of objects, using the ordinal
number words first through twentieth
b. Identify the ordinal positions first through twentieth,
using an ordered set of objects
c. Identify the ordinal positions first through twentieth,
using an ordered set of objects presented in lines or
rows from
1. left to right
2. right to left
3. top to bottom
4. bottom to top
d. Write 1st, 2nd, 3rd, through 20th in numerals
continued on next page
7
GRADE 2
Name: _________________________
page 2 of 2
2.3
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a. Recognize fractions as representing equal-size parts of
a whole
2 2 3
b. Identify the fractional parts of a whole or a set for 2, 3, 4,
2 7 7
, ,
, etc.
6 8 10
c. Identify the fraction names (halves, thirds, fourths,
2 2 3
sixths, eighths, tenths) for the fraction notations 2, 3, 4,
2 7 7
, ,
, etc.
6 8 10
d. Represent fractional parts of a whole for halves,
thirds, fourths, sixths, eighths, tenths using
1. region/area models (e.g., pie pieces, pattern blocks,
geoboards
2. sets (e.g. chips, counters, cubes)
3. measurement models (e.g., fraction strips, rods,
connecting cubes)
1 1 1 1 1
1
d. Compare unit fractions ( 2, 3, 4, 6, 8 and 10) using the
words greater than, less than or equal to and the
symbols ( , , =)
2.4
a. Determine patterns created by counting by twos, fives,
and tens on a hundred chart
b. Skip count by twos, fives, and tens to 100, using
manipulatives, a hundred chart, mental mathematics, a
calculator, and/or paper and pencil
c. Skip count by twos, fives, and tens to 100
d. Count backward by tens from 100
e. Use objects to determine whether a number is odd or
even
8
GRADE 3
Name: _____________________________
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – A Record of Understanding
Target for Understanding: Place Value and Fractions
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a. Investigate/identify place & value for each digit in a six-digit
numeral, using Base-10 manipulatives (e.g., Base-10 blocks)
b. Use the patterns in the place value system to read and
write numbers
c. Read six-digit numerals orally
d. Write six-digit numerals that are stated verbally or
written in words
e. Round a given whole number, 9,999 or less, to the
nearest ten, hundred, and thousand
f. Solve problems, using rounding of numbers, 9,999 or
less, to the nearest ten, hundred, and thousand
g. Determine which of two whole numbers between 0
and 9,999 is greater
h. Determine which of two whole numbers between 0
and 9,999 is less
i. Compare two whole numbers between 0 and 9,999,
using the symbols >, <, or =
j. Use the terms greater than, less than, and equal to
when comparing two whole numbers
3.2
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a. Use the inverse relationships between addition/subtraction
and multiplication/division to solve related basic fact
sentences. For example, 5 + 3 = 8 & 8 – 3 = __; 4 x 3 = 12 &12 ÷ 4 = __
b. Write three related basic fact sentences when given one basic
fact sentence for addition/subtraction and for
multiplication/division. For example, given 3 x 2 = 6, solve the
related facts __ x 3 = 6, 6 ÷ 3 = __, and 6 ÷ __ = 3
3.3
a. Name and write fractions (including mixed numbers)
represented by a model to include halves, thirds,
fourths, eighths, tenths, and twelfths
b. Use concrete materials and pictures to model at least
halves, thirds, fourths, eighths, tenths, and twelfths
c. Compare fractions using the terms greater than, less
than, or equal to & the symbols ( <, >, and =).
Comparisons are made between fractions with both
like & unlike denominators, using models, concrete
materials & pictures
9
GRADE 4
Name: _____________________________
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – A Record of Understanding
Target for Understanding: Place Value, Fractions, and Decimals
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a. Identify and communicate, both orally and in written
form, the placed value for each digit in whole numbers
expressed through the one millions place.
b. Read whole numbers through the one millions place
that are presented in standard format, and select the
matching number in written format.
c. Write whole numbers through the one millions place in
standard format when the numbers are presented
orally or in written format.
d. Identify and use the symbols for greater than, less than,
and equal to.
e. Compare two whole numbers expressed through the
one millions, using symbols >, <, or =.
f. Round whole numbers expressed through the one
millions place to the nearest thousand, ten thousand,
and hundred-thousand place.
g. Identify and communicate, both orally and in written
form, the placed value for each digit in whole numbers
expressed through the one millions place.
h. Read whole numbers through the one millions place
that are presented in standard format, and select the
matching number in written format.
i. Write whole numbers through the one millions place in
standard format when the numbers are presented
orally or in written format.
j. Identify and use the symbols for greater than, less than,
and equal to.
continued on next page
10
GRADE 4
Name: _________________________
page 2 of 3
4.2
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a. Compare and order fractions having denominators of
12 or less, using manipulative models and drawings,
such as region/area models.
b. Compare and order fractions with like denominators by
1 3
comparing number of parts (numerators) (e.g., 5 < 5 ).
c. Compare and order fractions with like numerators and
unlike denominators by comparing the size of the parts
(e.g.,
3
3
< ).
9
5
d. Compare and order fractions having unlike
denominators of 12 or less by comparing the fractions
1
to benchmarks (e.g., 0, 2 or 1) to determine their
relationships to the benchmarks or by finding a
common denominator.
e. Compare and order mixed numbers having
denominators of 12 or less.
f. Use the symbols >, <, and = to compare the numerical
value of fractions and mixed numbers having
denominators of 12 or less.
g. Represent equivalent fractions through twelfths, using
region/area models, set models, and measurement
models.
h. Identify the division statement that represents a
3
fraction (e.g., 5 means the same as 3 divided by 5).
continued on next page
11
GRADE 4
Name: _________________________
page 2 of 3
4.3
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a. Investigate the ten-to-one place value relationship for
decimals through thousandths, using Base-10
manipulatives (e.g., place value mats/charts, decimal
squares, Base-10 blocks, money).
b. Represent and identify decimals expressed through
thousandths, using Base-10 manipulatives, pictorial
representations, and numerical symbols (e.g., relate the
appropriate drawing to 0.05).
c. Identify and communicate, both orally and in written
form, the position and value of a decimal through
thousandths. For example, in 0.385, the 8 is in the
hundredths place and has a value of 0.08.
d. Read and write decimals expressed through
thousandths, using Base-10 manipulatives, drawings,
and numerical symbols.
e. Round decimals to the nearest whole number,
tenth, and hundredth.
f. Compare decimals, using the symbols >, <, =.
g. Order a set of decimals from least to greatest or
greatest to least.
h. Represent fractions for halves, fourths, fifths, and tenths
as decimals through hundredths, using concrete objects
(e.g., demonstrate the relationship between the fraction
1
4 and its decimal equivalent 0.25).
i. Relate fractions to decimals, using concrete objects (e.g.,
10-by-10 grids, meter sticks, number lines, decimal
squares, decimal circles, money [coins]).
j. Write the decimal and fraction equivalent for a
1
1
given model (e.g., 4 = 0.25 or 0.25 = 4 ).
12
GRADE 5
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – At a Glance
Target for Understanding: Factors and Multiples, Fractions, Decimals
5.1
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a. Round decimal numbers to the nearest whole
number, tenth, or hundredth.
5.2
a. Represent fractions (halves, fourths, fifths, eighths,
tenths, and twelfths) in their equivalent decimal form
and vice versa.
b. Recognize and name equivalent relationships between
decimals and fractions with denominators up to 12.
c. Compare and order from least to greatest and greatest
to least a given set of no more than five numbers
written as decimals, fractions, and mixed numbers with
denominators of 12 or less.
5.3
a. Identify prime numbers less than or equal to 100.
b. Identify composite numbers less than or equal to
100.
c. Explain orally and in writing why a number is prime
or composite.
d. Identify which numbers are even or odd.
e. Explain and demonstrate with manipulatives,
pictorial representations, oral language, or written
language why a number is even or odd.
13
GRADE 6
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – At a Glance
Target for Understanding: Relationships among Fractions, Decimals, and Percents
6.1
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a. Describe a relationship within a set by comparing part
of the set to the entire set.
b. Describe a relationship between two sets by comparing
part of one set to a corresponding part of the other set.
c. Describe a relationship between two sets by comparing
all of one set to all of the other set.
d. Describe a relationship within a set by comparing one
part of the set to another part of the same set.
e. Represent a relationship in words that makes a
comparison by using the notations
a
, a:b, and
b
a to
b.
f. Create a relationship in words for a given ratio
expressed symbolically.
6.2
a
b
c
d
e
f
g
h
i
Identify the decimal and percent equivalents for
numbers written in fraction form including
repeating decimals.
Represent fractions, decimals, and percents on a
number line.
Describe orally and in writing the equivalent
relationships among decimals, percents, and
fractions that have denominators that are factors of
100.
Represent, by shading a grid, a fraction, decimal,
and percent.
Represent in fraction, decimal, and percent form a
given shaded region of a grid.
Compare two decimals through thousandths using
manipulatives, pictorial representations, number lines,
and symbols (<, ,, >, =).
Compare two fractions with denominators of 12 or less
using manipulatives, pictorial representations, number
lines, and symbols (<, ,, >, =).
Compare two percents using pictorial
representations and symbols (<, ,, >, =).
Order no more than 3 fractions, decimals, and
percents (decimals through thousandths, fractions
with denominators of 12 or less), in ascending or
descending order.
14
continued
6.3
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a Identify an integer represented by a point on a
number line.
b Represent integers on a number line.
c Order and compare integers using a number line.
d Compare integers, using mathematical symbols (<,
>, =).
e Identify and describe the absolute value of an
integer.
6.4
a Demonstrate multiplication and division of fractions
using multiple representations.
b Model algorithms for multiplying and dividing with
fractions using appropriate representations.
6.5
a Recognize and describe patterns with exponents
that are natural numbers, by using a calculator.
b Recognize and describe patterns of perfect squares
not to exceed 20 2 , by using grid paper, square tiles,
tables, and calculators.
c Recognize powers of ten by examining patterns in a
place value chart: 104 = 10,000, 103 = 1000, 102 =
100, 101 = 10, 10 0 =1.
15
GRADE 7
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – At a Glance
Target for Understanding: Scientific Notation and Square Roots
7.1
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a Recognize powers of 10 with negative exponents by
examining patterns.
b Write a power of 10 with a negative exponent in
fraction and decimal form.
c Write a number greater than 0 in scientific notation.
d Recognize a number greater than 0 in scientific
notation.
e Compare and determine equivalent relationships
between numbers larger than 0 written in scientific
notation.
f Represent a number in fraction, decimal, and
percent forms.
g Compare, order, and determine equivalent
relationships among fractions, decimals, and
percents. Decimals are limited to the thousandths
place, and percents are limited to the tenths place.
Ordering is limited to no more than 4 numbers.
h Order no more than 3 numbers greater than 0
written in scientific notation.
i Determine the square root of a perfect square less
than or equal to 400.
j Demonstrate absolute value using a number line.
k Determine the absolute value of a rational number.
l Show that the distance between two rational
numbers on the number line is the absolute value of
their difference, and apply this principle to solve
practical problems.†
7.2
a Analyze arithmetic and geometric sequences to
discover a variety of patterns.
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.
16
GRADE 8
NUMBER AND NUMBER SENSE
Essential Knowledge Skills and Processes – At a Glance
Target for Understanding: Relationships within the Real Number System
8.1
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a. Simplify numerical expressions containing:
1. exponents (where the base is a rational number and
the exponent is a positive whole number);
2. Fractions, decimals, integers, and square roots of
perfect squares
c. Grouping symbols (no more than 2 embedded
grouping symbols)
Note: Order of operations and properties of operations
with real numbers should be used.
b. Compare and order no more than five fractions,
decimals, percents, and numbers written in scientific
notation using positive and negative exponents.
Ordering may be in ascending or descending order
8.2
a. Describe orally and in writing the relationships among
the sets of natural or counting numbers, whole numbers,
integers, rational numbers, irrational numbers, & real
numbers.
b. Illustrate the relationships among the subsets of the real
number system by using graphic organizers such as
Venn diagrams. Subsets include rational numbers,
irrational numbers, integers, whole numbers, and
natural or counting numbers.
c. Identify the subsets of the real number system to which
a given number belongs.
d. Determine whether a given number is a member of a
particular subset of the real number system, and explain
why.
e. Describe each subset of the set of real numbers and
include examples and nonexamples.
f. Recognize that the sum or product of two rational
numbers is rational; that the sum of a rational number
and an irrational number is irrational; and that the
product of a nonzero rational number and an irrational
number is irrational.†
Sample SOL Scrimmage
Dan Mulligan, flexiblecreativity.com
VERSION: 1.0
Benchmark/EKS: I can multiply a fraction or a whole number by a fraction and
use a visual fraction model to represent the equation.
Read carefully and follow the directions: SHOW YOUR THINKING!
1. Alice drank 3
4
of a
1
gallon of chocolate milk.
2
2. Multiply. Write your answer in simplest
form:
How much of a whole gallon did she drink? Write
your answer in simplest form.
1
x3=
4
Answer: ______
Answer: ______
Draw lines and shade in the rectangle to
represent the fraction.
Draw lines and shade in the squares to
represent the product.
3. Create a multiplication problem of a
fraction and whole number.
4. Multiply. Write your answer in simplest
form.
5 2
x
6 3
Problem: ___________
Shade the circles to represent the
problem.
Answer: ______
Use the rectangle below to represent the
problem.
Explain your thinking.
18
Sample SOL Scrimmage
Dan Mulligan, flexiblecreativity.com
VERSION: 2.0
Benchmark/EKS: I can
Read carefully and follow the directions: SHOW YOUR THINKING!
The chart below tells the lengths of six different rivers from around the world. Use the
lengths to complete the activities below the chart.
Name of
river
Length in
miles
Nile
Columbia
Mekong
Danube
Volga
Amazon
4,132 miles
1,450 miles
2,705 miles
1,795 miles
3,645 miles
3,976 miles
1. Fill in the blanks so each statement
is true.
2. Which of the expressions below is
equivalent to the 4 in the Columbus
River?
The value of 7 in the Danube’s length is
ten times the value of the 7 in which river’s Place a  next to all that apply.
length? _________________
___ 400 x 10 ___ 4,000 x 10
The value of 5 in which river’s length is ten
times the value of the 5 in the Volga’s
___ 4 x 100
___ 40 x 10
length.
The value of which river’s length is ten
times the value of the same digit in the
Danube’s length.
3. Circle each length that has a 6 that
is worth ten times as much as the 6
in the Volga’s length.
___ 400 ÷ 10
___ 4,000 ÷ 10
4 Explain how you used what you know
about place value to help answer the
questions in numbers 1 – 3.
26,175 miles 9,062 miles 64,582 miles
6,419 miles
40,678 miles
19
Sample SOL Scrimmage
Dan Mulligan, flexiblecreativity.com
VERSION: 1.0
Benchmark/EKS: I can
Read carefully and follow the directions: SHOW YOUR THINKING!
2. Compare the values of each 7 in the
number 771,548. Use pictures,
numbers and words to explain.
3. Norfolk State University’s Football
Stadium has a seating capacity of
31,452.
According to the 2010 census, the
population of San Jose, CA was
approximately ten times the amount of
people that NSU’s stadium can seat. What
was the population of San Jose in 2010?
Explain your reasoning.
4. How is the digit 2 in the number 582
similar to and different from the digit 2
in the number 528?
5. Tonya was practicing her
multiplication with the following facts:
9 x 10 = 90, 13 x 10 = 130, 456 x 10 = 4,500
She noticed that every time she
multiplied by ten there was a zero at the
end of each number.
Explain to Tonya why there is a zero at
the end of a number when it is multiplied
by 10.
20
Sample SOL Scrimmage
Dan Mulligan, flexiblecreativity.com
VERSION: 2.0
Benchmark/EKS: I can
Read carefully and follow the directions: SHOW YOUR THINKING!
The chart below tells the lengths of six different rivers from around the world. Use the
lengths to complete the activities below the chart.
Name of
river
Length in
miles
Nile
Columbia
Mekong
Danube
Volga
Amazon
4,132 miles
1,450 miles
2,705 miles
1,795 miles
3,645 miles
3,976 miles
2. Fill in the blanks so each statement
is true.
2. Which of the expressions below is
equivalent to the 4 in the Columbus
River?
The value of 7 in the Danube’s length is
ten times the value of the 7 in which river’s Place a  next to all that apply.
length? _________________
___ 400 x 10 ___ 4,000 x 10
The value of 5 in which river’s length is ten
times the value of the 5 in the Volga’s
length.
The value of which river’s length is ten
times the value of the same digit in the
Danube’s length.
4. Circle each length that has a 6 that
is worth ten times as much as the 6
in the Volga’s length.
___ 4 x 100
___ 40 x 10
___ 400 ÷ 10
___ 4,000 ÷ 10
4 Explain how you used what you know
about place value to help answer the
questions in numbers 1 – 3.
26,175 miles 9,062 miles 64,582 miles
6,419 miles
40,678 miles
21
In your own words:
1.
Share your current understanding of the terms below with your partner.
2.
In the second column, record a common understanding of each term.
3.
In the third column, record an example of each term from your school
or visits to a school.
Trait
Description
Example
Rigor
Relevance
Relationships
Results
Personal Reflections:
22
Rigor/Relevance Framework
The framework is a tool established by the International Center for Leadership in
Education to assist educators in examining curriculum, instruction, and assessment.
The Rigor/Relevance Framework is based on two dimensions of higher standards and
authentic student engagement.
1. Higher Standards – First, there is the knowledge continuum
that describes the increasingly complex ways in which we
think. The Knowledge Taxonomy is based on the six levels
of Bloom’s Revised Taxonomy.
Assimilatio
n of
Knowledge
Acquisition
(RIGOR)
of
Knowledge
The low end of this continuum involves acquiring knowledge and being able to recall or locate that
knowledge in a simple manner. The high end of the Knowledge Taxonomy is evident when the
learners takes several pieces of information and combine them in both logical and creative ways.
Students can solve multistep problems and create unique work and solutions.
Remembering
Understanding
Applying
Analyzing
2. Application Model – The second continuum describes
putting knowledge to use.. The five levels of this action
continuum are:
a)
b)
c)
d)
e)
Knowledge in one discipline
Apply in one discipline
Apply across disciplines
Apply to real-world predictable situations
Apply to real-world unpredictable situations
Thinking
Continuum
Evaluating
Creating
Action Continuum
Acquisition
of
Knowledge
Application
of
Knowledge
(RELEVANCE)
The Application Model describes putting knowledge to use. While low end is knowledge acquired for
its own sake, the high end signifies action – use of the knowledge to solve complex real-world
problems and to create projects, designs, and other works for use in real-world situations.
23
Rigor –
Rigor refers to academic rigor — learning in which students demonstrate a thorough, in-depth
mastery of challenging tasks to develop cognitive skills through reflective thought, analysis,
problem-solving, evaluation, or creativity. (think Bloom’s)
Rigor should be thought of as how often we require our students to solve complex problems,
apply what they have learned, and critically analyze the results. The focus of rigor should be
on helping the students develop a deeper understanding of the subject matter that goes
beyond memorizing, reciting and restating. The development of critical thinking skills is
paramount to "rigor". Teachers shouldn't take pride in the fact that a student has to do two
hours of homework per night and study three days for tests in order to pass their class. In
fact, absent the true "rigor" of higher-order thinking skills, this could be considered poor
teaching practice.
Relevance –
Relevance refers to learning in which students apply core knowledge, concepts, or skills to
solve real-world problems. Relevant learning is interdisciplinary and contextual. Student work
can range from routine to complex at any school grade and in any subject. Relevant learning is
created, for example, through authentic problems or tasks, simulation, service learning,
connecting concepts to current issues, and teaching others. (think student interest)
All educators have heard the phrase, "Why do I have to learn this? I'll never use it again." If
students have to ask this question, then "relevance" is missing in the classroom. Relevance
refers to how the subject matter relates to the student's interests and needs. Real relevance
cannot be developed unless students are allowed to utilize their learning in real-life situations
and contexts. When this is considered, it is easy to see how "rigor" and "relevance" begin to
overlap. When students are allowed to apply their learning to real-world situations (relevance),
they are required to use higher-order thinking skills (rigor). Therefore, true rigor is very
difficult to attain in the absence of relevance, and vice versa.
Relationships –
Relationships involve teaching a rigorous and relevant curriculum while understanding each
student’s needs and barriers to learning (think differentiation) Core Values – Myself, as your
teacher, taking the time to understand when you don’t.
Although "rigor" and "relevance" are keys to meaningful student learning, this learning cannot
occur in the absence of "relationships" in the school. Kids cannot learn if their social and
emotional needs have not been satisfied. We can have the most rigorous and relevant classrooms
in the country, but if our kids' affective needs are not being met, we will not be successful. In a
school focused on relationships, there is a caring, student-centered environment where students
feel a sense of connection to their school. Many schools have realized the importance of this
variable, and have tried to account for it through the development of the "school within a
school" concept. In this structure, interdisciplinary teams are developed and groups of students
are assigned to each team. Others have adopted an "advisory" structure, where each teacher is
assigned a small group of students.
Results – Results refer to accountability to each student to do all in our power to assist them reach
their true potential. The focus on the results of student learning using multiple indicators is nonnegotiable, so our teachers can adjust their practices and schools can offer personalized support to
students. (think college and career ready)
24
Rigor/Relevance Framework
Creating: “putting together”



Use old ideas to create new ones
Relate knowledge from several areas
Reorganize parts to create new original
things, ideas, concepts



Evaluating: “judge the outcome”



Compare and discriminate between
ideas
Assess values of theories,
presentations
Make choices on reasoned arguments



See patterns/relationships
Recognize hidden parts
Take ideas/learning apart
Find unique characteristics

Use the information
Use methods, concepts, theories in
new situations



Understand information
Translate knowledge into new context
Grasp meaning of materials learned,
communicate learnings, and interpret
learnings



Observation and recall of information
Knowledge of dates, events, places
Students extend and refine their
knowledge so that they can use it
automatically and routinely to
analyze and solve problems and
create solutions.
Student Thinking
Students Thinks and Works
(Relationships Important)
(Relationships Critical)
A
B
Acquisition
Application
Students gather and store bits of
knowledge and information and
are expected to remember or
understand this acquired
knowledge.
Students use acquired knowledge to
solve problems, design solutions,
and complete work. The highest
level of application is to apply
knowledge to new and unpredictable
situations.
3


Solve problems using
required skills and/or
knowledge
Make use of learning in new
or concrete manner, or to
solve problems
2



Order, group, infer causes
Interpret facts,
compare/contrast
Predict consequences
Remembering: “information
gathering”
Adaptation
Students have the competence that,
when confronted with perplexing
unknowns, they are able to use their
extensive knowledge base and skills
to create unique solutions and take
action that further develop their
skills.
Organize parts
Identify components
Separate into component
parts
Understanding: “confirming”


Assimilation
4
Applying: “making use of knowledge”

5
Verify value of evidence/
Recognize subjectivity
Make judgments/choices
based on criteria, standards,
and/or conditions
Analyzing: “taking apart”



C
Use innovation to make
something new
Generalize from given facts
Predict or draw conclusion


Mastery of subject matter
Gain specific facts, ideas,
vocabulary, etc.
Formula for Success:
Rigor x Relevance x Relationships = Meaningful Learning
(Note: if any one of these are missing, the equation equals
zero)
D
Studen
t
Driven
6
Teacher Works
(Relationships of Little
Importance)
Classroo
m
Student Work
(Relationships Important)
RELEVANCE
R
I
G
O
R
Teache
r
Driven
Real
Life
Rigor and Relevance Model Adaptation
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Creating: “putting together”



Use old ideas to create new ones
Relate knowledge from several areas
Reorganize parts to create new original things,
ideas, concepts
Assimilation
Adaptation
C
D
Student Thinking
Students Thinks and
Works
Evaluating: “judge the outcome”



Compare and discriminate between ideas
Assess values of theories, presentations
Make choices on reasoned arguments
Analyzing: “taking apart”




See patterns/relationships
Recognize hidden parts
Take ideas/learning apart
Find unique characteristics
(Relationships Important)
(Relationships Critical)

Use the information
Use methods, concepts, theories in new situations
Acquisition
Application
O
A
B
R
Teacher Works
Student Work
(Relationships of Little Importance)
(Relationships Important)
Understanding: “confirming”



Understand information
Translate knowledge into new context
Grasp meaning of materials learned, communicate
learnings, and interpret learnings
Remembering: “information gathering”


Observation and recall of information
Knowledge of dates, events, places
I
G
Applying: “making use of knowledge”

R
RELEVANCE
1
Knowledge in
one discipline
2
Apply
knowledge in
one discipline
3
Apply
knowledge
across
disciplines
4
Apply
knowledge to
real-life
predictable
situations
5
Apply
knowledge to
real-world
unpredictable
situations
26
C
D
R
I
A
B
G
O
R
RELEVANCE
27
Depth of Knowledge & Rigor/Relevance Quadrants
Level 1: Recall and Reproduction/DOK 1/Quadrant A – Acquisition (Low Rigor/Low
Relevance)
TEACHER WORKS
Curricular elements that fall into this category involve basic tasks that require the student to
recall or reproduce knowledge and/or skills. The subject matter content at this particular level
usually involves working with facts, terms and/or properties of objects. It may also involve use
of simple procedures and/or formulas. There is little transformation or extended processing of
the target knowledge required by the tasks that fall into this category. Key words that often
denote this particular level include: list, identify, and define. A student answering Level1/A
item either knows the answer or does not; that is, the answer does not need to be ‘figured
out” or “solved.”
Possible Products
Quiz
Test
Reproduction
Collection
Blog
Commenting
Social bookmarking
Definition
Label
Vocabulary Quiz
Explanation
Wiki
Bulleting
Searching
Fact
List
Recitation
Show and Tell
Podcast
Highlighting
‘Googling’
Worksheet
Workbook
Example
Outline
Categorizing/Tagging
Social networking
Roles
Teacher
Directs
Shows
Questions
Demonstrates
Compares
Tells
Examines
Evaluates
Listens
Contrasts
Student
Responds
Absorbs
Remembers
Recognizes
Memorizes
Describes
Explains
Translates
Restates
Demonstrates
Interprets
Examines
Reflections on Learning Opportunities:
28
Level 1: Recall and Reproduction/Quadrant A – Acquisition (Low Rigor/Low Relevance)
Potential Activities (DOK 1/Quad-A)

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

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
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























Develop a concept map showing a process or describing a topic
Make a timeline
Write a list of keywords you know about…
Make a chart showing …
Recite a fact related to…
Write in your own words…
Cut out or draw a picture that illustrates an event, process, or story
Report or present to the class
Make a cartoon strip showing the sequence of events, process, or story
Write and perform
Write a brief outline and explain the event, process, or story
Write a summary report of the event
Prepare a flow chart that illustrates the sequence of events
Paraphrase a chapter in the book
Retell in your own words
Outline the main points
Recall, restate, remember, or recognize a fact, term, or property (recognizing, listing,
describing, identifying, retrieving, naming, locating, finding
Use basic calculation tasks involving only one step (i.e., addition, subtraction, etc.) to complete
the following…
Locate or retrieve information in verbatim form
Straight-forward recognition tasks related to identifying features, objects, and/or steps that
don’t vary greatly in form (i.e., recognizing features of basic tools)
Writing tasks that involve applying a standard set of conventions and/or criteria that should
eventually be automated (i.e., using punctuation, spelling, etc.)
Basic measurement tasks that involve one-step (i.e., using a ruler to measure length)
Use this simple formula where at least one of the unknowns is provided t …
Locate information in maps, charts, tables, graphs, and drawings
Recall details of a story (events, character, plot, setting, etc.)
Identify specific information contained in graphics
Conduct basic mathematical calculations
Label locations on a map
Represent in words or diagrams a scientific concept or relationship
Preform routine procedures like measuring length or using punctuation marks correctly
Describe the features of a place or group of people
Identify who, what, where, when of a particular event or issue, list attributes, or define the
meaning of terms
29
Depth of Knowledge & Rigor/Relevance Quadrants
Level 2/B includes the engagement of some mental processing beyond recalling or
reproducing a response. This level generally requires students to contrast or contrast people,
places, things, events, and concepts; convert information from one form to another; classify or
sort items into meaningful categories; describe or explain issues and problems, patterns,
cause and effect, significance or impact, relationships, point of view or processes. A level 2
“describe or explain” would require students to go beyond a description or explanation of
recalled information to describe or explain a result of “how” or “why.” The learner should make
use of information in a context different from the one in which it was learned. Elements found
in a curriculum that fall in this category involve working with or applying skills and/or concepts
to tasks related to a field of study in a laboratory setting. The subject matter content at this
particular level usually involves working with a set of principles, categories, and protocols. At
this level students are asked to transform/process target knowledge before responding.
Example mental processes that often denote this particular level include: summarize,
estimate, organize, classify, and infer.
Possible Products
Photograph
Demonstration
Diary
Blog Reflecting
Illustration
Presentation
Journal
Moderating
Simulation
Interview
Mind Maps
Validating
Sculpture
Performance
Blog Commenting
Linking
Roles
Teacher
Shows
Observes
Organizes
Facilitates Solves problems
Evaluates Calculates
Questions Completes
Constructs
Student
Demonstrates use of Knowledge
Compiles
Illustrates
Reflections on Learning Opportunities:
30
Level 2: Working with Skills & Concepts/ Quadrant B – (Low Rigor/High Relevance)
Potential Activities (DOK 2/Quad-B)


























Classify a series of steps
Construct a model to demonstrate how it looks or works
Practice a play or perform in class
Make a diorama to illustrate an event
Write a diary/blog entry
Make a scrapbook about an area of study
Make a topographical map
Make a puzzle or game about a topic
Write an explanation about a topic for others
Make a model..
Perform routine application tasks (i.e., Applying a simple set of rules or protocols to a
laboratory situation the same way each time)
Explain a meaning of a concept and/or explaining how to perform a particular task
State relationships among a number of concepts and/or principles
Perform more complex recognition tasks that involve recognizing concepts and processes that
may vary in how they “appear”
Perform more complex calculation tasks (i.e., multi-step calculations like such as standard
deviation)
Complete research projects and writing activities that involve locating, collecting, organizing,
and displaying information (i.e., write a report with the purpose to inform; meeting all steps of
the writing process
Complete measurement tasks that occur over a period of time and involve
aggregating/organizing data
Identify and summarize the major events of a narrative
Use context clues to identify the meaning of unfamiliar words
Solve routine multi-step problems
Describe the cause-and-effect of a particular event or issue
Identify patterns in events or behavior
Compare/contrast people, places, events, and concepts
Convert information from one form to another form
Formulate a routine problem/issue given data and conditions
Organize, represent, and interpret data
31
Depth of Knowledge & Rigor/Relevance Quadrants
Level 3: Short-term Strategic Thinking/DOK 3/Quadrant C – Assimilation (High
Rigor/Low Relevance)
STUDENT THINKS
Items falling into this category demand a short-term use of higher-order thinking processes,
such as analysis and evaluation, to solve real-world problems with predictable outcomes.
Stating one’s reasoning is a key marker of tasks that fall into this particular category. The
expectation established for tasks at this level tends to require coordination of knowledge and
skill from multiple subject-matter areas to carry out processes and reach a solution in a
project-based setting. Key processes that often denote this particular level include: analyze,
explain, and support with evidence, generalize, and create.
Possible Products
Graph
Outline
Abstract
Report
Program
Podcast
Spreadsheet
Survey
Report
Evaluating
Film
Publishing
Checklist
Database
Debate
Investigating
Animation
Wiki-ing
Chart
Mobile
Panel
Concluding
Video cast
Roles
Teacher
Probes
Observes
Acts as a resource
Dissects
Clarifies
Guides
Guides
Evaluates
Organizes
Clarifies
Accepts
Discusses
Debates
Questions
Judges
Assesses
Justifies
Argues
Selects
Student
Uncovers
Thinks deeply
Examines
Disputes
Decides
Calculates
Tests
Compares
Reflections on Learning Opportunities:
32
Level 3: Short-term Strategic Thinking/Quadrant C – Assimilation (High Rigor/Low
Relevance)
Potential Activities (DOK 3/Quad-C)





























Use a Venn Diagram that shows how two topics are the same and different
Design a questionnaire to gather information
Survey classmates/industry members to find out what they think about a particular topic
Make a flow chart to show the critical stages
Classify the actions of the characters in a book
Prepare a report about an area of study
Conduct an investigation to produce information to support a view
Write a letter to the editor after evaluating a product
Prepare and conduct a debate
Prepare a list of criteria for a judge
Write a persuasive speech arguing for/against…
Make a booklet about rules you see as important. Convince others
Forma panel to discuss viewpoints on…
Write a letter to…advising on changes needed
Prepare a case to present your view about…
Short-term tasks and projects placing a strong emphasis on transferring knowledge to solve
predictable problems
Explain abstract terms and concepts
Tasks when the environment observed is real-world and often contains extraneous information
which must be sorted through
Solve complex calculation problems presented that draw upon multiple processes
Write or explain tasks that require altering a message to ‘fit’ an audience
Create graphs, tables, and charts where students must reason through and organize the
information
Support ideas, thesis, or predictions with specific evidence, details, and examples
Use voice appropriate to purpose and audience
Identify research questions and design investigations for a scientific problem
Develop a scientific model for a complex situation
Determine the author’s purpose and describe how it affects the interpretation of a reading
selection
Apply a concept in another context
Draw conclusions from a variety of sources of information
Make connections across time and place to explain a concept or big idea
33
Depth of Knowledge & Rigor/Relevance Quadrants
Level 4: Extended Strategic Thinking/DOK 4/Quadrant D – Assimilation (High Rigor/High
Relevance)
STUDENT WORKS and THINKS
Curricular elements assigned to this level demand extended use of higher order thinking
processes such as creating, reflecting, assessing, and adjusting plans over time. Students are
engaged in conducting investigations to solve real-world problems with unpredictable
outcomes. Employing and sustaining strategic thinking processes over a longer period of time
to solve the problem is a key feature of curricular objectives that are assigned to this level.
Key strategic thinking processes that denote this particular level include: create, reflect,
conduct, and manage.
Possible Products
Film
New Game
Story
Song
Project
Newspaper
Plan
Media Product
Roles
Teacher
Facilitates
Reflects
Evaluates
Extends
Analyzes
Designs
Takes risks
Proposes
Creates
Student
Formulates
Modifies
Plans
Reflections on Learning Opportunities:
34
Level 4: Extended Strategic Thinking/Quadrant D – Assimilation (High Rigor/High
Relevance)
Potential Activities (DOK 4/Quad-D)
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Apply information to solve ill-defined problems in novel situations
Solve tasks that require a number of cognitive and physical skills in order to complete
Write and/or research tasks that involve formulating and testing hypotheses over time
Conduct tasks that require students to make multiple strategic and procedural decisions as
they are presented with new information throughout the course of the event
Complete tasks that require perspective taking and collaboration with a group of individuals
Create graphs, tables, and charts where students must reason through and organize the
information without instructor prompts
Writing centered tasks having a strong emphasis on persuasion
Devise a way to…
Develop a menu for a new restaurant using a variety of healthy foods
Sell an idea
Write a jingle to advertise a new product
Conduct an internship in industry where students are faced with real-world, unpredictable
problems
Conduct a project that requires specifying a problem, designing and conducting an experiment,
analyzing its data, and reporting results and/or solutions
Analyze and synthesize information from multiple sources
Describe and illustrate how common themes re found across texts from different cultures
Design a mathematical model to inform and solve a practical or an abstract situation
Design a thesis, conduct an investigation using multiple sources, analyze and synthesize the
evidence in a written report (essay) or multimedia presentation, and present to an audience
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