BACKWARDS DESIGN BASICS
• Begin with the end in mind
• Develop a clear understanding of where
you want to go
• Map out the steps to get there
www.arps.org
STAGES TO BACKWARD DESIGN
Stage 1: Identify Desired Results
• What do you want the students to know and be able to do?
• Standards
Stage 2: Determine Acceptable Evidence of Learning
• Assessment tools that will act as evidence for student
understanding
• Formative to drive instruction
Stage 3: Design Learning Experiences and Instruction
• Unit/lesson plans that include activities, plans for meeting
needs for all students and instructional tools
BUILDING ASSESSMENT QUESTIONS
• Types of Assessments
• Power of the Question
• Common Core Aligned Tasks
Oakland Schools Website
Summative Assessment
Formative Assessment
Measure full range of CCSS
Repository of tools available to teachers
Computer Adaptive Testing for precision
to support quick adjustment and
Timely results
differentiated instruction
Engage Institutions of Higher Education
Help define student performance along
to ensure achievement standards reflect
the CCSS learning progressions
college and career readiness
Concrete strategies for immediate
Scale scores help inform growth mode
feedback loops
Interim Benchmark Assessment
Allow for finer grain of measurement (e.g., end
of unit)
Inform teachers if students on track to be
proficient on summative assessments
Multiple opportunities for students to
participate
Scale scores help inform growth model
Oakland Schools Website
Are you assessing
content or practice? OR
BOTH?
Maine West Department of
Education webiste
POWER OF THE QUESTION
Understand and use ratios, proportions and percents in a variety of
situations. –NJ Core Curriculum Content Standards for Mathematics
Grade 7, 4.1.A.3
1. 50% of 20:
2. 67% of 81:
3. Shawn got 7 correct answers out of 10 possible answers on his science
test. What percent of questions did he get correct?
4. J.J. Redick was on pace to set an NCAA record in career free throw
percentage. Leading into the NCAA tournament in 2004, he made 97 of 104
free throw attempts. What percentage of free throws did he make?
5. J.J. Redick was on pace to set an NCAA record in career free throw
percentage. Leading into the NCAA tournament in 2004, he made 97 of 104
free throw attempts. In the first tournament game, Redick missed his first five
free throws. How far did his percentage drop from before the tournament
game to right after missing those free throws?
6. J.J. Redick and Chris Paul were competing for the best free-throw shooting
percentage. Redick made 94% of his first 103 shots, while Paul made 47 out
of 51 shots.
Which one had a better shooting percentage?
In the next game, Redick made only 2 of 10 shots while Paul made 7 of
10 shots. What are their new overall shooting percentages? Who is the
better shooter?
Jason argued that if Paul and J.J. each made the next ten shots, their
shooting percentages would go up the same amount. Is this true? Why
or why not?
Hand Out
Driven By Data
What are the differences between these
six assessments of the same
standard?
What conclusions do you draw about
assessments when looking at these
examples?
POWER OF THE QUESTION
Driven By Data
1. What is the main idea?
2. This story is mostly about:
A. Two boys fighting
B. A girl playing in the woods
C. Little Red Riding Hood’s adventures with a wolf
D. A wolf in the forest
3. This story is mostly about:
A. Little Red Riding Hood’s journey through the woods
B. The pain of losing your grandmother
C. Everything is not always what it seems
D. Fear of wolves
INK WHILE YOU THINK:
WHAT ARE THE DIFFERENCES BETWEEN THESE THREE
QUESTIONS?
WHICH QUESTION IS MORE RIGOROUS?
Driven By Data
POWER OF THE QUESTION ACTIVITY
Why? Analyze the questions based on rigor
How?
 Silently look through the sample test
 Pick out 2-3 questions that could be asked in a more rigorous way
 Pick 2- 3 questions that are rigorous for third grade. Why?
 Be ready to discuss your questions
Concepts to Remember:
• Options define the rigor of multiple
choice questions
• Rubric defines the rigor in open ended
questions
• Good assessments will combine
multiple choice forms to achieve the
best measure of mastery
POWER OF THE QUESTION
Driven By Data Page
21
ASSESSING CONTENT AND PRACTICE
• Separating the practices from the content is not
helpful and is not what the standards require.
• The practices do not exist in isolation; the vehicle for
engaging in the practices is mathematical content.
• The K-6 math standards already have implied practice
standards embedded
ORIGINAL TO BETTER
3.OA.3
QUALITIES OF ALIGNED TASKS
• Intentionally selected to elicit one or more particular
practices for specified content
• Each targeted practice is observable in the student’s
response, directly in student work or indirectly
through an incorrect answer to a problem in which
the practice would have made a correct answer
much more likely.
• It must be unlikely or impossible to earn full credit
on the task without engaging in the practice.
PERSEVERANCE TASKS
MP.1 Make sense of problems and
persevere in solving them
•
Word problems involving ideas that are
currently at the forefront of the
student’s developing mathematical
knowledge (e.g., multiplication in
Grade 3 or addition or subtraction of
fractions in Grade 4).
•
Are designed to take a typical student
a long time to solve;
•
Require a large number of routine and
fairly easy steps;
•
May involve steps which in turn lead to
a more difficult problem
•
May have a rubric that awards
increasing points to each part
There will be 58 people at a
breakfast and each person will eat 2
eggs. How many cartons of eggs will
be needed for the breakfast. There
are 12 eggs in each carton.
ABSTRACT & QUANTITATIVE REASONING TASKS
MP.2 Reason abstractly and quantitatively
Contextual problems in which the student can
gain insight into the problem, and earn points
on the task, by relating the algebraic form of
an answer or intermediate step to the given
context.
ARGUMENT & CRITIQUE TASKS
MP.3 Construct viable arguments and critique the reasoning of others
• Basing explanations/reasoning on concrete referents such as diagrams
(whether provided in the prompt or constructed by the student’s response)
• Distinguishing correct explanation/reasoning from that which is flawed , and
– if there is a flaw in the argument – explaining what it is.
 For example, flawed ‘student’ reasoning is presented and the task is to
correct and improve it.
 Tasks presenting students with common errors are valuable because
students are often better at explaining why something is wrong that why
something is right.
CRITIQUE EXAMPLE
Joan wrote “2/3 + 2/3 = 4/6”
Critique her work.
Is it correct? Explain why or why not.
Tell what Joan did right or wrong.
TOOL USAGE TASKS
MP.5 Use appropriate tools strategically
Diagrams as a tool: Problems that are fairly easy to solve or to
answer correctly if you first draw a diagram, very hard to
solve or to answer correctly if you don’t – yet no direction
is given to the student to draw a diagram.
(Note: MP.5 is not a coded statement to ‘use a calculator.’ If
the student is not being strategic, the student is not
meeting the standard. Tasks must create circumstances
for poor use of tools, misuse of tools, and/or mistakenly
not using tools.
Sara built the block tower with 1-foot cubes.
How many cubes did she use?
PRECISION-ORIENTED TASKS
MP.6 Attend to precision
Tasks requiring the student to present solutions to
multi-step problems in the form of valid chains of
reasoning, using symbols such as equals signs
appropriately.
For example, rubrics award less than full credit for
the presence of nonsense statements such as
1 + 4 = 5 + 7 = 12, even if the final answer is
correct.
STRUCTURE-ORIENTED TASKS
MP.7 Look for and make use of structure
• Mathematical and real-world problems that reward seeing
structure in an expression and using the structure to rewrite it
for a purpose
• Numerical problems that reward or require deferring
calculations steps until one sees the overall structure.
Examples such as 357 + 17999 + 1 or 37 × 25 × 4
REPEATED REASONING TASKS
MP.8 Look for and express
regularity in repeated reasoning
• Problems in which a tedious and
repetitive calculation can be
made shorter by observing
regularity in the repeated steps
CONCEPTUAL UNDERSTANDING
•
•
Assess where the standards
explicitly call for it
These should mostly be short tasks
that are:
 computationally non-intensive and
easy to answer quickly,
 if the student understands the
concept in question,
 but difficult to answer at all if the
student doesn’t understand the
concept
Examples:
 Write four fractions that are all equal to 5:
 Which number is least and which is greatest?
¾
2
4/4
3/5
 Write a number that is greater than 1/5 and less
than ¼:
 Plot each of the following on the number line:
2
5/4
3× ½
¾+¾
2 – (1/10)
FLUENCY
Fluency assessment (with machine scoring of responses entered by
computer interface)
Fluency when the standards explicitly call for it
Fluency priorities:
3.OA.7, 4.NBT.4, 5.NBT.5, 6.NS.2
Mark each equation as true or false
8×
9 = 80 – 8
54 ÷ 9 = 24 ÷ 6
7 × 5 = 25
8× 3 = 4 × 6
49 ÷ 7 = 56 ÷ 8
MATHEMATICAL REASONING TASKS
Call for written arguments/justifications, critique
of reasoning, or precision in mathematical
statements (MP.3, MP.6)
May include other mathematics standards as
well
EXPRESSING MATHEMATICAL REASONING
3.OA.5
Amber doesn’t know what 7 × 5 equals,
but she knows 5 × 5 = 25 and 2 × 5 = 10.
Use drawings, words, and/or equations to
explain why Amber can add 25 and 10 to
find what 7 × 5 equals.
MODELING & APPLICATION
Tasks that involve real-world contexts or scenarios and require the student to:
 Apply knowledge and skills articulated in specified standards;
 Engage particularly in the modeling practice (MP.4)
 Make sense of problems and persevere to solve them (MP.1)
 Reason abstractly and quantitatively (MP.2)
 Use appropriate tools strategically (MP.5)
 Look for and make use of structure (MP.7) and/or
 Look for and express regularity in repeated reasoning (MP.8)
Each task consist of several related questions, or a single prompt with a longer
response required, or a combination.
MODELING TASKS
MP.4 Model with mathematics
•
Contextual word problems involving ideas that are currently at the forefront of the student’s
developing mathematical knowledge (e.g., multiplication or division in Grade 3 or addition and
subtraction of fractions in Grade 4)
•
Multi-step contextual word problems in which the problem is NOT broken into steps or sub-parts
•
Problems of either these two kinds may involve related practices, MP.1, MP.2, MP.5, MP.7, MP.8
Grade 3 example:
There are 60 straws in a package. How
many straws are in 7 packages?
MODELING
4th grade example of a multi-step contextual word problem in which the problem isn’t broken into
steps or sub-parts
A plate of cookies:
There were 28 cookies on a plate.
Five children each ate one cookie.
Two children each ate 3 cookies.
One child ate 5 cookies.
The rest of the children each ate two cookies.
Then the plate was empty.
How many children ate two cookies? Show your work.
INTEGRATIVE TASKS
Integrative tasks are:
• Tasks that are aligned to a cluster heading, domain
heading or grade/course title, rather than to one
specific standard
• These tasks are not intended to permit loose
interpretations of the standards.
INTEGRATIVE TASKS
The following problem codes straightforwardly to the cluster “Understanding place value” at Grade
5, but may not clearly code to any single standard in the cluster.
Write a number in each space to make true equations.
1 tenth =
100 tenths =
0.1 tenths =
0.01 tenths =
1/10 tenths =
hundredths
hundredths
hundredths
hundredths
hundredths
tenths = 0.1 hundredths
INTEGRATIVE TASKS
4.NF – Since this problem involves concepts and skills in fraction addition, multiplication
and equivalence, it does not clearly code to any single standard in the NF domain.
9 large trucks are carrying ½ ton of lumber each. 7 small trucks are carrying ¼ ton of
lumber each. How many total tons are being carried by all of the trucks?
INTEGRATIVE TASKS
The following grade 4 task involves place value understanding as well as elements of
place value computation, so it blends the first two clusters in 4.NBT
893,462
840,924
824,595
824,162
810,930
808,879
799,982
778,877
777,852
766,398
Find two numbers in this table
that differ by approximately
two thousand
INTEGRATIVE TASKS
Each of the following three illustrations is plausible as a direct implication of the
grade 5 standards for Number and Operations in Base Ten, but each involves place
value understanding as well as elements of place value computation, so each
blends the first two clusters in 5.NBT
Mark each statement true or false.
108 × 30 is equal to a four-digit number.
6731 × 23 is equal to a four-digit number.
2244 ÷ 11 is equal to a three-digit number.
INTEGRATIVE TASKS
A bakery made 3,200 cookies and needs to package them in bags for a
sale. 10 cookies will go in each bag. How many bags will be needed?
****
Compute each of the following
357 + 17,999 + 1
357 + 17,999
899 + 1343 + 101
37 × 25 × 4
1001 × 20
RESOURCES
Illustrative Mathematics
http://illustrativemathematics.org
Common Core Tools
http://commoncoretools.wordpress.com , a blog moderated by Dr. William McCallum,
distinguished professor and head of mathematics at the University of Arizona and
mathematics lead for the Common Core State Standards for Mathematics.
STEPS IN DESIGN
1. Choose a grade level
2. List the priority standards (using the Content
Emphases by Cluster Chart and Model Content
Framework).
3. Choose a priority standard or cluster, and an aligned
practice standard.
4. Follow the design recommendations from the
powerpoint for the practice standard(s) in question.
5. Use Illustrative Mathematics and your own
curriculum materials as resources.
6. Include an answer key &/or rubric.
TASK DESIGN EXAMPLE
Focus for Grade 3: Numbers and Operations-Fractions
Standard: Develop understanding of fractions as numbers
Cluster 3.NF 3d
Problem: A Letter to My Friend
 Some people think that ¼ is larger than 1/3.
 Some people think that 1/3 is larger than ¼.
 Which do you think is larger? Write a letter to a friend telling why you are
right.
 Use pictures, letters and numbers , or all of these to explain your thinking.
TASK DESIGN EXAMPLE
Focus for Grade 5: Numbers and Operations-Fractions
Standard: Use equivalent fractions as a strategy to add and
subtract fractions.
Cluster 5.NF 1 and 2
Problem: Mr. Jones bought a can of paint. He used 3/8 of it to
paint the garage door and ¼ of it to paint a wall.
 A. How much paint did he use altogether?
 B. How much paint was left?