Supporting Rigorous Mathematics
Teaching and Learning
Making Sense of the Number and Operations
– Fractions Standards via a Set of Tasks
Tennessee Department of Education
Elementary School Mathematics
Grade 5
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Rationale
Tasks form the basis for students’ opportunities to learn what
mathematics is and how one does it, yet not all tasks afford
the same levels and opportunities for student thinking.
[They] are central to students’ learning, shaping not only their
opportunity to learn but also their view of the subject matter.
Adding It Up, National Research Council, 2001, p. 335
By analyzing instructional and assessment tasks that are for
the same domain of mathematics, teachers will begin to
identify the characteristics of high-level tasks, differentiate
between those that require problem solving, and those that
assess for specific mathematical reasoning.
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Session Goals
Participants will:
• make sense of the Number and Operations –
Fractions Common Core State Standards (CCSS);
• determine the cognitive demand of tasks and make
connections to the Mathematical Content Standards
and the Standards for Mathematical Practice; and
• differentiate between assessment items and
instructional tasks.
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Overview of Activities
Participants will:
• analyze a set of tasks as a means of making sense of
the Number and Operations – Fractions Common Core
State Standards (CCSS);
• determine the Content Standards and the Mathematical
Practice Standards aligned with the tasks;
• relate the characteristics of high-level tasks to the
CCSS for Mathematical Content and Practice; and
• discuss the difference between assessment items and
instructional tasks.
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The Data About Students’
Understanding of Fractions
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The Data About Fractions
Only a small percentage of U.S. students possess the
mathematics knowledge needed to pursue careers in
science, technology, engineering, and mathematics (STEM)
fields. Moreover, large gaps in mathematics knowledge exist
among students from different socioeconomic backgrounds
and racial and ethnic groups within the U.S. Poor
understanding of fractions is a critical aspect of this
inadequate mathematics knowledge. In a recent national
poll, U.S. algebra teachers ranked poor understanding about
fractions as one of the two most important weaknesses in
students’ preparation for their course.
Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, Thompson, & Wray (2010).
IES, U.S. Department of Education
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The Data about Fractions:
Conceptual Understanding
A high percentage of U.S. students lack conceptual
understanding of fractions, even after studying fractions for
several years; this, in turn, limits students’ ability to solve
problems with fractions and to learn and apply computational
procedures involving fractions.
• 50% of 8th graders could not order three fractions from least
to greatest;
1
3
• 27% of 8th graders could not correctly shade of a rectangle;
• 45% of 8th graders could not solve a word problem that
required dividing fractions (NAEP, 2004).
• Fewer than 30% of 17-year-olds correctly translated 0.029 as
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1000
(Kloosterman, 2010).
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The Data about Fractions:
Conceptual Understanding
A lack of conceptual understanding of fractions has
several facets, including…students’ focusing on
numerators and denominators as separate numbers
rather than thinking of the fraction as a single number.
3
8
3
5
Errors such as believing that > arise from comparing
the two denominators and ignoring the essential
relationship between each fraction’s numerator and
denominator.
Siegler, Carpenter, Fennell, et al; U.S. Dept. of Education, IES Practice Guide:
Developing Effective Fractions Instruction for Kindergarten through 8th Grade.
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Analyzing Tasks as a Means of
Making Sense of the CCSS
Number and Operations –
Fractions
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Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
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Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
Setting Goals
Selecting Tasks
Anticipating Student Responses
Accountable Talk® is a registered trademark of the
University of Pittsburgh
Orchestrating Productive Discussion
• Monitoring students as they work
• Asking assessing and advancing questions
• Selecting solution paths
• Sequencing student responses
• Connecting student responses via Accountable
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Talk® discussions
Linking to Research/Literature:
The QUASAR Project
• Low-level tasks
• High-level tasks
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Linking to Research/Literature:
The QUASAR Project
• Low-level tasks
– Memorization
– Procedures without Connections
• High-level tasks
– Doing Mathematics
– Procedures with Connections
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The Cognitive Demand of Tasks
(Small Group Discussion)
Analyze each task. Determine if the task is a high-level
task. Identify the characteristics of the task that make it
a high-level task.
After you have identified the characteristics of the task,
then use the Mathematical Task Analysis Guide to
determine the type of high-level task.
Use the recording sheet in the participant handout to
keep track of your ideas.
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The Mathematical Task Analysis Guide
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction:
A casebook for professional development, p. 16. New York: Teachers College Press
.
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The Cognitive Demand of Tasks
(Whole Group Discussion)
What did you notice about the cognitive demand of the
tasks?
According to the Mathematical Task Analysis Guide,
which tasks would be classified as:
• Doing Mathematics Tasks?
• Procedures with Connections?
• Procedures without Connections?
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Analyzing Tasks: Aligning with the CCSS
(Small Group Discussion)
Determine which Content Standards students would
have opportunities to make sense of when working on
the task.
Determine which Mathematical Practice Standards
students would need to make use of when solving the
task.
Use the recording sheet in the participant handout to
keep track of your ideas.
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Analyzing Tasks: Aligning with the CCSS
(Whole group discussion)
How do the tasks differ from each other with respect to
the content that students will have opportunities to learn?
Do some tasks require that students use mathematical
practice standards that other tasks don’t require students
to use?
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The CCSS for Mathematical Content: Grade 5
Number and Operations—Fractions
5.NF
Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
5.NF.B.4
Apply and extend previous understandings of multiplication to multiply a
fraction or whole number by a fraction.
5.NF.B.4a
Interpret the product (a/b) × q as a parts of a partition of q into b equal
parts; equivalently, as the result of a sequence of operations a × q ÷ b.
For example, use a visual fraction model to show (2/3) x 4 = 8/3, and
create a story context for this equation. Do the same with (2/3) x (4/5) =
8/15. (In general, (a/b) x (c/d) = ac/bd.)
5.NF.B.4b
Find the area of a rectangle with fractional side lengths by tiling it with
unit squares of the appropriate unit fraction side lengths, and show that
the area is the same as would be found by multiplying the side lengths.
Multiply fractional side lengths to find areas of rectangles, and represent
fraction products as rectangular areas.
Common Core State Standards, 2010, p. 36, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 5
Number and Operations – Fractions
5.NF
Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
5.NF.B.5
Interpret multiplication as scaling (resizing), by:
5.NF.B.5a
Comparing the size of a product to the size of one factor on
the basis of the size of the other factor, without performing the
indicated multiplication.
5.NF.B.5b
Explaining why multiplying a given number by a fraction greater than 1
results in a product greater than the given number (recognizing
multiplication by whole numbers greater than 1 as a familiar case);
explaining why multiplying a given number by a fraction less than 1
results in a product smaller than the given number; and relating the
principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of
multiplying a/b by 1.
5.NF.B.6
Solve real-world problems involving multiplication of fractions and mixed
numbers, e.g., by using visual fraction models or equations to represent
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the problem.
Common Core State Standards, 2010, p. 36, NGA Center/CCSSO
The CCSS for Mathematical Content: Grade 5
Number and Operations – Fractions
5.NF
Apply and extend previous understandings of multiplication and division to multiply
and divide fractions.
5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions by
whole numbers and whole numbers by unit fractions.
5.NF.B.7a
Interpret division of a unit fraction by a non-zero whole number, and compute
such quotients. For example, create a story context for (1/3) ÷ 4, and use a
visual fraction model to show the quotient. Use the relationship between
multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 =
1/3.
5.NF.B.7b
Interpret division of a whole number by a unit fraction, and compute such
quotients. For example, create a story context for 4 ÷ (1/5), and use a visual
fraction model to show the quotient. Use the relationship between multiplication
and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.
5.NF.B.7c
Solve real-world problems involving division of unit fractions by non-zero whole
numbers and division of whole numbers by unit fractions, e.g., by using visual
fraction models and equations to represent the problem. For example, how
much chocolate will each person get if 3 people share 1/2 lb of chocolate
equally? How many 1/3-cup servings are in 2 cups of raisins?
Common Core State Standards, 2010, p. 36-37, NGA Center/CCSSO
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The CCSS for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning
of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
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A. Stretching a Rubber Band
Sadie has a rubber band that she stretches out. When
she stretches it, the rubber band is 4 feet long. That is
2
3
longer than its original length. What is the original
length of the rubber band?
Stretched = 4 feet
Original length = ?
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B. Multiplying Fractions
Calculate:
5x
3
4
3
4
Will of 5 be the same as or different from 5 x
3
?
4
How do you know?
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C. Dividing Fractions on a Number Line
Sheila says that she thinks it is easier to show some division
1
problems than others on number lines. She thinks that 4 ÷
3
1
3
is easier to show than ÷ 4. Use the number lines below to
illustrate each division problem.
4÷
1
3
1
3
÷4
Do you agree with Sheila? Which one was harder to show
and why?
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D. Time to do Jobs
Jill wants to get a few jobs done around the house
1
before she leaves in 2 hours. If each job takes of an
3
hour, how many jobs can she get done?
Write an equation using fractions to show how you
found your answer.
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E. Dividing Fractions Relationship
James found a relationship when he was dividing
fractions:
1
4
5 ÷ = 20
1
4
÷5=
1
20
Describe the relationship that James found.
Will this work with other numbers? Show your thinking
with a diagram or number line.
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F. Multiplying and Dividing Fractions
1
4
7x =
Use either a number line or an area model to show the
correct answer.
What division equations can also describe the SAME
1
model you created for 7 x ? Explain how they relate to
4
the model.
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G. Servings of Mashed Potatoes
Sam made 8 cups of mashed potatoes for
1
Thanksgiving dinner. Sam estimates that a serving is
2
cup. How many servings of mashed potatoes does
Sam have?
Write an equation and use a diagram to show your
thinking.
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H. Saving Money
1
3
Ravi received $60.00 for his birthday. He puts in the
1
5
bank. He gives to charity. How much does he have
left to spend?
Write fraction equations and use a diagram or number
line to show how much money Ravi has to spend.
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Analyzing Tasks: Aligning with the CCSS
(Whole Group Discussion)
How do the tasks differ from each other with respect to
the content that students will have opportunities to
learn?
Do some tasks require that students use Mathematical
Practice Standards that other tasks don’t require
students to use?
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Reflecting and Making Connections
• Are all of the CCSS for Mathematical Content in this
cluster addressed by one or more of these tasks?
• Are all of the CCSS for Mathematical Practice
addressed by one or more of these tasks?
• What is the connection between the cognitive
demand of the written task and the alignment of the
task to the Standards for Mathematical Content and
Practice?
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Differentiating Between Instructional
Tasks and Assessment Tasks
Are some tasks more likely to be assessment tasks
than instructional tasks? If so, which and why are you
calling them assessment tasks?
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Characteristics of Performance-Based
Assessments
• Each task is cognitively demanding. (TAG: require math
reasoning, an explanation for why formulas or procedures
work; analysis of patterns, formulation of a generalization,
prompt connection making between representations,
strategies, or mathematical concepts and procedures.)
• The task addresses several of the CCSS for Mathematical
Content.
• The task may require students to use more than one
strategy or representation when solving the task.
• The expectations in the task are clear and explicit regarding
the extent of the work expected.
• The task may ask students to (1) explain their reasoning in
words or to (2) use mathematical reasoning to justify their
response.
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Characteristics of Instructional Tasks
• A variety of tasks at the different levels of cognitive
demand are used. (TAG: require math reasoning, an
explanation for why formulas or procedures work;
analysis of patterns, formulation of a generalization,
prompt connection making between representations,
strategies, or mathematical concepts and
procedures.)
• The task addresses one or more of the CCSS for
Mathematical Content.
• The task may require students to use more than one
strategy or representation when solving the task.
• The task may be open-ended because the teacher
can guide the instruction.
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Instructional Tasks Versus Assessment Tasks
Instructional Tasks
Assessment Tasks
Assist learners to learn the CCSS for
Mathematical Content and the CCSS for
Mathematical Practice.
Assesses fairly the CCSS for Mathematical
Content and the CCSS for Mathematical
Practice of the taught curriculum.
Assist learners to accomplish, often with
others, an activity, project, or to solve a
mathematics task.
Assess individually completed work on a
mathematics task.
Assist learners to “do” the subject matter
under study, usually with others, in ways
authentic to the discipline of mathematics.
Assess individual performance of content
within the scope of studied mathematics
content.
Include different levels of scaffolding
Include tasks that assess both developing
depending on learners’ needs. The
understanding and mastery of concepts and
scaffolding does NOT take away thinking from skills.
the students. The students are still required to
problem-solve and reason mathematically.
Include high-level mathematics prompts.
(The tasks have many of the characteristics
listed on the Mathematical Task Analysis
Guide.)
Include open-ended mathematics prompts as
well as prompts that connect to procedures
with meaning.
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Reflection
• So, what is the point?
• What have you learned about assessment tasks and
instructional tasks that you will use to select tasks to
use in your classroom next week?
• How do we give students the opportunities during
instructional time to learn math so that they are
successful on the next generation assessment items?
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