Engaging Instruction:
Rich Problems & Tasks
TLQP 2013-14
Thomas F. Sweeney, Ph.D
The Sage Colleges
Session Goal
Developing a clear picture of the Common
Core State Standards by:
• Using rich problems to understand the
Standards for Mathematical Practice
• Digging into the content standards
through the Critical Areas of Focus
in order to create instruction based upon
the CCSSM and develop a foundation for
curriculum revision.
MP + CAF + Standards = Instruction
In order to design instruction that meets the
rigor and expectations of the CCSSM,
understanding the Mathematical Practices
and Critical Areas of Focus are essential.
Critical Areas of Focus
Critical Areas of Focus inform
instruction by describing the
mathematical connections and
relationships students develop in the
progression at this point.
Found in CCSSM on the first page of each new grade
Critical Areas of Focus
Grade
Priorities in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding
K–2
Addition and subtraction, measurement using
whole number quantities
3–5
Multiplication and division of whole numbers
and fractions
6
7
8
Ratios and proportional reasoning; early
expressions and equations
Ratios and proportional reasoning; arithmetic
of rational numbers
Linear algebra
Rich Problems: A Wealth of Benefits
A Problem or an Exercise?
Problem
• The answer is not
immediately known
• Requires persistence
• Engaging
• Feasible
• Valued
Exercise
• Computation “problem”
• Solution process is
recognizable
• Routine
• Contextual but not engaging
Let’s Warm-up!
Activity 1:
Complete these two puzzles
+
3
7
4
6
+
4
10
7
13
Which caused more thinking?
Activity 2:
375375
Think of a three digit number and write
it twice making a six digit number. Now
divide it by 7, the answer by 11 and the
answer by 13. What do you notice? Why
does this happen?
Your turn…
Rich Mathematical Tasks . . .
•Accessible to everyone
•Can be extended
•Let students do the thinking, speculating,
conjecturing, proving, explaining,
reflecting, reporting
•Are fun and enjoyable
Is it Rich?
• What are essential characteristics of rich
problems?
What Makes a Problem Rich?
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Significant mathematics
Mathematical Practices
Multiple layers of complexity
Multiple entry points
Multiple solutions and/or strategies
Leads to discussion or other questions
Students are the workers and the decision
makers
• Warrants reflection - Paired with discourse
Activity 3:
Connect Task to the
Standards for Mathematical Practice
• Individually work MARS task #3
• Identify Standards for Mathematical Practice
• Share with a partner:
– Solution(s)
– What makes the problem(s) rich?
– support 1-2 Mathematical Practices
http://map.mathshell.org/materials/tasks.php
Recall:
Standards for Mathematical Practices
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
Incorporating Rich Problems in Instruction
NOW
(Activity 4)
Pick one of the problems on the
accompanying sheet and work on it with
your table buddy.
1. Modifying a task.
Original Task
Writing the date on the board, say 25th
Modified task
Write today’s date using any operations and
only 2’s and 5’s.
2
5
2 25
5
2 5 2  5
52
5  2  5  2  5  2 
52
2.
Ask the Answer!
I have an area of 24cm2. What does the
shape look like?
This
is 1 fifth. What does the whole
shape look like?
(Activity 5)
Your turn
Work by grade level.
Take an ordinary problem for your grade & make
it rich
or
Create an entirely new rich problem for your
grade.
Report.
Repeat.
Rich Task Sources
Mathematics Assessment Resource Service
• http://map.mathshell.org/materials/index.php
Inside Mathematics
• http://www.insidemathematics.org
Balanced Assessment (MARS tasks)
• http://balancedassessment.concord.org
NCTM Illuminations
• http://illuminations.nctm.org/
Nrich Project (Univ. of Cambridge)
http://nrich.maths.org
Next Steps Rich Problems
• Identify “rich” mathematical problems
– Make sure to cite the source of the problem*
• Align this “rich” problem to:
– Grade level
– Critical Area of Focus
– Mathematical Practice(s)
• Share with colleagues
Find rich tasks at
http://www.illustrativemathematics.org/
Prepare to present one to the group.
Extra time:
Sub Problem
Grade 3-5
Grades 5 – 8 & up
Extra time ?:
Break this square into
11 smaller squares
that don’t overlap and
whose union is the
original square.
Check with numbers.
Generalize your
solution.
Extra time?
Grades 7-8
Farmer Brown
When Farmer Brown travels to town at 30km/hr
he arrives an hour early. When he travels at
20km/hr he arrives an hour late.
What is the question?
What can I find out?
The Puzzles
•How far was the return journey?
•How fast should he travel to arrive on time?
•How long did it take him to get to town?
•How fast should he travel to arrive 2hrs late?
There is a separate Farmer Brown PPT with several solutions
Pure Logic
No box is labeled correctly.
Select one sock from one box and re-label them
all correctly.
Black Socks
Black and White
Sock Mixture
White Socks
Happy Numbers
Think of a whole number. Square the digits
and add the results. This creates a new
number. Repeat this process. If the
sequence of numbers forms a cycle then the
original number is happy.
42 becomes 42 + 22 = 20, 4, 16, 37, 58, 89, 145, 42…aha a circle of happiness!
19 becomes 12 + 92 = 82,68, 100, 1,1 ,1 ….
Another happy number
Which positive integers are happy?
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Sources
http://www.corestandards.org/
http://www.parcconline.org/
http://engageny.org/
http://www.illustrativemathematics.org/
http://www.achievethecore.org/
http://commoncoretools.me/
http://insidemathematics.org/
https://www.teachingchannel.org/videos/
https:// www.OhioRC.org