De s i g n i n g L e s s o n s an d C o mmo n
C o r e : Do n ' t F o r g e t t he P r ac t i c e
( S t an d ar d s ) !
63rd Annual Meeting of the ICTM
Peoria, IL
October 19, 2013, Session 102
Dr. Janice Krouse
Dr. Donald Porzio
MDHWCS President 2013-14
630-907-5964
krouse@imsa.edu
ICTM President 2011-13
630-907-5966
dporzio@imsa.edu
http://staff.imsa.edu/~dporzio/
Mathematics Faculty
Illinois Mathematics and Science Academy
1500 Sullivan Rd., Aurora, IL 60506
3 lbs times 2 units away from lower cord equals 6.
1 lb times 4 units away from cord plus 2 lbs times 1 unit away from lower cord
also equals 6, so the sides of the lower bar balance.
6 total lbs times 2 units away from upper cord equals 12.
4 lbs times 3 units away from upper cord also equals 12, so the sides of the
upper bar balance.
Common Core State Standards for Mathematics
Standards for Mathematical Practice
MP1 - Make Sense of Problems and Persevere in Solving Them
Mathematically Proficient Students…



Explain the meaning of a problem to themselves and seek an entry point to its solution
Consider analogous problems, try special cases and simpler forms of the problem to gain insight
Check answers by asking themselves “Does this make sense?”
MP2 - Reason Abstractly and Quantitatively
Mathematically Proficient Students…



Make sense of quantities and their relationships in problem situations
Decontextualize – abstract a situation, represent it symbolically and manipulate the symbols
Contextualize – pause as needed during manipulation to probe referents for the symbols involved
MP3 - Construct Viable Arguments and Critique the Reasoning of others
Mathematically Proficient Students…



Understand and use stated assumptions, definitions and previous results to construct arguments
Analyze situations by breaking them into cases
Justify and communicate conclusions and respond to others’ arguments
MP4 - Model with Mathematics
Mathematically Proficient Students…



Apply mathematics to solve problems in everyday life, society and the workplace
Identify important quantities in a situation and map their relationships using mathematical tools
Interpret results in context of the situation and reflect on whether the results make sense
MP5 - Use Appropriate Tools Strategically
Mathematically Proficient Students…



Are familiar with and use grade appropriate tools when solving a mathematical problem
Detect possible errors by strategically using estimation or other mathematical knowledge
Use technological tools to explore and deepen understanding of concepts
MP6 - Attend to Precision
Mathematically Proficient Students…



Try to communicate precisely to others
Use clear definitions in discussion and in their own reasoning
Calculate accurately and efficiently
MP7 - Look for and Make Use of Structure
Mathematically Proficient Students…



Look closely to discern a pattern or structure
Are able to step back to look for an overview and/or to shift perspective
Can see complex things as single objects or as a composition of multiple objects
MP8 - Look for and Express regularity in Repeated Reasoning
Mathematically Proficient Students…



Note repetition and regularity in calculations, and look for both general methods and for shortcuts
Maintain oversight of the process while attending to details
Evaluate the reasonableness of their immediate results
Educator Modules; © 2012, Illinois Mathematics and Science Academy
Mod 1/ H 9
Standards for Mathematical Practice addressed by weight balancing
problem
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.
7.
Look for and make use of structure.
8.
Look for express regularity in repeated reasoning.
Were there others?
Some Content Standards addressed by weight balancing problem
Are there others?
Connecting the Standards for Mathematical Practice to the Standards
for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the
discipline of mathematics increasingly ought to engage with the subject matter as they grow in
mathematical maturity and expertise throughout the elementary, middle and high school years.
Designers of curricula, assessments, and professional development should all attend to the need to
connect the mathematical practices to mathematical content in mathematics instruction.
SO,….
How can you design lessons that target different Practice Standards
and also connect to Content Standards?
The Standards for Mathematical Content are a balanced combination of procedure and understanding.
Expectations that begin with the word “understand” are often especially good opportunities to connect
the practices to the content. Students who lack understanding of a topic may rely on procedures too
heavily. Without a flexible base from which to work, they may be less likely to consider analogous
problems, represent problems coherently, justify conclusions, apply the mathematics to practical
situations, use technology mindfully to work with the mathematics, explain the mathematics accurately
to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In
short, a lack of understanding effectively prevents a student from engaging in the mathematical
practices.
In this respect, those content standards which set an expectation of understanding are potential “points
of intersection” between the Standards for Mathematical Content and the Standards for Mathematical
Practice. These points of intersection are intended to be weighted toward central and generative
concepts in the school mathematics curriculum that most merit the time, resources, innovative energies,
and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional
development, and student achievement in mathematics.
Common Core State Standards for Mathematics, pg 8.
Key Point in all this:
Don't always let the content
drive your lessons.
Instead, think about the
practices first, then the content!
Or at the very least, think about
the practices and content at the
same time.
You'll be pleasantly surprised as
to how well the content will
take care of itself when the
practices are the focus and are
done well.
5
5
8
4
3
1
2
1
2
6
9
8
4
1
3
9
15
7
5
4
10
2
6
10
13
3
12
7
14
11
Geometric Play
The figure below consists of 20 identical
squares. Divide the figure into 5 pieces,
each consisting of 4 squares, so that no 2
pieces have the same shape. Note that each
square in each piece must share at least one
side with another square in that piece.
Geometric Play
The figure below consists of 20 identical
squares. Divide the figure into 5 pieces,
each consisting of 4 squares, so that no 2
pieces have the same shape. Note that each
square in each piece must share at least one
side with another square in that piece.
1
34 46
11
22 44
69
12
24
48
70
9
18
42 49
73
23
32 56 74
How Far to Zequop?
The Situation:
The seven towns of Tidville, Ubania, Vogton,
Wimpster, Xendic, Yunjar, and Zequop are
indicated on the map on the next page by the letters
A to G, not necessarily in order. The roads
connecting the towns run strictly north-south and
east-west. The distance between adjacent dots on
the map is exactly one mile. Thus, the town labeled
C is 5 miles (by road) from the town E. Each town
contains a sign that gives the length of the shortest
route to two other towns. These seven signs, one
from each town, are shown on the next page. The
last sign, though, has been left incomplete.
The Problem:
Discover which town is where, put each sign in its
correct town, and provide the correct distance to
Zequop that is missing on the last sign.
A
F
D
E
B
C
G
1
Tidville
Ubania
3
4
4
Wimpster 6
Xendic
3
2
Ubania
Vogton
7
3
5
Xendic
Yunjar
6
3
3
Vogton
5
Wimpster 6
6
Yunjar
Zequop
4
4
7
Tidville
Zequop
1
?
Wimpster
Sign 2 F
Tidville
Sign 6 E
A
Zequop
Sign 4 A
D
Yunjar
Sign 3 B
F
E
Xendic
Sign 7 D
B
C
G
Ubania
Sign 5 C
Vogton
Sign 1 G
7
Tidville
Zequop
1
3