THE IMPORTANCE OF
COHERENT LESSONS IN
MIDDLE SCHOOL
MATHEMATICS
Summer 2014
College and Career Readiness Conference
Based on RIGOR:
Reduce Side Chatter
Involve Yourself in the Process
Give Your Thoughts and Ideas
Open Your Mind to How You Can Change Instruction
Remember to Silence Electronic Devices
Focus
Coherence
Rigor
1. Conceptual Understanding
2. Procedural Skill
3. Modeling/Application
Coherence comes from the verb to cohere,
which means:
• To stick together in a way that resists separation;
• To be logically linked and naturally related.
Think of Rice Krispies Treats…
We want our mathematics curriculum to have coherence…
related concepts that stick together in a way that is obvious,
understandable, connected, and reliable… for teachers and
students alike!
A purposeful placement of standards to create
logical sequences of content topics that bridge
across the grades, as well as across standards
within each grade.
TODAY’S OUTCOME
Participants will:
Explore in depth the “shift” of
COHERENCE and its impact on
mathematics content, instruction,
and assessment of the Maryland
CCR Standards in mathematics.
CONTENT
The writers of the CCSS:


Built
standards.
into the content
Clarified and shared their vision of
grade-to-grade and within-grade
in the Progressions
documents.
http://ime.math.arizona.edu/progressions/
1. Compute the correct answers.
2. What strategy did YOU use to approach/solve this problem?
What other strategies might work, as well?
3. What are the mathematics concepts/skills that you must know
in order to solve the problem?
4. What are the grade levels for these concepts/skills.
5. In your own classroom, how would you handle a situation in
which students used a variety of strategies? What would you
do if some strategies did not match the strategy in your
textbook?
http://www.illustrativemathematics.org/illustrations/1564
Chichén Itzá was a Mayan city in what is now Yucatan, Mexico. This photo
shows the Pyramid of Kukulcán, which is located in the ruins of Chichén Itzá .
The floor of the temple at the top of the pyramid is approximately 24 meters
above the ground, and there are 91 steps leading up to the temple. How high
above the ground would you be if you were standing on the 50th step?
INSTRUCTION
Grade
From Grade to Grade
5th
Understanding of fraction equivalence and the skill to
generate equivalent fractions leads to the study of ratios.
6th
Skills with multiplication, division and fractions, contribute to
the study of ratios and unit rates
7th
Analyze proportional relationships and solve problems
involving unit rates associated with ratio fractions.
8th
Use similar triangles to prove that the slope of a line is welldefined and understand the points (x,y) on a non-vertical line
are solutions to the equation, y= mx + b, where m is the
slope of the line as well as the unit rate of a proportional
relationship.
INSTRUCTION
Scientists are
sending a rover to
the moon. Their
plan is to study a
rectangular area
of the moon using
the map shown.
On the grid, 1 unit
represents 1
kilometer (km).
http://www.parcconline.org/sites/parcc/files/FractionModelFINAL.pdf
INSTRUCTION
Grade 5
from Domain to Domain
NBT.B.7 Add, subtract, multiply, and divide decimals to
#1s
hundreds… using strategies based on place value and
properties of operations...
NF.B.4b Find the area of a rectangle with fractional side lengths
#2s
by tiling it with unit squares… and show that the area is
the same as would be found multiplying the side lengths.
MD.A.1
#3s
G.A.1
#4s
Convert among different-sized standard measurement
units within a given measurement system… and use
these conversions in solving multi-step, real-world
problems.
Graph points on the coordinate plane to solve real-world
and mathematical problems.
INSTRUCTION
Part A
The rover will land at (3.5, 1) to explore up to (3.5, 4),
and then over to (2, 4).
Plot these three points on the map.
Part B
What are the coordinates of the fourth vertex of the
rectangle that the scientists plan to explore?
( ____, ____ )
http://www.parcconline.org/sites/parcc/files/FractionModelFINAL.pdf
INSTRUCTION
Part C
What is the horizontal length of the rectangle?
___________ kilometers
What is the vertical length of the rectangle?
___________ kilometers
Part D
Find the area of the moon exploration. Show your work.
http://www.parcconline.org/sites/parcc/files/FractionModelFINAL.pdf
ASSESSMENT
Evidence of
in
assessment is not tremendously
different from evidence of
in classroom
instruction.
ASSESSMENT
Scientists are
sending a rover to
the moon. Their
plan is to study a
rectangular area
of the moon using
the map shown.
On the grid, 1 unit
represents 1
kilometer (km).
http://www.parcconline.org/sites/parcc/files/FractionModelFINAL.pdf
ASSESSMENT
•Type III task that calls for modeling and
application in a real-world context (MP.4)
•Which other Standards for Mathematical
Practice are assessed in this task?
• Content standards are assessed as
securely-held knowledge from grade 5.
ASSESSMENT
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 reasoning
ASSESSMENT
• 5.NBT.B (Major)
• 5.NF. B.4b (Major)
• 5.MD. A.1 (Major)
• 5.G.A.1 (Supporting)
In reality, the “key” to making the Shift of
Coherence a success rests in the arms of
curriculum writers and classroom teachers.
Instruction should take advantage of every
opportunity to build connections between topics
within a given grade, as well as between grades.
Illustrative Math https://www.illustrativemathematics.org/
• Bill McCullum, CCSS lead writer
• Sample Lessons that illustrate specific standards
Achieve The Core
https://achievethecore.org
• Jason Zimba, CCSS lead writer
• Multiple Resources – e.g. Lesson Plans, Assessments, Professional
Development courses, Grade-at-a-Glance
PARCC http://parcconline.org
• Information about PARCC Assessments
• Sample Lessons
• Practice Tests