Transcript Welcome High School Mathematics Educators

THE IMPORTANCE OF
COHERENT LESSONS IN
HIGH SCHOOL
MATHEMATICS
Summer 2014
College and Career Readiness Conference
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.
In the following task from Illustrative Mathematics,
complete the following steps:
1. Compute the correct answer.
2. List all of the mathematics concepts/skills you used.
3. Write down the grade at which the concepts/skills are
taught and the applicable domain, cluster, or standard.
4. List mathematics concepts/skills in later grades/courses
that build upon/expand the information from #2.
5. Reflect on what you just have written. What do you
notice about the task’s reliance on COHERENCE?
http://www.illustrativemathematics.org/illustrations/1564
Parking Lot!
A parking lot charges $0.50 for each half hour or fraction thereof, up to a daily maximum
of $10. Let C(t) be the cost in dollars of parking for minutes.
a. Complete the table below.
http://www.illustrativemathematics.org/illustrations/588
Do calculations here:
DS
http://www.illustrativemathematics.org/illustrations/588
Sketch graph here
DS
http://www.illustrativemathematics.org/illustrations/588
Questions:
Answers
http://www.illustrativemathematics.org/illustrations/588
INSTRUCTION
Grade
8th
From Grade-to-Grade
8.F.1: Understand that a function is a rule that assigns to each
input exactly one output. The graph of a function is the set of
ordered pairs consisting of an input and the corresponding
output (function notation is not required in Grade 8).
HS
ALG I
Analyze functions using different representations. Linear,
exponential, quadratic, absolute value, step, piecewise defined.
F.IF.7a, 7b, 7e, 8a, 8b, 9
HS
ALG II
Analyze functions using different representations. Focus on
using key features to guide selection of appropriate
type of model function F.IF.7b, 7c, 7e, 8, 9
Build a function that models a relationship between two
HS
Course quantities. Include all types of functions studied
F.BF.1b (NOT REQUIRED FOR ALL STUDENTS! This is a
4
PLUS Standard)
In the following task, complete the following steps:
1. Compute the correct answer.
2. List all of the mathematics concepts/skills you used.
3. Write down the grade at which the concepts/skills are
taught and the applicable domain, cluster, or standard.
4. List mathematics concepts/skills in later grades/courses
that build upon/expand the information from #2.
5. Reflect on what you just have written. What do you
notice about the task’s reliance on COHERENCE?
INSTRUCTION
An equipment rental company has 2 different penalty options for
returning a piece of equipment late:
Option 1: On day 1, the penalty is $10. On day 𝟐, the penalty is
$2𝟎. On day 3, the penalty is $30. On day 4, the penalty is $4𝟎
and so on, increasing by $10 each day the equipment is late.
Company 2: On day 1, the penalty is $𝟎.𝟎𝟏. On day 𝟐, the
penalty is $𝟎.𝟎3. On day 3, the penalty is $𝟎.09. On day 4, the
penalty is $𝟎.27 and so on, tripling in amount each additional day
late.
Jim rented a backhoe using option because he thought it had the
better late return policy. The job he was doing with the backhoe
took longer than he expected, but it did not concern him because
the late penalty seemed so reasonable. When he returned the
backhoe 12 days late, he was shocked by the penalty fee. What
did he pay, and what would he have paid if he had used Option 1
instead?
INSTRUCTION
Additional Questions for discussion:
1.
Write the formula for the sequence in option 1.
2.
Describe the sequence.
3.
Write a formula for the sequence in option 2.
4.
Describe the sequence.
5.
Which of the 2 options grows more quickly? Why?
INSTRUCTION
Do calculations here:
DS
INSTRUCTION
Do calculations here:
DS
INSTRUCTION
Grade
From Domain-to-Domain
HS ALG Interpret functions that arise in applications in terms of a
I
context. Linear, exponential, and quadratic F.IF.4, 5, 6
Create equations that describe numbers or
HS ALG relationships. Linear, quadratic, and exponential (integer
I
inputs only); for A.CED.3 linear only A.CED.1, 2, 3, 4
HS ALG Represent and solve equations and inequalities
I
graphically. Linear and exponential;
learn as general principle A.REI.10, 11, 12
HS ALG Construct and compare linear, quadratic, and exponential
models and solve problems. F.LE.1a, 1b, 1c, 2, 3
I
ASSESSMENT
Evidence of
in
assessment is not tremendously
different from evidence of
in classroom
instruction.
ASSESSMENT
Interpreting Functions (F-IF)

Major- Understand the concept of a function an due
function notation

Interpret functions that arise in applications in terms of
context

Supporting - Analyze functions using different
representations
Building Functions(F-BF)

Supporting - Build a function that models a relationship
between two quantities

Additional- Build new function from existing functions
ASSESSMENT
https://www.parcconline.org/samples/mathematics/high-school-functions
ASSESSMENT
https://www.parcconline.org/samples/mathematics/high-school-functions
Do calculations here:
DS
ASSESSMENT
•Which other Standards for Mathematical
Practice are assessed in this task? MP6Attend to Precision
• Content standards F-‐IF.9. Compare
properties of two functions each represented
in a different way (algebraically, graphically,
Numerically in tables, or by verbal
descriptions).
(According to PARCC)
ASSESSMENT
•Type I-Balance of conceptual understanding,
fluency, and application
•Type I- Machine scorable, innovative
computer-based formats
•Claim A- Major content with connections to
practices
•Scoring- Full credit requires selecting correct
answers from drop down menu. Partial credit
can be given for each answer that is correct.
(According to PARCC)
In reality, the “key” to making the Shift of
Coherence a success rests on the shoulders of
curriculum writers and classroom teachers.
Instruction must take advantage of every
opportunity to build connections between topics
both within a grade and from grade to grade.
INSTRUCTION
ASSESSMENT
COHERENCE
Grade-to-Grade Coherence Activity
What are the muddiest points?
Record any question
you still have after
today’s presentation
on your post-it note.
Please provide your
name and email
address.
Stick your post-it on the front wall as you
leave today. Thank you!