High School Mathematics and PARCC Presented by Don Biery

High School
Mathematics
and PARCC
Presented by Don Biery
Turn to page 1 of your hand out………
BLAH BLAH BLAH BLAH…….
Common Core
and The Modeling
System
Math is not a subject to
just fill curriculumn…….it
is life, it is all around us,
hidden in the fabric of
nature.
ParCC Modeling Questions
The real world is what
ParCC is after.
Life is biology- Biology is
Chemistry – Chemistry is
application of Physics – physics
is the application of Math
Math must be an interdisciplinary item
Departments must meet to work together…..
Classes must be aligned for content to interact i.e. ALG II
needs to teach systems of equations before chemistry get
to formulae balancing.
Math must have labs…………
Math Labs =
Math
Modeling
There are no x’s in the
world….
We must teach students to
be able to problem solve
not solve problems.
PARCC High Level Blueprints Mathematics
Math item counts
per form
Assess
me nt
EOY
Ite ms
Grade
3
Grade
4
Grade
5
Grade
6
Grade
7
Grade
8
Algebra
I
Geomet
ry
Math
II
Algebra
II
3
4
2
8
2
8
2
6
2
4
2
6
2
1
1
9
1
9
1
9
1
9
1
9
5
8
8
7
8
5
1
1
1
2
1
2
1
2
1
2
1
4
-
-
-
1
1
2
3
3
3
3
3
2
3
9
3
6
3
6
3
4
3
3
3
3
3
5
3
4
3
4
3
4
3
4
3
5
Type I
1Point
8
8
6
8
8
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Type I
2 Point
2
2
3
2
2
1
-
-
-
-
-
-
Type II
3 Point
2
2
2
2
2
2
2
2
2
2
2
2
Type II
4 Point
2
2
2
2
2
2
2
2
2
2
3
3
Type III
3 Point
2
2
2
2
2
2
2
2
2
2
2
2
Type III
6 Point
1
1
1
1
1
1
2
2
2
2
3
3
1
0
1
0
9
1
0
1
0
1
1
1
0
1
0
1
0
1
0
1
0
1
0
Type
II
4
4
4
4
4
4
4
4
4
4
5
5
Type
III
3
3
3
3
3
3
4
4
4
4
5
5
Type I
1 Point
Type I
2 Point
Type I
4 Point
E
O
Y
TOT
AL
PBA/M
YA
Type I
Type I
PBA/M
YA
TOTA
L
Math I
Math
III
Overview of Task
Types
•
The PARCC assessments for mathematics will involve three primary types of tasks: Type I, II, and III.
• Each task type is described on the basis of several factors, principally the purpose of the task in generating
evidence for certain sub claims.
Task Type
I. Tasks assessing concepts, skills
and procedures
Description of Task Type
•
•
Balance of conceptual understanding, fluency, and application
Can involve any or all mathematical practice standards
•
Machine scorable including innovative, computer-based formats
• Will appear on the End of Year and Performance Based Assessment
components
II. Tasks assessing expressing
reasoning, or precision in mathematical
mathematical reasoning
III. Tasks assessing modeling /
scenario (MP.4)
applications
•
Sub-claims A, B and E
•
Each task calls for written arguments / justifications, critique of
statements (MP.3, 6).
•
Can involve other mathematical practice standards
•
May include a mix of machine scored and hand scored responses
•
Included on the Performance Based Assessment component
•
Sub-claim C
•
Each task calls for modeling/application in a real-world context or
•
Can involve other mathematical practice standards
•
May include a mix of machine scored and hand scored responses
•
Included on the Performance Based Assessment component
•
Sub-claim D
Real World
Application
1. Financial
Literacy…..Leads to Better
Understanding and true
applicability
Real World
Application
2. Algebra: Why we learn
Real Analysis…..Be true to
our Students……
Real World
Application
3. Geometry: What can I do
with this junk?
Real World
Application
Labs and the real world….
1. Give them real problems
to solve
2. Don’t always tell them
what tool to use ---ParCC
Create a Lab:
1. Reason for Lab
(UBD)
2. Lab reporting out?
3.Lab Procedures
4. Possible
issues……
November 2013
HS
Type
Evidence
Statement
Most
Relevant
Standards for
Mathematical
Content
Popcorn Inventory
Type III 6 Points
HS.D.1-1: Solve multi-step contextual problems with degree of difficulty appropriate to
the course, requiring application of knowledge and skills articulated in 7.RP.A, 7.NS.3,
7.EE, and/or 8.EE.
7.RP.2: Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a coordinate plane and observing whether
the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of proportional relations hips.
c. Represent proportional relationships by equations. For example, if total cost t is
proportional to the number n of items purchased at a constant price p, the relationship
between the total cost and the number of items can be expressed as t = pn .
d. Explain what a point (x, y) on the graph of a proportional relationship means in
terms of the situation, with special attention to the points (0, 0) and (1, r) where r is
the unit rate.
7.RP.3: Use proportional relationships to solve multistep ratio and percent problems.
Examples: simple interest, tax, markups and markdowns, gratuities and commissions,
fees, percent increase and decrease, percent error.
These standards are major content in the seventh grade based on the PARCC
Model Content Frameworks.
Most
Relevant
Standards for
Mathematical
Practice
Item
Description
and
Assessment
Qualities
Students creating reasoned estimates must be able to reason abstractly and
quantitatively in order to build a model of the situation that is accurate enough for the
given situation (MP.2 and MP.4). Working with ambiguity is an important part of the
modeling skills expected at high school, and this requires students to productively
engage with the quantities given in the context. Modeling is a critical component of
the high school standards, and this item requires students to create a model from a
complex situation to make a real -world estimate given an unscaffolded situation where
a model is a useful tool. To make this model, students will have to reason with the
given quantities, their units and the proportional relationships between them. This will
require students to understand how to use the numbers mathematically, and then be
able to periodically check their own understanding of what those numbers mean.
This application task requires students to create a reasoned estimate in response to
solve a real-world problem. Students must first wrestle with the data displayed on the
Popcorn Inventory page . They should recognize that the amounts in the table are
decreasing because each day boxes and popcorn seeds are used. This means that
students should recognize that they are using about 15 medium boxes each day and
about 25-30 small boxes each day.
In order to address the amount of popcorn sold over the weekend, students must first
create a viable estimate of the number of cups of popcorn sold, then use the ratio
cups of popcorn seed:8 cups of popcorn. Students may choose to use this method to
estimate the amount of popcorn seed used Sunday through Thursday; however, other
methods could be used to determine the amount of popcorn seed used each day.
Students using this method must be sure to account for the amount o f popcorn seed
used on Sunday because the original information starts from end of day on Sunday.
The final estimate requires students to use the current amount of popcorn seed, the
amount used Friday and Saturday, and the amount used Sunday through Thursda y in
order to estimate the amount of popcorn seeds she should purchase so there are 100200 pounds left over next Friday morning.
Note that ratio and proportional relationships are key skills required for college and
career readiness, and this item provides a strong application of that content. Unlike
traditional multiple choice, it is difficult to guess the correct answer or use a choice
elimination strategy.
Scoring Rubric for HS.D.1-1
Task is worth 6 points. Task can be scored as 0, 1, 2, 3, 4, 5, or 6.
Scoring consists of 2 points for calculation and 4 points for modeling.
Structure (6 points total):
-
Scoring
Information
-
-
2 points for correctly addressing the cups of popcorn seed needed for SundayThursday
o 1 calculation point for adequate estimate
o 1 modeling point for adequate estimation strategy
3 points for correctly addressing the cups of popcorn seed needed for Friday
and Saturday.
o 1 modeling point for adequate estimation strategy for addressing two
sizes of boxes for both days.
o 1 modeling point for accurate use of the proportion of popcorn seed to
popcorn
o 1 calculation point for adequate estimate
1 point for correctly estimating the amount of popcorn seed that should be
ordered
o 1 modeling point for adequate estimation strategy (the calculation is
not as important as the strategy)
Example student response:
On Friday and Saturday, they will sell about 500 large boxes (250 + 250 = 500). I found
that they sold about 17 medium boxes (
(
–
) and about 30 small boxes
) each day in the table, so they would sell about 68 (2 x 17 for both
–
days) medium boxes and 120 (2 x 30 for both days) small boxes on Friday and Saturday
combined.
That means they need to pop:



Large:
Medium (approx.):
Small (approx.):
I added the three amounts of popcorn to find that they will need about 12,400 cups of
popcorn over the weekend.
Since -cups of popcorn seed makes 8 cups of popcorn, I know that 1 cup of popcorn
seed will make 24 cups of popcorn. That means that they need about
cups of popcorn seed for Friday and Saturday. So, they will need about 525 cups of
popcorn seed for the weekend.
According to the table, they used about 80 cups of popcorn seed each of the remaining
days of the week (
–
). They will need 80 x 5 or 400 cups of popcorn
seed for Sunday-Thursday.
I made this list to make sure she buys enough:
( 69.7 cups currrently)
525 cups of popcorn seed for Friday and Saturday
400 cups of popcorn seed for Sunday-Thursday
+ 100 extra cups to make sure she is between 100 and 200 cups on Friday morning
1,025 cups of popcorn seed to order in the morning
NOTE: There are a wide variety of estimation strategies that can receive full credit.
HS
Type
Evidence
Statement
Brett’s race
Type III 3 Points
HS.D.2-5: Solve multi-step contextual word problems with degree of
difficulty appropriate to the course, requiring application of course-level
knowledge and skills articulated in A-CED, N-Q, A-SSE.3, A-REI.6, A-REI.12,
A-REI.11-2, limited to linear equations and exponential equations with
integer exponents.
Clarification:
A-CED is the primary content; other listed content elements may be
involved in tasks as well.
Most Relevant
Standards for
Mathematical
Content
Most Relevant
Standards for
Mathematical
Practice
Item Description
and Assessment
Qualities
A-CED Creating Equations
A-CED.A Create equations that describe numbers or relationships
2. Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and
scales.
This standard is major content in the course based on the PARCC Model
Content Frameworks.
This item requires students to model the given situation using equations,
then students use that model to determine who will win the race and their
margin of victory (MP.4). In order to create and interpret these models,
students will have to decontextualize and contextualize the information at
various points in the solution process to create a mathematical model and
then to interpret the meaning and structure of that model (MP.2). Students
that choose to use the graph may create another model of the situation,
and look for and use structure within that model (MP.7).
This application task requires students to use content from widely applicable
algebra standards in order to solve a modeling problem with difficulty
expected in high school. Students first create equations that model the
situation described in the first paragraph. It is important for students to
define their variables when creating equations. Then, students reason with
their models, and perhaps the graphing tool, to interpret the model and
determine the margin of victory. There are a variety of solution methods
that students may use to successfully answer Part B.
Scoring Rubric for Sample Item HS.D. 2-5
Scoring
Information
Task is worth 3 points. Task can be scored as 0, 1, 2, or 3.
Task has 2 parts.
Scoring for Part A – Formulating the Model – 1 point
Student produces two equations to determine the distance in meters from
the starting line, of each person as a function of the time x, in seconds since
the Olympian starts running.
For example, Brett’s distance y, as related to time, x:
3
 = 8 1  + 20
. Or y = 12
100
x + 20
The Olympian’s distance y, as related to time, x:
 = 10 .
NOTE: All variables should be defined. The student may choose to define x
as time in seconds since Brett starts running.
Scoring for Part B
Student earns 1 calculation point for stating the correct winner and the
correct margin of victory.
Students earn 1 modeling point for providing an accurate justification using
the equations in Part A.
Sample Student Response 1:
• For Brett,  = 100 when
100 = 8 1  +
20
13
80 = 8 3 
 = 9.6
• For the Olympian  = 100 when
100 = 10
.
 = 10
• So, Brett wins the race by
seconds.
10 – 9.6 = 0.4
Sample Student Response 2 :
• When Brett finishes the race at 9.6 seconds, the Olympian is only
10(9.6) = 96 meters from the start. Therefore, Brett was 4 meters
ahead of the Olympian when he finished the race.
Note:
• If Part A contains incorrect equations, but Part B is correct based on
one or two incorrect equations in Part A, the student is still
awarded 1 or 2 points of the 3 possible points.
Task score: The task score is the sum of the points awarded in each
component.
Lessons Learned from
OOOPPPSSS
 1. Always do Lab Your Self…..
 2. Don’t always give them the tools,
allow them to ask for them
 3. Evaluate effectiveness of lab
 4. Evaluate effectivness of reporting
(group participation)
ParCC and CCSS
Curriculumn
 Discussion: Groups 5 ways to aid in
implementation
 5 hurdles you see
 5 ways to overcome those obstacles.
Questions ???????
 Contact Information
 Don Biery
 [email protected]