College and Career Readiness ConferenceSummer

2014

COHERENCETHE IMPORTANCE

OF COHERENT LESSONS IN HIGH

SCHOOL MATHEMATICS

Focus Coherence

Rigor1. Conceptual Understanding2. Procedural Skill3. Modeling/Application

SHIFTS

COHERENCECoherence 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!

COHERENCEA 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.

Four Corners

Teacher, Not HS Math

Not a Teacher

HS Math Teacher, taught

Functions HS Math Teacher, hasn’t taught

Functions

COHERENCE CONTENT

The writers of the CCSS:

Built COHERENCE into the content standards.

Clarified and shared their vision of grade-to-grade and within-grade COHERENCE in the Progressions documents.

GRADE-TO-GRADE COHERENCE

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

http://www.illustrativemathematics.org/illustrations/588

DS

Do calculations here:

http://www.illustrativemathematics.org/illustrations/588

DS

Sketch graph here

http://www.illustrativemathematics.org/illustrations/588

Questions:

Answers

GRADE-TO-GRADE

INSTRUCTION

Grade From Grade-to-Grade

8th 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).

HSALG 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 appropriatetype of model function F.IF.7b, 7c, 7e, 8, 9

HSCourse

4

Build a function that models a relationship between twoquantities. Include all types of functions studiedF.BF.1b (NOT REQUIRED FOR ALL STUDENTS! This is a PLUS Standard)

WITHIN-GRADE COHERENCE

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?

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 , the 𝟐penalty 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

INSTRUCTION

DS

Do calculations here:

INSTRUCTION

Do calculations here:

DS

WITHIN-GRADE COHERENCE

INSTRUCTION

Grade From Domain-to-Domain

HS ALG I

Interpret functions that arise in applications in terms of a context. Linear, exponential, and quadratic F.IF.4, 5, 6

HS ALG I

Create equations that describe numbers or relationships. Linear, quadratic, and exponential (integerinputs only); for A.CED.3 linear only A.CED.1, 2, 3, 4

HS ALG I

Represent and solve equations and inequalities graphically. Linear and exponential;learn as general principle A.REI.10, 11, 12

HS ALG I

Construct and compare linear, quadratic, and exponential models and solve problems. F.LE.1a, 1b, 1c, 2, 3

COHERENCEASSESSMENT

Evidence of Coherence in assessment is not tremendously different from evidence of coherence in classroom instruction.

High School ContentInterpreting 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

AS

SES

SM

EN

T

COHERENCE

COHERENCE

ASSESSMENT

https://www.parcconline.org/samples/mathematics/high-school-functions

COHERENCE

ASSESSMENT

https://www.parcconline.org/samples/mathematics/high-school-functions

Do calculations here:

DS

COHERENCEASSESSMENT

•Which other Standards for Mathematical Practice are assessed in this task? MP6- Attend to Precision

• Content standards F- ‐IF.9. Compareproperties of two functions each represented in a different way (algebraically, graphically,Numerically in tables, or by verbaldescriptions).

(According to PARCC)

COHERENCEASSESSMENT

•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.

Shift of Coherence

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!