College and Career Readiness Conference Summer 2014 THE IMPORTANCE OF COHERENT LESSONS IN HIGH SCHOOL MATHEMATICS

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

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?

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.

DS Do calculations here:

DS Sketch graph here

Questions: Answers

INSTRUCTION GradeFrom 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). 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 HS Course 4 Build a function that models a relationship between two quantities. Include all types of functions studied F.BF.1b (NOT REQUIRED FOR ALL STUDENTS! This is a 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?

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, the penalty is $2. On day 3, the penalty is $30. On day 4, the penalty is $4 and so on, increasing by $10 each day the equipment is late. Company 2: On day 1, the penalty is $.. On day, the penalty is $.3. On day 3, the penalty is $.09. On day 4, the penalty is $.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

DS Do calculations here:

INSTRUCTION Do calculations here: DS

INSTRUCTION GradeFrom 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 (integer inputs 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

ASSESSMENT

ASSESSMENT Do calculations here: DS

ASSESSMENT Which other Standards for Mathematical Practice are assessed in this task? MP6- Attend 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.

INSTRUCTIONASSESSMENT 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 address. Stick your post-it on the front wall as you leave today. Thank you!