© 2012 common core, inc. all rights reserved. commoncore.org nys common core mathematics curriculum...

© 2012 Common Core, Inc. All rights reserved. commoncore.org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios

Shifts that Support Teaching, Learning, and Monitoring

Student Progress in Mathematics


N YS C O M M O N CO R E M AT H E M AT I C S C U R R I C U LU M A Story of Ratios

Session Objective

• Understand the role of the eight Standards for Mathematical Practice in Common Core aligned instruction.

• Recognize evidence of the Standards for Mathematical Practice in classroom instruction.

• Refine tools for grading and communicating progress with content and practice standards to familes.



Group Tasks

• Read the MP and A Story of Ratios examples

• As a group, discuss:

1. Restating the MP in student friendly terms

2. What student behaviors does the MP elicit?

3. Teaching practices that support the MP’s

4. Teaching practices that suppress the MP’s

• Chart the results of your discussion



Sample Chart MP.1: Make sense of problems and persevere in solving them

Student Friendly : Don’t give up on tricky problems

Content: Multiple approaches to problem solving, toolbox of strategies from which to draw

Engagement: Persistence, risk-taking

Support Suppress

• xxxxx

• xxxxx

• xxxxx

• xxxxx

• xxxxx

• xxxxx

• xxxxx

• xxxxx



MP 1: Make sense of problems and persevere in solving them.

When presented with a problem I can make a plan, carry out my plan, and evaluate its

success.

Before During After

Explain the problem to myself Persevere Check

Organize information

Monitor my work

Change my plan if it isn’t working out

Ask myself “Does this make sense?”

• Have I solved a problem like this before?

• Is my answer correct?

• How do my representations connect to my algorithm?

• What is the question I need to answer?

• What is given? What is not given?

• What are the relationships between known and unknown

quantities?

• What prior knowledge do I have to help me?

Evaluate• What worked?

• What didn’t work?

• What other strategies were used?

• How was my solution similar to or different from my

peers?

http://www.insidemathematics.org/index.php/standard-2





MP1 in action – Grade 7Jake Disston works with his middle school students to deepen their

understanding of the connections between different representations of functions—graphs, t-charts, and equations. In this clip, he begins the period by challenging his students to work at their tables to group the cards they have in front of them, creating as many different groupings as they can, and devising language to describe each group. Disston's colleague Jesse Ragent then asks the students to do a "matching game." He passes out sets of tables and equations to the students, and challenges the students to "find a triple"-- an equation, a table, and a graph that all make up a "family, triple, or set."






MP 2: Reason abstractly and quantitatively.

I can use reasoning habits to help me contextualize and decontextualize problems.

Contextualize DecontextualizeI can take numbers and put them in a real-world

context.

I can take numbers out of context and work mathematically

with them.

Reasoning habits include: 1) creating an understandable representation of the problem solved, 2) considering the units involved, 3)

attending to the meaning of the quantities, and 4) using properties to help solve problems.



MP 2 in action – Grades 5/6Fran Dickinson leads a lesson on numeric patterning, helping students to

investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, Dickinson describes the importance of individual think time before he asks his students to discuss the relative strength of two different approaches to a patterning task. One pair discusses the numbers within the sample strategy, and Dickinson repeats back their conversation to the whole group, telling his students, “I’ve heard two really good questions about Learner B’s strategy. One was, what are all these 3’s? and Kelcey’s question was, what about this 4? Where’s the 4 coming from?” This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).







MP 3: Construct viable arguments and critique the reasoning of others.

I can make conjectures and evaluate the appropriateness of peer’s thinking.

I can construct, justify, and communicate

argument by

I can critique the reasoning of others by

considering context

using examples and non-examples

using objects, drawings, diagrams and

actions

listening

comparing arguments

identifying flawed logic

asking questions to clarify or improve

arguments



MP3 in action – 6th gradeJacob Disston leads a lesson on connections between ideas about equations,

inequalities, and expressions, helping students to use mathematical vocabulary for a purpose to describe, discuss, and work with these symbol strings. In this clip, his students have grouped themselves by the common features of the symbol strings. Disston then asks the students to explain their groupings, saying “Oliver, you had another whole category. Who are you standing with? … Why are you guys saying you guys are similar? You want to tell us?” He involves the whole class in evaluating the groups’ justifications. This clip is also indicative of standard 1 (make sense of problems and persevere in solving them) and standard 6 (attend to precision).






MP 4: Model with mathematics

I can recognize math in everyday life and use math I know to real world problems.

I can

make assumptions and estimate to make complex

problems easier

identify important quantities and use tools to show

their relationships

evaluate my answer and make changes if needed

Represent Math Using

Concrete models

Pictures

Real-world situations

Oral language

Symbols



MP4 in action – Grade 7/8Cecilio Dimas leads a lesson on making comparisons between three different

financial plans, helping students use multiple representations of mathematical problems: verbal, tabular, graphical, and algebraic generalization. In this clip, Dimas connects to the prior day’s lesson, in which the class “started a conversation about the economic status of our world [and] about making responsible decisions when we’re spending our money.” His students share that they were to represent various DVD rental plans using verbal and tabular representations.




MP 5: Use appropriate tools strategically

I know when to use certain tools to help me explore math and deepen my understanding.

I have a math toolbox and I

know HOW to use math tools

know WHEN to use math tools

can reason: “Was the tool I used appropriate for the

situation?”

can reason: “Did the tool allow me to find the correct

answer?”



MP5 in action – Grade 9/10Cathy Humphreys leads an extended exploration of a proof of the properties

of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, Humphreys circulates around groups of students as they use rulers, protractors, and sheets of paper to make different kinds of quadrilaterals and examine and “guarantee” their properties. A group of students debates whether or not one student’s assertion about the properties of a trapezoid hold in all circumstances, using the kite “sticks” to illustrate their points. 9th/10th grade first video This clip is also indicative of standard 1 (make sense of problems and persevere in solving them).





MP 6: Attend to precision.

I can use precision when solving problems and communicating my ideas.

Problem Solving

I can calculate accurately.

I can calculate efficiently.

My answer matches what the problem asked

me to do – estimate or find the exact answer.

Communicating

I can SPEAK, READ, WRITE, and

LISTEN mathematically.

I can correctly use

math symbols

math vocabulary

units of measure



MP6 in action – Grades 9/10 Cathy Humphreys leads an extended exploration of a proof of the properties

of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, she orients students to the task and explains how they are to communicate their ideas to one another in their group work. This clip is also indicative ofstandard 1 (make sense of problems and persevere in solving them).








MP 7: Look for and make use of structure.

I can step back and see the connections between concepts and look closely to see the

patterns.

Algebra Geometry

For Example:

Students recognize this expression as 5 minus a positive number

times a square and use that to realize that its value cannot be

more than 5 for any real number x and y.



MP7 in action – Grades 9 - 12Carlos Cabana works with his high school English language learning students

on algebraic reasoning and multiple representations around parabolas. In this clip, his students work together in a group, clarifying each other's process and thinking. The female students clarify accurate steps for the male student. They discuss how to use the x- and y-intercepts to support their process. They step back and ask themselves what they are seeking in their work.




MP 8: Look for and express regularity in repeated reasoning.

I notice when calculations are repeated. Then, I can find more efficient methods.

Noticing the regularity in the way terms cancel when expanding

(x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1)

might lead them to the general formula for the sum of a geometric series.

As they work to solve a problem, mathematically proficient students

maintain oversight of the process, while attending to the details. They

continually evaluate the reasonableness of their intermediate results.



MP8 in action – Grade 8Jesse Ragent asks the students to do a "matching game." He passes out sets of tables

and equations to the students, and challenges the students to "find a triple"-- an equation, a table, and a graph that all make up a "family, triple, or set." He asks students to consider distinctions, characteristics, attributes as they make their grouping decisions. Students jump right in and start the matching quite eagerly. They are justifying their matches in different ways. Some are using the math vocabulary from the prior whole class discussion, while others are just matching points from the graph with the t-chart, or checking to see if an equation will generate the points on the chart or graph. Either way, students are making connections among these different representations in ways that are meaningful to them. In the clips excerpted it seems that the students are listening to one another and letting the mathematics sway the argument rather than the force of a personality.






MP Sorting Activity

Use the posters to sort the behaviors/actions into Mathematics Practice groups.



Chalk Talk

What does this mean for Common Core aligned teaching and learning at my school/district?



Student Progress

What does/doesn’t the report card tell you about the student as a mathematician?

What questions does this raise for you?

Discuss with your table group:



Revisions

What are 3-5 characteristics of a high quality tool for reporting student progress with

Common Core aligned mathematics?



Moving Forward

• What do we need to do?

• What do they need to do?

• What do I need to do to help us/them get there?



http://commoncore.americaachieves.org/module/10





Biggest Takeaway

Turn and Talk:• What questions were answered for you?• What new questions have surfaced?

© 2012 common core, inc. all rights reserved. commoncore.org nys common core mathematics curriculum...

Documents

mathematics slide

story of ratios mp1

story of ratios examples

story of ratios group

sense of problems

discussion slide

xxxxx slide

middle school students