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Opposites Attract!Operating with Integers

Academic Coaches – Math MeetingDecember 7, 2012

Beth SchefelkerBridget SchockConnie LaughlinKevin McLeodHank Kepner

Looking Back…What’s in the future?

Expanding the number system from the previous work we’ve done with properties.

IntegersIntegers add to number the idea of opposite, so that every number has both size and positive or negative relationship to other numbers. A negative number is the opposite of the positive number of the same size.

-VdW p. 492

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How does the concept of integers grow through 6th and 7th grade?

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Learning Intentions and Success Criteria

• We are learning to apply and extend the operations of addition and subtraction to negative numbers.

• We will be successful when we can articulate how negative numbers behave when using the properties of addition and subtraction.

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Examining Two Problems and the Standards

Connections

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Charting Mathematical Connections

Problem #1 Standards Connection

Problem #2 Standards Connection

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Problem #1: Elevation

Denver, Colorado is called “The Mile High City” because its elevation is 5280 feet above sea level. Someone tells you that the elevation of Death Valley, California is −282 feet. How many feet higher is Denver than Death Valley?

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A Standards Progression

Read 6.NS.5, 6a & c Work in pairs to study the standards Share out your understanding and examples with

the your tablemates. Come to consensus on how the standards support

the cluster statement.

Making Connections: Where does the elevation problem connect to

standards 6.NS.5, 6a and c?

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Importance of Context

With integers, students often get confused as to which number is bigger or which direction they are moving when they are doing operations, so having a context is particularly important.

VdW, 479

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Common Contexts for Integers

Linear Representations Altitude Timelines

Quantity Representations Golf scores (above/below par) Debit/Credit

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Problem #2: When’s the Freeze in Antifreeze?

• Ocean water freezes at about −3∘C . Fresh water freezes at 0∘C . Antifreeze, a liquid used to cool most car engines, freezes at −64∘C.

• Imagine that the temperature is exactly at the freezing point for ocean water. How many degrees must the temperature drop for the antifreeze to turn to ice?

• Complete independently.• Share with your table• Add to your chart.

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A Standards Progression

Read 7.NS.1 Work in pairs to study the standards Share out your understanding and examples with

the your tablemates. Come to consensus on how the standards support

the cluster statement.

Making Connections: Where does the Antifreeze problem

connect to standards 7.NS.1?

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MP2. Reason abstractly and quantitatively

As you read Math Practice Standard 2 (p.6 CCSSM): Underline key phrases that identifies student

expectations.

How did MP2 surface when working on the Elevation and Antifreeze problems?

Use a different colored marker to add ideas of MP2 to the “standards box” of your chart for each problem.

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Making Sense of Addition and Subtraction of Integers:

Listening to students…

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Looking for Counterexamples

Decide if each statement will always be true.• If the statement is not always true, show an

example for which it is false ( a counterexample).

• If it is always true, present an argument to convince others that no counterexamples can exist. You will have 3 minutes for each question

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1. “I tried four different problems in which I added a negative number and a positive number, and each time, the answer was negative. So a positive plus a negative is always a negative.”

2. “I noticed that a negative number minus a positive number will always be negative because the subtraction makes the answer even more negative.”

Listening to Students Reasoning…

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3. “I think a negative number minus another negative number will be negative because with all those minus signs it must get really negative.”

4. “A positive fraction, like ¾, minus a negative fraction, like – ½ , will always give you an answer that is more than one.”

Listening to Students Reasoning…

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Connections to MP7 and MP8

MP7 Look for make use of structure.

MP8 Look for and express regularity in repeated reasoning.

How did MP7 and MP8 surface when working on the operations with integers and counterexamples?

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Learning Intentions and Success Criteria

• We are learning to apply and extend the operations of addition and subtraction to negative numbers.

• We will be successful when we can articulate how negative numbers behave when using the properties of addition and subtraction.

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Apply: Professional Practice

• As you work in classrooms, record examples of “rules” you hear students /teachers using that could lead to misconceptions when they are operating with integers?

• Bring two examples with you to the December 21st ACM meeting.

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A Time to Reflect…

• How is the study of the standard progressions helped deepen your understanding of properties as our number system expands?