Opposites Attract! Operating with Integers Academic Coaches Math Meeting December 7, 2012 Beth Schefelker Bridget Schock Connie Laughlin Kevin McLeod Hank Kepner 1

Looking BackWhats in the future? Expanding the number system from the previous work weve done with properties. Integers Integers 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 How does the concept of integers grow through 6th and 7th grade?

2 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.

3 Examining Two Problems and the Standards Connections 4 Charting Mathematical Connections Problem #1

Standards Connection Problem #2 Standards Connection 5 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? 6 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? 7 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 8 Common Contexts for Integers Linear Representations Altitude Timelines Quantity Representations

Golf scores (above/below par) Debit/Credit 9 Problem #2: Whens the Freeze in Antifreeze? Ocean water freezes at about 3C . Fresh water freezes at 0C . Antifreeze, a liquid used to cool most car engines, freezes at 64C. 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. 10 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? 11 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. 12 Making Sense of Addition and Subtraction of Integers:

Listening to students 13 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 14 Listening to Students Reasoning 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. 15 Listening to Students Reasoning 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 , minus a negative fraction, like , will always give you an

answer that is more than one. 16 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?

17 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.

18 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. 19

A Time to Reflect How is the study of the standard progressions helped deepen your understanding of properties as our number system expands? 20