The Number System (7.NS.1.C)

The Number System (7.NS.1.c) N.S. Apply and extend previous understanding of operations with fractions to add, subtract, multiply, and divide rational numbers. 1. Apply and extend previous understanding of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal and vertical number line diagram. c. Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

Goal: Students will be able to (SWBAT) explain why subtraction of rational numbers is the same as adding the additive inverse; and will be able to demonstrate it using a variety of mathematical models in other real world context.

Develop Understanding/Launch: a. Can you breathe? b. Patterns in Adding Goals: We want to surface ideas such as modeling adding and subtracting integers. We will highlight the fact that subtracting is the same as addition of the inverse.

Solidify: a. What’s your story? Goals: Expand their understanding into real world context.

Practice: a. Two Equal Expressions Goals: Students will demonstrate fluency in rewriting subtraction problems as addition problems.

Extension: Hook to the commutative property. Can You Breathe?

You dive off the diving board. You descend 15 feet. Then you rise 3 feet. Can you breathe?

You dive off the diving board. You descend 12 feet. Then you descend another 2 feet. Then you rise 4 feet. Can you breathe?

You dive off the diving board. You descend 5 feet. Can you breathe? Can You Breathe?

Anticipated Responses

You dive off the diving board. You descend 15 feet. Then you rise 3 feet. Can you breathe?

 No, you can’t breathe because you’re still under water. (Students may be more specific about how far under water they are.)  Maybe. (Students may question variables like position of their head relative to the body’s depth in the water, the height of the diving board relative to the pool deck or the surface of the water, etc.)  Yes. (Students don’t understand the task or are still questioning variables.)

You dive off the diving board. You descend 12 feet. Then you descend another 2 feet. Then you rise 4 feet. Can you breathe?

 Yes, your head is out of the water.  Maybe, students wonder if being at the surface enables them to breathe or not (are your face up or face down? is half of your mouth above water?)  No. (Students don’t understand task or miscalculated integers.)

You dive off the diving board. You descend 5 feet. Can you breathe?

 Yes, you haven’t reached the water yet.  Maybe. (Students wonder if this is a legitimate question because nobody can stop their dive in midair.)  No. (Students argue that this scenario is not possible.)

Questions to explore: What information is given? What variables do you think are important to consider? How can you compare being under water to negative integers?

Patterns in Adding 1.) Make the expression on the left hand side of the equation unique and then give the solution for the expression on the right hand side of the equation. You need to experiment with the signs and operations to achieve uniqueness. Be sure to only use addition and subtraction.

The first one is done for you.

+ + 2 + 3 Mod el

2 3 Mod el

2.) Now, represent each equation above using two different models.

3.) Which equations are equivalent? How do you know?

4.) Do you notice any patterns emerging?

5.) Can you make a generalization? What is it and why does it work? Can you find a case in which the generalization is not true? Patterns in Adding Editable Table 2 3 Model = 2 3 Model = 2 3 Model = 2 3 Model = 2 3 Model = 2 3 Model = 2 3 Model = 2 3 Model = Patterns in Adding

Teachers: Be sure to have physical manipulatives for your students to physically model the situations. Be sure that the students have experience using them. Students can use tables, chips, gummy bears, number line, thermometer, etc… to model positive and negative numbers.

Which equations are equivalent? How do you know?

 Because the answer the right side of the equation are the same. Do you notice any patterns emerging? If so, describe the pattern.

 The answers are some variation of one and five.  The order doesn’t matter (of the operation and the sign of the second number doesn’t matter).  Where both are negative, ends up positive.  Where both are positive, ends up positive.

Can you make a generalization?

 Adding a negative number is the same as subtracting that positive number.  Subtracting a negative number is the same as adding that positive number.  Adding a positive is the same as subtracting a negative.  Subtracting a positive is the same as adding that negative. Is adding the same as subtracting the opposite?

Anticipation:

Can make the left hand side unique, but then mess up the right hand side because doesn’t know the rules. Is this task effective???? They would need to feel very comfortable with adding and subtracting using models to get the sum right.

Needs to know vocabulary: addend, operation, sum.

Are the equations unique if the answer is the same? / What does it mean to be unique.

What if they never think of the word opposite?

What’s Your Story?

Part I

 Create your own “Can you breathe?” story with different distances and scenarios.  Write two equivalent expressions for your story.  Why do the two expressions yield the same answer? Model your justification.

Part II  Now write a story that matches the following expression: -3 + 6 + 5.7 – 8 – (- 2)  Model the expression in two different ways.  Rewrite the given expression into an equivalent expression.

What’s your story?

Students may use colored chips, positive and negative signs, number lines, thermometers, a table to track changes, number charts, etc…

Students may have difficulty coming up with more than one model.

They may have difficulty writing an expression for their story. For example, students may confuse subtracting a positive number in the negative region with, what they really want to do, add a positive or subtract a negative in order to move towards the surface of the water.

It might be hard for them to think of the expression. I believe they will be able to get the correct answer. They may think they need to subtract to move up, when they should add. Two Equal Expressions

Write two equivalent expressions that represents the situation. Demonstrate or model why these two expressions are the same.

1. Mt. Everest, the highest elevation in Asia, is 29,028 feet above sea level. The Dead Sea, the lowest elevation, is 1,312 feet below sea level. What is the difference between these two elevations?

2. In Buffalo, New York, the temperature was -14°F in the morning. If the temperature dropped 7°F, what is the temperature now?

3. A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its new position?

4. Roman Civilization began in 509 B.C. and ended in 476 A.D. How long did Roman Civilization last?

5. A submarine was situated 450 feet below sea level. If it descends 300 feet, what is its new position?

6. In the Sahara Desert one day it was 136°F. In the Gobi Desert a temperature of -50°F was recorded. What is the difference between these two temperatures?

7. The Punic Wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last? 8. Metal mercury at room temperature is a liquid. Its melting point is -39°C. The freezing point of alcohol is -114°C. How much warmer is the melting point of mercury than the freezing point of alcohol?

Write an equivalent expression to the following expressions.

9. 3 – 5

10. 7 – 12 + 6

11. 9 + 7

12. -2v + 7

13. 3x – 4

14. 2x – (-5y)

15. 3 – 2y