Integers

1. Integers Mathematical Systems 7th grade Pre-Algebra

2. Teacher Page • Content: Mathematical Systems Integers • Grade Level: 7th grade • Creator: Mary Anne Burton • Objective: Students will compute the sums, differences, products or quotients using integers in equation format. • Process: MA-1 1.6, 1.8 2.2, 3.3 Integers

3. Addition of Integers • You are probably familiar with a number line. Traditionally, zero is placed in the center. Positive numbers extend to the right of zero and negative numbers extend to the left of zero. In order to add positive and negative integers, we will imagine that we are moving along a number line. • Draw a number line on your paper. Integers

4. Adding Integers on a Number Line • If asked to add 8 and -2, we would start by moving eight units to the right of zero. Then we would move two units left from there because negative numbers make us move to the left side of the number line. Since our last position is six units to the right of zero, the answer is 6. Integers

5. Adding Integers on a Number Line • If asked to add -13 and 4, we start by moving thirteen units to the left of zero. Then we move four units to the right. Since we land up nine units to the left of zero, the answer is -9. Integers

6. Adding Integers on a Number Line • If asked to add -6 and -5, first move six units to the left of zero. Then move five units further left. Since we are a total of eleven units left of zero, the answer is -11. Integers

7. Rules for Addition • Positive + Positive = Positive • Positive + Negative = Depends • Negative + Positive = Depends • Negative + Negative = Negative Integers

8. How to solve positive + negative or negative + positive • Absolute Value: This is the number of spaces from zero on the number line. • Example: the absolute value of 10 is 10 the absolute value of -10 is 10 Once you find the absolute value of each number you subtract and take the sign of the greater number. Integers

9. Subtraction with Integers • The technique for changing subtraction problems into addition problems is extremely mechanical. There are two steps: • Change the subtraction sign into an addition sign. • Take the opposite of the number that immediately follows the newly placed addition sign. Integers

10. Subtraction with Integers • Let's take a look at the problem: 3 – 4 = x • According to step #1, we have to change the subtraction sign to an addition sign. • According to step #2, we have to take the opposite of 4, which is -4. • Therefore the problem becomes • 3 + (-4) = x • Using the rules for addition, the answer is -1. Integers

11. Subtraction with Integers • Here is another problem: -2 – 8 = x • Switching the problem to an addition problem, it becomes -2 + (-8) = x, which is equal to -10. • 6 - (-20) = x is equal to 6 + 20, which is 26. • -7 - (-1) = x is the same as -7 + 1, which is -6. Integers

12. Rules for Multiplication • Positive x Positive = Positive • Positive x Negative = Negative • Negative x Positive = Negative • Negative x Negative = Positive Integers

13. Understanding Multiplication Rules • The first rule is the easiest to remember because we learned it so long ago. Working with positive numbers under multiplication always yields positive answers. However, the last three rules are a bit more challenging to understand. Integers

14. Multiplication with Integers • The second and third rules can be explained simultaneously. This is because numbers can be multiplied in any order. -3 x 7 has the same answer as 7 x -3, which is always true for all integers. [This example demonstrated the commutative property.] Integers

15. Multiplication with Integers • The fourth rules follows the pattern that when two numbers of the same sign are multiplied the answer is always positive. • -8 x 3 = -24 • -8 x 2 = -16 • -8 x 1 = -8 • -8 x 0 = 0 • -8 x -1 = 8 • -8 x -2 = 16 • -8 x -3 = 24 Integers

16. Here are some examples: • 5 x -10 = ? • 0 x 8 = ? • -10 x 9 = ? • 6 x 6 = ? • 7 x -6 = ? Integers

17. Division with Integers • The rules for division are exactly the same as those for multiplication. If we were to take the rules for multiplication and change the multiplication signs to division signs, we would have an accurate set of rules for division. Integers

18. Rules for Division with Integers • Positive ÷ Positive = Positive • Positive ÷ Negative = Negative • Negative ÷ Positive = Negative • Negative ÷ Negative = Positive Integers

19. Division with Integers • -9 ÷ 3 = -320 ÷ (-4) = -5 • -18 ÷ (-3) = 6 Now you try the ones below: • 36 ÷ (6) = ? • 8 ÷ (-7) = ? • -63 ÷ (-9) = ? • 18 ÷ (-2) = ? • -70 ÷ (-10) = ? Integers

20. Now You Are the Teachers • Group Presentation demonstrating your knowledge of the rules of Integers. • 2-3 students per group • Choose one of the following presentations: Cartoon Creations Demonstration Dilemma Sing Us A Song Thriving Thespians Integers

21. Cartoon Creations •  This project must contain a comic strip that demonstrates your understanding of Integers. It should consist of eight panels, minimum. Elements such as artistic quality and mathematical accuracy will be checked. Cartoons that contain those items as well as a creative component will receive more points. Integers

22. Demonstration Dilemma • This project is appropriate for students who enjoy showing people how to do new things and/or creating models. It may also include a physical explanation on how to solve problems with Integers using mathematical tools. Students may also explain how and where Integers are used in our everyday world. This project, although open-ended, must involve student acquired props and models. The more involved the class gets, the more points will be awarded. It must involve at least a five minute presentation to the class. Integers

23. Sing Us A Song • This project is appropriate for those who enjoy using their voice, It involves creating lyrics, possibly an original work or a parody of something already in existence (like the theme to a television program), that demonstrates your understanding of Integers. The song must be presented to the teacher as a hardcopy and must be at least a page in length.     The actual song will be sung in class. The singer may choose to supplement this presentation with recorded music or a played instrument, which may be a team effort -- two people maximum. Part of this grade will be based on the emotional delivery of the piece. Integers

24. Thriving Thespian • A skit will be produced and it involves creating a script, possibly an original work or a play on something already in existence [pun intended], that demonstrates your understanding of Integers. The script must be presented to the teacher as a hardcopy and must be at least four pages in length.     The actual presentation must be at least five to ten minutes. This project can involve a team effort, up to three people maximum. The team may choose to supplement this presentation with props and music. Part of this grade will be based on the emotional delivery of the skit. Integers

25. Scoring GuideProjects will be worth a maximum of 100 points! Factor Weak ----Strong Creativity 0   1   2   3   4   5 Math 0 1 2 3 4 5 Content   0  1   2   3   4   5 Neatness 0   1   2   3   4   5 Presentation 0   1   2   3   4   5 • Once the factors are determined they will be added together and then multiplied by 5 to generate the points for the project Integers