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TRANSCRIPT
Name: ____________________________________ Block:____________ Teacher: _____________
Accentuating the Negative
Integers and Rational Numbers
Show what you know and are able to do…
Quiz _____________ Quiz____________ Unit Test _____________
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Mathematical and Problem-Solving Goals
Find the mean, median, mode and range of a data set.
Relate the percents or fractions in a circle graph to the actual numbers from a situation.
Read and answer questions about histograms.
Explore the use of positive and negative numbers in real life.
Interpret and write number sentences.
Locate positive and negative numbers on a number line.
Compare and order positive and negative numbers.
Explore the relationship between a positive or negative number and its opposite.
Write number sentences to find missing parts (start, change, and end).
Use a number line model for addition and subtraction.
Use a chip model for representing addition and subtraction.
Interpret and write number sentences.
Develop algorithms for adding integers.
Model addition of integers using the chip model or the number line model.
Determine whether or not the commutative property works for addition.
Develop algorithms for subtracting integers.
Model subtraction of integers using the chip model or the number line model.
Determine whether or not the commutative property works for subtraction.
Explore the connection between addition and subtraction, and how you can use it to simplify problems.
Use the algorithms we developed to add and subtract integers.
Explore the relationship between addition and subtraction using fact families.
Solve simple questions with missing facts by using related fact families.
Use your understanding of positive and negative numbers to graph points in all four quadrants.
Extend knowledge of the number line model to explore multiplication with +/- numbers.
Develop algorithms for multiplying with positive and negative numbers.
Use number patterns to check the algorithm for multiplying with positive and negative numbers.
Explore division of +/- numbers using the relationship between multiplication and division in fact families.
Develop algorithms for dividing with positive and negative numbers.
Practice multiplying and dividing integers using the integer product game.
Explore the use of order of operations to solve problems.
Model the distributive property using areas of rectangles.
Use the distributive property with multiplication over addition.
Use the distributive property with multiplication over subtraction.
Solve problems using the distributive property.
Apply the properties of operations to add, subtract, factor, and expand algebraic expressions.
Use equivalent expressions to simplify and solve problems.
Investigation 1: Extending the Number System
Vocabulary:
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Negative NumbersDefinition: ________________________________________________________________________
Example: ________________________________________________________________________
Postitive NumbersDefinition: ________________________________________________________________________
Example: ________________________________________________________________________
OppositesDefinition: ________________________________________________________________________
Example: ________________________________________________________________________
IntegersDefinition: ________________________________________________________________________
Example: ________________________________________________________________________
Rational NumbersDefinition: ________________________________________________________________________
Example: ________________________________________________________________________
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(Difference Distance on a number line… How far apart?)
Super Brains and Rocket Scientists: __________________
Super Brains and Know-It-Alls: ______________________
Rocket Scientists and Know-It-Alls: __________________
(Number Sentence Equation)
Super Brains: 1) ____________________________
2) ____________________________
Rocket Scientists: 1) ____________________________
2) ____________________________
Know-It-Alls: 1) ____________________________
2) ____________________________
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a) ____________________________________________________________________
b) ____________________________________________________________________
c) ____________________________________________________________________
Highest: _______________________ Lowest: ______________________
(Difference Distance on a number line… How far apart?)
Super Brains and Rocket Scientists: __________________
Super Brains and Know-It-Alls: ______________________
Rocket Scientists and Know-It-Alls: __________________
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A: ________ B: _________ C: __________ D: __________ E: _________
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A B C D
8
9
A B C D
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Equation:
Equation:
Equation:
Equation:
Equation:
Equation:
11
start value + change = end value
Equation:
Equation:
Equation:
Equation:
Equation:
Equation
Equation:
12
Equation:
13
14
# Start Change End Equation
1
15
2
3
4
16
# Start Change End Equation
1add 5 black chips
2remove 5 red chips
17
3remove 3 black chips
4add 3 red chips
Start Change End Equation
Start Change End Equation
# Start Change End Equation
1
2
3
4
5
18
6
Mathematical Reflection 1In this investigation, you learned ways to order and operate with positive and negative numbers. The following questions will help you summarize what you have learned.
Think about your answers to these questions. Discuss your ideas with other students and your teacher. Then write a summary of your findings in your notebook.
1. How do you decide which of two numbers is greater whena. both numbers are positive?
b. both numbers are negative?
c. one number is positive and one number is negative?
2. What does comparing locations of numbers on a number line tell you about the numbers?
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Investigation 2: Adding and Subtracting Integers
Vocabulary:Algorithm: ________________________________________________________________________
Example: ________________________________________________________________________
Use number lines and chip boards.
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Group 1: Group 2:
Group 1: Group 2:
Group 1: Group 2:
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A B C
A B C
22
23
A B C
24
25
1) 91/4 + -51/5 = _____ 11) 61/2 + -22/6 = _____
2) 3/6 + 21/2 = _____ 12) -74/5 + 32/3 = _____
3) 41/3 + -64/5 = _____ 13) 1/2 + 51/5 = _____
4) -51/2 + -82/4 = _____ 14) -61/6 + 82/4 = _____
5) 72/5 + -71/2 = _____ 15) 71/6 + -72/3 = _____
6) -1/6 + 51/3 = _____ 16) -43/4 + 91/6 = _____
7) -81/2 + -11/5 = _____ 17) 53/5 + -11/3 = _____
8) -23/6 + 53/4 = _____ 18) 21/2 + 93/5 = _____
9) -11/2 + 42/6 = _____ 19) -33/4 + 12/4 = _____
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10) 51/2 + 1/3 = _____ 20) 2/6 + -32/3 = _____
1) 74/5 + -74/6 = _____ 11) -51/3 + -42/6 = _____
2) 72/3 + 32/5 = _____ 12) -1/4 + 21/5 = _____
3) 21/2 + 21/3 = _____ 13) 22/5 + 54/6 = _____
4) 51/3 + -1/2 = _____ 14) -92/4 + -73/5 = _____
5) 1/3 + -72/4 = _____ 15) 43/6 + -11/2 = _____
6) -51/4 + 11/3 = _____ 16) -11/6 + 41/5 = _____
7) -81/2 + 31/4 = _____ 17) 61/3 + -72/6 = _____
8) 51/5 + 41/4 = _____ 18) 75/6 + 62/4 = _____
9) -22/3 + 1/2 = _____ 19) 74/6 + -83/6 = _____
10) 61/2 + 61/2 = _____ 20) 24/5 + 25/6 = _____
1)-8.8
+ -0.5
2)8.1
+ -2.5
11)-4.1
+ 2.1
12)4.8
+ 6.0
3)-3.8
+ 8.9
4)-3.9
+ -5.0
13)9.1
+ 8.7
14)-8.7
+ 8.2
5)6.7
+ 6.1
6)-1.7
+ 7.3
15)-5.7
+ -3.8
16)-2.7
+ 9.8
7)-2.4
8)8.3
17)-1.4
18)5.9
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+ 8.6 + -0.7 + 9.1 + -8.9
9)4.9
+ -5.5
10)-4.4
+ 8.6
19)3.1
+ -10.0
20)4.5
+ 8.3
Fact Family: 3 + 2 = 5 start + change = end
5 – 3 = 2 end – start = change
Absolute value: ____________________________________________________________________
Example: ________________________________________________________________________
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Group 1: Group 2:
Group 1: Group 2:
Group 1: Group 2:
A B C
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A B C
30
31
A B C
32
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1) a)_____________________ b) _____________________ c) _____________________
2) a)_____________________ b) _____________________ c) _____________________
3) a)_____________________ b) _____________________ c) _____________________
4) a)_____________________ b) _____________________ c) _____________________
1) -32/3 - -83/6 = _____ 11) -73/5 - -82/4 = _____
2) 41/2 - -2/3 = _____ 12) 22/5 - -1/6 = _____
3) -51/4 - -71/2 = _____ 13) -82/3 - 95/6 = _____
4) 72/3 - -72/5 = _____ 14) 91/6 - 61/2 = _____
5) -11/3 - -64/6 = _____ 15) -71/4 - -84/5 = _____
6) 21/2 - -32/6 = _____ 16) -83/4 - 81/3 = _____
7) 92/4 - 81/2 = _____ 17) 42/6 - 21/3 = _____
8) -21/2 - -42/4 = _____ 18) -81/5 - -62/6 = _____
9) 81/5 - -61/4 = _____ 19) 35/6 - -54/6 = _____
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10) 43/4 - -22/3 = _____ 20) 41/2 - 13/4 = _____
1) 61/3 - -91/5 = _____ 11) -71/4 - -91/6 = _____
2) 32/3 - -32/5 = _____ 12) 83/4 - 43/6 = _____
3) 63/5 - -52/3 = _____ 13) 4/6 - -61/4 = _____
4) 13/4 - -65/6 = _____ 14) -51/5 - -83/5 = _____
5) -41/3 - -53/4 = _____ 15) 71/2 - 62/6 = _____
6) 83/4 - 24/5 = _____ 16) -52/4 - -91/6 = _____
7) 62/5 - 44/5 = _____ 17) -82/4 - -23/5 = _____
8) -31/3 - -81/2 = _____ 18) 2/6 - -42/3 = _____
9) -63/5 - -82/4 = _____ 19) -51/2 - -74/6 = _____
10) 24/5 - -61/4 = _____ 20) 82/3 - 21/4 = _____
1)-2.7
- -4.8
2)-6.3
- -9.7
11)7.0
- -4.2
12)0.4
- -8.5
3)-2.7
- -7.9
4)-5.2
- -6.8
13)9.2
- -5.5
14)7.2
- -2.1
5)0.1
- -9.6
6)-6.2
- -7.9
15)9.1
- 8.7
16)3.5
- -5.0
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7)-1.6
- -7.0
8)0.0
- -7.2
17)8.3
- -9.8
18)6.8
- -6.0
9)-2.7
- -9.6
10)-6.5
- -7.6
19)-5.6
- -5.8
20)-3.6
- -7.9
Sum Difference
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Sum Difference
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A B C
38
39
A B C
40
41
A B C
42
A B C
43
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1) 61/3 - -91/5 = _____ 11) -71/4 - -91/6 = _____
2) 32/3 - -32/5 = _____ 12) 83/4 - 43/6 = _____
3) 63/5 - -52/3 = _____ 13) 4/6 - -61/4 = _____
4) 13/4 - -65/6 = _____ 14) -51/5 - -83/5 = _____
5) -41/3 - -53/4 = _____ 15) 71/2 - 62/6 = _____
6) 83/4 - 24/5 = _____ 16) -52/4 - -91/6 = _____
7) 62/5 - 44/5 = _____ 17) -82/4 - -23/5 = _____
8) -31/3 - -81/2 = _____ 18) 2/6 - -42/3 = _____
9) -63/5 - -82/4 = _____ 19) -51/2 - -74/6 = _____
10) 24/5 - -61/4 = _____ 20) 82/3 - 21/4 = _____
1)-2.7
- -4.8
2)-6.3
- -9.7
11)7.0
- -4.2
12)0.4
- -8.5
3)-2.7
- -7.9
4)-5.2
- -6.8
13)9.2
- -5.5
14)7.2
- -2.1
5)0.1
- -9.6
6)-6.2
- -7.9
15)9.1
- 8.7
16)3.5
- -5.0
7)-1.6
8)0.0
17)8.3
18)6.8
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- -7.0 - -7.2 - -9.8 - -6.0
9)-2.7
- -9.6
10)-6.5
- -7.6
19)-5.6
- -5.8
20)-3.6
- -7.9
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Accentuate the NegativeFact Family Notes and Extra Practice
You can rewrite 3 + 2 = 5 to make a fact family that shows how the addition sentence is related to two subtraction sentences.
3 + 2 = 52 + 3 = 55 – 3 = 25 – 2 = 3
An addition fact family has two addition sentences and two subtraction sentences. The two addition sentences are different because the order of the addends are reversed; the sum is the same in both addition sentences. Both subtraction sentences begin with the number that is the sum. The subtraction sentences are different because the order of the other two numbers are reversed. Fact families may be written when an addend is a variable.
1. Complete the fact family for each number sentence.
a. ¯3 + 12 = 9 b. n + 7 = ¯9 c. ¯6 + n = 12
____________________ ____________________ ____________________
____________________ ____________________ ____________________
____________________ ____________________ ____________________
2. Write two subtraction sentences to complete the fact family for −3 + n = 45.
a. b.
c. Use one of the fact family sentences to find the value of n.
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3. Write two subtraction sentences to complete the fact family for −8 + n = 25.
a. b.
c. Use one of the fact family sentences to find the value of n.
4. Write two subtraction sentences to complete the fact family for −4 + n = 36.
a. b.
c. Use one of the fact family sentences to find the value of n.
5. Write two subtraction sentences to complete the fact family for −7 + n = 12.
a. b.
c. Use one of the fact family sentences to find the value of n.
6. Write two subtraction sentences to complete the fact family for −28 + n = 25.
a. b.
c. Use one of the fact family sentences to find the value of n.
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7. Write two subtraction sentences to complete the fact family for −2 + n = 10.
a. b.
c. Use one of the fact family sentences to find the value of n.
1. x - -71 = -23 2. 6 + y = -49 3. -49 = a - 39
4. x + -82 = -127 5. 59 = a - -12 6. x - 29 = -65
7. -92 + y = -4 8. x + -79 = -138 9. x - -62 = 138
10. -11 + y = 23 11. x - -19 = 43 12. -14 = a - -8
13. x + -73 = 11 14. 98 = a - -30 15. -31 + y = -129
16. x + -56 = -84 17. x - 27 = -101 18. -78 = a - 3
19. -15 + y = -58 20. -113 = a - 41 21. x + -67 = -163
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22. 4 + y = -50 23. x - 20 = -58 24. x + -58 = 22
25. -15 = a - -50 26. x - -100 = 148 27. 52 + y = 43
28. x - 7 = -24 29. -138 = a - 87 30. x + -42 = -132
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A: ________ B: ________ C: ________ D: ________
E: ________ F: ________ G: ________ H: ________
Quadrant I: x-value: _________ y-value: __________
Quadrant II: x-value: _________ y-value: __________ Quadrant III: x-value: _________ y-value: __________
Quadrant IV: x-value: _________ y-value: __________
A’: ________ B’: ________ C’: ________ D’: ________
E’: ________ F’: ________ G’: ________ H’: ________
Graph both A and A’; B and B’; etc.
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Accentuate the NegativeProblem 2.5 Coordinate Graphing Extra Practice
Find each ordered pair. Write the letter for the point named by the ordered pair.
1. (4, -6)
(2, 9)
(-4, -4)
(7, 4)
(-5, 3)
2. (-6, -3)
(7, 2)
(-3, -3)
(-2, 6)
(5, -3)
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Draw the ordered points on the grid. Label each point.3.
T (-1, 4)A (3, -1)E (-3, -1)B (0, -2)X (1, 2)V (-3, 5)
4.
U (3, 0)E (2, -3)Q (0, -3)F (-4, 3)Y (-2, 0)Z (-1, 5)
Complete the function table and then graph the function.5.
y = 2 - 2x x y
0
1
2
3
6.
y = 8 - 4x x y
0
1
2
3
Find each ordered pair. Write the letter for the point named by the ordered pair.7. (4, 2)_______
(5, 9)_______
(-7, -7)_______
(-8, 1)_______
(-4, -9)_______
8. (-8, 1)_______
(-4, -1)_______
(-4, 2)_______
(3, -9)_______
(3, -6)_______
Name the quadrant or on which axis the point lies.
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9. S_______
X_______
R_______
C_______
B_______
10. L_______
R_______
E_______
F_______
M_______
Complete the function table and then graph the function.11.
y = x - 2 x y
0
1
2
3
12.
y = 2x x y
-3
-2
-1
0
Find each ordered pair. Write the letter for the point named by the ordered pair.13. (-2, 5)_______
(-3, -5)_______
(1, -2)_______
(7, -5)_______
(-1, 2)_______
14. (0, -3)_______
(-2, 0)_______
(4, 2)_______
(2, 2)_______
(-1, -4)_______
Draw the ordered points on the grid. Label each point.
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15.
M (0, 2)D (-4, -5)H (6, -3)P (-3, 0)L (-2, -1)G (-4, -2)
16.
Q (-9, -8)V (-2, -3)S (-9, 1)D (0, 5)M (1, -3)C (6, -7)
Complete the function table and then graph the function.17.
y = 4x + 8 x y
-3
-2
-1
0
18.
y = x + 7 x y
-1
0
1
2
Write the coordinates for each point.19. R_______
F_______
K_______
Z_______
H_______
20. U_______
S_______
Q_______
D_______
C_______
Name the quadrant or on which axis the point lies.
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21. D_______
F_______
C_______
E_______
N_______
22. Q_______
S_______
M_______
E_______
G_______
Complete the function table and then graph the function.23.
y = 3 - 3x x y
-2
-1
0
1
24.
y = 2 + 3x x y
-1
0
1
2
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Mathematical Reflection 2In this investigation, you applied your ideas about integers to develop algorithms for calculating any sums and differences.
Think about your answers to these questions. Discuss your ideas with other students and your teacher. Then write a summary of findings in your notebook.
1. a. How can you decide if the sum of two numbers is positive, negative, or zero without actually calculating the sum?
b. How can you decide if the difference of two numbers is positive, negative, or zero without actually calculating the difference?
2. a. What procedure(s) will find the sum a + b of two numbers where a and b represent any integer?
b. What procedure(s) will find the difference a - b of two numbers where a and b represent any integer?
3. How can any difference a - b of two numbers be restated as an equivalent addition statement?
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Investigation 3
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Group 1: Group 2: Group 3:
Group 1: Group 2: Group 3:
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1)7.64
2)1.04
3)6.75
4)-3.69
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x 5.24 x 5.49 x -5.14 x -6.22
5)7.18
x 7.39
6)1.85
x 7.00
7)-4.59
x -5.21
8)-8.23
x -2.98
9)-4.87
x -1.20
10)-7.03
x -1.14
11)-6.81
x -2.03
12)-5.96
x 2.28
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1) 22/3 x 33/4 = 3) 41/2 x -43/4 = 5) -11/2 x 1/2 = 7) -22/4 x 2/3 =
2) -32/3 x -42/4 = 4) 32/4 x -32/4 = 6) -23/4 x -42/4 = 8) 2/3 x 11/2 =
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Group 1: Group 2: Group 3:
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Can you predict the algorithm for dividing integers based on these
fact family examples?
Group 1: Group 2: Group 3:
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Mathematical Reflection 3In the problems of this investigation you studied ways to use multiplication and division of integers to answer questions about speed, time, distance, and direction of motion. You used the results of those calculations to develop algorithms for multiplying and dividing any two integers. The questions that follow should help you to summarize your findings.
Think about your answers to these questions. Discuss your ideas with other students and your teacher. Then write a summary of your findings in your notebook.
1. How do you find the product of two numbers whena. both are positive?
b. one is positive and one is negative?
c. both are negative?
d. one is 0?
2. How do you find the quotient of two numbers whenb. both are positive?
c. one is positive and the other is negative?
d. both are negative?
e. the numerator is 0?
3. Suppose three numbers are related by an equation in the form a x b = c where a, b, and c are not equal to 0. Write two equivalent number sentences using division.
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(4) (-3) (-6) (5)
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1. 24 + 6 x 22 - 82 2. 22 - 9 - 7
3. 47 - 27 - 25 + 27 4. 45 ÷ 9 + 23 + 40 + 41 x 11
5. 14 + 44 ÷ 44 6. 45 - 3 + 24 ÷ 12 x 29
7. 42 + 24 - 25 8. 77 ÷ (3 + 8)
9. 41 + 8 + 42 - 29 10. 24 + 198 ÷ 11
11. 4 - 28 ÷ 4 + 47 12. 32 x 16 - 26 - 6 + 3
13. 8 ÷ 4 + 7 + 43 - 2 + 49 14. 31 + 100 ÷ 10
15. 88 ÷ (5 + 6) 16. 41 - 42 + 14
17. 24 - 29 + 22 + 14 18. 15 ÷ 3 + 9
19. 28 - 38 + 17 + 47 x 8 20. 20 x 32 - 180 ÷ 10
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21. 19 + 28 + 16 22. 104 ÷ 8 x 19
Area=
A=
A=
A= A=
A= A=
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1) a: ____________________________ b: ___________________________
2) a: ____________________________ b: ___________________________
3) a: ____________________________ b: ___________________________
4) a: ____________________________ b: ___________________________
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1: _____________ 2: ____________ 3:____________ 4:______________
1: _____________ 2: ____________ 3:____________ 4:______________
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Mathematical Reflection 4In this investigation, you compared important properties of arithmetic with positive numbers to properties of arithmetic with negative numbers. The following questions will help you summarize what you have learned.Think about your answers. Discuss your ideas with other students and your teacher. Then write a summary of your findings in your notebook.
1. a. What is the order of operations? Why is it important for you to understand?
b. Give an example of an equation where the use of parentheses changes the result of the computation.
2. a. What does it mean to say that an operation is commutative?
b. Which operations on integers are commutative? Give numerical examples.
3. What does it mean to say that multiplication distributes over addition and subtraction? Give numerical examples.
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Mathematical and Problem-Solving Goals
Find the mean, median, mode and range of a data set.
Relate the percents or fractions in a circle graph to the actual numbers from a situation.
Read and answer questions about histograms.
Explore the use of positive and negative numbers in real life.
Interpret and write number sentences.
Locate positive and negative numbers on a number line.
Compare and order positive and negative numbers.
Explore the relationship between a positive or negative number and its opposite.
Write number sentences to find missing parts (start, change, and end).
Use a number line model for addition and subtraction.
Use a chip model for representing addition and subtraction.
Interpret and write number sentences.
Develop algorithms for adding integers.
Model addition of integers using the chip model or the number line model.
Determine whether or not the commutative property works for addition.
Develop algorithms for subtracting integers.
Model subtraction of integers using the chip model or the number line model.
Determine whether or not the commutative property works for subtraction.
Explore the connection between addition and subtraction, and how you can use it to simplify problems.
Use the algorithms we developed to add and subtract integers.
Explore the relationship between addition and subtraction using fact families.
Solve simple questions with missing facts by using related fact families.
Use your understanding of positive and negative numbers to graph points in all four quadrants.
Extend knowledge of the number line model to explore multiplication with +/- numbers.
Develop algorithms for multiplying with positive and negative numbers.
Use number patterns to check the algorithm for multiplying with positive and negative numbers.
Explore division of +/- numbers using the relationship between multiplication and division in fact families.
Develop algorithms for dividing with positive and negative numbers.
Practice multiplying and dividing integers using the integer product game.
Explore the use of order of operations to solve problems.
Model the distributive property using areas of rectangles.
Use the distributive property with multiplication over addition.
Use the distributive property with multiplication over subtraction.
Solve problems using the distributive property.
Apply the properties of operations to add, subtract, factor, and expand algebraic expressions.
Use equivalent expressions to simplify and solve problems.
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