Add and Subtract Rational Numbers - 7th Grade Math

1. What is the value of $(-\frac{3}{4})+\frac{3}{4}$ using additive inverses?

   $0$. $-\frac{3}{4}$ and $\frac{3}{4}$ are additive inverses.

2. What subtraction rule rewrites $a - b$ using addition and the opposite of $b$?

   $a - b = a + (-b)$. Subtraction is adding the opposite.

3. What is the equivalent expression for $a-(b-c)$ written using distribution?

   $a-(b-c)=a-b+c$. Subtracting a difference changes the sign of the second term.

4. What is the value of $3-(4-2)$ using $a-(b-c)=a-b+c$?

   $1$. $3-4+2=1$.

5. Find the value of $\frac{2}{3} - \frac{5}{3}$ by using $a - b = a + (-b)$.

   $-1$. Same denominator: $\frac{2}{3} + (-\frac{5}{3}) = -\frac{3}{3} = -1$.

6. What is the equivalent expression for $a-(b+c)$ written using distribution?

   $a-(b+c)=a-b-c$. Subtracting a sum means subtracting each term.

7. What is the additive inverse of a rational number $a$ (the number that makes the sum $0$)?

   $-a$. The opposite of $a$ that gives $a + (-a) = 0$.

8. Find the value of $-3 - 8$ by rewriting as $-3 + (-8)$.

   $-11$. Rewrite as $-3 + (-8) = -11$.

9. What property states that $a+(-a)=0$ for every rational number $a$?

   Additive inverse property. A number plus its opposite always equals zero.

10. What is the value of $-7+12$ using addition properties?

    $5$. Start at $-7$, move right $12$ units.

11. What is the value of $-\frac{3}{4}+\frac{1}{2}$ using a common denominator?

    $-\frac{1}{4}$. Convert to fourths: $-\frac{3}{4}+\frac{2}{4}=-\frac{1}{4}$.

12. What is the additive identity property written as $a + 0 = a$ called?

    Additive identity (identity property of addition). Adding zero to any number leaves it unchanged.

13. What is the value of $-4+(-9)$ using the associative property to group signs?

    $-13$. Same signs add: $-4+(-9)=-13$.

14. What property is shown by $a + b = b + a$ when adding rational numbers?

    Commutative property of addition. Order doesn't matter when adding rational numbers.

15. What is the sign of $-(a-b)$ when rewritten using distribution?

    $-(a-b)=-a+b$. Negative distributes as $-a$ and $+b$.

16. Identify the property used to rewrite $-3 + 7$ as $7 + (-3)$.

    Commutative property of addition. Order of addends can be switched.

17. Find the value of $-8 + 3$ by using addition of a negative and a positive.

    $-5$. Add: $-8 + 3 = -5$ (more negative than positive).

18. What is the value of $(\frac{2}{3}-\frac{5}{6})+\frac{1}{6}$ using associativity?

    $0$. Combine fractions: $\frac{4}{6}-\frac{5}{6}+\frac{1}{6}=0$.

19. What is the value of $\frac{2}{3}-\frac{5}{3}$ by rewriting as $\frac{2}{3}+(-\frac{5}{3})$?

    $-1$. $\frac{2-5}{3}=\frac{-3}{3}=-1$.

20. What is the value of $9-14$ by rewriting subtraction as addition?

    $-5$. Rewrite as $9+(-14)=-5$.

21. Find the value of $12 - 7$ by rewriting subtraction as addition.

    $5$. Rewrite as $12 + (-7) = 5$.

22. What does it simplify to: $-(-a)$?

    $a$. The opposite of the opposite is the original number.

23. What is the value of $a + (-a)$ for any rational number $a$?

    $0$. A number plus its opposite always equals zero.

24. What property states that $(a+b)+c=a+(b+c)$ for all rational numbers $a,b,c$?

    Associative property of addition. Grouping doesn't change the sum of rational numbers.

25. Which property justifies rewriting $a+(-b)$ as $(-b)+a$ when adding rational numbers?

    Commutative property of addition. Addition order can be switched for any two terms.

26. Identify the property used to rewrite $(2 + (-5)) + 9$ as $2 + ((-5) + 9)$.

    Associative property of addition. Parentheses can be moved when adding.

27. Find the value of $-4 + (-9)$ using properties of addition.

    $-13$. Adding two negatives: $-4 + (-9) = -13$.

28. Find the value of $-1 + \frac{3}{4} - \frac{1}{4}$ by rewriting subtraction as addition.

    $-\frac{1}{2}$. Combine: $-1 + \frac{3}{4} + (-\frac{1}{4}) = -1 + \frac{1}{2} = -\frac{1}{2}$.

29. What is the value of $\frac{5}{6}+0$ using the additive identity property?

    $\frac{5}{6}$. Adding zero doesn't change the value.

30. Find the value of $5 - (-6)$ by rewriting as $5 + 6$.

    $11$. Subtracting negative is adding: $5 + 6 = 11$.

31. What property states that $a+b=b+a$ for all rational numbers $a$ and $b$?

    Commutative property of addition. Order doesn't matter when adding rational numbers.

32. What is the most useful way to rewrite $a+(-b)$ as a subtraction expression?

    $a+(-b)=a-b$. Adding a negative is the same as subtracting.

33. What is the value of $-3-8+3$ by regrouping additive inverses?

    $-8$. Rearrange: $-3+3-8=0-8=-8$.

34. What property states that $a(b+c)=ab+ac$ for all rational numbers $a,b,c$?

    Distributive property. Multiplication distributes over addition.

35. Find the value of $-10 - (-4)$ by rewriting subtraction as addition.

    $-6$. Rewrite as $-10 + 4 = -6$.

36. What is the value of $8+(-13)$ using addition of a negative number?

    $-5$. Adding $-13$ to $8$ gives $-5$.

37. Find the value of $15 + (-6) + (-9)$ by using associativity to group terms.

    $0$. Group: $15 + (-15) = 0$ since $(-6) + (-9) = -15$.

38. What property states that $a+0=a$ for every rational number $a$?

    Additive identity property. Adding zero to any number leaves it unchanged.

39. What is the value of $-2+7$ by reordering to add positives first?

    $5$. Add $7$ to $-2$ to get $5$.

40. What is the value of $18+(-18)+7$ by rearranging terms?

    $7$. Use $18+(-18)=0$, then $0+7=7$.

41. What is the value of $\frac{5}{6}-\frac{1}{3}$ by rewriting as addition?

    $\frac{1}{2}$. Convert to sixths: $\frac{5}{6}-\frac{2}{6}=\frac{3}{6}=\frac{1}{2}$.

42. Find the value of $6 + (-11)$ using rational number addition.

    $-5$. Add: $6 + (-11) = -5$ (more negative than positive).

43. What is $-(-a)$ equal to for any rational number $a$?

    $-(-a)=a$. The opposite of an opposite is the original number.

44. What is the value of $-6-(-2)$ using the rule $a-b=a+(-b)$?

    $-4$. $-6-(-2)=-6+2=-4$.

45. What is the value of $-4-9$ by rewriting subtraction as addition?

    $-13$. Rewrite as $-4+(-9)=-13$.

46. What is the value of $-(-a)$ for any rational number $a$?

    $-(-a)=a$. The opposite of an opposite is the original number.

47. What is the value of $-2.5-(-1.8)$ using properties of subtraction?

    $-0.7$. Rewrite as $-2.5+1.8=-0.7$.

48. Find the value of $(-2) + 9 + (-7)$ using grouping to make an easy sum.

    $0$. Group: $(-2) + (-7) + 9 = -9 + 9 = 0$.

49. What is the sign of $-(a+b)$ when rewritten using distribution?

    $-(a+b)=-a+(-b)$. Distribute the negative sign to both terms.

50. What is the value of $(-1.5+4.5)+1.5$ by regrouping to make a zero pair?

    $4.5$. $-1.5+1.5=0$, then $0+4.5=4.5$.

51. What is the value of $3-(4+2)$ using $a-(b+c)=a-b-c$?

    $-3$. $3-4-2=-3$.

52. What property is shown by $(a + b) + c = a + (b + c)$ for rational numbers?

    Associative property of addition. Grouping doesn't change the sum of rational numbers.

53. Find and correct the error: $6-(-2)=6+(-2)=4$. What is the correct value?

    Correct value: $6-(-2)=6+2=8$. Error: $-(-2)=+2$, not $-2$.

54. What is the equivalent expression for subtracting $b$ from $a$, written using addition?

    $a-b=a+(-b)$. Subtraction is the same as adding the opposite.

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