Question

understand multiplication and division of rational numbers
model with mathematics for problems 1 and 2, model the situation with a multiplication and an addition expression involving negative numbers. then evaluate.

1. after a gymnastics competition, the coaches reviewed all the gymnasts scores to identify areas for improvement. a missed landing on an aerial cartwheel deducts 2 points from the score. the coaches found that aerial cartwheels were missed 5 times across the competition. how did these missed landings affect the scores overall?
2. the temperature fell by 3°f every hour during a 6 - hour period. what was the overall change in temperature during the 6 - hour period?
3. use repeated reasoning complete the statements. explain.

-4(5 + (-2)) = ( )(5)+( )(-2)
-4( ) = -20+
this means that (-4)(-2)=
-12 = -12
identify the sign of the product (positive or negative). do not evaluate.

1. -819×(-324)
2. -1,201×54
3. 10,005×(-84)
4. construct arguments dario said that if the dividend and divisor have the same sign, then the quotient also has that same sign. do you agree or disagree? explain.
5. use structure explain how you can use the fact that multiplication and division are inverse operations to help you determine whether the quotient -10÷(-5) is positive or negative.

Explanation:

Step1: Analyze problem 1

Each missed landing deducts 2 points (represented as - 2), and there are 5 missed landings. The multiplication expression is \(5\times(-2)\). Using the rule that a positive times a negative is negative, \(5\times(-2)=-10\). The addition expression would be \((-2)+(-2)+(-2)+(-2)+(-2)\), which also equals \(-10\).

Step2: Analyze problem 2

The temperature falls by \(3^{\circ}F\) (represented as - 3) each hour for 6 hours. The multiplication expression is \(6\times(-3)\). Using the rule that a positive times a negative is negative, \(6\times(-3)=-18\). The addition expression would be \((-3)+(-3)+(-3)+(-3)+(-3)+(-3)\), which also equals \(-18\).

Step3: Analyze problem 3

Using the distributive property \(a(b + c)=ab+ac\), for \(-4(5+(-2))\), we have \(-4(5)+(-4)(-2)\). Then \(5+(-2)=3\), so \(-4\times3=-12\), and \(-20 + 8=-12\). This means \((-4)(-2)=8\).

Step4: Analyze problem 4

For \(-819\times(-324)\), since a negative times a negative is positive, the product is positive.

Step5: Analyze problem 5

For \(-1201\times54\), since a negative times a positive is negative, the product is negative.

Step6: Analyze problem 6

For \(10005\times(-84)\), since a positive times a negative is negative, the product is negative.

Step7: Analyze problem 7

Agree. If \(a\) and \(b\) are non - zero numbers, and \(\frac{a}{b}=c\), then \(a = b\times c\). If \(a\) and \(b\) have the same sign (both positive or both negative), then \(c\) must be positive. For example, if \(a = 6\), \(b = 2\), then \(c = 3\); if \(a=-6\), \(b=-2\), then \(c = 3\).

Step8: Analyze problem 8

We know that if \(\frac{-10}{-5}=x\), then \(-10=-5\times x\). We are looking for a number \(x\) such that when we multiply \(-5\) by \(x\) we get \(-10\). Since \(-5\times2=-10\), and using the relationship between multiplication and division (inverse operations), the quotient \(\frac{-10}{-5}\) is positive.

Answer:

1. Multiplication: \(5\times(-2)=-10\), Addition: \((-2)+(-2)+(-2)+(-2)+(-2)=-10\). The overall effect is a deduction of 10 points.
2. Multiplication: \(6\times(-3)=-18\), Addition: \((-3)+(-3)+(-3)+(-3)+(-3)+(-3)=-18\). The overall change in temperature is a decrease of \(18^{\circ}F\).
3. \(-4(5+(-2))=(-4)(5)+(-4)(-2)\), \(-4(3)=-20 + 8\), \((-4)(-2)=8\).
4. Positive.
5. Negative.
6. Negative.
7. Agree. If \(a\) and \(b\) (non - zero) have the same sign, and \(\frac{a}{b}=c\), then \(a = b\times c\). Since \(a\) and \(b\) have the same sign, \(c\) must be positive.
8. If \(\frac{-10}{-5}=x\), then \(-10=-5\times x\). We know that \(-5\times2=-10\). Using the inverse relationship between multiplication and division, the quotient is positive.