8th Grade Common Core Math Worksheets - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1. Which expression has the smallest value?
* Analysis: We need to compare $-\pi$, $-\sqrt{10}$, $-\frac{16}{5}$, and $-3.02$. Since they are all negative, the number with the largest absolute value (farthest from zero) is actually the smallest.
* $-\pi \approx -3.14$
* $-\sqrt{10} \approx -3.16$ (since $\sqrt{9}=3$ and $\sqrt{16}=4$, it's a bit more than 3.1)
* $-\frac{16}{5} = -3.2$
* $-3.02$
* Comparison: Looking at the values: $-3.2$ is farther to the left on the number line than $-3.16$, $-3.14$, or $-3.02$. Therefore, $-3.2$ is the smallest.
* Match: $-3.2$ corresponds to option 3 ($-\frac{16}{5}$).

2. Which number has the greatest value?
* Analysis: Compare $1\frac{2}{3}$, $\sqrt{2}$, $\frac{\pi}{2}$, and $1.5$.
* $1\frac{2}{3} \approx 1.67$
* $\sqrt{2} \approx 1.41$
* $\frac{\pi}{2} \approx \frac{3.14}{2} = 1.57$
* $1.5$
* Comparison: $1.67$ is the largest number in this list.
* Match: Option 1 ($1\frac{2}{3}$).

3. In which list are the numbers in order from least to greatest?
* Analysis: The numbers are $3.2$, $\pi$, $3\frac{1}{3}$, and $\sqrt{3}$. Let's approximate them:
* $\sqrt{3} \approx 1.73$
* $\pi \approx 3.14$
* $3.2$
* $3\frac{1}{3} \approx 3.33$
* Order: Smallest to largest: $\sqrt{3}$ ($1.73$), $\pi$ ($3.14$), $3.2$, $3\frac{1}{3}$ ($3.33$).
* Match: This order matches option 2.

4. Which numbers are arranged from smallest to largest?
* Analysis: The numbers are $3.14$, $\frac{22}{7}$, $\pi$, $\sqrt{9.1}$.
* $3.14$
* $\pi \approx 3.14159...$ (so $\pi$ is slightly larger than $3.14$)
* $\frac{22}{7} \approx 3.1428...$ (this is slightly larger than $\pi$)
* $\sqrt{9.1}$: Since $\sqrt{9} = 3$, $\sqrt{9.1}$ is just barely above 3 (approx $3.016$).
* Order: $\sqrt{9.1}$ ($~3.01$), $3.14$, $\pi$ ($~3.141$), $\frac{22}{7}$ ($~3.142$).
* Match: This order matches option 4.

5. Which list is in order from smallest value to largest value?
* Analysis: The numbers are $\sqrt{10}$, $\frac{22}{7}$, $\pi$, $3.1$.
* $3.1$
* $\pi \approx 3.14159$
* $\frac{22}{7} \approx 3.1428$
* $\sqrt{10} \approx 3.16$
* Order: $3.1$, $\pi$, $\frac{22}{7}$, $\sqrt{10}$.
* Match: This order matches option 3.

6. Which list shows the numbers $|-0.12|$, $\sqrt{\frac{1}{82}}$, $\frac{1}{8}$, $\frac{1}{9}$ in order from smallest to largest?
* Analysis:
* $|-0.12| = 0.12$
* $\sqrt{\frac{1}{82}} = \frac{1}{\sqrt{82}}$. Since $\sqrt{81}=9$, $\sqrt{82}$ is slightly larger than 9. So this fraction is slightly smaller than $\frac{1}{9}$.
* $\frac{1}{8} = 0.125$
* $\frac{1}{9} \approx 0.111$
* Comparison:
* Smallest: $\sqrt{\frac{1}{82}}$ (approx $0.110$)
* Next: $\frac{1}{9}$ (approx $0.111$)
* Next: $|-0.12|$ ($0.12$)
* Largest: $\frac{1}{8}$ ($0.125$)
* Match: Option 3 shows this order: $\sqrt{\frac{1}{82}}, |-0.12|$... wait, let me re-check option 3 vs 4.
* Option 3: $\sqrt{\frac{1}{82}}, |-0.12|, \frac{1}{9}, \frac{1}{8}$. This puts $0.12$ before $0.111$, which is wrong.
* Option 4: $\sqrt{\frac{1}{82}}, \frac{1}{9}, |-0.12|, \frac{1}{8}$. This puts $0.110, 0.111, 0.12, 0.125$. This is correct.
* Match: Option 4.

7. In which group are the numbers arranged in order from smallest value to largest value?
* Analysis: Numbers: $\pi$, $3.14$, $\sqrt{9.86}$, $\frac{22}{7}$.
* $3.14$
* $\pi \approx 3.14159$
* $\frac{22}{7} \approx 3.1428$
* $\sqrt{9.86}$: Since $\sqrt{9.86}$ is close to $\sqrt{9.86}$, let's square the others to compare or estimate. $\sqrt{9.86} \approx 3.14$ (actually $3.14^2 = 9.8596$). So $\sqrt{9.86}$ is extremely close to $3.14$, but slightly larger because $9.86 > 9.8596$. However, it is smaller than $\pi$ ($3.14159^2 \approx 9.869$).
* Let's check the decimals more carefully:
* $3.14 = 3.1400$
* $\sqrt{9.86} \approx 3.14006$
* $\pi \approx 3.14159$
* $\frac{22}{7} \approx 3.14285$
* Order: $3.14$, $\sqrt{9.86}$, $\pi$, $\frac{22}{7}$.
* Match: Option 4.

8. Which is the correct arrangement of these terms in order of value, from smallest to greatest?
* Analysis: Terms: $3\sqrt{2}$, $4\frac{1}{8}$, $|-4.24|$, $\sqrt[3]{75}$.
* $3\sqrt{2} \approx 3 \times 1.414 = 4.242$
* $4\frac{1}{8} = 4.125$
* $|-4.24| = 4.24$
* $\sqrt[3]{75}$: We know $4^3 = 64$ and $5^3 = 125$. $75$ is closer to $64$. Let's estimate $4.2^3 \approx 74.08$. So $\sqrt[3]{75}$ is approx $4.217$.
* Comparison:
* $4.125$ ($4\frac{1}{8}$)
* $4.217$ ($\sqrt[3]{75}$)
* $4.24$ ($|-4.24|$)
* $4.242$ ($3\sqrt{2}$)
* Order: $4\frac{1}{8}$, $\sqrt[3]{75}$, $|-4.24|$, $3\sqrt{2}$.
* Match: Option 3.

9. Which inequality is true if $x = \frac{3.04}{1.48}$, $y = 1.99 + 0.33$, and $z = (1.3)^3$?
* Calculate x: $3.04 \div 1.48 \approx 2.054$
* Calculate y: $1.99 + 0.33 = 2.32$
* Calculate z: $1.3 \times 1.3 \times 1.3 = 1.69 \times 1.3 = 2.197$
* Compare:
* $x \approx 2.05$
* $z \approx 2.20$
* $y = 2.32$
* Order: $x < z < y$.
* Match: Option 3.

Final Answer:
1) 3
2) 1
3) 2
4) 4
5) 3
6) 4
7) 4
8) 3
9) 3