Number and Operations>Ordering Real Numbers in Mathematical and Real-World Contexts(TEKS.Math.8.2.D)
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Texas 8th Grade Math › Number and Operations>Ordering Real Numbers in Mathematical and Real-World Contexts(TEKS.Math.8.2.D)
Which shows these numbers in order from least to greatest? $-2.5$, $-\sqrt{7}$, $2.8$, $\frac{7}{2}$, $\pi$
$-\sqrt{7},\ -2.5,\ 2.8,\ \pi,\ \tfrac{7}{2}$
$-\sqrt{7},\ -2.5,\ 2.8,\ \tfrac{7}{2},\ \pi$
$-2.5,\ -\sqrt{7},\ 2.8,\ \pi,\ \tfrac{7}{2}$
$-\sqrt{7},\ -2.5,\ \pi,\ 2.8,\ \tfrac{7}{2}$
Explanation
$\sqrt{7}\approx2.646\Rightarrow -\sqrt{7}\approx-2.646$, which is less than $-2.5$. For the positives, $2.8<\pi\ (\approx3.142)<\tfrac{7}{2}=3.5$. On a number line: $-\sqrt{7}$ is left of $-2.5$, then $2.8$, then $\pi$, then $\tfrac{7}{2}$.
Order from greatest to least: $\sqrt{10}$, $3.1$, $\frac{22}{7}$, $-3.05$, $-\pi$
$\sqrt{10},\ 3.1,\ \tfrac{22}{7},\ -3.05,\ -\pi$
$\sqrt{10},\ \tfrac{22}{7},\ 3.1,\ -3.05,\ -\pi$
$\tfrac{22}{7},\ \sqrt{10},\ 3.1,\ -\pi,\ -3.05$
$\sqrt{10},\ \tfrac{22}{7},\ -3.05,\ 3.1,\ -\pi$
Explanation
Approximate: $\sqrt{10}\approx3.162$, $\tfrac{22}{7}\approx3.143$, $3.1=3.1$, $-3.05=-3.05$, $-\pi\approx-3.142$. From greatest to least: $3.162>3.143>3.1>-3.05>-3.142$, so $\sqrt{10},\ \tfrac{22}{7},\ 3.1,\ -3.05,\ -\pi$. On the number line, the less negative number ($-3.05$) is to the right of $-\pi$.
Water levels (meters) at five docks are: $-\sqrt{16}$, $3.4$, $\frac{11}{3}$, $\pi$, $-3$. Which shows these values in order from least to greatest?
$-3,\ -\sqrt{16},\ \pi,\ 3.4,\ \tfrac{11}{3}$
$-\sqrt{16},\ -3,\ \tfrac{11}{3},\ \pi,\ 3.4$
$-\sqrt{16},\ -3,\ \pi,\ 3.4,\ \tfrac{11}{3}$
$-\sqrt{16},\ -3,\ 3.4,\ \pi,\ \tfrac{11}{3}$
Explanation
$-\sqrt{16}=-4$ and $-3$ are the negatives, with $-4<-3$. For the positives: $\pi\approx3.142$, $3.4=3.4$, $\tfrac{11}{3}\approx3.667$. Thus $\pi<3.4<\tfrac{11}{3}$. So the correct order is $-4,\ -3,\ \pi,\ 3.4,\ \tfrac{11}{3}$.
Which number is closest to 3.5?
$\sqrt{12}$
$\pi$
$-\sqrt{10}$
$\tfrac{7}{2}$
Explanation
Compute distances to $3.5$: $\left|\sqrt{12}-3.5\right|\approx|3.464-3.5|=0.036$, $\left|\pi-3.5\right|\approx|3.142-3.5|=0.358$, $\left|-\sqrt{10}-3.5\right|\approx|-3.162-3.5|=6.662$, and $\left|\tfrac{7}{2}-3.5\right|=0$. The smallest distance is $0$, so $\tfrac{7}{2}$ is exactly $3.5$ and is closest.
Which shows these numbers in order from least to greatest? $-\pi$, $-3.1$, $-\sqrt{2}$, $3$, $\sqrt{11}$
$-\pi,\ -3.1,\ -\sqrt{2},\ 3,\ \sqrt{11}$
$-3.1,\ -\pi,\ -\sqrt{2},\ 3,\ \sqrt{11}$
$-\pi,\ -3.1,\ -\sqrt{2},\ \sqrt{11},\ 3$
$-\pi,\ -\sqrt{2},\ -3.1,\ 3,\ \sqrt{11}$
Explanation
Approximate: $-\pi\approx-3.142$, $-3.1=-3.1$, $-\sqrt{2}\approx-1.414$, $3=3$, $\sqrt{11}\approx3.317$. On the number line, more negative numbers are less: $-3.142<-3.1<-1.414<3<3.317$. So $-\pi, -3.1, -\sqrt{2}, 3, \sqrt{11}$.