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Simplify Expressions with Square Roots
In the following exercises, simplify.
$\sqrt{4}$
$\sqrt{169}$
$\sqrt{16}$
$\sqrt{144}$
$\text{\u2212}\sqrt{100}$
$\text{\u2212}\sqrt{121}$
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write as the ratio of two integers.
ⓐ 8 ⓑ 1.61
ⓐ $\text{\u2212}12\phantom{\rule{0.2em}{0ex}}$ ⓑ 9.279
ⓐ $\frac{\mathrm{-12}}{1}$ ⓑ $\frac{9297}{1000}$
ⓐ $\text{\u2212}16$ ⓑ 4.399
In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers
$0.75,0.22\stackrel{\text{\u2013}}{3},1.39174$
ⓐ $0.75,0.22\stackrel{\text{\u2013}}{3}$ ⓑ $1.39174\text{\u2026}$
$0.36,0.94729\text{\u2026},2.52\stackrel{\text{\u2013}}{8}$
$0.4\stackrel{\text{\u2013}}{5},1.919293\text{\u2026},3.59$
ⓐ $0.4\stackrel{\text{\u2013}}{5},3.59$ ⓑ $1.919293\text{\u2026}$
$0.1\stackrel{\text{\u2013}}{3},0.42982\text{\u2026},1.875$
In the following exercises, identify whether each number is rational or irrational.
ⓐ $\sqrt{44}$ ⓑ $\sqrt{49}$
ⓐ $\sqrt{225}$ ⓑ $\sqrt{216}$
In the following exercises, identify whether each number is a real number or not a real number.
ⓐ $\text{\u2212}\sqrt{81}$ ⓑ $\sqrt{\mathrm{-121}}$
ⓐ real number ⓑ not a real number
ⓐ $\text{\u2212}\sqrt{64}$ ⓑ $\sqrt{\mathrm{-9}}$
ⓐ $\sqrt{\mathrm{-36}}$ ⓑ $\text{\u2212}\sqrt{144}$
ⓐ not a real number ⓑ real number
ⓐ $\sqrt{\mathrm{-49}}$ ⓑ $\text{\u2212}\sqrt{144}$
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
$\mathrm{-8},0,1.95286\text{\u2026},\frac{12}{5},\sqrt{36},9$
ⓐ $0,\sqrt{36},9$ ⓑ $\mathrm{-8},\sqrt{36},9$ ⓒ $\mathrm{-8},0,\frac{12}{5},\sqrt{36},9$ ⓓ $1.95286\text{\u2026}$ ⓔ $\mathrm{-8},0,1.95286\text{\u2026},\frac{12}{5},\sqrt{36},9$
$\mathrm{-9},\mathrm{-3}\frac{4}{9},\text{\u2212}\sqrt{9},0.40\stackrel{\text{\u2013}}{9},\frac{11}{6},7$
$\text{\u2212}\sqrt{100},\mathrm{-7},-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},\mathrm{-1},0.77,3\frac{1}{4}$
ⓐ none ⓑ $\text{\u2212}\sqrt{100},\mathrm{-7},\mathrm{-1}$ ⓒ $\text{\u2212}\sqrt{100},\mathrm{-7},-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},\mathrm{-1},0.77,3\frac{1}{4}$ ⓓ none ⓔ $\text{\u2212}\sqrt{100},\mathrm{-7},-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},\mathrm{-1},0.77,3\frac{1}{4}$
$\mathrm{-6},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2},0,0.\stackrel{\text{\u2014\u2014\u2014}}{714285},2\frac{1}{5},\sqrt{14}$
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
$\frac{1}{4},\frac{9}{5},\frac{11}{3}$
$\frac{7}{10},\frac{5}{2},\frac{13}{8},3$
$\frac{3}{4},-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$
$\frac{3}{4},-\phantom{\rule{0.2em}{0ex}}\frac{3}{4},1\frac{2}{3},\mathrm{-1}\frac{2}{3},\frac{5}{2},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}$
$\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{2}{5},1\frac{3}{4},\mathrm{-1}\frac{3}{4},\frac{8}{3},-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$
In the following exercises, order each of the pairs of numbers, using<or>.
$\mathrm{-1}\_\_\_-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$
<
$\mathrm{-1}\_\_\_-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}$
$\mathrm{-1}\frac{3}{4}\_\_\_\mathrm{-2}$
$-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}\_\_\_-\phantom{\rule{0.2em}{0ex}}\frac{7}{12}$
>
$-\phantom{\rule{0.2em}{0ex}}\frac{9}{10}\_\_\_-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}$
$\mathrm{-3}\_\_\_-\phantom{\rule{0.2em}{0ex}}\frac{13}{5}$
<
$\mathrm{-4}\_\_\_-\phantom{\rule{0.2em}{0ex}}\frac{23}{6}$
Locate Decimals on the Number Line In the following exercises, locate the number on the number line.
$\mathrm{-0.9}$
In the following exercises, order each pair of numbers, using<or>.
$0.86\_\_\_0.69$
$0.415\_\_\_0.41$
$\mathrm{-0.1}\_\_\_\mathrm{-0.4}$
$\mathrm{-7.31}\_\_\_\mathrm{-7.3}$
Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.
ⓐ How many busses will be needed?
ⓑ Why must the answer be a whole number?
ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?
ⓐ 4 busses ⓑ answers may vary ⓒ answers may vary
Child care Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.
ⓐ How many teachers will be needed?
ⓑ Why must the answer be a whole number?
ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?
In your own words, explain the difference between a rational number and an irrational number.
Answers may vary
Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
ⓑ On a scale of $1-10,$ how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
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