Chapter 2 — Comparing and Ordering Real Numbers

Standard: 8.NS.1 — The student will compare and order real numbers and determine the relationships between real numbers.

By the end of this chapter you will be able to:

• Name both square roots of a number, positive and negative, for values from 0 to 400 (8.NS.1a, b)
• Estimate the two consecutive natural numbers a positive square root lies between, and justify which one is the better approximation (8.NS.1a)
• Write a rational approximation of an irrational number to the nearest hundredth (8.NS.1b)
• Locate irrational values on a number line, choosing a scale that shows the comparison you need (8.NS.1b)
• Compare and order up to five real numbers written as integers, fractions, decimals, mixed numbers, percents, scientific notation, radicals, and $\pi$, and justify the order (8.NS.1c)

Lessons: 2.1 Both Square Roots of a Number · 2.2 Between Which Two Natural Numbers? · 2.3 Rational Approximations to the Nearest Hundredth · 2.4 Locating Real Numbers on a Number Line · 2.5 Comparing and Ordering Five Real Numbers

Calculator note. A calculator is available on the Grade 8 test, and it will hand you $\sqrt{40} \approx 6.32$ without comment. That is exactly why this chapter trains the estimate by hand: 8.NS.1a asks you to name the two consecutive natural numbers a square root lies between and justify which is the better approximation, and a decimal you did not reason about cannot be justified. Every estimate here is reachable by squaring a candidate in your head or on paper. The perfect squares from $0^2$ to $20^2$, which you learned in Grade 7, are the only facts you need memorized. If you find yourself reaching for a calculator, you have usually skipped the step of asking which perfect squares the number sits between.

Numbering note. Item numbers run straight through the chapter, from 1 in Lesson 2.1 to 122 at the end of the review. They do not restart at each lesson.

Lesson 2.1 — Both Square Roots of a Number

What Grade 7 established, and what changes now

In Grade 7 you learned that a perfect square is a number formed by multiplying a whole number by itself, and that the positive square root of a perfect square is the whole number that was squared. You found $\sqrt{144} = 12$ because $12^2 = 144$, and you learned the perfect squares from $0^2 = 0$ through $20^2 = 400$.

All of that still holds. Grade 8 adds three things.

First, every positive number has two square roots, not one. Second, the radicand — the number under the radical sign — no longer has to be a perfect square, which means the root is often irrational. Third, you will be asked not just to name roots but to place them: between which whole numbers, at which decimal, in what order relative to other numbers.

Two numbers square to 144

Ask the question carefully: what numbers, when multiplied by themselves, give $144$?

$$12 \cdot 12 = 144 \qquad \text{and} \qquad (-12) \cdot (-12) = 144$$

Both work, because a negative times a negative is positive. So $144$ has two square roots, $12$ and $-12$.

The radical symbol is not ambiguous, though. The symbol $\sqrt{\ }$ always means the positive square root, also called the principal square root. To name the negative one, you write a minus sign in front:

$$\sqrt{144} = 12 \qquad -\sqrt{144} = -12$$

Read $-\sqrt{144}$ as "the opposite of the square root of 144." The minus sign is applied after the root is taken. This is the notation students most often misread, and the number line shows why the two values are genuinely different numbers.

The two roots are the same distance from zero in opposite directions. They are opposites, and $-\sqrt{81}$ is the smaller of the two — a fact that will matter constantly once you start ordering numbers.

Zero is the one exception to "two roots." Since $0 \cdot 0 = 0$ and no other number squares to $0$, we have $\sqrt{0} = 0$ and nothing else.

When the root is irrational

A rational number can be written as a fraction of two integers; as a decimal it either terminates or repeats. An irrational number cannot, and its decimal goes on forever without ever settling into a repeating block.

If the radicand is a perfect square, the root is rational — in fact an integer. If the radicand is a whole number that is not a perfect square, the root is irrational.

This chapter works with radicands from $0$ to $400$, so the perfect squares you are checking against are exactly the ones you already know.

Notice how the perfect squares thin out as you move right. Near zero they are one or two apart; near 400 they are almost forty apart. That spreading is why estimating $\sqrt{380}$ takes more care than estimating $\sqrt{8}$, and it is the reason Lesson 2.2 exists.

Worked examples

Example 1 — Both roots of a perfect square

Name both square roots of $225$.

Since $15^2 = 225$, the positive root is $15$. Its opposite also squares to $225$.

Answer: $15$ and $-15$

Example 2 — Reading the minus sign correctly

Evaluate $-\sqrt{196}$.

Take the positive square root first: $\sqrt{196} = 14$, since $14^2 = 196$. Then apply the minus sign.

Answer: $-14$

Example 3 — The root of zero

Evaluate $\sqrt{0}$.

Only $0 \cdot 0 = 0$, so there is exactly one square root here.

Answer: $0$

Example 4 — Rational or irrational

Classify $\sqrt{80}$ and $\sqrt{81}$.

$81$ is a perfect square, $9^2$, so $\sqrt{81} = 9$ is rational. $80$ falls between $8^2 = 64$ and $9^2 = 81$, so it is not a perfect square and $\sqrt{80}$ is irrational.

Answer: $\sqrt{80}$ is irrational; $\sqrt{81} = 9$ is rational.

Example 5 — A side length from an area

A square tabletop has an area of $289$ square inches. How long is each side?

Area is (side)$^2$, so the side is a square root of $289$. Since $17^2 = 289$, the two square roots are $17$ and $-17$. A length cannot be negative, so only the positive root is a possible answer.

Answer: $17$ inches. Both $17$ and $-17$ square to $289$, but a side length must be positive.

Guided practice

1. Find $\sqrt{144}$.
2. Find $-\sqrt{144}$.
3. Find $\sqrt{0}$.
4. Name both square roots of $225$.
5. Is $\sqrt{121}$ rational or irrational? Explain in one sentence.
6. Is $\sqrt{122}$ rational or irrational? Explain in one sentence.

Independent practice

7. Evaluate each. a) $\sqrt{169}$ b) $-\sqrt{196}$ c) $\sqrt{400}$ d) $-\sqrt{1}$
8. Name both square roots of $64$.
9. Which of these are irrational? $\sqrt{49}$, $\sqrt{50}$, $-\sqrt{81}$, $-\sqrt{80}$
10. True or false: $-\sqrt{36} = -6$. Justify your answer.
11. Reasoning. Both $\sqrt{100}$ and $-\sqrt{100}$ come from the number $100$. Explain why they are different numbers, and describe where each one sits on a number line.
12. Application. A square patio covers $289$ square feet. Find the length of one side, and explain why only one of the two square roots of $289$ answers the question.
13. Reasoning. Explain why $0$ has only one square root when every other perfect square has two.
14. Error analysis. A student writes $-\sqrt{49} = 7$, reasoning that "the negative signs cancel." Explain the mistake and give the correct value.

Exit ticket 2.1

15. Evaluate $\sqrt{256}$.
16. Evaluate $-\sqrt{324}$.
17. Name both square roots of $361$.
18. Explain the difference between $\sqrt{16}$ and $-\sqrt{16}$.

Lesson 2.2 — Between Which Two Natural Numbers?

Trapping a root between two perfect squares

$\sqrt{40}$ is not a whole number, because $40$ is not a perfect square. But you can trap it. Find the perfect square just below $40$ and the perfect square just above it:

$$36 < 40 < 49$$

Now take the positive square root of all three. Taking square roots preserves order for positive numbers — a bigger area means a bigger side — so the inequality survives:

$$\sqrt{36} < \sqrt{40} < \sqrt{49}, \quad \text{that is,} \quad 6 < \sqrt{40} < 7$$

$\sqrt{40}$ lies between the consecutive natural numbers $6$ and $7$. Consecutive means one right after the other, with nothing whole in between.

The whole procedure is: list the perfect squares you know, find the two that bracket your radicand, and take their roots.

Which of the two is the better approximation?

The standard asks a second question, and it is the more interesting one: is $\sqrt{40}$ closer to $6$ or to $7$?

You could try to guess from the picture, but there is an exact test that needs no decimals at all. Compare the radicand to the number halfway between the two perfect squares.

Halfway between $36$ and $49$ is $42.5$. Since $40 < 42.5$, the number $40$ sits in the lower half of that interval, so $\sqrt{40}$ sits in the lower half of the interval from $6$ to $7$. $6$ is the better approximation.

Counting the gaps says the same thing and is often faster: $40 - 36 = 4$ and $49 - 40 = 9$. Four is less than nine, so $40$ is nearer to $36$, and $\sqrt{40}$ is nearer to $6$.

Two cautions about this test.

Compare the radicands, not the roots — you do not know the root yet, which is the whole reason you are estimating. And do not eyeball "is $40$ close to $49$?" without doing the comparison. The gaps between consecutive perfect squares grow, so a radicand can look close to the upper square and still be nearer the lower one.

A closer look at a near-tie

Take $\sqrt{90}$. It sits between $\sqrt{81} = 9$ and $\sqrt{100} = 10$. Which is better?

$$90 - 81 = 9 \qquad 100 - 90 = 10$$

By one unit, $90$ is nearer to $81$, so $9$ is the better approximation. That answer surprises many students, because $90$ "feels" close to $100$. The arithmetic settles it, and it is right: $\sqrt{90} = 9.4868\ldots$, which is just short of the halfway mark $9.5$.

Boundary cases like this are common because halfway between $n^2$ and $(n+1)^2$ is $n^2 + n + 0.5$, and radicands are whole numbers. When the radicand lands exactly on $n^2 + n$ — as $90$ does for $n = 9$ — the root is just barely below the midpoint, and the smaller natural number always wins.

Worked examples

Example 1 — A straightforward estimate

Between which two consecutive natural numbers does $\sqrt{150}$ lie, and which is the better approximation?

$144 < 150 < 169$, so $12 < \sqrt{150} < 13$. Gaps: $150 - 144 = 6$ and $169 - 150 = 19$.

Answer: Between $12$ and $13$; $12$ is the better approximation.

Example 2 — Nearer the upper square

Between which two consecutive natural numbers does $\sqrt{75}$ lie, and which is better?

$64 < 75 < 81$, so $8 < \sqrt{75} < 9$. Gaps: $75 - 64 = 11$ and $81 - 75 = 6$.

Answer: Between $8$ and $9$; $9$ is the better approximation.

Example 3 — A small radicand

Estimate $\sqrt{8}$.

$4 < 8 < 9$, so $2 < \sqrt{8} < 3$. Gaps: $8 - 4 = 4$ and $9 - 8 = 1$.

Answer: Between $2$ and $3$; $3$ is the better approximation.

Example 4 — Near the top of the allowed range

Estimate $\sqrt{380}$.

$361 < 380 < 400$, so $19 < \sqrt{380} < 20$. Gaps: $380 - 361 = 19$ and $400 - 380 = 20$. The margin is one unit, but it decides the question.

Answer: Between $19$ and $20$; $19$ is the better approximation.

Example 5 — An application

A square vegetable garden has an area of $200$ square feet. Between which two whole numbers of feet is its side length, and which is the better estimate?

The side is $\sqrt{200}$. Since $196 < 200 < 225$, the side is between $14$ and $15$ feet. Gaps: $200 - 196 = 4$ and $225 - 200 = 25$.

Answer: Between $14$ and $15$ feet, and much nearer $14$ feet.

Guided practice

For items 19 through 26, name the two consecutive natural numbers the root lies between, then name the better approximation.

19. $\sqrt{40}$
20. $\sqrt{90}$
21. $\sqrt{150}$
22. $\sqrt{75}$
23. $\sqrt{20}$
24. $\sqrt{300}$
25. $\sqrt{8}$
26. $\sqrt{380}$

Independent practice

27. Name the two consecutive natural numbers each root lies between. a) $\sqrt{12}$ b) $\sqrt{55}$ c) $\sqrt{99}$ d) $\sqrt{210}$
28. For each root in item 27, name the better approximation and show the gap comparison that justifies it.
29. Estimate $\sqrt{125}$: between which two consecutive natural numbers, and which is better?
30. Estimate $\sqrt{250}$: between which two consecutive natural numbers, and which is better?
31. Estimate $\sqrt{170}$: between which two consecutive natural numbers, and which is better?
32. Reasoning. Explain why the question "between which two consecutive natural numbers does $\sqrt{400}$ lie?" has no answer.
33. Application. A square patch of lawn has an area of $130$ square meters. Between which two whole numbers of meters is one side, and which whole number is the better estimate?
34. Application. A square window has an area of $45$ square feet. Estimate the side length to the nearer whole foot, and say whether your estimate is a little too big or a little too small.
35. Reasoning. Explain why comparing $40$ to $42.5$ is enough to decide whether $\sqrt{40}$ is closer to $6$ or to $7$, even though you never compute $\sqrt{40}$.
36. Error analysis. A student says $\sqrt{90}$ is between $9$ and $10$ and is closer to $10$, "because $90$ is close to $100$." The first part is right and the second is wrong. Explain the error and give the correct better approximation.

Exit ticket 2.2

37. Between which two consecutive natural numbers does $\sqrt{60}$ lie? Which is the better approximation?
38. Between which two consecutive natural numbers does $\sqrt{110}$ lie? Which is the better approximation?
39. Between which two consecutive natural numbers does $\sqrt{45}$ lie? Which is the better approximation?
40. Describe the halfway test in your own words, as instructions someone else could follow.

Lesson 2.3 — Rational Approximations to the Nearest Hundredth

Why we approximate at all

$\sqrt{40}$ is an exact number. Its decimal expansion, $6.324555\ldots$, never terminates and never repeats, so no decimal you could finish writing is equal to it. What you can do is name a rational number that is very close, and say honestly that it is close. That is a rational approximation, and the symbol for "is approximately equal to" is $\approx$.

$$\sqrt{40} \approx 6.32$$

The equals sign would be a lie here; the wavy sign tells the truth. This standard asks for approximations to the nearest hundredth, which means two decimal places.

Squaring your way in

Lesson 2.2 got you to $6 < \sqrt{40} < 7$ and told you $6$ was the nearer whole number. Now narrow the trap. You are looking for the number whose square is $40$, so square candidates and see which side of $40$ they land on.

Step 1 — find the tenths. Try numbers between $6$ and $7$ by tenths, starting near $6.3$ since the root is in the lower half:

$$6.3^2 = 39.69 \qquad 6.4^2 = 40.96$$

$39.69 < 40 < 40.96$, so $6.3 < \sqrt{40} < 6.4$.

Step 2 — find the hundredths. Now try hundredths between $6.3$ and $6.4$:

$$6.32^2 = 39.9424 \qquad 6.33^2 = 40.0689$$

$39.9424 < 40 < 40.0689$, so $6.32 < \sqrt{40} < 6.33$.

Step 3 — round. Both endpoints are hundredths, so decide which one $40$ is nearer to: $40 - 39.9424 = 0.0576$, while $40.0689 - 40 = 0.0689$. The lower one is nearer.

$$\sqrt{40} \approx 6.32$$

Every multiplication in that process is by hand and small. The only skill it needs is multiplying a two- or three-digit decimal by itself — which is why you can carry out this whole method, justification included, faster than you can explain a calculator's decimal.

Here is the same reasoning for a radicand near the top of the range.

Negative radicals and $\pi$

A negative radical is approximated by approximating its positive partner and then attaching the minus sign:

$$\sqrt{8} \approx 2.83 \quad \Rightarrow \quad -\sqrt{8} \approx -2.83$$

Watch the rounding direction when you do this. The positive value $2.828\ldots$ rounds up to $2.83$, so the negative value $-2.828\ldots$ rounds down to $-2.83$. Both statements say the same thing: the approximation is the nearest hundredth, whichever side of zero you are on.

One irrational number in this chapter is not a radical at all. The number $\pi$, the ratio of a circle's circumference to its diameter, is irrational: $\pi = 3.14159265\ldots$, running forever without repeating. To the nearest hundredth, $\pi \approx 3.14$.

Be careful with the familiar approximations of $\pi$. Both $3.14$ and $\frac{22}{7}$ are rational numbers used in place of $\pi$; neither is $\pi$.

Under a fine enough zoom the three separate cleanly: $3.14$, then $\pi = 3.14159\ldots$, then $\frac{22}{7} = 3.142857\ldots$. So $3.14 < \pi < \frac{22}{7}$, which is worth remembering, because ordering problems like to include two of the three at once.

Worked examples

Example 1 — A familiar root

Approximate $\sqrt{2}$ to the nearest hundredth.

$1 < 2 < 4$, so the root is between $1$ and $2$. Tenths: $1.4^2 = 1.96$ and $1.5^2 = 2.25$, so it is between $1.4$ and $1.5$. Hundredths: $1.41^2 = 1.9881$ and $1.42^2 = 2.0164$. Since $2 - 1.9881 = 0.0119$ and $2.0164 - 2 = 0.0164$, the lower endpoint is nearer.

Answer: $\sqrt{2} \approx 1.41$

Example 2 — Rounding up

Approximate $\sqrt{30}$ to the nearest hundredth.

$25 < 30 < 36$, so it is between $5$ and $6$. Tenths: $5.4^2 = 29.16$ and $5.5^2 = 30.25$, so it is between $5.4$ and $5.5$. Hundredths: $5.47^2 = 29.9209$ and $5.48^2 = 30.0304$. Since $30 - 29.9209 = 0.0791$ and $30.0304 - 30 = 0.0304$, the upper endpoint is nearer.

Answer: $\sqrt{30} \approx 5.48$

Example 3 — A negative radical

Approximate $-\sqrt{150}$ to the nearest hundredth.

First $\sqrt{150}$. It is between $12$ and $13$. Tenths: $12.2^2 = 148.84$ and $12.3^2 = 151.29$, so between $12.2$ and $12.3$. Hundredths: $12.24^2 = 149.8176$ and $12.25^2 = 150.0625$. The upper endpoint is nearer, since $150 - 149.8176 = 0.1824$ and $150.0625 - 150 = 0.0625$. So $\sqrt{150} \approx 12.25$, and the opposite carries the minus sign.

Answer: $-\sqrt{150} \approx -12.25$

Example 4 — Using an approximation to compare

Which is greater, $\sqrt{50}$ or $7.1$?

$49 < 50 < 64$, so $\sqrt{50}$ is just above $7$. Hundredths: $7.07^2 = 49.9849$ and $7.08^2 = 50.1264$, so $\sqrt{50} \approx 7.07$. Compare $7.07$ with $7.10$.

You can also settle it in one step by squaring the competitor: $7.1^2 = 50.41$, which is bigger than $50$, so $7.1$ is bigger than $\sqrt{50}$.

Answer: $7.1 > \sqrt{50}$

Example 5 — $\pi$ against a radical

Which is greater, $\pi$ or $\sqrt{10}$?

$\pi \approx 3.14$. For $\sqrt{10}$: $9 < 10 < 16$, and $3.16^2 = 9.9856$ while $3.17^2 = 10.0489$, so $\sqrt{10} \approx 3.16$. Since $3.16 > 3.14$, the radical is greater. Checking against squares agrees: $\pi^2 \approx 9.87 < 10$.

Answer: $\sqrt{10} > \pi$

Guided practice

For items 41 through 48, write a rational approximation to the nearest hundredth.

41. $\sqrt{2}$
42. $\sqrt{5}$
43. $\sqrt{10}$
44. $\sqrt{30}$
45. $\sqrt{40}$
46. $-\sqrt{8}$
47. $\sqrt{200}$
48. $-\sqrt{150}$

Independent practice

49. Approximate to the nearest hundredth. a) $\sqrt{3}$ b) $\sqrt{7}$ c) $\sqrt{15}$ d) $\sqrt{20}$
50. Approximate to the nearest hundredth. a) $-\sqrt{12}$ b) $-\sqrt{50}$ c) $\sqrt{90}$ d) $-\sqrt{300}$
51. Approximate $\sqrt{110}$ to the nearest hundredth, showing the two hundredths you squared.
52. Approximate $\sqrt{250}$ to the nearest hundredth, showing the two hundredths you squared.
53. Which is greater, $\sqrt{50}$ or $7.1$? Justify with a squaring argument.
54. Which is greater, $\pi$ or $\sqrt{10}$? Justify.
55. Which is greater, $-\sqrt{26}$ or $-5.2$? Justify.
56. Application. A square rug covers $45$ square feet. Find the side length to the nearest hundredth of a foot.
57. Reasoning. Explain why $3.14$ is not equal to $\pi$, and what the symbol $\approx$ is claiming when we write $\pi \approx 3.14$.
58. Error analysis. Asked for $\sqrt{72}$ to the nearest hundredth, a student computes $8.4^2 = 70.56$, sees that $70.56 < 72$, and answers $8.4$. Explain what step is missing and give the correct approximation.

Exit ticket 2.3

59. Approximate $\sqrt{80}$ to the nearest hundredth.
60. Approximate $-\sqrt{130}$ to the nearest hundredth.
61. Which is greater, $\sqrt{99}$ or $9.9$? Justify.
62. Explain why a rational approximation can never be exactly equal to an irrational number, no matter how many decimal places you use.

Lesson 2.4 — Locating Real Numbers on a Number Line

Plotting is estimating, made visible

Every real number has exactly one home on the number line, irrational numbers included. Plotting one is a three-step routine that reuses everything from the last two lessons.

1. Bracket it. Find the two consecutive whole numbers it lies between.
2. Approximate it. Get a decimal to the nearest tenth or hundredth, depending on how fine the question is.
3. Choose a scale that can show the answer. This is the step students skip.

To place $\sqrt{55}$: it is between $7$ and $8$, since $49 < 55 < 64$. Tenths: $7.4^2 = 54.76$ and $7.5^2 = 56.25$, so it is between $7.4$ and $7.5$, and much nearer $7.4$. On a line marked in whole numbers, plot it a bit less than halfway from $7$ to $8$.

Choosing the scale

A number line marked in whole numbers from $0$ to $20$ is useless for deciding whether $\sqrt{200}$ or $14.1$ is greater. Both land in the same gap between $14$ and $15$, and no amount of squinting resolves them. A line from $14$ to $15$ marked in tenths puts $14.1$ on a tick and $\sqrt{200} \approx 14.14$ just to its right, and the question is answered.

The rule of thumb: the scale has to be finer than the difference you are trying to see. Comparing $\pi$ to $\frac{22}{7}$, whose difference is about $0.0013$, needs a line marked in thousandths — which is exactly what the figure in Lesson 2.3 does.

Negative values reverse your intuition

On the negative side, the number farther from zero is the smaller number. So $-\sqrt{50} \approx -7.07$ sits to the left of $-7$, which means $-\sqrt{50} < -7$.

Students who reason "seven point oh seven is bigger than seven" and place $-\sqrt{50}$ to the right of $-7$ have compared the distances from zero instead of the numbers. When in doubt, plot both and read left to right.

Worked examples

Example 1 — Plotting between whole numbers

Between which two tick marks on a whole-number line does $\sqrt{55}$ sit, and where in that gap?

$49 < 55 < 64$, so between $7$ and $8$. Gaps: $55 - 49 = 6$ and $64 - 55 = 9$, so it is in the lower half. To the nearest hundredth, $\sqrt{55} \approx 7.42$.

Answer: Between $7$ and $8$, a little less than halfway across, at about $7.42$.

Example 2 — A negative radical

Plot $-\sqrt{30}$ on a number line from $-6$ to $0$.

$\sqrt{30} \approx 5.48$, so $-\sqrt{30} \approx -5.48$. That is between $-6$ and $-5$, just left of halfway.

Answer: At about $-5.48$, between $-6$ and $-5$ and slightly nearer $-5.5$ than $-5$.

Example 3 — Choosing a finer scale

On a line from $3$ to $3.2$ marked in hundredths, plot $\pi$ and $\sqrt{10}$.

$\pi \approx 3.14$ goes on the tick labeled $3.14$. $\sqrt{10} \approx 3.16$ goes two ticks to its right.

Answer: $\pi$ at $3.14$ and $\sqrt{10}$ at $3.16$, with $\sqrt{10}$ to the right.

Example 4 — Reading an inequality off the line

Use a number line to decide whether $-\sqrt{50}$ is greater or less than $-7$.

$-\sqrt{50} \approx -7.07$, which lies to the left of $-7$. On a number line, left means less.

Answer: $-\sqrt{50} < -7$

Example 5 — Nearest tenth

$\sqrt{40}$ is plotted on a line marked in tenths. Which tick is it nearest?

$6.3^2 = 39.69$ and $6.4^2 = 40.96$, so it is between $6.3$ and $6.4$. Since $40 - 39.69 = 0.31$ and $40.96 - 40 = 0.96$, it is much nearer $6.3$.

Answer: The tick at $6.3$.

Guided practice

63. Between which two whole-number tick marks does $\sqrt{55}$ sit, and roughly where in that gap?
64. Plot $-\sqrt{30}$ on a number line from $-6$ to $0$. Give the value to the nearest hundredth.
65. On a line from $3$ to $3.2$ marked in hundredths, where do $\pi$ and $\sqrt{10}$ go?
66. Which lies farther right, $\pi$ or $\frac{22}{7}$? Explain what scale you need to see the difference.
67. Plot $\sqrt{2}$ on a number line from $1$ to $2$. Give the value to the nearest hundredth.
68. On a line marked in tenths, which tick is $\sqrt{40}$ nearest?

Independent practice

69. Plot $\sqrt{17}$, $\sqrt{45}$, and $\sqrt{95}$ on a number line from $0$ to $10$, and give each value to the nearest hundredth.
70. Plot $-\sqrt{12}$, $-\sqrt{50}$, and $-\sqrt{85}$ on a number line from $-10$ to $0$, and give each value to the nearest hundredth.
71. Between which two consecutive tenths does $\sqrt{60}$ lie?
72. Between which two consecutive hundredths does $\sqrt{200}$ lie?
73. On a line from $5$ to $6$ marked in tenths, plot $\sqrt{30}$ and $5.5$. Which is farther right?
74. Application. A carpenter needs to mark a length of $\sqrt{24}$ feet on a tape measure that shows tenths of a foot. Between which two tenths does the mark fall, and which tenth should the carpenter use?
75. Reasoning. Explain why a number line marked in whole numbers cannot show whether $\pi$ or $\frac{22}{7}$ is greater, and describe a scale that can.
76. Error analysis. A student plots $\sqrt{90}$ exactly halfway between $9$ and $10$. Explain why that is not quite right and say which side of halfway it belongs on.

Exit ticket 2.4

77. Plot $\sqrt{70}$ on a number line from $0$ to $10$. Which tenth is it nearest?
78. Plot $-\sqrt{45}$ on a number line from $-10$ to $0$. Give the value to the nearest hundredth.
79. Which lies farther right, $\sqrt{26}$ or $5.2$? Justify.
80. Explain how a number line shows that $-\sqrt{50} < -7$.

Lesson 2.5 — Comparing and Ordering Five Real Numbers

One list, many disguises

A single ordering problem can hand you an integer, a fraction, a decimal, a mixed number, a percent, a number in scientific notation, a radical, and $\pi$ — all at once, all in the same list. The numbers are not hard. The disguises are.

The dependable strategy is to convert every number to a decimal, rounding irrational values to the nearest hundredth, and then order the decimals. Here is the conversion table for the forms this standard uses.

Two habits make the conversions reliable. Line the decimals up with the same number of places before you compare — writing $4.5$ as $4.50$ makes the comparison to $4.47$ instant. And keep the original form written beside its decimal, because the answer must be given in the original forms, not the decimals you worked with.

A worked ordering

Order these ascending: $\sqrt{20}$, $4.5$, $\frac{21}{5}$, $430\%$, $4\frac{3}{8}$.

Ordering the right-hand column ascending gives $4.20, 4.30, 4.375, 4.47, 4.50$, so the answer, written in the original forms, is

$$\frac{21}{5} < 430\% < 4\frac{3}{8} < \sqrt{20} < 4.5$$

Ascending means least to greatest, left to right on the number line. Descending means greatest to least. Read the question twice; reversing the two is the most common way to lose an otherwise perfect answer.

Other strategies worth having

Converting everything is reliable but not always fastest. Three shortcuts:

Benchmarks. Sort into negatives, then values below $1$, then values above $1$, before doing any careful work. Anything negative is less than anything positive, immediately.

Common denominators. Comparing $\frac{5}{8}$ to $\frac{3}{5}$ is one step with a denominator of $40$: $\frac{25}{40} > \frac{24}{40}$. No decimals needed.

Squaring. To compare a radical with a decimal, square both. $\sqrt{50}$ against $7.2$: $50$ against $51.84$, so $\sqrt{50} < 7.2$. This is exact, while comparing rounded decimals is not.

Seeing it on the line

A number line is a justification, not just a picture. Plotting the five numbers and reading left to right proves the order.

Negatives need the same care as always. When every number in the list is negative and close together, zoom in — and remember that the number farthest from zero comes first in ascending order and last in descending order.

Worked examples

Example 1 — Comparing two numbers with a squaring argument

Compare $\sqrt{50}$ and $7.2$ using $<$ or $>$.

Squaring both: $(\sqrt{50})^2 = 50$ and $7.2^2 = 51.84$. Since $50 < 51.84$ and both numbers are positive, the order carries back to the originals.

Answer: $\sqrt{50} < 7.2$

Example 2 — A negative comparison

Compare $-\sqrt{30}$ and $-5.5$.

$\sqrt{30} \approx 5.48$, so $-\sqrt{30} \approx -5.48$. On the number line, $-5.48$ is to the right of $-5.5$.

Answer: $-\sqrt{30} > -5.5$

Example 3 — Scientific notation in the mix

Compare $2.5 \times 10^{2}$ and $249$.

Move the decimal point two places right: $2.5 \times 10^{2} = 250$.

Answer: $2.5 \times 10^{2} > 249$

Example 4 — Five numbers, ascending

Order ascending: $\sqrt{30}$, $5.4$, $\frac{26}{5}$, $545\%$, $5\frac{1}{2}$.

Convert: $\sqrt{30} \approx 5.48$; $5.4 = 5.40$; $\frac{26}{5} = 5.20$; $545\% = 5.45$; $5\frac{1}{2} = 5.50$.

Sorted: $5.20, 5.40, 5.45, 5.48, 5.50$.

Answer: $\frac{26}{5} < 5.4 < 545\% < \sqrt{30} < 5\frac{1}{2}$

Example 5 — Five numbers, descending, all negative

Order descending: $-\sqrt{10}$, $-3.1$, $-\frac{22}{7}$, $-3\frac{1}{4}$, $-305\%$.

Convert: $-\sqrt{10} \approx -3.16$; $-3.1 = -3.10$; $-\frac{22}{7} \approx -3.14$; $-3\frac{1}{4} = -3.25$; $-305\% = -3.05$.

Descending means greatest first, which on the negative side means closest to zero first: $-3.05, -3.10, -3.14, -3.16, -3.25$.

Answer: $-305\% > -3.1 > -\frac{22}{7} > -\sqrt{10} > -3\frac{1}{4}$

Guided practice

For items 81 through 85, compare using $<$ or $>$ and justify in one sentence.

81. $\sqrt{50}$ and $7.2$

82. $\pi$ and $3.2$

83. $-\sqrt{30}$ and $-5.5$

84. $\frac{3}{8}$ and $40\%$

85. $2.5 \times 10^{2}$ and $249$

86. Order ascending: $\frac{5}{8}$, $0.58$, $62\%$, $0.65$, $\frac{3}{5}$

87. Order ascending: $\sqrt{20}$, $4.5$, $\frac{21}{5}$, $430\%$, $4\frac{3}{8}$

88. Order descending: $-\sqrt{10}$, $-3.1$, $-\frac{22}{7}$, $-3\frac{1}{4}$, $-305\%$

Independent practice

89. Order ascending: $\sqrt{30}$, $5.4$, $\frac{26}{5}$, $545\%$, $5\frac{1}{2}$
90. Order descending: $\sqrt{72}$, $8.4$, $\frac{43}{5}$, $855\%$, $8\frac{3}{8}$
91. Order ascending: $-\sqrt{8}$, $-2.9$, $-\frac{11}{4}$, $-285\%$, $-2\frac{4}{5}$
92. Order ascending: $1.5 \times 10^{1}$, $15.2$, $\sqrt{200}$, $14\frac{1}{2}$, $1460\%$
93. Order descending: $\pi$, $\sqrt{10}$, $\frac{22}{7}$, $3.15$, $3.1$
94. Compare $\sqrt{99}$ and $9.9$, and justify with a squaring argument.
95. Compare $-\sqrt{145}$ and $-12$, and justify.
96. Application. Five square tiles have areas $5$, $6.25$, $6$, $5.76$, and $7$ square inches. Order the tiles by side length, from shortest to longest, and give each side length to the nearest hundredth of an inch.
97. Reasoning. Describe two different strategies for deciding whether $\frac{5}{8}$ or $62\%$ is greater, and carry both out.
98. Error analysis. Asked to order $-3.5$, $-3.05$, and $-3.2$ ascending, a student writes $-3.05, -3.2, -3.5$. Explain the error and give the correct order.

Exit ticket 2.5

99. Order ascending: $\sqrt{45}$, $6.8$, $\frac{33}{5}$, $690\%$, $6\frac{3}{4}$
100. Order descending: $-\sqrt{20}$, $-4.4$, $-\frac{21}{5}$, $-4\frac{1}{2}$, $-455\%$
101. Compare $7.05$ and $\sqrt{50}$ using $<$ or $>$.
102. Explain why converting every number to a decimal is a reliable ordering strategy, and give one situation where a different strategy is faster.

Chapter 2 Review

Vocabulary. perfect square · radicand · positive (principal) square root · negative square root · consecutive natural numbers · better approximation · rational number · irrational number · rational approximation · nearest hundredth · $\pi$ · scientific notation · ascending order · descending order

Part A — Estimating between consecutive natural numbers (8.NS.1a)

103. Between which two consecutive natural numbers does $\sqrt{55}$ lie? Which is the better approximation?
104. Between which two consecutive natural numbers does $\sqrt{210}$ lie? Which is the better approximation?
105. Between which two consecutive natural numbers does $\sqrt{130}$ lie? Which is the better approximation?
106. Between which two consecutive natural numbers does $\sqrt{85}$ lie? Which is the better approximation?
107. Reasoning. For $\sqrt{240}$, name the two consecutive natural numbers and the better approximation. The two gaps are close, so show the halfway comparison that decides it.

Part B — Rational approximations and the number line (8.NS.1b)

108. Approximate to the nearest hundredth. a) $\sqrt{11}$ b) $-\sqrt{29}$ c) $\sqrt{160}$ d) $-\sqrt{390}$
109. Locate $\sqrt{150}$ on a number line from $12$ to $13$ marked in tenths. Which tenth is it nearest?
110. Which is greater, $-\sqrt{82}$ or $-9.1$? Justify.
111. Between which two consecutive hundredths does $\sqrt{73}$ lie?
112. Reasoning. Explain how to place $\sqrt{2}$ and $\sqrt{3}$ on a number line between $1$ and $2$, using rational approximations, and say which is farther right.

Part C — Comparing and ordering real numbers (8.NS.1c)

113. Order ascending: $\sqrt{40}$, $6.4$, $\frac{31}{5}$, $635\%$, $6\frac{1}{4}$
114. Order descending: $-\sqrt{110}$, $-10.5$, $-\frac{53}{5}$, $-1040\%$, $-10\frac{3}{8}$
115. Order ascending: $2.5 \times 10^{-1}$, $0.3$, $\frac{7}{25}$, $26\%$, $0.245$
116. Order descending: $\pi$, $3.2$, $\sqrt{11}$, $\frac{17}{5}$, $315\%$
117. Order ascending: $\sqrt{0}$, $-\sqrt{16}$, $3.5$, $-\frac{7}{2}$, $400\%$
118. Which is greater, $\frac{5}{6}$ or $83\%$? Justify in writing.

Part D — Mixed application and reasoning

119. Application. Five square rooms have floor areas of $120$, $144$, $130$, $156$, and $121$ square feet. Order the rooms by side length from shortest to longest, give each side length to the nearest hundredth of a foot, and name the two rooms whose sides are whole numbers of feet.
120. Reasoning. A number line is marked in whole numbers from $0$ to $20$. Explain why that picture cannot show whether $\sqrt{200}$ or $14.1$ is greater, and describe what to do instead. Then answer the question.
121. Error analysis. A student writes $\sqrt{50} = 7.07$. Explain what is wrong with the equals sign, support your explanation with a computation, and write a correct statement.
122. Plot $-\sqrt{12}$, $-3.5$, $\pi$, $\sqrt{2}$, and $2.9$ on a number line, then list them in ascending order and explain how the picture justifies your list.

Standards coverage check — Chapter 2

Every radicand in this chapter lies between $0$ and $400$, every rational approximation is given to the nearest hundredth, and no ordering item asks for more than five numbers, as the standard requires.

Answer keys for every set in this chapter are in Appendix A.