Grade 8 Math

Chapter at a Glance

• Chapter: 1
• Strand: Pure — Numbers
• Difficulty: Level 3 of 5
• Builds on: Decimals: terminating and repeating, Negative numbers and absolute value, Whole numbers to one billion

Try This First

Try this short activity before reading the lesson.

> Draw a square on grid paper with sides $10$ little squares long. Call that $1$ unit. Draw the diagonal from one corner to the opposite corner. Measure the diagonal with a ruler, in units. Is it a whole number? Is it exactly $1.4$? Is it exactly $1.41$? That length is $\sqrt{2}$. Can you find a fraction that gives it exactly?

The Big Idea

A rational number is any ratio of two integers, $\frac{p}{q}$ with $q \neq 0$. That includes every whole number. It includes every decimal that stops, and every decimal that repeats. Each of those can be written as a fraction. An irrational number is a point on the line that no fraction lands on. $\sqrt{2}$ is one. It is the length of the diagonal of a unit square. You can draw it. But its decimal, $1.41421356\ldots$, never stops and never repeats. $\pi$ is another. They are not broken numbers. They are numbers that fractions miss.

A square root is the number that squares to give what is under the sign. So $\sqrt{49} = 7$ exactly, and $\sqrt{50}$ sits just above $7$. To estimate a root, trap it between two perfect squares. Cube roots work the same way with cubes.

This is also where negative numbers get their arithmetic. Adding and subtracting is walking along the number line. Multiplying follows a pattern that nobody voted on. $3 \times 2$, $3 \times 1$, $3 \times 0$ go down by $3$ each step. So $3 \times (-1)$ has to be $-3$. And $(-3) \times 2$, $(-3) \times 1$, $(-3) \times 0$ go up by $3$. So $(-3) \times (-1)$ has to be $3$. Very big and very small numbers get scientific notation, $a \times 10^k$ with $1 \le a < 10$. Exactly one nonzero digit before the decimal point is what makes the form standard.

Words to Know

• rational: Can be written as $\frac{p}{q}$ with integers $p$ and $q$, and $q \ne 0$. Its decimal stops or repeats. $0.75 = \frac34$.
• irrational: Cannot be written as a ratio of two integers. Its decimal never stops and never repeats. $\sqrt{2}$ and $\pi$.
• real number: Any point on the number line. Every rational and every irrational number together.
• square root, cube root: $\sqrt{n}$ squares to give $n$. $\sqrt[3]{n}$ cubes to give $n$. $\sqrt{81} = 9$ and $\sqrt[3]{27} = 3$.
• scientific notation: $a \times 10^k$ with $1 \le a < 10$. $4{,}560{,}000 = 4.56 \times 10^6$. $0.00072 = 7.2 \times 10^{-4}$.

Formulas and Theorems

• Scientific notation: $a \times 10^k,\quad 1 \le a < 10$

Outcomes and Milestones

• Students can: classify rational vs irrational; $\sqrt{n}$, $\sqrt[3]{n}$; estimate $\sqrt{50}$ between 7 and 8; operate on signed numbers; scientific notation $a\times 10^k$ with $1\le a<10$.
• Watch for: $\sqrt{16/25}$ called irrational; $\pi=\frac{22}{7}$; $3.2\times 10^3$ written $32\times 10^2$ as “scientific.”
• Ready when: they can classify $\sqrt{49}$, $\sqrt{50}$, $0.\overline{3}$, $\pi$ and write $4{,}560{,}000$ in scientific notation.

Worked Examples and Your Turn

Study each example, then try the paired problem.

Example 1: Sort them, and write the big number

Problem: Say whether each of $\sqrt{49}$, $\sqrt{50}$, $0.\overline{3}$ and $\pi$ is rational or irrational, and why. Then write $4{,}560{,}000$ in scientific notation.

Solution:

• Work out $\sqrt{49}$: it is $7$, a whole number. Rational.
• Trap $\sqrt{50}$: $49 < 50 < 64$, so it is between $7$ and $8$. It is not a whole number. $50$ is not a perfect square, so its root is irrational. Its decimal, $7.0710678\ldots$, never settles.
• Turn $0.\overline{3}$ into a fraction: it is $\frac13$. Why: a repeating decimal is always a fraction. Rational.
• Look at $\pi = 3.14159265\ldots$. It has no repeating block, and it is not any fraction. Irrational. ($\frac{22}{7} = 3.142857\ldots$ is close to $\pi$, and rational, and not $\pi$.)
• Write $4{,}560{,}000$. Put the point after the first non-zero digit: $4.56$. Count how many places it moved: six. So $4.56 \times 10^6$.

Answer: rational, irrational, rational, irrational; $4.56 \times 10^6$

> Notice: A square root of a whole number is irrational exactly when that number is not a perfect square. The root sign does not make a number irrational. What is under it does.

Your Turn:

• Prompt: Rational or irrational: $\sqrt{64}$, $\sqrt{65}$, $0.\overline{7}$, $2\pi$? Then write $8{,}200{,}000$ in scientific notation.
• Answer: $\sqrt{64} = 8$, rational. $\sqrt{65}$ is irrational. $0.\overline{7} = \frac79$, rational. $2\pi$ is irrational. $8.2 \times 10^6$.

Example 2: Trap a root between two squares

Problem: Between which two whole numbers is $\sqrt{40}$? Estimate it to one decimal place. Between which two whole numbers is $\sqrt[3]{100}$?

<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 400 170" width="400" height="170" role="img" aria-label="Two number lines. On the first, √40 sits between 6 and 7, closer to 6, at about 6.32, because 6² = 36 and 7² = 49. On the second, the cube root of 100 sits between 4 and 5, at about 4.64, because 4³ = 64 and 5³ = 125. Both lines are marked in tenths."><rect x="0" y="0" width="400" height="170" fill="#ffffff"/><line x1="50.0" y1="50.0" x2="350.0" y2="50.0" stroke="#333" stroke-width="1.5"/><line x1="92.9" y1="44.0" x2="92.9" y2="56.0" stroke="#333" stroke-width="1.5"/><line x1="114.3" y1="47.0" x2="114.3" y2="53.0" stroke="#333" stroke-width="1"/><line x1="135.7" y1="47.0" x2="135.7" y2="53.0" stroke="#333" stroke-width="1"/><line x1="157.1" y1="47.0" x2="157.1" y2="53.0" stroke="#333" stroke-width="1"/><line x1="178.6" y1="47.0" x2="178.6" y2="53.0" stroke="#333" stroke-width="1"/><line x1="200.0" y1="47.0" x2="200.0" y2="53.0" stroke="#333" stroke-width="1"/><line x1="221.4" y1="47.0" x2="221.4" y2="53.0" stroke="#333" stroke-width="1"/><line x1="242.9" y1="47.0" x2="242.9" y2="53.0" stroke="#333" stroke-width="1"/><line x1="264.3" y1="47.0" x2="264.3" y2="53.0" stroke="#333" stroke-width="1"/><line x1="285.7" y1="47.0" x2="285.7" y2="53.0" stroke="#333" stroke-width="1"/><line x1="307.1" y1="44.0" x2="307.1" y2="56.0" stroke="#333" stroke-width="1.5"/><text x="92.9" y="70.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle" font-weight="bold">6</text><text x="307.1" y="70.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle" font-weight="bold">7</text><text x="92.9" y="84.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">36 = 6²</text><text x="307.1" y="84.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">49 = 7²</text><circle cx="162.4" cy="50.0" r="3.6" fill="#1f5f8b" stroke="#1f5f8b" stroke-width="1.5"/><text x="162.4" y="38.0" font-family="Source Sans 3" font-size="12" fill="#1f5f8b" text-anchor="middle">√40 ≈ 6.3</text><line x1="50.0" y1="120.0" x2="350.0" y2="120.0" stroke="#333" stroke-width="1.5"/><line x1="92.9" y1="114.0" x2="92.9" y2="126.0" stroke="#333" stroke-width="1.5"/><line x1="114.3" y1="117.0" x2="114.3" y2="123.0" stroke="#333" stroke-width="1"/><line x1="135.7" y1="117.0" x2="135.7" y2="123.0" stroke="#333" stroke-width="1"/><line x1="157.1" y1="117.0" x2="157.1" y2="123.0" stroke="#333" stroke-width="1"/><line x1="178.6" y1="117.0" x2="178.6" y2="123.0" stroke="#333" stroke-width="1"/><line x1="200.0" y1="117.0" x2="200.0" y2="123.0" stroke="#333" stroke-width="1"/><line x1="221.4" y1="117.0" x2="221.4" y2="123.0" stroke="#333" stroke-width="1"/><line x1="242.9" y1="117.0" x2="242.9" y2="123.0" stroke="#333" stroke-width="1"/><line x1="264.3" y1="117.0" x2="264.3" y2="123.0" stroke="#333" stroke-width="1"/><line x1="285.7" y1="117.0" x2="285.7" y2="123.0" stroke="#333" stroke-width="1"/><line x1="307.1" y1="114.0" x2="307.1" y2="126.0" stroke="#333" stroke-width="1.5"/><text x="92.9" y="140.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle" font-weight="bold">4</text><text x="307.1" y="140.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle" font-weight="bold">5</text><text x="92.9" y="154.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">64 = 4³</text><text x="307.1" y="154.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">125 = 5³</text><circle cx="230.3" cy="120.0" r="3.6" fill="#b5452c" stroke="#b5452c" stroke-width="1.5"/><text x="230.3" y="108.0" font-family="Source Sans 3" font-size="12" fill="#b5452c" text-anchor="middle">³√100 ≈ 4.6</text></svg>

Solution:

• Find the squares on each side: $36 < 40 < 49$. So $6 < \sqrt{40} < 7$.
• Try a decimal. $40$ is closer to $36$ than to $49$, so try low: $6.3^2 = 39.69$ and $6.4^2 = 40.96$. So $\sqrt{40} \approx 6.3$.
• Find the cubes on each side: $4^3 = 64$ and $5^3 = 125$. Since $64 < 100 < 125$, we get $4 < \sqrt[3]{100} < 5$.

Answer: $6 < \sqrt{40} < 7$, about $6.3$; $4 < \sqrt[3]{100} < 5$

> Verification & Check: $6.3 \times 6.3 = 39.69$, a little under $40$. $6.35^2 \approx 40.3$, a little over. So $6.3$ is right to one place.

Your Turn:

• Prompt: Between which two whole numbers is $\sqrt{30}$? Estimate it to one decimal place. Between which two whole numbers is $\sqrt[3]{30}$?
• Answer: $25 < 30 < 36$, so between $5$ and $6$. $5.4^2 = 29.16$ and $5.5^2 = 30.25$, so about $5.5$. $27 < 30 < 64$, so $\sqrt[3]{30}$ is between $3$ and $4$.

Example 3: Adding and subtracting on the line

Problem: Work out $(-8) + 5$, $(-8) - 5$, and $4 - (-6)$.

<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 420 240" width="420" height="240" role="img" aria-label="Two number lines from −14 to 6 with jumps drawn as arcs. Blue: start at −8 and jump 5 right to −3, so (−8) + 5 = −3. Red: start at −8 and jump 5 left to −13, so (−8) − 5 = −13.

The chapter continues in the book.