Question

thursday 12/18 a set of signed numbers includes integers, fractions, decimals, and percents. explain how you would decide where each number belongs on a number line. in your response, describe the steps you take, how you compare different forms, and how you check that your placements make sense. use at least two different number forms in your explanation.

Explanation:

Step1: Convert to Decimal

To place numbers on a number line, first convert all numbers to decimal form. For example, take the fraction $\frac{3}{4}$ and the percent $75\%$. Convert $\frac{3}{4}$ to decimal: $\frac{3}{4} = 0.75$. Convert $75\%$ to decimal: $75\%=\frac{75}{100} = 0.75$. Also, take the integer $-2$, fraction $-\frac{1}{2}$, and decimal $-0.5$. Convert $-\frac{1}{2}$ to decimal: $-\frac{1}{2}=- 0.5$.

Step2: Compare and Order

Now, compare the decimal values. For positive numbers, $0.75$ (from $\frac{3}{4}$ and $75\%$) is greater than, say, $0.5$ (if we had $\frac{1}{2}$). For negative numbers, $-2$ is less than $-0.5$ (from $-\frac{1}{2}$). Order the numbers from least to greatest (or vice - versa) based on their decimal values. For example, with numbers $-2$, $-\frac{1}{2}$, $0$, $\frac{3}{4}$, $75\%$: convert all to decimals: $-2$, $-0.5$, $0$, $0.75$, $0.75$. Then order: $-2 < - 0.5=-\frac{1}{2}<0 < 0.75=\frac{3}{4}=75\%$.

Step3: Place on Number Line

Mark the positions on the number line. The number line has negative numbers to the left of $0$ and positive numbers to the right. For $-2$, we move 2 units to the left of $0$. For $-0.5$ (or $-\frac{1}{2}$), we move half a unit to the left of $0$. For $0$, it's the origin. For $0.75$ (or $\frac{3}{4}$ or $75\%$), we move three - quarters of a unit to the right of $0$.

Step4: Check Placement

To check, we can convert back to original forms. For example, the position of $0.75$ should represent both $\frac{3}{4}$ (since $\frac{3}{4}=0.75$) and $75\%$ (since $75\% = 0.75$). For the negative numbers, $-0.5$ should represent $-\frac{1}{2}$ (since $-\frac{1}{2}=-0.5$) and its position relative to $-2$ makes sense because $-0.5$ is greater than $-2$ (closer to $0$ on the number line). We can also use another number form, like fraction for positive numbers: $\frac{3}{4}$ is between $0$ and $1$, and $75\%$ is also between $0$ and $1$, so their position between $0$ and $1$ is correct. For negative numbers, $-\frac{1}{2}$ is between $-1$ and $0$, and $-0.5$ is also between $-1$ and $0$, and $-2$ is to the left of $-1$, so the placements are consistent.

Answer:

To place signed numbers (integers, fractions, decimals, percents) on a number line: 1. Convert to a Common Form (e.g., decimal): Convert fractions (e.g., $\frac{3}{4}=0.75$) and percents (e.g., $75\% = 0.75$) to decimals. Integers (e.g., $-2$) and decimals (e.g., $-0.5$) remain as is (or convert fractions like $-\frac{1}{2}=-0.5$). 2. Compare/Order Values: Use decimal values to order numbers (e.g., $-2 < - 0.5=-\frac{1}{2}<0 < 0.75=\frac{3}{4}=75\%$). 3. Place on Number Line: Mark positions (negative left of $0$, positive right; e.g., $-2$ is 2 left of $0$, $0.75$ is 0.75 right of $0$). 4. Check: Convert back (e.g., $0.75$ matches $\frac{3}{4}$ and $75\%$) or use another form (e.g., fraction for positive numbers) to verify consistency (e.g., $\frac{3}{4}$ is between $0$ and $1$, so its position makes sense).