Question

1. order \\(\sqrt{3}\\), \\(2\pi\\), and 1.5 from least to greatest. then graph them on the number line. (example 2)\\(\sqrt{3}\\) is between \\(\underline{quadquad}\\) and \\(\underline{quadquad}\\), so \\(\sqrt{3}\approx\underline{quadquad}\\)\\(\pi\approx3.14\\), so \\(2\pi\approx\underline{quadquad}\\)\\(\

$$\begin{array}{cccccccccccccc}&&&&&&&&&&&&&\\\\-\\infty&&0&0.5&1&1.5&2&2.5&3&3.5&4&4.5&5&5.5&6&6.5&7&\\infty\\\\&&&&&&&&&&&&&\\end{array}$$

\\)from least to greatest, the numbers are \\(\underline{quadquad}\\), \\(\underline{quadquad}\\), \\(\underline{quadquad}\\).10. four people have found the perimeter of a forest using different methods. their results are given in the table. order their calculations from greatest to least. (example 5)\\(\underline{quadquad}\\)\\(\underline{quadquad}\\)essential question check - in11. explain how to order a set of real numbers.\\(\underline{quadquad}\\)\\(\underline{quadquad}\\)\\(\underline{quadquad}\\)\\(\underline{quadquad}\\)

Explanation:

Problem 9

Step1: Estimate $\sqrt{3}$

We know that $1^2 = 1$ and $2^2 = 4$, so $\sqrt{3}$ is between $1$ and $2$. Calculating the approximate value, $\sqrt{3} \approx 1.732$.

Step2: Calculate $2\pi$

Given $\pi \approx 3.14$, so $2\pi \approx 2\times3.14 = 6.28$.

Step3: Order the numbers

We have the numbers $1.5$, $\sqrt{3} \approx 1.732$, and $2\pi \approx 6.28$. Comparing these values, $1.5 < 1.732 < 6.28$. So from least to greatest, the numbers are $1.5$, $\sqrt{3}$, $2\pi$.

Step1: Calculate Leon's value

$\sqrt{17}\approx4.123$, so $\sqrt{17}-2\approx4.123 - 2 = 2.123$.

Step2: Calculate Mika's value

$1+\frac{3}{2}=1 + 1.5 = 2.5$.

Step3: Calculate Jason's value

$\frac{12}{5}=2.4$.

Step4: Order the values

Comparing $2.5$ (Mika), $2.4$ (Jason), $2.3$ (Anna), and $2.123$ (Leon), we get $2.5 > 2.4 > 2.3 > 2.123$. So the order from greatest to least is Mika, Jason, Anna, Leon.

To order a set of real numbers, first, identify if the numbers are integers, fractions, decimals, or irrational numbers (like square roots). For irrational numbers, estimate their approximate decimal values (by finding between which perfect squares/cubes they lie). For fractions, convert them to decimals (either by dividing or finding a common denominator). For mixed numbers, convert to improper fractions or decimals. Then, compare the decimal values (or the estimated values for irrationals) and arrange them from least to greatest (or greatest to least as required) based on their numerical magnitude.

Answer:

$\sqrt{3}$ is between $1$ and $2$, so $\sqrt{3} \approx 1.732$; $\pi \approx 3.14$, so $2\pi \approx 6.28$; from least to greatest, the numbers are $1.5$, $\sqrt{3}$, $2\pi$.

Problem 10

First, we need to calculate or estimate the values of each expression:

• For Leon: $\sqrt{17}-2$. We know that $4^2 = 16$ and $5^2 = 25$, so $\sqrt{17} \approx 4.123$, then $\sqrt{17}-2 \approx 4.123 - 2 = 2.123$.
• For Mika: $1+\frac{3}{2} = 1 + 1.5 = 2.5$.
• For Jason: $\frac{12}{5} = 2.4$.
• For Anna: Let's assume the value is $2.3$ (from the table, though it's a bit unclear, but proceeding with the given).

Now, ordering these values from greatest to least: $2.5$ (Mika), $2.4$ (Jason), $2.3$ (Anna), $2.123$ (Leon). Wait, but let's check again. Wait, maybe the Anna's value is different? Wait, maybe I misread. Wait, the table has Leon: $\sqrt{17}-2$, Mika: $1+\frac{3}{2}$, Jason: $\frac{12}{5}$, Anna: let's see, maybe it's $2.3$? Wait, no, let's recalculate:

• $\sqrt{17}-2$: $\sqrt{16}=4$, $\sqrt{17}\approx4.123$, so $4.123 - 2 = 2.123$.
• $1+\frac{3}{2} = 2.5$.
• $\frac{12}{5} = 2.4$.
• Let's assume Anna's value is, say, if it's $2.3$ (as per the table's partial view). Then ordering from greatest to least: $1+\frac{3}{2}$ (Mika), $\frac{12}{5}$ (Jason), Anna's value (assuming $2.3$), $\sqrt{17}-2$ (Leon). Wait, but maybe the Anna's value is different. Wait, maybe the table has Anna's value as $2.3$? Alternatively, maybe I made a mistake. Wait, let's check again.

Wait, maybe the correct order is:

Mika: $2.5$, Jason: $2.4$, Anna: $2.3$, Leon: $2.123$. So from greatest to least: Mika, Jason, Anna, Leon. But we need to confirm the Anna's value. Wait, maybe the table shows Anna's value as $2.3$? Let's proceed with that.