Identify common misconceptions in learning number and algebra and classify numbers from a given set.

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Question 1

a) (i) Two common misconceptions in learning number and algebra are:

Misconception: "The equals sign means 'the answer is'."
- Explanation: Students often interpret the equals sign ( $=$ ) as a signal to perform a calculation and write down the result, rather than understanding it as a symbol of equivalence or balance between two expressions. This can hinder their ability to solve algebraic equations where the unknown is not isolated on one side, or to understand equations like $2 + 3 = 4 + 1$ .
Misconception: "Multiplication always makes numbers bigger, and division always makes numbers smaller."
- Explanation: This misconception is true for positive integers greater than 1. However, it breaks down when multiplying by fractions (e.g., $10 \times \frac{1}{2} = 5$ ) or decimals less than 1, or when dividing by fractions (e.g., $10 \div \frac{1}{2} = 20$ ). Similarly, multiplying by negative numbers changes the sign and magnitude in a way that contradicts this belief.

a) (ii) Given the set of numbers, $S = \{-3, -1, 0, 2, 4\}$ .

(α) Identify which numbers are natural numbers, which are whole numbers, and which are integers.
- Natural numbers: These are positive integers starting from 1. From $S$ : $\{2, 4\}$
- Whole numbers: These are natural numbers including zero. From $S$ : $\{0, 2, 4\}$
- Integers: These include all positive and negative whole numbers, and zero. From $S$ : $\{-3, -1, 0, 2, 4\}$

(β) Represent the set on a number line. Draw a number line and mark the positions of $-3, -1, 0, 2, 4$ with closed circles.

<-----•-----•-----•-----•-----•-----•----->
     -3    -2    -1     0     1     2     3     4
                                    •           •

(γ) Determine if the set is closed under addition. For a set to be closed under addition, the sum of any two elements in the set must also be an element of the set. Consider two elements from $S$ : 2 and 4.
$2 + 4 = 6$
Since 6 is not an element of $S$ ( $6 \notin S$ ), the set is not closed under addition. $\boxed{\text{No}}$

b) (i) You need $\frac{5}{6}$ cup of milk to make a cake, but you only have $\frac{1}{4}$ cup. How much more milk do you need?
To find out how much more milk is needed, subtract the amount you have from the amount you need.

\text{Milk needed} = \frac{5}{6} - \frac{1}{4}

Find a common denominator for 6 and 4.
The least common multiple (LCM) of 6 and 4 is 12.
Convert the fractions to equivalent fractions with the common denominator.
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$ $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
Perform the subtraction.
$\frac{10}{12} - \frac{3}{12} = \frac{10 - 3}{12} = \frac{7}{12}$
You need $\frac{7}{12}$ cup more milk.
$\boxed{\frac{7}{12} \text{ cup}}$

b) (ii) Express the number 987531.342 in standard form.
Standard form (scientific notation) expresses a number as $a \times 10^n$ , where $1 \le |a| < 10$ and $n$ is an integer.
Move the decimal point to the left until there is only one non-zero digit to its left.
$987531.342 \rightarrow 9.87531342$
Count how many places the decimal point was moved.
It was moved 5 places to the left. This means $n=5$ .
$987531.342 = 9.87531342 \times 10^5$
$\boxed{9.87531342 \times 10^5}$

b) (iii) Round the number 3826.117 to the nearest tenth and the nearest hundredth.
- To the nearest tenth: The tenths digit is 1. Look at the digit immediately to its right (the hundredths digit), which is 1. Since $1 < 5$ , we round down, meaning the tenths digit remains unchanged. $3826.117 \approx 3826.1$ $\boxed{3826.1}$
- To the nearest hundredth: The hundredths digit is 1. Look at the digit immediately to its right (the thousandths digit), which is 7. Since $7 \ge 5$ , we round up, meaning the hundredths digit increases by 1. $3826.117 \approx 3826.12$ $\boxed{3826.12}$
b) (iv) Find the sum of $5\sqrt{2}$ and $3\sqrt{2}$ .
These are like terms because they both have $\sqrt{2}$ as their radical part. To add them, simply add their coefficients.
$5\sqrt{2} + 3\sqrt{2} = (5 + 3)\sqrt{2} = 8\sqrt{2}$
$\boxed{8\sqrt{2}}$

c) The question is incomplete. Please provide the full question for part (c).

Identify common misconceptions in learning number and algebra and classify numbers from a given set.

Step-by-step solution

Find a common denominator for 6 and 4.

Convert the fractions to equivalent fractions with the common denominator.

Perform the subtraction.

Move the decimal point to the left until there is only one non-zero digit to its left.

Count how many places the decimal point was moved.

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