Number and Operations
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Texas 6th Grade Math › Number and Operations
Whole numbers are 0, 1, 2, 3, ... Integers include the negative whole numbers as well as 0 and the positive whole numbers. Rational numbers are any numbers that can be written as a fraction or as a terminating or repeating decimal. Consider this set of numbers: -5, 3.2, 7, -1, 0, $4/3$, 2.5, -8.
How would these numbers be organized in a Venn diagram showing the relationship Whole ⊂ Integers ⊂ Rational?
Whole: 0, 7; Integers not whole: -5, -1, -8; Rational not integer: 3.2, 2.5, $4/3$
Whole: 0, 7, -1; Integers not whole: -5, -8; Rational not integer: 3.2, 2.5
Whole: 0; Integers not whole: -5, -1, -8, 7; Rational not integer: 3.2, 2.5
Whole: 0, 7, 3.2; Integers not whole: -5, -1, -8, $4/3$; Rational not integer: 2.5
Explanation
Whole numbers are 0 and positive counting numbers: 0, 7. Integers not whole are the negative integers: -5, -1, -8. Rational not integer numbers include decimals/fractions that are not whole/integer values: 3.2, 2.5, $4/3$.
Temperature readings: Monday 2.5°F, Tuesday -1.3°F, Wednesday -0.5°F, Thursday 3°F. Order the days from coldest to warmest.
Thursday, Monday, Wednesday, Tuesday
Tuesday, Wednesday, Monday, Thursday
Wednesday, Tuesday, Monday, Thursday
Monday, Thursday, Tuesday, Wednesday
Explanation
Coldest means the lowest number. On a number line: -1.3 < -0.5 < 2.5 < 3, so Tuesday (coldest), then Wednesday, Monday, Thursday (warmest). A common mistake is to look at absolute values and think -0.5 is colder than -1.3 because 0.5 < 1.3, but real temperatures get colder as the value becomes more negative.
Which statement correctly explains the relationship between $7/3$ and $7 \div 3$?
They are different because $7 \div 3$ is 2 remainder 1.
They represent the same number because the fraction bar means division with $7$ as the dividend and $3$ as the divisor.
$7/3$ is the same as $3 \div 7$.
$7/3$ means $7 \times 3$.
Explanation
By definition, $a/b$ represents $a \div b$ (with $b \ne 0$). Here $7/3 = 7 \div 3$; $7$ (the numerator) is the dividend and $3$ (the denominator) is the divisor. Reversing to $3 \div 7$ or changing to multiplication is incorrect.
Which expression has the same value as $8 \div \frac{2}{3}$?
$8 \times \frac{3}{2}$
$8 \times \frac{2}{3}$
$8 \div \frac{3}{2}$
$8 \times -\frac{3}{2}$
Explanation
The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ because $\frac{2}{3} \times \frac{3}{2} = 1$. Dividing by a fraction equals multiplying by its reciprocal: $8 \div \frac{2}{3} = 8 \times \frac{3}{2}$. In general, for any nonzero fraction $\frac{b}{c}$, $a \div \frac{b}{c} = a \times \frac{c}{b}$.
A recipe calls for 3 cups of flour. If you make $\frac{5}{4}$ of the recipe, do you need more or less than 3 cups of flour?
Less than 3 cups
Exactly 3 cups
Not enough information
More than 3 cups
Explanation
$\frac{5}{4}>1$, so multiplying 3 by a number greater than 1 increases the amount.
Golf scores relative to par: Alex -2, Bella +1, Carlos -3, Dana 0. Order the players from best to worst performance.
Bella, Dana, Alex, Carlos
Dana, Alex, Carlos, Bella
Alex, Carlos, Dana, Bella
Carlos, Alex, Dana, Bella
Explanation
In golf, lower scores are better. Order the numbers from least to greatest: -3 < -2 < 0 < +1 gives Carlos (best), Alex, Dana, Bella (worst). A trap is to think 0 is automatically best or to compare absolute values; but performance aligns with the actual order on the number line, not closeness to zero.
What does the fraction bar represent in the expression $15/4$?
Subtraction, so $15/4$ means $15 - 4$.
Multiplication, so $15/4$ means $15 \times 4$.
It makes both numbers positive, like absolute value.
Division, so $15/4$ means $15 \div 4$.
Explanation
The fraction bar represents division: $a/b = a \div b$. In $15/4$, $15$ (numerator) is divided by $4$ (denominator). It is not subtraction or multiplication, and it does not act like absolute value.
A store changes a price by multiplying it. Option 1 multiplies by $\frac{4}{5}$ and Option 2 multiplies by $\frac{6}{5}$. Which option increases the original price?
Multiply by $\frac{4}{5}$
Both options increase the price
Neither; both decrease the price
Multiply by $\frac{6}{5}$
Explanation
$\frac{6}{5}>1$ increases the price, while $\frac{4}{5}<1$ decreases it.
Which multiplication by a reciprocal is equivalent to $-6 \div \left(-\frac{3}{4}\right)$?
$-6 \times -\frac{3}{4}$
$-6 \times -\frac{4}{3}$
$-6 \times \frac{4}{3}$
$-6 \div \frac{4}{3}$
Explanation
The reciprocal of $-\frac{3}{4}$ is $-\frac{4}{3}$ because $\left(-\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = 1$. So $-6 \div \left(-\frac{3}{4}\right) = -6 \times \left(-\frac{4}{3}\right)$. This follows the rule $a \div \frac{b}{c} = a \times \frac{c}{b}$.
Whole numbers are 0, 1, 2, 3, ... Integers include the negative whole numbers as well as 0 and the positive whole numbers. Rational numbers are any numbers that can be written as a fraction or as a terminating or repeating decimal. Consider this set of numbers: -9, $4/3$, 2.5, 0, -7.0, 5, $-1/4$.
Which numbers from this set are rational numbers but not integers?
2.5, $4/3$
2.5, $4/3$, -7.0
2.5, $4/3$, $-1/4$
$4/3$, $-1/4$, 0
Explanation
Integers in the set are -9, 0, -7.0 (which equals -7), and 5. The numbers that are rational but not integers are 2.5, $4/3$, and $-1/4$.