Justifying Mathematical Ideas with Precise Language(TEKS.Math.6.1.G)
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Texas 6th Grade Math › Justifying Mathematical Ideas with Precise Language(TEKS.Math.6.1.G)
Jake claims that $\frac{3}{4}$ is greater than $\frac{7}{8}$ because 3 and 4 are smaller numbers than 7 and 8. How would you correctly explain which fraction is larger?
Smaller numbers make smaller fractions, so $\frac{3}{4}$ is smaller than $\frac{7}{8}$ because 3 < 7 and 4 < 8.
You cannot compare fractions with different denominators, so there is no way to tell without long decimals.
Rewrite with a common denominator: $\frac{3}{4} = \frac{6}{8}$ and $\frac{7}{8}$ stays the same. With the same denominator, compare numerators: $6 < 7$, so $\frac{3}{4} < \frac{7}{8}$.
$\frac{3}{4}$ is close to 1 and $\frac{7}{8}$ is not, so $\frac{3}{4}$ must be larger.
Explanation
Comparing fractions is valid by using a common denominator. Convert $\frac{3}{4}$ to $\frac{6}{8}$ and compare it to $\frac{7}{8}$. Because both fractions have denominator 8, the one with the greater numerator is larger. Since $6 < 7$, $\frac{6}{8} < \frac{7}{8}$, so $\frac{3}{4} < \frac{7}{8}$.
Mia says $-5$ is greater than $-2$ because 5 is greater than 2. What is wrong with Mia's reasoning, and what is the correct comparison?
For negative integers, the number with the greater absolute value is less. On a number line, $-5$ is to the left of $-2$, so $-5 < -2$.
Negatives do not follow comparison rules, so you should ignore the signs and say $-5 > -2$.
They are equal because both numbers are negative.
You cannot compare negative numbers without converting them to fractions.
Explanation
Absolute value measures distance from 0. Between two negative numbers, the one with the greater absolute value lies farther left on the number line and is smaller. Since $|{-5}| = 5$ and $|{-2}| = 2$ with $5 > 2$, it follows that $-5 < -2$.
Liam states that $3(x+4)=3x+4$ because you only multiply the $x$ by 3. What is the correct justification?
Only the variable part gets multiplied, so constants stay the same.
Parentheses mean you add first and then attach the 3, so $x+4$ becomes $x4$.
Try $x=1$ once; if it does not work for that value, then it never works.
By the distributive property, multiply 3 by each term inside the parentheses: $3(x+4)=3x+3\cdot4=3x+12$.
Explanation
Multiplication distributes over addition. The distributive property states $a(b+c)=ab+ac$. Here, $3(x+4)=3x+3\cdot4=3x+12$, not $3x+4$.
Ava evaluates $8-3\times2$ as $(8-3)\times2=10$. What is wrong with Ava's reasoning, and what is the correct value?
Subtraction must be done before multiplication, so $8-3=5$ and then $5\times2=10$.
According to the order of operations, perform multiplication before subtraction: compute $3\times2=6$, then $8-6=2$. Parentheses would be needed to change this order.
Subtraction and multiplication are the same priority, so you can do either first and still get 10.
The order does not matter because you will always get the same result.
Explanation
Order of operations requires multiplication before subtraction. Evaluate $3\times2=6$ and then subtract: $8-6=2$. The original expression had no parentheses around $8-3$, so $(8-3)\times2$ is not equivalent.
Noah says, "A recipe uses 6 cups of flour for 4 people. For 8 people, just add 3 cups to get 9 cups." How would you correctly justify the amount of flour needed for 8 people?
Because 8 is double 4, subtract half the flour: $6-3=3$ cups.
Add 2 cups since 8 is 2 times 4, so $6+2=8$ cups.
Use proportional reasoning. The unit rate is $\frac{6\text{ cups}}{4\text{ people}}=1.5$ cups per person. For 8 people: $1.5\times8=12$ cups. Equivalently, doubling the servings means multiply 6 cups by 2 to get 12 cups.
Add 4 cups because the number of people increased by 4.
Explanation
The situation is proportional. Either find the unit rate, \$1.5$ cups per person, then compute $1.5\times8=12$ cups, or use the scale factor: going from 4 to 8 doubles the recipe, so multiply 6 cups by 2 to get 12 cups.