Mathematical Process Standards>Justifying Mathematical Ideas with Precise Language(TEKS.Math.7.1.G)
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Texas 7th Grade Math › Mathematical Process Standards>Justifying Mathematical Ideas with Precise Language(TEKS.Math.7.1.G)
Claim: $3(x+4)=3x+12$. Which explanation best justifies this claim?
Because multiplication comes before addition, multiply 3 and x first and then bring down +4.
$3x=3x$, so keep the +4 the same to get $3x+4$.
You can cancel the 3 with the +4 to get $x+1$.
By the distributive property of multiplication over addition, $a(b+c)=ab+ac$. With $a=3$, $b=x$, and $c=4$, distributing gives $3(x+4)=3x+3\cdot 4=3x+12$.
Explanation
The distributive property states that for all real numbers $a$, $b$, and $c$, $a(b+c)=ab+ac$. Taking $a=3$, $b=x$, and $c=4$, we distribute 3 to each term inside the parentheses: $3(x+4)=3x+3\cdot 4=3x+12$.
Claim: For any finite set of real numbers, the mean lies between the minimum and maximum values. Which explanation best justifies this claim?
Exactly half the data values are above the mean and half are below it, so it must be in the middle.
The mean is the middle value when the data are ordered, so it is between the smallest and largest values.
The mean is the total sum divided by the number of values, which is a weighted average with positive weights that sum to 1. A weighted average of numbers cannot be less than the minimum or greater than the maximum, so the mean must be between the minimum and maximum.
Outliers do not affect the mean, so the mean stays between the smallest and largest values.
Explanation
If the data are $x_1, x_2, \dots, x_n$ with minimum $m$ and maximum $M$, then $m \le x_i \le M$ for all $i$. The mean is $\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$, which is a convex (weighted) average with nonnegative weights summing to 1. Such an average cannot be less than $m$ or greater than $M$, hence $m \le \bar{x} \le M$.
Claim: If two triangles have two pairs of corresponding angles congruent, then the triangles are similar. Which explanation best justifies this claim?
If two angles match, the triangles must be congruent, not just similar.
By the Angle-Angle (AA) similarity criterion, if two pairs of corresponding angles are congruent, the third pair must also be congruent because the angle measures in a triangle sum to $180^\circ$. With all corresponding angles congruent, the triangles are similar.
The triangles look the same shape, so they are similar.
Two equal angles are not enough; you also need a matching side between those angles to be sure.
Explanation
In any triangle, the interior angle measures sum to $180^\circ$. If two pairs of corresponding angles are congruent, then the third pair is automatically congruent. With all corresponding angles congruent, the triangles satisfy the AA similarity criterion and are therefore similar.