Mathematical Process Standards>Analyzing Mathematical Relationships to Connect Ideas(TEKS.Math.6.1.F)
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Texas 6th Grade Math › Mathematical Process Standards>Analyzing Mathematical Relationships to Connect Ideas(TEKS.Math.6.1.F)
Consider these two procedures: 1) Finding 10% of a number. 2) Dividing the number by 10. How are these two mathematical ideas related?
Both decrease numbers by 10 each time.
Finding 10% of a number is the same as dividing the number by 10.
Finding 10% is the same as subtracting 10 from the number.
Dividing by 10 is the same as taking 10% more.
Explanation
Because 10% equals $\frac{1}{10}$, finding 10% of a number means multiplying by $\frac{1}{10}$, which is equivalent to dividing by 10. Recognizing percent–fraction equivalences builds mental math flexibility (e.g., 10% is divide by 10).
Consider these two ideas: 1) Subtracting $b$ from $a$. 2) Adding the opposite of $b$ to $a$. How are these ideas related?
$a - b$ equals $a + (-b)$.
$a - b$ equals $(-a) + b$.
Subtracting always makes a number smaller, but adding the opposite always makes it larger.
They are only equal when $a$ and $b$ are positive integers.
Explanation
Subtraction is defined as adding the additive inverse: $a - b = a + (-b)$. Understanding this connection unifies subtraction and addition of integers and supports consistent strategies with signed numbers.
Consider these procedures: 1) Use the distributive property on $5(n+7)$. 2) Find the sum of $5n$ and $35$. What connection exists between these procedures?
$5(n+7)$ equals $5n+7$.
The distributive property changes the value of the expression.
By the distributive property, $5(n+7) = 5n + 5\cdot 7 = 5n + 35$.
$5(n+7)$ is only equal to $5n + 35$ when $n=7$.
Explanation
Distributing multiplies $5$ by each term inside the parentheses: $5(n+7)=5n+5\cdot 7=5n+35$. Recognizing this structure connects factoring and expanding, aiding equation solving and mental computation.
Consider these procedures: 1) Finding the unit rate of a ratio like $x$ miles in $y$ hours. 2) Writing an equivalent ratio with denominator 1 by dividing both terms by $y$. How are these procedures related?
The unit rate is found by subtracting the two numbers.
Finding the unit rate is the same as dividing the first quantity by the second to get a per 1 unit measure (e.g., miles per 1 hour).
To get a unit rate, multiply both terms of the ratio by the second term.
A unit rate only exists when both numbers are whole numbers.
Explanation
For $x$ miles in $y$ hours, the unit rate is $\tfrac{x}{y}$ miles per 1 hour. Dividing both terms of the ratio by $y$ gives an equivalent ratio $(\tfrac{x}{y} : 1)$. Seeing this link clarifies ratio equivalence and speeds up rate calculations.