Mathematical Process Standards
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Texas 7th Grade Math › Mathematical Process Standards
A circular garden has a diameter of 9.85 meters. You need its circumference using $\pi \approx 3.14$ and must round to the nearest tenth. Which tool/technique is most efficient and accurate?
Mental math
Paper and pencil computation
Calculator
Spreadsheet
Explanation
Multiplying a decimal by $\pi$ requires precision and careful rounding; a calculator is fastest and least error-prone. Mental math risks accuracy, paper-and-pencil is slower and more error-prone, and a spreadsheet is overkill for one calculation.
A fundraiser sold 350 bars at $2 each toward a $1500 goal.
What is the first step to determine how much more money is needed to reach the goal?
Multiply 350 by 2
Subtract 350 from 1500
Divide 1500 by 2
Add 350 and 1500
Explanation
First compute the amount raised in dollars: $350 \times 2 = 700$. Then compare to the goal by calculating $1500 - 700$. Subtracting 350 from 1500 mixes units (bars vs. dollars), and the other options do not set up the needed comparison.
A cell phone plan costs 40 per month plus 0.12 per text over 300 texts. Alex sent 520 texts this month. Which expression gives Alex's total bill for the month?
40 + 0.12(520 - 300)
40 + 0.12(520)
0.12(520 - 300)
40 + (520 - 300)/0.12
Explanation
Monthly fee 40 plus overage: 0.12 per text for (520 − 300) = 220 extra texts, so total is 40 + 0.12(520 − 300). Choice B charges for all 520 texts. Choice C omits the fixed 40. Choice D divides by 0.12 instead of multiplying.
Which equation matches this table? x: -2, 0, 1, 3; y: -1, 3, 5, 9
$y = 2x + 3$
$y = 3x + 2$
$y = 2x - 3$
$y = -2x + 3$
Explanation
When $x=0$, $y=3$, so $b=3$. From $x=0$ to $x=1$, $y$ increases from 3 to 5, a change of 2, so $m=2$. The matching equation is $y=2x+3$. The other choices change the slope or intercept.
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: $3(x+4)=3x+3\cdot 4=3x+12$.
The library tracked book checkouts by grade and genre last week. 6th grade: fiction 28, nonfiction 15, graphic novels 19. 7th grade: fiction 22, nonfiction 20, graphic novels 14. 8th grade: fiction 25, nonfiction 18, graphic novels 21. You want to compare grade and genre together.
Which representation best organizes this data to compare groups?
A two-way table with grades as rows and genres (fiction, nonfiction, graphic novels) as columns
A bar chart of the total checkouts for all grades combined by genre only
A single total number of all checkouts for the week
A list of random book titles checked out
Explanation
A two-way table neatly displays counts for each grade–genre pair, making it easy to compare across both categories. A genre-only bar chart loses the grade breakdown. A single total and a list of titles do not support comparing groups by two variables.
Which equation matches this table? x: -2, 0, 2, 6; y: 7, 4, 1, -5
$y = 1.5x + 4$
$y = -x + 4$
$y = -1.5x + 4$
$y = -1.5x - 4$
Explanation
When $x=0$, $y=4$, so $b=4$. From $x=-2$ to $x=0$, $y$ changes from 7 to 4 (down 3) while $x$ increases by 2, so $m=\frac{-3}{2}=-1.5$. Thus $y=-1.5x+4$.
A ride-share app charges a base fee of 2.50 plus 1.80 per mile. Jen's trip was 12 miles. Which equation correctly models the total cost $C$ of this trip?
$C = 1.80(12)$
$C = 2.50 + 1.80(12)$
$C = 2.50 + 1.80 + 12$
$C = 2.50(12) + 1.80$
Explanation
Base fee 2.50 plus mileage: 1.80 per mile for 12 miles, so $C = 2.50 + 1.80(12)$. Choice A omits the base fee. Choice C adds 12 instead of multiplying by the rate. Choice D multiplies the base fee by miles, which is incorrect.
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 $m \le \bar{x} \le M$.
A water tank holds 800 liters. It currently has 275 liters. A pump adds 12 liters per minute.
Which calculation best checks whether running the pump for 44 minutes will fill the tank?
$275 + 44$
$800 - 275 \div 12$
$275 + 12 \times 44$
$800 \times 44 - 12$
Explanation
To check reasonableness, model the final amount after 44 minutes: $275 + 12 \times 44$. If this is at least 800, the time is reasonable. The other options do not represent adding 12 liters each minute to the current amount.