Mathematical Process Standards
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Texas 8th Grade Math › Mathematical Process Standards
A lawn service charges a 12 dollar travel fee plus 7.50 dollars for each lawn bag of leaves hauled away. If $b$ is the number of bags, which expression gives the total cost in dollars?
$12 + 7.50b$
$12b + 7.50$
$7.50(b - 12)$
$12(7.50 + b)$
Explanation
12 is the fixed travel fee, and 7.50 is the cost for each bag. With $b$ bags, the variable part is $7.50b$, so total cost is $12 + 7.50b$. Option B multiplies the fixed fee by $b$. Option C subtracts the fixed fee inside the bag count. Option D multiplies the fixed fee by the sum, overcounting.
Find the intersection of $y=1.6x-12.3$ and $y=-0.25x+18.4$ to the nearest hundredth. Which tool would you choose as the best option?
graphing calculator
spreadsheet
mental estimation
paper-and-pencil algebra
Explanation
A graphing calculator quickly graphs both lines and computes their intersection to decimal accuracy. Solving by hand with these decimals is slower and prone to arithmetic errors. A spreadsheet is not ideal for solving two equations graphically, and mental estimation cannot deliver coordinates to the nearest hundredth.
Mateo biked 84 miles in 2 hours at a constant speed. He wants to know how far he would travel in 3.5 hours at the same speed. What is the first step in an efficient plan to solve this?
Compute the unit rate by dividing 84 by 2.
Multiply 84 by 3.5 immediately.
Subtract 2 from 3.5 to find the extra time.
Add 84 and 2 to combine the data.
Explanation
Analyze: speed is constant with 84 miles in 2 hours. Plan: find miles per hour, then multiply by 3.5. Solve: 84 ÷ 2 = 42 mph, then 42 × 3.5 = 147 miles. Check: 3 hours at 42 mph is 126 and half an hour adds 21, totaling 147, which is reasonable. Computing the unit rate first is essential.
Claim: For any two distinct points on the graph of $y=3x+1$, the slope between them is $3$. Which explanation best justifies this claim?
Choose two points on the line, such as $(0,1)$ and $(1,4)$. By the slope formula, $m=\frac{4-1}{1-0}=3$, so the slope is $3$.
Let two generic points on the line be $(x_1,,3x_1+1)$ and $(x_2,,3x_2+1)$ with $x_2\ne x_1$. By the slope formula, $$m=\frac{(3x_2+1)-(3x_1+1)}{x_2-x_1}=\frac{3(x_2-x_1)}{x_2-x_1}=3,$$ using factoring and cancellation (multiplicative inverses).
In $y=3x+1$, the $1$ is the $x$-coefficient, so it must be the slope.
Divide both sides of $y=3x+1$ by $x$ to get $\frac{y}{x}=3+\frac{1}{x}$; the $3$ is the slope since it is next to $x$ after dividing.
Explanation
B uses the slope formula with arbitrary points $(x_1,3x_1+1)$ and $(x_2,3x_2+1)$, then factors and cancels $(x_2-x_1)$ to conclude $m=3$, correctly naming the slope formula, factoring, and multiplicative inverses. A verifies only one example and does not justify the claim for all points. C misidentifies the slope: in $y=mx+b$, the slope is the coefficient of $x$ (here $3$), not the constant term $1$. D performs an invalid step for determining slope; dividing by $x$ produces $\frac{y}{x}$, which is not the definition of slope between two points and varies with $x$.
A gym charges a \$29 signup fee plus \$19 per month. If the total paid was $181, let $m$ be the number of months. What is the first algebraic step to solve \$29 + 19m = 181$ for $m$?
Divide 181 by 19 first.
Multiply 29 by 19 to combine fees.
Subtract 29 from both sides to isolate the term with $m$.
Add 29 to both sides to combine constants.
Explanation
Analyze: total cost equals fixed fee plus monthly fee. Plan: isolate the $m$ term in $29 + 19m = 181$. Solve: subtract 29 to get $19m = 152$, then divide by 19 to get $m = 8$. Check: $29 + 19(8) = 29 + 152 = 181$. Subtracting 29 first is necessary to isolate $m$.
A recipe uses 3 cups of pancake mix to make 12 pancakes. How many cups of mix are needed to make 18 pancakes if the relationship is proportional?
3
5
6
2
Explanation
3 cups corresponds to 12 pancakes. 18 is 1.5 times 12, so the cups also scale by 1.5: $3 \times 1.5 = 4.5$ cups. 3 ignores scaling, 6 doubles instead of multiplying by 1.5, and 1.5 reverses the ratio.
While shopping, decide quickly which is the better buy: 24 oz for \$4.59 or 28 oz for \$5.19. What technique is best for a fast, reasonable decision without exact calculation?
paper-and-pencil algebra
graphing calculator
spreadsheet
mental estimation
Explanation
Mental estimation is fastest and sufficient: compare unit prices by rounding and cross-multiplying. For example, $4.59\times 28\approx 4.6\times 28=128.8$ and $5.19\times 24\approx 5.2\times 24=124.8$, so the 28 oz option is slightly cheaper per ounce. A calculator or spreadsheet is slower for an aisle decision, and paper-and-pencil is unnecessary.
Claim: $4(x-3)+2x$ is equivalent to $6x-12$. Which explanation best justifies this claim?
Factor out $2$: $4(x-3)+2x=2\big(2(x-3)+x\big)=2(2x-6+x)=2(3x-6)=6x-6$; this matches.
Subtract inside first: $x-3=1x$, so $4(1x)+2x=6x$, and the $-12$ cancels because $+2x$ balances it.
Distribute $4$ to $x$ only: $4(x-3)+2x=4x-3+2x=6x-3$; then lower the constant to $-12$ because there was a $-3$ three times.
Apply the distributive property and combine like terms: $4(x-3)+2x=4x-12+2x=(4x+2x)-12=6x-12$, using distribution and combining like terms.
Explanation
D correctly uses the distributive property to get $4x-12$ and then combines like terms $4x$ and $2x$ to obtain $6x-12$. A's factorization changes the expression incorrectly and yields $6x-6$, which is not equivalent. B incorrectly treats $x-3$ as $1x$ and ignores the constant term $-12$. C misapplies distribution (omits multiplying $-3$ by $4$) and then makes an unjustified adjustment to the constant.
A school fundraiser sold T-shirts for 12 dollars each and wristbands for 4 dollars each. They sold 25 items in all and collected 220 dollars. How many T-shirts did they sell?
10
25
15
5
Explanation
Let $t$ be T-shirts and $w$ be wristbands. Then $t + w = 25$ and $12t + 4w = 220$. Substitute $w = 25 - t$: $12t + 4(25 - t) = 220 \Rightarrow 12t + 100 - 4t = 220 \Rightarrow 8t = 120 \Rightarrow t = 15$. Here, 12 is dollars per T-shirt, 4 is dollars per wristband, 25 is total items, and 220 is total dollars.
A map has a scale of 1 cm to 50 km. The distance between two towns on the map is 7.2 cm. A student calculates the real distance as 360 km. Which check best verifies this result?
Compare 360 to the car's gas mileage to see if the trip is possible.
Subtract 50 from 360 to see if the remainder is 7.2.
Double 7.2 to see if it equals 360.
Divide 360 by 7.2 to confirm it equals 50 km per cm.
Explanation
Analyze: scale is 50 km per 1 cm. Plan: multiply measured length by 50. Solve: $7.2 \times 50 = 360$ km. Check: $360 \div 7.2 = 50$, matching the scale. Confirming the original ratio verifies the calculation.