Mathematical Process Standards>Using a Problem-Solving Model to Solve and Justify Solutions(TEKS.Math.8.1.B)
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Texas 8th Grade Math › Mathematical Process Standards>Using a Problem-Solving Model to Solve and Justify Solutions(TEKS.Math.8.1.B)
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.
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 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.
A 24-pack of markers costs $9.60. At the same rate, how much would 15 markers cost? What is the necessary first step?
Multiply 9.60 by 15 to get the cost.
Find the unit price by dividing 9.60 by 24.
Subtract 15 from 24 to see what remains in a pack.
Add 24 and 15 to find the total markers.
Explanation
Analyze: cost is proportional to number of markers. Plan: find cost per marker, then multiply by 15. Solve: $9.60 \div 24 = 0.40$ per marker; $0.40 \times 15 = 6.00$. Check: 12 markers would be $4.80$, so 15 markers at $6.00$ is reasonable. Finding the unit rate first is essential.
After an 8% sales tax, the total for a bike was $162. Let $p$ be the pre-tax price. What is the first step in an efficient plan to find $p$?
Divide 162 by 1.08 to undo the 8% increase.
Multiply 162 by 0.08 to get the tax amount.
Subtract 8 from 162, since the tax is 8.
Add 0.08 to 162 to remove the tax.
Explanation
Analyze: the total is 108% of the pre-tax price. Plan: model with $1.08p = 162$. Solve: divide both sides by 1.08 to get $p = 150$. Check: $150 \times 0.08 = 12$, and $150 + 12 = 162$. Dividing by 1.08 first directly isolates $p$.