Mathematical Process Standards
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Texas 6th Grade Math › Mathematical Process Standards
A class trip costs 180 dollars. The class has 24 dollars saved but must pay a one-time 12-dollar materials fee before selling bracelets. They will sell bracelets for 6 dollars each. How many bracelets are needed to reach the goal?
What is the first calculation you should do to plan the solution?
$180 \div 6$
$24 + 12$
$180 - 6$
$180 - 24 + 12$
Explanation
Analyze: Goal $=180$, current savings $=24$, one-time fee $=12$ reduces savings, each bracelet adds $6$. Plan: First find the remaining amount needed after accounting for savings and the fee: $180 - 24 + 12 = 168$. Then determine bracelets: $168 \div 6 = 28$. Determine solution: $28$ bracelets. Justify: The sequence mirrors the situation (apply fee and savings before dividing by price per bracelet). Evaluate: Check total: $24 - 12 + 6\times 28 = 12 + 168 = 180$, exactly the goal, so reasonable.
A bike shop recorded helmets sold by size in one day: small 7, medium 12, large 9, extra-large 2.
What representation would best organize this data to quickly compare sizes and compute the total sold?
A frequency table with columns for size and number sold, plus a row for the total
A sentence describing the sizes and numbers in one long list
A collage of helmet drawings without counts
One bar showing only the total number of helmets sold
Explanation
A frequency table organizes each size with its count and allows adding a total, making comparisons and calculations clear. The other options are incomplete or do not support analysis.
A student needs the exact total cost for 8 items priced 3.49, 7.95, 12.59, 4.39, 9.75, 6.88, 2.49, and 5.27, plus 8.25% sales tax. Which tool would be most appropriate to get an accurate answer efficiently?
Mental math
Estimation
Calculator
Paper and pencil
Explanation
This requires adding many decimals and then applying a percentage for tax. High accuracy is needed, and a calculator reduces place-value and computation errors compared with mental math or paper/pencil.
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).
Jake claims that $\frac{3}{4}$ is greater than $\frac{7}{8}$ because 3 and 4 are smaller numbers than 7 and 8. How would you correctly explain which fraction is larger?
Smaller numbers make smaller fractions, so $\frac{3}{4}$ is smaller than $\frac{7}{8}$ because 3 < 7 and 4 < 8.
You cannot compare fractions with different denominators, so there is no way to tell without long decimals.
Rewrite with a common denominator: $\frac{3}{4} = \frac{6}{8}$ and $\frac{7}{8}$ stays the same. With the same denominator, compare numerators: $6 < 7$, so $\frac{3}{4} < \frac{7}{8}$.
$\frac{3}{4}$ is close to 1 and $\frac{7}{8}$ is not, so $\frac{3}{4}$ must be larger.
Explanation
Comparing fractions is valid by using a common denominator. Convert $\frac{3}{4}$ to $\frac{6}{8}$ and compare it to $\frac{7}{8}$. Because both fractions have denominator 8, the one with the greater numerator is larger. Since $6 < 7$, $\frac{6}{8} < \frac{7}{8}$, so $\frac{3}{4} < \frac{7}{8}$.
A community cleanup is making snack bags. Each of the 96 volunteers gets 1 granola bar, and each of the 7 team leaders needs 2 bars. Boxes contain 18 bars. What expression should you calculate first to find the total number of bars needed before dividing by 18?
$96 + 7 + 2$
$96 \times (7 + 2)$
$96 + 2 \times 7$
$(96 + 7) \times 2$
Explanation
You need the total bars required: 96 bars for volunteers plus 2 bars per team leader, so $96 + 2 \times 7$. The other choices either add everything, group incorrectly, or double the entire group.
Words: Three-fourths of a class of 24 students passed the test. Which representation shows the same relationship?
Words: One-fourth of the class passed, which is 6 students.
Symbols: $\frac{3}{4} \times 24 = 18$
Diagram: A bar divided into 4 equal parts with 1 part shaded to show those who passed.
Graph: A line $y=1.25x$ with the point (24, 18) highlighted.
Explanation
Taking three-fourths of 24 gives $\frac{3}{4}\times 24=18$, so 18 students passed. The other choices mismatch the proportion or show a different relationship.
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.
Mia says $-5$ is greater than $-2$ because 5 is greater than 2. What is wrong with Mia's reasoning, and what is the correct comparison?
For negative integers, the number with the greater absolute value is less. On a number line, $-5$ is to the left of $-2$, so $-5 < -2$.
Negatives do not follow comparison rules, so you should ignore the signs and say $-5 > -2$.
They are equal because both numbers are negative.
You cannot compare negative numbers without converting them to fractions.
Explanation
Absolute value measures distance from 0. Between two negative numbers, the one with the greater absolute value lies farther left on the number line and is smaller. Since $|{-5}| = 5$ and $|{-2}| = 2$ with $5 > 2$, it follows that $-5 < -2$.
A recipe makes 12 muffins using 3 cups of flour. Carla wants to bake 20 muffins. One bag has 8 cups; she has half a bag at home.
Which expression will correctly find the cups of flour needed for 20 muffins?
$\left(\frac{3}{12}\right) \times 20$
$\left(\frac{12}{3}\right) \times 20$
$\left(\frac{3}{20}\right) \times 12$
$3 + 12 + 20$
Explanation
Analyze: Flour per muffin is $\frac{3}{12}$ cup. Plan: Multiply cups per muffin by the desired muffins: $\left(\frac{3}{12}\right)\times 20 = 5$ cups. Determine solution: Needs $5$ cups. Justify: Proportional reasoning keeps the recipe ratios. Evaluate: Carla has half a bag $=4$ cups, so she needs $5 - 4 = 1$ more cup. The calculations are consistent and reasonable.