Translate quantities into expressions and preserve signs, grouping and units.
Loading your saved responses…
Try
A fictional print shop charges 8 dollars to set up a job and 0.50 dollars for each page. Write a rule for the cost of p pages before calculating a 12-page job. Which part of the cost changes when the page count doubles?
Predict
The rule is C = 8 + 0.50p, where C is dollars and p is the number of pages. Predict whether doubling p doubles C. Keep the fixed fee visible: it is charged once, not once per page.
Pages p
Page charge
Total C
0
0 dollars
8 dollars
12
6 dollars
14 dollars
24
12 dollars
20 dollars
Zero pages is an algebraic input; whether the shop accepts an empty job is a separate business rule.
Experiment
Build two more rows for 6 and 18 pages. The totals are 11 and 17 dollars. Compare changes: six more pages always add 3 dollars. Then evaluate 8 + 0.50(−2). The arithmetic gives 7, but a negative page count is outside this story’s domain. A computable expression is not automatically a sensible model input.
Understand
An expression names a value using operations. An equation asserts that two expressions are equal. In C = 8 + 0.50p, p is an input variable and 0.50 is its coefficient. Its unit is dollars per page, so multiplication by pages produces dollars, which can be added to the setup fee.
Substitution replaces the entire variable with its value. For 5 − 3x at x = −2, write 5 − 3(−2) = 5 + 6 = 11. Parentheses preserve the sign. For (5 − 3)x, the same input gives 2(−2) = −4: grouping changes the expression.
Evaluate grouping first, then powers, then multiplication/division, then addition/subtraction. Operations at the same priority proceed from left to right. In −x² the square is evaluated before the outside minus. In (−x)² the minus belongs to the quantity being squared.
Apply
Complete the practice below. For each numeric answer, first write the substituted expression. In the written diagnosis, an example disproves an “always” claim, but identifying every equality case needs an argument. Use exact fractions when requested; the checker does not interpret arithmetic expressions.
Continue
Next, ask whether two differently written expressions always produce the same output. A few matching table rows can suggest equivalence; distribution can explain it. Return here whenever a later algebra error might be a sign, grouping or unit error.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Evaluate 3 − 2x when x = −4.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Evaluate (2/3)y + 5 when y = 9/2.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Evaluate 2(5 − 3)² − 7.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A fictional bike rental charges $6 once and $4 per hour h. Which expression gives dollars?
A learner says −x² and (−x)² always agree. Use x = 3 to diagnose the error and explain when they do agree over the real numbers.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
