Vermont Grade 7 Math

Every term inside the parentheses matters

Rewrite 3(2x + 4) − x without parentheses and combine like terms. A student writes 6x + 4 − x. Explain the error and justify the correct expression for any rational value of x. Test x = 1/2 as a check.

Follow the reasoning

The factor 3 multiplies both terms inside the parentheses: 3(2x + 4) = 6x + 12. Subtracting x leaves 5x + 12. The student multiplies 2x by 3 but leaves the 4 unchanged, so the written expression is not equivalent. Distribution and combining like terms establish equality for every rational x; a few substitutions can illustrate that reasoning but cannot replace it. At x = 1/2, the original gives 3(1 + 4) − 1/2 = 14.5, and 5(1/2) + 12 also gives 14.5. The student’s version gives 6.5, exposing the error in that case.

Check the explanation

An equivalent expression has the same value for all permitted substitutions. Combining 6x and −x is valid because both terms have the same variable part. The constant 12 cannot be combined with 5x as though both were multiples of x.

Try the next step

Rewrite 3(2x − 4) − x as 5x − 12. Explain how the negative constant changes the distribution. At x = −1/2, both expressions give −14.5. Use the properties to justify the identity rather than treating that one check as a proof.

Vermont reference for this worked activity

The Vermont mathematics standards directory identifies Common Core State Standards adoption in 2010. The linked publisher PDF is the June 2010 CCSS mathematics edition; the upload directory date does not establish a new adoption. Read the complete requirements and the school’s current curriculum sequence. The reference below selects one worked focus.

This selected reference identifies the activity focus. The complete standard and grade curriculum include additional work. These labels do not certify alignment of every independent lesson, worksheet, or quiz in the broader directory.

Choose a next task from the student’s explanation

Use a different set of numbers to check whether the student can explain the same relationship. If the explanation is unclear, return to the model; if the reasoning is sound but a calculation is wrong, work through a shorter calculation and check it against the model. Use the related learning resources alongside the teacher’s assigned goals.

The Vermont Agency of Education Academic Proficiency overview lists general VTCAP English and mathematics assessment in Grades 3–9. Grade 7 is included in that mathematics range. Follow the school’s current directions and accommodations. Independently published practice follows its own timing, tools, question count, and scoring; its result does not establish a state performance level. This original activity is not an official state assessment question. Read Vermont assessment guidance.

State sources and the selected complete PDF page checked October 6, 2026. Confirm later guidance and local assignments before planning assessment preparation.