New Hampshire Grade 7 Math

Circumference measures the length around a circle, while area measures the region it covers. Both calculations use the same radius, but their formulas and units answer different questions. This original diagram labels the radius and the full diameter.

Try this original activity

A circular paper design has radius 4 cm. Find the length of ribbon needed to trace its boundary and the area of paper inside it. Give exact answers using π and approximate answers using π ≈ 3.14. Ignore overlaps, trimming, and extra ribbon.

Work through the reasoning

The diameter is twice the radius: d = 8 cm. Circumference is C = 2πr = 8π cm, approximately 25.12 cm. Area is A = πr² = 16π cm², approximately 50.24 cm². The first answer counts length around the edge, so its unit is centimeters. The second counts covered surface, so its unit is square centimeters. To explore the formulas’ relationship, cut a separate paper circle into many equal sectors and alternate them into a near-rectangle. Its height approaches r and its long side approaches half the circumference. This suggests A = (C/2)r = (2πr/2)r = πr². A small number of curved sectors makes only an approximation, which improves as the sectors become narrower.

Check the common mistake

Using eight as the radius doubles the stated radius; the eight-centimeter segment is the full diameter. Squaring the radius belongs in the area formula, not the boundary formula. Equal-looking numerical results in some examples do not make area and circumference interchangeable: their units differ. Real ribbon work would need allowance for joins or overlaps, which this simplified task explicitly excludes.

Try a related case

Double the radius to 8 cm. The circumference becomes 16π cm, twice the original. The area becomes 64π cm², four times the original. Explain why the linear factor doubles but the squared factor grows fourfold. Use two paper circles or calculations rather than an unscaled image comparison.

Selected Common Core references for this activity

New Hampshire’s current school-approval rules, Ed 306.34, allow local boards to use the adopted 2010 Common Core mathematics standards or approved alternative academic standards. Confirm the school’s standards choice and local competencies first. The selected Common Core references below were checked in the original publisher’s 2010 document. They identify this activity’s focus, rather than a complete local course. Read the full requirements, mathematical practices, examples, and progression with the school’s curriculum.

Sources checked October 6, 2026. Read complete requirements, clarifications, limitations, and mathematical practices alongside the school’s sequence. This selected activity is an entry point for instruction.

Connect the explanation with practice

Use the lesson, worksheet, and quiz directory to choose a related skill. The existing resource links remain available; open each activity’s directions to understand its timer, tools, question count, and scoring.

The 2025–2026 public NH SAS manual describes general mathematics testing in Grades 3–8, including Grade 7, using a computer-adaptive assessment. Confirm the learner’s assigned pathway, tools, and the school’s current administration directions. The reviewed manual is the 2025–2026 edition; confirm later-year directions with the school. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration specifications or a prediction of a state performance level. Read the public NH SAS manual resource page.

Read New Hampshire local curriculum rules for later guidance. Keep one example of the student’s reasoning with the practice result. Choose a smaller model, a related question, or a new context according to the difficulty revealed in the worked activity.