Connecticut Grade 8 Math

Two panel designs change in different ways

An imaginary display project has two designs. The rectangular panel has a fixed height of 3 centimeters and a width of x centimeters. Its area is A = 3x square centimeters. The square panel has side length x centimeters, so its area is B = x² square centimeters. Use positive lengths for actual panels. Compare the rules without assuming that both areas change at a constant rate.

For x values 1, 2, 3, and 4, the rectangular areas are 3, 6, 9, and 12. The square areas are 1, 4, 9, and 16. Each input gives one area for each design, so both rules define functions. The table gives four samples; the rules explain the relationships beyond those samples.

Compare changes over equal input intervals

When x increases by one centimeter, the rectangular area increases by three square centimeters each time. Its rate is constant, so the rule is linear. The square’s increases over the same intervals are three, five, and seven square centimeters. Those growing differences show that its rate is not constant; the square-area rule is nonlinear.

Draw a new one-centimeter-wide strip added to a rectangle: its fixed height makes the added area three square centimeters. Increasing both sides of a square changes two dimensions. That geometric change helps explain why adding one to x does not always add the same amount to the area.

Interpret an agreement at one input

Both designs have area nine square centimeters when x = 3, but that one agreement does not make the rules identical. At x = 4, their areas are twelve and sixteen. A student who treats equal outputs at one input as proof of equal functions needs to compare another input and then explain the rules.

A student who says a curved graph cannot be a function is confusing function with linear function. The key function requirement is one output for each input. A graph can be nonlinear while meeting that requirement. Use the table to point out the unique area assigned to each chosen length.

Connect the rule, table, and graph

Plot length on the horizontal axis and area on the vertical axis for each design. Label the units and use equal spacing on each scale. The rectangular points lie on a straight line; the square points do not. For x = 5, predict fifteen square centimeters for the rectangle and twenty-five for the square, then explain the differences from the x = 4 values. The follow-up checks the relationship, not just whether the four original pairs were memorized.

Selected Connecticut Grade 8 references

The Connecticut mathematics standards directory links the Core Standards publisher. The state’s Core Standards overview identifies adoption in 2010 and explains that districts develop local curricula. These labels summarize the activity’s focus; read the complete expectations and mathematical practices in the publisher’s mathematics document.

Sources checked October 5, 2026. This selected activity is an entry point for instruction, not a complete curriculum checklist.

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.

Grade 8 is within Connecticut’s Smarter Balanced mathematics testing range. The linked learning resources have their own directions and scoring; a practice score does not establish an official state performance level. Read Connecticut’s official summative assessment overview.

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.