Michigan Grade 6 Math

Which quantity depends on the other?

A paper-strip model starts with a strip five centimeters long. Each step attaches a three-centimeter piece with no overlap and no gap. Let n count the attached pieces and L give the total length in centimeters. Write an equation, make a table for zero through four pieces, and describe the plotted points. This ideal classroom model ignores any extra joining material.

Explain the solution

The initial five centimeters remain part of the strip. Attaching n pieces adds 3n centimeters, so L = 5 + 3n. The length depends on the chosen number of pieces: n is the independent quantity and L the dependent quantity in this model. For n equal to 0, 1, 2, 3, and 4, the lengths are 5, 8, 11, 14, and 17 centimeters. Plot n on the horizontal axis and L on the vertical axis. The point (2, 11) means that two attached pieces produce an eleven-centimeter strip. Every additional piece raises the length by three centimeters, which the equation, table, and plotted points all show.

Notice the reasoning

The rule L = 3n misses the starting strip. Check n = 0: the model already has five centimeters before any pieces are attached. Also distinguish the constant increment of three from the total length, which changes. Reversing the axes is possible if clearly labeled, but the stated graph uses pieces horizontally and length vertically. This model allows whole numbers of complete pieces, including zero; points for half a piece do not describe the specified construction. The plotted points lie on a line, but connecting them would need an explanation of whether partial pieces are allowed. An equation alone does not supply that context.

Try a related task

Predict the length after six pieces: 5 + 18 = 23 centimeters. Then ask whether a length of sixteen centimeters is possible with complete three-centimeter pieces in this model. Subtracting the initial five gives eleven centimeters, which is not a whole number of three-centimeter pieces. A sixteen-centimeter length therefore does not fit the stated construction. Explain the restriction using both the equation and the table’s pattern. Compare the model with a second starting strip to see how changing the initial length shifts the output without changing the increment per piece.

Selected references from Michigan mathematics standards

The Michigan mathematics standards directory links the current mathematics document, adopted by the State Board of Education in 2010 and currently linked in the state’s academic standards directory. These selected references identify the worked activity’s focus. The expectations include numbered requirements, examples, and mathematical practices. Read the full document and confirm the school’s current instructional sequence.

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 currently linked 2025–26 state guide identifies M-STEP English language arts and mathematics in Grades 3–7, including Grade 6. Follow the school’s current assigned assessment pathway and directions. 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 currently linked state assessment guide.

Read MDE assessment updates 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.