Massachusetts Grade 6 Math

What mass does the model predict?

An illustrative table describes two samples of the same uniform material: a sample with volume 4 cubic centimeters has mass 12 grams, and a sample with volume 7 cubic centimeters has mass 21 grams. Assume the material has the same mass per unit volume throughout. Predict the mass of a 9-cubic-centimeter sample and explain the assumption. These invented values are a mathematical model, not measurements of an identified substance.

Explain the solution

For the first sample, divide 12 grams by 4 cubic centimeters to obtain 3 grams per cubic centimeter. The second sample gives 21 divided by 7, also 3 grams per cubic centimeter. The two supplied pairs are consistent with a constant rate. Under the stated uniform-material assumption, a 9-cubic-centimeter sample has predicted mass 9 times 3, or 27 grams. The equation m = 3v represents this model when m is measured in grams and v in cubic centimeters. The rate carries units; the number three is not a volume or a mass by itself. A ratio table can show how each mass corresponds to its volume.

Notice the reasoning

Subtracting the two recorded values can find the same rate here: the mass grows by 9 grams as the volume grows by 3 cubic centimeters. Adding three grams to every new sample regardless of its volume would confuse a per-unit rate with a fixed increment. The model assumes the same uniform material and comparable conditions. Two data pairs alone do not prove that every unknown sample follows it. A different material, an empty space included in a volume measurement, or inaccurate measurement could change the relationship. Keep the mathematical prediction separate from claims about an actual substance.

Try a related task

Predict the volume corresponding to 30 grams under the same model: divide 30 by 3 to obtain 10 cubic centimeters. Then consider a third hypothetical record of 5 cubic centimeters and 20 grams. Its rate is 4 grams per cubic centimeter, so it does not agree with the model. Explain what would need checking before using one constant rate for all samples. Massachusetts explicitly includes mass-and-volume problems within Grade 6 ratio reasoning. Read the full requirements and choose related unit-rate practice with clear quantities and units.

Selected references from Massachusetts mathematics standards

The Massachusetts mathematics standards directory links the current mathematics document, adopted in March 2017 and currently listed in the state’s frameworks 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.

Grade 6 is within the statewide MCAS mathematics assessment range. MCAS is untimed within a regular school day. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration rules or a prediction of a state performance level. Read Massachusetts’s 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.