Vermont Grade 8 Math

A graph with three kinds of change

A fictional height model connects the points (0, 0), (3, 6), (5, 6), and (7, 2) with straight segments. Time is in minutes and height is in centimetres. Describe where the height increases, stays constant, and decreases. Is the entire graph one straight-line relationship?

Follow the reasoning

From 0 to 3 minutes, the height increases from 0 to 6 cm. From 3 to 5 minutes, it stays at 6 cm. From 5 to 7 minutes, it decreases from 6 to 2 cm. Each individual segment is straight, but the entire graph changes slope, so the whole relationship is not a single linear function. The segment slopes are 2 cm per minute, 0, and −2 cm per minute. Every time in the model’s stated interval has one height, even though several times can have the same height. The graph describes how height changes; it does not identify what causes that change.

Check the explanation

This is an invented continuous piecewise model, not measured data. Its straight segments specify the behavior between the named points. If only four observations were supplied without that connecting assumption, the behavior between observations would be unknown. The model gives no values outside 0–7 minutes.

Try the next step

Sketch a different graph that starts at height 2 cm, increases steadily to 5 cm over three minutes, remains there for one minute, and then decreases steadily to 1 cm over two minutes. Label the points (0, 2), (3, 5), (4, 5), and (6, 1), and explain why each time still has only one height.

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 8 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.