CCSS.Math.Content.HSG-GMD.A

CCSS.Math.Content.HSG-GMD.A is a cluster heading. These are the individual standards under it.

• CCSS.Math.Content.HSG-GMD.A.1

  Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

• CCSS.Math.Content.HSG-GMD.A.2

  (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

• CCSS.Math.Content.HSG-GMD.A.3

  Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

01

What Students Need to Do

Students connect volume formulas to the shapes and measurements that create each solid. They use those formulas to find missing volumes or dimensions and justify why the formulas make sense.

02

What Mastery Looks Like

Students choose the correct formula for a prism, cylinder, pyramid, cone, or sphere and label units as cubic units. They explain key formula features using base area, height, cross sections, or comparisons between solids.

03

Common Misconceptions

Students may use surface area when the problem asks for volume. They often forget the one-third factor for pyramids and cones or confuse radius with diameter. Some assume equal heights guarantee equal volumes without checking base areas or cross sections.

04

How to Assess It

Give students a cylinder and cone with equal base radii and heights. Ask them to find both volumes and explain why one volume is one-third of the other.

1. 01

   Fill a cone with rice three times and pour it into a matching cylinder to show the one-third relationship.

2. 02

   Ask students to write why doubling a cylinder’s radius changes its volume more than doubling its height.

3. 03

   Have pairs match solid cards, dimension cards, and volume expressions, then check each set with a calculator.

4. 04

   Compare two drink containers by measuring their dimensions and calculating which holds more liquid and how much more.

• CCSS.Math.Content.8.G.C

  Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

• CCSS.Math.Content.5.MD.C.5

  Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

• CCSS.Math.Content.8.G.C.9

  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.