CCSS.Math.Content.HSG-GMD.A.1
The standard
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
Common Core State Standards for Mathematics
What this standard means
Students need to explain where measurement formulas come from, not just plug numbers into them. They should use pictures, cutouts, stacking, and comparisons to argue why circle circumference and area formulas make sense, and why cylinders, pyramids, and cones have their volume formulas.
Mastery looks like a student pointing to a diagram or model and saying, for example, why a cone has one third the volume of a matching cylinder. Students often get stuck by memorizing formulas without connecting radius, height, base area, and volume. They also confuse surface area with volume.
Ways to teach it
- Have students cut a paper circle into wedges, rearrange them into a near-rectangle, and label why the area approaches pi r squared.
- Ask students to write: How does a stack of same-size circles help explain the volume of a cylinder?
- Show a cone and matching cylinder, then ask students to predict and verify how many cones of rice fill the cylinder.
- Connect circle area to pizza pricing by comparing two pizza sizes and asking which gives more food per dollar.
Plan a lesson for CCSS.Math.Content.HSG-GMD.A.1
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Related standards
- CCSS.Math.Content.HSG-GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- 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.
- 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.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumf...