CCSS.Math.Content.HSN-VM.B.5
The standard
(+) Multiply a vector by a scalar.
Common Core State Standards for Mathematics · Vector and Matrix Quantities
What this standard means
Students need to scale a vector by a number. They should multiply each component by the scalar, describe how the length changes, and know when the direction stays the same, reverses, or becomes the zero vector.
Mastery looks like moving between component form, arrow diagrams, and real situations such as speed or force. Students often multiply only one component, confuse scalar multiplication with dot product, or miss that a negative scalar flips the vector direction.
Ways to teach it
- Hands-on activity: Give students grid paper and vector cards, then have them draw 2v, 0.5v, and -v from the same starting point.
- Writing prompt: Explain what happens to the length and direction of vector (3, -4) when it is multiplied by -2.
- Quick assessment: Ask students to compute 3(2, -5), draw it, and write one sentence comparing it to the original vector.
- Real-world connection: Use a wind velocity vector and ask how doubling wind speed changes the arrow without changing its direction.
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Related standards
- CCSS.Math.Content.HSN-VM.C.7
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- CCSS.Math.Content.HSN-VM.B.4
(+) Add and subtract vectors.
- CCSS.Math.Content.HSN-VM.B.5a
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(...
- CCSS.Math.Content.HSN-VM.C.11
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations...