CCSS.Math.Content.HSN-VM.B.5b
The Standard
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students determine how multiplying a vector by a number changes its magnitude and direction. They use the scalar's absolute value for length and its sign for direction.
What Mastery Looks Like
- Students correctly find that scaling by c changes the magnitude by a factor of |c|. They state whether the result points with the original vector, against it, or has no direction.
Common Misconceptions
- Students may multiply the magnitude by a negative scalar and report a negative length. They may also miss that a negative scalar reverses direction, while the zero vector has no direction.
How to Assess It
- Give students a vector with magnitude 6 and ask for the magnitude and direction of 3v, -2v, and 0v.
Lesson moves
Ways to Teach It
Use tape arrows on the floor to build v, 2v, and -0.5v, then compare their lengths and directions.
Ask students to explain why multiplying a vector by -3 changes its length and reverses its direction.
Draw scalar and vector cards, then race to calculate each new magnitude and label its direction as same, opposite, or none.
Model a cyclist's velocity, using positive scalars for forward speed changes and negative scalars for movement in the opposite direction.
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Printable HSN-VM.B.5b Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSN-VM.B.5b, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
Download the worksheetLearning progression
Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSN-VM.B.5a
Understanding scalar multiplication as scaling and reversing vectors is needed to determine the resulting vector’s magnitude and direction.
- CCSS.Math.Content.HSN-VM.A.1
Students must understand vector magnitude, direction, and notation before interpreting how scalar multiplication changes length and orientation.
- CCSS.Math.Content.7.NS.A.2
Signed rational multiplication and absolute value support understanding how a scalar changes a vector’s length and reverses direction when negative.
Keep exploring
Related Standards
- CCSS.Math.Content.HSN-VM.B.4a
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the mag...
- CCSS.Math.Content.HSN-VM.B.4b
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
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