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
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students multiply every component of a vector by the same number. They connect the calculation to a scaled arrow, including changes in length and direction.
What Mastery Looks Like
- Given a vector and a scalar, students correctly multiply every component. They can sketch the result and explain how the scalar changes the vector’s length and direction.
Common Misconceptions
- Students may multiply only one component or add the scalar to each component. They may think a negative scalar creates a negative length, rather than reversing direction. They may overlook that multiplying by zero produces the zero vector.
How to Assess It
- Exit ticket: Let v = (3, -2). Find -2v, sketch both vectors on the same grid, and compare their lengths and directions.
Lesson moves
Ways to Teach It
Give students paper arrows and rulers, then build 2v, 0.5v, and -v from one labeled vector.
Ask students to explain why multiplying a vector by -3 triples its length and reverses its direction.
Run a card match with vector cards, scalar cards, component products, and arrow diagrams.
Model a wind velocity vector, then use scalars to show doubled speed, half speed, and reversed direction.
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Printable HSN-VM.B.5 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSN-VM.B.5, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSN-VM.A.1
Scalar multiplication depends on knowing a vector as a directed quantity whose length changes and direction may reverse.
- CCSS.Math.Content.HSN-VM.A.2
Finding vector components from points supports scalar multiplication by giving the coordinate form whose components are scaled.
- CCSS.Math.Content.7.NS.A.2c
Scalar multiplication scales each vector component, so rational multiplication rules and properties help students compute and interpret component changes.
Keep exploring
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...
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