CCSS.Math.Content.HSF-BF.A.2
The Standard
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students write arithmetic and geometric sequences as term-by-term rules and as formulas for any term. They use both forms to represent contexts and convert one form into the other.
What Mastery Looks Like
- Students correctly identify a sequence as arithmetic or geometric and find its common difference or ratio. They write both forms with a clear starting value and translate between them without changing the sequence.
Common Misconceptions
- Students often use a common difference for geometric sequences or a common ratio for arithmetic sequences. They may omit the starting term in a recursive rule or shift an explicit formula by one index. They may also confuse repeated addition with repeated multiplication.
How to Assess It
- Exit ticket: For 5, 15, 45, ... and 22, 18, 14, ..., write recursive and explicit formulas, state whether n starts at 0 or 1, and name a situation each could model.
Lesson moves
Ways to Teach It
Build a tile staircase, record the number of tiles in each stage, then write recursive and explicit rules.
Explain how the first term and common difference or ratio appear differently in recursive and explicit formulas.
Match card sets showing term lists, recursive rules, explicit rules, and contexts, then justify each match to a partner.
Model two savings plans, one adding a fixed amount monthly and one growing by a fixed percent, and compare their formulas.
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Printable HSF-BF.A.2 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSF-BF.A.2, 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.HSF-BF.A.1a
Turning a context into an explicit rule or recursive process supports modeling sequences in both formula forms.
- CCSS.Math.Content.HSF-IF.A.3
Seeing sequences as functions with integer inputs supports writing recursive and explicit rules for arithmetic and geometric patterns.
- CCSS.Math.Content.HSF-LE.A.2
Constructing arithmetic and geometric rules from tables or descriptions supports finding common differences or ratios for explicit and recursive sequence formulas.
What This Unlocks
Mastery here sets students up for these next.
- CCSS.Math.Content.HSA-SSE.B.4
Finite geometric series require recognizing geometric terms, common ratios, and explicit nth-term structure from geometric sequences.
- CCSS.Math.Content.HSF-LE.A.1c
Geometric sequences give the repeated multiplication idea needed to recognize constant percent growth or decay across equal intervals.
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