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G.5(A)·Geometry·Patterns & TransformationsReadiness

Patterns to Algebraic Expressions

Use numeric and geometric patterns to write algebraic expressions.

Students translate growing dot arrangements and shrinking arc measures into nth-term expressions, distinguishing constant differences from constant ratios and checking the starting index.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice50 minutes

Learn the concept

Constant differences give arithmetic rules

If a pattern starts at a1a_1 and changes by d each stage, then an=a1+d(n−1)a_n=a_1+d(n-1). For example, a fixed group plus n equal groups can also be counted directly as dn+bdn+b. The two forms may describe the same dot arrangement.

Constant ratios give geometric rules

If each term is multiplied by the same ratio q, then an=a1qn−1a_n=a_1q^{n-1}. A ratio between 0 and 1 produces a decreasing sequence. Arc measures can follow this pattern even though the arcs come from congruent circles.

Check the expression, not just its appearance

Substitute the displayed stage numbers into each candidate rule. Stage 1 checks the starting value, stage 2 checks the first change, and a later stage helps distinguish rules that happen to agree at the beginning.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 13 · 1 point

Four figures of dots. Figure 1 has 2 dots. Figure 2 has 5 dots. Figure 3 has 8 dots in two rows of 4. Figure 4 has 11 dots in rows of 3, 4, and 4.
Dot pattern, Figures 1 to 4Original visual from the released STAAR test.

Archived released STAAR Geometry EOC · 2013 · Item 35 · 1 point

Four arcs from congruent circles. Stage 1 is a 180-degree arc, Stage 2 is 120 degrees, Stage 3 is 80 degrees, and Stage 4 is 53.3 repeating degrees.
Arc pattern, Stages 1 to 4Original visual from the released STAAR test.

Worked example

A dot design has one fixed center dot and adds a group of four dots at each stage. Stage 1 has 5 dots. A separate arc pattern begins at 144° and keeps two-thirds of the previous measure. Write both nth-term rules.

  1. The dot counts are 5, 9, 13, and 17, with common difference 4.
  2. The dot rule is Dn=5+4(n−1)=4n+1D_n=5+4(n-1)=4n+1.
  3. The arc measures begin 144°, 96°, and 64°, with common ratio 2/3.
  4. The arc rule is An=144(23)n−1A_n=144\left(\frac{2}{3}\right)^{n-1}. Check that n = 1 gives 144.

Dn=4n+1D_n=4n+1 dots; An=144(23)n−1A_n=144\left(\frac{2}{3}\right)^{n-1} degrees

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 1 Recursive Sequences

Five Minute Math · Aaron Daffern, Dallas ISD

Compare repeated addition and multiplication in sequences, then connect recursive descriptions to explicit nth-term rules.

Watch on YouTube
Texas public school video

Algebra 1 Writing Equations to Describe Functional Relationships

UT IPSI · Kid2Kid / Texas Education Agency

Translate patterns and relationships into algebraic equations.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. A dot pattern has 8, 13, 18, and 23 dots in stages 1 through 4. Write a rule and find stage 12.

    Show answer and reasoning

    Dn=5n+3D_n=5n+3; stage 12 has 63 dots.

    The common difference is 5, so Dn=8+5(n−1)=5n+3D_n=8+5(n-1)=5n+3. Then D12=60+3=63D_{12}=60+3=63.

  2. 2. Arc measures are 128°, 64°, and 32° in stages 1 through 3. A student proposes 128(1/2)^n. Correct the expression.

    Show answer and reasoning

    128(12)n−1128\left(\frac{1}{2}\right)^{n-1}

    There have been no halvings at stage 1. The exponent n − 1 gives zero at the first stage and one at the second.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 13 · 1 point

Four figures of dots. Figure 1 has 2 dots. Figure 2 has 5 dots. Figure 3 has 8 dots in two rows of 4. Figure 4 has 11 dots in rows of 3, 4, and 4.
Dot pattern, Figures 1 to 4Original visual from the released STAAR test.

A pattern of dots is shown in the four figures. If the pattern continues, which expression can be used to find the total number of dots in Figure nn?

  1. 3n−53n - 5
  2. 2n+22n + 2
  3. 2n+32n + 3
  4. 3n−13n - 1
Show STAAR answer and explanation

3n−13n - 1

The figures have 2, 5, 8, and 11 dots. Starting with 2 and adding 3 each time gives 2+3(n−1)=3n−12+3(n-1)=3n-1.

3n−53n - 5: At n = 1 this gives −2 rather than 2.

2n+22n + 2: This starts at 4 and increases by 2, not 3.

2n+32n + 3: This starts at 5 and increases by 2, not 3.

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Archived released STAAR Geometry EOC · 2013 · Item 35 · 1 point

Four arcs from congruent circles. Stage 1 is a 180-degree arc, Stage 2 is 120 degrees, Stage 3 is 80 degrees, and Stage 4 is 53.3 repeating degrees.
Arc pattern, Stages 1 to 4Original visual from the released STAAR test.

The first four stages of a pattern of arcs from congruent circles are shown. If this pattern continues, which expression can be used to find the degree measure of the arc in Stage nn?

  1. 20(10−n2)20(10 - n^2)
  2. 180(13)n−1180\left(\frac{1}{3}\right)^{n-1}
  3. 60(4−n)60(4 - n)
  4. 270(23)n270\left(\frac{2}{3}\right)^n
Show STAAR answer and explanation

270(23)n270\left(\frac{2}{3}\right)^n

The measures 180°, 120°, 80°, and 53.3‾°53.\overline3° form a geometric sequence with ratio 2/32/3. Its nth term is 180(2/3)n−1=270(2/3)n180(2/3)^{n-1}=270(2/3)^n.

20(10−n2)20(10 - n^2): This matches the first two stages but gives 20° rather than 80° at stage 3.

180(13)n−1180\left(\frac{1}{3}\right)^{n-1}: This uses ratio 1/3; stage 2 would be 60° instead of 120°.

60(4−n)60(4 - n): This subtracts 60 each stage; stage 3 would be 60° instead of 80°.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Recognize arithmetic and geometric patterns from figures or tables.
  • Write an explicit expression for the nth term with the first stage indexed by 1.
  • Verify a proposed expression against several displayed stages.

Instructional sequence

  1. 1

    Engage

    5 min

    Build a dot pattern with a fixed group and a repeated added group. Ask how to count a distant stage without drawing it.

  2. 2

    Arithmetic patterns

    12 min

    Connect constant differences to a first-term-plus-growth expression and to a structural count of the dots.

  3. 3

    Geometric patterns

    10 min

    Use arc measures with a constant ratio to introduce exponential expressions and the n − 1 exponent.

  4. 4

    Expression testing

    15 min

    Students match expressions to dot and arc patterns, testing stages 1, 2, and a later stage before choosing.

  5. 5

    Closure

    8 min

    Complete the exit ticket and explain whether the pattern is additive or multiplicative.

Misconceptions to address

““A constant decrease means the rule must subtract the same number.””

Teacher move: Compare differences and ratios. A sequence can decrease by multiplying by a fraction even though its differences change.

““The nth stage always uses n copies of the growth factor.””

Teacher move: From stage 1 to stage n there are n − 1 changes. Test n = 1 to catch an extra addition or multiplication.