Inductive Reasoning
Use patterns and inductive reasoning to formulate and test conjectures.
Students extend patterns in tables, make precise conjectures, and test whether an apparent trend continues, using rectangular layouts whose areas can increase and then decrease.
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.
Learn the concept
Describe the observed pattern
Inductive reasoning begins with examples. State exactly what changes from one case to the next before predicting another case. A table may contain separate patterns for its inputs and a different pattern for its outputs.
Test beyond the visible rows
For rectangles, calculate for every new row. If width rises by 2 and length falls by 2, their sum stays constant, but their product need not change by a constant amount.
Revise a conjecture responsibly
A conjecture is a proposed generalization supported by observations. New cases can strengthen it or reveal a counterexample. Continuing a displayed pattern is different from proving that every possible case obeys a rule.
Worked example
A table lists rectangle dimensions 4 by 20, 6 by 18, 8 by 16, and 10 by 14. Test the conjecture: “Every later plan has more area than the previous plan.”
- The width increases by 2 while the length decreases by 2.
- The listed areas are 80, 108, 128, and 140 square units.
- The next plan is 12 by 12, with area 144.
- The following plan is 14 by 10, with area 140. This area is less than 144.
The conjecture is false; the 14-by-10 plan is a counterexample to continued growth.
Watch the lesson
Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.
5th Grade Math Recognizing Patterns in Tables
Five Minute Math · Aaron Daffern, Dallas ISDReview how to identify and test relationships in tables before extending more complex geometric patterns.
Watch on YouTubeTry it yourself
TEKSForge practice · unscored
1. Continue the dimensions 5 by 19, 7 by 17, and 9 by 15 for two more rows. Give both areas.
Show answer and reasoning
11 by 13 and 13 by 11; both areas are 143 square units.
Add 2 to the width and subtract 2 from the length each time. Reversing the factors produces the same product.
2. A student checks four rectangles and claims that all rectangles have area less than perimeter. What should you do?
Show answer and reasoning
Test additional rectangles; a 6-by-6 square disproves the claim.
Its area is 36 square units and its perimeter is 24 units. Four supporting examples do not prove a universal claim about the numerical measures.
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 49 · 1 point
A town wants to fence in a rectangular section of a park. The table shows five possible plans for the dimensions of this fenced section. The changes in the width and the length of these plans follow a pattern. If six additional plans are added to the table and follow the same pattern, which conclusion is not correct?
| Plan | Width (feet) | Length (feet) | Area (square feet) |
|---|---|---|---|
| 1 | 16 | 34 | 544 |
| 2 | 18 | 32 | 576 |
| 3 | 20 | 30 | 600 |
| 4 | 22 | 28 | 616 |
| 5 | 24 | 26 | 624 |
- The area of one of the additional plans exceeds 624 square feet.
- The area of one of the additional plans is less than 544 square feet.
- The area in Plan 6 is the same as the area in Plan 5.
- The area in Plan 7 is less than the area in Plan 6.
Show STAAR answer and explanation
The area of one of the additional plans exceeds 624 square feet.
Continue by increasing width 2 and decreasing length 2. Plans 6–11 have areas 624, 616, 600, 576, 544, and 504 ft². None exceeds 624, so the first conclusion is not correct.
The area of one of the additional plans is less than 544 square feet.: Plan 11 has area , which is less than 544; this conclusion is correct.
The area in Plan 6 is the same as the area in Plan 5.: Plan 6 has area , equal to Plan 5; this conclusion is correct.
The area in Plan 7 is less than the area in Plan 6.: Plan 7 has area , so this conclusion is correct.
Learning objectives
By the end of the lesson, students can:
- Extend input patterns consistently across several additional rows.
- Test conjectures by calculating new cases rather than extending a visual trend.
- Distinguish evidence for a conjecture from a deductive proof.
Instructional sequence
- 1
Engage
5 minShow three rectangles with increasing areas but the same perimeter. Ask whether the areas must keep increasing.
- 2
Table investigation
12 minRecord the changes in width and length separately, then extend the table beyond the nearly square rectangle.
- 3
Direct instruction
10 minDefine inductive reasoning and conjecture. Model how one new case can disprove an always statement.
- 4
Test and revise
15 minStudents evaluate claims about repeated areas, constant perimeter, and continued area growth using added rows.
- 5
Closure
8 minComplete the exit ticket with a supporting calculation and revise the opening prediction.
Misconceptions to address
Teacher move: Continue both dimension patterns. The product can decrease after the dimensions pass the most balanced arrangement.
Teacher move: Examples support a conjecture, but do not establish every case. One counterexample is enough to disprove it.