Patsy Y. Saldivar Campos
Strategy Overview
Using patterns as a problem-solving strategy makes problems easier to understand and solve. Once you identify the rule or rules for a sequence, you can use the pattern to find missing numbers or predict future terms.
To begin, it is important to understand some key vocabulary that helps you work with patterns. Johnson et al. (2004) defined a sequence as “an ordered string of numbers tied together by a consistent rule or set of rules that determine the next term in the sequence,” and a term as “an individual member of a sequence” (p. 108). When working with sequences, you must apply the same rule or rules each time to get the correct answer.
A common question is, How do I find the rule in a sequence? This is a skill that gets better with practice. One way to start is by looking at the differences between the numbers. Sometimes the pattern is how much each number increases or decreases. In more difficult sequences, the pattern may be found by looking at the differences between those differences. In other cases, you may need to use multiplication or division instead of addition or subtraction.
Before looking for the rule, it helps to organize the information. You can make a chart or diagram to show the sequence more clearly. This makes it easier to see how the numbers are related. Once the information is organized, you can test different operations, such as addition, subtraction, multiplication, or division, to find the pattern. After you identify the rule, apply it to different terms in the sequence to make sure it works. Finally, use the rule to write an equation that can be used to find any term in the sequence and solve the problem.
For what types of problems is this strategy helpful?
- Works with problems that follow a pattern or rule.
- Helps when information repeats or changes in a predictable way.
- Can be used to find missing numbers or terms.
- Helps predict what comes next in a sequence.
- Works with patterns that use the same rule over and over, such as growth over time or repeated steps in a process.
When is it NOT very useful?
When information does not follow a clear or consistent rule, the finding patterns strategy is not very useful. It does not work well when the data is random or has no predictable order, because there is no pattern to identify. It is also not effective when the rules in a problem change in different ways and no single rule repeats throughout the process. In addition, this strategy is not helpful for problems that require the use of a specific formula instead of discovering a pattern.
Common mistakes or misconceptions to watch out for:
A common mistake is assuming a pattern exists when there is no clear rule. Another mistake is when students try to force a pattern even when the numbers are random or do not follow a consistent change. Students may also forget to check their rule across all terms in the sequence. Finally, another mistake is not organizing the information first, which makes it harder to clearly see the pattern.
How does this strategy help students become better mathematical thinkers?
Using patterns as a problem-solving strategy helps students become better mathematical thinkers because it teaches them to look for structure instead of just memorizing steps. When students learn to find patterns, they start noticing relationships between numbers and how values change from one term to the next. This helps them move from just guessing answers to actually understanding what is happening in the problem.
This strategy also helps students think more critically because they have to analyze the information, test possible rules, and check if their rule works for all the terms in the sequence. Instead of using only one method, students learn to try different ideas until they find the one that makes sense. Over time, this builds confidence when solving new and unfamiliar problems.
Finding patterns helps students turn confusing problems into something more organized and easier to understand. This is why it is such an important strategy for developing strong mathematical thinking skills.
Example Problems
Problem 1:
The Hat that Didn’t Sell
Unable to sell a hat for $20, a haberdasher lowered the price to $8. It still did not sell, so he cut the price again to $3.20, and finally to $1.28. With one more markdown, he will be selling the hat at cost. Assuming that he followed a system in marking his price cuts, can you tell what the next markdown will be? (Johnson & Herr, 2004, p.131)
Problem 2:
Comic of The Month
I subscribe to the Comic-of-the-Month Club. Each month I can buy any number of the 48 titles offered by the club. The first month I bought five comics for $3.07. The second month I bought two comics for $1.72. The next month I bought six of the club offerings for $3.52. In May I bought three more for $2.17. The club charges a fee for each comic and a handling fee for the entire order. How much would it have cost to buy all 48 titles at the same time? (Johnson & Herr, 2004, p.133)
Worked Examples + Think-Alouds
Problem 1:
Problem 2:
Student Self-Talk Questions
Now let’s put the finding patterns strategy into practice by using the following self-talk questions. These questions will guide you through the problem-solving process and help you check if your pattern and your answer are correct.
- Understand the Problem: What information is given, and how can I organize it to look for a pattern?
- Make a Plan: What rule do I think the sequence is following?
- Carry Out the Plan: Does my rule work for every term in the sequence?
- Look Back: Can I use my rule to find another term and prove that it is correct?
Teacher Notes and Instructional Tips
Lesson Launch Ideas
Begin the lesson with a simple number sequence that students can solve mentally, such as 2, 4, 6, 8, __ or 5, 10, 20, 40, __. Ask students to explain how they know the next number instead of only giving the answer. Then introduce a more challenging sequence that requires students to look for a different rule. Explain that finding patterns is a strategy that helps simplify problems by identifying a rule that can be applied to every term in the sequence.
Teacher Questions & Anticipated Student Responses
- Teacher Question: What do you notice about the numbers in this sequence?
- Anticipated Student Response: The numbers are increasing or decreasing by the same amount.
- Teacher Question: What rule do you think the sequence is following?
- Anticipated Student Response: I think the rule is adding, subtracting, multiplying, or dividing by the same number.
- Teacher Question: How can you prove your rule is correct?
- Anticipated Student Response: I can apply the rule to every term in the sequence and check if it works.
Differentiation Suggestions
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Support:
- Provide students with a table or graphic organizer to help them organize the information. Begin with simple patterns that use addition or subtraction before introducing more complex patterns.
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Extend:
- Challenge students with sequences that have more than one rule or ask them to write an equation that represents the pattern. Students can also create their own sequence and have a partner identifying the rule.
Formative Assessment Tools
Observe students as they explain their thinking while solving a sequence. Use exit tickets where students identify the rule, explain how they found it, and solve for a missing term. Ask students to justify why their rule works for every term in the sequence.
Student Reflection Prompts
- How did organizing the information help me find the pattern?
- What strategy helped me identify the rule?
- How did I know my rule was correct?
- What would I do differently if I could solve the problem again?
What prior knowledge do students need?
Students should understand basic mathematical operations such as addition, subtraction, multiplication, and division. They should also know how to identify numbers in a sequence and understand that a sequence follows a rule or set of rules.
What classroom challenges commonly arise?
Some students may assume there is a pattern when no consistent rule exists. Others may stop after finding a rule that works for only part of the sequence instead of checking all the terms. Students may also struggle to organize the information before looking for a pattern or may rely on guessing instead of testing different mathematical operations.
