Letter Patterns Look at the letter pattern below.
Draw what you think the 4th and 5th stages will look like. What would the 10th stage look like? Develop a rule that would tell you how many blocks would be in any stage of the pattern.
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Letter Patterns Suggested Grade Span 3–5 Grade(s) in Which Task Was Piloted 4
Task Look at the letter pattern below.
Draw what you think the 4th and 5th stages will look like. What would the 10th stage look like? Develop a rule that would tell you how many blocks would be in any stage of the pattern.
Alternative Versions of Task More Accessible Version: Look at the letter pattern below.
Draw what you think the 4th and 5th stages will look like. What would the 10th stage look like? What pattern do you notice? Can you develop a rule that would tell you how many blocks would be in any stage of the pattern?
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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More Challenging Version: Look at the letter pattern below.
Draw what you think the 4th and 5th stages will look like. What would the 10th stage look like? Develop a rule that would tell you how many blocks would be in any stage of the pattern.
NCTM Content Standards and Evidence Algebra Standard for Grades 3–5: Instructional programs from pre-kindergarten through grade 12 should enable students to ... Understand patterns, relations and functions. • NCTM Evidence: Describe, extend and make generalizations about geometic and numeric patterns. • Exemplars Task-Specific Evidence: This task requires students to observe a pattern of how the letter grows, determine how many blocks would be needed to make the fourth, fifth and tenth pattern, and then generalize the pattern.
Time/Context/Qualifiers/Tip(s) From Piloting Teacher This is a medium-length task. This task was part of an ongoing unit teaching students how to recognize patterns and develop these patterns into algebraic rules. It was used as a formative assessment to see what areas the students still needed practice with. Before doing problems on paper, we spent time doing many pattern-building exercises with blocks, unifix cubes and pattern blocks. Students would then create charts to show the number relationships and use these charts to find and to express those patterns. We did a number of whole-class activities around growing patterns and how rules can be developed to find any number-stage of a pattern by looking at the relationships of stages and total number of blocks. A helpful hint is to ask students what part of the letter (leg, side, bottom, etc.) is always the same and what part is growing.
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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To print a copy of the more accessible version of the task with the graphic image, refer to page 6. To print a copy of the more challenging version of the task with the graphic image, refer to page 7.
Links This task could be used when looking at graphic-art designs and which stages are pleasing to the eye and which stages are not. This task can link to Anno’s Magic Seeds, by Mitsumasa Anno, which shows how quickly numbers can grow.
Common Strategies Used to Solve This Task Because students could recognize patterns, create charts/tables and develop rules, these were the strategies they used to determine the solution. Students typically begin by drawing the fourth and fifth stage. From there they create a chart to show the number relationships between the stage and the total number of blocks. Developing the rule from the chart/table and from their drawings of the fourth and fifth stages is difficult. Some students are able to recognize how the pattern is growing but are not able to put it into algebraic form. This is a difficult concept for fourth graders, but with continued practice and application to numerous types of growing pattern problems, they will begin to come up with algebraic rules. Using blocks and other manipulatives makes it more concrete for these learners.
Possible Solutions nx2+3=t n = any stage number t = total number of blocks More Accessible Version Solution: n+2=t n = any stage number t = total number of blocks More Challenging Version Solution: nx5+2=t n = any stage number t = total number of blocks
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Task-Specific Assessment Notes General Notes If students develop a rule, they can substitute any letter for the number as long as they have a key that clearly shows which is the stage number and which is the total number of blocks. All charts/tables must be labeled. I also used the "Exemplars Functions" and "Algebra Concepts Mini-Rubric" for this task. Novice The Novice will show little understanding of variables and growing patterns. S/he will not draw the fourth and fifth stages correctly and will not be able to create a chart/table to show the growth pattern and what the 10th stage would look like. The Novice will be unable to come up with a rule to describe what any stage pattern would be. Apprentice The Apprentice will show evidence of discovering and describing patterns and may be able to represent these relationships with a table. (The student may draw stages four and five correctly, and accurately then describe what the 10th pattern would look like, for example.) There may be a labeled chart/table (although there may be no key) that shows the growth pattern (+2) and that across the top stays the same (3), but the student will not accurately describe the rule for this pattern. Practitioner The Practitioner will show evidence of discovering, describing and generalizing patterns. S/he will represent these relationships with tables and rules. The Practitioner will draw stages four and five correctly and through a table, show what the 10th stage will look like. There will be a labeled chart/table with a key. S/he will use a rule to prove that it works by writing the corresponding equations for each stage. Expert The Expert will show evidence of discovering, describing and generalizing patterns. S/he will represent these relationships with tables and rules: for instance (n x 2) + 3 and (n + 1) x 3 – n. The Expert will draw stages four and five correctly and describe what the 10th stage will look like. There will be a labeled chart/table with a key that shows a solid understanding of what stays the same (the top), what is changing (blocks going down) and the relationship between these. The Expert will use her/his rule to prove that it works by writing the corresponding equations for each stage.
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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More Accessible Version Worksheet Look at the letter pattern below.
Draw what you think the 4th and 5th stages will look like. What would the 10th stage look like? What pattern do you notice? Can you develop a rule that would tell you how many blocks would be in any stage of the pattern?
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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More Challenging Version Worksheet Look at the letter pattern below.
Draw what you think the 4th and 5th stages will look like. What would the 10th stage look like? Develop a rule that would tell you how many blocks would be in any stage of the pattern.
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Novice
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Apprentice
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Apprentice
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Practitioner
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Practitioner
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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Expert
Letter Patterns Copyright 2008, Exemplars, Inc. All rights reserved.
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