InputOutput 14 27 310 413 516 WHAT IS ADDITIVE THINKING? 123123 +3 When students use additive thinking, they consider the change in only one set of data. For instance, in the examples below, students can recognize that the pattern increases by 3 blue tiles each time, or that the value in the right column increases by 3 each time. Students who utilize only additive thinking do not recognize the co-variation between the term number and tiles, or between the two columns in the table.
MULTIPLICATIVE THINKING Understanding the co-variation of two sets of data For instance, in this pattern, the mathematical structure can be articulated initially by a pattern rule, number of tiles = term number x3+1 In older grades more formal symbolic notation can be used, y=3x+1 This allows students to confidently predict the number of tiles for any term of the pattern 123
Multiple Representations of Growing Linear Patterns
Tiles = position number x1+1 Tiles = position number x3+1 Tiles = position number x5+1 What is similar in the 3 rules? What is different? What is similar in the 3 patterns? What is different? What is similar about the trend lines on the graph? What is different?
Tiles = position number x3+2 Tiles = position number x3+6 Tiles = position number x3+9 What is similar in the 3 rules? What is different? What is similar in the 3 patterns? What is different? What is similar about the trend lines on the graph? What is different?
GENERALIZATION STRATEGIES StrategyDescription CountingDrawing a picture or constructing a model to represent the situation to count the desired attributes RecursiveBuilding on the previous term or terms in the sequence to determine subsequent terms (Additive thinking) Whole- object Using a portion as a unit to construct a larger unit by multiplying. There may or may not be an appropriate adjustment for over-or-undercounting. Guess- and-check Guessing a rule without regard to why this rule might work. Usually this involves experimenting with various operations and numbers provided in the problem situation. ContextualConstructing an explicit rule that expresses the co-variation of two sets of data, based on information provided in the situation. An explicit rule can allow for the prediction of any term number in the pattern.
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