How do our learners determine an equivalent expression to 4(x+3)-2(x+3)? How would they determine the zeros of y=x2-4? How might we provide opportunities for them to successfully look for and make use of structure? We want every learner in our care to be able to say I can make look for and make use of structure. (CCSS.MATH.PRACTICE.MP7) But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning? One of the CCSS domains in the Algebra category is Seeing Structure in Expressions. Content-wise, we want learners to
- “use the structure of an expression to identify ways to rewrite it. For example, see x4–y4 as (x2)2–(y2)2, thus recognizing it as a difference of squares that can be factored as (x2–y2)(x2+y2)”
- “factor a quadratic expression to reveal the zeros of the function it defines”
- “complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines”
- “use the properties of exponents to transform expressions for exponential functions”.
How might we offer a pathway for success? What if we provide cues to guide learners and inspire noticing? Level 4 I can integrate geometric and algebraic representations to confirm structure and patterning. Level 3 I can look for and make use of structure. Level 2 I can rewrite an expression into an equivalent form, draw an auxiliary line, or identify a pattern to make what isn’t pictured visible. Level 1 I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole. Illustrative Mathematics has several tasks to allow students to look for and make use of structure. We look forward to trying these, along with a leveled learning progression, with our students. 3.OA Patterns in the Multiplication Table 4.OA Multiples of 3, 6, and 7 5.OA Comparing Products 6.G Same Base and Height, Variation 1 A-SSE Seeing Structure in Expressions Tasks