7. Look for and make use of structure

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x  8 equals the well remembered 7 x  5 + 7 x  3, in preparation for learning about the distributive property. In the expression x²  + 9x  + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x  – y ) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x  and y.

Representative IMP Year 1 Lesson:

Continuing the Pattern (All About Alice), 438

Students explore the various powers of two in this activity. Throughout this unit, they have worked with positive exponents and see the growth with a base of two. They now explore negative exponents and examine the various patterns that emerge with negative exponents. Students explain how to find the result of a negative exponent using fractions as their results.

Representative IMP Year 3 Lesson:

Which is Which? and Formulas for nPand nC(Pennant Fever), 42, 43

Combinations and permutations are explored in these activities. Students examine their previous work with combinations and permutations and are asked to explain the difference between the two. They must use the proper notation for both. After looking at the patterns in their work, students find a general formula for permutations in terms of n and r. They are also asked to find a general equation expressing the relationship between permutations and combinations.