Classroom Sneak Peek - Mathematical Practice #7

Posted March 16th, 2012 by Michelle Flaming

The past couple of weeks I began blogging about the 8 Mathematical Practices from the Common core.  I have finished Mathematical Practice # 1, Mathematical Practice # 2, Mathematical Practice #3, Mathematical Practice #4, Mathematical Practice #5, and Mathematical Practice #6.  This week the focus is on CCSS Mathematical Practice #7 -Look for and make use of structure.  I'll address what this looks like in the classroom, what students will be doing, what teachers will be doing, and the most important, the type of questions teachers will be asking. 

 

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 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 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)2 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.

 

 

What does this really look like?  The chart below is a work in progress.  I've designed this with the expertise of many classroom teachers.  If you have other ideas, please don't hesitate to email me and share your expertise as well.  If you are interested in using this process with your staff, read What Do The Common Core Standards Look Like in the Math Classroom.

 

 

 

Mathematical Practice: Look for and make use of structure.

 

Student Actions:

Teacher Actions:

Open-Ended Questions:

 

  • Students share strategies and different algorithms with each other and discuss why different algorithms provide the same correct answer.

 

  • Look for connections between properties, such as 5 + 3 is the same as 3 + 5, why does this produce the same answer?

 

  • Look for patterns in numbers, operations, number of sides, attributes of shapes, side lengths, etc.

 

  • Apply a variety of strategies to solve the same problem.

 

 

 

 

 

  • Intentionally and purposefully help students make connections between different algorithms that incorporate different properties. For example:

            52 x 8 is the same as (50 x 8 )+

            (2 x 8).

 

  • Asking students to explain, show how, show another way, to solve problems.

 

  • 2 + 2 + 2 = 6, how does this relate to multiplication?

 

  • 10 - 2 - 2 -2 -2 -2, how does this repeated subtraction relate to division?

 

  • Allow time daily for students to formulate and explain their ideas to other students, to the class, and to the teacher.
  • Facilitate the discussions and explanations and use probing questions

 

  • How might you explain the problem in another way?

 

  • Why does this strategy work and so does yours?

 

  • How are the strategies the same? Why do they both work?

 

  • How can you solve the problem using a different operation?

 

  • How does repeated addition relate to multiplication? Can you solve the problem using a different method?

 

  • What pattern are you noticing?

 

  • What might be another way for you to sort?

 

  • What are you noticing?