Part 5 of 8, Strategies for Integrating the Mathematical Practices into Instruction

By Dr. Michele Douglass

Looking for and Making Sense of Structure means using deductive reasoning. In other words, I recognize the pattern and can apply it to solve a specific problem. This is the one practice that I am most often asked about by teachers in primary grades. They want to know what happened to all the standards about patterns. My response is always the same: Math is all about patterns, so it isn’t something that should be taught as a single standard, but rather as a practice that we use when thinking mathematically.

This practice is about how we work with students so that they are always looking for and making sense of repeated structures. For example, a sequence of numbers begins with 5. The next term is found by adding 4, and the next term is found by multiplying by -1. If this pattern continues, what is the 25^th term in this sequence? Do you have to write the first 24 terms in order to figure this out? Seeing and using patterns moves beyond the primary standards of the past of recognizing AB or ABBA patterns.

You might need to think about this practice by answering a few questions:

• How do we help students look closely for patterns?
• How do we use patterns to teach generalized ideas and concepts?
• In the same manner, do we help students seek these patterns in numbers, operations, attributes of shapes, and figures?
• How do we help them take complicated concepts and see them as a single entity so they recognize what to do?

“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. … 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.” —Common Core State Standards, MP7: Look for and Make Use of Structure

Here are a few ways that I have used this practice while teaching specific content standards over the past year:

Example 1: Use the following problems to help you subtract 13 – 4, 13 – 6, and 13 – 8. How might this set of problems help you subtract 14 – 5 or 16 – 8? What other facts could you use this strategy to solve?

As students are learning subtraction facts, there are patterns that can be used so students are NOT learning isolated facts. The pattern in this strategy subtracts down to a ten first and then subtracts the rest. In other words, a student is only subtracting part of the subtrahend at first and then subtracting the rest of it. This pattern works well with students, as many of them have been successful in learning the facts that make ten.

Example 2: What relationship or pattern do you see in the highlighted columns of this multiplication table? How could this help you with learning the multiplication facts for 7?   

Multiplication facts can be learned using the distributive property. When students have time to play with the multiplication table in this way, they learn that there are multiple patterns. For example, 5 x 7 is equal to (5 x 3) + (5 x 4), and it is also (5 x 2) + (5 x 5). Likewise, 8 x 7 = (8 x 3) + (8 x 4) or (8 x 2) + (8 x 5) as well as (8 x 6) + (8 x 1). Using patterns such as these are powerful ways to support students in learning about multiplication, the distributive property, and multiplication facts.

Example 3: Solve this proportion for x: What are at least two different approaches to solving this proportion? When would each method be useful?

One approach students have been learning for years is cross multiplying and solving the resulting equation. When proportions are first taught as being equivalent ratios, students are able to use this thinking as a strategy. In this problem, the two ratios are equivalent, which means one can multiply the first ratio (the one on the left of the equal sign) by 1 written as 2/2. Now numerators and denominators are equivalent. The denominators create a rather simple equation to solve, which is 2(x – 1) = x + 3. You know that students are proficient with fractions as well as proportions when they articulate something like, “This strategy can always work but is best to use when either numerators or denominators are even multiples of one another.”

Example 4: Use a graphing tool to graph each of the following equations. Describe what shape you see in your graph and what is changing on the graph each time.

1. x² + y² = 1                                 b. x² + y² = 4                                             c. x² + y² = 9

d.    (x + 2)² + y² = 4                       e. x² + (y + 3)² = 4                                  f. x² + (y – 3)² = 4

How might this help you to describe the graph of x² + y² = 16 and x² + (y – 6)² = 4 ?

I encourage you to look at the references and resources below for additional ideas about how to help students build structures and see patterns. In the elementary grades, think about how place value is a structure as well as the process of long division. It would include having students sort shapes to discover common attributes before we define them. It helps to provide time during lessons for students to formulate and explain their ideas to other students, to the class, and to the teacher.

These types of conversations need careful facilitation. You shape students’ thinking by the way in which questions are posed during instruction as well as by how questions are written on assessments. I will end with a list I like to incorporate as I am supporting students in becoming more proficient with this mathematical practice:

• Why does this strategy work, and can a solution be found using this strategy?
• What pattern do you find in ___?
• What are other problems that are similar to this one?
• How is ____ related to ____?
• Why is this important to the problem?
• What do you know about ____ that you can apply to this situation?
• In what ways does this problem connect to other mathematical concepts?
• How can you use what you know to explain why this works?
• What patterns do you see?
• Is there a structure? How can you describe the structure?

References and Additional Resources:

Common Core State Standards Initiative Website: http://www.corestandards.org/Math/Practice/

Implementing the Mathematical Practice Standards Website: http://mathpractices.edc.org/

Inside Mathematics Website: http://www.insidemathematics.org/index.php/mathematical-practice-standards

Mathematics Assessment Resource Service Website: http://map.mathshell.org/materials/stds.php

Michele Douglass, Ph.D., is the president of MD School Solutions, Inc., a company that contracts with school districts on content and pedagogy with teachers and leaders. Her experience ranges from math instructor to director of curriculum and instruction at Educational Testing Services. She has authored several math curricula, as well as technology programs and professional development, including NUMBERS® with John Woodward, Ph.D., and Mary Stroh.