What looks simple on the surface can be deceptively complex and elegant.

How might we teach our young learners to deepen their algebraic reasoning?

Let’s see what you think…

**Unit 8: Cartesian Coordinate Plane, Two-Variable Equations, Graphing, and Regularity in Repeated Reasoning**

- graph on the Cartesian coordinate plane,
- look for and make use of structure,
- look for and express regularity in repeated reasoning,
- use and connect mathematical representations?

Kristi Story, Trinity’s 6th Grade math teacher, set the above goals for student learning and selected what looks like a simple, yet is actually a deep task that aligns with these goals. Providing opportunities for students to learn important mathematics content and to engage in essential mathematical practices are at the forefront of this planning.

Tasks that provide the richest basis for productive discussions have been referred to as doing-mathematics tasks. Such tasks are nonalgorithmic—no solution path is suggested or implied by the task and students cannot solve them by the simple application of a known rule.(Smith, 16 pag.)

Day 1’s Task is modified from Fawn Nguyen‘s Visual Patterns, pattern #10 puppies. Starting as a simple number talk, how many do you see and how do you see them?

Obviously, 6th graders know that there are three puppies, but how do they see the three? Do they see two puppies in the top row and one puppy in the bottom row? Do they see two puppies in the first column and one puppy in the second column? Either way, they would write 2+1=3. To make their thinking visible, they circle the two and the one. Also, they might see a 2×2 square with one puppy missing and write 2×2-1=3. It is a quick check about attending to precision, making use of structure (making visible what isn’t readily seen), and writing an equation.

Continuing the number talk, how many do you see and how do you see them?

6th graders immediately know that there are five puppies, but how do they see the five? Do they see three puppies in the top row and two puppies in the bottom row and write 3+2=5? Do they see two puppies in the first two columns and one puppy in the third column and write 2+2+1=5 or 2×2+1=5? Do they see a 2×3 rectangle with one puppy missing and write 2×3-1=5? Additional practice attending to precision, making use of structure (making visible what isn’t readily seen), and writing an equation.

6th graders immediately know that there are seven puppies, but how do they see the seven? Do they see four puppies in the top row and three puppies in the bottom row and write 4+3=7? Do they see two puppies in the first three columns and one puppy in the fourth column and write 2+2+2+1=7 or 2×3+1=7? They might also see a 2×4 rectangle with one puppy missing and write 2×4-1=7.

The important reflection question is: Did I use the same structure for each of the figures, or did I make use of different structures with each figure?

Using previously discovered structures, students predicted the number of puppies in Figure 4 and in Figure 10. Connecting to the algebra in their previous unit, they wrote a generalization for any figure number using their structure and reasoning. We found the following different expressions.

(n+1)+n. where n is the figure number

2(n+1)-1, where n is the figure number

1+2n, where n is the figure number

“These all represent the same pattern. Are they equivalent expressions?” asked Kristi. Using the distributive property, and combining like terms, they proved equivalence.

Committed to deep understanding for our young learners, Kristi asked students to graph (Figure Number, Number of Puppies) on the coordinate plane.

Trained to notice and note, our students were surprised to discover a linear pattern.

JH said, “Hey, to go from one point to the next, all you have to do is go up 2 and over 1.”

When asked, CJ interpreted the point (6, 13) saying “that means that there will be 13 puppies in Figure 6.”

My #ObserveMe notes illustrate more of the details and flexibility.

Our students graphed points and a line on the Cartesian coordinate plane, made use of structure, expressed regularity in repeated reasoning, used and connected mathematical representations, and deepened algebraic reasoning.

That’s a lot of Algebra I for a 6th grader, don’t you think?

Deep learning. Empowered learners.

Never underestimate the power of a motivated learner.

Smith, Margaret (Peg) S.. The Five Practices in Practice [Middle School] (Corwin Mathematics Series). SAGE Publications. Kindle Edition.