# Simple, Yet Not: Growing Patterns – Growing Mathematicians

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.

# Common denominators – “Let’s see why”

Everybody knows that you must have common denominators to add fractions, right?  Do we know why? If asked to construct a viable argument, could we? Can we draw it (i.e., communicate why visually)?  How mathematically flexible are we when it comes to fractions? From Jo Boaler’s How to Learn Math: for Students:

…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.

Today’s Building Concepts lesson: Adding and Subtracting of Fractions with Unlike Denominators, had our young learners working to show their understanding of adding and subtracting fractions in multiple ways.

Kristi Story (@kstorysquared) used a phrase today that has really stuck with me is “Let’s see why…”  It immediately reminded me of Simon Sinek’s How great leaders inspire action.

And it’s those who start with “why” that have the ability to inspire those around them or find others who inspire them.

I wonder if, when young learners struggle with numeracy, it is because they do not see why.  Have they been so concerned with “getting the right answer” that they have missed the theory, reasoning, and geometry? This slideshow requires JavaScript.

What if we  leverage appropriate tools and use them strategically? What if we use technology to personalize learning and offer every learner the opportunity to see why?

#LL2LU draft for use equivalent fractions as a strategy to add and subtract fractions.

Level 4:
I can solve real-world and mathematical problems involving the four operations with rational numbers.

Level 3:
I can solve word problems involving addition and subtraction of fractions by using visual fraction models or equations to represent the problem.

Level 2:
I can add and subtract fractions with unlike denominators, including mixed numbers, by replacing given fractions with equivalent fractions.

Level 1:
I can understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

I can recognize and generate simple equivalent fractions, and I can explain why the fractions are equivalent using a visual fraction model.

#LL2LU for I can apply mathematical flexibility.

Level 4: I can analyze different pathways to success, find connections between pathways and add new strategies to my thinking.

Level 3: I can apply mathematical flexibility to show what I know using more than one method.

Level 2: I can show my work to document one successful  method.

Level 1: I can find and state a correct solution.

#LL2LU for I can construct a viable argument and critique the reasoning of others.

Level 4: I can build on the viable arguments of others and use their critique and feedback to improve my understanding of the solutions to a task.

Level 3: I can construct viable arguments and critique the reasoning of others.

Level 2: I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1: I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

# What is a Fraction? … be flexible, use appropriate tools strategically

What if we use technology to visualize new concepts and interact with math to investigate and learn? What if we pair a process learning progression with a content learning progression?

By the end of this lesson, we want every learner to be able to say:

I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.

AND

I can apply mathematical flexibility to show what I know using more than one method.

We have completed Jo Boaler’s two courses – How to Learn Math: For Students, and How to Learn Math: For Teachers and Parents.  As a team we are working on our math flexibility with math learners of all ages.  We challenge ourselves to offer more visuals and additional pathways for success. How might we leverage appropriate tools and use them strategically?

Enter: Building Concepts lessons from Texas Instruments.  Kristi Story (@kstorysquared) used What is a Fraction? to review and assess what is already known with our 6th graders.

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To offer a glimpse of the learning experience, a copy of my raw notes from this lesson are below.

Kristi starts with The Number of the Day to chalk talk a number talk.

It is obvious that our students have an understanding of fractions, decimals and percents.  Kristi encourages students to and modeled making connections between different representations of 2 1/5, the number of the day.  Many students answered aloud and enthusiastically moved to the board to draw or write a different representation.  By using the chalk talk method, this number talk encouraged number flexibility and creativity and the number talk offered all learners the opportunity to expand their understanding and fluency. Kristi launches the TI-Nspire software and the lesson What is a Fraction? and encourages our students to explore and investigate what the software will do and interpret the results.  This led to a side conversation about 1.5/3 and complex fractions.

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Kristi introduces the vocabulary of unit fraction. Interesting discussion and another chance for mathematical flexibility happens when students are asked to describe/illustrate what happens when the value of the denominator increases.  How does the number of equal parts in the interval from 0 to 1 change? What happens to the length of those parts?

Students clearly possess background knowledge of fractions, and Kristi challenges them to become more flexible in representing fractions.  Note: Many students are drawing circles to represent fractions.  In addition, we want them to draw number lines  and rectangles.

The discussion transitions to compare 3/5 to 7/5. Student answers included

3/5 is 3 copies of 1/5.
3/5 is a little more than 1/2
3/5 is 60% of the way between 0 and 1
3/5 is 2/5 back from 1
7/5 is 2/5 more than 1
7/5 is 3/5 less than 2
Both are 2/5 away from 1 but in different directions.

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Kristi and students use Think-Pair-Share to describe how they decided to explain their answer to the question Is 11/8 closer to 1 or 2? Kristi asks everyone improve their answer based on partner feedback. Kristi asks for volunteers to read their partner’s idea.

From me to Kristi:

I thought today was great! I love how you facilitated a discussion encouraging all learners to talk about math. My notes are attached.  Thank you for your willingness to pilot this software with our students.  I was glad to hear that you have enjoyed this start with fractions.

From Kristi:

Thank you for all the feedback. As I said yesterday, it was exciting to present fractions in a way that I think will make a difference in their understanding of fractions. I’m looking forward to continuing this series.

What if we use technology to visualize new concepts and interact with math to investigate and learn?

#LL2LU for What is a Fraction?

Level 4:
I can decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation, and I can justify decompositions by using a visual fraction model.

Level 3:
I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/from 0.

Level 2:
I can represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.

Level 1:
I can explain and illustrate that a fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts.

I can compare fractions by reasoning about their size.

Level 3:

#LL2LU for Mathematical Flexibility

Level 4:
I can analyze different pathways to success, find connections between pathways and add new strategies to my thinking.

Level 3:
I can apply mathematical flexibility to show what I know using more than one method.

Level 2:
I can show my work to document one successful  method.

Level 1:
I can find and state a correct solution.

What if we pair a process learning progression with a content learning progression?