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?

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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.