## Chocolate Arrays

Lately, I’ve been seeing arrays everywhere I go: at the grocery store, at the pharmacy, at the farmers’ market. And, of course, at Costco. The big, bad bulk retailer is bursting with interesting items arranged in perfect columns and rows.

Naturally, I made a beeline for the chocolate.

What follows is a quick-image string for exploring the associative property, the patterns that occur when multiplying, and the relationship between columns and rows. It supports the development of some key strategies for multiplying: doubling and halving to maintain equivalence, doubling a dimension to double to product, and using partial products to solve.

## The delight of disequilibrium

Disequilibrium is Piaget’s term to describe when what a learner already knows comes into conflict with new information. Learners must work through the confusion to reconstruct new knowledge. How does the process feel to a learner?  How as a teacher can we respond during a number string when students demonstrate disequilibrium?

## Challenge problems, helper problems

When designing strings, there are typically one or more helper problems before a challenge problem.  For example, you might start an addition string on compensation (a big idea) with:

50 + 20 = (helper problem)

48 + 22 = (challenge problem)

## Arrays on the West Side

We recently learned that the third grade team at Manhattan School for Children (Elizabeth Frankel-Rivera, Madelene Geswaldo, Alice Hsu and Marissa Denice) started a really cool Homework Page for their classes.  In addition to reading for 20-30 minutes a night and writing two entries in their ELA homework section, third graders are also expected to think about and solve a set of math problems.

## Closed to Open Array

After looking over beginning-of-the-year assessments, a new 4th grade teacher was concerned that half of her students were still unsure of the open array model. Some were simply still not convinced of the empty boxes! As her coach, we planned a number string together that would engage the students through questioning, get to know the students even more through open discussions (since it’s still September), and to help each student “hook in” and trust the open array. I modeled this string, hoping to model the questioning but to also model for the teacher who is coming across the open array for the very first time.

## Making Thinking Visible

I recently worked with a 6th grade teacher, Miss T, as she led a string for the very first time.  Twenty-eight middle school students quietly re-arranged desks and chairs and situated themselves at the front of her room — itself, no small feat — as she prepared to facilitate a messy multiplication string:

17 x 10 =

17 x 2 =

17 x 12 =

17 x 20 =

17 x 19 =

17 x 21 =

## Photo number strings for multiplication

Here are two photos I snapped as I walked by a 99 cent store in LA. Beautiful arrays, no?

I am thinking about how to use these kinds of images as the anchors for number strings, particularly for intervention work with older students.  Sometimes older kids need work thinking about multiplication, but in an age-appropriate way.  What kind of questions do you think of with this image?  One could most simply begin by asking what kids noticed about the image.  That would bring most of the interesting mathematics forward, I think. Beginning perhaps with how many boxes of hot chocolate do you see (nice numbers)?  And then, considering this is a 99 cent store, how much would it cost to buy all of this chocolate.  It reminds me of some work that Pamela Harris suggests in her book on Powerful Numeracy, in which she asks kids what is 99 plus any number?  A 99 cent store is a great way to think about what is 99 times any number?