# Basic Number Relationships….Is thinking your goal?

### Reading the paragraph above inspired me to write today’s post. Chunk-itZ are a thinking tool that have emerged and evolved across my work with learning to teach mathematics through thinking and reasoning.  The Chunk-itZ hook students because they fit together in puzzles.

As students puzzle, teachers observe & listen for opportunities to prompt lessons around number, space and shape.

Do you see 3 and 2? How about 5?

Do you see a staircase?

Do you see a decagon?

The best way to learn mathematics is to follow the road that the human race originally followed: Do things, make things, notice things, arrange things, and only then reason about things.

***WW Sawyer

### Ideas that form the foundations for multiplicative reasoning and what teachers refer to as  ‘place value’ emerge.

As he was exploring the ChunkZ, a Grade one boy called me over to ask: “Is this 1 “two?” Then turned it over and finished his question: “or two “ones?”

I said: “hmmmm, interesting. How much is each of them worth?”

He responded, “they both are worth 2 but one is a 1 two and the other is 2.”

I asked, “What would this be?”

He responded: “Two ‘twos’. That’s four.

I turned the pieces over and asked: “And now what do I have?”

He responded: ” Two ‘ones’, and that is still four.”

He responded: “three ones, that’s 6 cause each one is a 2, or if you turn them over three twos and that is still 6.”

He is demonstrating the ability to unitize. He recognizes that we can talk about numbers as single units or as one unit of… this idea is critical to understanding place value which is based on multiplicative reasoning.

### Ideas about what it means to be equal emerge.

Can you “see” 4 and 1 inside 2 and 3?

Do you see how 5 + 2 and 4 + 3 are related in this puzzle?

ChunkitZ engage learners through visual spatial reasoning.  As they turn and trace the pieces to make their puzzles, learners develop intuitions and insights into how parts are related into wholes. In the hands of a skilled teacher, those wholes can be identified and related using numbers and number expressions.  Those wholes can be described and related using the attributes and properties of shape and the vocabulary of space.

This three has 6 sides. What do we name any 6 sided shape? Are any of your puzzles hexagons? Can you use ChunkitZ to build a hexagon shaped puzzle?

This student counted the sides of the three Chunk and said it has 6 sides. He then built a ’12’ puzzle by putting 2 three ChunkZ together and was wondering how many sides it would have. Will it be twice as many sides?

Nearly a century of research confirms the close connection between spatial thinking and mathematics performance. The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related.

***Mix & Cheng, 2012

The connection does not appear to be limited to any one strand of mathematics. It plays a role in arithmetic, word problems, measurement, geometry, algebra and calculus. Researchers in mathematics education, psychology and even neuroscience are attempting to map these relationships.

Combined with dot collections, Chunk-it tasks differentiate to challenge ALL learners as they develop, adapt, practice and refine key skills and attitudes that are critical to developing more than just RECALL of facts. ChunkitZ engage young learners in mathematical REASONING that will make the difference to their success across all the strands and all future grades. In the hands of a skilled teacher, the discussions and drawings form the basis for teaching children to explain their thinking through oral and written communication. Reading, writing, symbolizing the mathematical relationships, those are the goals. Automatic recall of facts are just a given, it comes and it lasts.

ChunkitZ help develop proportional reasoning which forms the foundation for making sense of multiplication, division, fractions and decimals. Outcomes related to proportional reasoning include comparing size of units to number of units needed when measuring (Grade 2), understanding the relationship between minutes and hours, days and weeks, months and years (these are all ratios), cm to m and mm to cm and m (more ratios) making sense of how multiplication facts are related and how area grows, making sense of how fractions are related and how equivalent fractions are related… the list goes on.

Here is a task for proportional thinking: I show this puzzle on the overhead or on a task card. Students work with the full size Chunk pieces to reproduce it without coming up and direct comparing.

He is directly comparing. He is fitting the pieces in to make the puzzle. The beginning of proportional reasoning.

The task above is very different from the task below. I show this puzzle on the Smartboard or give it to students on a small task card and they build it with the Chunk pieces which are 3 tior four times the size.  The outline they create will be similar (the same shape, angles are same) but not congruent (same side lengths). This is a much more sophisticated version of the Grade one matching above. And Grade 5 to 8 students love it.

Once they get good at it they are able to also double and triple the dimensions of the shapes, using chunk “ones” as their referent.