Lesson Planning Case Study: Giving Kids a Concrete Sense of “One Million”

What would a pile of one million centicubes look like?

How big is a million? Students in grade 6 have to grapple with numbers up to a million (in Ontario). I would venture to say they have a very limited or abstract understanding of “millionness”. We do know that a sense of “5ness” and “10ness” are signpost along the way to understanding and using place value in base 10.

After “10ness” is developed, we can certainly get to using bigger numbers. Very young children can see that adding zeroes to the end of a number makes it bigger. Kids can get a sense of “100ness” very easily. Rather, the organization and logic of counting in English makes it easy to just “keep going”.

I am unsure if “one thousand” is concrete for kids. Actually, we have more than a thousand pieces of Lego in this house, so I could easily explain that to my five year old.

As numbers get bigger, kids can’t tie them to concrete objects- we simply don’t have that many objects. Do kids take it on faith that large numbers exist? There is something cultural about it- kids might learn made up big number words like “jillion” or “zillion” at a young age, for example.

Let’s focus on a million for a minute. Here is a bit of a lesson plan for giving kids a sense of “millionness” using concrete objects.

Consider this overall number expectation in Ontario, grade 6:

-read, represent, compare, and order whole numbers to 1 000 000, decimal numbers to thousandths, proper and improper fractions, and mixed numbers

I know we can easily represent 1 000 000 on a place value chart but I am concerned with concrete objects.

Let’s say our learning goal is this:

We are learning to get a sense of the size of one million.

This could be your task:

How many centicubes would it take to fill this room?

Hold up, wait up, you say- this is a measurement lesson!

Here are the measurement overall expectations:

estimate, measure, and record quantities, using the metric measurement system;

• determine the relationships among units and measurable attributes, including the area of a

parallelogram, the area of a triangle, and the volume of a triangular prism.

Here is a rough plan for what you could do with this task.

-kids struggle with square units, so, lay out a square metre on your classroom floor. Help students to see that it is a 100 centicube by 100 centicube square

-pose a few problems for practice with area, like, “How many kids could stand in the square metre”

-now that you have a sense of “10 thousandness” (one square metre), you could ask a related question like, “how many centicubes would cover the floor?”

Now add the third dimension.

-build a skeleton for the cubic metre using newspaper, or metresticks. I suppose you could put it in the corner and mark the walls for help.

-ask kids: “how many centicubes would fill the square metre?”

It may help to dump all the centicubes you can into the square, to get a sense of just how many it would take to fill ilt.

Thing you could add for context: Minecraft is based on cubic metre blocks!

Now you are ready for the main task (it may take a week to get there, depending on kids’ prior knowledge).

How many centicubes would fill this classroom?

You can have great fun estimating, establishing upper and lower bounds, maybe taking the mean of the answers, and so on. Let kids’ thinking loose on “millionness”!

My suggestion is to let kids loose on whiteboards, or chart paper. Let them pair up, or work in groups, or on their own. If an observer walks in, they should know right away that your classroom is an open space for kids’ mathematical thinking.

After, ask yourself this key assessment question to decide how to follow up in your instructional sequence:

One idea for a follow up: one billion is a big number. Can you think of a room or structure that would hold a billion blocks?