Making Sense of Division

When solving division problems, students are often able to produce a quotient without realizing what that quotient represents. For example:

The bag has 695 pieces of candy corn, and Alyssa wants to put them into bags of 50. How many bags will she need?

Students may be able to mechanically produce the quotient 13. But do they fully understand the referential meaning of "13"? And let's not even get started about what the remainder represents!

Explicitly teaching students, with a model, that there are two ways to think about division can help them develop powerful mental structures for analyzing their results. I'm going to suggest Cuisenaire Rods as an excellent model for representing division.

Sharing (partitive) division is the situation your students are most comfortable with because it can be solved by dealing out. In sharing division, we know the total and we know the number of groups. The unknown is the number of items in each group. We can solve this by dealing out, or sharing, all of our items one at a time until there are no more to be distributed.

Sarah has 10 puppies and 2 doghouses. How many puppies can be grouped evenly in each doghouse?

The Cuisenaire Rods offer a nonlinguistic representation of the situation. We have a given total divided into 2 groups. We must find the number in each group.

There will be 5 puppies in each doghouse.

In the second division situation, the action to solve the problem is making groups. This is called measurement division, and lends itself well to the idea of repeated subtraction - repeatedly removing groups of a particular quantity until there are no more to be removed.

Climb the Ladder!

Danylle, the 4th grade teacher at Hedog, is concerned about her students' fluency with base-10 concepts. We dug out her Kathy Richardson "Understanding Numbers: Place Value" kit and put together an effective (We hope!) set of activities for Menu.

It was funny, though, because our planning session began with me giving her an impromptu idea that seemed great at first, (Don't they all? Ha!) but looked more and more shabby the deeper we went into Kathy's materials.

Danylle proved her worth as a teacher, though, with unflagging encouragement! "I like your idea, Chris," she said. "I'm going to still use it."

Uh huh.

I left Hedog that afternoon determined to craft this idea into an activity that would produce results, and wouldn't look pitiful next to the other Menu items. I happened across some "Climb the Ladder" activities while browsing Mathwire.

Climb the Ladder was just the touch I needed! So I'm happy to share with you an activity that requires only a deck of digit cards and a work some activity sheets. This should be good for latter year second graders, third graders, and 4th graders who need further support or practice.

Note: The directions ask students to represent their number using squares, lines, and dots. This vocabulary was chosen for simplicity. These representations should be connected with the flats, rods, and cubes of the base-10 blocks, or to the sheets, strips, and singles of the Sticker Station.

Have fun!

Download: Climb the Ladder Base-10 and Place Value Concepts

Non Linguistically Speaking

When students struggle to interpret story problems into a mathematical statement they can tackle and solve, a nonlinguistic representation may lead to success. Explicitly engaging students in the creation of lonlinquistic representations increases activity in the brain (Gerlic & Jausovec, 1999)

Language challenged 3rd graders might find this problem difficult to navigate:

Riverside School had 517 students last year. This year, 60 students moved away before school started. How many students does the school have now?

A nonlinguistic representation to help students create a mental picture of, and organize, the essential information.

TOTAL
Part	Part

Substituting the information from the problem, our representation becomes:

517 students (Total)
60 students (Part)	Part

Using this representation, it is perhaps clearer to students that they are looking for the difference between 60 and 517. They could then solve the problem in one of two ways: by counting on from 60, or by subtracting 60 from 517.

Calculation Nation!

Calculation Nation^® uses the power of the Web to let students challenge opponents from anywhere in the world. At the same time, students are able to challenge themselves by investigating significant mathematical content and practicing fundamental skills. The element of competition adds an extra layer of excitement.

“The games on Calculation Nation^® provide an entertaining environment where students can explore rich mathematics,” said Jim Rubillo, Executive Director of the National Council of Teachers of Mathematics (NCTM). “Through these games, students are exposed to the same mathematical topics that they see in class as well as those that are recommended in Curriculum Focal Points.”

What Do Third Graders Know About Addition?

Mr. Birch's third graders have been putting together a list of statements they believe to be true about addition. Here is what they have so far:

What do we know about addition?

1.  Numbers usually get larger.

2.  You are putting things together.

3.  Adding sign is +

4.  Find the total.

5.  How many all together?

6.  What is the sum?

7.  Addends can be added up in any order to the sum.

8.           7   ----addend ------     5
            +5   --- addend ------- + 7
           12  ----  sum     -------  12

What do other third graders think about this list? If you'd care to add something, or have any questions for us, please leave a comment!




I have a question about the first item on their list. Mr. Birch's students suggest that numbers usually get larger with addition. Can you give an example of when the sum gets smaller???

Spring Creek Mathematicians Tell Addition Stories!

8 Sleeping Billy Goats + 2 Billy Goats Playing from (Chris) on Vimeo.

Mathematics is about far more than numbers - and certainly must include more than lists of rules. In her book What's Math Got to Do With It?, Jo Boaler describes math as

a human activity, a social phenomenon, a set of methods used to help illuminate the world

 Second graders at Spring Creek are exploring how mathematics can be used to illustrate simple stories. More specifically, they are crafting stories involving combinations of 10. Perhaps you can tell what time of day these were recorded by all the references to sleep.  /grin

Motorcycles, Math, and Mistakes

First Start from (Chris) on Vimeo.

Some might think that the only math in the minds of bikers heading to Sturgis revolves around miles per hour and motel rates. When things go wrong, though mathematics is often the culprit. And just as reliably - part of the solution.

Two weeks ago, my engine stuck a valve. When I lifted the rocker covers, I found enough evidence to warrant a complete top-end overhaul. There was plenty of good math involved!