Ants go marching two by two.

Delve into division and number grouping with these buggy problems

Even Steven and odd man out may be two colloquial expressions you hear kids use well in a sentence, but do younger children know what even and odd numbers are? Do older students know what remainders are? Two delightful books by Elinor J. Pinczes provide the context for children to think about and become more comfortable with these concepts.

STEPHANIE'S PRIMARY LESSON

About the Book

Elinor J. Pinczes's book One Hundred Hungry Ants (Houghton Mifflin, 1993) tells the story of hungry ants headed to a picnic, and one ant's frustration at their slow progress. In an attempt to speed things up, he repeatedly stops the troupe and rearranges them, first into 2 rows of 50, then 4 rows of 25, and so on up to 10 rows of 10. When the ants finally arrive at the picnic, alas, it's too late and the food is all gone.

Step 1

I read the story aloud, pausing so children can comment. Children talk about what the littlest ant is doing and what they notice in the illustrations. (The food is being carted off by other bugs.)

Step 2

I pose this question: Could our class line up in 2 even lines? I ask all the children to stand and try. I choose two students to start the lines, standing side by side.

Step 3

If we have an even number of students in the class, I might write on the board "2 rows of 12" and make a quick sketch of it using circles to represent children. If we have an odd number, I record, "2 rows of 12 and one extra." Then I ask, "What if we got a new student in our class tomorrow: Could we still line up in 2 even lines?" In kindergarten we solve this problem together, using counters to work it out. Older children can solve the problem in pairs by using counters or drawing pictures.

I use the numbers we investigate to tell students: "Twenty-four is called an even number because we can line up in 2 even lines, and 25 is called an odd number because we can't line up in 2 even lines."

Step 4

I wonder out loud whether other classes in our school can line up in 2 even lines. After the children make predictions, pairs choose one class to investigate.

Step 5

Pairs report back to the class and I record the information on the board. Then I use a hundreds pocket chart and insert the numbers of the class sizes that could be broken into 2 even lines. I reinforce the words even and odd by saying: "Twenty-six children can line up in 2 rows evenly, so 26 is an even number" or "Twenty-seven children can't line up evenly, so 27 is an odd number." I also ask the children to observe the pattern emerging on the chart.

Step 6

To take the activity further for second graders or interested first graders, over the next several weeks I have them test other numbers and notice the pattern that continues to emerge on the hundreds chart. It takes many experiences for young children to build and cement their understanding of even and odd numbers.

RUSTY'S INTERMEDIATE LESSON

About the Book

A Remainder of One by Elinor J. Pinczes (Houghton Mifflin, 1995) is a rhyming story of a troop of 25 bug soldiers who line up in different ways to satisfy their queen. The bugs first line up in 2 groups, leaving soldier Joe as the odd bug out, much to the chagrin of the queen who likes things tidy. Over the next several days, the bugs line up with 3 in each line, then 4, leaving out soldier Joe each time. Finally, the bugs line up with 5 in each row with no remainder, satisfying their queen and saving Joe from humiliation.

Step 1

After reading the first several pages to my fifth and sixth graders, I leave sentences unfinished so students can guess the rhyming word. For example, I read, "All 25 soldiers marched past the bug crowd, nervously hoping they'd make their queen __________.' Students respond in a chorus, "Proud!"

Step 2

When I finish reading the part where the 25 bugs line up in 4 rows of 6, I pose this question: How could 25 bugs line up so that poor Joe isn't left as the remaining bug? I listen while students explain that the bugs could line up either in 1 row of 25 or 5 rows of 5.

Step 3

While a student rereads the story, I record on the chalkboard the division sentences that describe the events. I write: 25 bugs divided into 2 groups=12 bugs in each group with 1 remaining. I continue recording the sentences: 25 divided by 3=8 R1, 25 divided by 4=6 R1, 25 divided by 5=5 RO.

Step 4

I remind the children that the 25 bugs in the story lined up in different ways. I then tell them that I'm thinking of another number of bugs between 1 and 25 and ask them to solve my remainder riddle. I write the following seven clues on chart paper:

1. When you divide my number by 1, the remainder is 0.

2. When you divide my number by 2, the remainder is 0.

3. When you divide my number by 3, the remainder is 1.

4. When you divide my number by 4, the remainder is 2.

5. When you divide my number by 5, the remainder is 0.

6. When you divide my number by 6, the remainder is 4.

7. When you divide my number by 7, the remainder is 3.

I ask students to discuss the clues with the classmates at their table. Without revealing the answer, the students then report to the class any deductions based on the clues.

Step 5

Once the students solve my riddle (the answer is 10), they write their own remainder riddles for any number between 1 and 25. I tell them that their riddles must have seven clues and that they may use my clues from the chart paper as a model.

Step 6

Students pair up to exchange riddles. Some are able to divide mentally, some need to write the numerals 1 through 25 on a piece of paper and eliminate numbers as they read the clues, and some need to write division problems to check their numbers. For students who have difficulty, I suggest using concrete materials for dividing numbers. Ask those who want an extra challenge to write a remainder riddle for any number between 1 and 100.

STEPHANIE SHEFFIELD, a first-grade teacher in Houston, Texas, and RUSTY BRESSER, a fifth/sixth-grade teacher in San Diego, California, are the authors of Math and Literature books, published by Math Solutions Publications, available from Cuisenaire. MARILYN BURNS is the creator of Math Solutions inservice workshops.