The goal of this lesson is to further develop the sharing concept of division by using objects. Students will have one and two digit quotients with and without remainders.
- base 10 blocks
- Remainder of One, Pinczes, Elinor J.
- centimeter cubes
- 2 sheets of paper per student (for book)
- bugs in an array handout
- hundreds board
- calculator (optional)
- glue, crayons, and scissors
- die (one for each pair of students)
Lessons for Introducing Division, by Maryann Wickett, Susan Ohanian,
and Marilyn Burns (Math Solutions Publications)
Background for Teachers
Students use division informally long before they receive any classroom instruction.
One type of division strategy is known as sharing or partitioning. Students
divide objects by sharing them one by one until there arent any more
or there arent enough to go around. For example, if they want to share
20 cubes in 4 rows, they place one cube in each row until each row has five
cubes. The goal of this lesson is to further develop the sharing concept of
division by using objects. Students will have one and two digit quotients with
and without remainders. These concepts will take several days to develop.
Prior to this lesson, the students should already know that multiplication
and division are inverse operations. They should have some experience with building
arrays by dividing individual cubes into equal rows with a "0" remainder.
(For example, if 15 cubes are divided equally into 3 rows, there will be 5 in
each row. I can check this answer because an array with 3 rows of 5 cubes have
a total of 15 cubes.) The students would also be able to interpret this information
in a simple story problem (e.g., if 15 pencils are divided equally among 3 students,
how many will each student get?).
Intended Learning Outcomes
1. Demonstrate a positive learning attitude toward mathematics.
3. Reason mathematically.
4. Communicate mathematically.
5. Make mathematical connections.
6. Represent mathematical situations.
Invitation to Learn
Read the book Remainder of One. Have students use 25 cubes to make
the rectangular arrays discussed in the story (e.g., 2 rows of 12, 3 rows of
8, 4 rows of 6, and 5 rows of 5).
- Record the various arrays from Remainder of One by creating an array book.
Use the bugs in an array handout. Have the students cut out the various arrays,
glue them into their book, and connect each array to the symbolic algorithm.
- Give the students a hundreds board. Ask, If you were marching 5
bugs in each row, how many leftover bugs would there be if there were only
16 bugs altogether? (1). Try building various numbers. Color the numbers
on the hundreds chart with one color for a remainder of 1, a second color
for a remainder of 2, and so on. Are there any patterns? Do you think the
pattern will change if the bugs marched in a row of six instead of five?
- Play the game of Leftovers in partners. Players start with 20 counters.
The first player rolls a number cube, divides the current number of counters
by that number, and states the division problem (e.g., if I take 20 counters
and divide them equally into 3 rows, there will be 6 in each row with 2 left
over). The player takes the remainder counters and tells the other player
how many counters to start with. The game continues until no counters remain.
To determine the winner, roll a number cube. If you get an odd number, the
player with the most counters wins. If you get an even number, the player
with the least counters wins.
- Remainder of One Riddles. Have students choose a number between 1 and 25.
Write Remainder of One Riddles (see handout).
- Have the students use base ten blocks to explore the following problem:
47 ÷ 3 = ?
- The student can show the dividend as 4 tens and 5 singles. The divisor
is the number of equivalent rows to be formed (e.g., 47 ÷ 3 = ?
Divide the blocks equally among 3 rows).
- Distribute 1 ten to each of the 3 rows.
- A record should also show that a total of 3 tens has been removed from
- The remaining ten cannot be distributed among three rows. They are
traded for 10 singles, and the other seven singles are joined with them.
This is shown by bringing down the 7.
- The 17 singles are then distributed equally among the three rows. Distribute
2 singles to each of the 3 row.
- A record should show that only 15 singles could be distributed in equal
- A remainder of 2 singles is left.
Answer: 15 R 2
Play Leftovers Game using larger numbers and a 4 - 9 number cube.
Homework & Family Connections
Play Leftover game. Teach a member of your family how to do long division. Return
a note indicting the shared mathematical experience between the family member
and the student.