This activity teaches students how to use formulas for dividing numbers. They will learn the formulas for multiples of 3, 2, and 5.
Invitation to Learn
Background for Teachers
The rules of divisibility are simple formulas for understanding how
fair shares can be created from large numbers without practicing long
or short division. Students usually come to fifth grade with an implicit
understanding about why numbers are divisible by 2, 5, and 10, but
it is important in fifth grade to make that understanding explicit.
Additionally, the formulas for dividing numbers by 3 and 9 must be
taught, since they are rarely discovered by children. It is helpful to
separate the formulas for 2, 5, and 10 (which depend on the digit in
the ones column) from the formulas for 3 and 9 (which depend on the
sum of the digits and the formula for 6, which combines the rules for
2 & 3). Note that there are simple formulas for divisibility by 4, and
8, (as well as more complicated formulas for larger numbers) but they
are not part of the Utah fifth grade Core Curriculum requirements.
Information about these formulas is included in the curriculum
extensions section for interested students.
This lesson should be sequenced after division with whole numbers
has been reviewed and practiced, division with remainders has been
reviewed and practiced, and students are familiar with vocabulary
terms dividend, divisor, and quotient. It may also be used to review
prime and composite, since every number greater than 2 that is
divisible by 2, 3, 5, 6, 9, or 10 is composite; also, when discussing
divisibility, students will probably remember that all numbers are
divisible by one and themselves.
Invitation to Learn
Divide the class into teams of three members each. One member
is the director, one the recorder, and one the materials coordinator.
Each team takes four index cards and writes a different digit from 0-9
on each card. Then, from the four choices of digits, the team makes a
list of all the possible four-digit number combinations using each digit
once. There will 24 possible number combinations. Next, have the
students each take a graphic organizer, Divisibility Test, with columns
for the numbers they created, plus the columns for 2, 3, 5, 6, 9, and 10
listed across the top. Using calculators if you wish, have the students
divide each of their 24 numbers by 2, 3, 5, 6, 9, and 10 to decide
if their numbers divide evenly without leaving remainders. If the
number divides evenly, have the students write "yes" in the column on
the graphic organizer. If the number does not divide evenly, have the
students write "no" in the column on the graphic organizer.
After the graphic organizer is complete, have each team record their
"yes" examples on chart paper hanging around the room, one piece for
each of the numbers 2, 3, 5, 6, 9, and 10. Once this is done, have each
team make a hypothesis about a "rule" for divisibility for each of the
numbers 2, 3, 5, 9, and 10. Have them record their hypotheses on the
graphic organizer labeled Divisibility Rules. It is important that each child have his or her own copy of the two graphic organizers because
the next part of the lesson is done as a whole class.
- After teams have completed their Divisibility Test graphic
organizer, recorded their numbers on the chart paper, and
made hypotheses about divisibility on their Divisibility Rules
graphic organizer, have them return to their individual seats
for a whole-class lesson.
- Using the chart paper lists as summaries of numbers
generated by the class teams, discuss each chart and have the
students share their hypotheses of divisibility rules. Guide their
discussions to the correct rules for each number, and have them
write them on the graphic organizer. Then have them trim the
edges of their graphic organizers and glue them into their math
journals for later referencing.
- Ask the students if it is possible to divide their rules into two
main categories, using a Venn Diagram to compare and contrast
the categories. Lead them to separate the numbers where
the ones digit determines the divisibility (2, 5, 10) from the
numbers that require adding all the digits (3, 9). Have them
complete a Venn Diagram in their math journals while you
model one on the board.
- Play Divisibility Rocks using students' journals as reminders of
the divisibility rules. Note: if this game is used as one station
in a variety of center activities, fewer sets of the game will need
to be produced.
How to play Divisibility Rocks
- Divide the class into groups of two to six students per game.
(An ideal size game is three students because each player will
always have a job.)
- Give each group one game set. Each set requires a deck of
cards, a bag of rocks, and a Divisibility Key.
- Divide the cards face down evenly among members of a group.
Discard any remaining cards. Pile the rocks in the center of the
- Decide which person will be the first Player. The person to his
or her right will hold the Divisibility Key and the person to his
left will be the Challenger.
- The first Player turns over his or her top card. The person
holding the Divisibility Key asks, "is it divisible by 2?" If the
Player answers, "yes," then he takes a rock from the pile. The
process is repeated with the numbers 3, 5, 6, 9, and 10, with the
Player taking a rock for each "yes" answer. (An example is a
student would receive three rocks for the number 10 because it
is divisible by 2, 5, and 10.)
- Then the person with the Divisibility Key turns to the
Challenger and asks, "do you want to challenge him?" If the
Challenger believes any answers were incorrect, he or she may
answer "yes," telling what numbers are believed to be incorrect.
- If the Challenger is correct, he gets all the rocks from the
Player. If the Challenger is incorrect, he forfeits the next turn.
- If the Player is wrong and the Challenger refuses to challenge,
the person with the Divisibility Key corrects the turn and
corrects the number of rocks taken.
- The play then moves clockwise to the left, with the past Player
now responsible for the Divisibility Key, and the Challenger
becoming the next Player.
- At the end of a round, the person with the most rocks collects
the cards used in the round and all the rocks are returned to the
center of the game. A new round is played.
- At the end of a round, if there is a tie, both Players involved in
the tie turn over their next card and collect the rocks for that
card. Whoever holds the card that earns the most rocks wins
- A player is out when he is out of cards; the Player with all the
cards at the end of the game is the winner.
- To shorten the game, the teacher may set a time limit; the
person with the most cards at the end of the allocated time is
Strategies for Diverse Learners
- Advanced learners may enjoy discovering the rule of divisibility
for 4 (last two digits are either 00 or are divisible by 4), and the
rule for 8 (last three digits are divisible by 8). Rules for higher
numbers are available on the web sites listed in the additional
- Why do the rules work? Advanced learners may enjoy
hypothesizing about the rules for 3 and 9--why are adding
digits meaningful? Explanations are given on the web sites
listed in the additional resources.
- Heterogeneous grouping for the invitation to learn and the card
game help struggling learners through cooperative processes.
- The scientific method is used to discover mathematical
absolutes. Children may recognize science vocabulary as the
rules for divisibility are discovered through the formation of
hypotheses, the gathering of data, the formation of conclusions,
etc. Explicit teaching of these vocabulary terms strengthens
both areas of science and mathematics.
- Can the rules of divisibility apply to real-life situations? Ask
the students to find at least one example after school where the
rules of divisibility shorten the task of creating equal shares. An
example: mom fries scones and makes 15 scones. She knows
they can be divided evenly among the five people in her family.
Repeat this assignment for a few days, until everyone has had a
chance to discover an example.
- Are the rules of divisibility for 3 and 9 unfamiliar enough to
mystify people? How many people can you surprise by asking
them to tell you a 10-digit number and then you telling them whether it is divisible by 3 and 9? Record their numbers and
their comments and report back to class for a discussion.
- Pre-assessment: Observe the children's hypotheses as they write
on their graphic organizers to see if their prior knowledge about
divisibility is accurate, especially with numbers 2, 5, and 10.
- Formative assessment: Check for accuracy as students write
correct rules on their graphic organizers, complete their Venn
diagrams, and verbalize their responses during the Divisibility
- Final assessment: Using the Divisibility Test graphic organizer
as a master, list ten numbers with a variety of divisibilities
and have the students complete the chart with "yes" or "no"
Furner, J.M., Yahya, N., Duffy, M.L. (2005). Teach mathematics: Strategies to reach all
students. Intervention in school and clinic, Vol. 41, No. 1, 16-23.
In 2000 the National Council of Teachers of Mathematics identified
"equity" as the first principle for school mathematics, meaning all
children have the right to understand mathematical principles. This
article offers 20 teaching strategies to reach the wide variety of learning
styles and ability levels in our classrooms as we aim to meet the equity
principle. Good lessons may incorporate several of these 20 strategies
at one time: we may draw, explain verbally, organize conceptually,
demonstrate manipulatively, and practice kinesthetically. Grouping
heterogeneously and connecting culturally helps our lessons cross
learning barriers and provide opportunities for children to help each
Ball, D., (1992). Magical Hopes: Manipulatives and the reform of math education: American
educator, Summer 1992.
Although this article is 15 years old, its concerns are still valid:
are we using manipulatives wisely when we teach mathematics to
children? What are the relative merits of different concrete objects?
Are lessons using manipulatives sensible to adults because we
already understand the concepts they are designed to represent? As
teachers it is important for us to understand the purpose behind
the manipulatives we use when we design instruction, and it is vital
for us to link the activities using manipulatives to the mathematical
concepts explicitly for children to make important connections.