Transcript Arithmetic Rack

Structuring Numbers
using the Arithmetic Rack

The Arithmetic Rack allows for a variety of
solution strategies. Students are able to
make use of the number relations they can
think of. These activities induce a shift in
children’s thinking from visual, or tacitly
countable items to an increasingly numerical
interpretation.
│----------------------- ОООООООООО│
│----------------------- ОООООООООО│
The beads that are shifted to the left are the
ones that count. The beads shifted to the
right don’t.
Students may think of setting up a given
number in different ways. Here are some
possibilities for the number 7.
1.
│ООО-----------------------ООООООО│
│ОООО-----------------------ОООООО│
Students may think of 7 as 3 + 4, or as
6 + 1 = 3 + 3 + 1.
2.
│ООООО-----------------------ООООО│
│ОО-----------------------ОООООООО│
Students may think of 7 as 5 + 2.
3.
│ООООООО-----------------------ООО│
│-----------------------ОООООООООО│
Students may think of 7 as 7 ones, 5 + 2,
or as 10 – 3.
4.
│ОООООО------------------ОООО│
│О------------------ООООООООО│
This does show the number 7; however, the
previous ways are encouraged over this
way because it does not reference 5’s or
doubles.
Determining the total number of beads
│ООООООО-----------------------ООО│
│ОООООООО-----------------------ОО│
This may be seen as 7 + 8, however the
students have to figure out how much that
is. The same configuration may be thought
of as 5 + 2 and 5 + 3, which can be totaled
as 5 + 5 = 10; 2 + 3 = 5; 10 + 5 = 15. Or, it
can be seen as (7 + 7) + 1 = 15.
To help students show one quantity:


Show a number (any way) without a
discussion of efficiency.
Show a number with a discussion of
efficiency.


Does the showing make it easy or hard to ‘read’
the number of beads?
Without counting, can you recognize a quantity?
Anticipate a quantity?
The Arithmetic Rack can be used as a way
to keep track of the number of people on
the upper and lower decks of a doubledecker bus.
Six people are sitting on a bus. Three are on
top and 3 are on the bottom. At the next stop,
9 more passengers get on. How many people
are now riding on the bus?
│ООО-----------------------ООООООО│
│ООО-----------------------ООООООО│
Nine could be added as 9 = 5 + 4.
│ООО-ООООО-----------------------ОО│
│ООО-ОООО-----------------------ООО│
10 + 5 = 15
Nine could be added as 7 + 2 (filling up the
upper 10 first).
│ООО-ООООООО-----------------------│
│ООО-ОО-----------------------ООООО│
10 + 5 = 15
Nine could be added as 8 + 1, with 8 = 4 + 4.
│ООО-ОООО-О----------------------ОО│
│ООО-ОООО-----------------------ООО│
Staggered Tasks

The teacher gives the first number, and the
students make it on their AR. Then the
teacher gives the second number (using the
story of the bus). Students complete the
problem on their AR’s. Discuss the results.
Anticipating the second step.

The teacher shows a first number. Students
make the same number in the same way on
their AR’s. The teacher gives the second part
of the problem but BEFORE students
complete it on their AR’s and discuss how
they plan to do so.
Reasoning without moving the beads.

The teacher shows a number and asks
questions such as how many people need to
get on (or off) to get another number of
people on the bus.
Imagination Bingo

The teacher gives the first number that
students make on their AR’s. Then they
IMAGINE completing the problem and show
the result on their Bingo cards.
Anticipating Efficient Ways

The teacher poses the entire problem (7
people are on the bus and 8 more get on).
BEFORE students solve the problem on their
AR’s they discuss how they plan to do it and
why they have the plan that they do.
Number Sentences without the Rack

The teacher presents written number
sentences which students are to solve
without using their AR’s. We want the
students to use rack-related strategies. They
may talk in terms of moving the beads on the
rack.