## Modeling (24÷3)÷2 = 24÷6

Modeling (24÷3)÷2 = 24÷6 in a way that makes sense to a fifth grader is tricky.  Using the definition of division is too pedantic.  A rectangular array of equal groups (dividing 24 into 3 equal rows, then subdividing the rows in 2), while visually appealing, does not accurately model (24÷3)÷2.  Such models generate more questions than they answer.  The video below is a possible solution to this modeling problem using a bar diagram model. Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.
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### 5 Responses to Modeling (24÷3)÷2 = 24÷6

1. Robin Ramos says:

I think it works beautifully. Very elegant.

When returning to the original bar, one way to see both the “larger” and “smaller” units (and I appreciated your use of your fingers to show the different measurements) is to use a dotted line to subdivide. It is one way to go. I can see the purpose in doing it both ways, depending on what your emphasis is with your students.

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2. Alistair Windsor says:

I also use word context problems.
There are 3 groups each of 2 students. A teacher has 24 counters to divide equally amongst the students. There are two ways he could do this.

1) He could divide his counters equally between the groups of students. Then a student in each group could divide the counters equally between the students in the group.

Each groups receives 24÷3 counters since there are 3 groups. Each student receives (24÷3)÷2 counters since there are 2 students in each group.

2) He could give each student the same number of counters.

There are 3 groups each of 2 students so there are 3 x 2 students. Each student receives 24 ÷ (3 x 2) counters.

Both ways lead to the counters being equally distributed between the students and so their answers must be the same (24 ÷ 3) ÷ 2 =24 ÷ (3 x 2).

This is possible to do in a classroom (without a prime number of students 😉 ) and matches experiences in distributing materials.

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3. howardat58 says:

I think it is high time that the division sign was retired, permanently. See my first post on my blog:
https://howardat58.wordpress.com/2014/05/07/the-first-victim-in-the-battle/
Viewing this problem as (24/3)/2 written out normally shows that it is a fraction calculation, as 1.2 of (1/3 of 24), and a quick evaluation shows this as 1/6 of 24.

You might like this as well:
https://howardat58.wordpress.com/2015/02/03/commutative-distributive-illustrative-ly/

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• Scott Baldridge says:

I’m curious what other people reading this post think. Chime in!

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• howardat58 says:

typo ” as 1.2 of ” is ” as 1/2 of “

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