Are you as smart as a 3rd grader? As was mentioned in the previous post, many of us did not do well in high school algebra class. One reason I suggested in that post was because we were taught rules and shortcuts rather than reasoning. Further, algebra was abstract for us, 2D at best. Today’s children attending up-to-date schools begin learning equalities and number sense through the use of math manipulatives.

One of the most important manipulative is the balance shown below.

Students attach blue rectangles to both sides so that the yellow stick ‘balances.’ Even kindergarten students can solve an equation like 3x = 9. Note that earlier, the students explored ways to balance 9 using combinations of two blues on the other side. Today they begin by placing one blue rectangle on 9. Then the teacher tells them to pick up 3 more blues and put all three on the other side on the SAME peg. Of course, they aren’t given 3x = 9 as the problem to solve, but they solve it nonetheless.

The state of Ohio has adopted the* common core state standards* for teaching as have 45 of 50 states. These standards challenge administrators and teachers to focus on core conceptual understandings and procedures rather than on rote learning. Fortunately because of retirements, newer public school teachers are moving away from rote paper and pencil mathematics and have been trained in both the new standards as well as manipulative mathematics.

Here is a problem for us to solve: Mike has some blocks then finds 5 more. Now he has 12. Write an algebraic expression for this problem.

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Did you remember your Algebra 1? Which of these did you write?

a. x + 5 = 12

b. x + 12 = 5

c. x – 5 = 12

Children in the 3rd grade in Ohio had this question on their ‘proficiency’ test and are expected to solve it.

If these 3rd graders had been using the balance for the past 4 years, then they could visually imagine the 12 on one side and one blue at 5 and another [unknown] blue on that same side. Was it that easy for us oldies?

How about another challenge. 18 + y < 22 y = ___

Here it is in 3rd grade representation:

Of course the 3rd grader is expected to give only one correct number, but his/her work with the balance helps to easily identify the 3 as the correct response.

Then there is this problem: x + 5 = y. Make an x, y table of 3 values of each unknown so that the statement is true.

The Ohio 3rd grade problem:

One can again suspect that the use of the balance may help speed up the delineation of the correct set of answers.

Beyond the physical balance, the student can use the virtual balance on the computer to solve equality problems. One such program allows students to choose increasingly more complex problems for personal enrichment such as the example below.

I’ll let you find x. To use this virtual program, the student drags four x’s to the left and one x onto the right. Then he/she places one 1 atop the 4 left x’s and seven 1’s atop the right x.

Do you remember your Algebra 1 teacher telling you to ‘*cross out* equal amounts on the left and right side of the equal sign?’ In fact, at this stop in the virtual problem, the student *puts back* equal cubes and blocks on the right and left. The next frame shows the result.

At this point the more mathematically astute student may jump to the answer by noting a 1:2 relationship of x to 1’s. If not the student removes one x and notes the imbalance; it re-balances only when two 1’s are removed.

Enough algebra for today. If nothing else, at least some old neurons were fired up while reading this post.

Did I mention how much I love math? I came up with a different answer to your first example:

” Mike has some blocks then finds 5 more. Now he has 12″

These are the steps I used in my head.

1. The number must be a whole number because the blocks are whole (aka no fractions or negative numbers).

2. I picture Mike’s original blocks all the same color, let’s say blue.

3. Then I picture the new blocks (yellow) added to the “original” line-up of blocks. This would be the “new” line-up made up of blue blocks to the left and yellow blocks on the right.

4. Now I take away each blue block one at a time until all the blue blocks are gone.

5. I count the remaining blocks, all yellow, to come up with the answer.

Sounds complicated, but if you’re working in pictures it just happens all at once. 12 – 5 = x

Oh, the reason is that all the formulas in the first example required more steps to solve.

You math sleuth! Good answer. In fact, although not on the state test, students are often asked to defend their thinking process.

And now a tribute to the N.R.A.:

Michele Bachmann is being investigated:

http://news.yahoo.com/michele-bachmanns-presidential-campaign-investigated-ethics-watchdog-203206309.html

Remember that Latta was heavily involved with her Presidential campaign. Golly gee, I hope Bob didn’t do anything bad. His father’s reputation was marred after the senior Latta obtained a large federal grant for someone, then immediately retired from Congress and went to work for that person as a lobbyist (with high pay & benefits). And because his father was a Congressman since 1958 he could “grandfather” (keep) his campaign funds, instead of being required to surrender the money after he retired as mandated by a new law.