## Monday, January 21, 2008

### 24x9 is one hundred more than 29x4. Why? Now generalize... An Investigation for Gr 4-9

Continuing with our digit and number relationships, the question in the title is meant to provoke the reader/student to probe more deeply and try to understand the reasons behind number observations. Asking students to explain these kinds of results leads to fruitful dialog.

Target Audience:
Upper elementary, middle school students through Algebra 1 (Grades 4-9)

Objectives:
(1) Develops understanding of and reinforces the distributive property (in both directions!) numerically and in algebraic form
(2) Develops understanding of and reinforces place value
(3) Develops meaning for multiplication, including multidigit operations
(4) Reinforces the meaning of multiplication as repeated addition
(5) Develops pattern recognition and generalization
(6) Motivates algebraic representation of number
(7) Practice with open-ended investigations

Correlation to Standards

Refer to the K-12 Benchmarks for Mathematics at the Achieve website.
Note: There are many links there. I will attempt to correlate to more specific expectations in the future.

Today we are going to investigate the effect of interchanging or switching the units' digits when multiplying a 2-digit number by a 1-digit number. There are many ways of thinking about this but our main focus will be on using the distributive property.

We will start by considering which is larger:
24x9 or 29x4?

So we've concluded that 24x9 is greater than 29x4. Makes sense doesn't it because 24x9 is the same as adding a set of NINE 24's and 29x4 is the same as adding only FOUR 29's. Obtaining the actual products, we see that
24x9 = 216
29x4 = 116
Thus 24x9 - 29x4 = 100
Is it possible to determine the difference of 100 without actually doing each separate product?
Today, you and your research team will devise such a method and explain why it works!!

Specific Instructions:
Your research project for today is to devise a method for finding the difference between two products in which we interchange or switch the units' digits. Make a table using at least 10 additional product pairs in addition to the ones we've given you as models. Make sure you include many different digits in the tens' places as well as the ones.

Product 1.......Result...........Product 2............Result............Difference
24x9...............216.................29x4....................116..................100
43x8..............344.................48x3....................144..................200

........

Conclusions:
To find the difference between PQ x R and PR x Q, do the following:

Explanation for Method:

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Aside to the Instructor:
The following is a guide for this research.

Here is one approach to the question that uses the distributive property in both directions. The algebraic form is given in italics to the right of each step. For each line, think about which version of the distributive property we are using. [Answers given in brackets to the right].
24x9 = 24x4 + 24x5 [a(b+c) = ab+ac]
29x4 = 24x4 + 5x4 [Same]
Gain: 24x5
Loss: 5x4
Net Gain: 24x5 - 4x5 = 20x5 = 100 [ba-ca = (b-c)a]

Mathfr3d said...

The value of PQxR is (P*10 + Q) * R = R*P*10 + Q*R
The value of PRxQ is (P*10 + R) * Q = Q*P*10 + R*Q

Since difference of QP and PQ are the same
We only have to look at the difference of
R*P*10 and Q*P*10 which is:

R*P*10-Q*P*10 =
10*(R*P-Q*P)

in the above example we get for
P=2,Q=4,R=9 the following:

10*(9*2 - 4*2) =
10*(18-8) = 100

Dave Marain said...

Nice analysis...
Another way to express the difference is 10P(R-Q).
10P is the 2-digit number whose tens' digit is P and whose units' digit is zero.
R-Q is the difference of the units' digits.
Thus the difference between 24x9 and 29x4 is 20(9-4) = 100.

Ok, now let's extend the investigation to 3-digit numbers; 232x5 vs. 235x2, for example.
The difference is 690. Can you extend the previous method?