Concept of average

I felt like giving away something again to help all of you who need math lessons... This one is taken from Math Mammoth Division 2 book and is a lesson on the concept of average... meant for initial teaching for 4th-5th graders who have mastered long division.

I've written it quite recently, and in fact improved it just last night by adding the bar graph problems into it. (That seems to be the "way of life" with these books: I constantly keep tweaking and improving them as I learn. Which means constant uploading of files here and there... keeping our bandwidth usage large...)

The lesson mainly uses word problems, BTW. Please post your comments below; I'm anxious to hear them!

Right click and choose save:
Average_from_Math_Mammoth_Division_2.pdf

Comments

mathmom said…
I like your lesson. But the one thing I would suggest changing is in your "backward" example you use average speed. Average speed is a topic that is full of pitfalls and deserves its own careful presentation. By casually presenting an average speed question here, among a set of "simple average" problems, you may feed the common misunderstanding that if one is given a bunch of speeds for different parts of a trip, one can find the average speed by simply averaging those numbers. For this reason, I'd stay away from "average speed" examples until you are ready to cover that topic more comprehensively.
4:07 PM, February 23, 2008
Maria Miller said…
Thanks; I changed it a bit now.
5:53 PM, February 24, 2008
Anonymous said…
I wanted to post a comment about averages but the verification symbols don't show up on my computer.

Here is a useful way to compute an average for some number lists. Let me use three of your problems.

20 32 27 37 24

Pick any number and make it your reference value. Let's take 20. Now find the differences between the other numbers and this value: +12 +7 +17 + 4. Sum these: 40
Divide by 5 => 8. Add 20 + 8 = 28

A better variant of this is to pick the median or a value close to it as the reference number, so that the differences are minimized and the running sum is easier to manage as positive and negative numbers partially cancel each other out.

So take 27. The differences here are: -7 +5 +10 –3 = 5 5 / 5 = 1 27 + 1 = 28.

If the student writes each difference above the number in the list, and includes 0 above the reference number, this makes it easier to remember to divide the sum of differences by the correct divisor, here 5.

78 87 69 86

Take 78. The differences are +9 -9 +8, and sum these = +8
Divide 8 / 4 = 2. 78 + 2 = 80.

.

234 178 250 198

Take 198. Our differences are +36 –20 +52 => 68 / 4 = 17 198 + 17 = 215


This is a nice "unconventional alternative" method for finding averages of short lists of 2- and 3-digit numbers with relatively small differences. The student doesn't need to write a column of numbers to sum, and in these examples s/he does not need to perform long division, which take more time and are prone to generating trivial arithmetic errors. At the same time, it reminds students that they don't have to "automatically" resort to a calculator for problems that would, if tackled conventionally, normally call for one.
10:10 PM, February 25, 2008
Maria Miller said…
I actually use exactly that method myself, if I find an average with mental math.

I think it is a good addition to some later lesson on average. It does require some knowledge of negative numbers though.

In this lesson, the calculations didn't even involve decimals or fractions. In reality that would be the norm.
10:13 PM, February 25, 2008
Post a Comment

Popular posts from this blog

Conversion chart for measuring units

Image
I was just sent a link to this site; all it is, is a handy one-page printable conversion chart for various US measures, metric measures, and US vs. metric measures. Includes even a comparative Fahrenheit vs. Celsius thermometer. http://metricconversioncharts.org/
Read more

Meaning of factors in multiplication: four groups of 2, or 4 taken two times?

Someone asked me about the interpretation or meaning of the two numbers in multiplication, such as 4 × 2. Do you view it as four groups of 2 , or as four, taken two times ? Here's their question: "...I noticed that your explanation about how to interpret a math phrase, such as 4 × 2, is different from how I have understood it and taught it to my older daughter who was using RightStart Math at this level. Let me explain:  I see that you are using the first factor to denote number of groups and the second factor tells how many are in each group (4 × 2 would be interpreted 4 groups of 2).  On the other hand, I have taught and thought of it reversed, with the first factor telling how many are in each group and the second factor telling you how many groups (4 × 2 would be 4 taken 2 times).  I know that the difference in explanations does not ultimately change the product, but I am wondering what you thoughts are.  When I conceptually picture what I'm doing I see i...
Read more

Geometric art project: seven-circle flower design

Image
I got inspired by the blogpost Art School | Geometric Design with Islamic Art where Deborah shows how to make a neat flower design with seven circles , using a compass, and then coloring it using 2, 3, or 4 colors (or however many of your own). I think it ties in neatly with mathematics, and lets students practice drawing circles with a compass.  My girls loved the art/math project. Here are pictures they made: Here are step-by-step instructions for the flower design: 1.  Draw a line and a circle so that the circle's center point is on the line. Then mark the points where this circle intersects the line. 2. Use those points as center points, and draw two more circles. The radius is the same all the time, so make sure you don't change it on your compass!We used 5 cm as the radius, and that made the whole design fit neatly on a regular letter size paper. 3. Now note the two points marked in the picture. They will be used as center points in the next...
Read more