sticking points in area and perimeter

I've finally found an image to explain one of the toughest concepts for kids:





Christopher has a dreadful time with figures like these, and so did my neighbor's son last year (no report on how he's faring this year).

Christopher simply can't 'see' that if all the angles are right angles, the right side equals:

8 ft + 5 ft

Nor can he see that the short horizontal line segment between the 8 & the 5 equals:

30 ft - 22 ft

I would like to have a few worksheets of figures like these.



taking a measure without starting at 0

Here's another category of problem that's incredibly hard for kids to do:




Doug, if you're around, and you feel like taking on another project, this is something I'll wager every grade schooler on the planet could use.

Christopher would be in much better shape today if he'd been given a bunch of 'simple' measurement problems in which the left side of the object being measured is placed somewhere other than 0 on the ruler.

For the life of me, I don't know why kids aren't bringing home such assignments as homework.

Some of you may remember that, last year, Christopher eked out a '4' on the TONYSS ('Test of New York State Standards,' a test schools in NY state can purchase from a private company to use in 'off' years). His score was one point above the cut-off.

The scale he flubbed was measurement!

I was shocked.

I'd been working around the clock with him (at an age when he was still willing to work with his mom) — and he flunks measurement! (Apparently, this was true of kids all over the state.)

Then we heard from teachers explaining that measurment is a difficult concept and skill to learn. Meanwhile the Singapore series takes measurement as one of its core subjects. They place huge emphasis on that topic.

Live and learn.

Now I see why measurement is a) difficult and b) incredibly valuable.

Think how much knowledge and skill goes into figuring out a problem like the line measurement above.

1
You can figure out the measure of the line either by adding or subtracting fractions.

2
The fact that you can figure it out by adding or subtracting reinforces the concept that addition and subtraction are inverse operations.

3
You can also figure out what the measurement is by counting-up using fractions instead of whole numbers. 'Counting by fractions' is an incredibly valuable activity. You almost can't not see that 'fractions are numbers' when you count by fractions. Saxon Math has numerous Mental Math tasks requiring students to count up (and, I think, back down) by fractions.

1/5, 2/5, 3/5, 4/5, 1, 1 1/5, 1 2/5, 1 3/5, 1 4/5, 2

btw, Schoolhouse Tech has a very nice sheet of fraction number lines available for download. (pdf file)







update from Doug

Doug recommends drawing simple perimeter problems on quadrille paper, like this one from Enchanted Learning (I think you have to be a member to download the sheet):








I'm going to print out the sheet and see if Christopher readily transfers from the quadrille problems to problems written on blank paper.



from last year

I just found a number of comments about kids and measurement that I'd forgotten:


measurement advice from Carl L:

My first year teaching high school freshman (I just finished my 3rd year at a urban neighborhood school) I was completely shocked that none, and I mean none, of the kids could measure using an inches ruler.

How can they get out of middle school, or even grade school, not knowing how to measure? I still have no clue. I doubt its the constructivists fault due to their fondess for hands-on, manipulatives, and project, which all lend themselves to measurement.

What I have observed:

• Metric OK, Inches Not -- While the kids can't (or won't) measure in inches, many (but not all) can measure using a centimeter ruler. Fractions rear their ugly head again.

• Estimation, Schmestimation -- The kids do not know when it is, or is not, appropriate to estimate. The kids have trouble estimating measurements between the lines of the ruler. But the kids are very willing to make bad estimates to avoid having to figure out what the little lines mean. 2 5/16 inevitably becomes 2 1/2.

• What is a protractor? -- The kids REALLY don't know how to use a protractor (except as a frisbee). Most don't even know that its purpose is to measure angles.

A side note related, I believe, to measurement. Each year I do a lesson where we compare the kids height in inches to their shoe size. The majority of the kids do not know how tall they are, let alone how to convert the height in inches.

So by all means get a ruler, protractor, some measuring cups and spoons, and a kitchen scale (or even better a pan balance) and start measuring everything around the house!

Barry's reaction:

Interesting observation and good advice. I just purchased the Saxon Math 76 book for 6th grade, and I notice that many of the problems have a scale on the page (in inches, sometimes divided into 8ths, 16ths, etc depending on the problem), with a line above it and students are asked to give the length of the line. I thought it strange to have such measurement practice but now I don't.

(Obviously that's part of our problem around here. We skipped Saxon 7/6 and went directly from 6/5 to 8/7.

from Interested Teacher:

Learning to read/measure from an 'inch' ruler has to be incremental. Younger students can't look at a ruler and automatically discern what all of those marks mean. They have to be taught to find the 'half' mark and measure using the 'half' marks. Then add the 'fourth' marks, (Don't be surprised that students don't automatically know that the 'half' mark also becomes a 'fourth' mark.) Then have students measure using the 'fouth' and half' marks. And so on, going into 'eighth' marks, etc. Practice between each incremental step.

Practice is necessary so students develop the skill of disregarding the smaller (16ths and 32nds) marks. For some students, with visual discrimination problems, this is horribly difficult.

Saxon 6/5 covers through 'fourths' and I add a little 'eighths' for more advanced students.

I was looking through Passport to Mathematics,Book 1, a text that I am previewing for personal reasons, and I see lots and lots of metric work, but little with feet and inches. On pg. 32, students measure to the nearest inch, and nothing else that I can see until pg. 318. With no review of 'half' and 'fourth' inches, it jumps to 'eights' -- there is one problem.

-- CatherineJohnson - 09 Mar 2006

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Perhaps you could get him to draw a lot of figures, with pencil and ruler?

Obviously work in inches or centimeters, not feet.

Unless you have a very large basement in your house.

-- TracyW - 09 Mar 2006

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Me too. I've been drawing simple, obvious ones on graph paper because when you finally get to worksheets covering this, the problems are much more complicated with bigger numbers. We need more practice with only missing one or two sides before we graduate on to several.

I even emailed edhelper about it, as well as having sheets with fewer problems written bigger.

-- SusanS - 09 Mar 2006

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There are some exercises on measuring from a non zero starting point in Saxon 54.

-- SamanthaRawson - 09 Mar 2006

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You might try building a figure like that out of square tiles (or drawing it on graph paper, though that might have a different effect), and counting units. Then point out that the short bit is the length of the long bit minus the medium bit. (Technical terms a trademark of, well, me.)

I'd suspect several different figures would be required to firm up the knowledge.

-- DougSundseth - 09 Mar 2006

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(That last comment was about your perimeter problem.)

For measurement, you might want to try breaking (or sawing) the end off of a wooden ruler, and asking Chris to measure with that. FWIW, I seldom measure from the end of a ruler (and especially not from the end of a tape measure), because I don't the end to be accurately at 0. (Ooh, real-world application.)

(For good reason: I've seen many rulers that have a 0 mark that isn't at the end of the ruler. And many others whose 0 mark would clearly be beyond the end of the ruler.)

-- DougSundseth - 09 Mar 2006

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I even emailed edhelper about it, as well as having sheets with fewer problems written bigger.

hey!

I DIDN'T REALIZE YOU COULD DO THAT!

I'm going to send them an email, too.

-- CatherineJohnson - 09 Mar 2006

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You might try building a figure like that out of square tiles (or drawing it on graph paper, though that might have a different effect), and counting units. Then point out that the short bit is the length of the long bit minus the medium bit. (Technical terms a trademark of, well, me.)

They do have these on edhelper, and they're easy for me to make....I bet you're right; I bet he'd transfer his knowledge that way!

-- CatherineJohnson - 09 Mar 2006

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Samantha

Thanks!

Unfortunately, I don't own 5/4....and those books are kind of expensive....

Of course, now I'm tempted.

-- CatherineJohnson - 09 Mar 2006

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I seldom measure from the end of a ruler

HEY!

ME, TOO!

FOR THE SAME REASON!

maybe i'll be a math brain yet

-- CatherineJohnson - 09 Mar 2006

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I was wrong; the perimeter sheets I saw are at Enchanted Learning (I think you have to be a member to download the sheet):








I'm going to print out the sheet and see if Christopher readily transfers from the quadrille problems to problems written on blank paper.

-- CatherineJohnson - 09 Mar 2006

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whoa.....Christopher SERIOUSLY can't do these problems!

-- CatherineJohnson - 09 Mar 2006

WebLogForm
Title:	sticking points in area and perimeter
TopicType:	WebLog
SubjectArea:	ElementaryMath, HomeSchooling, MiddleSchoolMath, ParentsTeachingKids, TeachersTeachingKids
LogDate:	200603091435

Attachment	Action	Size	Date	Who	Comment
PerimeterandAreaLshape1.pdf	manage	17.6 K	09 Mar 2006 - 22:32	SmartestTractor
PerimeterandAreaLshape2.pdf	manage	15.6 K	09 Mar 2006 - 22:33	SmartestTractor
PerimeterandAreaLshape3.pdf	manage	15.6 K	09 Mar 2006 - 22:33	SmartestTractor
PerimeterandAreaLshape4.pdf	manage	15.6 K	09 Mar 2006 - 22:33	SmartestTractor