Ideas drawn from Think Math!http://thinkmath.edc.org/Learn more
at Think Math! Information Exchange
****And you will see some of the ways*And you will see some of
the waysChildren have plenty of experience with things that grow
gradually: the height of water in the bathtub, their own height,
the temperature of the bathwater as warm water enters. Those
measurements do not step from 1 to 2 to 3, but move through all the
values in between as well.
Counting starts when there is one object, and it matches the
objects with whole numbers: seven animals, never seven and a
half.***This is the moving right and left image. Here, numbers
represent addresses (places) on the number line (where the child
starts and stops) and also distances (how far the child moves, and
which direction, for each instruction).
The addresses are also a kind of how far, in that they all say
how far the child is from zero.Why start at 0 instead of 1?Why
number from the bottom to the top?
On a conventional number chart that starts with 1, the sixties
are not all in one row, nor are the 20s or 30s. Each decade is
split, with the number that starts that decade at the end of one
row, and the rest of the decade in the next row down. In a
conventional number chart, 43 is not higher than 33; it is directly
*below* 33, lower.
That doesnt necessarily confuse children, but it does make
language more ambiguous. The question which is lower, 84 or 74 can
be interpreted two ways. Yes, it is more precise to ask which is
less, but we even mathematicians often use spatial imagery to talk
about numbers. Low and high prices. Low and high temperatures. Low
and high yield. And because that language is natural, and used, it
is helpful to support it with the image.
Moreover, graphing uses a bottom-to-top increase as well as a
left-to-right increase, so this image is consistent with general
mathematical usage.*Why start at 0 instead of 1?Why number from the
bottom to the top?
On a conventional number chart that starts with 1, the sixties
are not all in one row, nor are the 20s or 30s. Each decade is
split, with the number that starts that decade at the end of one
row, and the rest of the decade in the next row down. In a
conventional number chart, 43 is not higher than 33; it is directly
*below* 33, lower.
That doesnt necessarily confuse children, but it does make
language more ambiguous. The question which is lower, 84 or 74 can
be interpreted two ways. Yes, it is more precise to ask which is
less, but we even mathematicians often use spatial imagery to talk
about numbers. Low and high prices. Low and high temperatures. Low
and high yield. And because that language is natural, and used, it
is helpful to support it with the image.
Moreover, graphing uses a bottom-to-top increase as well as a
left-to-right increase, so this image is consistent with general
mathematical usage.*Why start at 0 instead of 1?Why number from the
bottom to the top?
On a conventional number chart that starts with 1, the sixties
are not all in one row, nor are the 20s or 30s. Each decade is
split, with the number that starts that decade at the end of one
row, and the rest of the decade in the next row down. In a
conventional number chart, 43 is not higher than 33; it is directly
*below* 33, lower.
That doesnt necessarily confuse children, but it does make
language more ambiguous. The question which is lower, 84 or 74 can
be interpreted two ways. Yes, it is more precise to ask which is
less, but we even mathematicians often use spatial imagery to talk
about numbers. Low and high prices. Low and high temperatures. Low
and high yield. And because that language is natural, and used, it
is helpful to support it with the image.
Moreover, graphing uses a bottom-to-top increase as well as a
left-to-right increase, so this image is consistent with general
mathematical usage.*Why start at 0 instead of 1?Why number from the
bottom to the top?
On a conventional number chart that starts with 1, the sixties
are not all in one row, nor are the 20s or 30s. Each decade is
split, with the number that starts that decade at the end of one
row, and the rest of the decade in the next row down. In a
conventional number chart, 43 is not higher than 33; it is directly
*below* 33, lower.
That doesnt necessarily confuse children, but it does make
language more ambiguous. The question which is lower, 84 or 74 can
be interpreted two ways. Yes, it is more precise to ask which is
less, but we even mathematicians often use spatial imagery to talk
about numbers. Low and high prices. Low and high temperatures. Low
and high yield. And because that language is natural, and used, it
is helpful to support it with the image.
Moreover, graphing uses a bottom-to-top increase as well as a
left-to-right increase, so this image is consistent with general
mathematical usage.***And subtraction is the operation that answers
how far one must move (and in which direction) to get from one
number to another.
The question How much is 76 47? is identical with how far is 76
from 47 on a number line?
The number line image is useful for mental subtraction. Put a
couple of useful rest stops along the way, at the nearest tens to
each end, then add up the distances.
43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step.
The total is 28 steps. This is, in essence, the shopkeepers way of
making change. Start at the cost and count up in convenient units.
Theres nothing wrong with borrowing, but its hard to do in ones
head. This method tends to be lots easier.
*Name a two digit number (e.g., 43) and ask what multiples of 10
are nearest to it. What tens is it between?
This question asks about place value and rounding/approximation,
but the language is spatial! BETWEEN suggests position The number
line again!
*In this context, it is natural to ask how far the number is
from those nearest multiples of 10.
The pair of distances is, of course, a pair that makes 10, which
theyve been studying to supercompetence since they started Think
Math!
*This is the distance image of subtraction.
See FractionsStoryline.ppt and DecimalsStoryline.ppt for
examples of subtraction of fractions and decimals on the number
line.
Notice that this method does not involve borrowing or even
notions of regrouping. It uses notions of rounding and distance
from the nearest 10, and the shopkeepers notion of adding up from
the smaller number to the larger.
37 + ___ = 62 Kids are *not* troubled by missing addends!
*Kids *not* troubled by missing addends!****To see fractional
numbers between the whole numbers, we could zoom in on the number
line with a magnifying glass. Depending on the strength of the
magnifying glass, we might see one new number () or two new numbers
( and ) or four new numbers (, , , ), and so on. The magnifying
glass that doubles what we can see, shows us the halves.
**And subtraction is the operation that answers how far one must
move (and in which direction) to get from one number to
another.
The question How much is 76 47? is identical with how far is 76
from 47 on a number line?
The number line image is useful for mental subtraction. Put a
couple of useful rest stops along the way, at the nearest tens to
each end, then add up the distances.
43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step.
The total is 28 steps. This is, in essence, the shopkeepers way of
making change. Start at the cost and count up in convenient units.
Theres nothing wrong with borrowing, but its hard to do in ones
head. This method tends to be lots easier.
*And it generalizes perfectly to decimals, and explains why
decimal algorithms for addition and subtraction are identical to
the algorithms for whole numbers.
Do the same thing with 4.3 instead of 43 and ask which ones
(instead of which tens) it is between.*How far from the nearest 1s.
Notice how familiar these answers are. The whole world of decimals
begins to feel strangely familiar.*And subtraction is the operation
that answers how far one must move (and in which direction) to get
from one number to another.
The question How much is 76 47? is identical with how far is 76
from 47 on a number line?
The number line image is useful for mental subtraction. Put a
couple of useful rest stops along the way, at the nearest tens to
each end, then add up the distances.
43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step.
The total is 28 steps. This is, in essence, the shopkeepers way of
making change. Start at the cost and count up in convenient units.
Theres nothing wrong with borrowing, but its hard to do in ones
head. This method tends to be lots easier.
*And subtraction is the operation that answers how far one must
move (and in which direction) to get from one number to
another.
The question How much is 76 47? is identical with how far is 76
from 47 on a number line?
The number line image is useful for mental subtraction. Put a
couple of useful rest stops along the way, at the nearest tens to
each end, then add up the distances.
43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step.
The total is 28 steps. This is, in essence, the shopkeepers way of
making change. Start at the cost and count up in convenient units.
Theres nothing wrong with borrowing, but its hard to do in ones
head. This method tends to be lots easier.
*And subtraction is the operation that answers how far one must
move (and in which direction) to get from one number to
another.
The question How much is 76 47? is identical with how far is 76
from 47 on a number line?
The number line image is useful for mental subtraction. Put a
couple of useful rest stops along the way, at the nearest tens to
each end, then add up the distances.
43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step.
The total is 28 steps. This is, in essence, the shopkeepers way of
making change. Start at the cost and count up in convenient units.
Theres nothing wrong with borrowing, but its hard to do in ones
head. This method tends to be lots easier.
*And subtraction is the operation that answers how far one must
move (and in which direction) to get from one number to
another.
The question How much is 76 47? is identical with how far is 76
from 47 on a number line?
The number line image is useful for mental subtraction. Put a
couple of useful rest stops along the way, at the nearest tens to
each end, then add up the distances.
43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step.
The total is 28 steps. This is, in essence, the shopkeepers way of
making change. Start at the cost and count up in convenient units.
Theres nothing wrong with borrowing, but its hard to do in ones
head. This method tends to be lots easier.
****