We need Group Names for +,–,x,÷
A couple of patterns?run through elementary math that we are not fully leveraging. If we give the arithmetic pairs group names early on, we will have unifying concepts and catchwords that span elementary math education.
1. The answer/step-towards-the-answer...time and again..involves doing The Opposite
2. Couples need the same Name before they unite
We need to use natural language to teach concepts until the student becomes the teacher. ?Then, refine these ‘layman’ terms with more technical terms. ?A parrot can recite words. ?The main goal is to teach concepts that transfer.
The summary below reviews most of the basic concepts of elementary math. ?It introduces a couple of age-appropriate group names. ?We need group names for the basic math operations early on to connect and integrate these topics:
Equation simplification (matching variables)
Why wait until fifth or sixth grades and use, ‘multiplicative operations’ and ‘additive operations’? ?The Egyptians were wrong. ?These group names are lengthy, confusing, redundant..and empty. ?Group names should be concise and memorable. ?They need cognitive hooks to prior knowledge, and they need to aid in analogical reasoning. ?We need the first group name the first time the inverse (The Opposite) relationship becomes a formal strategy for solving problems.
Group names are more efficient. ?They facilitate decision-making by reducing the number of options. Group names break down problems into smaller parts. They also streamline communications because we can address similar things simultaneously. ??Remembering two group names and their elements is easier than four individual operations.
These groups are pairs
+ connects to ?–
x connects to ÷ ?
Pairs because they Reverse one another. Pairs because they are Opposites. ?If the message is they are connected because they are Opposites, math educators can ask the same questions over and over - for years - to help guide students to the answer. Or just point to the poster -->
What is its pair?
? Why are they paired?
Catchphrases that can be used to answer questions on the exact same eight subjects listed above. Connecting operational pairs with group names integrates elementary math.
Singles/Repeaters could be a conceptual stepping stone for the pair names..or we could start with something more lasting..
Couplers ??+ ??–
Sizers ?? ?????x ??÷
Couplers Combine two digits.
Sizers do not combine. They change the Size of the original Base value.
Couples need matching names before they unite. That is why we line up Place Value positions. That is why fraction names (de-name-inators) need to match.
Sizers do not worry about matching names because they do not combine?with the Base. They simply MAKE COPIES?of it – or?– they SPLIT it. ??Sizers change the..size.
The Base value could be 12 (a value on a number line), 12 inches, or 12 pounds. Multipliers 'make copies' of the 12 inches, the 12 lbs, 12 goats...whatever you want to copy. Multipliers are Copy Machines?that copy more than just paper. They make things bigger by making copies & adding them up. Dividers slice & dice. Whatever you start with gets smaller.
So..it all depends on what you want to accomplish or what the problem asks: make something bigger or smaller or..keep it the same.?(0 and 1 misbehave as usual; Unit Conversion issue addressed later)
Couplers & Sizers?address the fundamental differences?between the operational pairs.
Couplers unite TWO digits. Just two.
Couplers need the same Name
- Name as in Place Value name
- Name as in fraction name (the de-name-inator)
?Sizers are carefree about the Place Value names issue. A single Sizer can be ‘distributed’ among multiple digits (even billions of digits). Here is an example of a Sizer (2), that names 'Ones' ....that interacts with BOTH the 'Ones' and the 'Tens'. Couplers don't do that. With 14 +?2, Couple the 4 +?2. With 14 x?2..the operation, x, is ‘set’ for 2 copies....of BOTH digits. ???
14 is composed of a 10 and a 4.
Two?copies of each, plz, then add ‘em up
(2?x 10) + (2?x 4)
note: number writing section at end
The Names issue comes up again when adding fractions. The top digits of the fractions (the numerators) are digits to add (just like always)..but you can not add them UNTIL they have the same de-name-inators.
The Names issue comes up again with decimals. The first instinct is to right-align the two values to be added (unmindful of decimal points/place values), but..you can not Couple two digits with different Names.
The Names issue comes up again with Unit Conversions. Names are a theme that runs through elementary math, and we need to leverage this tool. One can ask the same question for years: Do the digits have the same name? ?(You only need to know three questions to teach elementary math;)
Sizers change the SIZE of the Base/original Value. Multipliers always increase the size of the Base. Dividers always make the Base smaller. Suggesting that multiplying by a fraction is multiplication distorts the basic meaning of what it is to multiply. The basic math operations have a consistent meaning if we focus on the forest.?It is division, and it is represented it with a multiplication sign and referred to it as ‘multiplication’.
Focusing on smaller parts distorts the overall meaning and leads to mislabeling. Accurate, logical nomenclature gives consistent meanings to multiplication and division. When something is divided, it gets smaller, right? We need to be able to count on that conceptually..and the reverse.
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Multiplying by a fraction or decimal is dividing. Multiplication by a fraction is two steps: multiply by the top number and divide by the bottom one. The denominator is always larger. It has a more significant effect. If one gave a descriptive name to this process, one would call it..Division. Decimals - same principle as fractions. The decimal’s 'denominator' is conveyed by its Place Value and its ‘denominator’ is always larger.
Multiply = make copies of the Base/original value and add them up. At first, one at a time..then build the answer with partial totals, and ultimately, a memorized-total in one step.
Example: when learning the 7s, for 7 x 7, throw seven 7s on the table and straighten them. “Group/add-up the digits however you like. You know your fives, right?” (circle or take-away five of the 7s) “OK, we are at 35, how are we going to add the rest?” (one 7 at a time or a double-7 are the choices) This was an example of building the answer - a more important skill than simply memorizing 7 x 7. One could build that same answer with double-7s until there was only one 7 left.
Note: ?Digits 1, 9, 10, and 11 require neither memorization nor practice building answers/scaling. They leverage the scaling skills used to Size answers for digits 2 - 8. (It's witchcraft.)
Divide = separate the Base/original value into parts. At first, the Base value is the number of ‘cards in your hand’, and the divider is the number of ‘players’. Later, with larger Base values, it’s multiply and subtract, multiply and subtract..until there is no (or little) remainder.
Dealing cards to players is distribution. It is dividing cards among players. When there are too many cards to deal it's time to REVERSE thinking. Do the Opposite. The Opposite of division is..multiplication.
Division changes from, “one for you, one for me, one for joe” until the cards are gone to....multiplication. MULTIPLY-to-divide. Sounds crazy so say it again.
Multiply to divide.??Reverse division just like you reverse subtraction. ?Except..with subtraction, the decision to reverse is based on distance apart on a number line. ?With division you pretty much reverse it all the time.
ADD-to-subtract?and MULTIPLY-to-divide?have the EXACT SAME steps. ?Just do the COMPLETE opposite.
Do EVERYTHING the Opposite
that's everything
You can’t just Add-to-subtract. 8-5 would become 8+5. That's 13. Off by 10. The full term is, ‘add-to-subtract-AFTER-switching-the-starting-point’?
Simpler to understand with beans. Take two piles of beans—one with 5, one with 8. Point to the group of 5, “How can we make these equal if we start with this one?” Then reverse the 'equation', point to the group of 8 beans, “What if we start here instead?”
Both bean calculations yield the same digit. The difference. Changing the starting pile mirrors changing the starting digit of the equation. ?
To illustrate how The Opposites connect,?for 8?– 5, draw a curved arrow from the bottom of the 5 back to the 8 (no other symbols or digits). Label the line, +. ?That is how to reverse –
Same diagram for 8 ÷ 2 so illustrate side by side.
If everyone knows The Opposites, no need to label the arrows. Need a hint? Point to the 5 on a number line and ask, “How do we get to the 8?”
To understand why the Sizers are opposites, stop thinking about how to divide or distribute the cards. Forget about the cards. Instead, think about how to FILL a space with blocks, or COVER a canvas with stamps, or..fill a box with post-its.
To see (in 3D!) how multiplication & division are connected..
1. Place four small post-its together (forming a rectangular box).
2. Outline the box perimeter. Write 2 on each post-it, remove them, and write 8 in the box. (foreshadowing)
3. Separately, write down and discuss, 8 ÷ 2 = ?, and how one learns to answer that question using count-bys ('2, 4, 6, 8…there are four 2s in 8'). Then, discuss how count-bys are multi-addition, and multi-adds are (slow) multiplication because you are adding the copies ONE AT A TIME. We progress from adding the copies one by one, to adding the copies in groups, to adding them all at once. ?
4. Back to the Box & Post-its --> fill/cover the box with 2s and take turns explaining to one another what it means to ‘fill’ the box, adding the 2s one at a time. Hopefully, connecting Count-bys to (slow) multiplication. Then, reverse the process. As you remove the post-its, take turns explaining how removing a piece is subtraction (a take-away). ?Taking away Multiple pieces is Multi-subtraction...which is Division...IF you take the pieces away ONE AT A TIME. ?(far too slow)
The above still does not show why we MULTIPLY to divide. One can easily distribute something small among few. Large numbers are 'filled' not divided. ?
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Note on learning to write/assemble numbers
14 = 10 + 4, right? ?Where did the '0' come from? ?It is always there! ?Believe it or not, it is best to think of the 0s as always being beside the REAL digits they place into position. ?That 0s are 'spacers' for the original..the REAL digits (1 to 9).
The zero is like a space bar on a keyboard. ?The analogy is not a stretch since the zero evolved long after the 'real' digits. ?The zero started out as a blank space - like the spaces between these words. ?Except, in math, the spaces are also ‘placeholders’ for other digits. ?If there are no digits to represent in that PV position, the 0 ‘shows through’.
If you can count to 10 you can write numbers up to 999 in short order. ?Write 1 - 10 in a column and repeat the numbers together. Then, put a 0 after each digit using a different color. ?This new set of numbers 'rhymes'. ?Repeat together until..the student becomes the teacher. Next, add another 0, keeping the 0s the same color, and..more mimicry.
Time to pick a number and build it. ?Use toy digits if possible (3D!). Say the first digit, then wait until the number is assembled before saying the next number. Build each digit WITH its respective 0s (one color for each digit and its 0s). ?For 538, say,
? ??"Five hundred" (build 500)...
?? ?"thirty" (build the 30 OVER ?the two 0's that BELONG to the 5)...
??? "eight" (8 is placed over what is now two 0s).
Disassemble the 538 to show the 500, 30, and 8 separately. ?Repeat the cycle with 538, then ?build some other numbers. This exercise addresses number writing and introduces the concept that numbers are built with components. Legos.
Parrots can recite numbers. ?What do the digits mean? ?Assemble a 'flat' 538 (no 0s under the digits..but they are still there, right?). ?Point to the 3 and discuss the name of this Place Value position, how it can be represented/modeled, and how it relates to the adjacent digits.
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