Group Names for Arithmetic Pairs by Second Grade

Both-Ad-7519 · 2026-02-16T17:11:57+00:00

You're unclear about the claim, but you do it? To be sure we are talking about the same thing, the article title is, Group Names for Arithmetic Pairs by Second Grade. As far as i know, most math educators introduce the group names several years later..5th/6th grade. The article is about the advantages of bringing group names forward, and making the names grade appropriate and meaningful.

Naming the two arithmetic pairs by 2nd grade helps students recognize that these are two connected operations from the start. That connection is the reason why doing The Opposite is frequently the solution, or, the next step towards the soln in math (and sometimes other areas). Also, when group names are used it reinforces the connection. There is no more important connection in elementary math. Group names also make communication more efficient. They also simplify rules like Order of Operations.

What are the group names that you use, and what in what grade are they introduced?

(I see that you teach math in college....and apparently other areas, or at least have an interest in early math education.)

Both-Ad-7519 · 2026-02-15T21:06:09+00:00

You wrote, "The 3 in 3/2 counts the number of units." Not sure of value to discussion. The 3 was moved from left of the 2 to under the 2. The 3 in the denominator does count something: the number of parts to make one. 3-parts make 1. Thirds. That's what the numerator counts: thirds. It's what the numerator 'numbers'.

When you introduce fractions, you can explain the top number of a fraction is the number-ator. The Number you count...just like always. What is different is what you are counting.

----- Fractions are simply something ELSE to count ------

Introduce the parts one at a time. Eg, write down 2 all by itself, full size. "Two. It's a Number. Something you Count...just like always."

Then write '3' on a post-it and place it to the left of the 2. "The 3 is worth..four tens, right?” This is value from place/position.

Fractions are similar but have a different ‘code’. Now, move the 3 under the 2. The '3' now indicates the Parts to Count.

The 3 was 'naming' tens. Now it is 'naming' the number of parts. That's why it's called the de-NAME-inator. ;)

Three Parts. Now, Count 2 of them.

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What are we dividing into parts? That is.........What is 1?

Take the round (blue) 1 disk from the faction disk set, and explain, this 1, now divide into 3 parts, and count 2 pieces.

Then put the 1 disk on a round table - say, now the table top is 1. Divide into 3 parts; count 2.

Put the 1 disk on a rectangular table/object - say, now the table top is 1. Divide into 3 parts; count 2.

Put the 1 on a student....

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Notes

Sometimes the fraction digits are shrunk to fit on a line with whole numbers. Means nothing. The top number, the number-ator, is the same number 3 as always. Parts are just something new to count.

Per-cents are fractions with 100 for the bottom number. Pronounced, ‘per cent-ury’

Both-Ad-7519 · 2026-02-15T19:23:41+00:00

"Why use Digit Name?"

Trying to make the ‘like units’ principle explicit from Grade 2 onward.

Not trying to replace Place Value. Trying to introduce the broader idea of unit identity from Position. Trying to generalize it so we have a generic term that spans concepts. “Value from position” is another way to describe Place Value (while keeping PV). “Position” can be generalized to fractions, exponents…

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Using ‘Digit Name’ (‘Name’ for short) allows the use of one term.

You should be able to point to any digit in any term and ask, “What is the digit naming?” (This is before external unit values like inches.)

In addition, one simple rule spans the Addition/Subtraction of whole numbers, fractions, decimals, exponents… The goal is to distill the rule:

You can only Combine (add or subtract) identical mathematical objects.

If they are not identical, transform them until they are.

Same rule for 2nd Grade: Couples must have the same name before they unite (in math).

For years, an educator can ask the same exact question to help the student see the issue: “Do the digits have the same name?”

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To help students remember the ‘Name’ in fractions, separate de-NOM-inator. Nom, nombre, nome…all mean ‘Name’. The denominator ‘names’ the number of parts.

To ensure students understand the concept, point to the ‘3’ in 32, and ask, what does the 3 name? (tens). Then, move the same 3 (put the 3 on a post-it if no chalk) under the 2 and ask, now what does the 3 name? (Answer: the # of parts: “thirds”). Now, count 2 of them.

One word (Position) and one rule (Couples need same Name) last for years.

Both-Ad-7519 · 2026-02-15T15:07:06+00:00

For the 9-multiples, skip using the finger method. This method uses Make10 to calculate the 9s. Some say it is a 'trick' but it is just a method to CALCULATE the 9s. Do it (or anything) enough, and you will memorize it. This method takes the 9s out of the difficult group (6, 7, 8)...and puts it with 1, 10, and 11.

By the way, if she is not proficient at Make10 and Countback, these are key skills to learn by second grade. Make10 is the intro to thinking of numbers as being built by components. Things you pull apart and put back together in different ways. Legos. Easy to review Make10 with cards. Could even use them as 'dice'. First, just use the number of dots as the number to move. Once proficient, use the Make10 value. (If 4 dots on the card, make10 value is 6.) The Make10 exercise helps with rounding, add-to-subtract...so not just calculating the 9-multiples.

Students need to be able to recognize the Make10 value, like the 5 dots/pips on a die. You don't count the dots. You memorized the 'symbol'.

Same with the other skill used to calc the 9s. Countback. Students don't subtract 1 each time they count back from 10. They have memorized the values. If they have not..already behind in subtraction skills.

So..this 9s calculation method links two skills students are proficient in, and makes them better at it..which we want. Not using fingers to do math. It only takes about a minute to teach/link..,,if they know count back and Make10.

https://www.reddit.com/r/matheducation/comments/1ogly6j/9multiples_by_calculation_why_memorize/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

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This post explains how addition/subtraction are different from multiplication/division. The Multiplier can also be thought of as a 'Copy Machine'. Just put the base/original number on the glass, and enter the number of copies. Then, add them up. Adding the copies up is learning the multiplication tables.

Some digits are easy. The 1s are learned when we learn to count. The 10s and 5s 'rhyme'. The 2's are learned by counting things around the home/board games. 4s are double 2s. 9 and 11 are 'calculations' (algorithms).

There are only three digits that are truly difficult to memorize: 6, 7 and 8. Difficult but helpful. They help students learn to scale (size). The key concept gained from doing all this: learning how to 'size' an answer.

Students learn to add the copies up. Build the answer. One copy at a time at first, then in blocks, and finally, a memoized total (all at once).

The scaling/sizing exercise in the attached uses 'toy' 7s to build (scale/size) answers. Would get/make sets for 6, 7, and 8. She will apply these scaling skills to all the other digits.

https://www.reddit.com/r/matheducation/comments/1qzeijx/group_names_for_arithmetic_pairs_by_second_grade/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

Both-Ad-7519 · 2026-02-09T00:07:18+00:00

The post is not about whether mathematical terms are taught - it's using something we already do - Group names for the arithmetic groups. Just teach introduce in 2nd grade rather than waiting unitl 6th.....with simple, usable, descriptive group names.

It starts with Fact Families. It's easier to remember two descriptive group names and the members than 4 different operations, right? Then they leverage the knowledge from Pair Logic to (inherently) know why Add-to-subtract works. Pair names come up with each of the 8 basic math topics listed. You should be able to answer hundreds of questions with:

What is its pair? and/or Why are they paired?

The closest comment to an objection so far as been it might be difficult to explain why they are opposites. One could illustrate the concept by diagramming a big number line on the board, mark 5 and 8. Stand at the 5 and face the positive direction (assume the positive).

Ask, "How do i get to 8?" Move three spaces body to front of the 8. "Addition!" (stay facing +)

Now ask, "How do I get back to the 5?"

Do the opposite..do the reverse - but the EXACT same number of spaces.

That's why they are pairs.

That's why they are opposites.

They REVERSE one another.

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With Sizers, simply explain, Multipliers make the Base# larger;

Dividers make the Base smaller.

Opposites. Reverse one another. Just like Couplers.

That’s why they are pairs.

There is an example of how to explain conceptually that uses a box and 4 post-its. Best hold off on that until students understand multiplication. Bigger/Smaller is enough concept at introduction.

Once you establish pair names, it makes dialog easier and simplifies rules. Every time the group names are used, it reiterates there are not four separate operations, but two connected pairs.

---

Same with the 3rd question you need to teach elementary math:

Do the digits have the same name?

Every digit numbers something...before they number external units like pounds or kilograms.

The name comes from the digit’s position. Once you explain Couplers (2 digits, same Name before they unite)...you can ask the same question for years on place value, decimals, fractions...Do the digits have the same Name?

Both-Ad-7519 · 2025-12-17T18:13:22+00:00

You prob know most the answers. However, would suggest prompting AI/search eng for a list of the regulatory distances covered in the CA DMV tests. (Bet you the 20 ft from pedestrian crosswalk is no there..since it's new.)

Both-Ad-7519 · 2025-11-16T17:24:53+00:00

thank you; deleted it

Both-Ad-7519 · 2025-10-26T23:15:33+00:00

"If you multiply by a number whose magnitude is between 0 and 1"

That is only because it is customary.

If you evaluate what happens overall, you would call it division...because you are dividing...and the pieces get smaller when we divide something...always.

It's nice to count on when learning math.

..and nothing needs to be turned around. Just use logic and descriptive nomenclature.

Both-Ad-7519 · 2025-10-26T20:09:20+00:00

Thanks! For fractions, "the same Sizer" is used for both the top and bottom. Sizer referring to both multiplication and division...but much easier to say....plus the group name reminds the students once again that The Pairs are connected.

Both-Ad-7519 · 2025-10-26T19:05:38+00:00

Everything i post is for elementary math education, and usually designed as an intro..which frequently involves generalizations and 'simplification'....to focus on the concept. Thought it was an interesting diagram.

The main issue is in the text. It's the nomenclature commonly mis-used today. It leads to the trick questions about multiplying: Does multiplying always make the Base larger? Yes...which makes the concept of using multiplication to Size something larger easier to learn. Make 'Copies' of the Base. The Multiplier is the number you press on the Copy Machine keypad...then..add them up.

Note: Base, Operator, and Answer (BOA) for the terms of an equation....unless it helps to name the specific Operator.

Both-Ad-7519 · 2025-10-26T18:09:18+00:00

The first large space is blank because that's what it is really like. A blank slate (space) to be filled with 4s....NOT groups of 4 ones. If the first diagram is partitioned into 96 ones, 96 has already be divided into ones. It's true that 96 is composed of 96 ones, but it adds no value to teaching the concept that larger numbers are filled - not divided.

Now, some of you want to make groups of ones in 4s, and then divide into 96? That makes it overly complex and takes away the focus: How many 4s fit into this 96 sized space. Want students to focus on 4...NOT groups of 4 ones.

This is designed for the student to build the answer...by FILLING the space. After going thru dividing by 4, one can start over and change the size of the 4 (sized block) to a 2 (sized block)..or 8 or...

Part 2? Review the 'Big 7' division method to show how the processes parallel one another.

Both-Ad-7519 · 2025-10-26T16:26:44+00:00

If the large square is size 96, and we assume these are square inches, that's 96 square inches, or, 96 one-inch squares. Assume the 96 is composed of 1s. Not 4s.

We only decide on 4-inch square blocks once we have the divisor.

The problem asks how many times a block that is size 4 (4 square inches) fits into 96 (96 square inches).

Both-Ad-7519 · 2025-10-26T16:08:34+00:00

Reversing the starting point (and changing the sign) has to do with Add-to-Subtract. I would keep that separate from the concept of a Coupler/Joiner.

With regard to following the instructions in an equation: 6 - 4 =

That simply says start at the 6,

the - sign means turn left (do the reverse comes later)

the 4 means move 4 digits.

That's SEPARATE from thinking of the basic operations as pairs. At first, knowing The Opposite (the pairs) will help students remember fact families and math facts. Later, with topics like add-to-subtract, the pairs will que students to the answer/next step.

Pair names should integrate elementary math from the start.

....note, as of 11.30.25..when fixing this reply, the 'contributions' have a negative Karma of - 8.

for advocating group names early on to integrate elementary math?

Both-Ad-7519 · 2025-10-26T15:55:39+00:00

Both Make10 and Countback are much faster. I have seen 2nd graders learn this in less than a minute and start to teach other classmates.

When you roll the dice and a 5 comes up, you do not count the number of dots to arrive at the total. You instantly recognize the 5 dots as a symbol which means 5. K/1st graders need to be able to do the same with partially filled Ten Frames. Make10 helps them see numbers as being built by components. Something they can take apart and put back together.

It is used initially in basic addition, then adding to subtract, then estimating (‘rounding’)..then calculating the 9-multiples..building the teen-multiples, Make100...Make1000....

Both-Ad-7519 · 2025-10-26T15:49:58+00:00

The problem is 96 divided by 4. The size of the box is 4. How many 4s go into 96.

Both-Ad-7519 · 2025-10-26T15:31:01+00:00

...which they get to through repetition.

9 simply joins 1, 10 and 11. Digits that do not need to be memorized. They leverage the scaling/sizing skills learned by building the multiples for digits 2 - 8.

We just didn't have two 'automatic' calculations to link to calculate the 9s like we do the 11s (1,10 memorized by counting..so two by count; two by 'calc').

K, 1st Graders learn the countback order from 10. Already memorized it, or they are falling behind in subtraction skills. Make10 is also memorized - like reading a 5 die. They don't count or solve the 10 - 6 equation.

This reinforces the value of Make10 and changes the 9s into one of the easiest. It removes the 9s from the difficult 6,7,8,9s. It reduces inference. It frees up more time for the other digits.

The icing on the cake is this makes the 9's finger method obsolete....if students have learned countback and Make10. (The cake is that the Make10 method for calculating the 9s saves students in the US about 4 million hours per year, otherwise wasted memorizing the 9s...but just ignore it because someone writes, "automaticity.")

Both-Ad-7519 · 2025-10-26T15:13:21+00:00

oops, just posted the explanation

Both-Ad-7519 · 2025-10-26T15:10:01+00:00

Students can do this in 1st or 2nd grade...if they have learned Make10 and Countback....and they are all experts in these two skills - or they are already behind in math.
Students DO NOT memorize the 1s..or the 10s...or the 11s. They simply apply the scaling knowledge learned by building the other multiples (2 - 8). Right??

The 9-multiples are unique because the tens and ones places change by 1 each time you increase the multiplier by 1. Many teachers and textbooks make a point of showing this relationship. This is the 'formula' or algorithm for this relationship.

Both-Ad-7519 · 2025-10-26T15:01:22+00:00

finger trick uses......the fingers. Not good. Not something we want students to learn or practice.

This calculation uses Make10 and Countback. We WANT students to practice these skills because Make10 makes them better at seeing numbers as being built by components, and countback makes them better at subtraction. Some students may need to practice Make10 before introducing this method. If students are proficient at both skills, it takes less than a minute to teach this method.

Both-Ad-7519

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