PUFM 1.4 Subtraction

Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

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ace of spades card deck

Game: Target Number (or 24)

[Photo by stevendepolo.]

Math concepts: addition, subtraction, multiplication, division, powers and roots, factorial, mental math, multi-step thinking
Number of players: any number
Equipment: deck of math cards, pencils and scratch paper, timer (optional)

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Do Your Students Understand Division?

Cheerios by sixes
[I couldn’t find a good picture illustrating “division.” Niner came to my rescue and took this photo of her breakfast.]

I found an interesting question at Mathematics Education Research Blog. In the spirit of Liping Ma’s Knowing and Teaching Elementary Mathematics, Finnish researchers gave this problem to high school students and pre-service teachers:

  • We know that:
    498 \div 6 = 83.
    How could you use this relationship (without using long-division) to discover the answer to:
    491\div6=?
    [No calculators allowed!]

The Finnish researchers concluded that “division seems not to be fully understood.” No surprise there! Check out the pdf report for detailed analysis.

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yahtzee-dice

Contig Game: Master Your Math Facts

[Photo by Photo Mojo.]

Yahtzee and other board games provide a modicum of math fact practice. But for intensive, thought-provoking math drill, I can’t think of any game that would beat Contig.

Math concepts: addition, subtraction, multiplication, division, order of operations, mental math
Number of players: 2 – 4
Equipment: Contig game board, three 6-sided dice, pencil and scratch paper for keeping score, and bingo chips or wide-tip markers to mark game squares

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schoolyard

30+ Things to Do with a Hundred Chart

[Photo by geishaboy500.]

Are you looking for creative ways to help your children study math? Even without a workbook or teacher’s manual, your kids can learn a lot about numbers. Just spend an afternoon playing around with a hundred chart (also called a hundred board or hundred grid).

Here are a few ideas to get you started…

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cat-fight

Hit Me! (A Math Game)

Photo by paparutzi.

Math concepts: addition, subtraction, negative numbers, mental math, absolute value
Number of players: any number
Equipment: math cards (two decks may be needed for a large group)

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Math homework photo by MC Quinn.

How to Teach Math to a Struggling Student

photo by MC Quinn via flickr (CC BY 2.0)

Paraphrased from a homeschool math discussion forum:

Help! My daughter struggles with arithmetic. I guess she is like me: just not a math person. She is an outstanding reader. When we do word problems, she usually has no trouble. She’s a whiz at strategy games and beats her dad at chess every time. But numbers — yikes! When we play Yahtzee, she gets lost trying to add up her score. The simple basics of adding and subtracting confuse her.

Since I find math difficult myself, it’s hard for me to know what she needs. What’s missing to make it click for her? She used to think math was fun and tested well above grade level, but I listened to some well-meaning advice and totally changed the way we were schooling. I switched from using workbooks and games to using Saxon math, and she got extremely frustrated. Now she hates math.

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Calculations

Subtracting Mixed Numbers: A Cry for Help

Photo by powerbooktrance.

Paraphrased from a homeschool math discussion forum:

Help me teach fractions! My son can do long subtraction problems that involve borrowing, and he can handle basic fraction math, but problems like 9  -  5 \frac{2}{5} give him a brain freeze. To me, this is an easy problem, but he can’t grasp the concept of borrowing from the whole number. It is even worse when the math book moves on to 10 \frac{1}{4}  -  2 \frac{3}{7} .

Several homeschooling parents replied to this question, offering advice about various fraction manipulatives that might be used to demonstrate the concept. I am not sure that manipulatives are needed or helpful in this case. The boy seems to have the basic concept of subtraction down, but he gets flustered and is unsure of what to do in the more complicated mixed-number problems.

The mother says, “To me, this is an easy problem” — and that itself is one source of trouble. Too often, we adults (homeschoolers and classroom teachers alike) don’t appreciate how very complicated an operation we are asking our students to perform. A mixed-number calculation like this is an intricate dance that can seem overwhelming to a beginner.

I will go through the calculation one bite at a time, so you can see just how much a student must remember. As you read through the steps, pay attention to your own emotional reaction. Are you starting to feel a bit of brain freeze, too?

Afterward, we’ll discuss how to make the problem simpler…

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Nim game tokens (jellybeans)

Nim Revisited

Photo by Sister72.

Dave at MathNotations offers another version of Nim that will give your students something to think about:

[1,2]-3-[4,5]-6-[7,8]…21 Helping Children Devise and Understand Winning Strategies


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Euclid game

Euclid’s Game on a Hundred Chart

Math concepts: subtraction within 100, number patterns, mental math
Number of players: 2 or 3
Equipment: printed hundred chart (also called a hundred board), and highlighter or translucent disks to mark numbers — or use this online hundred chart

Set Up

Place the hundred chart and highlighter where all players can reach them.

How to Play

  • Allow the youngest player choice of moving first or second; in future games, allow the loser of the last game to choose.
  • The first player chooses a number from 1 to 100 and marks that square on the hundred chart.
  • The second player chooses and marks any other number.
  • On each succeeding turn, the player subtracts any two marked numbers to find and mark a difference that has not yet been taken.
  • Play alternates until no more numbers can be marked.

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Dirty numbers

Game: Hundred Chart Nim

Photo by Håkan Dahlström via flickr.

Math concepts: addition and subtraction within 100, logical strategy
Number of players: 2 or 3
Equipment: printed hundred chart (also called a hundred board) and beans, pennies, or other tokens with which to mark numbers — or use this online hundred chart

Set Up

Place the hundred chart and a small pile of tokens where both players can reach them.

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Junto books diagram

Ben Franklin Math: Elementary Problem Solving 3rd Grade

The ability to solve word problems ranks high on any math teacher’s list of goals. How can I teach my students to solve math problems? I must help them develop the ability to translate “real world” situations into mathematical language.

In two previous posts, I introduced the problem-solving tools algebra and bar diagrams. These tools help our students organize the information in a word problem and translate it into a mathematical calculation.

Working Math Problems with Poor Richard

This time I will demonstrate these problem-solving tools in action with a series of 3rd-grade problems based on the Singapore Primary Math series, level 3A. For your reading pleasure, I have translated the problems into the universe of a well-written biography of Ben Franklin, Poor Richard by James Daugherty.

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Number Bonds, Number Rainbows

Basic bar diagram

Many of us use the idea of number bonds with our young students. A number bond is a mental picture of the relationship between a number and the parts that combine to make it.

Now we have a new, colorful way to show these relationships, thanks to Maria at Homeschool Math Blog. If you teach math to young children, check this out:

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Bar diagram for Janie and Bill

Penguin Math: Elementary Problem Solving 2nd Grade

The ability to solve word problems ranks high on any math teacher’s list of goals. How can I teach my students to reason their way through math problems? I must help my students develop the ability to translate “real world” situations into mathematical language.

In a previous post, I analyzed two problem-solving tools we can teach our students: algebra and bar diagrams. These tools help our students organize the information in a word problem and translate it into a mathematical calculation.

Now I want to demonstrate these problem-solving tools in action with a series of 2nd grade problems, based on the Singapore Primary Math series, level 2A. For your reading pleasure, I have translated the problems into the universe of one of our family’s favorite read-aloud books, Mr. Popper’s Penguins.

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7 Things to Do with a Hundred Chart


This post has been revised to incorporate all the suggestions in the comments below, plus many more activities. Please update your bookmarks:

Or continue reading the original article…


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Egyptian Math in Hieroglyphs

Pyramids clip artEgyptians wrote in hieroglyphs, a type of picture writing, and in hieratics, which were like a cursive form of hieroglyphs.

Hieroglyphs came first. They were carved in the stone walls of temples and tombs, written on monuments, and used to decorate furniture. But they were a nuisance for scribes, who simplified the pictures and slurred some lines together when they wrote in ink on paper-like papyrus. This hieratic writing — like some people’s cursive today — can be hard to read, so we are only using hieroglyphic numbers on this blog.

Download this page from my old newsletter, and try your hand at translating some Egyptian hieroglyphs:

Then try writing some hieroglyphic calculations of your own.

Edited to add: The answers to these puzzles (and more) are now posted here.

To Be Continued…

Read all the posts from the September/October 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.


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Number Bonds = Better Understanding

[Rescued from my old blog.]

number bondsA number bond is a mental picture of the relationship between a number and the parts that combine to make it. The concept of number bonds is very basic, an important foundation for understanding how numbers work. A whole thing is made up of parts. If you know the parts, you can put them together (add) to find the whole. If you know the whole and one of the parts, you take away the part you know (subtract) to find the other part.

Number bonds let children see the inverse relationship between addition and subtraction. Subtraction is not a totally different thing from addition; they are mirror images. To subtract means to figure out how much more you would have to add to get the whole thing.

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negative number tree

Negative Numbers for Young Students

[Rescued from my old blog.]

Would you like to introduce your students to negative numbers before they study them in pre-algebra? With a whimsical number line, negative numbers are easy for children to understand.

Get a sheet of poster board, and paint a tree with roots — or a boat on the ocean, with water and fish below and bright sky above. Use big brushes and thick poster paint, so you are not tempted to put in too much detail. A thick, permanent marker works well to draw in your number line, with zero at ground (or sea) level and the negative numbers down below.

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Order of Operations

[Rescued from my old blog.]

Marjorie in AZ asked a terrific question on the (now defunct) AHFH Math forum:

“…I have always been taught that the order of operations (Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction) means that you work a problem in that order. All parenthesis first, then all exponents, then all multiplication from left to right, then all division from left to right, etc. …”

Many people are confused with order of operations, and it is often poorly taught. I’m afraid that Marjorie has fallen victim to a poor teacher — or at least, to a teacher who didn’t fully understand math. Rather than thinking of a strict “PEMDAS” progression, think of a series of stair steps, with the inverse operations being on the same level.

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qiki

The Game That Is Worth 1,000 Worksheets

[Rescued from my old blog. Image via Wikipedia.]

Math concepts: greater-than/less-than, addition, subtraction, multiplication, division, fractions, negative numbers, absolute value, and multi-step problem solving.

Have you and your children been struggling to learn the math facts? The game of Math Card War is worth more than a thousand math drill worksheets, letting you build your children’s calculating speed in a no-stress, no-test way.

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Kids Do the Craziest Things

[Rescued from my old blog.]

My youngest daughter wanted to do Singapore math today. Miquon Red is her main math text this quarter, but we add a bit of Singapore Primary Math 1B whenever she’s in the mood. We turned to the lesson on subtracting with numbers in the 30-somethings. The first problem was pretty easy for her:

30 – 7 = []

I reminded her that she already knows 10 – 7. She agreed, “10 take away 7 is 3.” Then her eyes lit up. “So it’s 23! Because there are two tens left.”

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