Introducing the Equals Sign ("Is the Same As") and Infinity

I learned many years ago in college (I think it was 1998) from reading a NCTM article on "Incredible Equations" about the importance of getting a clear message to children about the Equals Sign from the very beginning.

The equals sign means "is the same as." It does NOT mean "put the answer here." Yet, if you ask many elementary aged children what that sign means, they will say, it is where the answer goes.

Then, when you introduce algebra, you will find a problem that pops up, which stems from this long-ago misunderstanding. It is easy to check if your child has internalized this misunderstanding. Give your child the problem 3 + 7 = ___ + 5. And see if they write 10 in the blank. They disregard the + 5 as being some sort of strange teacher error. And they put the answer in the blank after the sign that they think means "put the answer here."

It's not hard to get this straight from the beginning. When we first show children how to write number sentences, and you say "and" when you write plus on the board, you say "is the same as" when you write the equals sign on the board. 5 + 5 = 10. Five and five is the same as 10.

Every year I introduce and explain this because every year I have new students in my classroom. I do this every day in the month of October. Halloween is the perfect opportunity! I draw a Ghostie Number problem. Instead of the blank or a variable such as x, I draw a math problem with a little ghostie in the place of one of the numbers. I tell my class that the number is wearing his Halloween costume, and we have to figure out what number is hiding under the sheet.

This is easy to differentiate to any age group. You can even put Ghostie in a fraction or decimal problem! Or have Ghostie equal to zero. That's tricky!

So, I have done this for years and years... and then this summer I ran into a new situation. I am tutoring a little girl who needed this lesson but working with Ghostie Numbers didn't really seem like a good fit. It's not Halloween! I know we could do it anyway, but it got me thinking. What else can you do to practice the concept "is the same as"?

I decided to do this lesson with the Gnome King and the Math Gnomes.

Each day the Gnome King would announce what quantity of gems he wanted brought to him, and each gnome would oblige, working in the way he/she knows how. We wrote them as long number sentences and said "is the same as" each time the equals sign was written.

10 = 5 + 5 = 20 - 10 = 1 x 10 = 100 ÷ 10

17 = 10 + 4 + 3 = 20 - 3 = 1 x 17 = 34 ÷ 2

Then I gave her a big challenge! She had to choose a number (she chose 40) and come up with ten different ways to make that number.

40 =

44 - 4

30 + 10

10 x 4

100 - 60

160 ÷ 4

50 - 10

0 + 40

36 + 4

60 - 20

70 - 30

This is interesting assessment, because you can see which operations they are most comfortable with. You can also ask the classic Waldorf question, "Which is the most beautiful way to make 40?"

Since this little girl also needed the Infinity Street lesson, I used our Equals Sign work to raise the question, "Is there a limit to how many ways we can make 40, or are there an endless number of ways?"

We read Infinity and Me, my new favorite book about Infinity since the author interviewed many children to hear their explanations of it.

We also talked about the Infinity symbol and made a Möbius strip. This works really well if you cut a long strip of white paper and then color one side of it with color first, before you twist and tape the two ends together.

The following day I presented the Infinity Street lesson.

I've written more about this on the Column Algorithms page on my website. Grade 2 is when Waldorf really gets into place value.

We have houses and mailboxes up to septillion, and slips of paper with the names of all of the families up to novemdecillion

Simple
Thousand
Million
Billion
Trillion
Quadrillion
Quintillion
Sextillion
Septillion
Octillion
Nonillion
Decillion
Undecillion
Duodecillion
Tredecillion
Quattuordecillion
Quindecillion
Sexdecillion
Septendecillion
Octodecillion
Novemdecillion

This is a fantastic extension for children who are learning about prefixes (which also happens in Second Grade). I like to give them the slips of paper after septillion and have them figure out how to put them in order. Of course, you can decide what the child you're working with is up for.

When it comes to talking through the prefixes, you can refer to the months of the year as a way to help but this can get surprisingly tricky since September is no longer month #7, October is no longer month #8, etc. I remember finding this irritating when I was a child! They find out the reason for this in grade 3 Calendars, so you can plant the seeds of, "I wonder..."

I have actually found Zin! Zin! Zin! A Violin by Lloyd Moss to be extremely useful in this lesson, especially for sliding under the radar the prefix < sex > as representing six (trio, quartet, quintet, sextet, septet, octet, nonet).

If you don't want to get into the whole prefixes lesson, you can still do Infinity Street as far as you'd like. If you want a book for WHY we might need really big numbers, Is a Blue Whale the Biggest Thing There Is? by Robert E. Wells is a good choice.

If you want to practice reading numbers to the hundred thousands place, you can't beat A Million Dots by Andrew Clements. It is excellent!

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