Category: Sets and Counting

  • Flash Cards and Marshmallows: Teaching One-to-One Correspondence

    Flash Cards and Marshmallows: Teaching One-to-One Correspondence

    My mom is a big fan of flash cards. So when she figured out how to use my School Zone Numbers 0-25 Flash Cards with MATH, she was so excited.

    We’ve been doing MATH a lot – but she says we can only do it once a day. I love doing it because I get to eat the MATH after we’re done.

    No fair that we can only do it once each day.

    So here’s what it is…

    There’s a bag full of them in the cupboard. The individual MATHs are tiny white things that taste SO good! They’re chewy and squishy and fun in my mouth.

    They’re about the size of Cheerios. (Oh – and comparing sizes is something called “math” – but Cheerios don’t taste like this, so it must be different).

    Taica took the picture at the top of this article so I could show you what I’m talking about. Look to the right of the book in the photo – that’s MATH. YUM!

    You can do it with flash cards.

    She pulled out my number flashcards (made by School Zone and given to me by Granfuffy) and let me pick a card. I chose the one with cars. I like cars.

    I put one MATH on each car. Taica counted as I did this. After a while I got really good at putting them on and taking them off. It’s called “one-to-one correspondence,” apparently. Or at least she kept saying that.

    Then she turned the card sideways!

    This was a little harder. But I managed.

    Then upside down – I was really wanting to eat the MATH at this point, but I continued to play along.

    At one point the marshmallow rolled off the car. So I left it there. I knew where it was supposed to be so it wasn’t a big deal.

    When I finished, I was out of MATHs and still had the yellow car left! Like this:

    Taica told me that if I didn’t make sure that the MATH was directly on the car, I would lose the “one-to-one correspondence” and end up with a leftover car.

    We figured it out and then it was time to eat.

    Do you want to try?

    I’m looking forward to the next time – there are some flash cards that have lots and lots of things to match!

    If you want to do it, you can get everything you need at a store. You have to “pay for them” before you take them home, but that’s easy for grownups.

    How do you like your MATH? Let me know in the comments so I can do more stuff. And share this on twitter, too!

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  • First Grade Math – Place Value Practice

    First Grade Math – Place Value Practice

    A number system is a highly advanced concept. And yet we throw it at first grade math students expecting them to immediately grasp it.

    Our number system is based on place value, like any number system (like clocks, years, etc.). Which means everything wraps around. Once you get to the “top” of the list of numerals, you have to start over, in a sense. This is crazy weird – it’s no wonder kids struggle at this point!

    I promised to help a teacher this weekend who was struggling teaching place value to her first grade math class. I dug out a MathRack, a brand of rekenrek, which was part of a set MathRack.com shared with me months ago. I peeked at their book Mastering the MathRack to Build Mathematical Minds to get an idea of how to teach place value using this amazing tool.

    The video above uses the MathRack 20 and some place value cards. I followed the Hidden Numbers activity on page 61 of the book. (As of writing this, I’m unable to find the book online. The site where it is supposed to be doesn’t seem to be functioning anymore.)

    Glenda, the first grade math teacher, specifically wanted help teaching the comparing numbers and ordering numbers. So here goes…

    Comparing numbers is easier when visualized.

    Children can see the value of two digit numbers better when they see the quantity of beads. Let them practice comparing numbers for a while using both the rekenrek and the place value cards. The more they practice, the better feel they’ll get for the place values in our number system.

    I’m not sure what the structure of a first grade math class is, but the more days they can “play” with their MathRack like this, the better they will get at comparing numbers. If you have limited time, do a few minutes each day for more days, rather than more time on fewer days.

    Ordering numbers is also easier when children see it.

    Once the children have played a while with the rekenrek, they will have some comparison skills. Ordering numbers is the next step. Teach them that the act of ordering numbers is just comparing numbers many times.

    Computers order numbers by comparing them one at a time to each of the other numbers. Let students try ordering numbers this way, as well as other ways. The one-at-a-time method might be slower, but it could be what the child needs.

    Keep trying and share what you learn.

    How about it – can you use this for your first grade number system lessons? Do you have a MathRack or can you make one? Share your successes in the comments!

    Click here to share this on Twitter/X.

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  • Teaching Little Ones Math with the Toddler Counting iPhone App

    Teaching Little Ones Math with the Toddler Counting iPhone App

    Daughter is addicted to the iPhone.

    It’s sad, really, because we’ve managed to keep her off TV and any screens for two years. And now she thinks the iPhone is the place for cartoons and all sorts of flashy lights and sound.

    But she can also learn math on the iPhone!

    Occasionally I’ll find an app that makes me glad she’s on the iPhone. Like Toddler Counting.

    This app does something grownups don’t think about – it teaches kids the one-to-one correspondence between numbers and objects. That’s a very advanced topic in math that we grownups take for granted.

    Here’s a demonstration of it:

    What do you think? Will you get it for your little one? At $0.99, Toddler Counting’s a deal!

    Un-Disclaimer: I paid for this app and don’t have any affiliation with the folks who created it. Heck – I haven’t even told them I’m writing this!

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  • Create Your Own Number Path – at Home!

    Create Your Own Number Path – at Home!

    I recently read about the difference between a number path and a number line in the book Mastering the Math Rack to Build Mathematical Minds.

    Walking up and down my hallway I noticed the tiles made a very nice number path. So I used some removable whiteboard wall decals and cut them in quarters.

    I wrote the numbers 0-11 on them with a permanent marker and slapped them down on the tiles.

    Later, while reviewing Carlito C. Caterpillar’s Math House Games for The Homeschool Post, I noticed Carlito suggested the same thing!

    My hallway is now a counting lesson!

    When we run down the hallway, now, we say the numbers as we step on them. This integrates counting, recognition of numbers and linearity all with body movement – which serves to solidify the learning.

    Not only that, but the removable decals don’t hurt the floor!

    And there’s more…

    When we were at the ice cream parlor, she noticed the tiles on the floor and started running along them and saying numbers! This was something I didn’t expect at all.

    Not only that, the three sets of three tiles created a number path of 11 when you included the two spacers. I don’t know if she recognized this, or if it was merely a coincidence. But it was fun to see.

    Will you do it?

    You can do this with anything that has a “block” pattern – at home, or in a classroom.

    Let me know if you try it – and the reaction of your kids – in the comments below.

    Disclaimer: The sweet folks over at MathRack.com sent me a bunch of MathRacks and the book, at no charge, for me to check out and report to you on, if I wanted to. You’ll be seeing more articles about these soon (they are really cool!).

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  • Counting with Big Numbers

    Counting with Big Numbers

    Last night I had the privilege to meet and teach Eddie*, an ESL student from Mexico, at Literacy Advance of Houston. He was there to learn in the “Math and Your Life” class, as part of the “Math and…” class series.

    I didn’t realize I was there to learn too.

    I walked in prepared to discuss just about anything math related. And I’m glad that was the preparation I did.

    Eddie was interested in something that I’ve long struggled with. And I’m guessing many children struggle with it, too.

    In English, the number 1600 is pronounced both as sixteen hundred and as one thousand six hundred. I still get these mixed up. Not when I stop and think about them, but when I casually and quickly throw them out.

    Husband is often stunned when I tell him I saw a new suburban at the low low price of thirty-five hundred dollars. Of course I mean thirty-five thousand dollars!

    It’s not just me, I guess.

    I wonder how many other grown-ups still struggle with this. And how often we neglect to teach this to children.

    We are quite accustomed, and comfortable, with teaching our youngsters to count from 1 to 10. Were amazingly proud when we can get them to count from 1 to 20.

    Is that enough? Based on my conversation with Eddie last night, no.

    Teach them skip counting with hundreds!

    Why not use the 1-20 model with hundreds? Like this:

    one hundred
    two hundred
    three hundred
    .
    .
    .
    eight hundred
    nine hundred
    one thousand
    eleven hundred
    twelve hundred
    thirteen hundred
    fourteen hundred
    .
    .
    .
    nineteen hundred
    two thousand
    twenty one hundred
    .
    .
    .

    Teach them all sorts of counting!

    I suggested in this article to count with your children by fractions. It never occurred to me to count by giant numbers.

    What other ways should we teach children to count? Share your ideas in the comments.

    *”Eddie” is used as a variable – i.e. his name has been changed because I didn’t ask his permission to talk about him.

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  • The 1-2-3 Song

    The 1-2-3 Song

    Part of the Count 10 Read 10 series to help parents connect with kids through math a little each day.

    Did you know that the alphabet song, Twinkle Twinkle Little Star and Baa Baa Black Sheep are all the same tune!?

    Well, now there’s another!

    Thanks to all the great folks who have public domain images out there that I could use for this.

    Specifically

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  • How to Teach Subitizing: 1 – 4

    How to Teach Subitizing: 1 – 4

    The comparison of numeracy to literacy is curious.

    Learning math is the opposite of learning to read. When you read, usually simultaneous to learning a language, you sound out words and then put meaning to them. When you learn to count and do math, you know the meaning inherently and then put a language to it.

    At some point we learn to recognize words without sounding them out. And at some point we learn to recognize quantities without counting them out. This is called subitizing.

    The Your Baby Can Read program uses the concept of subitizing to teach reading – you show your baby the word alongside the object. So the shape of the word car is as recognizable as a car itself.

    The children using Your Baby Can Read don’t learn to sound out words. They don’t understand the concept of letters any more than babies not using the program. But they instantly recognize the shapes of the words – giving them an (assumed) advantage.

    Aside: We didn’t use the “Your Baby Can Read” program, not because it was gimmicky (I love anything that looks gimmicky), but because there is a huge DVD element to it. We decided not to put Daughter in front of the TV for her first 2 years. A decision we stuck with, but sometimes was a struggle!

    This article contains a “your baby can count” type program. (And it’s a free download!)

    How did we learn subitizing?

    I don’t recall having been taught it directly. Although I could be wrong. The research on it has been happening since the early 1900s, so it might have been taught without being labeled “subitzing.”

    In a previous article about why learning to subitize is important, Christine Guest commented that she learned it out of frustration for counting with chanting.

    I wonder how many of us do that. Are grownups so adept at subitizing that they forget that’s how we assess quantity? Maybe we’re taught to chant-count because that’s the way we think counting is.

    But it isn’t!

    How do you teach subitizing?

    Images are accompanied by the written numeral as well as the number spoken aloud. The images would be printed on cards, done via video or “live” with 3D objects.

    I’m still working on the numbers 5-10 and up, but for the numbers 1-4, the following 8 styles of image sets would be done twice. Once using the same objects for each image set, and once using different objects for each image set.

    1. Organized in a row vertically.
    2. Organized in a row horizontally.
    3. Organized in a row diagonally.
    4. Organized in a row other way diagonally.
    5. Organized in a regular shape (triangle, square).
    6. Organized in a differently oriented regular shape.
    7. Organized in an irregular shape.
    8. Organized in a different irregular shape. (There will be more of these for 4 than 3, etc.)

    The objects could be blocks, cars, little dolls, just about anything. I created the set below from blocks I found left in Daughter’s block set.

    Each zip file contains a few .jpg files with 4″ x 6″ pictures. You can print them at home or ship them to Walmart, Target, CVS, etc. for printing. I left off the MathFour.com logo so the kiddos wouldn’t get distracted. Please share them along with links back here.

    What do you think? Can you use these? Did you?

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  • Toddlers Begin Counting at the Number 2 (not 1)

    Toddlers Begin Counting at the Number 2 (not 1)

    I had this grand idea when we got married and were hoping for kids – I would teach our children to count starting at 0.

    When Daughter was 15 months old, I decided we should start teaching to count with negatives.

    But I was wrong on both.

    And so is everyone else.

    Why do we teach toddlers to count?

    We practice counting 1-10 with our kids. We know (somehow) that before they’re official school age, they should know how to count to 10. And how proud we are as parents if they can count to 20!

    But these are just words.

    I can teach Daughter to memorize the Fibonacci sequence, but she’d no more know what that means than what counting to 10 means.

    In fact, I know this first hand because I used to count to 10 in Spanish. And I’d leave out ocho everytime!

    I saw a guy made fun of in Germany because he told a waitress he had fünf people in his party and held up four fingers. (She did it behind his back to another waitress – she wasn’t so rude to say it to his face. (Thank goodness; I would’ve had to go Texan on her.))

    We teach toddlers to count for the same reason that we teach them to say please, thank you, yes ma’am and no ma’am – because someday they’ll understand what it means. And in the meantime they can establish good habits.

    So where do they start understanding?

    Regardless if we teach a toddler to start counting with -5, 0 or 1, they start with 2.

    -5 to a toddler makes no sense. Teaching -5 to a toddler can only be dreamed up by a math teacher with no kids (i.e. me three years ago).

    0 is useless. Why would you even mention that you have zero? Maybe saying that there are zero cookies after she ate them all might work. But generally zero things can’t be seen and by the time you’re down to 0 cookies, there’s probably a meltdown in the works. And we all know there’s no learning during a meltdown.

    1 is just as useless. Why count things that are only one? They started with one mom, one dad, one dog, one couch, one bed, one bear,… Almost everything in their world is a single. The number “one” is just as useless to them as the words “the” or “a.”

    But 2 is interesting!

    Daughter was so amazed at the discovery that she had two SnackTraps. Not just the ordinary situation of a bowl of snacks but “TWO BOWLS!”

    As soon as multiple copies of things are in her world, she takes note. If you’re an identical twin, the first time your child sees you with your twin might be traumatic. My best friend is the daughter of a twin and she tells horrors stories of this discovery.

    This is an extreme, but consider all the pairs of things that kids can notice – two shoes (vs. only one that you can find when you’re freaking out and you’re late), two forks (when you’re begging for yours back from her because you’ve not eaten since breakfast), two cars (when you need to get in one and she insists on going in the other).

    And, toddlers really don’t start counting at 2. They don’t start their mathematical careers with counting at all! They start by recognizing multiples. And 2 is the first and fastest multiple.

    So what can you do?

    Keep teaching your kids to count – they still need this skill, just like they need to memorize math facts. But also teach them to subitize (recognize amounts without counting them out). Hold up two of the same items and exclaim “TWO ORANGES!” Then go to another two items and exclaim, “TWO RAISINS!” Stick with one number at a time.

    Daughter is on “two,” so we’ll stick with that for a few months. We’ve got plenty of time.

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  • Union, Intersection and the Empty Set

    Union, Intersection and the Empty Set

    The union is on the left and the intersection is on the right.

    I discussed empty sets in a previous article. I hadn’t yet listened to the recent podcast of The Math Dude, Jason Marshall, in which he introduced sets and subsets. Such a curious coincidence; I thought I would continue down our united path.

    This week, the sweet Math Dude talked about union and intersection of sets. So I’m heading there too – and incorporating the empty set.

    Union and intersection have symbols.

    These symbols represent “operations” (like addition or multiplication) that you do to a pair of sets. They are called the cup and cap and look like, well, a cup and a cap. Or a U and an upside down U. The Union is the U and the intersection is the other one – that’s how I remember it and teach it.

    For the sake of illustration, I’ll use Daughter’s handy SnackTraps like in the picture above.

    The union is everything.

    The sets I’m using are

    L ={green cube, blue half circle thingie}

    R = {green cube, red triangular prism, orange rectangular prism}

    I’m using the letters L and R for left and right in the picture.

    The union of the two sets results in all the pieces from each set, all crammed together. (And if there is a repeated item, you only take it once.)

    The union of the two sets is everything combined (without duplicates).

    Let’s take the set of all states that border Texas and the set of all states that border Oklahoma, like this:

    T = {Louisiana, Arkansas, Oklahoma, New Mexico}

    O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}

    Then the union of the two would be “The set of all states that border both Texas or Oklahoma or both” and would be T U O = {Louisiana, Arkansas, Oklahoma, New Mexico, Colorado, Kansas, Missouri, Texas}

    Notice we don’t keep duplicates.

    The intersection is only the common stuff.

    Using the same L and R sets from above, I can take the intersection.

    The intersection is only the stuff they have in common (we get rid of the duplicates here, too).

    Let’s go back to our set of all states that border Texas and Oklahoma:

    T = {Louisiana, Arkansas, Oklahoma, New Mexico}

    O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}

    Then the intersection of the two would be “The set of all states that border both Texas and Oklahoma, at the same time” and would be T ∩ O = {Arkansas, New Mexico}.

    What happens with the empty set?

    When you start throwing the empty set into the mix, you follow the same rules.

    For the union, you throw everything in both sets into one bag. Well, since there’s nothing in the empty set, “everything” is the original other set!

    The union of the empty set with anything… is that same set back!

    For the intersection, you take only the stuff that’s in common between the two. Since the empty set has nothing – there’s nothing in common.

    The intersection of the empty set with any other set it just the empty set – there’s nothing in common at all!

    Think about it…

    Does this remind you of anything? What similarities do you see between this and addition or multiplication (or subtraction or division)? Ask your children these questions too.

    Get out some snack traps and blocks (or a map) and go for it! Let me know how it goes in the comments.

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  • Empty Sets – When to Use Curly Brackets

    Empty Sets – When to Use Curly Brackets

    Are you teaching sets? Some of the stuff that you talk about in set theory  is pretty clear – you have a basket { } and you put stuff in it {♥, ✂, ☎, ✿}. No problem.

    But what’s the deal with the basket with no stuff in it?

    That’s the empty set. It’s written like Ø or simply as the “basket” like this: { }.

    But kids might be tempted to write {Ø} – and then they get it wrong. BUT WHY?

    The empty set is an object!

    Suppose you have four empty sets. Do you have nothing? Nope – you have four containers. Just check your cabinets – I’m guessing you have a handful of empty sets in there with the Rubbermaid® or Tupperware® logo on them.

    And {Ø} is an object inside a set.

    Supposed you put one inside the other, like this:

    This is the same as {Ø}. Would you be willing to put leftovers inside the bigger one while the smaller one is there? It would be something like this:

    Instead, wouldn’t you be more likely to remove one empty set from the other and have just Ø and Ø (no curly brackets) like this:

    What do you think? Does this help you understand the empty set? Will it help you teach it?

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