Tag: addition

  • Count 10 Read 10: Random Number Game

    Count 10 Read 10: Random Number Game

    Learning math isn’t just about being taught math. It’s about fun, discovery and experimentation. In the Count 10, Read 10! program, parents get to spend 10 minutes a night playing math with their children.

    Like many games you’ll find here, this is a version of Calvinball (from Bill Watterson’s Calving & Hobbes cartoon). You and your children make up the rules as you go along or as you see fit.

    This is merely a guideline or starting point.

    Random Adding

    Objective:

    Have fun with numbers, counting and quantities.

    Breakable rules:

    1. The leader starts by saying a number.
    2. The next player says “plus” and another number. Then adds them and says the result.
    3. The next player says “plus” and another number. She adds that to the previous result and says the new result.
    4. Play continues until a winner is determined.

    Example

    Leader: Five!

    Player 2: Plus three is eight!

    Leader: Plus one is nine!

    Player 2: Plus two is eleven!

    Leader: WINNER!

    End game, and how to choose a winner.

    The round ends when the youngest child reaches their limit of counting or adding. The winner is determined by a rule or random choosing. The older the children, the more “real rules” you’ll need to follow.

    Possible winning rules:

    • The first person to add up to 10 – or a number designated by the leader at the beginning of the game.
    • The person who noticies that another player is wrong in their calculations (this is perfect for the parent to “test” the kid).
    • At the whim of child or parent.

    Variations

    The point is to have fun with counting and math. As your children grow, you’ll have to adjust the rules to give them more challenge and to fit the “real game” model. Here are some options for variations:

    • Each player can only add a multiple of their age (grownups use one of the digits from their age).
    • Each player can only add a multiple of a roll of a die (get foam dice for bedtime).
    • Subtraction – instead of adding up, start with a higher number and add down.
    • Multiplication – instead of adding, multiply each new number. This one could get “fun” really quick!

    Will it work?

    All games created at MathFour.com are tested or will be tested on Daughter. The rub is that Daughter is almost 2 – we’ll have to wait a while to do this one. So your input is important.

    Will it work? Did it work? Try it and let me know how it goes in the comments, please. Also share your own variations.

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  • Order of Operations Explained: Addition, Subtraction and Conclusions

    Order of Operations Explained: Addition, Subtraction and Conclusions

    This is the 6th and last in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

    Cup Of Coffee by Petr Kratochvil

    I started this series over a month ago. In that time, I’ve gotten pretty deep in thinking, learning and reading about the order of operations. I’ve seen a variety of ways people view, use and teach it.

    Before I go too far into some conclusions, though, let’s look at addition and subtraction.

    Subtraction is the same as addition.

    Yup. You might remember that from the fourth article.

    Consider the problem . Moving from left to right, and doing both subtraction and addition as we come to them, we get 4. If we found a book, or person, that meant the full-on PEMDAS and wanted addition done strictly before subtraction, then we would end up with 0. The latter is because we would do the addition of 3 and 2 before we did the subtraction.

    Which is right?

    It depends on what you really mean. If you don’t know if you should go left to right or strictly addition before subtraction, either look in the textbook you’re using or demand parenthesis.

    The text will clearly outline the order of operations it’s following. Be careful, too because there isn’t always agreement among textbooks. I have seen some texts that instruct the learner to do multiplication first and then go back and do all the division signs. While others (and this is more common, today) have us do multiplication and division from left to right, simultaneously.

    If you compare contemporary texts to each other, you’re likely to find them all the same. But grab a math text from the 80s at Half Priced Books. I’ll bet you’ll find at least 50% of the time they put division strictly after multiplication. (I’ll verify this the next time I’m there.)

    The order of operations needs context.

    Jelly Donuts by Petr Kratochvil

    I have $5 in my bank account. Then I bought a coffee for $3 and a bagel for $2. I might accidentally write down . I still mean, “I need to add up the stuff I spent and subtract it from my balance.” I wrote it in error, though. What’s “mathematically” correct is .

    But you knew what I meant.

    This was a typo that was helped along by using the context.

    Until there’s a reason to do arithmetic, the order in which we do things is arbitrary. If we all agreed to do addition first, then multiplication, we would calculate  and come up with 35 (instead of 23).

    As long as we all come up with the same thing, we’re fine.

    “We” have agreed to do multiplication things before we do addition things. So “we” would come up with 23 in the example.

    Coach G noted it correctly: the order of operations is a convention. In other words, we’ve decided on it. We invented it.

    How can you use this to teach your children?

    The coolest thing is that you can let them play. Get dirty. Break it.

    Remember opposite day? Have that. Let your little one make new rules. Let them see what happens if you all decide one day to do multiplication before addition. If your child is older and doing some algebra, this will mean reversing the order in which you UNDO the operations too!

    This is a real brain stretcher. But it’s just math. You’re not building a bridge or balancing your checkbook. Let them break it. Let them see what happens if you make your own rules.

    And then they’ll really learn!

    Let me know how it goes – share your stories in the comments.

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  • The Order of Operations Explained: Exponents, Multiplication and Addition

    The Order of Operations Explained: Exponents, Multiplication and Addition

    This is the 4th in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

    The Order of Operations can be boiled down into three “real” operations.

    Parenthesis are merely a way to group things – they aren’t a real operation. So they doesn’t count as a real operation.

    Since division is really just multiplication turned upside down, we don’t need to include it separately, either. Likewise, subtraction is addition on its ear. So we throw him out, too.

    Now we have only three: exponents, multiplication and addition.

    Exponents are the shortcut for multiplication.

    In a previous article about remembering exponent rules, I recorded this video about exponents:

    You see that 3 x 3 x 3 x 3 is 34. The exponent is the shortcut for multiplication.

    Multiplication is the shortcut for addition.

    Likewise, when we get overwhelmed with adding up the same numbers over and over, like in the video, 4 + 4 + 4 + 4 + 4, we can use multiplication to shortcut it: 5 x 4.

    Follow the shortcut evolution.

    The shortcut evolution is like this

    1. Addition came first.
    2. Then we created multiplication to make addition easier.
    3. Then we created exponents to make multiplication easier.

    So when you do arithmetic, we should do the “recent” shortcuts first (exponents) and then the “older” shortcuts (multiplication) and then the “regular” arithmetic (addition). Remember that subtraction is addition and division is multiplication.

    And keep in mind that we need to watch the grouping or anything isolated with parenthesis, absolute value bars or in a fraction. The grouping/isolation tools always trump the other operation rules.

    What if your answer isn’t the same as the back of the book?

    Some textbooks actually have multiplication done before division. This means that the problem 9 ÷ 3 x 2 will show a different answer (3/2) in that textbook’s solution pages than what you would get following the OoO I’m describing here (6). These texts are rare, but I’ve seen them. So keep your eyes peeled.

    The Order of Operations is a set of rules that we’ve agreed on. Which means that as long as a textbook clearly defines their order that they will follow, they can do things like this.

    If you get confused as to which to do first, demand parenthesis. (Yes, you can do that.) Or put parenthesis into your child’s textbook to help them out.

    The problem sets should be there to enforce, not confuse.

    What do you think? Does this help or hinder the way you’ve always viewed the Order of Operations? Share your thoughts in the comments.

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  • Why Learning to Subitize Is Important

    Why Learning to Subitize Is Important

    Previously, I gave a definition and some resources for subitizing – assessing quantity without counting. The questions I posed in the previous article were

    • If you do it, how did you learn to do it?
    • How do we teach children to do it?

    Having thought a great deal about it, I wondered if it was even relevant to teach it. And Husband asked the question, “Don’t they need to count and know their numbers before learning to subitize?”

    You don’t have to learn to count to know “how many.”

    Although evidence indicates that subitizing and counting happen in the same part of the brain, you don’t have to know how to count to subitize.

    Subitizing up to 4 or 5 using recognition bypasses the “counting” of numbers.

    Daughter is currently subitizing 2 – without counting. I haven’t taught her how to count objects at home and they don’t do it at school yet. So she has no concept of one. But “two bows” (one for my hair and one for hers) is very important to her. Likewise “two bowls” (one with goldfish and one with Rice Chex) is an amazing thing.

    This article by Ernst von Glasersfeld explains how number words can be associated with the quantity without ever counting. Children give the appearance of counting because they’ve memorized the number words and they use the cadence to tap their finger on objects. But they’re usually doing this to be rewarded with praises and cheers. They don’t really understand the concept of counting. This is what I saw the little girl doing at the playground.

    It IS important for kids to subitize.

    What’s the point, anyhow? If I can quickly see 8 things  – does it matter? And is it important to tell the difference between 8 things and 9 things? If there are 8 hungry tigers heading for me and I subitize that there are 9, will it make a difference in how fast I run from them?

    I researched more and found out that  subitizing does matter to the understanding of numerical concepts.

    Subitizing 2 items and putting it with a subitzed 3 items can yeild the perceived 5 without ever knowing what addition is. And having the group of 4 items and seeing within it a group of 3 or 2, leads quickly to subtraction and division.

    Thus subitizing “teaches” various mathematical concepts without ever getting into the language or construct that grownups have put on it.

    So how did we learn it?

    My initial guess was that we’ve learned to do it the same way we’ve learned to read without sounding out the words. Over the years we’ve seen and counted many hundreds of bundles of 3 things that when we see another bundle of 3 things, we know there’s 3 of them.

    Upon further research I’m now of the opinion that I learned it from Ma saying to me “there’s three of them” while I looked at a collection of three objects. I bypassed the counting and went straight to learning the cardinality – with the help of grownups.

    The cliffhanger…

    So I’ve yet to answer the question, “How do we teach subitizing?” Alas, it’s coming. (UPDATE: it’s here, complete with downloadables!)

    In the meantime, where do you observe subitizing and does it seem important?

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  • The Order of Operations Explained: Parenthesis

    The Order of Operations Explained: Parenthesis

    This is the 2nd in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

    I mentioned in the introductory article for this series that “the order of operations isn’t best practice for expressions involving variables.”

    When you involve a variable, you can’t just “do” the arithmetic. Like in the case of 2(3x + 4)=11. You can’t add 3x and 4 to get a result before moving on. You have to use the distributive property.

    And even the distributive property won’t work sometimes – as in the case of absolute values shown below.

    Parenthesis mean isolation.

    When we say “parenthesis” in the Order of Operations (OoO for short), we mean anything that’s grouped together and isolated. This could mean with actual parenthesis. This could mean [square brackets] or {curly brackets}.

    Although grownups seem to know that square and curly brackets are the equivalent of parenthesis, children don’t. This has to be said out loud.

    “Parenthesis” in the OoO can also mean |absolute value bars|. This one’s not so clear.

    When you try to solve the equation 4|2x+3|=20, you have to start with dividing by 4 to isolate the absolute value chunk. There’s not a “rule” for distributing the 4.

    (Although, it would be worth it, and fun, to see if your children can come up with some rules for distributing within absolute value bars. This would be some real mathematics at work for them – experimenting and discovering.)

    Tops and bottoms of fractions are implied parenthesis.

    It’s also the case that the numerator (top) and denominator (bottom) of a fraction are isolated places. These fall under the OoO as parenthesis.

    Check out this older video I did. It shows how this works with fractions:

    Parenthesis are for deviation from the other rules.

    David Chandler of Math without Borders commented this in the previous article of this series:

    The rule is to do higher level operations first. Use parentheses whenever your intention is to deviate from this rule.

    If you can focus on this instead of a mnemonic device, you can get students to internalize what’s going on with the OoO. It’s important, however, to make sure they remember about other bracket shapes as well as isolation.

    Let us know your tips and thoughts on the P in PEMDAS!

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  • What is Subitizing?

    What is Subitizing?

    Suzanne at Guided Math Study Group emailed me about subitizing the other day.

    I used perceptual subitizing to “know” that there were three green hair curlers here.

    Alas, I had never heard this term. So I did some research and thought I’d share what I’d learned.

    Subitizing is a way of instantly counting. In fancy math terms it would be getting to the cardinal number of a set (how many) without without going through the ordinals (counting each one’s position).

    There are two types, perceptual and conceptual. Perceptual is perceiving the number of objects immediately. Conceptual is putting a little effort into it.

    For instance, I glanced at the hot rollers Daughter had scattered on my side of the bed. I perceived there were three on the nightstand. But I had to do a little conceptual subitizing to arrive at the number of curlers on the floor (see the pictures).

    I had to mentally “chunk” these hair curlers into smaller perceivable bundles to conceptually subitize the number here.

    Here is a list of resources I found about it:

    In the next article, I’ll attempt to answer the two questions I see associated with subitizing:

    • If you do it, how did you learn to do it?
    • How do we teach children to do it?

    What do you think of these questions? Do you do it? Do you teach it?

    (By the way, it is pronounced with ooo – like “Ooo! Subitizing is neat!”)

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  • The Order of Operations Explained: Intro and Mnemonics

    The Order of Operations Explained: Intro and Mnemonics

    The Order of Operations (OoO for short) is used everywhere in mathematics because it encompasses many of the foundational rules that we’ve agreed to follow.

    Alas, students have been given the cheap and dirty version of it for years. “Here, memorize this thing about your Dear Aunt Sally!” What the heck?!

    There are subtleties in the Order of Operations that every person over the age of seven should know.

    The series begins today.

    The order of operations is a set of rules – like the drivers’ handbook for math. If everyone follows the rules, we’ll all be safe. But if someone makes a bad turn, we could be looking at a crash.

    But the Order of Operations is only a set of rules for arithmetic! It isn’t even the best practice when it comes to expressions involving a variable like x. I’ll cover what I mean in this weekly series.

    Here are the proposed articles:

    1. Intro and mnemonics
    2. Parenthesis
    3. Exponents
    4. Exponents, Multiplication and Addition
    5. Multiplication and Division
    6. Addition, Subtraction and Conclusions
    7. Exponents of Negative Numbers
    8. Another Reason to Ban PEMDAS (aka parenthesis aren’t an operation)

    Mnemonics for PEMDAS

    Well, there’s one: PEMDAS (pronounced just like it looks). That’s what the cool kids in high school always said. It was the same kids who said “soh-cah-toa” – which I thought sounded really goofy.

    And then there’s “Please Excuse My Dear Aunt Sally.” And of course “Piranhas Eat Mostly Decayed Antelope Skin.”

    What’s your way to remember it?

  • Practicing Math Skills Early in Life Is a Brain Changing Event

    Practicing Math Skills Early in Life Is a Brain Changing Event

    Thanks to Will Summer who inspired this post with a retweet to me.

    Seems researchers have previously studied the differences in math skills among children vs. adolescents and grownups. Dr. Vinod Menon with has done new research on early math training that focused on kiddos who where merely one year apart.

    Turns out that in one year of math, the brain changes quite significantly!

    You can integrate early math learning into just about anything.

    Daughter is heavy into Brown Bear, Brown Bear by Bill Martin, Jr. and Eric Carle. This is not a math book. Nor does it pretend to be. But I’ve found a way to use it to develop math skills.

    When we arrive at the page with the children, we practice counting the kids. First in one direction, then in the other – as direction matters to kids. It isn’t inherent that if you count one way you’ll get the same number as if you count the other way.

    Sometimes we’ll count the top row of children and add it to the number of the bottom row. We can practice the commutative property by adding 5 + 4 = 9 and the adding 4 + 5 = 9.

    As an alternative twist, we’ll count and add the kids on the left page to the kids on the right page. This one gets fun, because that little blond girl is half and half!

    So we can now add fractions and practice the commutative property at the same time!

    What else?

    With Dr. Menon’s research, we now know how important it is to get started early with math learning. What other things can you do to start teaching math skills early?

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  • How Adding Evens and Odds is like Multiplying Positives and Negatives

    How Adding Evens and Odds is like Multiplying Positives and Negatives

    I was playing cribbage with Husband when I found my hand to be 8, 4, 2, 2 and the cut card was also a 2. I focused hard to see if I could come up with 15. (If any group of cards add to 15, you get two points.)

    My resolve was strong. Then I remembered that for any group of numbers to add up to an odd number, one of them must be odd.

    While shuffling for the next hand, I started thinking about adding evens and odds:

    • Even + Even = Even
    • Odd + Odd = Even
    • Odd + Even = Odd

    This looked familiar. Then I remembered the rules of multiplying positives and negatives:

    • Pos x Pos = Pos
    • Neg x Neg = Pos
    • Neg x Pos = Neg

    I was so excited to notice this. Alas, I’m pretty sure I’ve noticed it before. I love rediscovering patterns as much as discovering them!

    Look for patterns everywhere.

    Always keep your eyes peeled for similarities. When you see them, if your children are old enough, use leading questions to get them to observe the same thing. If they aren’t old enough, keep a journal to share things with them later. Encourage your kids to look for patterns in everything. Use toys and manipulatives to improve this, but also use nature and everyday activities. What patterns or nifty things in math have you noticed recently?

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  • Teaching Subtraction Using a Balance

    Teaching Subtraction Using a Balance

    I love finding nifty ways to use tools for teaching math. Especially tools that aren’t supposed to teach math. Or at least the math I’m trying to get it to teach.

    I have this very cool balance that I got from Discovery Toys that would normally be a science toy. But, alas, I’m a mathematician, Jim, not a doctor. So I’ve taken the fancy science toy and turned it into a way to teach subtraction.

    You can, of course, use it to teach addition and later I’ll do a post on using it to teach multiplication and division.

    If you have children who struggle with math concepts, teaching them with hands on bits (manipulatives) sometimes helps. Here’s how to teach subtraction using a balance:

    This nifty trick can be done with any balance as long as you have weights appropriately sized. Sometimes that’s not so easy to find. Order a colorful balance that’s similar to the Discovery Toys one in video here.

    Did it work? How did your children receive this method of learning arithmetic? Please share your experience with it in the comments!