parent support

Fluency in Multiplication: Why Your Child Needs To Know How to Distribute!

Jen Moffett, , , 4 Comments

This is the third in a series regarding fluency of basic facts. For the introduction to this series, please read Fluency and Memorization Are Not Synonyms!!!

I am a believer that all students can be successful in understanding mathematics. Maybe not memorizing erronerous facts, but truly understanding what the numbers mean and how they relate to one another. One of my favorite interviews to do with my sixth graders is on multiplication facts. You may want to ask your child this question as well…

If I know that 7 x 6 is 42, how does that help me figure out what 7 x 7 is?

Most students will say “49”, but will not find the relationship between the two facts. Here are the common answers for my sixth graders:

1. “I just know it.”

2. (Counting each one on his/her fingers) “7, 14, 21, 28, (usually begins counting by ones), 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 49…7 x 7 is 49.”

3. Gets the answer wrong, or says he/she does not know that one.

Now don’t get me wrong, I am appreciative when a student “just knows it”, because it usually makes her life easier. The middle school years are ALLLL about multiplicative thinking and proportional reasoning. If you do not know your multiplication facts fluently, you spend sooo much time figuring out the fact (such as student #2) you get lost in the actual work you are trying to do. This leads to frustration, feelings of failure, and a negative attitude towards mathematics.

The answer I am looking for is similar to the following.

“I can just add one more group of 7 to 42 and get 49.”

Why is this important? Because many children do not see multiplication of whole numbers as simply a certain number of equal groups. 7 x 6 means that I either have 7 groups of 6 OR 6 groups of 7 (Commutative Property. Please see my blog on fluency with addition facts.) So if I have 7 x 7, I have one more group of 7 than I did before.

Mathematically, it looks like this: 7 x 7 =(7 x 6) + (7 x 1) = 42 (the fact I was given) + 7 = 49.

This type of strategy, doing your multiplication in parts, is often called in elementary books the partial products. In middle school, this particular set of partial products is called the distributive property. It means you are going to distribute (pass out) a number, multiplying it by everything it has a relationship to. This is not new. We used this in Algebra back in the day. But to use it to help with multiplication facts is a novel idea for most, though it brings understanding to what they are learning.

Here is another example: 6 x 9. (One of the most commonly missed facts.)

Let’s say I do not know my 9’s. However, I do know my x 10’s and my x 1’s. I can think about multiplying 6 by 9 as multiplying 6 by one less than 10, since 10 – 1 = 9.

6 x 9 = 6 x (10 – 1) (true)

So now I need to know how much 6 x 10 is and 6 x 1 is. (You are distributing the 6.) Those are easy facts for me!

6 x (10 – 1) = (6 x 10) – (6 x 1) = 60 – 6 = 54.

I can do this with any of the facts. Say your student struggles with 7’s. However, they know their 2’s and 5’s. They can break the 7 into 2 + 5 and use the distributive property.

4 x 7 = 4 x (2 + 5)

Now multiply 4 by the 2 and 4 by the 5 (You are distributing the 4.):

4 x (2 + 5) = (4 x 2) + (4 x 5) = 8 + 20 =28. Trust me, it looks a lot freakier on paper than just doing the partial products in your head!

Here is one for you to try: 8 x 7. Try it on paper before sneaking a peak!

There are several options. Here are a few. If yours isn’t here, please add it in the comments for others to see!

Option 1 (what I do): Break apart the 7 into 2 and 5.

8 x 7 = 8 x (2 + 5) = 16 + 40 = 56.

Option 2: Break apart the 8 into 4 and 4.

 8 x 7 = (4 + 4) x 7 = 28 + 28 = 56.

Option 3 (what my son does): Break apart the 8 into 10 – 2.

8 x 7 = (10 – 2) x 7 = 70 – 14 = 56.

There is no counting on your fingers, no tapping of feet or bopping of heads (which is what your child does to keep count if they don’t have the facts fluent yet). They just break up one of the factors (one of the numbers you are multiplying together) and multiply by parts. I really only need to know my 1’s, 2’s, 5’s and 10’s. Every other factor can be broken into these parts. It takes the number of multiplication facts to memorize from 100 (if you count 10’s) to 36.

Here’s why you really want to use this strategy with your child. If you are reading this, I bet your child is struggling with memorizing the facts. This strategy isn’t drill and kill. It isn’t a slower method that makes your child feel less than intelligent. It is actually a more sophisticated way to do the math, as it shows where the relationships are among the numbers. It is algebraic thinking in an arithmetic expression. The more your child “plays around” with numbers, the more relationships he/she will see, and the easier algebraic thinking will become. And yes, most (after using the distributive property over and over) will find that the facts will become automatic over time.

Let’s show a problem with a multi-digit number. (Another favorite of mine to use when interviewing children.) 6 x 199. Try it out. How would you solve it?

I just did it in my head. And no, I am not brilliant! Just using relationships to make the problem easier!

6 x 199 is the same as 6 x (200 – 1) = 1200 – 6 = 1194. Sooo much easier than trying to carry and multiply a bunch of numbers by 9!!!

I have to end this with a final thought. I have two dear friends that blatantly state they do not have their multiplication facts memorized. “It’s a waste of necessary brain space when all I need to know are my simple facts and build up the others in my head”, claimed the first. “Why do I need them memorized when I can figure it out quickly using partial products???” stated the second. What to know what their occupations are? The first is a mathematician and professor. The second is a statistician and psychologist. Hmmmm….

Next Blog: On-line resources and games/activities to build multiplication fluency!!!

Fluency in Multiplication: Why Your Child Needs To Know How to Distribute!

Basic Addition Facts For Fluency: Beyond Flash Cards!!!

Jen Moffett, , , Leave a comment

This is the second of a series of blogs regarding fluency of basic facts.

Memorizing any operational facts can be easy for some (who have photographic memories) and difficult for others. I tried and tried every type of “flash card” with my oldest and all it did was provide frustration, tears, and dislike for the math. We tried flash cards, memory games, on-line shoot the sum games. And none of them worked. I freaked out, knowing that if he didn’t get these facts down, it would be an uphill battle. But is it really? Do we need to focus on memorization or understanding???

This blog is about understanding and a teeny tiny bit of memorization. Please feel free to disagree, as this is a blog after all and participation is encouraged! However, at least read on before fighting the fight for memorizing only.

Depending on whether you live in a Common Core state or not, typically we want children to know fluently all of their facts to five by the end of Kindergarten and all of their single digit facts (10 + 10 is cool too) by the end of grade 1. Note I wrote “fluently” versus “to automaticity”. So what’s the difference? It means I get three whole seconds to figure it out. I may not be able to see the answer in 0.5 seconds, but using some strategic thinking I can get it. That’s the goal. Anything beyond is cool, but this is for all children.

Fun fact: If all you are going to do is memorize the sums for two single digit numbers, the amount of facts you need to memorize is 81. 81! That is a LOT! Have fun making 81 flashcards! No thank you!

So let’s dwindle the amount to memorize. Below are two strategies to teach your kiddos to make that 81 significantly less. (There are others, but if you focus on these two you are waaay ahead of the game!)

  1. Commutativity. This is a property (a rule that ALWAYS works in mathematics) of equality. Think about the distance you drive to work. And the distance you drive home. Now if you take the same route both times, you go the same distance. No matter if you start from home, or you start from work, your commute distance is the same. Commutativity works the same for numbers. I can add 3 + 5 or 5+3 and my result (the sum) is the same, 8. So if I use my commutative property, I now only have to memorize one of the two facts! I am now down to 45 facts! (You may think, “Hey, why isn’t it half of 81?” Well, the doubles such as 2 + 2 and 3 + 3 do not numerically have a commutative fact, so you are stuck with them.)
  2. Making 10. This is a big stinking deal. Your kiddos need to get under their belt the ways to make ten. (1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5 and the reversals based on strategy #1.) Why? For starters, we live in a base ten world. Once you hit 9, the next number literally moves over one spot to the left and now you have a double digit value. Once you hit 99, you move over one spot to the left and you now have a triple digit. And so on. Really though, let’s be practical. Is it easier to add 9 + 8 or 10 + 7? 10 + 7, because you can add the 0 + 7 easily. So if I can manipulate my digits to make a 10 +, I am good to go. Example: 9 + 6. I know that I only need one more to make 9 a 10. So I am going to take it from the 6. 9 + 6 = 10 + 5 = 15. If I know how to make a 10 + expression from the addition fact I am working with, I now go from 45 memorization facts to 25!

What are those 25 facts? 1 + 1, 2 + 1, 3 + 1, 4 + 1, 5 + 1, 6 + 1, 7 + 1, 8 + 1, 9 + 1, 2 +2, 3 + 2, 4 + 2, 5 + 2, 6 + 2, 7 + 2, 8 + 2, 3 + 3, 4 + 3, 5 + 3, 6 + 3, 7 + 3, 4 + 4, 5 + 4, 6 + 4, 5 + 5 (Notice the largest sums are 10, because all others can be made using a 10 + strategy.) So could your child learn these facts? ABSOLUTELY! Just remember to include the others for them to practice their two cool strategies with!

Next Blog: Fun games to play to work on making ten!

HUGE NOTE: Please do not time your children. For many, it stresses them out and they lose focus. Also, if you time children before they master the strategies, it will encourage them to count on their fingers. Though that may work for these small values, it will not help them in the long run when adding much larger numbers.

Basic Addition Facts For Fluency: Beyond Flash Cards!!!

Which Is Bigger? Comparing Objects Without Quantifying

Jen Moffett, , Leave a comment

One of my threenager’s (yes, I said “threenager”) favorite things to do is go to the zoo. Particularly to see the alligators and caimans. As he watches these creatures I have a habit of asking, “Which is bigger?” or “Which is smaller?” These characterizations, called attributes, are often the first way toddlers learn to compare objects. From there, they begin to quantify (How much bigger? or How much smaller?) and realize there is a need for numbers when comparing.

Yet all too often, our smart little toddlers overgeneralize and decide that EVERYTHING has an attribute of “big” (or “small”). For example, my little will tell his older brother, “You are bigger than me.” But what does he mean? Does he mean he weighs more? Is taller? Is older? Has more body hair? Each of these measures has a different attribute. In fact, Vygotsky stated that the way in which we talk about quantities and the relations between them can have a strong impact on how children think about quantities. And quantifying each of these attributes requires a different type of measurement.

My point? How we communicate with our littles makes a HUGE difference in how they look at comparing and relating objects. Not everything is bigger or smaller. Start using different attributes and introduce them to a wonderful world of vocab that will help them articulate exactly what they want to compare. For example, you are at the grocery store getting apples. Yes, have her point out the colors. Yes, have her count as you put the apples in the bag. But take it one step further. As you are waiting in line, have her hold the bag of apples versus the bag of green beans and decide which is heavier. Or which has more items. Which item is longer. Which item is wider. You get my drift.

Mathematics is not just about numbers. It also relies heavily on one’s ability to compare, estimate and see relationships, with or without numbers. Number is simply an extension of more basic ideas about relationships between quantities. Even if your child doesn’t know their numbers, you can start comparing objects NOW.

Great reference for teachers of Pre-K and Kinder:Developing Essential Understanding of Number and Numeration, NCTM (2011).

http://www.nctm.org/store/Products/Developing-Essential-Understanding-of-Number-and-Numeration-for-Teaching-Mathematics-in-Pre-K-2/

Which Is Bigger? Comparing Objects Without Quantifying

Get Ready To LOVE Mathematics!!!

Jen Moffett, 8 Comments

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.  ~S. Gudder

My first math post. Well, actually, it is my seventh math post of the night, but the first one to go public. The others started with very colorful language expressing my fear and nervousness. (And those of you who actually know me, can hear me saying all of that right now!)

A little bit about me. My name is Jen Moffett, and I love math. I am actually addicted to mathematics. I read mathematical journals and books, dream about math, see mathematics in everyday life, and try to get everyone around me to love and appreciate it as much as I do. I have been in education for 20 years. In that time, I have taught elementary and middle school mathematics, worked at the county level as a curriculum specialist, and provided professional development to teachers and parents on a variety of math related ideas and concepts.

Recently, I was given the “gift” of being able to stay home with my littles. (You will hear about them more in later posts, as they are my guinea pigs for math strategies!) Emotionally rewarding, yet mentally I am needing more. I need to be teaching!!! It fulfills me in a way nothing else can. I have pimped my way into some classes through Skype, but it is still not the same. So what to do…what to do…

Yesterday, two public discussions arose about the mathematics children were bringing home, and the frustration (from the parent and child) in doing the homework. Or not doing the homework, because, really, what the heck are they asking my child to do??? These heated debates are not uncommon. Facebook has rant after rant about the struggles parents and children are feeling regarding the “new way” of doing math (Which is not new, but not the time to discuss.). I left both conversations feeling their pain. An idea sparked. I am pretty sure a lightbulb appeared above my head, but it was 1 am and no one was up to prove this.

What if I could support parents (and therefore students) through a blog?

So here it is. I am creating a blog to help you, my fellow parents. It will not be sequential, but I will provide tags so you can search for what you need. When I use great resources, I will reference them so you can check them out. There will be tasks for us to use to explore the difficult concepts, and the research supporting the strategies that may appear different than what we learned in school. Most important, I want to help you understand the “why” behind the mathematical strategies, and for you to gain an appreciation and a love (or a like for you haters) for mathematics.

Happy (math) reading!

Get Ready To LOVE Mathematics!!!