War!(HUH!) What is it good for? (Absolutely lots in Math!)

With my family on the go so much during the non-lazy days of summer, we need easy games to entertain the tiny humans that don’t require mass amounts of attention from the adults (who are often in conversation). Enter the game of War. This versatile game can be used for all age groups and can really keep your child’s skills in arithmetic in check during the “summer slump”.

How to Play (Basic Version)

  1. Grab a deck of cards (I keep one in my purse and in the car at all times). You don’t have to, but I prefer to take out the face cards and jokers. Shuffle the rest and divvy out to all who are playing.
  2. All players shove all of their cards into a “deck” and keeps the deck face down.
  3. All players (at the same time to avoid cheating) flip the first card. The player with the largest value is the winner and takes all of the cards in the round.IMG_5434
  4. If there is a tie (that is the largest value), those players place 3 cards on their original face down and flip the fourth card. Whichever player NOW has the largest value gets all of the cards from the round. IMG_5435
  5. Continue playing until either a) one player has all of the cards; or b) you get sick of playing. The player with the most cards is the winner.

Additional Versions

  • For younger players: Use only 2-5 from the decks and play with those. The game dot cardsgoes faster and they are working only with 2, 3, 4, and 5. You can use the aces as 1. Even better, use number cards or dot cards (see below for links). Print on cardstock (4 cards per number) or go online and buy a set.
  • For any age: You can also play and whoever gets the smallest value wins. This is great for preK-1st graders!
  • For students who need review with addition: Play two cards at a time and add them. The player with the largest sum is the winner of the round.
  • For students who need review with multiplication: Play two cards at a time and multiply them. The player with the largest product is the winner of the round.
  • For grades 5-7: red cards are negative values; black cards are positive values. Flip over one card. If I have a red 6 and you have a black 2, you are the winner since positive values are always greater than negatives. IMG_5434
  • For grades 6-8 (or 7-8 if using Common Core): Play two cards and add them, using reds as negatives and blacks as positives. The player with the largest sum is the winner of the round.
  • For grades 6-8 (or 7-8 if using Common Core): Play two cards and multiply them, using reds as negatives and blacks as positives. The player with the largest product is the winner of the round.
  • For grades 6-8, use only values ace (for 1) through 5. Flip the first card; that is your base. Flip the second card; that is your exponent. The player with the highest value wins  the round.

Different Sets of Cards:

  • You could probably look on Amazon for different card types, but I love the sets at 52 Pickup. They are of high quality and there are many different types ranging from dot cards to ten frames to cards that go through the thousands (so you can work on place value!)



War!(HUH!) What is it good for? (Absolutely lots in Math!)

Why Distribute in Third Grade?

I am blessed to work with dedicated teachers who care deeply for their students and are working hard to understand the conceptual shifts CCSS brings to the table in math. One such teacher emailed me this weekend distraught, not knowing how to respond to a frustrated parent. The premise was the mother did not understand why her third grader was being asked to learn the distributive property, when she herself hadn’t learned it until Algebra I.

The arithmetic properties (commutative, associative, identities, etc.) were not created for Algebra I, though many of us didn’t learn them until then. I remember thinking that a bunch of old guys must have made them up for the sheer joy of torturing me into memorizing random stuff. No, the properties are the rules that give us the freedom to simplify math problems to make them easier to calculate while keeping the value the same.

An example. Which would be easier to solve in your head, 15 + 29 then add 5  OR  15 + 5 then add 29? The second, because we can make tens (15+5 = 20) and easily add 29 to it rather than have to “carry the 1” on the first example. This illustrates the commutative property: when adding or multiplying, I can perform that operation with any numbers in the problem first. I can switch the numbers around to make the problem easier to add (or multiply).

These properties should be celebrated as early as Kindergarten. Students do not necessarily need to know the names, but should realize through exploration that they exist and help them find their values.

Back to the distributive property. We old-timers saw it used like this: 6(5x + 2) = 30x + 12. This is not what we are asking third graders to do! Since the CCSSM standards require single digit multiplication fluency in third grade (1×1 through 9×9), it is natural to teach the distributive property at this level. This property allows me to break up one of my bigger numbers into parts. I can then multiply those smaller parts by the other factor to make it easier.

6 x 7 arrayExample: 6 x 7   This is always a toughie. Is it 48? 42? 56? I don’t know! Even if I draw an array (the above pic), that is a whole lot of dots to count!

But if I know my 5’s and 2’s, I can figure it out using the distributive property!             Break up the 7 into 5 and 2.   So now 6 x 7 becomes 6 x (5 + 2).    distribute array 6 x 7

I know 6 x 5 = 30.  (This is illustrated in yellow.)

I know 6 x 2 = 12 (This is illustrated in red.)

So 6 x (5 + 2) = 30 + 12 = 42.

distributive property 6 x 7

You may also see the distributive property used as a “number bond”. This is when it is broken up into parts, either using boxes or circles to show the parts. In this case, 7 is broken up into two parts:  5 and 2.

Utilizing the distributive property is an amazing strategy for students who struggle with memorization! They can use the distributive property to break up larger values they don’t know the multiplication facts for, until they have time to build fluency. More important, it teaches kids the value of knowing that math isn’t just memorization. There are structures and patterns that I can use. If I know the rules that govern those patterns (the properties), I can change the structure to find the value in an easier way. 

Why Distribute in Third Grade?

Taking The Distributive Property to Middle School: Making Multiplying Mixed Numbers Easier!

The last couple of blogs have highlighted the importance of the distributive property in multiplication for grades 3-8.  Let’s refresh ourselves…

Example: 6 x 9    Try it out using the distributive property! (There are many, but I have highlighted two.)

Method 1: Think about the 9 as (10 – 1).  Distribute (multiply everything in the parenthesis by) the 6.

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

Method 2: Think about 9 as (4 + 5). Distribute the 6.

6 x (4 + 5) = (6 x 4) + (6 x 5) = 24 + 30 = 54

So you may think, “Jen, where the heck are you going with this?! This is a waste of my time and my child’s time.” Yet here was my son’s homework earlier this year.

multiplying mixed number

He was to multiply 17 2/5 by 5. Most of us learned you MUST convert the mixed number (17 2/5) to an improper fraction, then “multiply tops/multiply bottoms”. Finally change the improper fraction BACK TO A MIXED NUMBER. What a stinkin’ waste of time!

My oldest had the problem done in less than a minute. He got 87. Do you see it? What did he do? I have to admit I didn’t see it at first, and had to ask him. This is what he said (please insert a very monotone voice, since he wasn’t pleased with having to tell his mother his strategy…)

I thought about the 17 2/5 as two parts. 17 + 2/5. I mutliplied 5 times 17 and then 5 times the 2/5. Then I put it back together. (Duh, mom.)

So what does this look like mathematically?

5 x 17 2/5 = 5 x (17 + 2/5)

                                                 5 x (17 + 2/5) = (5 x 17) + (5 x 2/5) = 85 + 10/5 = 85 + 2 = 87.

In this case, using the distributive property is MUCH faster than the rules we were taught! And it makes sense! I am using partial products. I break up my number into easier parts, multiply them by the given value, and put it back together.

Here is another example. This came from one of my intervention students (students who are brilliant but don’t excel with the traditional book method in mathematics…) :

Import 9.3.14 709

You can admire all of his fabulousness, but the piece I want to focus on is 9 1/11 x 7. In red, notice he rewrote it for everyone to understand. Please note he had done it in his head (the blue). All red work was written after to let his class know what he was thinking.

                                  9 1/11 x 7 = 7 x 9 1/11 (commutative property)

7 x 9 1/11 = 7 x (9 + 1/11)

                                     7 x (9 + 1/11) = (7 x 9) + (7 x 1/11) (distributive property)

                         = 63 + 7/11 = 63 7/11%

Why am I showing you this? As parents, you WILL see the distributive property used with fractions, decimals, and percents. It makes it faster to compute, as students can do most of it in their heads. Most important, when students use the distribute property, I have noticed less “silly” errors. They get it right because it makes sense!

What can you do to support this kind of thinking? Play with numbers. Use a whiteboard. Use the sidewalk with chalk. Use the bathroom mirror with a dry erase pen. Give one problem a day and see how easy it gets to think about multiplication this way.

Here are some starter problems…

7 x 6, 7 x 7, 7 x 8, 7 x 9, 7 x 12 (break the 12 into 10 and 2)

8 x 4, 8 x 6, 8 x 7, 8 x 9, 8 x 12

9 x 4, 9 x 6, 9 x 7, 9 x 8, 9 x 12

3 x 16 (break into 10 + 6), 3 x 17, 3 x 18, …You could go on forever!

2 x 4 1/2, 2 x 5 1/2, 2 x 6 1/2, ….and so on

3 x 4 1/3, 3 x 4 2/3, 3 x 5 1/3, 3 x 5 2/3, …and so on

4 x 3 1/4, 4 x 3 1/2, 4 x 3 3/4, … and so on

If you need more, just look up free worksheets that multiply a mixed number by a whole number. Use one a day!

Taking The Distributive Property to Middle School: Making Multiplying Mixed Numbers Easier!

Learning Multiplication Facts Fluently: Resources

I  was hoping to have this post up a few days ago. However, I have been deeply disappointed with the lack of resources for students to truly gain an understanding of what multiplication IS. My least favorite involved rhymes for EVERY fact that had absolutely nothing to do with multiplication! (4 door x 6 chicks= denty floor, 24. Are you kidding me????)

There are a ton of “just fact” games: give the fact, type in the product, and so on. And these are fine for students who already know their facts. But if your child is struggling to learn their multiplication facts, that is the LAST place you want to go! So I am going to provide some alternatives for you to explore.

Before I do, I want you to truly ask yourself this question. Does your child REALLY know the facts, or is she merely skip counting quickly? This question could also be asked of your first grader with addition. (Does he really know his addition facts, or is he counting by ones really fast?) When you ask your child a multiplication fact, does she tap it out, count on her fingers, or (what my middle school students would do ) bounce through the multiples? Let me give you an example: 3 x 7. Does your child bounce through (3, 6, 9, 12, 15, 18, 21…21!)? If so, she was timed too early in learning her multiplication facts. Your child should not be timed until she can use a strategy or just knows the fact within three seconds. And that is one-one thousand, two-one thousand, three-one thousand. Once they can multiply that quickly, you can go to some of these fun blast ‘em multiplication games.

So how can you help get her to three seconds? Here are three games I used with my son. I used them in this order, as they get more difficult and begin to have a sort of time crunch to them. If you have others, especially great apps, please include them in the comments section. If you add a resource, please let us know if it is to review mastery or to build strategies towards mastery. Thanks!

1. Circles and Stars. I have seen this in a number of mathematics resources, but the one I love is from Marilyn Burns (see link below). This is slightly adapted. All you need are two dice, paper and a pencil (or whiteboard and marker, which is my personal choice).

Roll two dice. The first number tells you how many circles to draw. The second number tells you how many stars to draw in EACH circle. The total number of stars is your product. Play 5 rounds. The person with the most stars at the end of 5 rounds wins.

Example: Sam rolls a 3 and a 4. He draws 3 circles, with 4 stars in each circle. There are twelve stars in all. 3 x 4 = 12.

Take it one step further. Roll two dice. You choose which number represents the circles and which number represents the number of stars in EACH circle. At some point, you want to ask your child, “Does it matter?” “Will I get the same amount of stars in the total?” The answer is yes! However, 5 circles with 2 stars in each circle looks different than 2 circles with 5 stars in each circle. Yet they both give me 10 stars in all. This is a great place to discuss the commutative property. For multiplication (and addition), when I reverse the order of my factors (the numbers I am multiplying together), I will get the same result. So if your child knows 3 x 7, but thinks he doesn’t know 7 x 3, think again!

2. Math Boggle! I loved Boggle growing up. Shake up the letters, and try to find as many words as you could in the time allotted. This helped me a great deal with learning to spell. Well, we need to play with numbers just like we play with letters. We are just going to start with one fact, and try to figure out as many strategies in the time allotted to find the product (the answer to a multiplication problem). Your kids may know this as a “Number Talk” if their teacher uses them as warm-ups. (If not, encourage them to do so!)

Example: 5 x 6. You have five minutes to find as many different ways to get the value. Go!

  1. 5 + 5 + 5 + 5 + 5 + 5= 30 (Still reliable and true, but not the fastest method.)
  2. 6 + 6 + 6 + 6 + 6 = 30 (Great to have a discussion as to which method, a or b, is more efficient and discuss the commutative property yet again.)
  3. 5 x (2 x 3) = 10 x 3 = 30 (I factored the 6 into 2 x 3. You know it as taking the prime factorization of 4. It’s actually a useful tool when working with multiplication and division.)
  4. 5 x (3 + 3) = (5 x 3) + (5 + 3) = 15 + 15 = 30. This is using the distributive property. Please see prior post for details! Also notice it is 15 twice, which could lead to the next strategy.
  5. 5 x (3 x 2) = 15 x 2 = 30. Similar to c, but using the commutative property. Also, similar to d, as I have 15 two times. This leads to a nice discussion about why we break apart the 6 instead of the 5. The number 6 is composite; it can be broken into the product of primes. 5 is prime, so it cannot be broken down into smaller whole numbers through multiplication.
  6. 5 x (4 + 2) = (5 x 4) + (5 x 2) = 20 + 10 = 30. Again, using distributive property, but I broke up the 6 differently.
  7. 5 x (5 + 1) = 25 + 5 = 30. Distributive property. Notice I stopped writing in the middle step. Once kids understand what is going on (in this case, I am multiplying the 5 times the 5 and then the 5 times the 1), they can omit this step. The goal is to get as many as you can. Further, we are eventually wanting them to do most of this in their head so they know their facts.
  8. (2 + 3) x 6 = (2 x 6) + (3 x 6) = 12 + 18 = 30. You can use distributive property by breaking up the 5, but notice the partial products you get (12 and 18) are tougher to add together in your head. Great place to talk about why we naturally break apart the 6 instead of the 5.
  9. There are many others, but I am hoping you get the idea.

You can play this all day long! Just use a whiteboard and work as a team. Take 5 minutes to work alone. Discuss all of the different strategies each of you used. Perhaps work together to find one more. Go to a different fact. Play in the car (Obviously the kids are playing, not you while driving!), while waiting for an appointment, at a restaurant. The more you play with numbers with your kids, the more fun they have, and the more important they see numbers playing a role in their lives.

3. The Product Game: http://illuminations.nctm.org/Lesson.aspx?id=5729

This is the first of four different multiplication activities your child can do with you to learn multiplication fluently. I highly recommend playing this after your child has really worked with the first two activities. The directions are included on the site as well as a pdf of the game you can print out. You are basically taking two numbers and multiplying them together to earn a spot on the grid. The first person to get four in a row (like tic-tac-toe), wins. Have a whiteboard or scratch paper handy in case your child wants to use some of their strategies to figure out the products. The other activities can be found through this link: http://illuminations.nctm.org/unit.aspx?id=6104

If you only have time or energy for one of these three activities, please use #2. It gets you the most bang for your buck. Happy number playing!!!

Teachers: For a great resource for multiplication please check out Lessons for Introducing Multiplication, Grade 3 by Marilyn Burns (2001)

Learning Multiplication Facts Fluently: Resources

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

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!