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The Language of Multiplication

Multiplication (2 groups of 3)Recently, my kindergartener, Lilly,  came home and declared, “I’m bad at times.”  Let me clarify a couple of things.  First, when she said that, she didn’t mean that she was occasionally evil.  She meant she was bad at multiplication.  Second, we don’t do, “I’m bad at ___.” in our house.  We work pretty hard to send the message that we’re all capable of most things if we only apply ourselves.  Lilly knows this, but as the youngest kid in her class, and likely her entire grade district-wide, she’s constantly comparing herself to classmates who are much older and wiser.  She’ll tell you all day long, “You just have to practice, practice, practice.”  And at times, she actually believes this.  Over spring break she was determined to skip a bar on the monkey bars and by jove, she did it.  Other times, she needs some convincing.

So your first thought at a kindergartener declaring her deficiency in multiplication might be, “Why are they doing multiplication in kindergarten?”  I assure you, they’re not (at least not formally).  This statement came from a classmate, whose older brother just went through third grade.  Third grade is the year students are supposed to know from memory (and by the end of the year) the product of all single digit numbers.  Needless to say, her classmate spent a year listening to his brother complete math homework and picked up a little – perhaps a lot – here and there.

My first thought was, “You’re not bad at times.  You’re using the wrong language.”  Most of us learned our multiplication facts from drill and practice.  We learned tricks to remember the nines and were always stumbling over the sevens and eights.  No one bothered to explain the meaning behind multiplication until after we’d memorized our facts.  Turns out, if we do the reverse, understand before we practice, memorization comes a whole lot faster.  And better.  Part of understanding is knowing what “times” means.  I sat my daughter down and explained that “times” just meant “groups of”.  We practiced this way:

Me: “2 times 3 just means 2 groups of 3.”Multiplication (4 groups of 2)

Lilly: “Oh, I know that!  That’s just 3+3.  It’s 6!”

Me: “Let’s try another one.  4 times 2.”

Lilly: “4 times 2 . . . so that’s four groups of 2.  So 2, 4, 6, 8!”

When multiplication language is understood, especially the language of operations, competency and fluency increases.  Learning multiplication requires an understanding of the language (times, factor, product, etc.) and an ability to think flexibly about numbers.  Most kids are comfortable multiplying by 1, 2, 5, and 10.  That’s because they all have had practice skip counting in these number patterns.  If they’ve had good instruction, they understand that counting by 2s is just adding 2 each time.  This connects to student understanding of multiplication as repeated addition.  The other facts anchor around knowing these facts and are based on students being able to decompose (break) numbers into parts.  There are classroom appropriate posters that are useful for parents to use as references for helping their children at home too.

Practice with multiplication strategies and work with word problems allows third graders to develop an understanding of multiplication, which in turn supports recall of “basic facts”.  Because kids have had so much practice developing a foundation of multiplication in K-2 in looking for patterns, working with repeated addition and arrays, they are able to more quickly develop fluency for multiplication.

If your little one isn’t yet learning multiplication, don’t hesitate to pose questions that support multiplicative thinking.  As I was writing this post, I took a break to put my daughter to bed.  We finished reading The Lion the Witch and the Wardrobe earlier in the day.  At bedtime, Lilly asked how many pages we’d read.  I wasn’t sure, and since I was in the middle of writing this post, I couldn’t help but respond, “I’m not sure how many pages but, if we read 4 chapters and each chapter was about 9 pages, how many pages do you think?”  She said, “So 9+9+9+9 . . .”  She understood the situation and was thinking multiplicatively.  I’m assuming, with more conversations like this, she’ll be ready to go when she gets some formal instruction in multiplication in three years!

The simplest math tool you never knew existed . . .

How are you feeling about your child’s knowledge of addition and subtraction facts?  Are you worried that he hasn’t memorized them yet. Worried that his teacher isn’t focusing on facts enough?  Well, if you’re older than 18, you probably learned math very differently than your child is learning math today.  Here are a couple of tidbits that differ greatly from math instruction in “our day” and math instruction today:

1. The Common Core State Standards for Mathematics specify that children should learn their addition and subtraction facts through strategy work, not from drill and practice.

2. The standards outline that, by the end of kindergarten, kids should “add and subtract fluently within 5” which means that they should be able to add or subtract any numbers up to 5 and get the correct answer with relative speed.  By the end of first grade kids should be able to do this within 10 and that not until the end of second grade should they “know all sums within 20 from memory.”

So how can you help your child with math homework if you don’t know how they’re supposed to learn their facts?  I’m just going to lay it out on the table – talk to your child’s teacher first if you’re not sure what to do or how to help.  Also, remember that your job is to help your child “figure it out” not to do the thinking for him.  Beyond all that, here’s an amazing tool you can create and use to help your child at home.

Ready for it?  It’s called a . . . Rekenrek.  If it’s easier, call it a number rack.  Literally translated it means arithmetic rack and it was created in the Netherlands.  It was invented a good time ago to help children understand the number system, develop number sense and improve computational fluency (learn addition and subtraction facts).

Rekenreks are easy to build.  All you need is 2 black pipe cleaners, 10 red pony beads, 10 white pony beads and a piece of foam board (a piece about 4 inches by 8 inches is plenty big).  Cut 2 slits on the short sides of the board, evenly spaced between each other and the edges of the board.  String 5 red and 5 white beads on each pipe cleaner and pull through the slits on the foam board.  It is important to make sure that the bead color lines up on both pipe cleaners.  When completed, all beads should be pushed to the right and the beads on the left should be red and the beads on the right should be white.  See the images below.

The simplest math tool you never knew existed . . .The simplest math tool you never knew existed . . .The simplest math tool you never knew existed . . .

 

These are easy to build and there is value in having your child build it (with guidance).  On a Rekenrek, each bead represents 1 so by building it himself, your child will see the embedded structure of 5 red and 5 white to make 10 and the overall structure of 10 and 10 to make a total of 20.

So here’s how the Rekenrek can help your child.  When working on facts, typically your child only sees the numeral (the symbol for the number).  Using the Rekenrek allows your child to see the quantity attached to each number.  Through practice your child will eventually come to just know these facts.  What’s more, your child is likely learning strategies for adding and subtracting.  For example, many children learn doubles  (4+4, 5+5) automatically.  Children can use this knowledge to solve near doubles.  Children recognize that in the problem 6+7, 7 can be split into a 6 and 1 to make the problem 6+6+1.  The problem with this strategy is that because children easily memorize doubles they don’t “think” about what the fact means.  So asking them to use the double to solve a close fact isn’t always intuitive.  Demonstrating this on the Rekenrek lets your child literally see the double plus 1.

 

The simplest math tool you never knew existed . . .

Work with Rekenreks support much more basic understanding of number as well.  Rekenreks help children subitize (recognize small quantities without counting) and work with composing and decomposing numbers.

The simplest math tool you never knew existed . . .

Below are links to 2 activity guides (meant for the classroom) you could use with your child to support understanding of numbers and operations.  They are free!

Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics

From the Math Learning Center – an in-depth series of lessons that builds from most basic number sense skills to addition and subtraction activities.

Mathrack Activities and Directions

From Mathematically Minded – a brief description of activities as well as assembly instructions.

Bedtime Math

My principal has started sharing all the math related articles he finds with me.  On the one hand, it means I’ve got more reading to do.  On the other, I find out about awesome resources like . . . Bedtime Math!

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So the idea is this: everyone knows about the importance of reading a bedtime story.  Why not some bedtime math problems?  This nonprofit, working in partnership with the Overdeck Foundation, has developed bedtime math problems associated with a short story.  Each story has 3 levels of problems based on age and skill.

daily math problem

 

They’ve even got a book for sale that builds on the same ideas you find on the website.

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Upload the app for bedtime math on the go!

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Perhaps my favorite at bedtimemath.org is the Crazy 8s Club.  This club is designed to meet after school for about 8 weeks for varying grade levels/ages.  Anyone can lead the club and the materials are free.  What an amazing resource!

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Restaurant Math

I was never amazing at packing enticing activities for my little one while waiting at a restaurant.  I’d always remember, right as we handed our waitress the menus, that prepared moms packed a goodie bag to keep their children occupied.  Luckily, I learned how to make due with what we had.  As my daughter hit toddlerhood, we began playing games with the sugar and jelly packets often found at breakfast diners.  From this, restaurant math was born.

The idea of this activity is to support a child’s ability to subitize (seeing a quantity without having to count it).  This is a vital skill and kids are able to subitize small quantities from a very early age.

So the idea is this . . .

Select a few different sugar or jelly packets.  Quickly scatter them in groups for your child to see.  Initially, put out one or two items and ask, “How many?”  Your child will likely be able tell you without having to count.  Here’s the most important part.  After your child tells you how many, always follow up with, “How do you see it?”  This will seem silly with just 1 or 2 packets because it’s pretty obvious, even if you’re 3!  Even so, it’s ok for your child to learn to say, “I can just see it!”  When you start to increase the quantity, it helps your child develop a  verbal pathway to explain his/her thinking about the connection between quantity and adding.  So if you put out 5 packets, your child might say, “I see 3 and 2.” or “I see 4 and 1.”  The colors you use and the arrangement you use will impact what your child sees.  When you first start increasing the quantity, separate the items with a lot of space and by color.  This will help your child “see” the groups.  As your child gets stronger at subitizing, you can put the items in one group.  Think about dice patterns.  Here are some photos of what this might look like early on.

 

"How many?" "5!" "How did you see it?" "I see 2 purple and 2 red.  That's 4 and 1 orange is 5!"

“How many?”
“5!”
“How did you see it?”
“I see 2 purple and 2 red. That’s 4 and 1 orange is 5!”

 

"How many?" "Um, 6!" "How did you see it?" "I saw 2 blue and 1 pink.  That's 3.  And 3 yellow are 6!"

“How many?”
“Um, 6!”
“How did you see it?”
“I saw 2 blue and 1 pink. That’s 3. And 3 yellow are 6!”

Even though your child may be able to see the groups, don’t expect him/her to know the total without counting.  Early on, children will need to count to determine the total.  In the picture above, your child might say, “I see 2 blue and 1 pink, that’s 3.  There are 3 yellow!  So 1, 2, 3, 4 ,5, 6!”  As children become more familiar with the different groupings, he/she will begin to recognize how smaller groups are combined to create a larger total.

 

For more information on early subitizing, read the article, Beyond Counting by Ones, by Deann Huinker.  This article is full of activities you can do with your child.

What activities have you invented to keep your kiddo occupied?  I’d love to hear about it!

Counting by Twos

Counting by twos supports a variety of other mathematical tasks.  Counting by twos allows students to count more quickly.  More importantly counting by twos supports doubling, pairs and grouping for multiplication.  Here are some activities you can do to help your child learn to count by two numbers and understand what it means to count by two.
 
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  • Take turns counting.  You say “1”, your child says “2”, you say “3”, your child says “4”, etc. through 20.  Do the counting again but this time you whisper (your child still says the number out loud).  The third time you just mouth the number but your child still says the number out loud.  This will reinforce the counting by twos skills.
  • Think of a toy that is both small and interesting, that already belongs to your child.  Legos, blocks, cars, etc.  Play a game where you set out a pile and ask your child to figure out how many.  Your child should touch two and count two as he/she pushes them away from the pile.  Also, when he/she is done counting, ask if every item had a partner.  Ask if the group was even (each item had a partner) or odd (one item was left without a partner).  Take turns selecting and then counting piles.

Single Digit Subtraction

Subtraction is always more difficult than addition.  Students inevitably struggle with this concept more so than they do with addition.  Here’s the good thing – subtraction is directly related to addition.  In fact, it is the opposite of addition.  Think back to school when you had to do fact families (creating 2 addition and 2 subtraction facts using the same 3 numbers).  That concept, in fact, is how teachers have tried to help kids understand  the relationship between addition and subtraction.  When you subtract, you are “undoing” addition.

Sometimes it’s easier to think about adding and subtracting as parts and a whole.  When adding, you know the parts (usually 2 of them) and your task is to determine the whole.  For example, my collection contains 4 red marbles and 5 blue marbles so how many is my whole collection?  Sometimes you know only one part but you also know the whole.  I have 4 red marbles and some blue marbles and my whole collection consists of 9 marbles.  You can consider subtracting the exact same way.  When subtracting, you typically know the whole, are then given a part to separate from the whole in order to determine the other part.

Success with subtraction depends on 2 key skills – understanding what the action of subtraction is (and the associated symbols in equations) and having effective strategies to solve the problems.

Subtracting is about finding the difference between 2 numbers.  Young children frequently think about subtracting as “take away”.  In fact, most children will read 4-1 as, “Four take away one.”  It is important for kids to gain understanding about the multiple meanings of subtraction.  Subtracting is also about splitting or separating a group into parts and determining the size of the parts.  For example, you have 10 coins.  If they are pennies and nickels and 4 are pennies, how many are nickels?  Finding the difference can, perhaps, be the most difficult form of subtracting.  We find the difference when measuring.  We also find the difference when comparing 2 separate groups.  Problems are frequently phrased, “How many more?” or “How many fewer?”

There are many strategies children can use to subtract.  Young children may have to represent each problem with some kind of object.  Cubes and counters are great tools for this.  linking cubes or unifix cubes can be a great tool for subtracting.  With these cubes you can link them together into a stick and then physically break it into the appropriate parts.  This tool very clearly demonstrates the “parts” and “whole” of adding and subtracting.  You can ask your child’s teacher to borrow a few cubes or purchase them online at any teaching resource site.  If children understand what the symbols mean when working with equations, and understand the action behind the symbols, cubes or counters can be excellent tools to solve unknown subtraction facts.

If kids have moved beyond the need for concrete representations, they should be using a counting strategy.  You’ll find that counting up and back are both effective strategies, depending on the specific fact.  For example, given the problem 9 – 2 = ? a child can begin at 9, count backwards 2 counts and “land” on the answer.  Counting backwards comes with a few hiccups which you can read about in the counting backwards post.  For other problems, counting up makes more sense than counting back.  Problems in which the difference between the total and the part is small (i.e. 9 – 8 = ?) do not lend themselves to counting backward.  Keeping track of 1, 2 or 3 counts is relatively simple but having to track much more than that requires a tracking device . . . otherwise known as fingers.  A more efficient strategy is for a child to start at the smaller number and count up to the larger number.  The number of counts is the difference (or answer) between the 2 numbers given.  Counting up (or counting forward) is typically easier and more accurate than counting down or counting backward.

A Good Place to Start

Before sitting down to a sheet full of subtraction problems.  Gather up some of your child’s little toys (cars, marbles, coins, etc).  Play with these items by identifying how many you have in the group, then remove some, say how many you have removed, and ask your child to determine how many are left.  You can reinforce the idea of splitting or separating by sorting the group as well.  Identify the group size, sort the items into 2 groups, tell how many are in 1 group and ask how many are in the other group.

You can also support your child’s understanding of subtracting in your daily conversations.  When running errands, talk about how many errands you will be running throughout the day.  After completing a few, ask how many tasks you have left.  When setting the table, putting groceries away, folding clothes, talk about how many you have, what you’ve finished and then ask your child to determine how many are left.

By engaging in these activities before completing subtraction equations, you are giving your child a reference.  You’ve created a real world situation for him to reference when thinking about subtracting.  If your child can associate these kinds of equations with experience, helping your child with subtraction homework will be less about you telling your child the “steps” to completing a problem, and more about him deciding what strategy to use.  Working with simple problems in this way will help your child with more difficult, multi-digit problems.  It will also increase fluency (speed) and recall.

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