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Category Archives: Subtraction

Includes posts regarding all forms of subtraction (basic facts, multi-digit and word problems).

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.


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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