Techniques for Subtracting with Ease
By the end of this lesson, it will be easy to do that in your head. Really, with a couple tricks and a little practice you can do it! Have faith!
In the last lesson, we talked about using benchmarks as resting points in addition problems involving a singledigit value. We can apply benchmarks to subtraction too.
Example 1
The benchmark value is 10 because it is the closest multiple of 10 less than 15. From 15 to 10 is 5 units. Therefore, we have subtracted 5 of the 9 needed.
We need to subtract 4 more to have taken away a total of 9.
Mentally, you can take the subtraction one step at a time: subtract 5 from 15, then 4 from the result.
Counting Up
Another popular technique for mental subtraction is counting up. In this method you begin with the smaller number and count up to the larger number, using benchmarks as needed.
Begin with 9. It takes 1 unit to get from 9 to 10.
From 10 to 15 is 5 units.
We’ve added a total of 6 units, therefore 15 – 9 = 6.
Example 2
Step 1: Begin with the smaller number and count up to the first benchmark. In this case, the smaller number is 18 and the first benchmark is 20.
Step 2: There are 10 units between 20 and 30, so add 10 to get to 30.
Step 3: Lastly, count up from 30 to 33.
How to Solve Math Problems Faster: 15 Techniques to Show Students
“Test time. No calculators.”
You’ll intimidate many students by saying this, but teaching techniques to solve math problems with ease and speed can make it less daunting.
This can also make math more rewarding. Instead of relying on calculators, students learn strategies that can improve their concentration and estimation skills while building number sense. And, while there are educators who oppose math “tricks” for valid reasons, proponents point to benefits such as increased confidence to handle difficult problems.
Here are 15 techniques to show students, helping them solve math problems faster:
Addition and Subtraction
1. TwoStep Addition
Many students struggle when learning to add integers of three digits or higher together, but changing the process’s steps can make it easier.
The first step is to add what’s easy. The second step is to add the rest.
Let’s say students must find the sum of 393 and 89. They should quickly see that adding 7 onto 393 will equal 400 — an easier number to work with. To balance the equation, they can then subtract 7 from 89.
Broken down, the process is:
 393 + 89
 (393 + 7) + (89 – 7)
 400 + 82
 482
With this fast technique, big numbers won’t look as scary now.
2. TwoStep Subtraction
There’s a similar method for subtraction.
Remove what’s easy. Then remove what’s left.
Suppose students must find the difference of 567 and 153. Most will feel that 500 is a simpler number than 567. So, they just have to take away 67 from the minuend — 567 — and the subtrahend — 153 — before solving the equation.
Here’s the process:
 567 – 153
 (567 – 67) – (153 – 67)
 500 – 86
 414
Instead of two complex numbers, students will only have to tackle one.
By making math engaging, students that use Prodigy consistently outperform those that don’t on standardized assessments
3. Subtracting from 1,000
You can give students confidence to handle fourdigit integers with this fast technique.
To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.
Let’s say students must solve 1,000 – 438. Here are the steps:
 9 – 4 = 5
 9 – 3 = 6
 10 – 8 = 2
 562
This also applies to 10,000, 100,000 and other integers that follow this pattern.
Multiplication and Division
4. Doubling and Halving
When students have to multiply two integers, they can speed up the process when one is an even number. They just need to halve the even number and double the other number.
Students can stop the process when they can no longer halve the even integer, or when the equation becomes manageable.
Using 33 x 48 as an example, here’s the process:
 66 x 24
 132 x 12
 264 x 6
 528 x 3
 1,584
The only prerequisite is understanding the 2 times table.
5. Multiplying by Powers of 2
 This tactic is a speedy variation of doubling and halving.
 It simplifies multiplication if a number in the equation is a power of 2, meaning it works for 2, 4, 8, 16 and so on.
 Here’s what to do: For each power of 2 that makes up that number, double the other number.
 For example, 9 x 16 is the same thing as 9 x (2 x 2 x 2 x 2) or 9 x 24. Students can therefore double 9 four times to reach the answer:
 9 x 24
 18 x 23
 36 x 22
 72 x 2
 144
10 Tricks for Doing Math Quickly in Your Head
You don’t have to be a math teacher to know that a lot of students—and likely a lot of parents (it’s been awhile!)—are intimidated by math problems, especially if they involve large numbers. Learning techniques on how to do math quickly can help students develop greater confidence in math, improve math skills and understanding, and excel in advanced courses.
If it’s your job to teach those, here’s a great refresher.
10 tricks for doing fast math
Here are 10 fast math strategies students (and adults!) can use to do math in their heads. Once these strategies are mastered, students should be able to accurately and confidently solve math problems that they once feared solving.
1. Adding large numbers
Adding large numbers just in your head can be difficult. This method shows how to simplify this process by making all the numbers a multiple of 10. Here is an example:
644 + 238
While these numbers are hard to contend with, rounding them up will make them more manageable. So, 644 becomes 650 and 238 becomes 240.
Now, add 650 and 240 together. The total is 890. To find the answer to the original equation, it must be determined how much we added to the numbers to round them up.
 650 – 644 = 6 and 240 – 238 = 2
 Now, add 6 and 2 together for a total of 8
 To find the answer to the original equation, 8 must be subtracted from the 890.
 890 – 8 = 882
 So the answer to 644 +238 is 882.
2. Subtracting from 1,000
 Here’s a basic rule to subtract a large number from 1,000: Subtract every number except the last from 9 and subtract the final number from 10
 For example:
 1,000 – 556
 Step 1: Subtract 5 from 9 = 4
 Step 2: Subtract 5 from 9 = 4
 Step 3: Subtract 6 from 10 = 4
 The answer is 444.
3. Multiplying 5 times any number
When multiplying the number 5 by an even number, there is a quick way to find the answer.
For example, 5 x 4 =
 Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 4 become the number 2.
 Step 2: Add a zero to the number to find the answer. In this case, the answer is 20.
 5 x 4 = 20
 When multiplying an odd number times 5, the formula is a bit different.
 For instance, consider 5 x 3.
 Step 1: Subtract one from the number being multiplied by 5, in this instance the number 3 becomes the number 2.
 Step 2: Now halve the number 2, which makes it the number 1. Make 5 the last digit. The number produced is 15, which is the answer.
5 x 3 = 15
4. Division tricks
Here’s a quick way to know when a number can be evenly divided by these certain numbers:
 10 if the number ends in 0
 9 when the digits are added together and the total is evenly divisible by 9
 8 if the last three digits are evenly divisible by 8 or are 000
 6 if it is an even number and when the digits are added together the answer is evenly divisible by 3
 5 if it ends in a 0 or 5
 4 if it ends in 00 or a two digit number that is evenly divisible by 4
 3 when the digits are added together and the result is evenly divisible by the number 3
 2 if it ends in 0, 2, 4, 6, or 8
5. Multiplying by 9
 This is an easy method that is helpful for multiplying any number by 9. Here is how it works:
 Let’s use the example of 9 x 3.
 Step 1: Subtract 1 from the number that is being multiplied by 9.
 3 – 1 = 2
 The number 2 is the first number in the answer to the equation.
 Step 2: Subtract that number from the number 9.
 9 – 2 = 7
 The number 7 is the second number in the answer to the equation.
 So, 9 x 3 = 27
6. 10 and 11 times tricks
The trick to multiplying any number by 10 is to add a zero to the end of the number. For example, 62 x 10 = 620.
There is also an easy trick for multiplying any twodigit number by 11. Here it is:
11 x 25
Take the original twodigit number and put a space between the digits. In this example, that number is 25.
 2_5
 Now add those two numbers together and put the result in the center:
 2_(2 + 5)_5
 2_7_5
 The answer to 11 x 25 is 275.
 If the numbers in the center add up to a number with two digits, insert the second number and add 1 to the first one. Here is an example for the equation 11 x 88
 8_(8 +8)_8
 (8 + 1)_6_8
 9_6_8
 There is the answer to 11 x 88: 968
7. Percentage
Finding a percentage of a number can be somewhat tricky, but thinking about it in the right terms makes it much easier to understand. For instance, to find out what 5% of 235 is, follow this method:
 Step 1: Move the decimal point over by one place, 235 becomes 23.5.
 Step 2: Divide 23.5 by the number 2, the answer is 11.75. That is also the answer to the original equation.
8. Quickly square a twodigit number that ends in 5
Let’s use the number 35 as an example.
 Step 1: Multiply the first digit by itself plus 1.
 Step 2: Put a 25 at the end.
 35 squared = [3 x (3 + 1)] & 25
 [3 x (3 + 1)] = 12
 12 & 25 = 1225
 35 squared = 1225
9. Tough multiplication
When multiplying large numbers, if one of the numbers is even, divide the first number in half, and then double the second number. This method will solve the problem quickly. For instance, consider
20 x 120
Step 1: Divide the 20 by 2, which equals 10. Double 120, which equals 240.
 Then multiply your two answers together.
 10 x 240 = 2400
 The answer to 20 x 120 is 2,400.
10. Multiplying numbers that end in zero
Multiplying numbers that end in zero is actually quite simple. It involves multiplying the other numbers together and then adding the zeros at the end. For instance, consider:
 200 x 400
 Step 1: Multiply the 2 times the 4
 2 x 4 = 8
 Step 2: Put all four of the zeros after the 8
 80,000
 200 x 400= 80,000
 Practicing these fast math tricks can help both students and teachers improve their math skills and become secure in their knowledge of mathematics—and unafraid to work with numbers in the future.
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Your Complete Guide to Teaching Your Child the Subtraction Facts
Everything you need to know to teach your child the subtraction facts, without hours of rote memorization, counting on fingers, or flash cards.
My rookie teacher mistake
When I was a brandnew teacher, I devoted weeks to making sure that all my fifthgrader students fully mastered the addition facts.
I knew that the addition facts were an essential foundation, and that my students would never feel confident in math without them.
But I didn’t spend a single day reviewing the subtraction facts. I figured that once my students knew the addition facts, they’d be able to figure out subtraction.
I was wrong.
All year long, the subtractionstrugglers had trouble anytime we hit a topic involving subtraction. Long division. Decimals. Fractions. For each of these topics, my students spent so much effort figuring out basic subtractions they didn’t have much mental energy left over for learning the new concepts.
So why couldn’t my students readily apply their addition fact knowledge to figure out subtraction facts. After all, if they knew 9 + 5 = 14, shouldn’t they also know that 14 – 9 = 5?
What I learned
I was making two mistakes: one about subtraction, and one about how kids think.
First, I was assuming that related addition facts are always the best way to figure out subtraction facts. That’s true of some of the subtraction facts, but often a different thinking strategy works better.
Fast Arithmetic Tips
 Addition of 5 When adding 5 to a digit greater than 5, it is easier to first subtract 5 and then add 10.
For example,
7 + 5 = 12.Also 7 – 5 = 2; 2 + 10 = 12.
 Subtraction of 5 When subtracting 5 from a number ending with a a digit smaller than 5, it is easier to first add 5 and then subtract 10.
For example,
23 – 5 = 18.Also 23 + 5 = 28; 28 – 10 = 18.
 Division by 5 Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.
 For example,
 1375/5 = 2750/10 = 275.
 More examples and explanation
 Multiplication by 5 It's often more convenient instead of multiplying by 5 to multiply first by 10 and then divide by 2.
 For example,
 137×5 = 1370/2 = 685.
 More examples and explanation
 Division by 5 Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.
 For example,
 1375/5 = 2750/10 = 275.
 More examples and explanation
 Division/multiplication by 4 Replace either with a repeated operation by 2.
For example,
124/4 = 62/2 = 31. Also, 124×4 = 248×2 = 496.
 Division/multiplication by 25 Use operations with 4 instead.
 For example,
 37×25 = 3700/4 = 1850/2 = 925.
 More examples and explanation
 Division/multiplication by 8 Replace either with a repeated operation by 2.
For example,
124×8 = 248×4 = 496×2 = 992.
 Division/multiplication by 125 Use operations with 8 instead.
For example,
37×125 = 37000/8 = 18500/4 = 9250/2 = 4625.
 Squaring two digit numbers.

You should memorize the first 25 squares:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 4 9 16 25 36 49 64 81 100 121 144 169 196 15 16 17 18 19 20 21 22 23 24 25 225 256 289 324 361 400 441 484 529 576 625 
If you forgot an entry. Say, you want a square of 13. Do this: add 3 (the last digit) to 13 (the number to be squared) to get 16 = 13 + 3. Square the last digit: 3² = 9. Append the result to the sum: 169.
7 practical tips for mental math (that ANYONE can use!)
You have most likely heard about mental math — the ability to do calculations in one's head — and how important it is for children to learn it.
But why is it important? Because mental math relates to NUMBER SENSE: the ability to manipulate numbers in one's head in various ways in order to do calculations. And number sense, in return, has been proven to predict a student's success in algebra.
Essentially, what we do with variables in algebra is the same as what students can learn do with numbers in the lower grades.
People with number sense use numbers flexibly. They are able to take them apart and put them together in various ways in order to do calculations. It is quite similar to being able to “PLAY” with words in order to make interesting sentences, or being able to play with chords and melodies in order to make songs.
But mental math/number sense isn't just for “math whizzes” — quite the contrary! EVERYONE can learn the basics of it, and it will make learning math and algebra so much easier! We expect our children to learn lots of English words and to be able to put those words together in many different ways to form sentences, so why not expect them to do the same with numbers? And they can, as long as they are shown the basics and shown examples of how it happens. So let's get on to the practical part of this writing: mental math strategies for EVERYONE.
 The “9trick”.
To add 9 to any number, first add 10, and then subtract 1. In my Math Mammoth books, I give children this storyline where nine really badly wants to be 10… so, it asks this other number for “one”. The other number then becomes one less. For example, we change the addition 9 + 7 to 10 + 6, which is much easier to solve.
But this “trick” expands. Can you think of an easy way to add 76 + 99? Change it to 75 + 100. How about 385 + 999?
How would you add 39 + 28 in your head? Let 39 become 40… which reduces 28 to 27. The addition is now 40 + 27. Yet another way is by thinking of compensation: 39 is one less than 40, and 28 is two less than 30. So, their sum is three less than 70.
 Doubles + 1.
Encourage children to memorize the doubles from 1 + 1 through 9 + 9. After that, a whole lot of other addition facts are at their fingertips: the ones we can term “doubles plus one more”. For example, 5 + 6 is just one more than 5 + 5, or 9 + 8 is just one more than 8 + 8.
 Use addition facts when adding bigger numbers.
Once you know that 7 + 8 = 15, then you will also be able to do all these additions in your head:
 70 + 80 is 15 tens, or 150
 700 + 800 is 15 hundreds, or 1500
 27 + 8 is 20 and 15, which is 35. Or, think this way: since 7 + 8 is five more than ten, then 27 + 8 is five more than the next ten.
 Subtract by adding.
How to Subtract
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1
Write down the larger number. Let's say you're working with the problem 32 – 17. Write down 32 first.

2
Write the smaller number directly below it. Make sure that you line up the tens and ones columns, so that the 3 in “32” is directly above the 1 in “17” and that the 2 in “32” is directly above the “7” in 17.

3
Subtract the number in the ones column of the bottom number from the number in the ones column of the top number. Now, this can get a little bit tricky when the bottom number is larger than the top number. In this case, 7 is larger than 2. Here's what you need to do:[1]
 You'll need to “borrow” from the 3 in “32” (also known as regrouping), in order to turn that 2 into a 12.
 Cross off the 3 in “32” and make it a 2, while making the 2 a 12.
 Now, you have 12 – 7, which is equal to 5. Write a 5 below the two numbers you subtracted, so it lines up with the ones column in a new row.

4
Subtract the number in the tens column of the bottom number from the number in the tens column of the top number. Remember that your 3 is now a 2. Now, subtract the 1 in 17 from the 2 above it to get (21) 1. Write 1 below the numbers in the tens columns, to the left of the 5 in the ones column of the answer. You should have written 15. This means that 32 – 17 = 15.

5
Check your work. If you want to be sure that you correctly subtracted the two numbers, then all you have to do is to add the answer to the smaller number to make sure that you get the larger number. In this case, you should add your answer, 15, to the smaller number in the subtraction number, 17. 15 + 17 = 32, so you've done your work correctly. Well done!

1
Identify which number is larger.
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