Adding or Subtracting Fractions Quickly

Adding or subtracting fractions can be done quickly by utilizing some of multiplication techniques discussed in earlier blogs.


Example:
A particular cookie recipe called for 3/4ths of a cup of sugar. When the baker wasn't looking a young child added multiple sugar packets to the dough, approximately 2/5ths of a cup worth of sugar. How much total sugar was there in the dough? 

Example illustrated:

3             2
--    +      --  =    ?
4             5


Step 1) Multiply diagonally to get the numerator. So 3 * 5 and 4 * 2
        15 + 8 = 23
        Add the two products, 15 and 8, together because it is an addition problem. If however the problem was subtracting fractions we would subtract the products instead to get the numerator.

Step 2) To get the denominator multiply the two denominators together. So 4 * 5
        4 * 5 = 20

Step 3) The sum of 3/4 and 2/5 is 23/20, reduce as necessary.

Answer: 1 and 3/20ths cups of sugar (1.15)

Social and psychological math challenges

Many students struggle with math, but there isn't a single solution to the problem. As we saw in the last post there are multiple ways to solving 95 * 95. Some great suggestions that were provided by readers:

1) HannahCitizenKane suggested to simplify the problem to one that is easier to solve; Subtracting (95 * 10)/ 2 from the product of 95 * 100.

2) Alexander suggested utilizing a square root property for numbers that add in five, similar to the post on Squares ending with five

3) Maria Miller suggested setting up the problem so that it can be solved using the FOIL (First, Outer, Inner and Last) method.

Two other methods that can be used are: 

• Numbers near 100 
• Tens are mirrors and ones sum to zero 
• Similar, but more broadly applied than the squares ending in five method

Just as there isn't one solution to solving 95 * 95 there isn't one solution in helping students with math. As the problems get more complex students need to enhance and upgrade their problem solving tools, while still utilizing the fundamental principles that worked earlier. If there are flaws in the foundational skills then there will be struggles with the more complex problem solving tools.

An important and potentially foundational aspect of why students might struggle with math is the social and psychological challenges. Some examples are:

• Math is hard / I'm not good at math 
• Math is uncool and boring / I don't want to stand out for knowing the right answer or for liking math
• Math isn't used in the real world / I am only being taught this math lesson so I can pass a test
• Most people aren't good at math / Nobody expects me to be good at math, especially if I'm a girl

What do you think are some other social and psychological challenges related to math? Any solutions that you might suggest?

How do I love thee? Let me count the ways... of solving 95 * 95

The famous quote can be borrowed on this Valentine's Day to look at the beauty of math. In life there are few problems that can only be solved one way, even though only one option may appear to us at first. It takes creativity to see potential solutions that are not obvious. If I am hiking and need water there are several potential options available besides for a body of water. Depending on your environment some creative options for getting water are cutting a green bamboo tree, melting non-salted snow or collecting morning dew with a cloth.

We can creative in solving math problems. Below are four different approaches to solving 10 * 11:
1) Add 10 to the product of 10^2
2) Use the multiply by 11 method
3) Subtract 11 from the product of 11^2
4) Double the product of 5 * 11

How many ways can we solve 95 * 95?
Reply back with how many, and which ways, you would solve the problem 95 * 95.

Multiplying Numbers Near 100

For numbers close to 100 we can utilize this method to quickly calculate the product. 

We will write our example problem vertically and solve our answer working left to right :

   96  

 * 94  

 -----

Step 1) Subtract each number from 100 and write the difference to the right of the number

            1A) 100 – 96 = 4

            1B) 100 – 94 = 6

    96    4

 *  94    6

 ----------

Step 2) Subtract the top number in the example (96) by the result of 1B (6).

96 - 6 = 90

Note you can also do the reverse, subtracting the bottom number in the example (94) by the result in 1A (4) such that 94 - 4 = 90  

The difference represents the two far left digits in the answer

                       96    4

                    *  94    6

                   ----------

Answer:       9 0  _  _

Step 3) Multiply the results of Steps 1A (4) and 1B (6)

4 * 6 = 24

The product represents the two far right digits in the answer

                       96    4

                    *  94    6

                   ----------

Answer:       9 0 2 4 

Multiplying Two Digit Numbers - Tens are mirrors and ones sum ends in zeroes

Step 1: Confirm whether the numbers you want to multiply meet both of the requirements listed. If either requirement is not met then this method can not be used.
  Requirement 1: For two digit numbers, the tens place digit needs to be the same. 72 * 82 would not work since the tens place digits (7 and 8) are not the same.
  Requirement 2: The sum of the ones place digits is equal to 10. If the sum is greater than or less than 10 then this method can not be used. 74 * 74 and 76 * 79 each would not work since the sum of their ones place digits (4+4) and (6+9) are not equal to 10.

Two examples where this method can work:
Example 1:    28 * 22  Both numbers, 28 and 22, have the same digit in the tens place, 2, and the ones place digits (8 + 2) sum to 10
Example 2:    59 * 51  Both numbers, 59 and 51, have the same digit in the tens place, 5, and the ones place digits (9 + 1) sum to 10

Step 2: Since there will not be any carryover we will solve our answer working left to right. Add 1 to the tens place digit.

Example 1:   2 + 1 = 3
Example 2:   5 + 1 = 6

Step 3: Multiply the tens place digit by the result in Step 2, the number that is one greater than the tens place digit. The product represents the two digits to the far left in the answer.

Example 1:    2 * 3 = 06    Answer 1:    0 6 _ _
Example 2:    5 * 6 = 30    Answer 2:    3 0 _ _


Step 4:  Multiply the ones place digits together. The '9' in 59 and the '1' in 51 are multiplied together in example 2. The product represents the two digits to the far right in the answer.

Example 1:   8 * 2 = 16     Answer 1:  0 6 1 6
Example 2:   9 * 1 = 09     Answer 2:  3 0 0 9

Notes:
1) We use four digits for the answer when multiplying a two digit number by a two digit number.
2) You can drop the zero if it is the left most digit in the answer. We can restate Answer 1 as 616, dropping the leading zero. We need to keep the zero if it is not the far left most digit, as shown in Answer 2 where we keep the zeroes in the answer's tens and hundreds place digits.

Squares ending in 5 - Two Digit Numbers

You can use this shortcut when multiplying a two digit number by itself that ends with a five.

Example: 75 * 75 (this can be shown as 75² and pronounced 75 squared)

Step 1) The last two digits of the answer will be 25 since the number we are squaring ends in a 5
_ _ 2 5

Step 2) To get the first two digits of the answer we need to do first have to to add one to the the non-five digit (7).
7 + 1 = 8

Step 3) Next, multiply the non-five digit by the result from Step 2.
7 * 8 = 56

Step 4) The first two digits in the answer is the product from Step 3. Our final answer is:
5 6 2 5

Multipying By 11 - Three digit numbers

We utilize the same foundation for multiplying three digit numbers by 11 as we do for two digit numbers, with one additional step. When multiplying a two digit number by 11 we calculate the middle digit in the answer by adding each of the two digits in the non-11 number together. When multiplying a three digit number by 11 we need to get an additional digit in our answer so we add another set of digits together. In two digit numbers there is only one set of digits next to each other (neighbors), whereas in three digit numbers there are two sets of pairs - (7 & 2) and (2 & 6) in the example below.

Example:   627 * 11 =  6897

Step 1) Take the digit to the far right (7) in the ones place as the first digit in our answer 
_ _ _ 7

Step 2) Add the digit in the far right (7) to the digit to its left in the tens place (7); (7 + 2) 
_ _ 9 7

Step 3) Add the digit in the tens place (2) to the digit to its left (6); (6+2)
_ 8 9 7

Step 4) Take the digit to the far left (6) as the last digit in our answer
6 8 9 7