Problem Solving, Sets, Addition, & Subtraction
Methods for Solving Addition
Give and Take
Using the associative and/or commutative property of addition, split up one number to make it ‘nice’.
34+28=(32+2)+28=32+(2+28)=32+30=62
Compensating
Basically, one number is rounded to a ‘nice’ number to make the problem easier and then the excess is taken off or added back on in a second equation.
34+28=? 34+30=64 64-2=62
Through a child’s eyes: “I added 30 to 34 and got 64. Then I took 2 away and got 62”.
Decomposing
Using a number line, split up one of the numbers into ‘nice’ numbers and then keep adding to or with nice numbers until the desired number is reached. For 34+28, you could choose to split 28 up into a 20, 6, and 2 or 6, 10, 10, and 2, and so on.
Through child’s eyes: “I did 34+6 is 40. Then I added 10 to get to 50. Then I added 10 to get to 60. Then I added 2 to get to 62…and that’s my number”.
Traditional
Stack the numbers, add the ones first and carry over to the tens if the total is 10 or more.
Partial Sums/Instructional Algorithm
You add the ones and then the tens, or vice versa and then add those two totals together.
34+28=50+12=62
Example of what a student might say: “30 and 20 is 50. 4 and 8 is 12. Now I can do the 50+12 and get 62”.
Maturation of Thinking about Addition
(4 being the highest level)
1. Direct modeling: physical representation
2. Counting on or back with fingers
3. Derived Facts: using what you know to help find an answer to a problem. Ex. 7+9=6+(1+9), it makes it easier to solve.
4. Fact/memorize: good to have, but not as useful as derived facts
Properties
Identity Property of Zero
When adding 0, the identity of the number stays the same.
Ex. a+0 = a
Associative Property
Can associate or group numbers together in different ways; think of as give and take
Ex. a+(b+c) = (a+b)+c; (8+7)+3 = 8+(7+3)
Commutitive Property
When adding, the order of the numbers does not matter.
Ex. a+b = b+a; 8+7+2+3 = 8+2+7+3
Models
Measurement/Number Line/Active
The word problem has something happen, there is a verb involved that indicates what needs to be done with the problem. This way is easier for a student to understand. Ex. Eroll has 7 chocolate chip cookies. He bakes 8 more, how many cookies does he have now?
Set Model
A word problem is presented in a way that makes it seem like there are two separate sets that do not necessarily have to do with one another. Ex. Eroll has 7 chocolate chip cookies and 8 oatmeal cookies. How many cookies does he have altogether?
Number Relationships
Part-Part-Whole
To conceptualize a number as being made up of two or more parts is the most important relationship to develop. To understand why things are broken up each time in the way they are; it is not random but to make the problem nicer.
Ex. 8+6=8+(2+4)=(8+2)+4
Benchmarks of 5 and 10
Since 10 plays such an important role in our number system (and that two 5’s makes 10), students must know how numbers relate to 5 and 10.
One and Two, More and Less
This is not the ability to count on two or back two, but rather knowing which numbers are two more or two less than any given number. It is about making the connection that addition and subtraction are the same as counting.
Spatial Relatinoships
Recognizing how many there is without counting by seeing the visual pattern. Cannot recognize more than 5 in chunks. It is a very abstract way of thinking about things.
Ex. 7 is just a symbol not a visual pattern.
Sets
Universal Set
All of the elements that are represented. This would include set A, set B, and everything that may not fit nicely into these sets.
Complement
This refers to everything that is NOT in the
set that is represented within the universal set. This is shown by a certain element having a line over it. This couldbe the entire union or intersection or either set A or set B.
Intersection
An intersection is shown as A ∩ B. This is
represented by what set A AND set B have
in common.
Union
A union is shown as A U B. This means
that all of the contents of set A OR set B are
represented. It is the 'union' of the two sets.
Bases
We are used to adding and subtracting with a base 10, to add or subtract in a different base the same idea is applied. Whatever the base is set to, let's say 6, none of the numbers within the equation can be greater than or equal to that number. If you are using base 6, all of the numbers involved would be less than 6, just like all of the numbers in base 10 are less than 10.
Addition
When adding, anytime the base number is reached
a number (most likely 1) must be carried over into the
next column. For example, if you are adding 3+2 in a base
4, you would end up with 11. Since you get 5, which is larger than 4, 4 must be taken away and moved as a 1, since you have base 4, to the next one over leaving you with a 1 in the ones column and a 1 in the tens column.
When subtracting in an unusual base, the main difference that you will see is when borrowing takes place. If you are working in a base 7 and have to borrow from the next column over, you would be borrowing 7. This is easy to imagine when you think about how this compares to base 10; when you borrow using base ten, you are borrowing ten of something. It is the exact same thing just with a different maximum number.
Problem Solving
The progression in which a child
develops the ability to understand
as well as solve a problem.
3. Abstract
-Represent the problem using symbols on paper.
More commonly recognized as alegbra. To get an equation, it helps to think about the different numbers in what stays the same(constants) and what changes (variables).
2. Representational
-This may look like drawing on a piece of paper; a visual aid to help solve the problem.
1. Concrete
-Using 3D, physical objects to help visualize and solve a problem; using your hands.
Main topic
Subtraction
Word Problems/Setup
Number Line/Measurement/Distance
Setting up a word problem so that a distance is always involved. Ex. Eroll hiked up the mountain trail 8 miles. 5 of these miles were hiked after lunch. How many miles did Eroll hike before lunch?
Comparison
Comparing two sets to see the difference Ex. Eroll has $8 and Kylie has $5. How much more does Eroll have than Kylie.
Take Away
When a word problem is set up to say that something needs to be taken away from a number. Ex. Eroll has $8 and he spends $5 for a movie ticket. How much money does
Missing Addend
The word problem is subtraction, but set up in such way that it is thought of as an addition problem with something missing. Ex. Eroll had $5. He found some more lying on the ground. Now he has $8, how much money did he find?
Strategies
Using a Number Line
-You can use 'friendly numbers' to add up to or
take away from in order to reach what the
problem was asking for.
Stacking and Borrowing
-The old-fashioned idea of how to subtract. If
things do not line up borrowing must take place, further complicating things.
Tens Place, Then the Ones Place
-If the subtraction involves double digits,then
the tens place is taken away first, followed by
the ones place, then combined back together.
Rounding
-Round to a 'friendly number',
then adjust the outcome accordingly
to make up for what was done when rounding
Methods for Subtraction
Decomposing
Starting with the first number and working your way back, taking away the second number. For 52-37, 37 might be split up into 30, 2, and 5.
“Holy Shift”
Making both numbers bigger or smaller so that they subtract nicely.
52-37=__-40 --> the number is 55 because 40 moved up 3, so 52 had to as well. 55-40=52-37=15
Compensating
This is easier for kids to understand in concrete terms, like everything else.
52-37=? 52-40=12 since you subtracted 3 too many you have to add it back, 12+3=15
Partial Differences
Splitting up the problem into tens and ones and then putting them back together to find the answer. The borrowing method can also be used to prevent negative numbers.
52-37=(50-30)+(2-7)=20+(-5)=15
52-37=(40-30)+(12-7)=10+5=15
Through a student’s eyes: “First ii know that 50-30 is 20. Then I saw that 2-7 would give me ‘less’ than zero… it would be -5. 20-5 is 15”.
“I know 50-30 is 20. When I went to do 2-7, I realized I can’t do that so I broke 52 into 40, 10, and 2. I did 40-30 is 10… 10-7 is 3… and then I knew there was 2 I hadn’t used so I added it to the 10 and 3 and got 15”.
Traditional
Stacking and borrowing method. Starting at the ones and moving to the left.
Distance/Difference
When using this method, it is nice to think of the subtraction problem in terms of addition to find the distance.
52-37=? --> 37+__=52, and then a number line can be used to jump up to 52.
