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ADDITION is not just about adding two numbers. You don't even need numbers. Addition happens when any two, or more, sets (groups anything) combined. The things being combined are called addends. The answer is the called the sum.
Example of addition of sets:
4 cats + 3 dogs = 7 pets
However, the set (groups), can't share any members. Their intersection (things in common) must be an empty set (contains nothing).
*Think: These groups of pets are disjoint sets since no pet can be in both groups. The intersection (things in common) is an empty set (contains nothing).
If we know 7 pets are brown and 3 pets have collars, we cannot add 7 and 3 since there may be some pets in both groups. The intersection (pets in common) of those sets (collections), may or may not be an empty set.
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SUBTRACTION is the difference between sets.
Example of subtraction of sets:
7 pets - 3 dogs = 4 cats OR 7 pets - cats = 3 dogs
Tools for Practicing Adding & Subtracting
CONCRETE OBJECTS: Use real objects like cubes, tiles, paperclips, pebbles, or any similar objects to show the problem. *This also lets young students practice fine motor skills. Base 10 blocks work well for representing larger numbers.
PICTURES: Draw the problem using dots, different colors, or anything that can be drawn easily. Pictures could also be items cut out of magazines. *This is another way for students to practice fine motor skills.
10's FRAMES: These work especially well for students with strong visual learning tendencies. The consistent arrangement of objects helps students to see grouping and not have to count out each item each time.
Thinking Blocks uses blocks to represent problems. Word problems can be challenging but Thinking Blocks shows how to illustrate and solve each problem, step by step.
This could also be used as a class activity with each student using their own concrete objects to solve the problems. If the topic hyperlink doesn't work, use this URL.
http://www.mathplayground.com/tb_addition/thinking_blocks_addition_subtraction.html
Properties of Addition
There are four basic mathematic properties that are used in addition. The ADDITIVE IDENTITY PROPERTY is used only in addition. It says that the sum of any number and zero is the original number.
Example: 5 + 0 = 5.
The COMMUTATIVE, ASSOCIATIVE, and DISTRIBUTIVE properties also apply to addition. To see an explanation and examples all of the properties that apply to addition, follow the hyperlink for this topic or use this URL http://www.aaamath.com/b/pro74ax2.htm.
Different Ways to Think About
Adding & Subtracting
Using all of these concepts shows the relationship between
adding and subtracting.
MISSING ADDEND:I have 5. How many more do I need to make 9?
Example: 5 + __ = 9
COMPARISON: I have 5. He has 9. How many more does he have?
Example: 9 - 5 = ___
TAKE AWAY: I had 9, but 4 ran away. How many do I have now?
Example: 9 - 4 = ___
NUMBER LINE / MEASUREMENT: I walked 9 blocks home. For 4 blocks it rained on me. How many more blocks did I have to walk after it quit raining?
Example: |---|---|---|---|---|---|---|---|---|
This activity illustrates the Take Away model and allows practice of addition/subraction facts. Use the topic link or this URL.
http://www.rabbittakeaway.co.uk/activity/
NUMBERS are used to describe, compare, and identify.
NUMBERS are used to describe and count many, many things including time, money, weights, and distances. When numbers are used for counting and measuring they are called cardinal numbers.
NUMBERS are used for ordering and for comparing. When numbers describe order, like first and second, they are called ordinal numbers.
NUMBERS are also used to identify people and things. When numbers are assigned as a name, like a bus number or a library number, they are called nominal numbers.
NUMBERS can be represented by symbols such as 1, 2, 3, ... or tally marks. They can be represented with physical objects, pictures, and number lines.
Tools for Practicing Working With Numbers
CONCRETE OBJECTS: Use real objects like cubes, tiles, marbles, noodles, or any similar objects to show sets, cardinal (counting) and ordinal (ordering) numbers. *This also lets young students practice fine motor skills.
PICTURES: Pictures and drawings can be used to represent sets, cardinal (counting), ordinal (ordering), and nominal (naming) numbers.
NUMBER LINES AND STRIPS: These are useful for counting and showing one to one correspondence.
Definitions of Numbers Terms
CARDINAL NUMBERS: Numbers used for counting are called cardinal numbers. The tell the number of items in a set, whether that is students, years, inches, or apples.
NOMINAL NUMBERS: Numbers used for naming or identifying are called nominal numbers.
ORDINAL NUMBERS: Numbers used to organize and order things are ordinal numbers.
FINITE/INFINITE: Finite numbers describe a set (group) that has an end. Infinite numbers are a set that has no end, like all of the whole numbers.
GREATER THAN/ LESS THAN: When numbers are used to describe sets (groups) that are larger or smaller than at least one other.
ONE TO ONE CORRESPONDENCE: Every thing in one set (group) can be matched exactly to everything in another set.
Example: Three cubes can be matched exactly with the whole numbers 1, 2, and 3 with nothing left over in either group and nothing paired more than once.
EQUIVALENT: Sets (groups) that are equal, that is, they have one to one correspondence.
An illustrated mathematics dictionary can be found by using the topic hyperlink or this URL https://www.mathsisfun.com/definitions/index.html.
MULTIPLICATION is repeated addition. Both x and * can be used to indicate multiplication. The numbers being combined are called factors, the total is called the product. A multiplication problem can be thought of as a group called a, happening b times.
Example: If a is 3 and b is 4, then 3+3+3+3, or 3 * 4, or 3 happening 4 times.
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DIVISION is repeated sharing or grouping. Both ÷ and / can be used to indicate multiplication. It may take practice for students to understand that 15/3 is not only a fraction is also the division problem 15÷3.The number being divided is called the dividend, the number of groups is called the divisor, the number in each group is called the quotient.
Example: If you own are 6 hungry rabbits and have 12 carrots to feed them, the carrots can be shared evenly by creating 6 equal groups.
Example with remainder*: If you own are 6 hungry rabbits and have 14 carrots to feed them, the carrots can be shared evenly by creating 6 equal groups and there will be 2 carrots remaining for you to eat!
*Having students share concrete objects is useful for explaining the concept of remainder.
Properties of Multiplication
There are four basic mathematic properties that are used in multiplication. The MULTIPLICATION IDENTITY PROPERTY is used only in multiplication. It says that the product of any number and one equals the original number.
Example: 8 * 1 = 8
The COMMUTATIVE, ASSOCIATIVE, and DISTRIBUTIVE properties apply to multiplication, as well as addition and sets. To see an explanation and examples all of the properties that apply to addition, follow the hyperlink for this topic or use this URL http://www.aaamath.com/b/pro74bx2.htm.
Tools for Understanding Multiplying & Dividing
CONCRETE OBJECTS: Use real objects like cubes, tiles, paperclips, pebbles, or any similar objects to show smaller problems. Base 10 blocks work well for representing larger numbers.
DIAGRAMS: Draw the problem using dots, different colors, or anything that can be drawn easily. Older students may find it helpful to use sets of tally marks.
** Once students understand the concepts, they will need to memorize facts using tables, apps, flash cards, and games. Students may feel overwhelmed by this but should understand that is much like memorizing ABC's and there sounds, and it's less information than the ABC's since numbers don't have upper and lower case. Visit Factorville© at the attached link!
Thinking Blocks uses blocks to represent problems. Word problems can be challenging but Thinking Blocks shows how to illustrate and solve each problem, step by step.
This could also be used as a class activity with each student using their own concrete objects to solve the problems. If the topic hyperlink doesn't work, use this URL.
http://www.mathplayground.com/tb_multiplication/thinking_blocks_multiplication_division.html
Different Ways to Think About
Multiplying & Dividing
There are many ways to model multiplication and division problems.
SKIP COUNT MODEL: This works well for introducing multiplication as repeated addition and for solving smaller problems.
ARRAYS MODEL: A grid or graph paper are used to draw a square or rectangle with sides as long as each factor in the problem. Squares are counted to find the product, which is the area of the rectangle.
RECTANGLES MODEL: This method combines using arrays, part-part-whole, and addition and the Associative Principle. Two digit numbers are broken apart and each problem is shown on its own array before combining the answers of each array. *A video is attached to demonstrate this method.
CARTESIAN PRODUCT MODEL: This is used to show the combination of two sets, such as 2 pair of pants in different colors and 3 shirts of different patterns. Each pair of pants can be combined with shirt 1, and with shirt 2, and with shirt 3. It can be thought of as 2 outfits + 2 outfits + 2 outfits, or 2 * 3.
SETS are categories of things. Sets might also be thought of as collections, classes, groups, or families. Those categories can be large, small, or even empty.
• Sets can be used to sort information.
• Sets can be combined (union), they can have things in common (intersection).
• Sets are made from a universe of things. The universe can be as large or small. A large universe could be the number of school children in the world. A small universe could be the number of socks in your drawer.
• A set can be described with a list, a description of its properties, or with images.
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For more information: To learn more about sets, including examples and illustrations, visit Math Is Fun using the hyper link on this topic or use this URL.
https://www.mathsisfun.com/sets/sets-introduction.html
Properties of Sets
There are four basic properties of sets working with whole numbers. Most of them are also properties of addition/subtraction or multiplication/division. This make sense since the both are ways to combine sets or groups of things.
TRANSITIVE: A ⊂ B and B ⊂ C, therefore A ⊂ C.
Example: Dallas is in Texas and Texas is in the United States, therefore Dallas is in the United States.
COMMUTATIVE: A U B = B U A and A ∩ B = B ∩ A
Think: The commute is the same or equal, backward or forward.
ASSOCIATIVE: A U (B U C) = (A U B) U C and
A ∩ (B ∩ C) = (A ∩ B) ∩ C
Think: It doesn't matter who associates with whom, it's still the same group.
DISTRIBUTATIVE: A ∩ (B U C) = (A ∩ B) U (A ∩ C)
Think: It doesn't matter if A is distributed to the group or separately, the result is the same or equal.
Symbols & Definitions For Sets
BRACES: A list of members or items in a set are written within brackets called braces. Written: { at the beginning and end of the set }
The symbol looks a little like a smile with braces.
UNION: Joining sets together. Written: A U B
The symbol looks like U for Union.
INTERSECTION: What sets have in common. Written: A ∩ B
The symbol looks a little like it has a "foot" in each area, where the two sets connect.
PROPER SUBSET: One set is completely with a larger set.
Written: A ⊂ B
SUBSET: One set is completely within another set and they are equal.
Written: A ⊆ B
The symbols look similar but the second one adds a half of the equal sign, showing that A is not only a subset of B, it is also equal to B.
ELEMENT: An item or number is member of the set. Written: x ∈ A
The symbol looks like E for Element.
EMPTY SET: There is nothing in the set. Written: ∅ or { }
The symbol looks like a zero or empty arms.
An extensive list of set symbols, their definitions and examples can be found at Rapid Tables.
Follow the topic hyperlink or use this URL, http://www.rapidtables.com/math/symbols/Set_Symbols.htm.
UNIVERSE: All of the items being considered for sets. Written: U
The symbol looks like U for Universe.
Working With Sets
DESCRIBING SETS:
• A set can be described with a list: {1, 2, 3, . . . 99, 100} or {red, blue, yellow, green, purple}.
• A set can be described by its characteristics or properties with set builder notation. Set builder notation can save time in describing large or complicated sets. This example {x | x < 5} would be read, x such that x is less than 5. In other words, this set contains all numbers less than 5.
SHOWING SETS:
• A Venn Diagram is very effective for illustrating a small number of sets. The circles can be separate (disjoint sets), overlap (intersection), or show one inside another (subset).
•There are nearly unlimited nonstandard ways to show sets using models, drawings, or concrete objects. Sets could be shown using models of corrals, shoeboxes, transparent colored film, sides of a room, etc. Follow the topic hyperlink to practice sorting using corrals to hold wacky animals, or copy and paste this URL. http://pbskids.org/cyberchase/math-games/logic-zoo/
