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**Addition and Subtraction**

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Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each. Plato 2

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Arithmetic Today Arithmetic has generally been learned through basic algorithms, but it has great potential through problem solving techniques. 3

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**Current Traditional Algorithm**

Addition 1 47 +28 75 “7 + 8 = 15. Put down the 5 and carry the = 7” Subtraction 7 13 83 - 37 46 “I can’t do 3 – 7. So I borrow from the 8 and make it a 7. The 3 becomes – 7 = 6. 7 – 3 = 4.” 4

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**Expanded Column Method**

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**Add on Tens, Then Add Ones**

Number Line Method Add on Tens, Then Add Ones = 76 = = 80 = 84 6

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**Partitioning Using Tens Method**

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Nice Numbers Method 8

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**Lattice Method First arrange the numbers in a column- like fashion.**

Next, create squares directly under each column of numbers. Then split each box diagonally from the bottom-left corner to the top-right corner. This is called the lattice. Now add down the columns and place the sum in the respective box, making the tens place in the upper box and the ones place in the lower box. Lastly, add the diagonals, carrying when necessary. 9

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**Counting Down Using Tens Method**

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**Partitioning Using Tens Method**

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Nice Numbers Method 12

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**The Counting-Up Method**

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**The Counting-Up Method**

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**Nines Complement 827 → 827 - 259 → 740 (nines complement)**

827 → → (nines complement) (to get the ten's complement) 1568 568 (Drop the leading digit) 15

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**Benefits of Alternative Algorithms**

Place value concepts are enhanced They are built on student understanding Students make fewer errors 16

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**Suggestions for Using/Teaching Traditional Algorithms**

We are not saying that the traditional algorithms are bad. The problems occur when they are introduced too early, before students have developed adequate number concepts and place value concepts to fully understand the algorithm. Then they become isolated processes that stop students from thinking. 17

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Integers Integers can be easily approached by thinking in regards of basic addition/subtraction and determining its position on the number line Is the final result positive or negative? 18

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**Integer Addition Rules**

Rule #1 – If the signs are the same, pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer. 9 + 5 = 14 = -14

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**Integer Addition Rules**

Rule #2 – If the signs are different pretend the signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer. = Larger abs. value Answer = - 4 9 - 5 = 4

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**One Way to Add Integers Is With a Number Line**

When the number is positive, count to the right. When the number is negative, count to the left. - + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

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**One Way to Add Integers Is With a Number Line**

= -2 + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -

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**One Way to Add Integers Is With a Number Line**

= +4 - 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 +

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**Adding Integers with Tiles**

We can model integer addition with tiles. Represent -2 with the fewest number of tiles Represent +5 with the fewest number of tiles.

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ADDING INTEGERS What number is represented by combining the 2 groups of tiles? Write the number sentence that is illustrated. = +3 +3

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**= -5 + ADDING INTEGERS Use your red and yellow tiles to find each sum.**

= ? = -5 +

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ADDING INTEGERS = ? + = - 4 = ? = +1 +

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**SUBTRACTING INTEGERS +3 +3**

We often think of subtraction as a “take away” operation. Which diagram could be used to compute = ? +3 +3

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SUBTRACTING INTEGERS This diagram also represents +3, and we can take away +5. When we take 5 yellow tiles away, we have 2 red tiles left. We can’t take away 5 yellow tiles from this diagram. There is not enough tiles to take away!!

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**SUBTRACTING INTEGERS -2 - -4 = ?**

Use your red and yellow tiles to model each subtraction problem. = ?

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**-2 - -4 = +2 SUBTRACTING INTEGERS Now you can take away 4 red tiles.**

2 yellow tiles are left, so the answer is… This representation of -2 doesn’t have enough tiles to take away -4. Now if you add 2 more reds tiles and 2 more yellow tiles (adding zero) you would have a total of 4 red tiles and the tiles still represent -2. = +2

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SUBTRACTING INTEGERS Work this problem. = ?

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**+3 - -5 = +8 SUBTRACTING INTEGERS**

Add enough red and yellow pairs so you can take away 5 red tiles. Take away 5 red tiles, you have 8 yellow tiles left. = +8

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**Why is adding fractions a difficult concept for students to grasp?**

Although children learn addition of whole numbers with ease, addition of fractions — though conceptually the same as addition of whole numbers — is much harder. It requires knowledge of fraction equivalencies. To add two fractions, you have to know that they must be thought of in terms of like units. We take this for granted when we add whole numbers: is really 3 ones + 5 ones — but not when we add fractions: 3 halves + 5 fourths is, for purposes of addition, 6 fourths + 5 fourths.

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**Let’s Eat Pizza The pizza is currently 8 pieces**

What if I wanted to eat one eighth of the pizza? One fourth of the pizza? One sixteenth of the pizza? One twelfth of the pizza?

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**Addition of Fractions The objects must be of the same type**

We combine bundles with bundles and sticks with sticks. Addition means combining objects in two or more sets In fractions, we can only combine pieces of the same size In other words, the denominators must be the same

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Addition of Fractions Example: + = ? Click to see animation

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Addition of Fractions Example: + =

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**Addition of Fractions = + Example: The answer is**

which can be simplified to

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**Addition of Fractions with equal denominators**

More examples

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**Addition of Fractions With different denominators**

In this case, we need to first convert them into equivalent fraction with the same denominator. Example: An easy choice for a common denominator is 3×5 = 15 Therefore,

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**Addition of Fractions With different denominators**

When the denominators are bigger, we need to find the least common denominator by factoring. If you do not know prime factorization yet, you have to multiply the two denominators together.

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More Exercises: = = = = = = = = =

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**Subtraction of Fractions**

Subtraction means taking objects away Objects must be of the same type we can only take away apples from a group of apples In fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.

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**Subtraction of Fractions**

equal denominators Example: This means to take away take away

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**Subtraction of Fractions**

More examples: Did you get all the answers right?

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**Adding/Subtracting 3 3 1 1 2 1 - - = = = 8 8 8 8 8 4 8**

Fraction Simplification 3 3 1 1 2 1 - - = = = 8 8 8 8 8 4 8 Fraction Addition/Subtraction

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**Adding/Subtracting 28 11 2 28 11 2 4 28 11 28 8 + = + = + 28 7 7 4 +**

19 = = 28 Common Denominator = ?????? Fraction Addition/Subtraction 28

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**Fractions: Steps for Success**

Know the fraction rules and how to apply them Show your work and write out each step Check your work for errors or careless mistakes

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Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that.

Adding and Subtracting Fractions Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that.

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