2 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.Plato2
3 Arithmetic TodayArithmetic has generally been learned through basic algorithms, but it has great potential through problem solving techniques.3
4 Current Traditional Algorithm Addition147+2875“7 + 8 = 15. Put down the 5 andcarry the = 7”Subtraction7 1383- 3746“I can’t do 3 – 7. So I borrow fromthe 8 and make it a 7. The 3becomes – 7 = 6.7 – 3 = 4.”4
9 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
15 Nines Complement 827 → 827 - 259 → 740 (nines complement) 827 →→ (nines complement)(to get the ten's complement)1568568 (Drop the leading digit)15
16 Benefits of Alternative Algorithms Place value concepts are enhancedThey are built on student understandingStudents make fewer errors16
17 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
18 IntegersIntegers can be easily approached by thinking in regards of basic addition/subtraction and determining its position on the number lineIs the final result positive or negative?18
19 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
20 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. valueAnswer = - 49 - 5 = 4
21 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.-+123456-1-2-3-4-5-6
22 One Way to Add Integers Is With a Number Line =-2+123456-1-2-3-4-5-6-
23 One Way to Add Integers Is With a Number Line =+4-123456-1-2-3-4-5-6+
24 Adding Integers with Tiles We can model integer addition with tiles.Represent -2 with the fewest number of tilesRepresent +5 with the fewest number of tiles.
25 ADDING INTEGERSWhat number is represented by combining the 2 groups of tiles?Write the number sentence that is illustrated.= +3+3
26 = -5 + ADDING INTEGERS Use your red and yellow tiles to find each sum. = ?= -5+
28 SUBTRACTING INTEGERS +3 +3 We often think of subtraction as a “take away” operation.Which diagram could be used to compute= ?+3+3
29 SUBTRACTING INTEGERSThis 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!!
30 SUBTRACTING INTEGERS -2 - -4 = ? Use your red and yellow tiles to model each subtraction problem.= ?
31 -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
33 +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
34 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.
35 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?
36 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 setsIn fractions, we can only combine pieces of the same sizeIn other words, the denominators must be the same
37 Addition of FractionsExample:+= ?Click to see animation
39 Addition of Fractions = + Example: The answer is which can be simplified to
40 Addition of Fractions with equal denominators More examples
41 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 = 15Therefore,
42 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.
44 Subtraction of Fractions Subtraction means taking objects awayObjects must be of the same typewe can only take away apples from a group of applesIn fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.
45 Subtraction of Fractions equal denominatorsExample:This means to take awaytake away
46 Subtraction of Fractions More examples:Did you get all the answers right?