Conceptual MathematicsHow does it all work together?

Lincoln County SchoolsAlycen WilsonMath Lead TeacherK-8

Addition

25

+4368

20 + 5

40 + 360 + 8

• It is important to understand the place value of each digit.

• The value of 25 is 2 tens and 5 ones.

• The value of 43 is 4 tens and 3 ones.

• Tens are added to tens and ones are added to ones.

Addition with regrouping

48

+34

82

40 + 8

+ 30 + 4

• When 8 ones and 4 ones are added together, there are 12 ones.

• We can decompose 12 and make a new ten out of the 12 ones.

• The new 10 will be moved to the ten’s place.

• The two ones will remain in the one’s place.

• Compose the tens and ones place.

10 + 2

80 + 2 = 82

10

12

280 +

Subtraction

243- 122

200 + 40 + 3100 + 20 + 2

100 + 20 + 1

• Each number is decomposed into place values.

• Subtract each place value • Compose the difference

= 121

Subtraction using regrouping

242- 127

200 + 40 + 2100 + 20 + 7

100 + 10 + = 115• Each number is decomposed into place

values. • There are 2 ones in the first number.

Are there enough ones to take away 7 ones? No.

• Make an equal exchange of one ten into ten ones.

• Move ten ones into the ones place to create 12 ones.

• 12 ones is enough to subtract 7 ones.

4 tens-1 ten = 3 tens = 30

10 ones +2 ones = 12 ones

5

Multiplication as repeated addition

3 x 7 = 3 x 7 can be seen as 3 groups of 77 + 7 + 7 = 21

Or 7 groups of 33+3+3+3+3+3+3 = 21

21 is the product, no matter how they are added or multiplied.

Area Model of Multiplication

24 x 15 =

• Decompose each number and represent the value with lines.

• Multiply each of the area sections.

• Add each product to determine the final product.

• Each product represents the area of the square that it is in.

• Add all of the products together to get the final product of 24 and 15.

20 + 4

10 +

5

20 x 10 =

200

4 x 10 =

40

4 x 5 =

20

20 x 5 =

100

200 + 100 + 40 + 20 = 360

Multiplication as Partial Products

24 x 36

Decompose each number in to tens and ones.(As students get comfortable with this method they will begin to do this step in their head.)

20 + 4x 30 + 6

Multiply by each place value, then add each product.6 x 4 =

246 x 20 = 12030 x 4 =

12030 x 20 = 600 864

Division as Repeated Subtraction

35 ÷ 7 =

Dividing means splitting into equal parts. How many groups of 7 can be made from 35.

35 – 7 = 2828 – 7 = 2121 – 7 = 1414 – 7 = 77 – 7 = 0There are 5 groups of 7 in 35, so 35 ÷ 7 = 5

Division as “Giving out” into equal Groups

Division can also be worked through the “giving out equal shares” method.84 ÷ 4 = 84 is given out into 4 equal groups. This can be done in a variety of ways. Counting by 10 is a friendly way to give 84 out.

1010

10 10

+ 1

10 10 1010

+ 1 + 1 + 1

21 21 21 21

84 is put into 4 equal groups, with 21 in each group.

21

Division as partial products. (Box Method)

147 ÷ 6 =

1476

10

- 60 87

876

10

- 60

27

276

4

- 24 3

10 10+ 424 r. 3

24 r. 3

There is not enough in the dividend to pull out another group of ten, I was able to use 4. I pulled out 6 groups of 4. I have a remainder of 3.

Divide with friendly numbers to make the math easier to manage. I used groups of 10 because I can easily multiply and divide by 10. I subtracted 6 groups of 10 from my total.

Always circle the number divided by to total in the end.

Move the remaining dividend to a new box and divided by 10 again

Division as Partial Products

27614

10

-140136

5

- 70

66

4

- 56

10

Add the partial quotients to find the final quotient.

19 r. 10

Subtract 14 groups of ten because 10 is easy for me to calculate.I don’t have enough to subtract another group of ten so I will subtract 5 groups of 14. I know 5 is half of ten so I can calculate that easily.

I do not have enough to subtract another group of 5 so I will subtract a group of 4.

I have 10 left and that is not enough to subtract another group of 14 so 10 is my remainder