4th Grade - Break A Problem Into Simpler Parts

 
     
 
     
 
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4th
Problem Solving
Break a Problem into Simpler Parts
Determine when and how to break a problem into simpler parts.
 

Sample Problems

(1)

Break apart numbers to add tens and ones separately:

47 + 23

40 + 20 = _____

7 + 3 = _____

_____ + _____ = ______


(60 + 10 = 70)

(2)

Use the factor tree to find prime factors:

30

/ \

__ x __

/ \

__ x __


Write the prime numbers ___________________

(2 x 3 x 5 = 30)

(3)

Write equations for the arrays shown:

 

 

 

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


_______________ = _______

2 x (4 x 3) = 24

(4)

273

x 8


8 x 3 = ___


70 x 8 = ____

200 x 8 = _____


Add the products: _____ + _____ + _____ = ________


( 24 + 560 + 1,600 = 2,184)

(5)

Find the area of the complex polygon:


Learning Tips

(1)

Break Apart Strategy

Helps the child compute mentally without using pencil and paper, worrying about borrowing and carrying number rules.


Addition (Mental Math): Find the sum of 56 + 28

  1. Add the tens 50 + 20 = 70

  2. Add the ones 6 + 8 = 14

  3. Add the sums 70 + 14 = 84


Subtraction (Mental Math): Find the difference for 87- 26

  1. Subtract the tens 80 – 20 = 60

  2. Subtract the ones 7 – 6 = 1

  3. Add the differences 60 + 1 = 61


Multiplication (Solve simpler problems)- find the product for 372 x 4


1. Multiply by the ones 4 x 2 = 8

2. Multiply by the tens 70 x 4 = 280

3. Multiply by the hundreds 300 x 4 = 1,200

4. Add the products 8 + 28 + 1,200 = 1,236

(2)

Break apart complex polygons

Divide figure to find area:

Example


 

 

 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Divide the polygon into two rectangles. The larger rectangle measures as (4) length and (7) at the width. Use the formula for finding the rectangle (A = l x w) A = 4 x 7 = 28. Next, find the area of the smaller square which is (3) length and (3) width. A= 3 x 3 = 9. Finally, add the two areas together 28 + 9 = 37 sq. units.



(3)

Factor trees

Reminder: There can be various ways to begin the factor tree, as long as children get the same result of prime numbers at the end of the tree (highlighted numbers).


12

/ \

6 x 2

/ \

2 x 3


Write the product of prime numbers as: 2 x 2 x 3 = 12


OR



12

/ \

4 x 3

/ \

2 x 2


Write the product of prime numbers as: 2 x 2 x 3 =12


(4)

Use models to break apart multiplication for equations

Make arrays, using graph paper. Shade in the number of squares you want the product to equal to. Cut the array in half (2 arrays) with the same value will appear. You can keep going as long as there are equal parts to the arrays.


For example,

Start with 6 x 8 = 48

 

 

 

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Next step, cut the array in half. Now there are (2 arrays)

2 x (6 x 4) = 48

 

 

 

 

 

 

 

 

 

 

 

 

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The process can continue to 4 x (3 x 4) = 48

(5)

Base ten blocks –Website you can use to get base ten block cut-outs for ones, tens, and hundreds) http://mason.gmu.edu/~mmankus/Handson/b10blocks.htm


This is the 1-block or unit block the smallest of all the blocks.

This is the 10-block corresponding to 10 units.  It is also referred to as a rod or long.

This is the 100-block and corresponds to 100 units. It is also called a flat.


Use base-ten blocks to model a multiplication problem in order to find the product.

You can make a model to find 142 x 3:

  1. Model 3 groups of 142 (using (1) hundred block (4) ten- blocks (2) one-blocks)

  2. Combine the groups

-(6) one-blocks

-(12) ten-blocks

-(3) hundred-blocks

3. Combine the ten-blocks to make one hundred-block (with two ten-blocks remaining)

4. Count the total number of blocks to find the product (4-hundred-blocks; 2- ten-blocks; and 6-one-blocks) = 426

5. Therefore, 142 x 3 = 426

Extra Help Problems

(1)

Use the “Break-Apart” strategy to find the sums and differences:

27 + 84

_____ + _____ = _____ (20 + 80 = 100)

_____ + _____ = _____ (7 + 4 =11)


_____ + _____ = ______ (100 + 11 = 111)



(2)

Use the “Break-Apart” strategy to find the sums and differences:

91 + 58

_____ + _____ = _____ (90 + 50 = 140)

_____ + _____ = _____ (8 + 1 = 9)


_____ + _____ = ______ (140 + 9 = 149)


(3)

Use the “Break-Apart” strategy to find the sums and differences:

58 – 26

_____ + _____ = _____ (50 – 20 = 30)

_____ + _____ = _____ (8 – 6 = 2)


_____ + _____ = ______ (30 + 2 = 32)


(4)

Use the “Break-Apart” strategy to find the sums and differences:

99 – 37

_____ + _____ = _____ (90 – 30 = 60)

_____ + _____ = _____ (9 – 7 = 2)


_____ + _____ = ______ (60 + 2 = 62)


(5)

Use the “Break-Apart” strategy to find the sums and differences:

134 + 623


_____ + _____ = _____ (100 + 600 = 700)

_____ + _____ = _____ (30 + 20 = 50)

_____ + _____ = _____ (3 + 4 = 7)

_____ + _____ = ______ (700 + 50 + 7 = 757)


(6)

Use the “Break-Apart” strategy to find the products:

142

x 6


____ x ____ = ______ (2 x 6 = 12)


____ x ____ = ______ (40 x 6 = 240)

_____ x ____ = ______ (100 x 6 = 600)


Add the products: _____ + _____ + _____ = ________ (12 + 240 + 600 = 852)



(7)

756

x 2


____ x ____ = ______ (6 x 2 = 12)


____ x ____ = ______ (50 x 2 = 100)

_____ x ____ = ______ (700 x 2 = 1,400)


Add the products: _____ + _____ + _____ = ________ (12 + 100 + 1,400 = 1,512)



(8)

444

x 4

____ x ____ = ______ (4 x 4 = 16)


____ x ____ = ______ ( 40 x 4 = 160)

_____ x ____ = ______ (400 x 4 = 1,600)


Add the products: _____ + _____ + _____ = ________ (16 + 160 + 1,600 = 1, 776)




(9)

529

x 70


____ x ____ = ______ (70 x 9 = 630)


____ x ____ = ______ (20 x 70 = 1,400)

_____ x ____ = ______ (500 x 70 = 35, 000)


Add the products: _____ + _____ + _____ = ________ (630 + 1,400 + 35,000 = 37,030)




(10)

372

x 50


____ x ____ = ______ (50 x 2 = 100)


____ x ____ = ______ (70 x 50 = 3,500)

_____ x ____ = ______ (300 x 50 = 15,000)


Add the products: _____ + _____ + _____ = ________ (100 + 3,500 + 15,000 = 18,600)





(11)

Write equations for the arrays shown: ____________________


 

 

 

 

 

 

 

 

 

 

 

5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


(2 x (5 x 3 ) = 30)

(12)

Write equations for the arrays shown: _______________________


 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

2

 

 

 

2

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

(4 x (2 x 2) = 16)

(13)

Break up the array and write the equation: ____________________


 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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(2 x (4 x 4) = 32)

(14)

Break up the array and write the equation: ___________________


 

 

 

 

 

 

 

 

 

 

 

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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(2 x (3 x 7) = 42)

(15)

Break up the array and write the equation:


 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(3 x (3x 3) = 27)

(16)

Use the factor tree to find prime factors:

20

/ \

__ x __

/ \

__ x __


Write the prime numbers ___________________ (2 x 2 x 5 = 20)


(17)

Use the factor tree to find prime factors:

120

/ \

__ x __

/ \ / \

__ x __ __ x __


Write the prime numbers ___________________(2 x 2 x 2 x 3 x 5= 120)


(18)

Use the factor tree to find prime factors:

39

/ \

__ x __


Write the prime numbers ___________________ (3 x 13 = 39)


(19)

Use the factor tree to find prime factors:

42

/ \

__ x __

/ \

__ x __


Write the prime numbers ___________________ (2 x 3 x 7 = 42)


(20)

Use the factor tree to find prime factors:

18

/ \

__ x __

/ \

__ x __


Write the prime numbers ___________________ (2 x 3 x 3 = 18)


(21)

Find the area of the complex polygon:

Rectangle A : __________ (8 x 3= 24)

Rectangle B: ___________(3 x 4 = 12)

Area = ______________ (24 + 12 = 36 sq. units)


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

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3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

(22)

Find the area of the complex polygon:

Rectangle A : __________ (2 x 7= 14)

Rectangle B: ___________( 3 x 2= 6)

Area = ______________ (14 + 6 = 20 sq. units)


 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

(23)

Find the measurements and solve the area of the complex polygon:


Rectangle A : __________ (8 x 4 = 32)

Rectangle B: ___________ (4 x 5 = 20)

Area = ______________ (32 + 20 = 52 sq. units)






 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(24)

Find the area of complex polygon:


Rectangle A : __________ (2 x 8 = 16)

Rectangle B: ___________ (6 x 2 = 12)

Area = ______________ (16 + 12 = 28 yd2)

(25)

Find the area of the complex polygon:


Rectangle A : __________ (3 x 4 = 12)

Rectangle B: ___________ (3 x 7 = 21)

Area = ______________ (12 + 21 = 33 cm2)


 

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