Arrays, Area Models, and Repeated Addition

Some Extra Help for Those Who Are Still Struggling with One or

More of These Concepts

Created by Dr. Peggy L. MochOctober 2010

MORE HELP WITH ARRAYS

Arrays• This represents 3 x 6 as an array of

forks• Rows are horizontal (3)• Columns are vertical (6)• Some folks use “RC Cola” to

remember the order• Some folks use “RCMP” (Royal

Canadian Mounted Police)

Rows

Columns

Arrays• This represents 6 x 3 as

an array of forks• Rows are horizontal (6)• Columns are vertical

(3)• This concept of rows

and columns is used in– Arrays– Area Models– Repeated Addition– Matrices

Rows

Columns

Arrays• Your Turn!• Provide arrays for each

of the following using O:– 4 x 3– 5 x 2– 2 x 3– 3 x 2– 2 x 5– 3 x 4

• Do the arrays on your own paper

• Label each array with the problem it represents

• Answers are provided on the next slide

Array Answers• 4 x 3

• 5 x 2

• 2 x 3

• 3 x 2

• 2 x 5

• 3 x 4

MORE HELP WITH AREA MODELS

Area Models• Uses place value of two

linear measures to create areas

• Areas are summed together

• Sum of areas is the product of the two linear measures

• Follows rows then columns idea of arrays for set up purposes– 12 x 37 is a short wide

rectangle– 37 x 12 is a tall narrow

rectangle

Area Models• Modeling 37 x 12

– 37 units (u) is the length of the left side of the rectangle (red)

– 12 units (u) is the length of the top of the rectangle (blue)37 u

12 u

Area Models• Modeling 37 x 12

– The rest of the rectangle is constructed by adding the congruent parallel parts (green)

37 u

12 u

Area Models• Modeling 37 x 12

– Next divide the rows by place value (estimate the size – it should be reasonable)

– Rows will break apart 37 into 30 units + 7 units

– The 30 unit length will be much larger than the 7 unit length

37 u

12 u

Area Models• Modeling 37 x 12

– Next divide the columns by place value (estimate the size – it should be reasonable)

– Rows will break apart 12 into 10 units + 2 units

– The 10 unit length will be much larger than the 2 unit length

37 u

12 u

Area Models• Modeling 37 x 12

– At this point the areas of each of the internal rectangles is calculated mentally and written into the their respective areas

• 30 x 10 for the top left rectangle• 30 x 2 for the top right rectangle• 7 x 10 for the bottom left rectangle• 7 x 2 for the bottom right rectangle

37 u

12 u

300 u2 60 u2

70 u2 14 u2

Area Models• Modeling 37 x 12

– The last step is to sum up the areas, which will equal the product of 37 units x 12 units

37 u

12 u

300 u2 60 u2

70 u2 14 u2

300 u2 + 60 u2 + 70 u2 + 14 u2 = 444 u2

Area Models• Your Turn!• Provide area models for

each of the following using an appropriately sized rectangle:– 14 x 35– 53 x 27– 32 x 177– 43 x 239

• Do the area models on your own paper

• Label each area model with the problem it represents

• Answers are provided on the next slides

Area Model Answers• 14 x 35

300 u2 + 50 u2 + 120 u2 + 20 u2 = 490 u2

14 u

35 u

300 u2

120 u2

50 u2

20 u2

Area Model Answers• 53 x 27

1000 u2 + 350 u2 + 60 u2 + 21 u2 = 1431 u2

53 u

27 u

1000 u2 350 u2

60 u2 21 u2

Area Model Answers• 32 x 177

3000 u2 + 200 u2 + 2100 u2 + 140 u2 + 210 u2 + 14 u2 = 490 u2

32 u

177 u

3000 u2

200 u2

2100 u2

140 u2 14 u2

210 u2

Area Model Answers• 43 x 239

8000 u2 + 1200 u2 + 360 u2 + 600 u2 + 90 u2 + 27 u2 = 10277 u2

43 u

239 u

8000 u2

600 u2

1200 u2

90 u2 27 u2

360 u2

MORE HELP WITHREPEATED ADDITION

Repeated Addition• Repeated addition is an

alternate way of defining multiplication

• The first factor tells how many sets or addends that will be needed (5)

• The second factor tells the cardinality of each set (3)

5 Sets

3 elementsin each set

5 x 3

Repeated Addition• In terms of Vygotsky

self-talk repeated addition (n x m) is thought of as “n sets of m each” (5 sets of 3 each)

• Repeated addition is usually written horizontally

• 3 + 3 + 3 + 3 + 3 = 15

3+3+3+3+3=15

5 x 3

Repeated Addition• Your Turn!• Provide repeated

addition models for each of the following:– 4 x 3– 5 x 2– 2 x 3– 3 x 2– 2 x 5– 3 x 4

• Do the repeated addition models on your own paper

• Label each model with the problem it represents

• Answers are provided on the next slide

Repeated Addition Answers• 4 x 3

3 + 3 + 3 + 3 = 12

• 5 x 22 + 2 + 2 + 2 + 2 = 10

• 2 x 33 + 3 = 6

• 3 x 22 + 2 + 2 = 6

• 2 x 55 + 5 = 10

• 3 x 44 + 4 + 4 = 12

The End