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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
