Concrete-Picture-Abstract: Let us Logical Sequence. Use of integer as content

1. Concrete-Picture-Abstract: Use a logical sequence
Muhammad Yusuf
Pedagogical Expert
August 20, 2020

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3. Concrete-Picture-Abstract (CPA): Instructional Approach
“CPA is a three-part strategy where each part
building of the previous instruction to
develop students learning and retain and to
address conceptual knowledge.”
Don’t Memorize, Understand Mathematics

4.  It is the first part to introduced mathematical concepts.
 The teachers start by demonstrating the concept with concrete
materials.
 We can call this part as “doing” part by using concrete material to
model problems.
Concrete:
Don’t Memorize, Understand Mathematics

5. Thousands, Hundreds, Tens and Ones
10 Ones
1 ten
10 tens
1 hundred
1 thousand
10 hundreds

6. How many counter did Hira collected?
2 thousands 1 hundred 3 tens 7 ones
Two thousands, one hundred thirty seven
Don’t Memorize, Understand Mathematics

7. In this part, the teacher transforms the concrete model into
a pictorial (semi-concrete) from by drawing pictures, using
circles, dots, and tallies.
We can call this stage the “Seeing” by using the picture to
model problems.
Picture:
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8. Count the thousands, hundreds, tens and ones in this chart.
Thousands Hundreds Tens Ones
1000 1000 1000
1000
100100
100100
100
10 10 10 10
10 10 10
1 1 1 1
1 1
4000
500
70
9
Four thousand, five hundred,
seventy six.
Don’t Memorize, Understand Mathematics

9.  In this part, the teachers demonstrate the concept at a symbolic level by
using only numbers, symbols, and notations.
 Operation symbols (+, -, ×, ÷) are used to indicate addition, subtraction
or division.
 This is the “symbolic” part, where students are able to use abstract
symbols to model and solve mathematical problems.
Abstract:
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10. Write the numbers in words.
a) 1347 b) 5900 c) 7058
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11. Let us apply CPA model in
Learning of integer

12. It is a summer day. Can you tell me the temperature?

15. Now, can you tell the difference of temperature from summer
day to winter day shown in the pictures?
36 0C -22 0C
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16. Sea Level
3500m above the sea level
2500m below the sea level
3500m
-2500m
0m

17. Let us remind ourselves
Natural Numbers
1,2,3,4,5,6,7, …..
Whole Numbers
0,1,2,3,4,5,6,7, …..
Integers
….,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7, …..
Don’t Memorize, Understand Mathematics

18. Addition of Integers
The zero principle: an equal number of positive and negative tiles add to give zero.
When adding, put tiles onto the table.
Positive Tile
Negative Tile
Using a green tile as +1, a red tile as – 1, and interpreting “neither red nor
blue” as 0

19. Concrete Form Result in Number
and Color of Tiles
Result in
Symbol Form
(+2) + (+3)
(+1) + (+3)
(+4) + (+5)
(+2) + (+6)
4 Green Tiles
5 Green Tiles
9 Green Tiles
8 Green Tiles +8
+5
+4
+9
= Add the numbers and the answer is positive(+) + (+)

20. Concrete Form Result in Number
and Color of Tiles
Result in
Symbol Form
(-3) + (-4)
(-6) + (-6)
(-5) + (-2)
(-2) + (-1)
7 Red tiles
3 Red tiles
7 Red tiles
12 Red tiles
-7
-12
-7
-3
= add the numbers and the answer is negative(‐) + (‐)

21. Concrete Form
Result in Number
and Color of Tiles
Result in
Symbol Form
(+2) + (-3)
(-7) + (+5)
(-2) + (+5)
(+4) + (-4)
(+6) + (-5)
1 red tile
2 red tile
3 green tiles
1 green tile
0 tile
-1
+1
0
+3
-2
=subtract the numbers and take the sign of the bigger number
(+) + (-)
(-) + (+)

22. Subtraction of Integers
Let us recall, What Subtraction is…
Take away,
reduce,
remove
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23. Concrete Form
Result in Number
and Color of Tiles
Result in
Symbol Form
(+6) - (+4)
(-5) - (-3)
(-2) - (-5)
2 Green Tiles
2 Red Tiles
+2
-2
3 Green Tiles +3
(+4) - (-2)
(-2) - (+3)
6 Green Tiles +6
5 Red Tiles -5

24. The sign of the first number stays same and change the
sign of second number. Follow rules of adding integers.
Symbolic Form Pictorial Form
Result in Number
and Colour of Tiles Result in symbols
(+5) - (+2)
(−3) - (−1)
(+2) - (+5)
(−5) - (+1)
(−3) - (-2)
(−4) - (+1)

25. Multiplication of Integers
Let us recall, What Multiplication is …
4+4+4+4+4+4 = 6X4 Add 6 groups of 4
5+5+5+5 = 4X5 Add 4 groups of 5
The order in which you multiply does not matter
A Quick way of doing repeated additions is to multiply
6X4 = 4X6 4X5 = 5X4 + × + = +
+ × - = -
- × + = -
- × - = +

26. Example 1: 4(3)
4(3) means the same as adding 4 groups of 3. In the concrete model, you start with
zero and put four groups of three white tiles on the table.
12
Don’t Memorize, Understand Mathematics

27. Example 2: 6(-2)
Similarly, 6(−2) means add five groups of negative two. The model would
have five groups of two red tiles
Try yourself
-3(2), 4(-5)
-2(6)
(−2)(6) may seem harder to visualize than the example above.
However, once we recall that the order of multiplication does not
matter, we see that (−2)(6) = (6)(−2) = – 12.
-12

28. How can be read -2
(-3)(-2)
Let us recall Negative 2
So, (-3)(-2) could be interpreted
“Subtract two groups of negative three.”
Opposite of 2
Subtract 2
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29. (-3)(-2) So, (-3)(-2) could be interpreted “Subtract two groups of
negative three.
+6
(-4)(-3)
+12
Don’t Memorize, Understand Mathematics

30. Now, it is your Turn
Q/A