Representing Algorithms K –2
(ADDITION)
Concrete / Pictorial Representation / Verbalization / Tabular (Abstract)Addition (with like things)
/ + / Two soccer ballsplusfour more soccer balls give a total of six soccer balls. / 2 + 4 = 6
Addition (with different things)
/ + / Two soccer ballsplusfour pumpkins give a total of six things. / 2 + 4 = 6
PONDER:
- Are there other representations? I mean, what if you have 2 soccer balls and 4 people, would we say that we have 6? Six what?
- How does this relate to the Commutative Property of Addition (turn-around facts)?
- How might young children get confused when learning about addition? How might multiple representations help such confusions?
- How might students connect these representations to algebra thinking?
NOTES:
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Representing Algorithms K – 2
(SUBTRACTION)
Concrete / Pictorial Representation / Verbalization / Tabular (Abstract)Subtraction(Take-away)
/
/ I have three soccer balls. I take away two. I have three left. / 5 – 2 = 3
Subtraction as a Comparison (Difference)
Bills Soccer Balls
Toms Soccer Balls / Bills Soccer Balls
Toms Soccer Balls / Bill has five soccer balls. Tom has tow soccer balls. Bill has three more soccer balls than Tom. / 5 – 2 = 3
PONDER:
- Are there other representations? I mean, what if the question where: How many more balls does Tom need to equal the number of balls that Bill has?
- Why doesn’t subtraction have a Commutative Property (turn-around facts)?
- How might young children get confused when learning about subtraction? How might multiple representations help such confusions?
NOTES:
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Representing Algorithms2 – 3
(ADDITION & SUBTRACTION w/ REGROUPING)
Concrete / Pictorial Representation / Verbalization / Tabular (Abstract)Addition
/
100 + 20 + 20 + 3 + 8
+10 +10
+2 +5 +1
123 133 143 145 150 151 / 123 plus 28 is 100plus 20, plus 20 to give 140. Then 3 plus 8 is 11, which is 10 plus 1.So 140, plus 10 is 150. And 150, plus 1 is 151.
(Verbalization of the Concrete Representation) / 123 + 28 = 151
1
123
+ 28
151
Subtraction
/ - 10
-2
-2 -2 -2
105 113 123
100 + 20 + 3
100 + 10 + 5 = 105 / To solve 123 take away 18, I took 10 from the 20 in the tens place and added it to the 3 ones. So now I have 13 ones and I have to take away 8 ones, which is a difference of 5. In the tens place I have a 10 and have to take away 10, so I have zero in the ones place. In the hundreds place I have 100 and nothing to take away, so I still have 100. / 13
123
- 18
105
PONDER:
- Are there other representations? We have used an open number line and base-10 representations, what else might we us?
- Can you develop an alternative method for solving and showing how you can solve regrouping problems? How might you solve regrouping problems without regrouping? What does regrouping mean?
- How might young children get confused when learning about regrouping? How can you connect the decomposition of numbers to regrouping?
NOTES:
Representing Algorithms3 – 4
(MULTIPLICATION)
Concrete / Pictorial Representation / Verbalization / Tabular (Abstract)Multiplication
/
XXX XXX / Two groups of three soccer balls produce six soccer balls. / 2 x 3 = 6
Multiplication
25¢ 25¢ 25¢ 25¢ 25¢ 25¢
25¢ 25¢ 25¢ 25¢ 25¢ 25¢ /
/ Twelve groups of 25 things (cents) will produce 200 + 40 + 50 + 10 or 300 things (cents). / 12 x 25
1
12
x 25
60
+240
300
PONDER:
- Are there other representations? How does the partial products algorithm relate to the traditional algorithm?
- How is the Distributive Property of Multiplication modeled by the traditional algorithm or array model?
- Is 12 x 25 the same as 25 x 12? Why or why not?
NOTES:
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Representing Algorithms4 – 5
(DIVISION)
Concrete / Pictorial Representation / Verbalization / Tabular (Abstract)Division
/ 20
5
0.2
0.2 / 20
5
0.2
0.2 / 20
5
0.2
0.2 / 20
5
0.2
0.2 / 20
5
0.2
0.2
25.4 per group
or
/ I looked at the number 127 and saw 100+20+7 and said, “Each of these numbers has to be shared equally into 5 groups.” So I split the 100 into twenties. I had 27 left. I turned that into 25 + 2. I split the 25 into 5s. I have a remainder of 2, so I looked at it like money and asked, “If I had $2.00, could I split it into 5 groups?” I could not figure that out, so I asked the same question for a dollar. And yes, each section got 0.2 or 20 cents. And I did the same thing for the last dollar I had. Within each group I have 25.4 things. / 127 5 = 25.4
or
25 R 2 or 25
PONDER:
- Are there other representations? Is the partial quotients model represented here?
- How are multiplication and division related?
- Why doesn’t division have a Commutative Property (turn-around facts)? Does 30 5 = 6 mean 5 groups with 6 things in each or 30 things split into in 5 things per group making 6 groups in all?
- How might students get confused when learning about division? How might multiple representations help such confusions?
NOTES:
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Representing Operations6 - 7
(Order of Operations)
Concrete / Pictorial Representation / Verbalization / Tabular (Abstract) / Symbolic (Abstract)/ with the same items
/ with the same items:
Two soccer balls plus three groups of five soccer balls equals seventeen soccer balls. / with the same items:
2 + 3 * 5
3 * 5 = 15
2+ 15 = 17 / 2s + 15s = 17s
2s + 3(5s)
/ with different items
/ with different items:
Two soccer balls added to three groups of five pumpkins equals seventeen things. / 2 + 3 * 5
3 * 5 = 15
2+ 15 = 17 / 2s + 15p
2s + 3(5p)