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U n t er r i ch t spl a n
Quo t at ive Divis io n
Altersgruppe: 3 r d Gr ade , 4 t h Gr ade
Virginia - Mathematics Standards of Learning (2009): 1.12, 3 .5 ,
4 .4 d, 5 .19
Virginia - Mathematics Standards of Learning (2016): 3 .4 .a, 3 .4 .b,
4 .2.c , 4 .4 .a
Fairfax County Public Schools Program of Studies: 1.12.a.4 ,
3 .5 .a.1, 3 .5 .a.2, 4 .4 .d.3 , 4 .4 .d.4 , 5 .19.a.1, 5 .19.a.2
Online-Ressourcen: C o v e r by R e c t angl e s
Opening
T eacher
present s
St udent s
pract ice
Class
discussion
8
12
14
10
3
min
min
min
min
min
Closing
M at h Obj e c t i v e s
E x pe r i e nc e covering a rectangle with smaller rectangles
P r ac t i c e calculating area
L e ar n to use repeated subtraction to calculate quotients
De v e l o p a conceptual understanding of division
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Ope ni ng | 8 min
Display the following expression. Ask students to write in their
notebooks a word problem that the expression could be used to
solve.
24 ÷ 6 =
When the students are done writing, share. Ask a number of students
to share their work. Say: Please read the problem you wrote that
could be solved by dividing 24 by 6.
Responses will vary. As students share their word problems, write
them on the board. Separate the problems into two categories:
ones that demonstrate par t i t i v e di v i si o n and ones that
demonstrate q uo t at i v e di v i si o n . An example of each follows:
Partitive division: There are 24 pencils. If 6 students want to share the
pencils fairly, how many pencils does each student get?
Quotative division: There are 24 cookies. A baker wants to package them in
groups of 6. How many packages can the baker make?
A sk : What is the difference between the two types of division
problems?
One type of problem has the 24 items shared equally into 6
groups of 4. The other type of problem has 24 items separated
into 4 groups of 6. The difference is whether the 6 determines
the number of groups or the number of items in each group.
T e ac he r pr e se nt s M at h game : C o v e r by R e c t angl e s Di v i de A r e a: L e v e l I | 12 min
Present Matific ’s episode C o v e r b y R e c t a n g le s - Div id e A r e a : L e v e l I
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to the class, using the projector.
The goal of the episode is to develop an understanding of quotative division
by covering a rectangle with smaller rectangles of a given area.
E x a m p le :
S ay : Please read the instructions.
Students can respond based on the episode.
A sk : How can we cover the blue rectangle with smaller rectangles
of the given size? Is there more than one way?
Students can respond based on the episode.
Create smaller rectangles as the students suggest. If you make an
error, you can move the incorrectly-sized rectangle into the white
part of the grid to remove it. When you have finished covering the
blue rectangle, click .
If the answer is correct, the episode will proceed to the next problem.
If the answer is incorrect, the instructions will wiggle.
The episode will present a total of five problems.
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S t ude nt s pr ac t i c e M at h game : C o v e r by R e c t angl e s Di v i de A r e a: L e v e l I | 14 min
Have the students play C o v e r by R e c t angl e s - Di v i de A r e a:
L e v e l I and C o v e r by R e c t angl e s - Di v i de A r e a: L e v e l I I on
their personal devices. Circulate, answering questions as necessary.
C l ass di sc ussi o n | 10 min
Run the episode. Display and discuss the given problem. For
example:
A sk : What is the ar e a of the blue rectangle? How do you know?
The rectangle has area 63 square units. We can multiply its width,
9, by its height, 7, to get the area.
Write on the board:
9 × 7 = 63
A sk : What division problem are we illustrating when we cover this
rectangle with rectangles with area 3? How do you know?
We are illustrating the problem 63 divided by 3. The area of the
blue rectangle is 63 square units. We are determining how many
smaller rectangles with area 3 fit inside the larger rectangle.
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A sk : How many rectangles with area 3 fit inside the rectangle with
area 63?
Twenty-one rectangles with area 3 fit inside the blue rectangle.
Write on the board:
63 ÷ 3 = 21
A sk : How can we write this fact as a multiplication problem?
We can write that 3 times 21 is 63.
Write on the board:
3 × 21 = 63
S ay : Let’s compare these two facts: 9 times 7 is 63 and 3 times 21
is 63. How are these two statements related? Why does it make
sense that if you know one fact, you can figure out the other?
If we start with 9 times 7, we can rewrite the 9 as 3 times 3. So
now we have 3 times 3 times 7 is 63. If we multiply 3 by 7, we get
21. So we can change the expression from 9 times 7 to 3 times 3
times 7 to 3 times 21. Since we are just f ac t o r i ng and
multiplying, we haven’t changed the expression. All of the
expressions must equal 63.
Write on the board:
9 × 7 = 63
(3 ×3 ) × 7 = 63
3 × (3 × 7) = 63
3 × 21 = 63
Place 21 rectangles with area 3 on the blue rectangle and click
to advance to the next problem. Display and discuss the next
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problem. For example:
A sk : What division problem is this problem illustrating? How do
you know?
This problem is illustrating 36 divided by 2. The blue rectangle
has area 36 square units. (We can multiply its dimensions, 6 by 6,
to get the area.) We are illustrating 36 divided by 2 because we
are determining how many rectangles with area 2 fit inside the
rectangle with area 36.
A sk : How many rectangles with area 2 fit inside the rectangle with
area 36?
Eighteen rectangles with area 2 fit inside the rectangle with area
36.
Place 18 rectangles with area 2 on the blue rectangle.
Write on the board:
6 × 6 = 36
36 ÷ 2 = 18
A sk : How can we rewrite the division problem as a multiplication
problem?
We can write that 18 times 2 is 36.
Write on the board:
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6 × 6 = 36
18 × 2 = 36
S ay : Let’s compare these two facts: 6 times 6 is 36 and 18 times 2
is 36. How are these two statements related? Why does it make
sense that if you know one fact, you can figure out the other?
Let’s consider the first fact: 6 times 6 is 36. We can rewrite the
second 6 in the problem as 3 times 2. So now we have 6 times 3
times 2 is 36. If we multiply 6 by 3, we get 18. So we can change
the expression from 6 times 6 to 6 times 3 times 2 to 18 times 2.
Since we are just factoring and multiplying, we haven’t changed
the expression. All of the expressions must equal 36.
Write on the board:
6 × 6 = 36
6 × (3 × 2) = 36
(6 × 3) × 2 = 36
18 × 2 = 36
S ay : We can use this type of manipulation of numbers to find
pr o duc t s that initially seem difficult. For example, let’s consider
14 times 5.
Write on the board:
14 × 5 =
A sk : How could we change this problem into a problem we could
do in our heads?
We could factor 14 into 7 times 2. Then we can multiply the 2 by
the 5. So the problem becomes 7 times 10, which we know is 70.
Write on the board:
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14 × 5 =
(7 × 2) × 5 =
7 × (2 × 5) =
7 × 10 = 70
S ay : Let’s look at a new problem. Let’s consider 75 times 4.
Write on the board:
75 × 4 =
A sk : How could we change this problem into a problem we could
do in our heads?
We could factor 75 into 3 times 25. Then we can multiply the 25
by the 4. So the problem becomes 3 times 100, which we know is
300.
Write on the board:
75 × 4 =
(3 × 25) × 4 =
3 × (25 × 4) =
3 × 100 = 300
S ay : Let’s look at a new problem. Let’s consider 45 times 8.
Write on the board:
45 × 8 =
A sk : How could we change this problem into a problem we could
do in our heads?
We could factor 45 into 9 times 5. Then we can multiply the 5 by
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the 8. So the problem becomes 9 times 40, which we know is 360.
Write on the board:
45 × 8 =
(9 × 5) × 8 =
9 × (5 × 8) =
9 × 40 = 360
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C l o si ng | 3 min
Display the following:
A sk : Suppose the blue rectangle is covered by smaller green
rectangles. What division problem is shown? How do you know?
The division problem shown is 12 divided by 2 is 6. The original
blue rectangle has area 12, because its dimensions are 3 by 4. The
green rectangles have area 2. 6 green rectangles are needed to
cover the blue rectangle.
S ay : The expression 12 divided by 2 can mean that 12 is divided
into 2 equal groups. That is not what is being shown here. What
does 12 divided by 2 mean in this context?
Here, 12 divided by 2 is asking how many times does 2 fit inside
12.
A sk : Instead of covering the blue rectangle with rectangles of area
2, what size rectangles could we have used?
We could have used rectangles with area 1, 3, 4, 6, or 12. Those
are all the factors of 12 besides 2.
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