1
U n t er r i ch t spl a n
Divis io n Wit h Quo t ie nt - M ixe d
Numb e rs
Altersgruppe: 4 t h Gr ade , 5 t h Gr ade , 6t h Gr ade
Virginia - Mathematics Standards of Learning (2009): 3 .3 a, 4 .5 b,
6.4 , 6.6a, 6.6b
Virginia - Mathematics Standards of Learning (2016): 6.5 .a
Fairfax County Public Schools Program of Studies: 3 .3 .a.1, 4 .5 .b.6,
6.4 .a.1, 6.4 .a.2, 6.6.a.1, 6.6.b.2
Online-Ressourcen: A P i e c e o f C ake
Opening
T eacher
present s
St udent s
pract ice
Mat h
Pract ice
5
10
18
10
2
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 visual model of fractions.
P r ac t i c e division of whole numbers with a quotient bigger
than one.
L e ar n to represent fractions.
De v e l o p an understanding of the influence of the numerator
and denominator on the value of the fraction.
Copyright 2015 www.matific.com
2
Ope ni ng | 5 min
Draw on the board, one apple, and say the apple represents a w h o le .
Then, draw on the board the following groups of apples:
a.
b.
c.
d.
A sk : What is the numerical value drawing “b.” represents? How do
you know?
Drawing “b.” represents the number 3, because it shows 3 whole
apples.
A sk : What is the numeric value drawing “d.” represents? How do you
know?
Drawing “d.” represents a quarter, because it shows the quotient
we get when dividing a whole apple into 4 equal parts. Drawing
“d.” is a half of a half of an apple.
A sk : What is the numerical value drawing “a.” represents? How do
you know?
Drawing “a.” represents a half, because it shows the quotient
when dividing the whole apple into 2 equal parts.
Copyright 2015 www.matific.com
3
S ay : (to the teacher: the next theme is a little bit complicated. In
weak classes it is recommended this less be skipped) Observe that
each one of the drawings gets its numerical value only after we
have determined the whole. If, for example, we determined a half of
an apple is the whole, drawing “a.” would represent 1, because in
this drawing,there is 1 half of an apple. Drawing “c.” represents 4,
because 4 halves of an apple are 2 apples. All the numbers, wholes
or fractions, relate to a specific whole.
A sk : If half of an apple is the whole, what is the numerical value
drawing “b.” represents? How do you know?
Drawing “b.” shows 3 apples. Every whole apple is 2 halves, so 3
apples are 6 halves, therefore drawing “b.” represents the number
6.
A sk : If half of an apple is the whole, what is the numerical value
drawing “d.” represents? How do you know?
Drawing “d.” shows a quarter of an apple. A quarter is half of a
half. So drawing “d.” represents the fraction a half.
T e ac he r pr e se nt s M at h game : A P i e c e o f C ake - Quo t i e nt
M o r e T han One | 10 min
Using preset mode, on the projector, present Matific ’s episode A Pie c e o f
C a k e - Qu o t ie n t M o r e T h a n On e to the class.
This episode presents whole-number division, with fraction and mixednumber quotients. The task is to divide a given number of cakes evenly
between a given number of plates - for example divide 5 cakes evenly
between 3 plates. Each cake can be partitioned into portions of equal sizes,
using a knife. Starting with the second question, a corresponding word
sentence has to be completed.
Copyright 2015 www.matific.com
4
E x a m p le :
S ay : In each screen there are a number of plates, cakes, and a knife.
We need to divide the cakes between the plates equally. We can
use the knife to cut each cake into a number of slices (2, 3, 4, or 5
equal slices).
Use the knife and cut one of the cakes into a number of slices.
Explain that one can use the cancel button
return the situation as it was before.
. Click the cancel button and
A sk : We need to divide 5 cakes into 2 plates equally, how should
we do it?
Answers may vary. Let’s simply try a few options and see which of
them works.
Place one cake in each plate. Show that we left with 3 more cakes so we can
continue the division. Place another cake in each plate. Now we have 2 cakes
on each plate and one cake on the table.
E x a m p le :
Copyright 2015 www.matific.com
5
A sk : How should we continue?
We divide the cake, left on the table, into 2 equal slices using the
knife, and place half of the cake on each plate.
Divide the cake, that is left on the table, into 2 halves and put each half on a
plate.
A sk : How much of the cakes are now on each plate?
Each plate has 2 whole cakes and a half.
S ay : Observe that when we divide 5 cakes onto 2 plates, we
actually are solving the problem 5 ÷ 2.
Write on the board:
Click on
and present the next question.
E x a m p le :
Copyright 2015 www.matific.com
6
S ay: Please read the instructions on the bottom of the screen.
Students can read the instructions.
S ay : We need to complete the equation 2 ÷ 3 =. We can use the
cakes, plates, and knife.
A sk : How many slices should we cut each cake, so the slices will
divide equally between the 3 plates? How do you know?
There are 3 plates,so we cut each cake into 3 slices and place 1
slice on each plate.
Cut the first cake into 3 slices and place each slice on a different plate, while
emphasizing the distribution manner. Repeat the same process on the next
cake.
A sk : So what is the answer to the problem 2 ÷ 3 ?
. We divide each cake into 3, so each slice equals of a
cake. On each plate there are 2 slices, so on each plate there are
2 times , which means on every plate there is
Copyright 2015 www.matific.com
of a cake.
7
Enter
and present the next question.
E x a m p le :
S ay: Please read the instructions at the bottom of the screen.
Students can read the instructions.
A sk : How would we complete the equation 5 ÷ 3 = using the cakes,
plates and the knife?
We put a whole cake on each plate and we are left with 2 cakes
on the table. Now we need to divide the remaining 2 cakes onto 3
plates equally. We cut each cake into 3 slices and put each slice
on a different plate.
Demonstrate the process.
E x a m p le :
Copyright 2015 www.matific.com
8
A sk : So how much is 5 ÷ 3 ? How do you know?
. On every plate there are 1 whole cake and 2 more
slices of cake. We cut each cake into 3, so each slice equals
a cake. Therefore, on each plate there is
of
of a cake.
Write on the board:
S t ude nt s pr ac t i c e M at h game : A P i e c e o f C ake - S har e s
M o r e T han W ho l e | 18 min
Have students play A Pie c e o f C a k e - S h a r e s M o r e T h a n W h o le and
A Pie c e o f C a k e - Qu o t ie n t M o r e T h a n On e on their personal
devices.
Circulate, answering questions as necessary.
Copyright 2015 www.matific.com
9
M at h P r ac t i c e : Di v i si o n w i t h Quo t i e nt B i gge r T han One
W o r kshe e t | 10 min
Hand students the first printable, and guide them to paint the
shapes so they represent the mixed numbers on every line. If the
students have difficulty, present an example on the board. Circulate,
answering questions as necessary. When done, review the answers.
Hand students the second printable, and guide them to write, on
each line, the fraction that the painted shapes are representing.
Circulate, answering questions as necessary. When done, review the
answers.
Pr in t a b le h a n d o u t : Div is io n W o r k s h e e t A
Paint the shapes so that they represent the mixed numbers and improper
fractions:
Copyright 2015 www.matific.com
10
Pr in t a b le h a n d o u t : Div is io n W o r k s h e e t B
Write the mixed numbers that are being represented in each line:
Copyright 2015 www.matific.com
11
Copyright 2015 www.matific.com
12
C l o si ng | 2 min
S ay : The number above the fraction line is called the nume r at o r
and the number below the fraction line is called de no mi nat o r .
A sk : What is the denominator?
The denominator represents the number of equal parts we have
divided the whole into. The denominator actually determines the
size of each part, because the larger the denominator, it means
we divided the whole into more parts, and that means that each
part is smaller.
A sk : What is the numerator?
The numerator identifies the number of parts we choose, out of
the parts that the whole was divided to.
Copyright 2015 www.matific.com