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U n t er r i ch t spl a n
Numb e r R id d l e s - Ad d it io n And
Divis io n
Altersgruppe: 4 t h Gr ade , 5 t h Gr ade
Virginia - Mathematics Standards of Learning (2009): 4 .4 a, 4 .4 c ,
5 .4
Virginia - Mathematics Standards of Learning (2016): 3 .4 .a, 3 .4 .b,
4 .2.c , 4 .4 .a, 4 .4 .d
Fairfax County Public Schools Program of Studies: 4 .4 .a.1, 4 .4 .c .2,
5 .4 .a.3 , 5 .4 .a.8
Online-Ressourcen: M agi c T r i angl e
T eacher
present s
St udent s
pract ice
Mat h
Pract ice
5
8
20
10
2
min
min
min
min
min
Opening
Closing
M at h Obj e c t i v e s
E x pe r i e nc e problem solving.
P r ac t i c e solving addition and division number riddles.
L e ar n that there are a number of ways to approach a riddle, and
there may be multiple solutions for a riddle.
De v e l o p problem solving skills.
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Ope ni ng | 5 min
Bring to class 25 markers and show them to the students.
A sk : Suppose we want to have 13 markers in two groups together,
and in one group we already have 4 markers, how many markers
should we have in the second group?
In order to have 13 markers altogether, in the second group we
should have 9 markers, because 4 + 9 = 13.
Demonstrate the steps, using the markers, to find the solution. First, add 1
marker to the 4 and show that we did not reach 13, then add 2 markers and
show that we again did not reach 13.
A sk : Suppose we have one group with 5 markers (show the students
5 markers in your hand), and suppose we want to form another group
of markers so that the sum of the two groups together is divide by
4, how many markers should I place in my second hand?
There are several possible answers. For example: I could take, in
my second hand, 3 markers so that the sum (8) will divide by 4, or
I could take 7 markers so that the sum (12) will divide by 4.
A sk : What is the connection between these two answers?
The first answer (with the minimal number of markers - 3) added
to the 5 markers, results in the number (8), that divisible by 4. All
other potential answers are repeats of the first answer, adding 4
more markers. Therefore, the possible answers are 3, 7, 11, 15
and so on.
Demonstrate the different solutions using the markers. First add 3 markers
to the 5, then add 7 markers to the 5, and so on.
S ay : Therefore, observe that there is a connection between sums
of groups to their property of dividing into a certain number.
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S ay : Today we will solve riddles with numbers. When we tackle a
problem, consider:
There may be a number of ways to solve the riddle.
Sometimes it is good to break down the riddle into smaller parts,
and solve each part separately.
Trial and error - don’t be afraid to try! (and maybe make a mistake).
We can learn from mistakes (for example, what is the r i ght way to
proceed).
T e ac he r pr e se nt s M at h game : M agi c T r i angl e Di v i si bi l i t y P uz z l e s | 8 min
Present Matific ’s episode M a g ic T r ia n g le - Div is ib ilit y Pu z z le s to the
class, using the projector.
The goal of episode is to solve number riddles in the context of divisibility by
dragging numbers to the edges and vertices of a triangle, such that the sum
of the numbers on each edge is divisible by a given number.
E x a m p le :
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S ay : We need to place the numbers from the bottom of the screen
on the vertices and edges of the triangle, so the sum of the
numbers, on each edge, is a multiple of 2.
A sk : If the sum is a multiple of 2, what does it indicate about the
property of the sum? What kind of a number is it?
If the sum is a multiple of 2, then the sum is an even number.
S ay : In order to get to an even sum, we need to use three even
numbers or two odd numbers and one even number.
S ay : Let’s start placing numbers according the rule we have found.
Because we have three odd numbers, they cannot be on the same
edge.
Place the odd numbers on a different edges, so that two of them are on the
same edge.
E x a m p le :
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A sk : How should we proceed?
Now we can place the three even numbers, no matter the order.
E x a m p le :
A sk : Does the sum on each edge represent a multiple of 2?
Yes. the sums are 14, 16 and 12.
S ay : Notice that we managed to solve the problem, regardless of
the exact value of each number and each sum, but only by
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categorizing the numbers into even\odd numbers. We could have
solved the problem in a different way, where we would calculate the
sums of the edges. But, in this case, the method we used is easier
and faster. So sometimes there are several ways to solve a
problem.
Click on
and present the next question.
E x a m p le :
S ay : We need to place the numbers so the sum on each edge is a
multiple of 5.
A sk : How should we start?
Answers may vary. We have to think of sums that can be divide by
5. For example, the number 6 has to be on the same edge with the
numbers 9 and 5, because any other combination leaves us with a
sum which is not a multiple of 5 (6 + 5 + 1 = 12, 6 + 9 + 1 = 16).
Place 6, 5, 9 on the lower edge.
E x a m p le :
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A sk : How should we proceed?
9, 5 and 1 also have a sum which is a multiple of 5. So we place 9
and 5 on the left edge, and the number 1 on the right one.
E x a m p le :
A sk : Is the sum on each edge a multiple of 5?
Yes. the sums are 20, 20 and 15.
S ay : Consider an appropriate plan for the placement of the
numbers, before we start to place them, can be very helpful in
solving the problem.
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S ay : In case you don't know how to proceed, you can get help by
clicking
.
S t ude nt s pr ac t i c e M at h game : M agi c T r i angl e Di v i si bi l i t y P uz z l e s | 20 min
Have students play M a g ic T r ia n g le - Div is ib ilit y Pu z z le s on their
personal devices.
Encourage students to try various options, and not to “get stuck” in their
approach of the riddle.
Circulate, answering questions as necessary.
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M at h P r ac t i c e : N umbe r R i ddl e s W o r kshe e t | 10 min
Write the following riddle on the board:
S ay : Choose three numbers and show how they are connected by
multiplication and division. For example: 2 × 5 = 10; 10 ÷ 2 = 5; 10 ÷
5 = 2.
A sk : How many groups of three did you find?
After the students finish, share the answers.
A sk : Which methods did you use, in order to solve the riddle?
The main method is trial and error. We take every pair of numbers
(in an order) and check whether the number that connects them (if
any), by multiplication or division, is in the rectangle.
Divide the class into pairs. In each pair, each student should prepare a riddle
(like the one we just solved) by drawing a rectangle containing 12 numbers.
After students are finished, exchange the riddles and solve.
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C l o si ng | 2 min
A sk : Repeat the different methods we discussed to solve a
problem.
Making a form of the problem by using objects (for examples with
markers).
Break down the riddle into smaller parts, and solve each part
separately.
Trial and error.
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