Download Basic Negative Number Addition on Number Line with Unknown

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U n t er r i ch t spl a n
Bas ic Ne g at ive Numb e r Ad d it io n
o n Numb e r Line wit h Unkno wn
Altersgruppe: 6t h Gr ade
Virginia - Mathematics Standards of Learning (2016): 6.13 , 6.14 .a,
6.14 .b
Online-Ressourcen: S av e t he B al l
Opening
T eacher
present s
St udent s
pract ice
Class
discussion
8
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12
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4
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Closing
M at h Obj e c t i v e s
E x pe r i e nc e visual interactive addition model on number line.
P r ac t i c e addition of negative numbers with unknowns, using
the number line.
L e ar n to use a number line to find unknown quantities.
De v e l o p addition strategies for negative numbers.
Ope ni ng | 8 min
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Display a number line with both positive and negative numbers.
S ay : This number line has both positive and negative numbers. As
you know, zero is at the center all numbers to the right of 0 are
positive, with numbers to the left being negative.
A sk : If a ball on -3 bounces to -5. How many places has it moved?
What direction?
It has moved 2 places towards the left (negative direction).
Or we can say that to reach -5 from -3 we add 2 places in the
negative direction, or -2.
So, the equation would be -3 + (-2) = -5
S ay : Let us figure out if the ball bounces from -3 to positive 5.
How many places will it move? What direction?
Positive 5 (or simply 5) is to the right of -3, so the ball will move
towards the positive direction.
The equation would be -3 + (+8) = 5
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Students may come up with an equation like: -3 + (-3) + (+5) = 5
(-3) for covering 3 negative places and (+5) for covering 5 positive places.
Explain that the sign (either -ve or +ve) is decided by the direction of the
movement. Since the ball moved towards right (or positive direction), the sign
attached to number 8 is positive.
A sk : I am standing on 0 and have to move to -7. The blocks that I
may use are -5, -3, 4, -4, 2. Which block or blocks should I use to
reach -7?
Display the same question on the board.
S ay : You may draw a rough number line to find the solution.
A sk: What should be our strategy for choosing the blocks?
We choose a block which will take us 7 blocks to the left
(negative side). Since there is no block for -7, we find blocks,
when combined, total to -7.
-3 and -4 is the only solution.
Explain that though 5 and 2 also add up to 7, these numbers have different
signs and will not give the desired result.
The negative 5 will move 5 blocks left while positive 2 will move 2 blocks
towards right and reach -3. (Show it on the number line)
A sk : Which blocks should I choose to reach -9?
Blocks -5 and -4.
The numbers total to 9 and both blocks move left.
Show that -5 and 4 is not a solution as it will result in -1.
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T e ac he r pr e se nt s M at h game : S av e t he B al l - A dd
N e gat i v e | 10 min
Present Matific’s episode S a v e t h e B a ll - A d d N e g a t iv e to the class,
using the projector or interactive whiteboard in a preset mode.
This episode practices addition of negative numbers using a number line.
There is a beam with numbered holes. You have, at your disposal, springs
that make the balls bounce either to the left or to the right an indicated
number of notches. Place springs in the correct places in order for the ball to
bounce into the net. You may have to place more than one spring for the ball
to reach the net.
S ay : Please read the instructions at the bottom of the screen.
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Students can read the instructions.
A sk: What do you need to do?
Place springs in the correct holes so the ball reaches the net.
A sk: When the ball is released where would it fall? Where is it
intended to fall? Where is the net?
The ball will fall in hole #0 when released, so we need to put a
spring on #0, so the ball bounces and falls in hole #(-5).
We need to place a spring which will move the ball 5 places in
negative direction (left).
We select a spring with number 5 with an arrow pointing left.
Place the spring and press “Go”. The ball will bounce to hole #(-5)
The equation is: 0 +(-5) = -5 Or 0 - 5 = -5
Press the arrow key to go to the new screen.
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A sk: Where should the ball reach? What should be done.
The ball has to reach “-7” or 7 places to the left. So, we have to
select springs pointing left.
There are two springs #5 and #2, both pointing left and totaling
#7. So, we need to place both the springs.
S ay : Which spring do we place first? Can we place either one first
or do we have to place a particular spring first? Let us find out.
Place #5 at 0.
S ay : This spring will make the ball bounce to ‘-5’. So we will place
spring #2 here at -5. The ball will then bounce to “-7” into the net.
-5 + (-2) = -7
S ay : Let us check by placing #2 on 0. The spring will take the ball to
“-2”.
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Place #5 at “-2”. Show that the ball will bounce to “-7” into the net.
-2 + (-5) = -7
S ay : We conclude that we can place either of the springs first (on
0).
So, -5 + (-2) = -7 and also -2 + (-5) = -7
Press “Go”. Either of the above equations will show on screen, depending on
the order of the placed springs. Press the arrow key. A new screen will
appear.
S ay : Press the left or right arrow to look for the net.
The net is on “-12”.
S ay: Where should the ball reach? What should be done?
The ball has to reach “-12” or 12 places to the left. So, we have to
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select springs pointing left which will add up to 12.
A sk: Which springs should we select? Why?
We will select springs #5 and #5, both pointing left, which will
add up to 10, and the #2, with the arrow pointing left. All the three
springs add up to 12. We need to place the three springs in a
manner that the ball reaches “-12”.
A sk : What are the various possible sequences, or orders, in which
these springs can be placed? Will all these take the ball to “-12”?
Encourage students to come up with the various sequence and write all the
possible solutions on board.
#5 (spring1) + #5 (spring 2) + #2 (spring 3) equation would be (-5) + (-5) + (-2)
= -12
#5 (spring1) + #2 (spring 3) + #5 (spring 2) equation would be (-5) + (-2) + (-5)
= -12
#5 (spring 2) + #2 (spring 3) + #5 (spring1) equation would be (-5) + (-2) + (-5)
= -12
#5 (spring 2) + #5 (spring1) + #2 (spring 3) equation would be (-5) + (-5) + (-2)
= -12
#2 (spring 3) + #5 (spring1) + #5 (spring 2) equation would be (-2) + (-5) + (-5)
= -12
#2 (spring 3) + #5(spring 2) + #5 (spring1) equation would be (-2) + (-5) + (-5)
= -12
S ay :Mathematically, if the numbers add up to -12 then we have the
correct answer.
Let us check two sequences to find if they are the correct
solutions.
Check:
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Place #5 at 0.
S a y: This spring will make the ball bounce to “-5”. So we will place
spring #2 here at “-5”. The ball will reach “-7”. Place #5 at “-7”, the
ball will then bounce to ‘-12” into the net.
-5 + (-2) + (-5) = -12
S ay : Let us check by placing #2 on 0. The spring will take the ball to
“-2”.
Place #5 at “-2”. Show that the ball will bounce to “-7”, place #5
at “-7”, the ball will bounce to “-12”, into the net.
-2 + (-5) + (-5) = -12
S ay : We conclude that we can place the springs in any any order.
-5 + (-2) + (-5) = -12 and also -2 + (-5) + (-5) = -12
A sk : What did you observe?
Both are correct solutions.
We conclude that the springs can be placed in any order.
(Commutative Law of Addition)
Press ‘Go’. Either of the above equation will show on screen depending on the
order of placed springs.
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S t ude nt s pr ac t i c e M at h game : S av e t he B al l - A dd
N e gat i v e | 12 min
Have the students play Matific’s episode S a v e t h e B a ll - A d d N e g a t iv e
on their personal devices. Circulate, answering questions as necessary.
Students may need help to decompose numbers. Give them hints rather then
the answer.
C l ass di sc ussi o n | 10 min
Ask the students what they thought was most challenging in this game, and
reiterate the steps to ease their anxiety.
Students should be able to relate the game with Associative and
Commutative Properties of Addition, of negative numbers.
Hand over the attached worksheet and discuss the first problem. The rest
can be assigned as homework.
Ask students to solve the first question.
Discuss the first question to check if the concept is clear by asking
questions such as:
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On what number was the net clamped?
What was the first hole the ball would fall? Did everyone use the
same spring first? Did everyone use all the springs in the same
order?
What was it they found challenging about finding the numbers,
that helped to reach the net?
Different combinations to reach the net.
Strategies used to find the right combination.
Pr in t a b le H a n d o u t : N u m b e r lin e , N e g a t iv e N u m b e r s W o r k s h e e t
Arrange the springs such that the ball falls in the net. You may use different
color pencils to trace the ball.
Example:
This is one of the solutions: -1 + (-2) + (-5) = -8
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C l o si ng | 4 min
A sk: What did we learn today?
Addition of negative numbers on the number line to find more
than 1 unknown number (addend).
Partitioning numbers
When two or three numbers are be added in any order, we would
still get the same result.
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