Grade 6: The Integers and the Temperature Tank Model
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Temperature Tank Model: A model for the integers is the temperature tank. The tank,
when it has zero chips in it is at zero degrees. If you add a PLUS chip, then the
temperature in the tank goes up one degree. If you add a MINUS chip, the temperature
in the tank goes down one degree.
Instructions: Please model what you are doing on the open number line AND express
each situation numerically.
1. The tank starts off with no chips inside it and is at zero degrees. Explain
what happens to the temperature of the tank, if you add three PLUS chips.
2. The tank starts off with no chips inside it and is at zero degrees. Explain
what happens if you add five MINUS chips.
3. The tank starts off with 6 PLUS chips and 5 MINUS chips. What is the
temperature of the tank?
4. The tank starts off with 6 PLUS chips and 5 MINUS chips. Explain what
happens to the temperature of the tank if you remove 3 PLUS chips.
5. The tank starts off with 6 PLUS chips and 5 MINUS chips. Explain what
happens to the temperature of the tank if you remove 3 PLUS chips, and then
remove 4 MINUS chips.
6. The tank starts off with 8 PLUS chips and 17 MINUS chips. Explain what
happens to the tank if you take out 3 MINUS chips.
7. The tank starts with 7 MINUS chips in it. Then Francisco adds 3 more MINUS
chips into the tank. At what temperature is the tank?
Definition: A zero pair is a pair of numbers that add up to zero.
8. List as many zero pairs as you can, but organizing them into a table. After
you create a table of all the zero pairs, explain any patterns you notice.
9. How many possible different ways are there to represent using zero pairs the
temperature zero degrees? Explain.
10. Mr. Q is wondering what would happen if you add a billion (1,000,000,000)
PLUS chips into the tank and then add a billion MINUS chips into the tank.
Mr. Q thinks that because the numbers are so large the result must be
REALLY big! Help Mr. Q figure out what is going on.
11. How many different ways can you express the number 5 using the
temperature tank? List at least 7 different ways.
12. How many different ways can you express the number (-3) using the
temperature tank? List at least 7 different ways.
13. The temperature tank is at 5 degrees, and you take out of the tank 3 MINUS
chips. What is the resulting temperature of the tank?
14. The temperature of the tank is -3 degrees. You reach into the tank and take
out 4 MINUS chips. Find what the temperature of the tank will be.
15. The temperature of the tank is 10 degrees, and you take out 11 MINUS chips.
What will the temperature of the tank be? Explain.
Connecting the model to numbers.
16. Using the temperature tank, explain why subtracting a negative is equivalent
to adding the absolute value of that number. For instance 5-(-7)=12.
Multiplication is conceptually a shorthand way to represent repeated
addition. Some ways to think about multiplying three times four is
3x4=3+3+3+3=4+4+4
Using the notion of repeated subtraction, explain why 3 x -4 = -12.
17. Explain why (-6) x 4 is equal to -24.
18. Using the notion of repeated subtraction, explain why (-3) x (-4) = 12. That
is, why is it the case that a negative times a negative is a positive?
19. Mr. X said to Mr. Y, I wonder why when you multiply (-6) times (-6) that you
get 36 instead of -36. Figure out way to help Mr. X figure this out.
20. Mr. D and Mrs. I need your help. They have a tank that is at -5 degrees. They
want to take out from the tank 25 MINUS chips in order to raise the
temperature of the tank to a warmer temperature. But they are having some
trouble. You see, the way they got to -5 degrees (after a long day of moving
chips in and out) is that they have 2 positive chips in the tank and 7 negative
chips in the tank. There are not enough negative chips in the tank, so your
teachers are stuck. They do have a big bag full of PLUS and MINUS chips,
however, that might be useful. Write a short letter describing what Mr. D and
Mrs. I need to do.
21. Looking Back: Make a list of the big ideas you learned in this unit. Then
explain why these ideas are important mathematically.
22. Ask your parents? Do your parents know why subtracting -7 is equivalent to
adding 7? Find out and see what they say. Then show them how you think
about it using the temperature tank model.
23. Practice!
a. 12-4=
b. -12 + 4
c. -12 -4=
d. -12-(-11)=
e. -12-(-13)=
f. -100-(-201)=
g. 1001-2001=
h. 2001-1001=
i. 2001-(-1001)=
j. -2001+1001=
k. -2001-(-1001)=
l. -1001-(-2001)=
m. -3 x 5 =
n. (-3) x (-5) =
o. 3 x (-5) =
p. (-11) x 7 =
q. 11 x (-7) =
r. (-7) x (-11) =
s. (-3) x (-5) x (-4) =
t. (-2) x (-2) x (-2) x (-2) =
24. If you multiply 17 numbers together, and you know that all numbers are
negative, then will the result be greater than zero or less than zero? Explain.
25. In each pair of numbers below figure out the larger number.
a. 3, 7
b. 0, -1
c. -2, 2
d. -3, -2
e. -3, -7
f. -1, -1,000,000
26. In this problem you are asked to explain why -7 is smaller than -3. This looks
strange at first, because 3 is less than 7. Write an explanation why -7 is less
than -3, using the temperature tank model.
27. We know that division is equivalent to repeated subtraction. Explain why
-60/-6 = 10
28. Based on your work from the previous problem, what is the sign of A/B,
where both A and B are negative numbers?