Download Math 107A Exam 2 Solutions 1 1. Consider the product 281 × 47. (a

Math 107A
Exam 2 Solutions
1
1. Consider the product 281 × 47.
(a) Use the expanded array model to compute 281 × 47. Please show all work and circle your final
answer.
200
80
1
40
8000
3200
40
7
1400
560
7
8000
3200
1400
560
40
+
7
13207
Thus 281 × 47 = 13207 .
(b) Use the standard algorithm to compute 281 × 47.
3
5
281
× 47
1967
11240
13207
(c) Look at the number in the bottom left corner of your array model. Where does that number
show up in the standard algorithm?
The 1400 is hidden within the 1967 in the standard algorithm. The 19 came from 7 times 2 plus
5. It is really 1400 because the 4 contributed to the 9 which is in the hundreds place and the 1
is in the thousands place.
2. Max computes 45 × 8 in his head in the following way. First he multiplies 45 by 10 and he gets 450.
He then subracts 90, to get 360. So 45 × 9 equals 360.
(a) Does Max get the correct answer? Circle the appropriate answer.
YES
NO
(b) Use Max’s method to compute 35 × 8. Be sure to show all work so that I can tell you are using
Max’s method. Please circle your final answer.
35 × 10 = 350
35 × 2 = 70
350 − 70 = 280
Thus 35 × 8 = 280 .
(c) Explain why Max’s method works.
Max is trying to compute 45 × 8, but this just means he needs to figure out what 8 45’s added
together equals. Instead he decides to add 10 45’s together, but this is 2 45’s too many so he
subtracts 2 45’s (which is 90) off which leaves him with 8 45’s.
Math 107A
Exam 2 Solutions
2
3. (a) Use the definition of division to compute 56 ÷ − 8. Be sure to show enough work so that I know
you are using the definition to compute this quotient. Please circle your answer.
By the definition of division we need to solve the following equation.
−
8×? = 56
Thus ? must equal − 7.
So 56 ÷ − 8 =
−
7.
(b) What rule of signs for multiplication tells us that a positive divded by a negative will be negative?
Explain.
If we take the division problem (+) ÷ (−) =? and turn it into the multiplication problem
(−)×? = (+). Thus ? must be negative. Hence the rule of signs (−) × (−) = (+) is what tells
us that a positive divided by a negative is a negative.
4. Consider the division problem 173 ÷ 8.
(a) Before doing any computations, list all possible remainders.
Since we are dividing by 9 the only possible remainders are 0,1,2,3,4,5,6,7.
(b) Compute 173 ÷ 8 and fill in the boxes below according to your results.
173
8
=
×
21
+
5
5. (a) State a word problem whose solution is found by computing 432 ÷ 18.
(b) What model of division does your word problem represent?
6. Use colored counters and the take away model to compute the following. Please circle your final
answer.
(a)
−5
− −2 =
(b) 4 − 7 =
−
3
−
3.
Math 107A
(c)
−
Exam 2 Solutions
−
5−3 =
8
7. Recall our list of KNOWs shown below.
KNOW#1: All arithmetic, properties, etc. with whole numbers.
KNOW#2: a + − a = 0
KNOW#3: The answer to a + − b when a is bigger than b
KNOW#4: The answer to a + − b when a is smaller than b
(a) Using only KNOW#1 and KNOW#2, compute 41 + − 23.
41 + − 23
= 18 + 23 + − 23
= 18 + 0
= 18
(b) Using only KNOW#1, KNOW#2, and KNOW#3, compute 32 + − 49.
32 + − 49
= − 17 + 49 + − 49
= − 17 + 0
=
−
17
(c) Using only KNOW#1, KNOW#2, KNOW#3 and KNOW#4, compute − 17 + − 29.
−
17 + − 29
= − 46 + 29 + − 29
= − 46 + 0
=
−
46
3
Math 107A
Exam 2 Solutions
4
8. (a) Use the definition of multiplication to compute 5 × − 8.
5 × − 8 = − 8 + − 8 + − 8 + − 8 = − 40
(b) Explain why you cannot use the definition of multiplication to compute − 5 × − 8.
If we were to use the definition of multiplication this would say to add − 8 to itself negative 5
times, but doing something negative 5 times doesn’t make sense.
(c) Assuming that you now know how to multiply a positive and a negative, compute − 5 × − 8. Be
sure to show all work so that I can see how you computed this product. Please circle your final
answer.
−
−
−
5×0 = 0
5 × (8 + − 8) = 0
5 × 8 + −5 × − 8 = 0
−
40 + ? = 0
Thus ? must equal 40, so − 5 × − 8 = 40 .