PoW-TER Problem Packet
Calculator Key-In (Author: Glenys Martin)
1. The Problem: Calculator Key-In [Problem #3200]
One evening Lisa was doing her math homework when Kristina came by to visit. “The exercise I
have to do,” said Lisa, “is this one.”
Convert this expression to its calculator key-sequence form, then use your
calculator to find its value.
“So far I have this much done,” she added.
12 [x] 5 [-] 8 [/] 4 [+] 7 [x] 2 [=]
“That looks fine to me,” said Kristina. “Let’s evaluate it now. ”Each girl took her own calculator and confidently entered the numbers and symbols as Lisa had
given them. But their smiles quickly turned to puzzled looks when they realized that different
results came up. “How can that be?,” they asked almost simultaneously. “Let's do it again. Maybe we pressed a
key incorrectly.” Once again they entered the expression, only to have the first results be confirmed. “Ah, I think I
know the trouble,” said Kristina. “Let me see your calculator.” Lisa showed her this:
Kristina then said, “Here’s mine.”
As you can see, they were using different kinds of instruments.
1. Explain how the calculators got different answers.
2. Explain which answer is the correct answer. Calculator Key-In 1 Note: This problem, Calculator Key-In, is one of many from the Math Forum @ Drexel's
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2. Saskatchewan Math 6 Number Outcomes N6.3
Outcome N6.3
Demonstrate understanding of the order of operations on whole numbers (excluding
exponents) with and without technology. [CN, ME, PS, T]
3. About the Problem This problem allows the exploration of order of operations within a mathematical sentence.
Since each of the calculators shows a different number, this tests the students’ depth of
understanding. In the next sections the student examples show various misconceptions that
interfere with the intended purpose of the order of operations. Through assigning this problem,
teachers are able to identify students with the various misconceptions and create class dialogues
to guide students in their personal reflection of their mathematical understanding. This problem allows for hands-on observation by using a variety of calculators. By allowing
students to use the two types of calculators they need to determine why the two calculators are
arriving at different answers. This creates a dilemma in many students’ current understanding
since they expect calculators to always calculate the same answer. The lack of clear conceptual understanding is apparent in many of students’ answers. The
students may refer to order of operation or PEDMAS but they incorrectly use the information in
addressing the problem. 4. Common Misconceptions
Students assume that someone made a human error while punching in the numbers. This is a
common occurrence for many students and they assume that is what happened.
Students did not have a clear understanding of order of operations. They assumed that there
would need to be parentheses if you needed to do multiplication and division before addition.
Parentheses appear to confuse the students.
Students do not have knowledge of order of operations or they do not see how it relates to this
problem.
Calculator Key-In 2 5. Sample Student Solutions 1. Students assume that someone made a human error while punching in the numbers. This
is a common occurrence for many students and they assume that is what happened.
Lisa has the correct answer because her and Kristina used different instruments - Kristina's was the professional one – so Kristina may have used the buttons that she
thought meant the same as what she was supposed to press, which of course was
w
12 [x] 5 [-] 8 [/] 4 [+] 7 [x] 2 [=] 40
Krisina used a professional calculator. Lisa used an ordinary calculator. Lisa's calculator is
much, much easier to use. Kristina may have got confused with all of the buttons meaning
complicated things. Lisa's calculator is straight forward. Quick and simple.
For 1 they are two different brand calculator and one is casio and one is texas instrument
and the casio calculator is the right answer
i think the reason why it is lke that becaue either one of them entered a wrong thing or different
calculators and also that i used acasio calulator and i got 72
The calculators are both diffrent.The one to the left is a regular school caluclator that does
not have any special parts toit.The one to the right hasmany diffrent thing that she could
have pressed to mess it up or the calculator could have been for multipucation.
12 x 5 - 8 / 4 + 7 x 2
60 - 8 / 4 + 7 x 2
60 - 8 / 4 + 14
52/4+14 -my pizza pie
52 / 18
2.88
2. Students did not have a clear understanding of order of operations. They assumed that
there would need to be parentheses if you needed to do multiplication and division before
addition. Parentheses appear to confuse the students.
Lisa's anwser on her calculater is correct.
We solved this problem by using the PEMDAS method. We wrote out the equation like what
Lisa did above. Then we followed the equation to find the answer. Lisa was the person who got
the right anwser and who solved corecctly. Kristina did not get the right anwser because she
thought that 8/4 was two, but the / really means divied by in the number sentece. So that is where
Kristina messed up also she did not use the parentheses in her equation. Therefore Lisa was
correct!
Calculator Key-In 3 Lisa is right and Kristina is wrong because she didn't use perentheses.
We found our anwser by doing the full equation. which was 12(x)5=60 60(-)8(=)52. 52(/)4=13.
13(+)7(=)20 20(x)2(=)40. Kristina was wrong because she didn't use parentheses. So she didn't
use prentheses and that changes the anwser because sometimes when you don't use parentheses
so she got the wrong anwser.
Kristina used parentheses but Lisa did not. Lisa is correct.
1. I started doing the first problem by putting in just one set of parentheses. Then I took out that
one set of parentheses and I put another set in that was in a different position. Next I tried the
same method but with two sets of parentheses instead of one. When I got two three sets of
parentheses I knew that there was only one possible way to put three parentheses on this one
problem. This is what I did (12*5)-(8/4)+(7*2)=72. After I did that problem I knew
that if you put three sets of parentheses in 12*5-8/4+7*2 you will get 72.
2. Lisa is correct because 12*5=60, 60-8=52, 52/4=13, 13+7=20, and 20*2=40. Lisa'sanswer was
40.
The girls got different answers because Kristina used parentheses and Lisa did not. Lisa's
answer is correct.
First, I figured out the problem.I got 40. Then I checked on a calculator.Next,I tried using
parentheses and I got 72 on both my calculator,and my paper. The question Lisa was solving
used no parentheses so I knew 40 was correct.
The correct answer is 40. The correct answer is 40 because I did 12 times 5 and got 6o. Then i
went 60 - 8 and got . Next, I went 52 divided by 4 and got 13. Then, I went !3 plus 7 and got 20.
Lastly, I went 20times 2 and got 40, which was Lisa's answer.
1. The calculators got diffrent awnsers becauseKristina put parenthis in. 2. Lisa's
calculator has the right awnser.
Lisa's calculator is the right awnser because 12 * 5 = 60 - 8 = 52 / 4 = 13 + 7 = 20 * 2 = 40.
Kristina looks like she put parenthsis in.
3. Students do not have knowledge of order of operations or they do not see how it relates to
this problem.
Lisas answer is right.
FIRST I DID THE PROBLEM ON THE CACLUATOR. THEN i knew that lisas answer was
right.
Calculator Key-In 4 The calculators got different answers because Kristina added up the answers to the
problems and Lisa didn't. The correct answer is 40.
The calculators got different answers because Lisa did 12*5- 8/4+7*2=40. Then Kristina, instead
of doing 12*5-8/4+7*2=40 she added/multiplied the answers of 12*5 8/4 and 7*2. 60-2+14=72.
This is how they got different answers. However Lisa's answer is correct because on the problem
it didn't say to add up the prouducts and quotients and then do the problem with the answers. So
Lisa is right the answer is 40 not 72.
kristina is right and I think how lisa got a different answer is beacause she got the symbols
mixed up.
I know that 12 times 5 makes 60 minus 8 fourths makes 58 add 14 makes 72.So how I think lisa
got it wrong is beacause she did 12 tmies 5 makes 60 then did 60 minus 14 makes 46 then got 8
forths mixed up with 6
The answere is 40 on Lisa's calculator.
She followed the order of operations.
6. Supporting Mathematical Conversations 1) Students assume that someone made a human error while punching in the numbers. This is a
common occurrence for many students and they assume that is what happened.
a) Have students punch in sequence into their own calculators.
b) As a class determine what answers their calculators reveal? (This may be set up by the
teacher having a calculator which does not do Order of Operations and one that does.)
Pass the calculators around and have a variety of students try.
c) What reasons would one calculator consistently get 40 and the other 72 for the answer?
d) How could you check your calculator to determine if it has the Order of Operations
feature?
2) Students did not have a clear understanding of order of operations. They assumed that there
would need to be parentheses if you needed to do multiplication and division before addition.
Parentheses appear to confuse the students.
a) Order of operations is a convention; this means that there is a general agreement in math
that number sentences are to be solved in a certain order. If the order of solving is not
important, how many answers can you find to this problem?
i) 12×5−8/4+7×2
b) If there are no parentheses in a number sentence does that mean you work it left to right?
c) What other things are considered in the convention of Order of Operations?
d) What is helpful with the acronym PEDMAS? What things do you find confusing?
3) Students do not have knowledge of order of operations or they do not see how it relates to
this problem.
Calculator Key-In 5 a) Concept Formation Flash Activity to help students define the correct Order of
Operations. The activity quickly reveal to students if they have selected the correct
operation to complete next. Through the 5 levels the students should be able to define
what the order of operations that the flash is revealing.
i) If this is done as a class on an interactive board the following questions may help
develop conversations.
(1) When you get an X in the flash activity, what is it telling you?
(2) When the operation you selected is correct, does that mean that should be the first
operation in all number sentences?
(3) How can you use this activity to determine what rules are defined when working
with operations?
7. Additional Support 1. Some students need to ground the math concept with a story to apply real world meaning to
the numbers.
12 × 5 −
€
8
+7×2
4
What could the number sentence be referring to?
Example story: For a sporting event 5 parents brought 12 oranges each. The oranges were
quartered. Eight players each took a quarter of an orange. Then two parents added an additional 7
oranges each. Before anyone takes any more, how many whole oranges are in the bowl?
2. Some students may benefit from the story and visual using a simpler problem.
2× 3−
Mom made three pumpkins pies €
and three blueberry pies for her
restaurant.
She cut the pies into 6 pieces
each, for lunch 12 people order
a piece of pie.
She knew that the supper orders
would require at least 10 pies
and she didn’t what to run short
so she made two lemon pies,
two chocolate pies, two apple
pies and two plum pies.
12
+4×2
6
€
Number sentence
Number of pies
2× 3
6
12
6
4
−
€
+4 × 2
12
€
Calculator Key-In 6