Use the given number game to answer the following question.

[eddy2017]

The instructionS are kind of fuzzy on this.
Pick a number.
Add 5 to the number.
Next, multiply 4.
Subtract 12.
Divide by 4. ????? what, the last operation or the whole numerator?.

 

[lev888]

The instructionS are kind of fuzzy on this.
Pick a number.
Add 5 to the number.
Next, multiply 4.
Subtract 12.
Divide by 4. ????? what the last operation or the whole numerator?.

I see absolutely no difference between multiply by 4 and divide by 4 regarding to what they are applied. With multiplication you decided to multiply the entire expression. Why do you think there is an ambiguity with division?

 

[eddy2017]

I see absolutely no difference between multiply by 4 and divide by 4 regarding to what they are applied. With multiplication you decided to multiply the entire expression. Why do you think there is an ambiguity with division?

I am following the instructions given in the exercise.

 

[eddy2017]

(x+5)⋅4−12/(4)
So, can i continue simplifying this?
[math]4x+20−12 / (4)[/math]
[math]y=4x+8/(4)[/math]
[math]y=x+2[/math]

 

[eddy2017]

okay y= x + 2

I will take a look now at the choices I have
y=2y equals 2
y=2xy equals 2x
y=x − 2y equals x minus 2
y=x+2y equals x plus 2

y=x+2

If x represents the starting number, and y represents the final number, which equation is equivalent to the number game?
D
Hooray!

 

[lev888]

(x+5)⋅4−12/(4)
So, can i continue simplifying this?
[math]4x+20−12 / (4)[/math]
[math]y=4x+8/(4)[/math]
[math]y=x+2[/math]

You are not answering my question.
In case of multiplication you added parentheses so that it applied to the whole expression we had so far. Why is it not clear that we need to do the same in case of division?

 

[eddy2017]

You are not answering my question.
In case of multiplication you added parentheses so that it applied to the whole expression we had so far. Why is it not clear that we need to do the same in case of division?

Don't understand what you're asking. Can you exemplify it for me in an equation . Don't tell me what to do but express your question with a mathematical notation, if you will. It sounds too abstract for me.
I think because we distribute using multiplication, not division. We don't distribute division. You foil only when you multiply, hence, the need for parentheses.

 

[lev888]

Don't understand what you're asking. Can you exemplify it for me in an equation . Don't tell me what to do but express your question with a mathematical notation, if you will. It sounds too abstract for me.
I think because we distribute using multiplication, not division. We don't distribute division. You foil only when you multiply, hence, the need for parentheses.

1. Whether or not the distributive property can be used with division does NOT matter for the current task. We apply the distributive property when we are evaluating or simplifying expressions. Right now we are writing the expression. Do you see the difference?

When writing an expression we are concerned with the order of operations. For the first 2 steps do we add x and 5 and then multiply the result by 4? Or do we multiply 5 by 4 and then add the result to x?
The answer is, we add first, because that's what the problem tells us to do. And since multiplication is done before addition we MUST use parentheses to tell the world that we want addition done first:
(x+5)*4.

Now, division by 4. What is the order of operations? Do we want to calculate the expression we have so far and then divide the result by 4? Or divide something by 4 and then do something else?

2. Regarding the distributive property. Note that division by a number is the same as multiplication by its reciprocal?
E.g. 6/2 = 6 *(1/2). If this is the case, why couldn't we use the distributive property?

 

[eddy2017]

Now, division by 4. What is the order of operations? Do we want to calculate the expression we have so far and then divide the result by 4? Or divide something by 4 and then do something else?

I have learned it this way. Multiplication, then division, going from left to right,
Caveat: It may come in handy, while working an equation to do the division first even if it comes after the multiplication. That I learned while watching a video about Order of Op. Don't remember now the source.
In the case I hand, I will multiply what I have and then divide the 4.

 

[JeffM]

because if I want to multiply 4 to the entire term x+5 then I have to use parentheses.

Eddy

Remember what I said about grammar and translation.

The problem says add 5 to the number.

Then it says multiply by 4. What is the intended English meaning? Pretty clearly what is implied is to multiply the sum calculated in the previous step by 4. (I admit the sentence is not explicit.) If I want to add before multiplying, I MUST USE PARENTHESES because the standard order of operation says to do multiplications and divisions before additions and subtractions UNLESS GROUPING SYMBOLS SAY OTHERWISE.

Then it says to subtract 12 from the result in preceding step. (Again, the sentence is not as explicit as it should be, but that is the sensible interpretation.)

Finally, it says to divide the result from the preceding step by 4.

If x is the number chosen, how do you translate into math notation the instructions

Compute the sum of x and 5.
Multiply that sum by 4.
Subtract 12 from the product computed in the preceding step.
Finally, divide the difference computed in the previous step by 4.

Call the resulting number y. What does y equal?

I have removed the slight ambiguities in the original statement of the problem.

You have to do the translation into mathematical notation remembering its grammar abbrevisted by PEMDAS.

Finally, you have to simplify the resulting expression.

 

[eddy2017]

Excellent, Jeff. I'll start working on it. Allow me some time. Still in bed. Thanks.

 

[eddy2017]

Excellent, Jeff. I'll start working on it. Allow me some time. Still in bed. Thanks.

One doubt, Jeff, before I start. The problem says to pick a number. I do pick a number, right?. that number is not x, right?. I have to pick a number, let's say I pick 3, and then I continue to follow the computation, is it correct?. Because at the beginning I was confused about the 'pick a number' bit. I thought of that number as being a variable, let's say x, so it is not a variable , I have to pick an actual number?. Confirm this when you see this, please.

 

[eddy2017]

I dont understand this. it is contradictory. thye ask me to choose a number. choosing a number is picking one let's say 3
then you wrote this:

If x is the number chosen, how do you translate into math notation the instructions
Compute the sum of x and 5.

by x, should I take it mean the number i choose?, eg, x=3, is that it?

 

[JeffM]

If this be madness, yet there is method in it.

You are a teacher and know that part of teaching is looking for the clearest way to explain.

Also, YOU are trying to learn algebra.

The way to do that is to use algebra every chance you get.

What is 3 more than 8? 11 obviously.

What is 3 more than x? x + 3.

Expressions are necessary in algebra. They are not necessary in arithmetic. We teach PEMDAS in algebra or pre-algebra rather than arithmetic.

It is true that there are many, many problems in arithmetic that are most naturally done in steps. First you do one computation. Then you do a second computation using the result of the first computation. Then you do a third computation using the result of the second computation. But, using math notation, it is possible to summarize such a process in an arithmetic expression using PEMDAS, but it is not natural. It is not the way we have learned to do arithmetic.

I really, truly, sincerely hate this problem. To introduce PEMDAS, it uses arithmetic, which is not where PEMDAS is necessary or routine. And it does not even make explicit that it is specifying a process where the result of each step is to be used as the starting point for the next step. And then it asks you to generalize with x and y.

I want you to begin thinking algebraically, which frequently involves translation from a natural language into mathematical notation, the language of mathematics. I also want you to learn abstraction and generalization, which are central aspects of ALL mathematics past arithmetic. So I changed the problem. I made explicit that the process involves using the result from one step in a later step. The problem involves order, what is done first, what is done second; in short, the order of operations. That sequence of operations can be described generically using algebra. I asked you to translate the process into generic form. When you use algebra, you will find a short-cut.

 

[eddy2017]

I really, truly, sincerely hate this problem. To introduce PEMDAS, it uses arithmetic, which is not where PEMDAS is necessary or routine. And it does not even make explicit that it is specifying a process where the result of each step is to be used as the starting point for the next step. And then it asks you to generalize with x and y.
I made explicit that the process involves using the result from one step in a later step. The problem involves order, what is done first, what is done second; in short, the order of operations.?
Here I go for better or worse!.
choose any number
I am picking 3
following the instructions in the order of operations

Pick a number.
Add 5 to the number.
Next, multiply 4.
Subtract 12.
Divide by 4.

I will order the steps to fit the order of OPS.
Choose the number.

I choose a number =5
multiply by 4= 20
divide by 4 =5
add 5 =10
sub 12
-2

That is what I get following the correct order of OPS.

 

[JeffM]

I really, truly, sincerely hate this problem. To introduce PEMDAS, it uses arithmetic, which is not where PEMDAS is necessary or routine. And it does not even make explicit that it is specifying a process where the result of each step is to be used as the starting point for the next step. And then it asks you to generalize with x and y.
I made explicit that the process involves using the result from one step in a later step. The problem involves order, what is done first, what is done second; in short, the order of operations.?
Here I go for better or worse!.
choose any number
I am picking 3
following the instructions in the order of operations

I will order the steps to fit the order of OPS.
Choose the number.

I choose a number =5
multiply by 4= 20
divide by 4 =5
add 5 =10
sub 12
-2

That is what I get following the correct order of OPS.

No.

Pick a number.

You pick 5.

Add 5 to what you got picked. You get 10.

Multiply what you got first, namely 10, by 4. You get 40.

Subtract 12 from what you got second, namely 40. You get 28.

Divide what you got third, namely 28, by 4. You get 7.

The problem gives you (a bit unclearly) a purely arithmetic process in English.

Now generalize the description of this process in algebraic notation using PEMDAS and simplify.

 

[Deleted member 4993]

We teach PEMDAS in algebra or pre-algebra rather than arithmetic.

I was taught BODMAS (that is what it is called in India - with a wink) in 6th grade - before algebra or pre-algebra. We did simplification of "number expression" (like 3 + 45 ÷ 9) - literally thousands of those. Thus when the simplification for algebra was needed, I did not have to think twice.

 

[eddy2017]

You can't hate this exercise more than me, Jeff. I can only guess what you want me to do here, so
I can only guess what you should actually have to do. The inly thing I can think of is that you want me to put the neccesary grouping symbols, so I did.
Maybe that ?
{[(5+5)*4]-12}:4=7

 

[JeffM]

You can't hate this exercise mroe than me, Jeff. I can oinly guess what you want me to do here, so
I can only guess what you should actually have to do.
Maybe that ?
{[(5+5)*4]-12}:4=7

YES although the colon should be a slash.

No generalize it.

If the result is y and the number picked is x. How do you write it in mathematical notation?

 

[eddy2017]

I'll get back to you. I'm on my cell because the site went down on my desktop. The page wouldn't load.