Starting # Ending # Table

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Mrs. J Beck

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5th grade homework problem -- it is a table with starting numbers and ending numbers. I am trying to find the rule to finish the table and I cannot think of a rule that is consistent


starting 2, ending 5
starting 10, ending 25
starting 14, ending 35
starting 16, ending 40
starting 20, ending ?
starting 32, ending ?

At first I thought it was double the starting number and then add but the adding was not consistent so I am lost
 
"Double the starting number" is a good start. Now observe what you need to add in each case. Look very carefully at those values, and after they marinate in your brain for a bit, a pattern may emerge.
 
Mrs. J Beck said:
5th grade homework problem -- it is a table with starting numbers and ending numbers. I am trying to find the rule to finish the table and I cannot think of a rule that is consistent


starting 2, ending 5
starting 10, ending 25
starting 14, ending 35
starting 16, ending 40
starting 20, ending ?
starting 32, ending ?

At first I thought it was double the starting number and then add but the adding was not consistent so I am lost
Click to expand...
starting 2, ending 5.................................. 2*1 \(\displaystyle \ \ \to \ \ 1*5\)
starting 10, ending 25............................. 2*5 \(\displaystyle \ \ \to \ \ 5*5\)
starting 14, ending 35............................. 2*7 \(\displaystyle \ \ \to \ \ 7*5\)
starting 16, ending 40............................. 2*8 \(\displaystyle \ \ \to \ \ 8*5\)
starting 20, ending ?.............................. 2*10 \(\displaystyle \ \ \to \ \ ?*5\)
starting 32, ending ?.............................. 2*16 \(\displaystyle \ \ \to \ \ ?*5\)
 
There are two great answers preceding mine.

Couple of additional points.

There are an infinite number of solutions to this kind of open-ended question. In that regard, I find them a bit unfair because there is no one answer that is correct. There may be, however, an answer that is relatively simple, and that simple answer is what the teacher expects. So it's best to assume as you did that the answer expected is fairly straightforward.

There is no RIGHT way to find answers to this kind of problem. You have to try things, experiment. That's actually what is being taught. You are told to find a relatively simple pattern, but the answer does not come from a fixed way of thinking. Subhotosh noticed that all the ending numbers were divisible by 5. Clever. He then tried to build an answer from that. But that is not the only path. You noticed that doubling got you close. Let's follow that path for a bit.

[MATH]2 * 2 = 4 \text { and } 5 - 4 = 1 \text { relates to 2 how?}\\ 2 * 10 = 20 \text { and } 25 - 20 = 5 \text { relates to 10 how?}\\ 2 * 14 = 28 \text { and } 35 - 28 = 7 \text { relates to 14 how?}[/MATH]Hmm. I may see a pattern. Let's check. Does the fourth entry in the table fit that pattern?

(Actually the pattern I see and the one the Khan of Khans noticed are mathematically interchangeable, but I wanted to show you that there are different ways to skin this particular cat.)
 
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Subhotosh Khan said:
starting 2, ending 5.................................. 2*1 \(\displaystyle \ \ \to \ \ 1*5\)
starting 10, ending 25............................. 2*5 \(\displaystyle \ \ \to \ \ 5*5\)
starting 14, ending 35............................. 2*7 \(\displaystyle \ \ \to \ \ 7*5\)
starting 16, ending 40............................. 2*8 \(\displaystyle \ \ \to \ \ 8*5\)
starting 20, ending ?.............................. 2*10 \(\displaystyle \ \ \to \ \ ?*5\)
starting 32, ending ?.............................. 2*16 \(\displaystyle \ \ \to \ \ ?*5\)
Click to expand...
Thank you so much for your help!!! I appreciate the visual and can see how to teach it to the students.
 
JeffM said:
There are two great answers preceding mine.

Couple of additional points.

There are an infinite number of solutions to this kind of open-ended question. In that regard, I find them a bit unfair because there is no one answer that is correct. There may be, however, an answer that is relatively simple, and that simple answer is what the teacher expects. So it's best to assume as you did that the answer expected is fairly straightforward.

There is no RIGHT way to find answers to this kind of problem. You have to try things, experiment. That's actually what is being taught. You are told to find a relatively simple pattern, but the answer does not come from a fixed way of thinking. Subhotosh noticed that all the ending numbers were divisible by 5. Clever. He then tried to build an answer from that. But that is not the only path. You noticed that doubling got you close. Let's follow that path for a bit.

[MATH]2 * 2 = 4 \text { and } 5 - 4 = 1 \text { relates to 2 how?}\\ 2 * 10 = 20 \text { and } 25 - 20 = 5 \text { relates to 10 how?}\\ 2 * 14 = 28 \text { and } 35 - 28 = 7 \text { relates to 14 how?}[/MATH]Hmm. I may see a pattern. Let's check. Does the fourth entry in the table fit that pattern?

(Actually the pattern I see and the one the Khan of Khans noticed are mathematically interchangeable, but I wanted to show you that there are different ways to skin this particular cat.)
Click to expand...

Thanks for your help!! I am going to see what my students figure out and see if any of them obtain this answer.. It should be an interesting discussion.
 
R.M. said:
"Double the starting number" is a good start. Now observe what you need to add in each case. Look very carefully at those values, and after they marinate in your brain for a bit, a pattern may emerge.
Click to expand...

Great way to explain this to the kids. I appreciate your help.:)
 
R.M. said:
"Double the starting number" is a good start. Now observe what you need to add in each case. Look very carefully at those values, and after they marinate in your brain for a bit, a pattern may emerge.
Click to expand...
5/2 = 2.5, 25/10 = 2.5. Maybe multiply the starting number by 2.5 to get the ending number. The problem is linear so should not be that hard to figure out.
 
Jomo said:
5/2 = 2.5, 25/10 = 2.5. Maybe multiply the starting number by 2.5 to get the ending number. The problem is linear so should not be that hard to figure out.
Click to expand...
I greatly doubt RM found it difficult. But then he was not trying to give away the answer.
 
Jomo said:
I was not referring to RM's post.
Click to expand...
Sorry. My mistake. RM's was the post you quoted. I see now that you were responding to the original post.

In any case, I'm not sure that suggesting a linear function is helpful to fifth graders.
 
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