Singapore 6A-6B, Specific Lessons

Julie in MN
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Joined: Mon Jun 28, 2004 3:44 pm
Location: Minnesota

Re: Singapore math help please 6b

Unread post by Julie in MN » Fri Feb 20, 2015 10:30 am

Alright, someone showed me the HIG answer key, with a diagram. There are a few of us Singapore nerds around ;)

I still had trouble reasoning through how to mesh a ratio with real numbers in this case, but I think I finally see what they did, so I thought I'd share. (I think visualizing the "after" diagram was the part that tripped me up, but once I could see the "after" Roses bar, the rest fell into place.) So here’s how I’d expand on their answer.

There's a reason this problem is an optional challenge problem %|
Julie
SarahP wrote:There were twice as many carnations as roses in a flower shop. After selling 50 carnations and 10 roses, there were 3 times as many roses as carnations left in the shop. How many roses were there in the shop at first?
BEFORE:
/----------R----------/
/----------C----------/----------C----------/

AFTER:
/-R-/-R-/-R-/
/-C-/

We don't know the value of R or C, but we know the /R/ and /C/ bars are equal in each pair of diagrams.

TO GO BACK TO THE "BEFORE" AND INCORPORATE INFORMATION FROM THE "AFTER":
If we add back the amount sold, we go back to the original ratio (2:1):
/-R-/-R-/-R-/ + 10 SOLD = before
/-C-/ + 50 SOLD = before

so this...
/----------R----------/
/----------C----------/----------C----------/
becomes this...
/-R-/-R-/-R-/-10-/
/-C-/---------------50 SOLD--------------/ (twice as long)
or we could look at 2:1 like this...
/-R-/-R-/-R-/-10-/ (roses)
/-R-/-R-/-R-/-10-/-R-/-R-/-R-/-10-/ (carnations)

MAKE USE OF CONCRETE NUMBERS:
We could compare the two "after" carnation diagrams, making use of all concrete numbers given (50 and the 10),
and I start to ignore one bar in each, because I know they are equal, and focus on the red info:
/-C-/--------------50 SOLD--------------/ (carnations)
/-R-/-R-/-R-/--10--/-R-/-R-/-R-/-10-/ (carnations)

Then, rearranging a bit:
/-C-/--------------50 SOLD--------------/ (carnations)
/-R-/-R-/-R-/-R-/-R-/-R-/-10-/-10-/ (carnations)

COMPARING NUMBERS TO BARS:
Okay at this point, we can see that 5 of the unknown bars are equal to 30, right?
/-C-/-------30 SOLD-------/-20 SOLD-/ (carnations)
/-R-/-R-/-R-/-R-/-R-/-R-/-10-/-10-/ (carnations)


So one unknown bar is 6 (because 5x6=30 in the purple area),
which can be plugged back into every /R/ and /C/ in the "after" diagrams.
/-R-/-R-/-R-/-10-/ (roses) = 28
/-R-/-R-/-R-/-10--/-R-/-R-/-R-/-10-/ (carnations) = 56
OR
/-R-/-R-/-R-/-10-/ = 28
/-C-/---------------50 SOLD---------------/ = 56
Julie, married 29 yrs, finding our way without Shane
(http://www.CaringBridge.org/visit/ShaneHansell)
Reid (21) college student; used MFW 3rd-12th grades (2004-2014)
Alexandra (29) mother; hs from 10th grade (2002)
Travis (32) engineer; never hs

Julie - Staff
Moderator
Posts: 1028
Joined: Fri Nov 03, 2006 11:52 am

Singapore 6B help

Unread post by Julie - Staff » Wed Mar 14, 2018 11:45 am

MuzzaBunny wrote:
Wed Mar 14, 2018 11:26 am
DD and I are determined to finish this book, but we've got two problems that are killing us. I can solve them through trial and error but I'm at a total loss to show her how using the correct methodology. My hubby and his crew of electrical engineers 8] cannot do it, nor can my brother with a bachelor's in math. I know I'm overlooking something simple, but I've poured over this for hours and even called mfw to no avail. Please, help! :~

First problem:
The ratio of Emily's money to Alyssa's money was 2:3 at first. After Alyssa spent $30, the ratio became 3:4. How much money did Alyssa have at first?

and

Second Problem:
There were twice as many carnations as roses in a flower shop. After selling 50 carnations and 10 roses, there were 3 times as many roses as carnations left in the shop. How many roses were in the shop at first?

Thank you Thank you!!!
Vicki
Hi Vicki,
I cheated and checked the archives ( viewtopic.php?f=23&t=13775 ). I found a conversation about the second problem, so I'll copy that here first. If no one chimes in on the other problem, I'll take that challenge on later :)

Also, I like to say that Singapore throws in a challenge problem here or there over the years. No worries if a couple of problems aren't solved.

So about the 2nd problem, in this first post, I showed what we did at our house - perhaps not ideal, but we can have a variety of tools in our tool belt.
Julie in MN wrote:
Fri Feb 20, 2015 12:42 am
When that happened to us, we just tried good old fashioned plug-n-chug. There may be better methods, but the most I can say is that plug-n-chug requires a bit of logic. So here is what I ended up doing:

I need to subtract 50 carnations, so I know the number of carnations must be more than 50. I'll start with 60.
60 – 50 = 10
And the number of roses has to be half of 60, so 30 roses to start, then subtract 10.
30 – 10 = 20

No, the ration of 10:20 is not what I want. That is 1:2 and I'm looking for 1:3.
I will try starting with 70 carnations instead.
70 – 50 = 20
35 – 10 = 25


That went in the wrong direction. The ratio of 20:25 is farther from the goal (1:3) than 10:20 (in other words, 20 and 25 are closer to one another than 10 and 20 were, and I want them farther apart, like 10 and 30). So I'll try using "less" than 60 carnations. I like even numbers, so I'll go down to 58.
58 – 50 = 8
29 – 10 = 19


Okay, that's getting better, 8:19 is better than 1:2, but it's still not 1:3 (8:24 would be 1:3). I'll try going down 2 more:
56 – 50 = 6
28 – 10 = 18

Aha, 6:18 is 1:3, I won! (Woops, I mean I figured it out :) )

Again, hoping someone has a better way. Otherwise, you might tell your student that one mom used the guess-and-check method and see what happens. I wouldn't worry if a student started with 100 or even 1,000, as long as he or she can narrow it down fairly quickly. (And hopefully the student won't wear himself out starting with difficult numbers like 342.75 LOL.)
Next (from page 2 of the link above) is a more Singapore-like method with bar diagrams:
Julie in MN wrote:
Fri Feb 20, 2015 10:30 am
BEFORE:
/----------R----------/
/----------C----------/----------C----------/

AFTER:
/-R-/-R-/-R-/
/-C-/

We don't know the value of R or C, but we know the /R/ and /C/ bars are equal in each pair of diagrams.

TO GO BACK TO THE "BEFORE" AND INCORPORATE INFORMATION FROM THE "AFTER":
If we add back the amount sold, we go back to the original ratio (2:1):
/-R-/-R-/-R-/ + 10 SOLD = before
/-C-/ + 50 SOLD = before

so this...
/----------R----------/
/----------C----------/----------C----------/
becomes this...
/-R-/-R-/-R-/-10-/
/-C-/---------------50 SOLD--------------/ (twice as long)
or we could look at 2:1 like this...
/-R-/-R-/-R-/-10-/ (roses)
/-R-/-R-/-R-/-10-/-R-/-R-/-R-/-10-/ (carnations)

MAKE USE OF CONCRETE NUMBERS:
We could compare the two "after" carnation diagrams, making use of all concrete numbers given (50 and the 10),
and I start to ignore one bar in each, because I know they are equal, and focus on the red info:
/-C-/--------------50 SOLD--------------/ (carnations)
/-R-/-R-/-R-/--10--/-R-/-R-/-R-/-10-/ (carnations)

Then, rearranging a bit:
/-C-/--------------50 SOLD--------------/ (carnations)
/-R-/-R-/-R-/-R-/-R-/-R-/-10-/-10-/ (carnations)

COMPARING NUMBERS TO BARS:
Okay at this point, we can see that 5 of the unknown bars are equal to 30, right?
/-C-/-------30 SOLD-------/-20 SOLD-/ (carnations)
/-R-/-R-/-R-/-R-/-R-/-R-/-10-/-10-/ (carnations)


So one unknown bar is 6 (because 5x6=30 in the purple area),
which can be plugged back into every /R/ and /C/ in the "after" diagrams.
/-R-/-R-/-R-/-10-/ (roses) = 28
/-R-/-R-/-R-/-10--/-R-/-R-/-R-/-10-/ (carnations) = 56
OR
/-R-/-R-/-R-/-10-/ = 28
/-C-/---------------50 SOLD---------------/ = 56

MuzzaBunny
Posts: 63
Joined: Wed Aug 04, 2010 2:52 pm

Re: Singapore 6B help

Unread post by MuzzaBunny » Wed Mar 14, 2018 12:59 pm

Oh I'd like to hug you! That problem kept me up til 2am thinking it through! lol Thanks so so much! :-)
Bunny

Julie - Staff
Moderator
Posts: 1028
Joined: Fri Nov 03, 2006 11:52 am

Re: Singapore 6B help

Unread post by Julie - Staff » Wed Mar 14, 2018 10:39 pm

You're welcome :)

So, here's one way to approach the other problem. I'll be interested to hear whether there are other approaches that worked!
The ratio of Emily's money to Alyssa's money was 2:3 at first. After Alyssa spent $30, the ratio became 3:4. How much money did Alyssa have at first?
BEFORE
/----------------/----------------/ Emily
/----------------/----------------/----------------/ Alyssa
AFTER
/----------------/----------------/----------------/ Emily
/----------------/----------------/----------------/----------------/ Alyssa

NEXT, CREATE SOMETHING IN COMMON BETWEEN THE TWO SO WE CAN COMPARE APPLES TO APPLES – I.E. find a common factor.
SINCE EMILY’S AMOUNT NEVER CHANGES, I’LL FIND A COMMON FACTOR TO MAKE EMILY HAVE THE SAME # OF BARS BEFORE & AFTER -
I.E. I'll give her 6 bars in both cases

BEFORE, ratio of 2:3
/----/----/----/----/----/----/ Emily {2 bars x 3 = 6}
/----/----/----/----/----/----/----/----/----/ Alyssa (3 bars x 3 = 9)
AFTER, ratio of 3:4
/----/----/----/----/----/----/ Emily (3 bars x 2 = 6)
/----/----/----/----/----/----/----/----/ Alyssa (4 bars x 2 = 8 )

SINCE EMILY IS THE SAME BEFORE & AFTER,
AND ALYSSA IS THE SAME BEFORE & AFTER EXCEPT SHE SPENT $30,
PLUG IN ALL THE INFO WE HAVE ON ALYSSA
/----/----/----/----/----/----/----/----/----/ Alyssa BEFORE
/----/----/----/----/----/----/----/----/$30/ Alyssa AFTER SHE SPENT $30

HOW MUCH DID ALYSSA HAVE BEFORE? 30 X 9 = 270

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