I was going to write this blog post about moving from guess and check to an algebraic solution method, and why you might want to use algebra.

Then I was reading through some of the Guess and Check submissions we got, and thinking, “wow, that looked pretty effortless.” One student, Jack M. from Rosemont School of the Holy Child, pointed out that the solution has to be a multiple of 3 between 20 and 40. If you start your guess and check with some of that reasoning, you know you won’t have to do very many guesses. And if you’re good at adjusting a guess up or down based on your results, you can get the answer pretty quickly. Jack only needed two guesses: he started with the first multiple of 3 above 20, which is 21, and got to 24 on his second guess!

Rashmi R. from West Woods Upper Elementary School got the answer in 5 systematic guesses (he didn’t focus on the multiple of 3 idea, but was still very efficient). He guessed 10 CDs were sold, then 20, then 30, then 25, then 24. Each time he used the data from the previous guess to think about if he needed to increase his guess or decrease to something between the previous two guesses.

Adam S. from Highlands Elementary School may have had a lucky first guess or he may have thought hard about a reasonable starting number… he doesn’t say. Either way, he started with 25 as first guess, realized it was too high, adjusted to 23 for his second guess, realized it was too low, and got to 24 on his third guess. Take a look at his work, below:

Read More→

In this week’s gift-themed Pre-Algebra Problem of the Week, students had to figure out an original cost given some percentages of the original cost. Specifically, they knew that five children contributed to pay for one big gift for their mother. Two of the (relatively broke) children paid $75 each. Of the remaining three, one paid 20% of the cost, one paid 25% of the cost, and the last paid 1/3 of the cost. Then the question was, how much did the whole thing cost? (A more realistic question might have been, “if the whole thing cost $692.31, did the last person to pay really contribute 1/3 of the cost like she planned to?” but it’s more algebraic reasoning this way).

Anyway, for me, the one reading the student solutions to the problem, the interesting challenge was to figure out what went wrong when students got answers other than $692.31. Were there conceptual struggles? Problems thinking up methods to get to the answer? Or problems executing the chosen paths? I always look for common wrong answers, because those usually show me conceptual struggles. This week, the common wrong answer was $681.82 (or sometimes just $680, or $681). I thought, “huh, I wonder why students are getting that answer…” Why do you think they might be? Read More→

I had such fun with this the table gallery we made with the Falling Leaves PoW, that I thought I’d do another one. This time, rather than share my thoughts, I’m curious to hear what you notice and wonder about each table. Students worked to solve the logic problem “Happy Thanksgiving” (to view the problem, you can sign up for a free Math Forum trial account, or if you’re already a member, just log in).

Enjoy!

From Jakub Z. at IS 141Q The Steinway [his classmates Ricardo M. and Ahbab A. also submitted great versions of similar tables]

Grandchildren	Who they can work with	Who they want to work with
Travis	Michael,Vinolia,Nicholas,Jen,Clark	anybody
Michael	everybody except for Stephen	will work with anybody if Vinolia helps too
Vinolia	Jen,Nicholas,Vinolia,Michael	everybody except for Jen
Nicholas	everybody	wont work without Funmi
Jen	everybody except for Funmi,Stephen,and Vinolia	anybody
Clark	everybody	Stephen and Nicholas only if stephen works
Stephen	everybody	everybody except for Travis,Michael,Vinolia and Jen
Funmi	everybody	Travis,Jen and Vinolia unless Stephen works too.

From Shayden S. and Avery B. at W. P. Sandin

	T	M	V	N	J	C	S	F
T	A	A	A	A	A	A	A	A
M	A	A	O	A	A	A	A	A
V	A	A	A	A	X	A	A	A
N	A	A	A	A	A	A	A	O
J	A	A	A	A	A	A	A	A
C	A	A	XUN	XUS	A	A	A	A
S	X	X	X	A	X	A	A	A
F	X	OOC	XUS	A	X	OOM	A	A

Legend:

A- yes

X- no

O- needs person to work

XU(initial)- no unless (initial helps)

OO(initial)- needs person or someone else

From Gaetano I. at IS 141Q The Steinway

funmi	clark	nicholas	stephan
micheal + clark	nicholas +stephan	funmi	micheal
just clark	just stephan	travis	vinolia
vinolia	vinolia	jen	funmi
stephan	stephan	clark	clark
nicholas	funmi	vinolia	nicholas
vinolia	clark	micheal
	travis
	jen	stephan

From Jacob D. at Patriot Elementary

Washing Dishes	Michael	Vinolia	Jen	Nicholas	Funmi	Clark	Stephen	Travis
Travis>	+	+	+	+	+	+	+	+
Michael		+
Vinolia			-
Jen	+	+	+	+	+	+	+	+
Nicholas					x
Funmi	x					x
Clark		@		/@			/
Stephen	-	-	-					-

From Jogdand A. at Seminary Hill School

The Algebra Problem of the Week that just finished up was about that classic playground game of trying to balance two different-sized people on a teeter totter. How many of you had an “aha!” moment on the playground one day when you realized that by sitting farther away from the fulcrum (the point where the seesaw balances on the base) means you can balance with a kid a lot heavier than you? A grown-up even? Or that you could leave your equal-sized friends dangling off the ground by sitting as far away from the fulcrum as possible?

In this PoW we asked students to work with the mathematized version of the situation. Instead of borrowing some small children and heading out to the playground to find a balancing point, we gave submitters a relationship between weights and distances from the fulcrum:

(weight 1)*(distance 1) = (weight 2)*(distance 2)

Most submitters chose to work with the equation we gave them in some fashion. Henry S. from Stony Point Elementary used guess and check to find the missing distances:

M	32*36 = 1,152	32*30 = 960	32*20 = 640	32*100 = 3,200	32*80 = 2,560	32*90 = 2,880
L	40*18 =  720	40*12 = 480	40*2 = 80	40*82 = 3,280	40*62 = 2,480	40*72 = 2,880

Other students, like Natalia N. from Sekolah Ciputra, thought of the problem as having two variables:

x=3/2+y
balance:
32x=40y
32(3/2+y)=40y
48+32y=40y
48=8y
y=6+t
x=7.5+t

I don’t know what the +t represents in Natalia’s solution. Perhaps it has to do with the fact that we don’t know how long the teeter-totter is. What do you think she might be using the t for?

Most submitters immediately thought of writing the two distances as x and x + 1.5, which impressed me because I don’t always think to ask, “how else could I write this quantity? What relationships could I use to write it in terms of another quantity?”

All of the submissions above mathematized the situation in the way we had suggested. But those weren’t the only kinds of responses. Some submitters did a lot of hard thinking about the physics involved, without getting into the mathematization, like Aspen K. from East Middle School:

They can both be balanced on it even though they don’t weigh the same. They can do this by the girl that is heavier sit on more of the end of the seesaw. So then the girl on the other end who must be lighter will sit closer to the middle. By doing that they should both be balanced out.

If I were Aspen’s teacher, I don’t think I would start my conversation with the mathematizing. Aspen is wrestling with the physics involved, and so I would head out to the playground or get out some see-saw models and start trying to balance lighter and heavier things. Once we had established the indirect relationship between weights and distances, then it would be time to start asking, “what are the quantities involved? How do they seem to be related?”

Finally, there were the students who chose to mathematize the problem in unexpected ways. These led to the solutions that were the most interesting and challenging for me, forcing me to look at the problem in totally new ways!

For example, Jeremy Z. and Steven M. from Cold Springs Elementary School used ratios to think about the problem, and never mentioned the equation with weights and distances. They may have used it to help them think of why if the weights are in a 4:5 ratio, the distances must be in a 5:4 ratio, but they didn’t explain that step of their reasoning so I’m not sure…

the ratio of the weights of them are Marnie:Lex=4:5. So the distance must be 5:4. since Marnie is sitting 1 1/2 feet further away from the middle of the seesaw then Lex. if 1 1/2 feet is 1 in difference in the ratio, then it is 90:72(inches).4*90=5*72. They both equal 360.

There were also some submissions that mathematized with some inaccuracies, or different theories about how the problem was working. Taylor H. from East Middle School focused on where to position students on a board of a given length so that one is 1.5 feet closer to the center. That was some good thinking! Several students focused on the length and how to arrange the students so they are 1.5 feet from the middle, ignoring the weight information. As Taylor worked the weight information in, I got a little confused. I think it’s neat that Taylor was trying to account for all of the quantities in the problem, and I would love to hear more from Taylor about if the weights were especially hard to think about and what they noticed about the weight information in the story:

Lets say the seesaw was 15 ft. long i subtracted 1 1/2 from that and got 13.5…. Then i had to divide that by two to get 6.75… It says that the girls weights are 32, and 40… I had to divide those by 6.75…… for Marnie it was 270.00 in. ……….. and finally for Lex it was 216.00 in. ………

Last but certainly not least, is this submission from Karrie P., also at East Middle School.

How i figured out this problem is I knew that Marine is 32 pounds and Lex is 40 pounds….. wich is a 8 pound difference. So, If Marine is 1 1/2 away from the fulcrum, than if you subtract 1/8, because there is a 8 pound differnce, from 1 1/2 it would be 1/2 away from the fulcrum for Lex. So…… over all, Marine is 1 1/2 away from the fulcrum and Lex is 1/2 away from the fulcrum.

I think it’s really neat that Karrie has the idea of reciprocal fractions (an 8 pound difference should result in a 1/8 difference in the distance) and I wonder where she got that idea. Many students wonder about the different ways to compare two quantities (should I subtract or divide? Why?) and I think that would be a neat conversation to have with Karrie (“how can you compare 32 pounds and 40 pounds? What’s the difference?”) Clearing up the fact that Marnie is 1 1/2 feet further away from the fulcrum seems like the last step to me, although it is still related to the whole idea of comparing numbers and the language we use with different types of comparisons.

Fiona O. at Wilson Middle School began her solution to this week’s Pre-Algebra Problem of the Week by listing all of the possible strategies she thought she might use:

• draw a diagram
• make a table
• list possibilities
• simple computation/logical reasoning

I think Fiona’s list covers almost every single solution submitted. Good call, Fiona! The only other strategy I noticed was that some students used “Solve a Simpler Problem” to help them with their diagrams, tables, possibilities, or logical reasoning.

The problem, Trick-or-Treat Route, is about the Anderson children’s plan for trick or treating in their neighborhood (the students also had information about how many minutes it would take to walk between each pair of houses): Reading the solutions, I was really fascinated by thinking about how the strategies Fiona named (plus Solve a Simpler Problem) are interrelated. Here are some things I’m wondering about:

• Would making organized lists help students who jumped right to logical reasoning, but whose answers didn’t match the story?
• What are different ways to organize lists and tables?
• Do some ways of organizing lists and tables make it easier to see patterns and simplify the problem?
• How can you tell if you’ve found all the possibilities?
• How can you tell if you’ve over-counted?
• How do students learn to check their own work on “find all the possibilities” problems?

The first thing I noticed was that of the students who said, “I kept track of all the possibilities” and then over- or under-counted only two submitted their lists. The rest talked about their lists but didn’t type them up. I wonder… did those students not make written notes? Were they too hard to read to re-type? Is there a correlation between not wanting to write down the possibilities and not finding the right number?

Read More→

The most recent Algebra Problem of the Week, Count Your Change, contained the following challenge:

[S]how that your answer is true for any case where there are three times as many nickels as dimes.

About half of the submissions that we received showed that the answer was true for any case, while the other half showed one or more examples but did not show that their answer was true for any case.

Showing your answer is true for any case means that you somehow have to represent the “any-ness” on one or more quantities in the problem. Then you have to look at the relationships in the problem and see if you can draw conclusions even with the “any-ness” represented.

What intrigued me reading the submissions is how students who worked mostly with special cases tried to “variablize” those cases without really connecting to the idea of a variable as representing an unknown (or varying or all possible) value(s) of a quantity.

Here’s an example from Sin Y. at Sekolah Ciputra:

n=5 cents
d=10 cents

number of n = 3x number of d
if number of d = 1
I’ve got 3n and 1d, which is
=(3×5)+(1×10) = 25 cents

reversed = 1n and 3d
so, I’ve got (1×5)+(3×10) = 35 cents

There’s so much interesting thinking going on here! First of all, I’m really excited about how organized and patterned the calculations are. I wonder how Sin would write out the calculations for beginning with two dimes. If they looked like this: (6×5)+(2×10) and (2×5)+(6×10) then I think Sin would be well set up to represent the work as (3*_*5)+(_*10) and (_*5)+(3*_*10).

Another thing that stands out for me about Sin’s work is the use of the letters n and d. I wouldn’t call them variables because they seem to represent the value of a nickel and a dime (which don’t vary, at least in our culture). I’ve noticed that students often use variables in the place of labels or words that they say. So Sin may have written 3n and 1d because out loud they would have said 3 nickels and 1 dime. Many students aren’t thinking of the letters they use in equations as varying quantities, they think of them as labels or abbreviations for words.

Let’s take a look at another example of that from Sin’s classmate Charina.

n = 3d = for every 10c (dime) you have 15c (nickels).
the total = 10 +15 = 25
when d = n, for every 5c (nickel) you have 30c (dimes)
The total = 5 +30 = 35

Again, there’s great thinking here. I love the idea of for every __ you have __. I think Charina has almost created a general solution here, because they recognized that for every 10 cents in dimes you have 15 cents in nickels and so Charina might be thinking that you’ll have groups of 25 cents and if you switch the number of dimes and the number of nickels you will have the same number of groups of 35 cents.

n groups of 25 cents converted to n groups of 35 cents is n increases of 10 cents per n groups of 35 cents. Proportionally, n 10′s out of n 35′s is the same as 10/35, or a 40% increase.

However, I wonder if Charina’s use of “c” which could be thought of as a variable, is actually a unit, “cents”? Charina’s work could be interpreted where c represents the number of dimes, and the value of dimes is 10c and nickels is 15c. But I think it’s more like that Charina is thinking about one dime and three nickels, and is using “c” for the units of cents.

Therefore, it’s hard to tell how much Charina has generalized the “for every 10c there are 15c” and if she is really on her way to a general solution.

Here’s one more example from Matthew at Sacajawea Middle School:

1 dime + 3 nickels = $0.25.   3 dimes + 1 nickel = $0.35

to represent this equation the algorithm d + d (3n) = T in other words dimes plus 3 nickels per dime equals total.

Here Justin has said in words the important relationship he is trying to represent: “dimes plus 3 nickels per dime equals total.” I have some questions about that relationship, especially “Total what? Total value? Total number of coins?”

Then I have some questions about his expression. He has d + d (3n) = T. Again I’m wondering if he is using the “d” and “n” partly as labels. His algorithm says to me, “write the number of times, then add that to that number times three which will be the number of nickels.”

Another possibility is that like Sin, Justin is thinking about d = 10, n = 5, with d and n representing the value of a nickel and a dime. In that case, the equation isn’t quite right, but 10 cents + 3 * number of dimes * 5 cents is an expression for the total value that’s not far from what Justin has…

In the Make a Mathematical Model activity from the Problem Solving and Communication Activity Series, we encourage students to name quantities and write relationships among them in words. It’s possible that may have helped Sin, Charina, and Justin translate their work, but I think all three students could articulate their thinking it words. The struggle seems to be figuring out what the heck a letter in an equation means. Is it a label? An abbreviation for a word?

In fact, it stands for a value of a quantity and so it works like a number, but a weird kind of number that can have any value. That’s a hard concept to grasp and I’m not sure what experiences our students need to have to grasp it. What do you think?

Some Count Your Change links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing “Count Your Change” (and a selection of all our PoWs) for 21 days with a free Math Forum trial account.
• Information about becoming a Math Forum Problems of the Week Member. Consider starting with a $25 membership, which gives you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!

Note: In a previous version of this post, Matthew’s submission was incorrectly attributed to Justin at Rosemont School of the Holy Child. Sorry Matthew!

The scenario for the Pre-Algebra Problem of the Week Building Fences has this paragraph:

We decided to put the fencing around a rectangular garden that we have been planning. The twenty-foot side of our shed will be along one of the edges of the garden.

We later learn that the author has bought 36 feet of fencing and wants to build a garden with the largest possible area.

Because the description of how the garden is fenced is not described with mathematical precision, just everyday talk, there are a lot of possible assumptions to make.

At first, the ambiguity in how the garden is fenced bothered me. I thought, “this isn’t a good math problem. We might trip students up. We are tricking them. It isn’t fair.”

But this problem comes from real life. Our colleague really did describe wanting to fence her garden this way, and really did want to figure out the maximum possible area she could fence.

I started realizing that in math problems that don’t come neatly packaged in textbooks, but rather arise from conversations about day-to-day life, the language isn’t always perfectly precise. We need to make assumptions, and know when we are making assumptions, to cope with the messy, ill-defined problems that happen outside of math books.

So I was thrilled to see how many students said things like, “at first I thought the shed would be one full side of the garden, but then I realized the fence could stick out from both sides of the shed,” or, “I had an aha! moment when I realized they didn’t have to use fencing along the 20 feet of the shed.” The students realized they had been making assumptions which made the possible maximum area smaller.

Ultimately, the assumptions that lead to the largest maximum area are to assume that the shed is only part of one side of the garden, and that no fencing is used along the shed. Other answers aren’t wrong, but they could be optimized.

Which then makes me wonder, what routines do we have in our classroom for ferreting out and challenging assumptions? There are some math problems (such as this one) that can’t even be solved without challenging assumptions! So how do we learn to do that?

I think some of the questions in the Get Unstuck and Wonder strategies are good starting places. What works for you in your classes?

Some Building Fences links in case you are interested

• The problem [requires a Math Forum PoW Membership].
• Information about accessing “Building Fences” (and a selection of all our PoWs) for 21 days with a free Math Forum trial account.
• Information about becoming a Math Forum Problems of the Week Member. Consider starting with a $25 membership, which gives you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!

Last week’s AlgPoW, Don’t Be Square, was (I think) a simple problem hidden behind lots of complications. It is a perfect problem to think about different ways to look at a hard situation and think about how to use the Solve a Simpler Problem strategy to eventually crack the whole, hard problem.

The PoW went something like this:

√((3/2)*(4/3)*(5/4)*…*(a/b)) = 10√10.

If the whole problem were completely simplified, it might look something like this:

a/2 = 1000

b = a – 1

Find a and b.

Easy, right? But that’s not the problem we gave, on purpose. We gave a problem where the solver’s job is to look at a mess and say to themselves,

• What’s making this problem hard?
• What patterns can I find and write more compactly?
• Are there other, simpler ways to write the same thing?
• What can I learn from replacing hard numbers with easier ones?
• Could I do the problem if…?
• What can I learn from doing a few sample of this long pattern?
• What happens if I try specific examples?

Those are all questions that are really valuable in all kinds of hard problem situations, whether the problem is hard because someone made it tricky for you, or because it’s one you stumbled across in daily life (like the problem I was stumped on for a while: how many times can 36 people in groups of 6 change groups before someone has to be in a group with someone they were with before?).

So, what happens when students ask and answer these questions? Here are some examples from the Don’t Be Square PoW.

Read More→

In the Make a Table strategy activities, we suggest making a gallery of all different types of tables you could make for one math problem. That’s because different tables can show different patterns and help with different kinds of noticing. Here are some ways tables can be different:

• Some tables show calculations, others just show values/results
• Some tables are labeled, some are just numbers
• Some tables are horizontal, some are vertical
• Some tables have lots of columns, some have just a few
• Sometimes the columns in a table are in one order, sometimes they are in another

This week’s Pre-Algebra Problem of the Week, Falling Leaves, inspired many students to make tables. Lots of different tables were made. In a table gallery walk, we encourage you to share one thing you noticed about each table that you really like, and things that you wonder about the table. Here are four different tables from this week’s PoW, and a noticing and wondering about each. In the comments I’m excited to hear what you noticed and wondered, and how you might compare/contrast the different tables.

Table 1, Avery and Nolan from W.P. Sandin:

I noticed: the “Drop per minute” number is always one more than the “Total leaves on the ground” number in the previous row. The 3 columns made it easy for me to see that pattern.

I wondered: what formulas were used to calculate each value. For example, if they used a computer to make a spreadsheet, what calculations would they tell it to do.

Table 2, Griffin from Patriot Elementary:

Mins.
1: 1 leaf
2: 1(leaves on ground)+1=2
3: 3(leaves on ground, 1+2)+1=4
4: 7(leaves on ground, 3+4)+1=8
No need to continue, the pattern is doubling
5: 15(laves on ground, 7+8)

I notice: I could see the calculations being done to get each new value, and it really showed the pattern that each row you were adding a power of 2 + one less than a power of 2 + 1, and so it makes sense you would get a power of 2 as the final sum.

I wonder: if adding a column for the leaves on the ground quantity would make the table a little easier to read? Right now I am a bit confused which calculations are referring to which value.

Table 3, Henri-Michael, Goshen Elementary:

1 min = 1 leaf = 20
2 min = 1+1=2 leaves = 21
3 min = 1+2+1= 4 leaves =22
4 min = 1+2+4+1=8 leaves =23

I notice: writing the power of 2 that each row sums to helps me see that the exponent is always one less than the number of minutes, which isn’t always easy to see. I also noticed that the sums, written this way, are all sums of powers of 2 as well!

I wonder: if labeling more of the quantities might make it clear what the middle calculation represents? For example labeling it “leaves already on the ground” and the calculations labeled leaves as “leaves falling this minute”

Table 4, Inteshar, Ravita, Ishnad, and Vasiliki from IS 141Q The Steinway School:

Minute On the Ground

1. 1
2. 2
3. 4
4. 8
5. 16
6. 32

I notice: The doubling pattern is super clear in this table since all we see is the column with the powers of two.

I wonder: This table is great for seeing the way the minutes relate to the end results of the calculations, but it’s not as easy to show how the end results were achieved. What other quantities were used to calculate the number of leaves falling? How could we organize those quantities in other columns or by showing them in a formula or calculation?