In the Math Fundamentals problem Frog Farming, the goal is to make at least four different rectangular pens, each of which uses 36 meters of fence. Many students thought of this the same way I did, which was to consider half the necessary perimeter as the sum of two adjacent sides.

Rachel B, Seven Bridges Middle School

I know that the problem was finding perimeter. I know first you divide 36 by 2 and get 18. Then you find addends of 18 and they are the length and width. I added 12 plus 6 which equals 18 and 12 times 2 plus 6 times 2 equals 36.

Sarah G, Laurel School

First I decided come up with a length and width for a rectangle that would equal 18 because 18 is half of 36 and you have to multiply that number by two to get the perimeter. I decided on 2 and 16. I checked it by doing 16+16+2+2= 36. One could by length=16, width=2.

Rachel and Sarah and I were thinking about perimeter, in the context of this problem, like this:

Another way I thought of this was as 2(L + W). Hmm…..

Then I was mentoring a few students in this problem and noticed that they were thinking about the problem differently.

Ethan Z, Lorne Park Public School

I thought of a rectange which has 4 sides and 2 sides are equal and the other 2 sides are equal because Farmer Mead wants a rectangular pen that uses 36m of fencing. First I got the answer by thinking of 2 equal numbers that add up to less than 36. Then, the last 2 equal numbers are the difference of 36 to the first 2 equal numbers. That’s how I got all the numbers of the first question.

Emily G, Laurel School

I used 2 numbers, and doubled 1 number by two (ex. 6×2=12). 36-12 is 24. 24 is an even number that can be split into 2, which is 12 (ex. 24÷2=12.) 24+12 is 36!

Maybe because I had “seen” the problem differently, it took me a few minutes to figure out what these other kids were doing. Then I realized they were “seeing” the problem like this:

This seems more like 2L + 2W! These students tended to use more of a Guess and Check strategy to find solutions, whereas kids who used the first method were a little more systematic from the start. But it was fun to me to see these two different methods to what is a pretty simple idea. I like when simple things are done different ways.

I wonder how you “saw” the problem when I first described it. And how did your kids tend to see it?

Some Frog Farming links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing Frog Farming (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!

Our last Geometry Problem of the Week, Voulez Vous des Voussoirs?, deals with building arches using trapezoidal blocks. Maybe you’ve had a chance to build such an arch at a museum. I’ve done it most recently with my sister’s family at the Minnesota Children’s Museum. In our problem, if you’re told the angle measure of the obtuse angles of the “trapezoid”, how can you figure out how many identical blocks are needed to build a semi-circular arch?

What is most interesting to me about this problem is that different people seem to “see” it in different ways. That is, they get one particular model in their head of the situation. When I first solved it, I “saw” it like this:

A few students saw it that way, though it wasn’t the most common method we saw. Here’s an excerpt from one solution:

Renuka D

I knew if the obtuse angles were 96 degrees, the supplement would be 84. Because of symmetry, the two sides of the voussoir must meet at the center of the semi-circle. In this way a triangle is formed, so therefore the angle at the center of the circle must be 180 – (84+84).

Here’s the second idea I “saw” when I solved this:

This was the basis of the most common method that students used this week, and is explained in excerpts from two solutions:

Gavin T, Highlands Elementary School

I knew that if all the angles were 90 degree angles, the arch wouldn’t be an arch, it would go straight up. 96 degrees meant that each angle angled the arch 6 more degrees.

Jed M, Waterford Elementary School

If there is a 6° increase on each angle. (I know that because there 96° angles, and if it was a 90° angle it would go straight up.) And there’s two angles on each voussoir. So theres a total of a 12° increase. I’ll use math sentences to get a total of 180 and what ever number times twelve equals 180 is my answer.

Then there’s the image conjured up by a group of students from Conners Emerson School in Maine. They made it a problem about angles of regular polygons. That had never occurred to me, so it was exciting to get their submission. Here’s my version of what they “saw”:

Here are some excerpts from their solution, including a hint and their picture:

Xingyao C, Tarzan M, and Tom G, Conners Emerson School

We drew a diagram of two of the voussoirs adjacent to each other. If the voussoirs make it all the way around, they would form a regular polygon.

The two adjacent acute angles of the trapezoids make an interior angle of the regular poylgon.

Hint: Try not to look at the voussoirs as solitary pieces, but as part of a convex polygon. You are not trying to find the information of each individual voussoir, but the information of the arch made by many.

(I can’t help but point out that they named their picture “French Words in Math”, which I think is awesome!

I wonder how you think your students “saw” this situation. Was one solution method more common, or was there a variety of models used in your classroom? How did you see the situation? Please let us know!

Some Voulez Vous de Voussoirs? links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing Voulez Vous de Voussoirs? (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!

The last Math Fundamentals Problem of the Week was a puzzle of sorts. You’re told there are three chips with a number on each side. The six numbers are consecutive one digit positive numbers. One way to throw the chips gives you 6, 7, and 8, for a sum of 21. You’re also given four other sums between 16 and 23 that are possible. Aside from adding three numbers together, there isn’t any other math concept that’s central to this problem. It isn’t about “fractions” or “proportional reasoning” or “measurement”. It’s about understanding what’s going on, and coming up with a method to figure out what numbers are on the other sides of the chips.

Since solving a problem like this often lends itself to running around in circles for a bit before you hit upon the answer, the resulting submissions are usually of two different types. The first is the solution that states the answer, and then confirms that the answer is correct. Here’s an example:

On the back of the 6 there is a 5, on the back of the 8 there is a 4, and on the back of the 7 there is a 9

for 16 i added: 7 plus 5 plus 4.

for 17 i added: 6 plus 7 plus 4.

for 20 i added: 6 plus 5 plus 9

for 23 i added: 9 plus 6 plus 8

That example doesn’t give me any idea how the student approached the problem, what sorts of things they tried or did, whether they made any mistakes and found them, or anything that seems systematic and efficient. We could do some work, using the sums that they provided, to figure out the correct combinations. But the student doesn’t provide any insight into the process they used to find the answer.

It’s not uncommon to see a lot of submissions that confirm results. Sometimes it’s hard to keep track of the different paths you traversed on your way to the answer. Good recordkeeping is a must. I have to say, however, that reading the solutions for this problem, I was excited to see so many students telling the “story” of how they solved the problem. Most used some form of the Guess and Check strategy. Some used Logical Reasoning. Here are a couple of examples:

Told 6+7+8=21. Number are consecutive, and to add to 23, one number has to be 9, for 6+9+8. 6 and 8 stay- 7 and 9 are on the same chip. 4 and 5 are the other numbers. To add to 16 you have 4+5+7. 17 needs 4+6+7. 5 and 6 have to be on the same chip. 8 and 4 are the remaining numbers and are on the same chip.

First I thought, since the toss of 23 is larger than the 6, 7, 8 toss, it must contain a number greater than 8. Therefore 9 is needed. The toss must be 9, 8, 6 because no other combination gives 23. So the 7 and 9 are on the same chip. This also means the other two numbers must be 4 and 5. Next I thought the toss of 16 must be made of 4, 5, 7. Then, the toss of 17 couldn’t be 4, 5, 8 because it is not possible since the 7-9 chip is missing.
So 4 must be on the 8 chip to have all 3 chips in the toss. This gives the 3 chips: 5/6, 4/8, 7/9.
Then I concluded that the 20 roll is possible with 5, 7, 8 so it all checked out.

However they did it, we really love reading stories about students’ problem solving adventures. How do you support your students in telling this story? Is it something that gets reported orally in the classroom a lot (which is great practice for eventually writing it down)? Do they read aloud to others? Do you just keep prodding them to say more about their process? We’d love to hear from you, since all the kids submitting to this problem didn’t do a good job by luck or accident!

Some Let the Chips Fall… links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing Let the Chips Fall… (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!

In the last Geometry Problem of the Week, Lotsa Popcorn!, we presented the World’s Largest Popcorn Ball (as of 2006), and explained that it weighed 3,415 pounds and was 8 feet in diameter. It was claimed to be “almost 50,000 times larger than the normal popcorn balls distributed for retail consumption.”

We went on to ask students to find the diameter of a popcorn ball whose diameter is 50,000 times smaller than this, as well as the diameter of a ball whose volume is 50,000 smaller. We also asked about the weight. We received a number of solid answers to this, including these two:

Obviously we don’t need to give that much direction to students. We just wanted to make sure they would do something reasonable, and we figured we might need to support them in that, especially if they were working on their own without the benefit of a class or partner conversation. What we really did was make all the really interesting math decisions for the students and make it easier for us to read the solutions. We should have just presented them with the facts of the giant popcorn ball, and asked them if the 50,000 times bigger thing was reasonable. What would they compare? I’m thinking back to the So Many Salmon problem we used in Math Fundamentals earlier this year. We presented students with some facts and then asked a question, but let them determine how they would answer that question.

It makes me wonder how many people used the Scenario Only, which did only present the facts, and how many used the whole problem with the three specific questions. What would you have done? What do you think your students, whatever age they are, would have done with just the facts and the question of whether or not the 50,000 times bigger statement was reasonable?

Some Lotsa Popcorn! links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing Lotsa Popcorn! (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!

Ratios is a concept that doesn’t always go smoothly in the classroom, and I know many teachers who grumble about the fact that their students don’t understand it better. I’d suggest that, as often is the case, this is at least partly attributable to the fact that we focus on procedures before the students have an intuitive idea of what ratios really represent.

I was reminded of this when I was reading the submissions to recent Math Fundamentals Problem of the Week, Anthony’s Famous Butter Rolls. Students are asked to figure out how many rolls Anthony has to make based on the instructions his mother gave him. One instruction is, “His mom wants him to make enough so that there will be 3 rolls for every 2 adults and 4 rolls for every 3 children.”

A ratio, you say! We can use proportions to do something! Yup, we could. But the problem doesn’t mention ratios, and only one kid even used the word “ratio” in their submission. Nobody used “proportion”. Instead, they solved the problem from a completely conceptual perspective, using four different methods. Below are excerpts from nine student solution.

By far the most common method was what these three kids did:

Mitchell Z, Highlands Elementary School

Since there will be 12 adults and each 2 adults gets 3 rolls, you first divide 12 by 2 because the answer is the number of the groups of adults. The answer to that is 6. Then you multiply 3 rolls by 6 groups of 2 adults. That would be 18 so now you know that Anthony needs to make 18 rolls for the adults.

Elisabeth M, Fred W. Miller School

[T]o find out how many rolls the adults will need, 12 adults in all,3 rolls for every 2 adults, I divided 12 by 2 and got 6 groups of adults. Then, I multiplied 6 by 3 and got a total of 18 rolls for adults.

Evan S, Fred W. Miller School

I figured out how many groups of 2 adults there were in 12 people.
12 divided by 2 = 6. Now that I know there are 6 groups of 2 and each group gets 3 rolls, there must be 18 rolls for the adults. (6×3 = 18)

Slightly less popular was the idea of making a table:

Molly M, Highlands Elementary School

In the problem, it said that for every two adults, Anthony should make three rolls. I wrote down numbers to twelve counting by two’s to start my graph, because there is twelve adults and for every two of them, Anthony needed to make three rolls. Here is my adult chart with my work.

Adult Rolls
Adults	Rolls
2	3
4	6
6	9
8	12
10	15
12	18

I knew now that the adults needed 18 rolls.

Andrei C, Lorne Park Public School

Since there were 3 rolls for every 2 adults and there were 12 adults:

3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
+____________
18 rolls 12 adults

Another method that is really close to making a table is to draw a picture. We didn’t get any actual pictures from submitters (other than some yummy-looking pictures of rolls!), but some students did a nice job of explaining their picture. Here are two examples:

Adabelle W, Steele Elementary

I made a chart with 27 circles. There is 27 because 12 adults and 15 children = 27 people all together. I put an “A” in 12 circles for Adult and an “C” in the other 15 for children. Then I put a big circle around 2 of the adults and wrote a 3 with them for how many rolls they needed. I did that until there were no more adults left to group. Then I did the same thing with the children except instead of 3 rolls for every 2 adults I grouped 4 rolls for every 3 children. Anthony needed to make 18 rolls for the adults. 3*6=18. He needed to make 20 for the children.

Annabel L, Steele Elementary

I first of all, drew a picture of 12 adults and 15 children. I then split the adults into groups of 2 and put a 3 beside them because they’re 12 adults and they’re 3 rolls for every 2 adults. I then did the same thing to the children, only I split them into groups of 3 and put a 4 beside them because they’re 15 children and they’re 4 rolls for every 3 children. I then, added all of the rolls up and got 38.

The final method we saw wasn’t used by very many submitters, but it works just fine! These two students figured out how many rolls needed to be made for each individual adult (and child).

Nick L, Highlands Elementary School
I knew that for every 2 adults there should be 3 rolls so I divided 3 by 2 to get 1.5 rolls per adult. Next, I knew that there was 12 adults so I multiplied 12 by 1.5 to get 18 rolls for all the adults.

Elise B, Elgin Street School
3 rolls every 2 adults. Each adult will get 1 1/2 rolls
One more roll for each person: [that's part of the directions]
An adult will get 2 1/2
A child will get 2 1/3
For 12 adults:
12*2=24
12*1/2=6
24+6=30 rolls for all adults

These students aren’t solving this problem using a procedure that someone has taught them. They’re solving it by doing something that makes sense to them. They understand the concept of what it means for there to be three rolls for every two adults, and they’ve come up with methods that will find the correct answer every time. Two of these methods (the second and third) aren’t something you would want to use with a big number of people, because it would take a while, but the students who used those methods are ready for a conversation about what it might look like if you had 120 people. The other two methods (the first and the fourth) could be used for any number of people. All four methods were used in the context of the problem.

What sorts of methods would your students use to solve a problem like this? How would you move students from this sort of conceptual understanding to developing the efficient procedures that they’ll eventually be expected to execute? We’d like to know!

Some Anthony’s Famous Butter Rolls links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing Anthony’s Famous Butter Rolls (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!

In the recent Math Fundamentals problem Count the Discount, students calculated the price of a jacket after it had been on the sale rack for five days. Each day on the sale rack, the price was discounted an additional 20%. Submitters used four different methods to figure out the new price each day.

The most common method by far was to multiply the current price by 0.2 to find the amount of the discount, and then subtract that from the current price to get the new price. (Many students also knew that 20% of 100 is 20, without doing any arithmetic.)

Charlotte C, Steele Elementary School

On the first day the jacket is $100.

Percent means out of a hundred, since it was 20% of 100, It was obviously $20. To make sure I multiplied 100 by 0.20 to figure out the percent. On the first day $20 was taken off of the jacket. 100-20=80

On the second day the jacket costs $80. Since The jacket isn’t $100 anymore My first method is harder so I will stick to the second method. 80 multiplied by 0.20=16. Now the jacket costs $64 because the discount was $16. 80-16=64

Breanna C, East Middle School

If the coat is originally $100, and you are taking 20% off the item, then you will use multiplication. To make the equation, you will end up having to convert the percent into a decimal. The equation will be $100 times .20 which equals $20.00. Then you will subtract that amount from the total.

Some students calculated the 20% by first finding 10% and then doubling the result. I wonder if the students who used this method didn’t have a calculator and thought this would be a little easier to do in their heads.

Natalie V, Seminary Hill School

1. I found 10% of $100.00 ($10.00), and multiplied it by 2.($10.00 x 2=$20.00)
2. I subtracted the first day’s discount ($20.00) from $100.00, and the answer was $80.00.( $100.00 – $20.00= $80.00.)
3. I found 10% of $80.00 ($8.00) and multiplied it by 2. ($8.00 x 2 =$16.00) That is the second day’s discount.
4. I subtracted $16.00 from $80.00 and the answer was $64.00. ($80.00 – $16.00 = $64.00)

Ria D., Laurel School

The steps i used to find the percents were

- first i divided the numbers by 10
- next i multiplied that # by 2 to get 20%
-then i subtracted the 20% from the original #

Another method to find 20% is to divide by 5, which this next student did because he wasn’t sure how else to figure it out. (Aside: What experiences do we need to provide students so that they develop the understanding that multiplying by 0.2 and dividing by 5 are the same thing? I don’t mean teaching procedures for converting from percents to decimals to fractions, but really developing true understanding.)

Praeek D, Highlands Elementary School

I subtracted $20 from $100, because the text said the price would go down 20% every day, and 20% of $100 is $20. I got $80, so I put it down under price, and beside 1st day, because it was the price on the first sale day.

Next I divided $80 by 5, because I didn’t know how to get 20% of $80, but I knew 100% dived by 5= 20 %. I got $16.00, so I subtracted $16.00 from $80, because the price had to go down 20%, and 20% of $80 was $16.

The last method we saw is the one that I would use, since it involves one step instead of two to find the new price each day.

Zoe B, Birch Wathen Lenox School

To solve the problem, I multiplied the total by .8 each ‘day.’ I did this because after the 20% discount, the item would be 80% of the price.

Developing facility with percents gets a lot of emphasis in elementary school, and it would be super if kids could use all these methods equally well and decide which one they like best, or which is best suited for a specific situation. If your students tended to use one of these methods, what would they think of the others? Did you have any conversations with them about different methods and how they’re related? What method is most popular in your classroom?

Some Count the Discount links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing “Count the Discount” (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!

In the most recent Math Fundamentals Problem of the Week, Peanut Butter and Jelly, students were told that a 48 ounce jar of jelly makes 60 sandwiches and that a 16 ounce jar of peanut butter makes 12 sandwiches (among other things). The Extra question asked whether each sandwich would contain more peanut butter or more jelly. Much like the So Many Salmon problem, this question can be answered in a number of different ways.

One method involved figuring out how many ounces of peanut butter or jelly are required to make a particular number of sandwiches. (The answer to the main part of the problem was 120 sandwiches, so it makes sense that some students would figure out how much peanut butter and jelly was used to make those 120 sandwiches.)

Breanna, East Middle School: If I combine 3 regular 16 ounce peanut butter jars together, it will be equal to 1 regular 48 ounce jar of jelly. Now I will combine each 3 peanut butter jars together so that both peanut butter and jelly jars will have equals amounts of 48 ounces in them. So, if I try to find if a sandwich has more peanut butter or jelly, the jars will have the same amount in them.

Now to try to figure this question out, I have to take the info. I had before and change it so that it fits with our new amount in the peanut butter jars.

Two 48 ounce jars of jelly= 120 sandwiches
≈ three 48 ounce jars of peanut butter= 120 sandwiches</p

So now to make a sandwich I know that it takes more peanut butter than jelly because it takes approximately 1 more 48 ounce jar of peanut butter to make 120 sandwiches than it does jelly.

 

Samantha, Birch Wathen Lenox School: 16 oz of peanut butter is needed to make 12 sandwiches (16/12) and converted it to 80 oz for 60 sandwiches (80/60) in order to compare to 48 oz of jelly making 60 sandwiches (48/60). So there is more peanut butter per sandwich.

 

Eric, Sacajawea Middle School: There is more peanut butter than jelly in each sandwich because 1 jar of jelly makes 60 sandwiches and 5 jars of peanut butter makes 60 sandwiches. The peanut butter jar is 16oz and the jelly jar is 48oz. so if you multiply 16oz by 5, it will become an equal comparison and the larger number has more of it per sandwich. So 16oz x 5 =80oz 80oz is greater than 48oz so there is more peanut butter per sandwich.

Other students figured out how many sandwiches can be made with a particular amount of peanut butter or jelly.

Twentyseven, Caughlin Elementary School: A 48 ounce thing of peanut butter will make 36 sandwiches and the same amount of jelly makes 60

The most popular method was to figure out exactly how much peanut butter and how much jelly are in a single sandwich.

Cat, Mount Desert Elementary School: For the peanut butter, I divided 16 ounces by 12 sandwiches because there are 16 ounces in a jar of peanut butter that makes 12 sandwiches. The product was 1.3 ounces of peanut butter repeating. For the jelly, I divided 48 ounces by 60 sandwiches because there are 48 ounces in a jar of jelly that can make 60 sandwiches. The product was 0.8 ounces of jelly. 1.3>0.8 therefore, there is more peanut butter in each sandwich.

 

Emma, Sacajawea Middle School: Each sandwich would contain more peanut butter. Since each jar of peanut butter contains 16 ounces of peanut butter, and makes 12 sandwiches, you do 16/12, which is 1.33 ounces. You do the same with the jelly. 48/60=0.8 ounces. (48 ounces in 1 jar of jelly/60 sandwiches.) So that means that each sandwich would contain more peanut butter than jelly.

 

Michael, Rosemont School of the Holy Child: Each sandwich will contain more pb than jelly because each sandwich will have 4/3 oz of pb (16 oz pb/12 sand=4/3 oz per sand) and 4/5 oz of jelly (48 oz jelly/60 sand=4/5 oz per sand).

 

Angi, Hanover Street School: Each sandwich will have more peanut butter because there is 75oz. in each sandwich of Peanut Butter. In a jar of jelly there is 8oz. in a sandwich. I did 48oz ÷60=.8oz (jelly)then I did 16oz ÷12oz=.75oz

Each of the methods above required some calculations. Nothing wrong with that, but do we actually need to do any calculations to answer this question? Consider these two explanations:

G, McDougle Elementary School: A 48 oz. of jelly makes 60 sandwiches. Because 48 is less than 60, you must use less than an ounce of jelly on each sandwich. A 16 oz. jar of peanut butter makes 12 sandwiches. Since 16 is greater than 12, you use more than ounce on each sandwich. So there must be more peanut butter than jelly on each sandwich.

 

Shawn, Lakewood Elementary School: There will be more peanut butter in 1 sandwich because 1 peanut butter jar contains 16 ounces but can only make 12 sandwiches, which means that there is more than 1 ounce of peanut butter in a sandwich because 16>12. But a jelly jar holds 48 ounces but can make 60 sandwiches, so that means that there is less than 1 ounce of jelly in a sandwich because 48<60.

Many comparisons don’t require calculations. Understanding such situations makes students more agile as mathematicians. They learn to think about what they really need to know in order to answer a question, and tend to ponder situations a little more before diving in and making calculations. How did your students solve this question? If they solved it making calculations, what do they think of the last two solutions? Let us know!

Some Peanut Butter and Jelly links in case you are interested:

• The problem [requires a Math Forum PoW Membership].
• Information about accessing “Peanut Butter and Jelly” (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!