Last week, my sixth graders helped me make connections between a puzzle I used in third grade and their current work with place value, specifically multiplying and dividing by powers of ten.
Sarah Carter (@mathequalslove/Math Equals Love) introduced me to Naoki Inaba’s mathematical puzzles via her blog. She shared Zukei puzzles for geometric reasoning. Afterwards, I saw a few other people posting about other puzzles by Inaba. (Apparently some have even been featured in the NYT… I feel so behind the times!)
On Inaba’s website, I found a puzzle Google Translate identified as “Zero Zero Expression,” I suppose because it involves zeroes…? Otherwise, this seems like a terrible name, and so I have re-christened them “Place Value Puzzles.” (I assume the actual, untranslated Japanese name is better, but…)
The original puzzles (found here) gives a series of one digit numbers and a sum, and asks the student to multiply the one digit numbers by 1, 10 or 100 to balance the equation. For example
1 + 2 + 3 = 231
1 + 200 + 30 + 231
The puzzles increase in difficulty. The first page requires no regrouping, and only deals with two digit numbers. The second page has three digit numbers, the third requires regrouping, etc. It works up to problems like this:
In this case, in order to get the 700, one can’t simply multiply a 7 by 100. Multiplying the 9 or 8 by 100 would result in a sum that is too large, and multiplying the 2 by 100 would be too small. It would need to be paired with another number in the hundreds. Thus, 6 must be multiplied by 100, leaving us with 2, 9 and 8.
Third Graders at Work: “Where Do We Start?”
I worked on this problem with some third graders I see for intervention work. We have been working on multi digit addition strategies and understanding place value. Students quickly realized that we needed to start with multiplying the 6 by 100 (600), but then felt stuck.
“How do we get all the zeroes? We need the 2, 9 and 8 to make 100, but you can’t add them together to make that,” Selena said.
Bella continued: “yeah, 2 + 9 + 8 is the same as 2 + 8 + 9, so it’s only 19. And 90 + 80 is way too big.” At least my work on benchmarks of 10 seemed to be paying back dividends…
Mohammed quickly realized that, if we want the resulting sum to have a 0 in the ones place, we need numbers that will regroup to 10 or 20. “2 + 8 equals 10! We should leave those as ones,” he said.
Eventually, the team came up with this:
2 + 90 + 600 + 8 = 700
The group agreed that Mohammed’s “ones place” strategy is a good starting place for many of these problems — or, as Bella summarized, “start either at the front or the back.”
Sixth Grade: The Cascading “What Ifs”
Meanwhile, I was preparing to launch a unit on decimals with my sixth grade class. I wanted to re-active their thinking about our Base 10 system, and extend 5.NBT.2 to include division of powers of 10 (e.g. ÷ 0.1), as well as describing the inverse nature of multiplication and division (e.g. multiplying by 100 gives the same result as dividing by 0.01) .
What if I designed puzzles that asked students to divide by 10 or 100, instead?
Then, this puzzle would transform from content for third grade intervention into re-activation of some fifth grade ideas around place value and decimals.
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Typically, I address these standards through number talks and discussion. It seems unlikely that a student will master them through two lessons, so instead I try to weave in this idea throughout many sessions. If nothing else, using a puzzle felt novel.
First, I wrote a series of puzzles that required minimal adjustments and regrouping — only to the tenths place. For example:
From there, I developed 6 additional problem sets of increasing difficulty: multiplying dividing by a power of 10 up to 100, regrouping, using pairs of 0.01/0.1/1 to regroup, using an additional addend, etc.
(Examples from problem set versions B, C, D, E F and G)
Document available here.
Then I gave the sixth graders Problem Set C (Set C, Problem 5 is listed above: 9 + 1 + 8 = 9). After a minute of individual work time, students were given time to talk and collaborate.
Almost instantaneously, Seth and Matthew launched into a debate over this problem.
SETH: First you multiply by 1, then divide by 10, then divide by 10.
MATTHEW: I think it has to be all multiplication or all division. You can’t multiply one of the numbers and then divide the others.
SETH: But my way works: 5 + 0.9 + 0.6 = 6.5
MATTHEW: Yeah, but that’s if you allow multiplication and division to mix. I did all multiplication, so I did multiply by 1, then 0.1, then 0.1 .
SETH: Well, why don’t you just divide by 1, then 10, then 10? Dividing by 10 is easier than multiplying by 0.1.
MATTHEW: But they’re the same thing. How can one be harder if they’re the same?
“The Same, But Different”
At this point, other students wanted to enter the arena for math debate.
RAYNA: I think they’re the same, but dividing by 10 is definitely easier.
CALEB: Maybe it’s because we learn how to divide by 10 at a young age and we only just started working with multiplying 0.1s.
MATTHEW: But they’re the same thing.
CALEB: They’re like equivalent fractions. 1/2 is easier to understand than… like… 99/198. But they end up the same.
I hadn’t thought about equivalent operations. My third graders had talked a little about multiplying by 1, 10 or 100, but they were certainly in no place to think about the inverse (dividing by 1, 0.1 or 0.01). In fact, they did not even label the multiplicative changes. They just attached zeroes so that the equation would be true. I assumed my sixth graders would hand in work that looked similar, but with more decimal points.
Instead, Caleb showed me this:
And Matthew continued to argue that #3 could be either ÷0.01, ÷100, ÷ 10 OR x100, x0.01 x0.1 . So… what if we did only use all multiplication or all division?
Noticing and Wondering: Connections Between Multiplication and Division of Powers of 10
With third grade, I had focused on strategies for addition based on place value. In sixth grade, we were delving into a conversation about the very structure of place value. I wrote the two “Matthew-style” solutions on the board.
We did some noticing and wondering.
“Multiplying by one hundred and dividing by one hundredth are the same.”
“They sound the same, but one has a fraction or decimal in it.”
“If the division has a decimal in it, then the multiplication won’t. Like you can divide by 100 or multiply by one hundredth, but you won’t get the same amount if you multiply by a decimal and divide by a decimal.”
“I wonder… what if we used bigger numbers?”
There’s that beautiful “what if” again.
Caleb gave the group this to solve:
All of my original puzzles were all single digits that were multiplied or divided by a power of 10. Now we were looking at starting with numbers with more than one digit, including decimals. Because Caleb’s solution did not involve regrouping, the most complicated part was keeping track of the zeroes.
A New Connection: Exponents and Scientific Notation
“Is it okay if I just write it like x10^5 instead of x100,000?” Seth asked.
I posed the question back to the students, who quickly reached consensus that this was a clever, more efficient notation.
“Wait, what if we wrote the problems on the board like this?” Isabel mused.
“Oh! No wonder they ‘sound’ the same! They’re opposites! One is a negative exponent and the other is the positive version,” Rayna summarized.
The Students are Hooked
I scrapped my original plans for homework (and class), and instead assigned students to design their own “Inaba Place Value puzzles.” Students later worked through them when we had a spare moment in class. A few examples are shown at right.
As you may have predicted, the sixth graders insisted on using more digits, and greater powers of 10. (My puzzles were limited to 10 to the power of 2 or -2.) Whether in first grade, or sixth, or eleventh, students seem to reach some consensus that problems are more complicated if the numbers in them are very large or very small. This is true for a few of the problems, but not all. This brings me back to Robert Kaplinsky’s “complex” vs “complicated.” Many of these problems are no more complex than the ones I created that have 4 single-digit addends, but they are a lot more complicated, and sometimes tedious.
Reflecting on the Power of “What If”
We do a lot of “noticing” and “wondering.” Sometimes, this routine can fall a little flat — a lot of noticing, and not as much wondering. The wonder is what brings the class to life. In this case, students pushed us down a rabbit hole of “what if” musings. We went from thinking about keeping uniform operations (all multiplication, all division) to thinking about and connecting different notation, and exploring “more extreme” numbers. These puzzles worked well in third grade, where the driving question was “what strategies will help us puzzle through these problems?” They worked so much better in sixth grade, where students propelled the inquiry forward.
Sometimes I wonder about the class demographics. One of these groups was an small group pulled from their elementary world language class to attend math intervention. The other was a daily section of highly motivated students. How do we ignite this “what if waterfall” for students that are struggling in class, and may be early on their journey to developing a positive math mindset? Certainly, I have had some successes. I’ve had some brilliant 3 Act Math experiences with a fifth grade intervention group. This third grade group also fell in love with Marilyn Burns’ “1001 Animals to Spot” investigation. Still, how much does peer motivation affect the tone of a group, and thus the nature of the “wonderings” students pose?
- original whole number puzzles (with answer key)
- “Inaba’s Place Value Puzzles – decimals” — my 8 decimal problem sets (answer key at end of document) — updated 2.23.17
- Sample puzzles created by some of my students