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Category: Problem-Solving

The Estimation Jar – Number Sense in Action

The Estimation Jar – Number Sense in Action

I was cleaning up the Estimation Table at my last Family Math Night event when I noticed a slip of paper next to the Hershey’s jar.  Taking a closer look at it, I realized I was looking at the thinking behind someone’s guess as to the number of Hersheys in the jar.

estimation table slip

This piece of paper is priceless to me as an educator.  It allows me to clearly understand the steps this child took to arrive at his/her answer – an answer that turned out to be exactly two Hersheys kisses off!

It starts with a multiplication problem:  4 x 27.  It’s hard to see from this photo, but if you counted the number of Hersheys that can be seen on the side of the jar, I’m guessing this student got ’27’.  Then, if you look at the number of rows of 27 that could be made from one side of the jar to the other side, I’m guessing that that’s where the ‘4’ came from.  From there, the student knew s/he had to multiply the 27 Hersheys by the 4 rows.  Since, from this point forward the student uses addition, I’m going to guess that the student either wasn’t comfortable with double-digit by single-digit multiplication or simply did not know how to do it.

So, instead, s/he used number sense by breaking down 4 x 27 into a simpler problem:  (27 x 2) + (2 x 27) which s/he wrote as (27 + 27) + (27 + 27).  From there it was simply finding the answer of ’54’ and adding that twice to get 108.

This is an amazing example of a student that has a clear mastery of number sense – breaking a multiplication problem down into a more manageable addition problem.  It’s also a great example of the distributive property of multiplication, although there’s a good chance the student has no idea what ‘distributive property’ means.  It doesn’t really matter; it’s the concept that’s important.

And this is what I love about the estimation jar – it gives kids an opportunity to practice number sense within the context of something fun…candy.  And because there’s a sense of excitement and anticipation over who will get the closest and win all the candy, kids want to participate.

From now on I’m going to make sure I include scrap paper at all of my Estimation Tables.

Family Math Night What Do You Notice? Poster

Family Math Night What Do You Notice? Poster

So for this What Do You Notice? poster, I decided to tie in rectangular arrays with prime and composite numbers.  That said, whatever math-y thing students notice is totally acceptable.  For example, one student noticed that the “buildings” had square windows.  Great.  That’s a little bit of geometry.  Another student noticed that each set of colored rectangles included the same number of squares.  Again, great, as that required some counting and comparison.  Just like the student who noticed that there are more yellow squares than any other color square.

As far as the prime and composite numbers…yep, we got that covered, too!  Here’s how David described it on his post-it, “The first 3 are composite, the last 2 are prime.  There are different arrays for each number.”  And he’s right.  The composite numbers have more than one rectangular array while the prime numbers only have one array.

But here’s an observation I totally didn’t notice.  “The columns go one higher, one lower.”  And, indeed, they do!

What Do You Notice

Family Math Night What Do You Notice? Poster

Family Math Night What Do You Notice? Poster

Here’s my latest What Do You Notice? poster from a recent Family Math Night event.  The nice thing about these posters is that they’re open-ended which allows for anyone to respond.  For this one, I decided to make a venn diagram.  But I didn’t draw the usual circles.  Instead I drew two hexagons.  This created the rhombus (parallelogram) in the center.

The categories I used for the numbers were even numbers and multiples of 5.  That said, anything appropriate would have been an acceptable answer.  For example, one kindergartner noticed a ‘diamond’.  Another student noticed that 12 x 4 = 48 and 15 x 5 = 75.  And yet another student noticed that the numbers on the left are all multiples of 4; the numbers in the center are all multiples of 10; and the numbers on the left are all multiples of 5.  And check out this observation (which never occurred to me):  The total area of the entire hexagon is 150 square inches so they would put ‘150’ in the middle (although they used the word ‘volume’, I’m pretty sure they meant ‘area’).

What Do You Notice?

Math Tricks

Math Tricks

I hired my youngest son, Ryan, for the summer. When your kids get older they don’t hang out with you as much so we have to find ways to keep them around. I find money works. :-)But hiring him was really a win/win. He needed a job and I needed help. So he’s been keeping track of the hours he’s worked and every couple of weeks we square things up.

Yesterday he reminded me that I owe him $100 for rent. We help him with his rent while he’s in college and he was able to sub-let his room for the summer for a fraction of what he usually pays.

“Just add it to what I already owe you,” I told him. About five seconds later he said, “Okay, that’s 6 hours and 40 minutes of work.”

Now, I do not subscribe to the idea that the faster someone can do math, the smarter they are. Math is not about speed. It’s about thinking. But that said, I was pretty impressed with how quickly he came up with the number of hours and minutes he needed to earn $100. So I asked him how he figured it out so quickly.

“I get $15 an hour. So $15 times 6 hours is $90. I had $10 left so I had to figure out how many minutes $10 of $15 per hour is so I thought how long it would take to make $10. 10 divided by 15 simplified is 2/3. And 2/3 of 60 minutes is 40 minutes. So, $100 is 6 hours and 40 minutes of work.”

I so badly wanted to be in a classroom and have him explain to the group how he solved the problem. This is exactly the type of thinking and sharing out that is encouraged in the Common Core and NCTM* Standards.

But that’s not what this newsletter is about. It’s about what happened next.

My son continued, “I found this really cool app called Math Tricks. I have to solve problems in a certain amount of time. But what I really like about it is that it shows me tricks that can help.”

He pulled it up on his cell phone and solved a problem within seconds. 111 x 105 = 11655. “Look, here’s how it works. Subtract 100 from 111 and add the 11 to 105 to get 116. Add two zeros after 116 to get 11600. Then multiply the 11 by the 5 and add the result to get the final answer of 11655.” He then showed me how he learned the “trick” by accessing the training mode where the trick is described.

So this started an argument, uh, discussion on math “tricks”. I insisted they not be called tricks. These tricks, I told him, are not tricks. A trick is intended to create mystery and that’s exactly what we don’t want to do in math. These are strategies based on mathematical structure (MP7). Some of them may be pretty complicated and difficult to understand but, regardless, they’re not tricks.

“Well, it doesn’t make sense but it works. It’s not like I really know what I’m doing but I know how to get the answer,” he argued.

So to illustrate my point I showed him how he could find the answer to 5 + 7 by simply finding the number in between 5 and 7 and doubling it. In other words, 6 – the number in between 5 and 7 – doubled is 12 so the answer to 5 + 7 is 12. A young child may call this a trick because they simply haven’t had the experience yet in working with numbers. But it can be easily modeled by taking 5 beans on one side and 7 beans on the other side, moving one bean from the 7 over to the 5 side so both sides have 6. So now it’s easy to see how this “trick” works.

He seemed impressed but I wasn’t done. “Let’s take 8 times 6,” I told him. “If you can’t remember 8 x 6, but you do know 4 x 6 then how about breaking it up into (4 x 6) + (4 x 6). Is that a trick or a strategy?” I was about to gather my inch tiles so I could create an array to prove that it would work when he acquiesced.

What I find so exciting about the Common Core State Standards in Mathematics is the push towards getting our students to understand – really understand – the math they are doing. When this happens, math will no longer be full of tricks. Students will begin to develop confidence in their ability to do math and we’ll move away from generations of I’m not good at math to generations of I can do math. And part of that comes from taking away the math tricks and replacing them with the math strategies they are.

I am happy to say that none of our Family Math Night kits include tricks. Just lots of fun. So if you haven’t calendared your Family Math Night event, there’s still time. As you know, our number one priority is to help you host the best event of the school year!

Family Math Night Make-N-Take Kit

Family Math Night Make-N-Take Kit


We all know that when parents get involved in their child’s education, student learning increases.  What better way to get parents actively involved in important skills than through fun and engaging games that reinforce classroom learning?  That’s the idea behind the newest addition to our Family Math Night kit series:  Make-N-Take Station Kit.

Designed to give students practice in important number skills, our Make-N-Take Kit is the perfect way to make sure your K-5 students continue the learning at home.

Similar to our Play-N-Take Station Kit, families will go home with game boards and the game pieces needed to play the games over and over.  And all games come INDIVIDUALLY packaged saving you a lot of time and effort!

What’s fun and unique about our Make-N-Take Kit is that families play a role in making their game boards and game pieces!  They love that!  And teachers and parents love the extra skills practice kids will receive in a fun and creative way – all aligned to the Common Core State Standards in Mathematics.

I invite you to check out this short video where I describe the kit contents and the mathematical learning involved.  As always, if you have any questions please do not hesitate to contact me.  Our number one priority is to help you host a fun and successful event.


Beginning Level

Balloon Bunch Capture and Go 4 the Win reinforce beginning place value, addition and subtraction to 20, and making a 10.CCSSM:  K.NBT.A.1; K.OA.A.1; K.OA.A.2; K.OA.A.4; K.OA.A.5; 1.OA.A.1; 1.OA.B.4; 1.OA.C.5; 1.OA.C.6; 1.NBT.B.2

Intermediate Level

Place Value Shuffle and Number Shuffle reinforce place value to the hundreds place, greater than/less than, even/odd numbers, and basic addition and subtraction skills.CCSSM:  2.NBT.A.1; 2.NBT.A.3; 2.NBT.A.4; 2.OA.B.2

Advanced level

Number Shuffle and Cake Walk reinforce place value to the hundreds place, even/odd numbers, multiples, factors, prime and composite numbers, 1- and 2-digit multiplication, division facts, and fractions on a number line.CCSSM:  4.OA.B.4; 4.NBT.A.1; 4.NBT.A.2; 4.NBT.A.3;4.NBT.B.5; 4.NBT.B.6; 4.NF.A.1; 4.NF.A.2; 4.NF.B.3; 5.NBT.A.1