# Build It = Deep Conceptual Understanding of Multiplication

Being a math interventionist is one of the hardest jobs I’ve ever had. It’s like a constant cycle of diagnose…teach…teach…teach…light bulb on…light bulb off…diagnose…teach…teach…teach…

I’ve been working with a group of fourth graders that were just struggling BIG TIME with multiplication. They weren’t able to see how a fact like 3×7 could help them solve 6×7 by just doubling. The numbers were too abstract, and they had nothing to connect it to.

Of course as I dug deeper I found that they simply had zero understanding of what 3×7 really means.  So as usual, I tried to find a way to connect it to real life, my favorite thing ever. I was trying to think of a way to help them remember the difference between rows and columns.  And then it occurred to me as I was waiting in line at the theater, no one wants to wait in line to sit in the first COLUMN for a popular movie. We all want the best seats in the house, so we wait to sit in the first ROW.

Enter the Movie Theater Multiplication Project.  I went home over the weekend, turned off all distractions and poured days and days into authoring this project. I needed to really think for myself what 3×7 represented.  I was not taught this way and I know that it doesn’t come easily to me.  I also had to think of a way that would be meaningful and that would STICK, since that seems to be the biggest challenge facing students in intervention. The last thing that I really needed to think about was the Concrete-Representational-Abstract instructional approach. Kids LOVE to build. The second they walk into my math lab, their hands are all over my cubes, blocks, counters, etc.  Starting with building means that they can usually connect a pictorial representation to it, and then connect that to numbers.

So I made a movie theater or two or three as I wrote the project. I made theaters that were 3 rows of 4 seats, making 12 seats total (3 x 4=12). I made tickets placing them in the correct row and seat number. And finally I made some mega theater designs so they could learn to use known facts to solve harder facts.  There is a fourth stage to the project also that involves some open ended challenges to calculate profits and revenue. Say hello to my little friends:

Seemed like it might actually work!

So I brought the idea to school. I got out some tools and some little tiny bears and THAT got their attention.  Tiny bears! Seriously, that is all it took?!

The first three days of the project were brutal.  They were making columns instead of rows, they were making rows of 20 instead of 20 seats total.  With probing questions they started to see what was happening.  And THEN, light bulbs turned on…and for several days now the light bulbs have stayed on! Is it sticking?! I hope so, and we’ll find out when they get to the mega theaters and can break down more difficult multiplication. Wish me luck!

If you want to try this out, in the preview you will find three parts of the four stages of the project that are free.  You could totally continue the project by giving them your own specifications!

# You are NOT Bad at Math

Did you know that there is not a “math gene” that makes us good at math? If you haven’t read this yet: Why Do Americans Stink at Math? I would highly suggest taking the time to read.  The long and short of it, is that during the industrial revolution, we took the fast track and tried to teach math the fastest way possible.  We taught shortcuts instead of conceptual understanding.  This method of teaching allowed our students to go directly from their education into a factory job, but it did a great disservice to a generation.  This is why people believe that they are “bad at math”.

The world has changed and we can no longer do this to our students, but according to the article above we still are.  Despite many attempts at changing our math practices, we still find the majority of U.S. teachers using traditional methods. Of students attending 2 year colleges, 60% of them are placed in remedial math classes, and only 25% of those students pass those classes! (Silva & White, 2013)

I would strongly recommend that you watch this video by Jo Boaler, of Stanford University. It is 20 minutes long, so if you can’t watch it all, try the first 8 minutes.  In it she talks about how we need to make math a learning subject (exploratory, messy, open ended and challenging), not a performance based subject (math is only about answering questions correctly).

To return math to being a learning subject, we can use rich open ended tasks, inquiry activities, real world projects and problems that encourage math talk and discourse!  Please check out my free Reasoning Puzzle Set to try out an activity that will really get your students thinking and talking and most importantly, learning at high levels.

Reasoning Puzzles to promote student to student math talk.

These are most appropriate for 3 and 4th graders, but even could be beneficial for fifth graders that are not used to thinking this way!  If you end up using them, I’d love to hear how it goes.

Today was a totally delightful 20 minutes of math play. It ended up being almost 40 minutes as we created our play situation.

I have to admit that sometimes pretending can be exhausting.  Maybe as I’ve gotten older I’ve lost that spontaneous creativity. So I was happy to find a way that I could “pretend” a real life situation…ordering food at a restaurant. Today, it was almost lunch time and my 5 year old and I decided we were going to make our own restaurant…PB & J. We built the menu, with her telling me the items and prices.  It was fun to think of the different categories and to put together the menu. We grabbed a notepad, marker and some cash and we were ready to go. (Notice some of her choices, “Soda is not very healthy so we can make it very expensive.”)

Right away she wanted to order a Hawaiian Punch juice box, and an appetizer of crackers.  That was when I asked her how much money she had.  She counted her cash, “I have 8 dollars.”  I asked her if she had enough money to buy a lunch.  This was one of those moments where I wish she would think out loud, because she immediately changed her drink order for milk. I bet all kinds of good mathematical thinking was going on there! Now, if I know my daughter, it’s because she wanted enough money for dessert!

Here she is counting her money after ordering her milk and crackers, to be sure she would have enough.

Sure enough, I took her order and it came out something like this:

She quickly realized that she was NOT going to have enough money for dessert, so she sprinted off to go and get her piggy bank, coming back with coins.  She asked me so innocently, “How many of these do I need for a dollar… one?” That was the perfect moment to tell her that a dollar is ten dimes, or four quarters…the perfect intro as to why she needs to know about coins and their values.

Which will lead to many more fun money play sessions!  We can use this same menu, but change up the ways to pay, the amounts and combinations of money.  All of which she will have a strong reason to want to know how to do it.

The best part? I told her that she needed to make sure to leave the server (me) a tip. As I left the room with her dirty plates, she secretly wrote this on a piece of paper and presented my “tip” to me when I came back:

I’ll take that tip over 20% any day!

# Classifying Shapes Using Tangrams

Since our geometry unit is so full of vocabulary, I am always looking for ways to have students apply the vocabulary to new situations. After learning how to make our own tangrams yesterday, we took it to another level today. We decided to sort and classify the shapes that make up the tangrams. Because we’ve spent a few days describing two types of polygons: triangles and quadrilaterals, we had some really nice anchor charts up on the wall.

Their task today was to classify and describe the shapes from their tangrams using sides and angles. This meant that some of their shapes could be described in more than one way. This was a really cool opportunity for students to participate in math talk, and it was even better to see them using the anchor charts we made together:

The student on the right is trying to classify his triangle by looking at angles.  The student on the left quickly realized that he was holding a shape with four sides.

This student is checking to see if all four of her sides are equal, as well as if there are four right angles in her shape.

There is nothing better in a math class than hearing students argue over whether something is a right angle or not. Hearing words being used authentically was really cool, especially since they could actually touch and manipulate the shapes! I’ve really taken geometry a long way since I started teaching (ahem…I used to use worksheets only…please don’t judge me!), and the terms and vocabulary has started to stick because of it.

# Earth Day Water Pollution Activity: A Cross Curricular Inquiry Study

Today was one of those magical days at school. It was the kind of day that makes me LOVE my job, where all the pieces go together very nicely and the worldview expands for the students in the classroom. Today was one of those “teach like a pirate” days, where we took risks and immersed ourselves fully into an important issue.

We began our Earth Week (not just Earth Day) by learning about the Great Pacific Garbage Patch, something that is largely unknown to the general population.  We read about the issue with a two page informational text today, while responding in writing with our new understandings.

Then we watched this video:

After lots of important discussion, I asked them a question, “How would you clean it up?”

That led to an awesome water pollution experiment, which ended up both engaging and frustrating the students.  The groups had to try to clean up the water using the least amount of money possible.  They had to write out their plan as a group together, figure out the cost and bring their proposal to my supply table to collect the materials they needed to clean the water. They had to be detailed and precise, use math in a very real way, and had to work under time pressure. Here is a peak into what it looked like:

Buckets were prepared with water and biodegradable items like coffee grounds, shredded paper, food that the kitchen hadn’t served but had to throw away, and soil.

Reused or recyclable items were for sale for students to use to clean up the water. (I wash out the cups and forks to reuse each year for this experiment! We reuse ziplock bags collected throughout the week.  We recycle other plastics and aluminum foil.)

The students cleaned the water after “purchasing” materials.

Many methods were used to clean the water.

We compared water samples at the end, to see who was most successful. We also compared budgets, to see who was able to keep costs down.

We’ll continue this work for the rest of the week:

• We’ll be writing a letter to explain what we learned today about the Great Pacific Garbage Patch and the difficulties of cleaning up water.
• I plan to immerse them in lots of good Earth Day literature like: The Great Kapok Tree by Lynne Cherry, Oil Spill by Melvin Berger, The Lorax by Dr. Seuss, The Wump World by Bill Peet.
• We’ll be learning about alternative energy to make wind powered cars later in the week, and we’ll even try to harness the sun by measuring and making solar ovens.
• We’ve also got some tough math problems in the works as part of the Great Pacific Garbage Patch unit to help us think about conserving!

I love Earth Day…Earth Week, and more importantly introducing important issues to our young learners, encouraging critical thinking in meaningful ways.  I’d love to hear what you do in your classroom! The more ideas we have to share, the more we can teach our children to be environmental stewards.

# Crank It Up a Notch: Add Something They Can Touch When Problem Solving

One of my favorite ways to amp up problem solving is to throw something into the mix that they can touch.  This makes the project or problem so much more interesting to students in one instant.  We are working on the Design a Dream Bedroom project, so I picked up some free samples from the hardware store:

Give them stuff to touch when they are working on real world math activities.

Of course you can be sneaky about introducing the materials.  Before I even went over the problem during math, I spent the morning organizing the materials on a common table when they were arriving for the day. I got about a million questions, and hands were reaching out to touch the carpet and flooring samples before I could even get them in the bucket.

That is all I needed to do to get them interested in the problem.  After I read through the introduction with them during math problem solving time, the students literally leaped out of their carpet spots to run up and grab the problem from me.

That is what we want problem solving to be like…exciting, engaging, rigorous and motivating! Putting things in their hands to make it real world has worked every time.

# 3 Hands On Earth Day Activities that Integrate Math

Here are three really amazing Earth Day activities for your elementary classroom that all include math. I’ve done them all and they’ve been memorable, educational, and fun! The best part is they always lead to deep moral and ethical conversations.

1.  Hold a Trash Free Lunch Picnic:  This is a two day project.  The first day, you ask the students to keep track of how many pieces of trash they have as they eat lunch.  For the second day, you send home a note asking parents to pack a trash free lunch, (as trash free as possible) to see if you can cut down on the amount of trash.  On the day of the trash free lunch, you ask the students to count how many pieces.

Here is an example of a student’s trash free lunch.

We kept track on a tally chart and realized the impact we can have if we change one simple thing, how we pack our lunches!  Then, the students draw a bar graph or a pie chart to show the results of the tally chart.

Record the data of a trashy lunch vs. a trash free lunch!

2.  Build a Solar Oven and Bake S’Mores:  Show a tutorial for how to make a solar oven a few days before Earth Day.  Here is one that could be made from a pizza box!

Tell the students to bring their own supplies from home (cardboard boxes, plastic wrap, aluminum foil, tape) and give them time to make them when they first get to school.  It took our class about 2 hours. The math involved is awesome, measurement, measurement and more measurement! I brought the graham crackers, chocolate, and marshmallows and they melted like crazy in the sun. It was super fun! Here are a few photos.

Students bring in their own materials, but you may want to have some extras on hand.

This student even put a skewer in the middle of the oven, too bad they decided not to use it in the end.

Some students learned the hard way that you need to CLOSE the solar oven!

3.  Study an Important Environmental Issue and Act on it: Perhaps the best Earth Day activity we’ve done is something that felt meaningful, like we could make change happen!  We studied the Great Pacific Garbage Patch by watching videos, reading about it, doing some math problems surrounding conservation, and by writing persuasive letters.  We ended the project by doing a water pollution science experiment.

Here is a video about the Great Pacific Garbage Patch if you haven’t heard about it (Depending on the age of your students, you may be able to show it to your class.):

Here are some photos of us trying to “clean” water, so students could find out how truly difficult it was.  As they work, each tool they borrow from me costs them money. They have to keep track of the cost of their clean up.

Students are trying to clean out a polluted basin of water using different tools (all of which cost different amounts of money). They keep track of their successes and the cost of cleaning out their basin of water.

A student is trying to remove vegetable oil from their basin of polluted water. NOT easy!

The bottom line, is there are so many things that students can do to learn about alternative energy, and to study current environmental issues. Instead of encouraging them to recycle with a coloring sheet or a worksheet, engaging them in these issues will help them feel an authentic push to do it!

I love Earth Day and the awareness it brings to young people! What kinds of things do you do with your class on Earth Day? Share below in the comments. 🙂

# Discovering Numerator and Denominator with a Pan of Brownies

The way to a child’s stomach heart brain is most definitely with sweet treats. While I don’t like to sugar up my students, I do like when they can connect math to the real world.  That was exactly my mission when I brought in a pan of brownies.

If all else fails, capture their interest with food!

So far at this point, we had examined the definition of a fraction, and thought about things that come in halves and quarters.  It was time to move into some more new vocabulary, the numerator and denominator of a fraction.

In came the pan of brownies.  I brought it over to a large rectangle table and had them all gather around me.  As they were salivating I asked them how I could split this pan into fractions so that we’d all get an equal amount.  I asked them to draw what that looked like in their math journals knowing that we had 25 students in the room.  This was easier said than done.

For some reason, a bunch of them abandoned the hard work we’ve done with arrays, and started drawing diagonals and squiggly lines all over their papers.  It was like they heard the word “fraction” and felt they needed to abandon everything they knew for this brand new concept.

*Sigh*

Then, I asked them to start sharing solutions, and we started to get somewhere. Arrays popped up on the chalkboard, 2 x 13 arrays (“I didn’t want to leave the teacher out!”), a 5×5 array and a 3 x 10 array.  I asked them which one would get them the best deal.

The settled on the 2 x 13 model so that I could get a brownie (how kind!).  That was when I began cutting.  I handed out the first one and asked them to think about what fraction of the brownie pan they were getting. That was when I introduced the fraction in number form and explained the difference between the numerator and denominator. The numerator was the number of pieces they were going to get to eat, and the denominator was the total pieces in the pan. For example (Hint: This is not the actual pan of brownies I used, since the cuts became VERY small and very messy…they were super gooey! So…I had to whip up another batch tonight for this picture, YUM!):

The numerator and denominator suddenly became clear!

They didn’t REALLY get it though, until the last person got their brownie. At that moment, I gave her my piece, telling her how proud I was that she was so patient to wait and be last. That was when the numerator part really sunk in, because I announced that she was getting 2/26 of the brownie, while everyone else only got 1/26. It was a lesson in patience as well as a lesson in math.

It was a pretty sweet mini lesson!

# Concrete Number Lines Teach The Value of Numbers

During one of my recent 20 minutes of math play sessions with my 5 year old daughter, we played around with a number line.  A number line is a really abstract thing, and without something to connect it to, it is nearly impossible for a young learner to understand. So we built this fun thing:

The conversation afterward was awesome! She could see that the more legos we had, the longer the line got, which meant we could talk about the value of the numbers.

That got me thinking that maybe we could build one that didn’t use similar sized objects for a number line, to help her understand that a number’s value will still be the same even if the object is a different size. That was when we came up with this one:

All of this building led to talk about sizes of things, grouping things, ordering and reordering, the value of numbers and so much more.  It was a ton of fun to make both with the number representations and actual concrete objects!

# Push Past the Tears: Peers Teach Perseverance

Today was the final stage of our Elementary Architects Project.  Students were about to turn in their design plans for approval (and classroom money), and were feeling the pressure as the clock was winding down.  With 5 minutes to go, one student (“Corey”- I certainly don’t want to cyber embarrass anyone with their real name) came to me, tears welling up and blurted, “I don’t want to turn this in, I don’t want to be paid, and I just want to throw it in the garbage!”

I was a little taken aback, because Corey’s blueprint of his dream bedroom was beautiful!  It had all the right things, and was designed with all kinds of creativity.  Miraculously it was to scale (many third graders struggle with this a bit!) and it was a totally perfect birds eye view.

So I sat him down at my table where four other students were sitting. Corey was fighting back tears as we went through the specifications sheet.  As we named each specification the other four students at the table gave him encouragement, pointing to each item on his blueprint as I read it off.  When we got to the part where it said to calculate the area of the space, that was when he really broke down in tears.

Now I know I should never underestimate the power of anchor charts, because at that moment, one of the other boys at the table looked at him and said, “Look up on the Smartboard!” We had worked through calculating area in three ways just a few days prior, so I’ve kept this slide up every time we worked on the project.

Then, he proceeded to help Corey figure out which way would be the best way to calculate the area of the space. The student who was helping used a calm voice, and his method was methodical.  Once Corey had the help, the tears went away and he handed me his final page with a very proud smile.

I learned today that a kind classroom community will support all learners. Students can stay calm in the face of obstacles if those around them are calm and supportive!  I think sometimes I feel like I have to be the one to help them, but their peers can be just as helpful if not more so.  As I reflected on my day today, I want to think more about how to foster that type of community so that everyone can learn to persevere together.