Check the trash first! Whenever teaching dimensional solids, I look around the school building for large boxes that may be thrown out. Especially in the teacher workroom, there are always bulletin board paper boxes, toner boxes etc. that are being thrown away. This is where I have found some of the best trash for treasure pieces for my 3D solids collection. When I have found one, I wrapped it in colored bulletin board paper with the name on each one to help students have a constant visual of prism pieces. At the time I teach solids, I also have the students bring in items they find at home that may be prisms, cubes, spheres, or other solids. They relish sharing their found items with the class. When they share them with the class, they must ask the students how many faces, edges, and vertices there are. Students get extra credit for bringing in solids. The best solid that I ever had a student bring in was an almost perfect triangular pyramid made out of rock! Below are pictured my recycled trash 3D solids.
After years of seeing students mix up math operations in word problems, I have finally figured out how to help students understand what operation to use in word problems. This little word is causing students much of the confusion–EACH. Haven’t we all taken for granted that students understand what this word means. The word ‘each’ is in nearly every multiplication and division problem, but many students don’t know what it means–every one in the group. If we teach students to read a word problem and replace the word each with its meaning, every one in the group, students somehow have a light bulb experience.
In conjunction with teaching students to understand the word each, also asking them questions about the problem helps facilitate understanding. For example when you ask, “Is this a joining or a separating situation,” students start to make sense of word problems. Students generally understand that words like altogether and in all mean that they are joining groups. The word total may need to be taught as a word that means in all, but total isn’t a difficult term for students to become comfortable with.
To help students further differentiate between multiplication and addition, ask questions like: are we adding the same amount over and over or are we adding two different sized groups? If the answer is adding the same amount over and over, then multiplication is repeated addition of equal sized groups. If students are confusing division and subtraction, ask, “are we subtracting different amounts or are we subtracting the same sized amounts over and over. If the answer is subtracting the same amounts over and over, then teach students that division is repeated subtraction of equal groups.
I’ve been wanting to incorporate counting collections at school for a while, but I haven’t had the understanding of how to organize counting collections effectively. I recently attended a colleague visit where a kindergarten teacher showed the procedures she used for teaching counting collections. So, after attending this training, I initiated counting collections at our school with the 1st and kindergarten teachers. In the meantime, one of the kindergarten teachers shared with me at school that she realized her students didn’t know what a group was– much less know what a group of ten was. She began her instruction with just discussing groups and what kinds of things can come in groups. They talked about groups of three, four, or six etc. They made groups of different amounts in whole group discussion under the document camera. Students were able to have a foundation to understand a “group of ten.” Then the teacher was able to place a different amount of counters underneath the camera to ask if she had a group of ten. First, she placed less counters under the camera like 8 and asked if she had a group of ten. After that she placed more counters under the camera, like a group of ten and 3 more, and asked if she had a group of ten. Doing all of these seemingly common sense-ical counting procedures before hand led to a much more successful counting collections lesson for students to count their collections effectively. These are the rudimentary things that no college or textbook teaches you!
To read the valuable counting collections article from Teaching Children Mathematics, click here.
When having second grade students explore patterns in number charts which were in increments of 300, it dawned on me to cover up some of the numbers to show students how the numbers repeated. I did this on the document camera. For those students who weren’t able to see the number patterns explicitly, this proved to be very helpful.
The number chart is shown above uncovered.
First, I left one column uncovered except for the hundreds place. Students were easily able to see how the hundreds place repeated.
Then I uncovered all but the tens place. Students saw that the tens place goes up by one ten going down each row.
Finally, I uncovered all except the ones place and students were able to see that the ones place remained the same ALL the way down the chart.
In case you are interested, these number chart printables to 1,200 are available here. There are fill in number charts too.
Smart Board lessons that match the printables are available here which may work even better for showing the patterns with the screen shade.
Another base ten realization occurred to me today! Working with one of my intervention groups I had them build the number 199. I initially had the intention for them to add one more unit after they had counted to 199 for them to cross a century. This would make the number 200. While working with the students in the group, only half of them could actually count the number they had built. Then I realized that students can easily build a number with blocks recognizing the pattern of hundreds, tens, and ones without actually understanding the number they have built.
While I know it may take a while, I suggest that while students are building or representing base ten blocks that students actually show their counting numbers underneath, which I had never thought to make students do before. I had always taken for granted that students understood the counting numbers if they could build the numbers with blocks, but this regretfully isn’t always the case.
Students who struggle with number sense aren’t sure how many 10′s are in 100, how many 100′s are in 1000 and so on. Because of this I work on this skill often with students in my intervention groups. On more than one occasion, I have found that students even as old as fifth grade have a misconception about the thousands block. Now that we have math tools made from plastic instead of the vintage wood ones, some students are confused when they lift the thousands block. They realize the plastic thousands block is hollow, so when I ask them how many hundreds are in a thousand, they count the sides and say six. I have to correct them and have them just stack the hundreds blocks until they are the same size. Then they realize that 10 hundreds make 1000.
To help 5th graders understand decimals last week, I built this number line using an old roll of fax machine paper. I measured off a little over two meters and then marked every two centimeters to put another number, so I would have room to write the numbers and for them to actually be seen. Students don’t usually have much of a problem ordering decimals to the hundredths place because they can visualize pennies and dimes, but past that students struggle. Also, thousandths are a bit daunting to teach…after all they don’t make “thousandths” manipulatives….at least that I am aware of. This coming week, students are going to build their own number line between two hundredths and we are going to connect all of the number lines and put them somewhere…I am not sure where because it will be VERY LONG because 100 numbers are written on it. Another something I did to the number line is I glued hundredths blocks down underneath the hundredths numbers, so students could see the concrete representation of these.
In case you aren’t familiar in decimal base ten block world:
a flat = 1 whole
a rod = 1 tenth
a unit= 1 hundredth
When explaining hundredths and thousandths to students I do the unthinkable. I take a blue foam base tenth block and a pair of scissors in front of the class and SNIP a hundredth goes flying a few feet away. This grabs students attention because #1, I just cut a holy math manipulative, and #2 something just went flying across the room for those students who may have just momentarily zoned out . No worries, I have had tubs and tubs of these math manipulatives (oh we are calling them “tools” now) that I could build a shrine to them with lit candles. In other words I have plenty that if I cut one it isn’t a big deal. THEN, I take the itty bitty hundredth that I just cut and SNIP another slice goes flying. I tell students that this slice is one thousandth. This visual really helps students to see how tenths, hundredths, and thousandths are related. A speck can even be cut off of the thousandth so that students can see what a ten thousandth looks like. After I have cut all of these pieces off, I put them underneath the document camera so students can see them up close.
I love it when I’m right. The other day I was having a friendly debate with another teacher about whether or not to teach the cent sign with the new common core standards. After all, sometimes students use the dollar sign at the same time along with the decimal point and get them confused. I argued, however, that you still see the cent sign at times in stores , but this person argued that you don’t see the cent sign anymore…well, here you go…the cent sign at back to school time! Seventeen cents for a spiral bound notebook. My proof that IT IS STILL IN USE, so we still need to teach students how to read them!
I’ll let you in on my little secret. Now beware it is a little simple and silly, but kids love silly and so my story works.
The cent sign at one time was the dollar sign’s girlfriend, but they broke up. Then the dollar sign and the decimal point got married, so they are seen together almost always. The cent sign got her feelings hurt when the dollar sign got married to the decimal, and so she ran away. THE END.
Adapt and embellish the story to fit your personal style. Now just remind your kids of this story any time they get confused about the notation of dollars and cents, and they will remember which sign to use.
Never take your blank classroom wall space for granted. Below I have pictured what I placed around my classroom clock. Then I added some extra items instead of just the numbers around the clock with the quarters of an hour as I saw students having misconceptions about several concepts. I noticed students struggling with all of the different words that meant before (to, til, until), so I posted an anchor chart showing this around the clock. I also posted an anchor chart showing how each fifteen minute increment adds up as quarters of an hour. This is a picture from a classroom I had several years ago, and it isn’t the most beautiful, perfect clock display. If you use this idea or have done something similar, post a link in the comments to show how you displayed yours in a more aesthetically pleasing way since mine is lacking in this department .
When I stepped into a classroom yesterday I was so intrigued that I couldn’t leave. Before I spill the beans on what I saw, I must say this. There has been a lot of emphasis at my school about having students share their work for a lesson closing. This idea could also spill over into the common core mathematical practices in which students must “construct viable arguments and construct the reasoning of others”. Now I understand that when students share their work in front of the class that this does promote other students’ higher levels of thinking as other students decide whether they agree or disagree. On the other hand at this late point in the school year the downfall of student sharing is that even with a doc camera and students’ micro phoned voices other students attention spans are likened to a fly hovering over a summer picnic buffet.
Now, onto what I saw. Ms. T was showing students a flip cam video of herself talking to a student named ‘Briana’, who was solving a double digit addition problem with base ten blocks which she had taped during the students’ work time. She showed the video to students after their work time and paused it after the questions she asked Briana in the video. Then Ms. T would ask the class what the answer was to the question in the video. The class would respond. Then Ms. T would un-pause the video to allow the class to see if Briana answered, counted, or exchanged blocks correctly. I absolutely loved this–so much more engaging than regular sharing!
Thanks to the literacy people who ordered these flip cams with literacy money! They were originally bought for students to do book talks. Using them for math sharing–so much better in my unbiased opinion .