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 ;).
Several years ago I worked at a charter school the first year it opened. The school implemented Singapore math, so that was my first year to test the waters of Singapore math. Our trainer instructed the 3rd grade teachers to go ahead and teach branching even though it was a skill the students should have learned in second grade. To teach children the procedure of branching, it took about four weeks total, and then not all of the students perfected the ‘procedure’ of addition and subtraction branching. The students had more success learning addition than subtraction branching. With the mandates of testing, we weren’t able to solely use Singapore math, but I had to supplement with other materials. Then as you are all familiar with, testing approached and likewise the pressure along with it. Then we didn’t have ‘time’ to teach number sense SO deeply since other skills are tested. Unsurprisingly, the teaching of Singapore Math somewhat fell apart midyear. Please don’t take this wrong I LOVE Singapore math because it works, but the conditions of testing hindered us from teaching it wholly.
Fast forward to four years later. After teaching small groups today, I have reflected on the year that I taught branching and its effectiveness. Yesterday I pulled small groups of average math students to teach them regrouping for the second day in a row, I had them build double digit numbers with base ten blocks blocks. I repeated this process today with the same group of students. After that I started notating their thinking with branching representation on a small white board. Students intently watched and helped me notate the thinking they had done with the blocks in (abstract) numbers . They began to understand grouping with tens and how to decompose numbers to build more tens or hundreds. Then I told them that they couldn’t use paper or blocks, but could only look at the addends I was about to write on the board. I asked them to whisper the answer in my ear so that others could still think. I was amazed! Half of them could answer the question correctly doing mental math. The other half were only 1 away from the correct answer. I was so proud.
I shared the above to really say the kids taught me something in just two days because of their adept ability to add mentally. Teaching branching worked so much better four years later–all I had to do was provide an experience with branching directly after building with base ten blocks. Why didn’t I start out with the concrete blocks first before I threw abstract numbers at them…duh me! Branching made so much more sense to them after building a concrete foundation. Reflection is priceless!
After sitting in a week’s worth of meetings about Common Core instruction and best math practices, I feel as if my way of instructing students is gradually morphing into a new beast, which I am not sure I fully can describe yet. In the midst of all of this, I went back to the fifth graders I have been working with on the day after these meetings. I decided to try out some of the teacher practices I had been learning. Sidenote: This set of fifth graders isn’t yet proficient at long division or ladder division for that matter. I posed a long division problem to the students, but instead of giving it to them in “naked numbers” alone, I gave it to them in word problem form as well. In reflecting, I should have started out with the word problem alone and let the students figure out that they were supposed to divide. After watching the students struggle with solving using ladder division or the traditional long division algorithm, I encouraged the students make equal groups to see if that helped them. One student in particular that I worked closely with (whose work is pictured below) managed to find a correct answer using equal grouping, however he could not find his mistake when he used ladder division. Another student…one I might add who typically doesn’t have much perseverance to solve problems…related completely to the contextual situation and persevered through to solve the problem to the end. To get to my MAJOR DIVISION REVELATION, let’s look at this student work below:
So, you ask, what is the big AH-HA? If you notice that the numbers in the student’s circles (equal groups) are the same numbers to the side of the division problem (or the ladder). The numbers to the side of the ladder are actually the chunks that students would naturally break off if they were naturally dividing cookies among people etc.! Now I know according to my PD classes that I shouldn’t have just told and shown the student this relationship, but I should have allowed him to figure it out. I was just a little too excited to hold in my own personal discovery I suppose.
Another discovery I had in experimenting with having students discover their own strategies, I learned that students actually do better and persevere more when the problem is in a context familiar to them and it isn’t just a naked numbers problem. Having a contextual situation actually gives the students an entry point into the problem, and so they don’t give up as quickly. I have always thought backwards. Students MUST UNDERSTAND THE NAKED NUMBERS FIRST and THEN they can SOLVE HARD WORD PROBLEMS. But in reality after my experiment with these fifth graders, they were much better at solving this division problem in their own way when they had a context. I have had a complete paradigm shift, and you were all here to see it above– that is.
Students often struggle with labeling the hands correctly on a clock with quarters of an hour. This is for several reasons:
1. Students hear quarter and immediately think 25 cents, so they label the minute hand on the 5 to represent 25 minutes. To redirect students from this misconception remind them that quarter means four parts. Quarts of a gallon means four quarts equal one gallon. A quarter of a dollar means 25 cents because 4 twenty-fives equal 100 cents or a dollar.
2. Students may understand that a quarter is fifteen minutes, but they hear literally, “Show me a quarter after 5.” Students will literally find the 5 on the clock and find fifteen minutes after the five. Then they will put the minute hand on the 8. To correct students explain that it is fifteen minutes after the minute hand hits the new hour or the twelve.
3. Students sometimes are confused with the language a quarter after, a quarter past, a quarter to, a quarter til, a quarter until, a quarter before. To help students with the language, create a chart that shows after and past mean the same things with an arrow showing counter clockwise. Likewise, chart a poster that shows that to, til, until, and before mean the same things with a hand pointing counter clockwise.
Seeing the same problem, students continuing to mix up area and perimeter questions, reoccur with our 4th and 5th grade students on their unit tests, we decided to try something new to help them differentiate between the two. With the questions already cut out, we took all of the released area and perimeter questions from our previous state tests and had students do a sort with them. Pairs of students sorted the questions underneath an area or perimeter heading. To add a little challenge to the activity, we added some volume, capacity, weight, multiplication, and division questions without telling them that these questions weren’t area or perimeter. As teachers, we learned during the students’ sort that students were thinking of area as the space inside of anything so that they were confusing volume and capacity with area. This led to students gaining a deeper understanding of the meaning of area. The students also learned from one another as Bloom’s higher order thinking on the evaluation level was in place. Students had to discuss each question and agree or disagree with one another about the decision to place it underneath a heading. See below for a look at our activity.
The common core standards for first grade state that students must become fluent with adding and subtracting tens. To promote fluency, students need to discover patterns and how they change numbers. Only when students have examined patterns and become comfortable with them should they be given timed “naked number” problems to assess and improve their efficiency in recalling the patterns. Finding the answer to a “naked numbers” problem needs to be merely a by product of knowing the pattern.
When trying to figure out how to teach fluency with adding and subtracting multiples of ten to first graders, I struggled to find resources to do this. Because of the lack of resources, I developed these sheets for next week to lead students and their teachers into guiding discussions about patterns on the hundreds chart. They are simplistic activity sheets but necessary. I will be trying them out this week. I am providing a link to them below. If you would like to try out one of the lesson sheets, the preview file will give you a sample for free at TPT.