If kids aren’t building with base ten blocks to add and problem solve FOR WEEKS initially, they will get no where with their number sense understanding for regrouping. Under common core standards, we are heading towards the understanding of the traditional algorithm in the 4th grade standard–not just being quick at a procedure of crossing out numbers and writing new ones above them. Mistakenly 2 years ago we tried to rush the traditional algorithm with our second graders. As a result they are still struggling with this in 4th grade. So here is the success story of what we did in second grade last year. Nearing the end of the year, there were several skills that hadn’t been taught to the degree that they needed to be such as geometry etc. I knew without the foundation of addition, counting, and problem solving we would be up against a wall again in 3rd grade, so we focused on these skills. Throughout the year we spent a lot of time filling out number charts and discussing pattens on the hundreds chart above 100 using these:
Counting you ask? Yes, we spent time counting and looking at patterns in numbers. I know it is in the standard so that is part of why we counted, but counting is so much more important than teaching because it is the standard. How can students reason about whether their answer makes sense if they can’t count? Reasoning about math is in the mathematical practices several times. Students who can’t count, can’t estimate and can’t round because they have NO idea about where the number comes in the whole sequence of numbers.
Second graders last year solved a CGI word problem each day while they were learning addition and subtraction. Students spent several weeks using base ten blocks to solve their addition and subtraction regrouping problems. When students weren’t permitted to use the actual blocks, we prompted them to draw illustrations of the blocks to help them solve their problems. Even after students were shown how the traditional algorithm worked with their blocks, most of them tended towards drawing a picture of the blocks to solve the problem. Most were successful doing this. I was satisfied with this progress because I knew in 3rd and 4th grade that they would again have an opportunity to learn the traditional algorithm and other addition/subtraction strategies.
So here is how we are beginning with the kiddos in 3rd and 4th grades this year to teach addition regrouping. The kids are still given the opportunity to use blocks if needed to formulate understanding. Now I know that in showing them how to regroup the kids aren’t really “discovering” or “constructing” the algorithm themselves, but they are gaining an understanding. I just don’t think we have enough time in the year for the kids to discover everything and they must be shown some things. I haven’t arrived at that place yet where I think in CGI utopia…maybe I will get there someday?? (Don’t get me wrong, I find value in CGI) For right now the kids are getting this method of teaching addition regrouping and making sense of it. I’m happy and the kids are learning.
Now what I’m about to show you is the students’ first experience with regrouping like is pictured above. It isn’t cute at all…not worthy of for sale anywhere…but it is real and handwritten. To make it on a handwritten page was just so much faster than doing it on computer so it is what it is. I wanted to create columns so the students wouldn’t get their numbers confused. This worked well. I didn’t have the kids put pluses between the numbers like true expanded form to keep them from confusion later on when we do subtraction regrouping similar to this.
We discovered that students had a difficult time in the hundreds column when they had a number regroup to the thousands place. They weren’t used to putting two numbers together that weren’t zeros so this seemed to confuse them. If we had three digit adding to do over, we would have the kids include a thousands column so that they could regroup their thousands there at first until they made the connection that they could put two digits other than zeros in the left hand column. In other words, we would have them add one column more than the number of digits that there were in the number. For example…
Later on last week, we taught the kids to regroup without the columns drawn and without the numbers being decomposed into hundreds, tens, and ones. We continued to have the kids draw the arrows and to estimate their answer. It was rocky at first and about half of the class got regrouping with numbers written in standard form (just normal). They will be working on regrouping again early next week.
Today I was at a professional meeting in which we were working on building a pacing guide for the 3rd-5th common core standards. The presenter for our meeting showed us this fabulous website which has cutting edge materials pertinent to the common core standards. There are a few sample tasks for each grade level with rubrics. Not only are there tasks, but there are videos of teachers, coaches, and principals discussing student thinking before and after a video taped lesson as well as teachers teaching lessons. If you just want to view a part of the video sessions, they are broken up into parts: before the lesson, the different sections of the lesson, and the debriefing after the lesson. I really enjoyed hearing the teachers discuss why and how they were planning because I had a sense of the fact that I am not alone in the problems I am having with children’s mathematical thinking. These problems are found across our country–the one aspect about the second grade video that grabbed my attention is that this second grade teacher was discussing how her students see a word problem and want to immediately add the two numbers within the problem even though the problem may not be an addition problem. All in all this website is a fabulous find especially if you are responsible for doing any professional learning meetings at your school–or even if you just want to learn yourself. (FYI: As of the time of this post, there are a lot more upper grade videos than elementary, but the site is worth a visit no matter what grade you teach.)
The first grade teachers at school absolutely love introducing subtraction and addition number sentences to their kids using the book Ten Flashing Fireflies by Philemon Sturges. I discovered this book in a lesson recorded in a Math Solutions book entitled Minilessons for Math Practice K-2. There is also a similar lesson (I think…not positive) in another Math Solutions book entitled Teaching Arithmetic. In the lesson students model the action of gathering fireflies into a jar using snap cubes. In the book there is a jar printable to use or the lesson suggests using a sheet of blue construction paper to represent the night sky. Not only is this lesson good for introducing the action of subtraction and addition, but it is also good for discussing one more and one less. Because this is such a beloved book that builds a great foundation for addition and subtraction, I worked on building this free SMART Board lesson to accompany the book this weekend, and so here is an example of this lesson. Just click to download the SMART Board lesson for free.
I learned this simple but powerful game–Make Ten from Melissa Conklin of Math Solutions at NCTM two years ago. The first and second graders at school have successfully played this game for several days to help strengthen their number sense. They have already become much more fluent in recognizing the sums (bonds) of ten. Make a deck of ten frame cards. Downloadable for free right here (Free Ten Frame Cards). Copy the printables four times so you have enough to make a deck. Students lay out four cards from the deck face up on the table between two to four partners. (I think the game works best with pairs). Then students take turns to pull two cards that have a sum of ten. If there are not two cards that have a sum of ten then students may pull one more and place it face up in the middle of the table until there are a set of two cards that will make ten. When students pull the pair of cards from the center of the table, they say the equation that matches, for example, three and seven make 10 or three plus seven equals 10. After students have played the game once or twice, have them record their equations in their journal. I highly recommend playing this game to build number foundations to ten.
I am also posting a clip here of a ten frame SMART Board slide I made for my K-2 teachers to adapt to their specific needs. This slide has all of the ten frame cards on it from 0-10 and would be great to adapt for many Math Solutions lessons such as this one.
Marilyn Burns (my math hero) co-authored a book called Extending Division, which has many lessons for students who are in the process of learning division. One of the lessons is called “The Division Game”. In short the students make two decks of cards–a pile of multiples cards and a pile of factor pairs. Students draw one factor pair card and five multiple cards. The object of the game is to eventually gain a hand of multiples cards (by drawing and discarding) that match the factor pair card. During an actual experience with fourth graders, the students needed most of the hour class period to actually understand the game and didn’t have quality time to experience the game in full the first day. Once the students had understanding of the game, they really enjoyed playing. The other quibble I have with the game is the amount of time that students must take to make their own multiple and factor pair cards. While making multiple and factor pair cards is another opportunity for students to become familiar with these concepts, again this becomes a time factor spent making cards when students, in my opinion, could spend their time doing a richer activity.
One of my favorite math games for elementary math students to play is “Close to 100”. This lesson and game can be found in TERC math Investigations books for third grade. The game instructions, number card blackline masters, and score sheets are in the unit Mathematical Thinking. In this game one student of a pair draws six numeral cards (0-9) without looking from a deck and selects four cards to use. With these four cards students are to build two two-digit addends to find a sum as close to 100 a possible. The player’s score is how far away from 100 the sum is. For example if the sum is 102, the score is 2. If the sum is 95, the score is 5. Each player totals up his scores at the end of the game, and the player with the least score wins. What I like so much about this game is that students are practicing facts, learning place value, buildingnumber sense, adding, and subtracting while they are engaged in cooperative learning. The only quibble I have with this lesson is that the only assessment to be gathered is informal teacher observation. To make up for the lack of assessment provided in the lesson, I recommend that after playing the game to give students a scenario in which they draw six cards. Have students write about which of the cards they would chose to use to get as close to 100 as possible. To differentiate this lesson, struggling students can pull only 5 cards and choose 3 number cards to make 20 (First grade Mathematical Thinking book), or advanced learners can pull 8 cards and choose 6 to build 1000 (Fifth grade Mathematical Thinking book).
See an example of this game here: http://www.pearsonschool.com/live/images/custom/investigations/Investigations_widget1.html.
One of my favorite lessons to teach comes from comes from Marilyn Burns’ Lessons for Algebraic Thinking, Grades K-2. This lesson is called Two of Everything. While this lesson is very wordy to read, as are most of Marilyn Burns’ books, the heart this lesson is very valuable. In this lesson the teacher reads to the students the book called Two of Everything by Lily Toy Hong which is about a couple that drops items into a magic pot and they double (excellent book). This book provides an solid foundation for students to conceptually understand input/output tables because items are being put into a magic pot (input) and items are being taken out of (output) a magic pot. Then students create their own magic pot patterns on their own input/output charts. When teaching this lesson, I like to get students attention for guided practice by bringing in a magic container of my own and already having some items stowed inside to pretend its magic as I show them other possible inputs and outputs for the table. In this lesson students do work on blank paper and draw their own pots and t-charts. However, I like to have prepared sheets for students to use especially when I have limited class time. Math wire happens to have a sheet that fits perfectly with this lesson–just follow the link.http://www.mathwire.com/algebra/magicpotworkmat.pdf.
If your students have difficulty counting coins a few things will help. A few things I have tried work well.
- Practice adding 10 more and 10 less on the 100’s chart with any number so that students recognize the patterns when adding 10 to a number. Discuss these patterns.
- First, begin counting money values with students without coins. For example, add 25, 5, and 10 randomly on the board and discuss with students how they decided to add the numbers and relate this to money. Some students may see 25 and 10 make 35 and then add 5 to make 40. Others will add 25 and 5 to make 30 and then add 10 to make 40. Having the class discuss the ways students add numbers and the most efficient ways strengthens students’ abilities to add coin values. (from Van DeWalle’s Elementary and Middle School Mathematics)
- Use coin antennas for students who struggle with counting. Antennas are the marks you make on each coin that stands for a value of 5. A quarter would have five antennas, a dime would have two, and a nickel one. If students struggle too much with number sense, then they can count by 5’s after they mark each coin. (from http://www.mathwire.com/money/money.html)
Visiting with a colleague, she passionately shared with me the amazing difference that a book called The Daily Five had made in her literacy instruction. My colleague, who is a veteran teacher, learned about this at a differentiation conference, and she tried using the methods for the first time this school year. Other teachers have come into her room during her reading workshop and see the quality of student engagement. They insist that she share what she is doing to have students so engaged. The Daily Five improves students’ stamina to read voraciously on their own and to independently work on their own while the teacher pulls small groups. The small groups are more targeted to individual needs because the teacher spends less time with each student and doesn’t have to attend to students who are working independently. To read more about “The Daily Five” visit this link: http://www.thedailycafe.com/public/department38.cfm.
I recently observed my mentee teach an engaging lesson that reached auditory, visual, and kinesthetic learners. She used the Yarn-Length Hunt lesson in Vicki Bachman’s Sizing Up Measurement: Activities for Grades K-2 Classrooms. In this lesson students compare string to their group members’ strings. They also form a line and compare their string with each student in the class. Then students are supposed to go on a hunt for objects that are shorter and longer than their string. Instead of students going on a length hunt my mentee had tubs of objects awaiting students at their tables for them to compare with their string. We decided on this instead of the length hunt to keep the students more focused. Students compared the objects with their string and traced or drew them on paper underneath the headings shorter or longer (than their string). All of these activities were very beneficial and led to students using rich math vocabulary with their peers. In reflection we thought instead of students drawing their objects, it would be better for them to trace them. Alternatively, students could use clip art objects that represented the items in their tubs, and then students could glue them down underneath the headings shorter or longer. When students draw their objects, they are difficult to decipher and to compare with the string for assessment. This lesson is excellent for its simplicity to reach kindergarten children. With just a few tweaks, we found it to be an excellent lesson that resulted in mastery of the vocabulary shorter, longer, and compare.