After using doubles and tens facts to learn one more and two more than all of those sums, that only leaves students with two facts to learn!!!
I allow students to tell me ways that these two facts can be easy to learn for them. Most students say that 3+6 is 9 so it is close to a ten fact. Some may be more comfortable with 3 +5, but after spending time talking about patterns with students, they will easily be able to discuss a way to get to this fact using a known fact. The thing that is uncomfortable about using 6+4 is that students have to go backwards and it is uncomfortable for them to go backwards in counting. Students favor going forward…cause more is better (like the commercial! :)) Also, 8+5 to me is an oddball. I can think of no easy way to get to this fact, but most students will say that 8 + 5 is close to 8+4 which is 12 so they know that 8+5 makes 13. Other students will say they know 8+2 is 10 so they can count up 3 more. I really don’t like that they have to count up 3 more, but at least it is better than counting up 5 from eight.
I will have to say that after working with intervention groups with all of these fact strategies, their answers aren’t as immediate as I would like, and at times they still use their fingers. I believe they still use their fingers because it is comfortable to them—more comfortable than thinking. After a strategy is learned it is imperative that they still practice with flash cards so that the facts remain fresh in their minds. I don’t work with a student population that readily has parents practicing with them at home on flashcards so the only extra practice they get is with me.
I plan on posting some of the materials I used to practice facts with the kids soon.
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If you have been following the previous posts, then you will see the progression of teaching number facts strategies. Nearly the last part of teaching addition facts focuses around doubles and doubles plus 2. I think this is one of the hardest strategies because kids may not readily see the double when it is two numbers away. With a little thinking and prodding, however, they will see the fact without using their fingers. Line the numbers up side by side so that students can see examples of both sets of numbers–the helping fact and the double plus 2.
Again, when having students recognize patterns and see relationships, I like to write them out of order so that students don’t say that the numbers are counting by 2’s etc. If students struggle to see the patterns, underline numbers to help them focus on what you want them to see such as underlining the second addends on both sets of equations. Then underline the sums on both sets of equations. Step back for a few moments and let the prolonged silence aid students in thinking about the relationships in the two sets of numbers. Give students time enough to generalize about how doubles can be a helping fact. * Note that students have already learned sums of 10 and 10s plus 2 more so they have strategies for 5+7, 4+6, and 6+8.
After this, students only have a few more facts to learn!!
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Thank you Erin Cobb! Frames Courtesy of Lovin Lit.
(Thank you Erin Cobb from Lovin’ Lit for the pretty border!)
Now after I have taught everything that I previously blogged about in Parts 1, 2, 3, and 4, which includes tens and tens plus one. Learning the sums/bonds of 10 is the foundation for this discussion. One of the tens plus 2 will already have been learned because it is a double, but there is no harm in learning multiple strategies to reach one fact. Also, doubles plus two facts will be learned later and doubles plus two will also give students a strategy to reach 7+5=12 and 5+7=12. Allow students to recognize this on their own when you reach that lesson. The more ownership students can have of the strategies without you telling it to them, the more they will remember the strategies and feel smarter for being able to discuss the strategies.
Again when you introduce these facts write them out of order on the board. Step back, wait, have children quietly look at the number facts and find relationships or patterns in their head. I use the Number Talks strategy and have them put their thumb on their chest when they find a pattern. This keeps everyone attentively looking for more patterns without the dramatic hand raisers flailing their arms in the air. If students say that they see lots of tens and twelves acknowledge this and then ask students to look for more. Eventually you will get what you are looking for if you have the foundation built from the previous lessons. If no students say that one of the addends goes up by 2 and the sum goes up by two, offer a hint by underlining these numbers so that they are focusing their attention there. Follow this up by fact (flashcards if you prefer) practice over the sums they have just discovered a strategy for and over previously learned facts.
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I thought I KNEW how to teach even and odd numbers until I saw this! Knowing that our third graders always miss the simple skill of even and odd numbers with two and three digits, I thought I would target this misconception. I told them that even numbers have partners and odd numbers have a lonely someone left out. To teach even and odd numbers,I asked if several small numbers like 4, 5, 6, 7, 8 were even or odd and drew pictures of counters to ask if they had a partner. With this idea, I asked if 227 was even or odd. I told the kids to write this down on a scrap of paper and cover it over with their hand so no one could see what they wrote. The teacher and I surveyed the room as kids secretly moved their hand. Just about half of the class thought it was even and half thought it was odd. Thinking of the 8 Mathematical Practices, I didn’t want to spill the beans. I wanted the students to really think about whether this number would come out with an even number of partners. I told them to start drawing 227 counters on their paper to see if each counter would come out with a partner. I knew this could be a tedious task, but to my astonishment, I saw a few kids actually drawing base ten blocks….WOW! I couldn’t believe they had thought of this. I had never thought of drawing base ten blocks!!!
Through all of their work building numbers in second grade, they were so comfortable and flexible with base ten blocks, they actually saw them as tools!! After we did this little exercise, which took about 15 minutes, we resurveyed the class and only 3 of the 25 students still thought the number was even. Success through taking the time for students thinking!
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.
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.