annzats Sep 21, 2011 1:15 PM (in response to wtaylor)
The lattice (or box) method is a great way to keep the places straight. Have the students create a 2 x 2 grid and put the numbers they are multiplying on the top and side. To fill in the grid, multiply the number above and to the left.
It's hard to show by type, but I'll try. For example: 23 x 14:
1 | 2 | 3 |
4 | 8+1 | 2 |
Then draw diagonal lines (top right to bottom left) to show how to add
the digits are 2, 8+1+3, 2 or 2, 12, 2 which becomes 322
here's a slightly more complicated version:
jciolli Sep 21, 2011 1:31 PM (in response to annzats)
This is good when trying to get this intially taught to a student, but I caution you in using this as the only means. I have had 6th graders that have come to me that know only how to use lattice to multiply and have no idea why they are doing what they are doing. Make sure to teach/show them both ways:)!
burtr Sep 21, 2011 9:00 PM (in response to annzats)
I don't allow my fifth grade students to use the lattice method as too many of them become dependent on the method and, in my opinion, it does not develop any kind of number sense at all. In addition, our middle school teachers do not allow their students to use the method either. My preference is partial products which is basically what Anna and Beth have discussed in their posts. In class, we call Beth's method Blank Array as it is basically empty boxes that would have been filled with dots in an actual array, but since the numbers are too large to actually do this, we just draw the boxes and put the numbers on the top and side to show how the numbers are expanded, then multiply and write the products inside each box, then add the products together.
The other method I teach because of its wide acceptance with middle school teachers and parents is the traditional method. After teaching the two methods I allow students to choose between the methods.
b.pendleton Sep 21, 2011 1:37 PM (in response to wtaylor)
lucynsky Sep 21, 2011 2:00 PM (in response to wtaylor)
Along with the area method, I teach my students to break it into two seperate problems multiplying by one digit. Then add the two partial products.
25 X 42
x 40 x 2
------- plus -------
It seems to help slower learners. It is imortant to teach them the standard algorithm as well when they are ready as Ann suggests. This is 4th grade.
dtrigiani Sep 21, 2011 7:29 PM (in response to wtaylor)
I teach the same way noted in Anna's post. It really reinforces expanded form and I constantly work on making them speak math language by explaining their answer as they model their strategies on the smart board. To piggy back off of Anna's post....I have taught my students to say 25 x 42 is solved by expanding the factor 42 into two mathematical problems. 25 x 40 and 25 x 2. We talk about how we all know our basic facts. We solve 25 x 40 as drop the zero to the basement. Multiply 5 x 4 in the ones place and then cross multiply the 4 in the ones by the two in the tens place add the two we regrouped. We do the same for 25 x 2. Then to end the problem we say we have to add the products of our two multiplication problems.
This is just another way to keep the mathematical language expanding and reinforcing expanded form and place value.
liztrimaloff Jan 19, 2012 12:12 PM (in response to wtaylor)
wgmccallum Jan 19, 2012 8:44 PM (in response to liztrimaloff)
In the Common Core, the fourth grade standard 4.NBT.5 includes "... multiply two two-digit numbers, using strategies based on place value and the properties of operations". This is well illustrated by the method for 25 x 42 described in Anna's post. We first use place value to understand 42 as 40 + 2, and then use the distributive property to write 25 x (40 + 2) = 25 x 40 + 25 x 2. Of course, we don't expect fourth graders to use the term "distributive property". Area or array models like the one in Beth's post are useful in understanding this property.
Ann's array for 23 x 14 suppresses the place value, by just using the digits themselves without the 10 or the 100. I like Beth's diagram better because it writes out the 20 x 40, 5 x 40 etc. I would want to make sure students really understood that the 2 in 23 stands for 20 or 2 x 10 before using the lattice method with just digits in the boxes.
I took a look at the turtle video, and I have to say I was really worried by it, because it seemed to be all about seeing a visual arrangement of digits without thinking of the meaning of them in the base 10 system. We want to encourage students to see numerical expressions as meaning something about operations on numbers, not just as pictures.
topdog Jan 19, 2012 10:06 PM (in response to wgmccallum)
Perhaps you may have heard of Howard Gardner and Multiple Intelligences. I have many visual-learners who understand partial products, but have a difficult time making the transition to the traditional algorithm. I find the visual of the video to be especially helpful with ADD or dyslexic children who are overwhelmed easily and need information to be decoded, packaged or organized. It's just another tool in my differentiation toolbox.
Your comment about being "worried" assumed that the video is used to teach the skill, which is not true. We all know how to teach this skill. But, that is not what this thread is about. Wendy is looking for different approaches to help her children succeed when they are struggling. Your evaluation of each person's post only serves to make participants shy away from participating. It best to help, not judge. After all, this is a blog, not a lecture hall.
wgmccallum Jan 20, 2012 6:03 AM (in response to topdog)
Sorry if it sounded like I was evaluating peoples' posts, that was not my intention. In the morning light with a cup of coffee in my hand I can see how my post might read that way! I found the comparison between the different methods interesting as I read through the thread, and used people's names to identify the methods, not to judge people themselves. That said, I do think it is appropriate to try to judge the methods. This judgement should certainly include thinking about how different children learn, as you describe, but it also includes thinking about the mathematical ideas. I can see how the turtle algorithm helps children execute the algorithm, but I worry about the price paid in understanding. I don't think it is a "strategy based on place value and the properties of arithmetic", but maybe it is a harmless mnemonic and I should stop worrying. I'd be interested to hear other peoples' opinions.
liztrimaloff Jan 20, 2012 6:43 AM (in response to wgmccallum)
WILLIAM MCCALLUM thank you so much for your inaugural post and your response to Diane.
We are all here to improve how we engage students and learn from each other. Everyone is absolutely welcome to share their opinions, especially with supporting rationale. Diane Hubacz was able to expand on her suggestion of the video, it is only part of what she uses, and I'm glad you raised your concern giving her the chance to really share more fully what she does in the classroom.
Let's definitely hear from others, and welcome your input with a different perspective, as we prepare for the Common Core. From my understanding, math is the subject expected to change the most. Maybe that isn't the case?
burt9172 Jul 4, 2012 9:39 AM (in response to wgmccallum)
Response to WG McCallum's request for opinions. I understand your concern over the turtle method, however, if it is used as just another way to reach those students that have not yet mastered the skill of multi digit multiplication, then I don't think there is any harm in it. I certainly would not start with this method, but with the area (blank array), partial products, and traditional methods. I too have used the turtle video and it clicked with some of my students that were still struggling with multiplication long after they should have mastered the skill. I do believe in teaching methods that will build number sense, and I also use appropriate vocabulary to go with any given method such as the distributive property as I want students to recognize and understand the term when they see it. I'm not sure why this is such a sensitive issue, but I know I have gone through a whole thought process on this topic when we were using Everyday Math which included the lattice method. Many of my students clung to this method and were reluctant to learn partial products, blank array, or the traditional method after learning lattice. This casued much discussion in our district as our 5th graders moved forward to middle school where their math teachers became frustrated with students that didn't know efficient methods. In support of the turtle method, it is much more efficient than lattice and it is my opinion that once students get it, the turtle falls away leaving the traditional algorithm in place.
wgmccallum Jul 4, 2012 10:02 AM (in response to burt9172)
Thanks, this all sounds quite reasonable to me. Some students need to understand what they are doing before they can make the moves (my youngest daugher, for example) and others want to learn the moves first before they are asked to explain why they work (my middle daughter, for example). So I guess the turtle method could be seen as a useful way of teaching the moves to the second sort of student. I feel some aversion to doing this, and as a result used to drive my second daughter crazy by always asking her why, but I respect the need for imperfect tools in imperfect situations.
community_manager Jan 24, 2013 10:26 AM (in response to wtaylor)
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