A Fun Way to Check Multiplication Problems
How often have you gone to a conference and been super-impressed by what a speaker shared? Has it happened often? It happened to me when I went to a conference as a very new teacher (in my second year, I believe), more than 20 years ago. At that conference, I was lucky enough hear Dr. Lola May speak. She was a great presenter, and certainly made an impression on me. I still have the book that was given at that conference and have referred to it many times over the years.
It was at this conference that I first learned how to use "casting out nines" to check the answers to multiplication and division problems. I had never heard of this method when I was a student, but being a new teacher, I kind of assumed it was a method well-known to other teachers.....
Have I mentioned that I love Jo Boaler’s books and site, Youcubed.org? Well, I do! She shares so much fantastic research and so many wonderful ideas.
So, I was reading her book Mathematical Mindsets this week, and read about the “array game” (called How Close to 100), which I’ve seen all over Pinterest and thought was very cool. I tried it with my classes last year during a little bit of down time, and they liked it. I hadn't really thought of using it this year, but last week I noticed the baggie of polyhedral dice that I've had for a looooong time and thought it would be cool to use the dodecahedron dice for the array game. With these dice, the students could use numbers up to 12, rather than 6.
To set up their game, students each outlined a 20 by 20 area on their own graph paper. They took turns rolling their dice and creating arrays to represent the multiplication problem they had rolled. It was very interesting to observe the way students arranged their arrays. Some started in the corner and worked their way out, while others started on one side and worked their way across. Some made the arrays touch, if possible, while others left a row between each one. Some just drew their first few arrays anywhere and then discovered that they didn't have a lot of room to fit additional ones. The "winner" was the student with the fewest number of boxes left (some did get to zero left). The students really had fun with this!
Of course, some finished their games earlier than others. In these cases, I asked students to create arrays that used different numbers than the numbers they rolled, but represented the same area. For example, if they rolled 12 and 5, their arrays could be 10 by 6, 15 by 4, or 20 by 3 (not 30 by 2, we discussed, because the grid is only 20 by 20). If they rolled a number that couldn't be represented by a whole-number array, they could then use an irregular shape, or a triangle - anything they could find the area of. It was interesting to see how some students got stumped when they tried to draw an irregular shape to represent a number like 81.
Most students enjoyed this twist (we continued it the next day so they all got to play this version), but a few complained that it made their heads hurt! That's ok...I know they were really thinking and growing mathematically!
The next extension for early finishers (only a few) was to use the icosahedron (20-sided) dice, and have students create area models to cover their grids and find the answer to the multiplication problems. This required a larger grid, so I had them tape 2 pieces of graph paper together and create 20 by 40 grids. Using the icosahedron dice gave a mix of 1-digit by 1-digit, 1 by 2-digit, and 2 by 2-digit problems to model and solve. Most students didn't get very far with this before we ran out of time, but I think this is a great way to them to visualize what multiplying by a two-digit number means. I'd like to revisit this one!
I'm so glad I thought about using those polyhedral dice!
Have you used polyhedral dice in your math classroom? If so, please share how!
Math rules. How often have you found that students are taught "tricks" to remember math rules? How often do they make procedural mistakes even though they've "learned" the rules?
I have taught decimal operations for more than 20 years, and I have seen, time and again, students who know how to add and multiply decimals but then follow the wrong "rule" for the operation they are completing. Line up decimal points when adding or when multiplying? "Jump" the decimal point over when adding and subtracting? Or is that multiplying? They don't remember when to use which method to place the decimal point.
Students' Thinking About Decimal Operations
So, this year, as we approach the decimal unit, I've been feeling like I don't want to talk about the rules for where/how to put the decimal point. I want to focus on logic. Today that feeling was reinforced when I asked my students to solve 35.2 + 7.489 and then explain why their answer made sense. Here are some of the answers and reasons (I didn't teach this yet, but they learned it last year):
"0.11009 makes sense because I tried my best and if I remember correctly, addition problems you don't need to line the decimals together"
"0.7838 makes sense because when I added I knew that it doesn't matter how it's lined up"
"78.42 - I added 9 and 2, then 1, 8 and 5. Next I added 1, 3, and 7. Finally I added 7 and 0 and I put the decimal in the middle."
"7.841 makes sense because with adding you only have to add the decimals on the top. Then you add and finally add the decimal back in."
"426.89 because I put the decimal point four spaces back because there are four numbers behind it"
"79.41 makes sense because you do it just like an addition problem (that's how I remember it anyway)"
"7.841 makes sense because you add like normal and take the decimal from the farthest out and put it with the answer"
A few correct answers, with reasons:
"42.689 - this makes sense to me because this is how I learned it. You do simple addition, but line up the decimal points"
"42.689 makes sense because I used what my fifth grade teacher taught me, line up decimals, add zeros so everything is lined up and then solve."
"42.689 - I don't know how it makes sense, but it's how I learned to do it."
Of the 120 students in my classes, only 8 said the answer made sense because "35 + 7 is 42" or because "I estimated" or "when we're doing addition, we know we end up with a bigger number."
Now, that doesn't mean that they didn't think about those things, but to them answers seemed to "make sense" when they followed the rules - even if the rules are remembered incorrectly; students got right and wrong answers and they all made sense because that's "how they learned it."
So, what is the point of teaching rules? Especially to those students who are a little weaker in math - if they can't remember the right rule, they can't tell if their answer is reasonable! They need to develop their number sense.
In the past, I have asked students to estimate the answer first, so they know if their answer is reasonable, and I have required them write these estimates on their tests. But we've also talked about the rules. I'm thinking that if I take the focus off the rules and put extra focus on the estimating/reasonable answer idea, students will be better able to identify reasonable answers and will feel less dependent on the rules.
Multiplication and Division of Decimals
I know that multiplication and division logic will be more difficult. Problems like 23.5 x 4.428, won't be as bad because there are whole numbers involved. This could be estimated as 25 x 4 = 100. So when placing the decimal point in 104058, students should place it so that the answer is about 100 - not 10, or 1, or 1000.
Multiplying 23.5 and 0.7 may be more confusing, but this will be the time to help them understand why the answer should be smaller than 23.5....but more than half of 23, since 0.7 is more than 0.5.
I think division will be the most challenging, as far as determining reasonable answers, and I need to think about this one a bit more. However, we have already done this activity I found on YouCubed -"Too Big or too Small Maze Board." In attempting to create the largest number possible (using a calculator to compute), many students have already made the discovery that dividing by a number less than one gave them a larger number, while multiplying by a number less than one gave them a smaller number. No rules were taught - they found this "secret" on their own. This will be great to reference and discuss when we begin working on the multiplication and division of decimals.
We'll see how it goes!
To Read Next:
I used this week's problem in class today (6th grade), for early finishers. Because we haven't gotten too "into" a particular topic, I made the problem a mix of operations - mostly division and multiplication, but I saw students using addition as well.
I really enjoy talking with my students about what they are thinking when they try to solve problems, for a few reasons - because 1) they think about problems in a different way than I do; 2) it makes me rethink the wording of the questions I ask (which makes me improve); and 3) I learn that there will be several ideas to share with class.
I noticed a few different things when the students were solving the different parts of this week's problem:
For part A, I multiplied 85 times 3 to get the total number of cookies and then divided by 24 (when I wrote the problem, I wanted the students to have to interpret the quotient, so I approached it with a desire to use division). And most students did the same thing (except for the few that multiplied 24 x 3 - that gave me some good info: -), but one student was just sitting and thinking, so I asked him what he was thinking. He started to say he divided 24 by 3 and then paused - I almost interrupted his thinking to redirect him to my way, but I successfully restrained myself, and asked why. He said he was thinking about how many baggies could be filled with one batch, and since the numbers worked nicely, he could definitely say that one batch would fill 8 baggies. I really liked his thinking process, because it hadn't occurred to me to do it that way. Now, if the numbers hadn't worked out evenly, it might not have been the best approach, but we can expand our class discussion to explore that. After deciding he could fill 8 baggies per batch, he added on sets of 8 until he reached the correct number of batches.
As some students worked on part C (below), I started to think that I should adjust the wording of the problem. When I wrote the problem, I thought it would be clear that the number of cookies for part C was the same as part A, but some students thought of the part C as using 85 baggies of 2 cookies (same number of baggies), instead of using the same number of cookies. As more students worked on it though, other students seemed to understand that the number of cookies should be the same as the original number they were working with, so I haven't changed it yet. If you use the problem, please let me know what you think.
Again, a few students approached this part in a different way than I did - they said that in both cases, the cookies cost 25 cents each. Using this reasoning, some students said the cost was the same, while others did not - again, a great opportunity for discussion, both in small groups and as a whole class.
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Have a great week!