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Description
This page is more of a description of a “continuous effort” than a specific activity. When I started the school year, I set a goal of expecting my students in grades 7-12 to use number sense when performing calculations of all kinds. I knew this would take time from the “normal” lesson, but since I have always expected it (and have been frustrated when I didn’t see evidence of it), I resolved to make it worth the time. Knowing what to expect (estimation) is a major tenet of number sense. It became my responsibility to prompt the students, “What do you expect?”, over and over, day after day. I also made a conscious effort to “think out loud” and model estimation each time I did a problem at the board. Part of the “out loud” thought process included a decision of whether to do the calculation mentally, by hand, or with a calculator.
Grade 7 received the brunt of my campaign, but by making it a habit in that class, it became second nature in my other classes as well. For most addition and subtraction problems (unless it was adding a large list of numbers), I modeled the adding by parts and “make 10” strategies that I discussed in No Need to Borrow. For long division I modeled the repeated subtraction algorithm, but at this level the calculator is appropriate for most long division (however, they should be able to estimate an answer).
Multiplication became an often-discussed topic. Squares and square roots, area and volume, percent and consumer problems—all examples that require multiplication savvy. A common question was “What do you know the answer is between?” If the problem was 27 x 63, I encouraged the students to think “20 x 60 = 1200 and 30 x 70 = 2100, so the answer is between 1200 and 2100.” Granted, you can narrow it down much further than that, but that served as a starting point that let me assess their multiplying by tens skills. Later we emphasized benchmark values, so 27 x 63 could be estimated by thinking ¼ of 64 x 100 = 1600 (since 27 is close to 25 and 25 is .25 x 100).
We talked about looking at the ones place when answering multiple-choice questions. 27 x 63 must end in a 1 since 7x3 ends in a 1. We looked at the patterns in the squares table through 9: only 12 and 92 end in 1; only 22 and 82 end in 4; etc. Sometimes I did long multiplication using the left-to-right rather than right-to-left method. The looks on the students’ faces were priceless when the light bulb came on about what I was doing and how it related to the traditional algorithm for multiplication. The distributive property is invaluable for mental math. I believe students who learn mental math will have less trouble with the distributive (and other) properties when they study formal algebra.
Since this was not a one-time
activity, I did not feel it was appropriate to use the rubric to evaluate
it. Sometimes the students were
engaged, sometimes not. However, I did
begin to see students using some of the methods independently. Several of the students showed a marked decrease
in calculator use.
The task looks longer, but the addition is easy and there is no carrying when multiplying. It also demonstrates place value. There’s no mystery about moving your numbers to the left one place each time you go to the next multiplier.
27 27
x 63 x
63
1200 60 x 20 1200 Or keep a running total
420 60 x
7 1620 as you multiply.
60 3 x
20 1680
__21 3 x 7 1701
1701
This problem could also be solved using the make “10” method:
27 x 63…think 27 x 60 = 1200 + 420, so 27 x 60 = 1620. We need 3 more 27s. Three more 20s is 60, which makes 1680. Three more 7s is 21, which makes 1701. So, 27 x 63 = 1701.
Similarly, you could think 30 x 63 = 1800 + 90, so 30 x 63 = 1890. By multiplying by 30, we included 3 too many 63s, so we need to take them away. Three 60s is 180, which leaves 1710. Three 3s is 9, which leaves 1701. So, 27 x 63 = 1701.