Whether you work with Common Core Standards, or your own state standards, students are being required to communicate mathematical ideas using appropriate language to justify and defend mathematical solutions. Some of the "tips and tricks" we have taught in elementary mathematics are now coming back to bite us because the "tips and tricks" are not rooted in sound mathematical practices or concepts. We must be so very careful teaching absolute truths in elementary math. If we teach an absolute truth in 2nd grade (the larger number always goes on top, you can't subtract a larger number from a smaller number, etc.), that absolute truth should still hold true in 6th, 7th, or 12th grade.
In the August, 2014 edition of Teaching Children Mathematics, NCTM published an article entitled 13 Rules That Expire. We thought we would highlight these rules and their implications so that we can get to work with math truths that don't change as students move through their math careers. In this post we will talk about three of the rules that expire and then continue with the remaining rules that apply to elementary math in upcoming blog posts.
Rule #13: When multiplying by 10 (or by 100 or 1,000), just add a zero (or two zeros or three zeros) to the end of the number. So why doesn't this rule hold true anymore? 5th graders are now multiplying decimals. Consider this problem, 2.14 x 10. If we follow the old trick of just adding a zero to the end of 2.14 our answer would be 2.140? No, 2.14 x 10 = 21.4. You will create a better understanding of our base 10 system if you help students understand that when you multiply a number by 10, the product results in an answer that is 10 times more than the factor and our decimal point moves to the right.
We want students to be efficient in their thinking and ability with multiplying by 10, 100, 1000, etc. so be specific in your instruction/investigation. Beginning in 3rd grade, students should be able to see the pattern that occurs every time they multiply a whole number by 10. Have students multiply several numbers by 10 and let them discover what occurs each time and then write a rule. A rule they can count on to work now and when they are 90 years old. The rule I help students write goes something like this... "Every time a whole number is multiplied by 10, the product (or the result) is the whole number with a zero on the end."
Rule #12: Key words indicate which operation to use. Most states have purposefully removed "key words" from assessments. We don't want students mindlessly looking for key terms to determine which mathematical operation to use. Let's look at an example. It has often been taught that each indicates multiplication or division, but consider this problem:
Mary's mother gave each of her three children some money. Sammy earned $5 for separating the recyclables and $2 for taking out the trash. Ronan walked and fed the dog every day of the week and received $3 for each day. Mary washed each of the two family cars and earned $10 for both. How much money did each child earn?
In the problem above, each was used four times, but only once does it indicate multiplication. Instead of using "key words" have students look at the action that is happening in the problem. Students could use strip diagrams to illustrate this problem to further help clarify their thinking.
Rule #11: The longer the number, the larger the number. This rule doesn't need much explanation. Think about decimal numbers, the length of the number has nothing to do with the value. Consider the following comparison:
53.1 ______ 53.069
Students will want to say that 53.1 is less than 53.069 because it has fewer digits. You can help students dispel this myth by reinforcing place value and comparing the number place by place. It often helps students to create numbers with equal number of decimal places. We can add zeros to the hundredths and thousandths place (because it doesn't change the value of the number) and then students usually see the difference in values right away.
I will end this post now and look forward to going through the next 10 rules in the coming days. Remember to teach students with understanding, not "tips and tricks". "Tips and tricks" often lead to misconceptions that become hard to undo as they move through more complex mathematical concepts.