Lesson: Math Magic
By Michael Naylor, Teaching PreK-8
Here are some surprising tricks to build number sense and algebraic reasoning
Mathematics truly is magical, especially for students with strong number sense and algebra skills. Here’s a variety of mathematical surprises that will capture your students’ interest and motivate exploration of mathematical ideas. While the tricks themselves are fascinating, push your students to think about the reasons why these stunning feats work – each one is built on a mathematical idea even more amazing than the trick itself!
Math dice (Grades K-2)
On a standard six-sided die, opposite numbers always add to seven. This forms the basis for several tricks and can help your students learn the fact family for 7. dice and numbers
On a standard six-sided die, opposite numbers always add to seven.
Ask your students to each roll a six-sided die and write down the number they’ve rolled so they’ll remember it. Then ask them to flip their dice over so the number that was on the bottom is now on top. They should now add the new number to their first number. After a dramatic pause, “guess” that the total they now have is 7. You’ll generate a list of the additive fact family for 7: 1+6=7, 2+5=7, etc.
Now ask them to roll their die and just by looking at the top number, they can predict which number is on the bottom.
“Magical” calculations (Grades 3-5)
These mental math tricks are so slick, they seem like magic.
Multiplying by 5: Divide the number by 2 and multiply by 10. For example, to multiply 18 × 5, divide 18 by 2 to get 9, then multiply 9 by 10 to get 90. Give your students a series of numbers to multiply by 5 – most will be able to do this quickly in their heads.
What about odd numbers that don’t divide evenly by 2? Just think in terms of decimals. For example, 41 × 5. First take half of 41, which is 20.5, then multiply by 10 to get 205. It works every time.
Multiplying by 11: To multiply a two-digit number by 11, write the two digits with a space between them and, in that space, write the sum of the two digits. For example, to multiply 52 × 11, write 5__2. What goes in the blank? 5 +2, or 7, so the answer is 572.
Give a few more examples, and have your students check with a calculator or pencil and paper to verify their results. Make sure that the two digits when added together are 9 or less, because it’s a little more complicated when they sum to 10 or greater.
Can your students explain why this works? Having them calculate with pencil and paper can help explain. Be sure to emphasize the place value concepts during your discussion.
Mystery numbers (Grades 4-8)
Read the following instructions to your students:
1. Choose a number from 1 to 20
2. Double it
3. Add six
4. Divide by two
5. Subtract the number you originally started with
Pause for dramatic effect and then announce: “The number you now have is 3!” Allow your students to try the trick again with different starting numbers.
To explain this to your students, draw a box on the board to represent the starting number. We are then asked to double that number, so then write “x 2” after the box. Now we add 6, so write a big “6” on the board also.
Now we divide everything by 2. Cross out the “6” and write “3” and ask what happens to the “__ x 2.” Since that part stands for twice the number, when you divide it by two, you “undo” the doubling and end up with just the starting number. Cross out the “x 2” part.
At this point, we now have just the original number plus 3. What happens when we subtract the original number? Only the 3 is remaining – and it doesn’t matter what the original was!
Ask your students to try to create their own instructions to predict a mystery number. Remind them that at some point in the calculations they need to subtract the original number.
This problem will build toward some rather sophisticated algebraic reasoning for your students.
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