1.5 Algebraic generalizations

• A

  Pick a number:_____

• B

  Add three:_____

• C

  Subtract three from your answer in part (b):_____

• D

  What happened?_________________________________

• E

  Write an algebraic generalization to represent this rule._____

• F

  Is there any number for which this rule will not work?_____

• A

  Pick a number:_____

• B

  Subtract five:_____

• C

  Double your answer in part (b):_____

• D

  Add ten to your answer in part (c):_____

• E

  Divide your answer in part (d) by your original number (a):_____

• F

  Now, repeat that process for three different numbers. Record the number you started with (a) and the number you ended up with (e).

• Started with:_____

• Ended with:_____

• Started with:_____

• Ended with:_____

• Started with:_____

• Ended with:_____

• G

  What happened?

• H

  Write an algebraic generalization to represent this rule.

• I

  Is there any number for which this rule will not work?

Here are the first six powers of two.

• $2^1=2$
• $$ $2^2=4$
• $2^3=8$
• $2^4=\text{16}$
• $$ $2^5=\text{32}$
• $2^6=\text{64}$

• A

  If I asked you for $2^7$ (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term?________________

• B

  Write an algebraic generalization to represent this rule.________________

Look at the following pairs of statements.

• $8\times 8=64$
• $7\times 9=63$

• $5\times 5=25$
• $4\times 6=24$

• $10\times 10=100$
• $9\times 11=99$

• $3\times 3=9$
• $2\times 4=8$

• A

  Based on these pairs, if I told you that $30\times 30=900$ , could you tell me (immediately, without a calculator) what $29\times 31$ is?________________

• B

  Express this rule—the pattern in these numbers—in words.

• C

  Whew! That was ugly, wasn’t it? Good thing we have math. Write the algebraic generalization for this rule.________________

• D

  Try out this generalization with negative numbers , with zero , and with fractions . (Show your work below, trying all three of these cases separately.) Does it always work, or are there cases where it doesn’t?