Tuesday, October 13, 2015

Distributive Law Worksheet

The Distributive Law

Multiplication distributes over addition: a(b +c) = ab + ac. We sometimes talk about factoring out a number using the distributive property ie. 81 + 45 = 9 * (9+5)


Compute $51 \cdot 9 + 51 \cdot 31$.

What is the value of $17 \cdot 13 + 13 \cdot 51 + 32 \cdot 13$?

What is $a(b + c + d +e)$?

what is -(6 + 8)?

What is $-1 \cdot (5 - 9)$ also written $-(5 - 9)$?

What is $-(x + 1)$?

What is $−1 \cdot(a - b + c - d)$?

Find numbers a, b, and c such that $a + (b \cdot c)$  is not equal $(a +b)\cdot(a+c)$ In other words show we can NOT go the other way and distribute addition over multiplication.

Take a look at the first perfect squares from $1^2$ to $10^2$. Do you see a pattern to what number is in the units place? 

Can you use the use distributive law to show why $8^2$ and $2^2$ for example end in the same digit. Hint: 8 = 10 - 2.  Try the same technique with another pair of numbers.

We can also use the distributive law to show why a negative times a negative is a positive. Pick any two negative numbers x and  y then try using the distributive law to find the sum $x \cdot y + x \cdot -y$.

Factor out x from $6x + 9x$.

Factor out x from $6x + 9x^2$

What is $x\cdot(9 +x)$?

What is $(x+1)\cdot(x + 1)$? (Hint use the distributive property multiple times)

What is $(x - 1)(x - 1)$?

*What is $(x -1)(x + 1)$?

The above answer lets you do a neat trick. Can you multiple $19 \cdot 21$ in your head?  Using the above  formula we can think of it instead as $(20 - 1)\cdot(20 + 1)$. What does that equal?

Try doing some other examples with a partner without writing anything down.

What is $(a + b)(c +d)$?


We say a number x is a multiple of another one n if we can find another number such that
$x = y \cdot n$. Example: 20 is a multiple of 5 since $20 = 5 \cdot 4$. Can you use the distributive law to prove that if you add two multiples of the same number n that the result is also a multiple of n.
Example: 20 is a multiple of 5, 35 is a multiple of 5.  20 + 35 = 55 is a multiple of 5. How about if a number is not a multiple? For example 11 is not a multiple of 5. Is there a pattern for what numbers you can add to 11 to get a multiple of 5?


  1. Note: you can select just the inner frame and print it without all the superfluous text.

  2. Another idea to explore adding on area modelling in the early part of the sequence.