Number Talks

As inspiration strikes, Mrs. Olson jots a quick post discussing a mathematical question or challenge. When inspiration strikes YOU, please email punnymathteacher@gmail.com to further comment or suggest mathematical questions YOU have!

Posts: (ctrl + F to jump to the post)

• What does division really mean?

• How can we use number lines in our mind?

• Challenge: Four 4's

• Challenge: Create Your Own Brain Teaser

• Why do we use the terms "sum", "difference, "product", and "quotient" for answers to addition, subtraction, multiplication, and division questions?

What does division of fractions really mean?

We use division to, of course, DIVIDE amounts of things. Common examples are dividing pizza or candy bars to share with friends. Let's try some problems and see how it works.

Basics: If I have 1 super large Hershey's chocolate bar, and my mom is making me share with my brother, I will break the bar in HALF. 1 divided by 2. I am making it into TWO pieces. I started with ONE piece of chocolate and end up with TWO pieces that are each 1/2 of the original size.

Extension: If I have 2 large pepperoni pizzas and 10 friends each eating one slice, what fraction of the pizza will each person eat? Remember in the end, I am determining not how MANY pieces I will need; instead, I am finding how large EACH PIECE will be. So, I have two whole pizzas, and 11 people need to eat. Remember that I am hungry, too.  I will take 2 wholes and divide it into 11 pieces. So, each person will be served 2/11 of pizza. Note, that 10/11 will feed 5 friends while I, as the host, will eat 1/11 from EACH pizza to make up my 2/11 portion. Draw a picture if you do not believe me.

Advanced: We all know the instruction to do (2/3) divided by (4/5) --> "multiply by the reciprocal". WHY does that work? Why is (2/3)(5/4) the same as (2/3) divided by (4/5)??? Let's break this down to understand. "Dividing (2/3) by (4/5)" means I am splitting (2/3) into (4/5) pieces. Rephrasing this means I am filling in the blank as follows: 2/3 is 4/5 of _____. I can write this as an equation with "?" filling in the blank: (2/3) = (4/5)*?. To solve for ?, we will UNDO the multiplication of (4/5) and ? by multiplying by the reciprocal on both sides. Thus, (2/3)(5/4) = ?. To calculate ?, (2/3)(5/4) could be stated as (2/3) OF (5/4). To find TWO-thirds, what is ONE-third of five fourths? I will start off by dividing the amount of candy into twelfths. Then, five fourths is equivalent to 15/12. One-third is 5/12. Two-thirds is 10/12 which simplifies to 5/6. 

What details did I miss? How would you explain division?

How can we use number lines in our mind?

Horizontal: When we all learn numbers lines in elementary school, we see the horizontal left-to-right poster stapled above the whiteboard in our classrooms. They begin at 0 and typically end around 100. They are helpful when learning about values of numbers. 23 is greater than 21 since 23 is farther right from 0 than 21. 78 is less than 99 since 78 is farther left from 0 than 21. Horizontal number lines also make sense for addition. 21 + 78 = 99, and I can show my work by starting at the first number 21 and moving 78 places to the right from 0, to add MORE. Subtraction is the INVERSE of addition and means we move left. With positive numbers, this means we move towards 0. For example, 78 - 21 means 78 - 20 = 58 and 58 - 1 = 57. What if we include numbers LESS than 0? NEGATIVES! How do we find 21 - 78? We move left, of course! Yet, "left" means we also go left PAST 0. 21 - 21 = 0 and since 78 - 21 = 57, we now move 57 LEFT BEYOND 0 to land at -57.

Vertical: Instead of moving left and right, what about an ever "moving" number line that moves from up to down? Now, adding 21 and 78 means you move the number line up like watching the red line move up in a thermometer. We land at 99 degrees outside with this heat wave of 78 degrees. For subtraction, we move from 78 down to -57 degrees outside (okay, not so realistic). I find that moving a number line up and down is more convenient for my mind to process addition and subtraction. I end up using my hands to "mark" locations in the air. To do 37 + 56, I mark 37 with my hand, move it up 50 to 87, and then move it up 6 more (knowing that 7 + 6 = 13 = 10 + 3) to land at 93. Subtraction is the opposite. I mark in the air where I am imagining the number line to start at before moving my hand down the line to take away. Thus, 75 - 54 becomes 75 - 50 = 25. And, 25 - 4 = 21 as my final answer.

Which do you prefer?

Challenge: The Four 4's

Source: https://www.youcubed.org/tasks/the-four-4s/ 

Go grab some paper and a pencil! Yes, a working eraser is recommended, too. Using FOUR 4's, how many numbers can you make?

For example, 4 + 4 + 4 + 4 = 16, 4 - 4 + 4 - 4 = 0, and by adding four square roots of 4, we can make 8. Can you find ALL the numbers from 0 to 20 using ONLY four 4's?

Listen to my podcast to hear my thought process as I try to get through as many as I can in only 5 minutes. How many can you find BEYOND 20?

Challenge: Create Your Own Brain Teaser

Similar to the previous challenge and the trend of "Which one does not belong" (see here for more https://wodb.ca), think of four numbers. The more random they seem, the better! Now, there are two challenges you can do:

1) Find all numbers from 1-20 using ANY operation or exponent or root, etc while still using all four numbers.

OR

2) Find a reason that EACH number COULD be the one that does not belong.

Listen to the podcast to hear my attempt at challenge #2 with 45, 67, 93, and -2! 

Why do we use the terms "sum", "difference, "product", and "quotient" for answers to addition, subtraction, multiplication, and division questions?

This questions struck me my sophomore (or second) year of college. On my way to a math class, I saw a poster in the hall that declared "Make a difference". The purpose and context was to motivate students to do more STEM activities and increase that academic field. As a punny math person, I took it as - oh! Subrtraction! Make a DIFFERENCE. Ha! This makes total sense, 4 and 17 are different numbers. How different? They are difference by 13, or, their difference is 13.

How can you explain the other three terms we usee for answer to the basic operations?