Proportional Thinking Put Right!

Three of the hardest concepts to grasp in mathematics for students of any age are proportional thinking, ratio and rate. However, they are quite simple once we know a little bit more about them.

Small explains proportional thinking or proportional reasoning as the way two amounts relate to each other multiplicatively (pp. 312). That means, the relationship between two amounts, when the value of one amount is always a constant multiple of the corresponding value of the other. It is the main topic with the subgroups ratio and rate.

Ratio describes the relationship between quantities of two values with the same unit. This relationship indicates how many times the first number contains the second. There are two different types of ratios:

            a) Part-to-Part ratios, which compare two parts of something

            b) Part-to-Whole ratios, which compare a part of something to its whole.

Zapanta, D. (2015, September 22). Determining Ratios
[Online Slide]. Retrieved from http://bit.ly/2yLXsOR 

Rate, on the other hand, is a comparison of two values with different units. Check out Math Gains for more information. One of the most useful abilities in life is being able to determine unit rate. Small describes a unit rate as an equivalent rate where the second term is 1 of that unit (pp. 320).

Ms. Smith. Unit Rates R' Us [Online Slide]. 

Retrieved from http://bit.ly/2y3EeAl

Now let's compare. Both ratio and rate are often expressed as fractions, contain two terms, and can be used to create equivalencies, such as equivalent ratios or equivalent rates.

And they both are both constantly used in everyday life. For me, I use ratios when I am baking muffins. I need to know the ratio of eggs to flour, so that when I make more or less than the recipe calls for, the muffins still come out fluffy and delicious. Rates are great when determining how many calories I am eating per serving of pizza, or my hourly wage, or determining my running pace in meters per hour. I am also constantly finding myself comparing products in the grocery store about the better; which item can I buy more of for less cost.

Ratio of an igloo's circumference to its diameter = Eskimo Pi

References:

Small, M. (2017). Making Math Meaningful to Canadian Students K-8, (3rd ed.). Nelson Education.

Integer Intelligence

Koya79. "Thermometer"
[Online Image].
Retrieved from
http://bit.ly/2wwl6eu

          Integers are everywhere in our daily lives. They are present in the temperature, financial statements (hopefully, more positive than negative), in terms of sea level, losing or gaining weight, etc. Clearly, they are important to understand and work with.

            This week's activity presentations shared a lot of great ideas on how to understand integers and make them challenging and fun. One idea that struck me as most interesting was "Integer Battle". One colour of toy soldiers represented the positive integers, and a different colour of toy soldiers represented the negative integers. Using their toy soldiers, students would answer integer addition questions, such as "-3 + +5 = ?" The colour and number of the most soldiers standing would "win", determining the correct answer. In this case, two positive soldiers would be left, so +2 is the answer. I found this game to be very intriguing and I especially like how it gamifies students' learning.

            After the presentations, we learned more in depth about what integers are. Integers are whole numbers that can be positive or negative. The negative integers are the opposite of the positive integers and vice versa, and this opposition explains the zero property all integers have. This means that opposite integers are the same distances away from 0 as the other. For example, (-4) + (+4) = 0. Interestingly enough, 0 is neither positive nor negative; therefore it is not considered an integer. To build a student's understanding of integers, teachers often use number lines, which are very effective manipulatives.

OnlineMathLearning.com (2015). Integer Number Line [Online Image]. Retrieved from http://www.onlinemathlearning.com/integer-number-line.html

            In search of some more ways to engage students in learning about integers, I found a song on YouTube by NumbaLumba. It is a simple video with cute animations, most likely aimed at younger middle school children, but it's a catchy song that will have your students thinking about integers all the time. The song provides some real life situations where students can expect use integers, which I think is extremely important to understanding them. If students cannot see how a certain math topic is applicable to their own lives, they are more likely not going to be eager to learn about it. I promise this song will change that attitude for students.  

If two negatives make a positive, how come two wrongs don't make a right!