We practice communicating mathematical explanations through fractions and percents.

In this section, we’re going to practice two key aspects of mathematics:
(a)
Using drawing to reason and justify mathematics
(b)
Communicating mathematical explanations to others.

Fractions

Before reading further, try this exercise. Explain fractions to a young child who has not learned about them in school yet. Think about what you might say. What might you write.

Fractions are writen in the form where and are numbers. For example, let’s consider and let’s say we know this is a fraction of sweet rolls. To interpret this fraction, we need to know what the ‘whole’ is. Are we talking about a fraction of sweet rolls? sweet roll? Let’s say sweet roll is the whole. Now we’ll divide the sweet roll into equal size pieces because the denominator is . Each of those pieces is of the sweet roll. So shading pieces of size gives us of sweet roll.

Explain to a young child how to solve the following problems. These may require some creativity to solve.
  • Camila has invited 9 guests over for breakfast. However, she only made 8 churros. On top of that, only 7 guests like cinnamon sugar on their churro. Draw a picture to illustrate how Camila can ensure each guest gets the same amount of churro and gets their desired cinnamon sugar/none on their churro. Then use the picture to determine what fraction of the churros should be coated with with cinnamon sugar.
  • Chris made sweet potato pies to split between him and friends. But he accidentally used to different sized pie dishes so they’re different sizes! How can Chris split up the pies so each person gets the same amount of pie?

Percentages

Percentages are a special type of ratio that can be expressed as a fraction in the form . For example,

The word ‘percent’ comes from the Latin ‘per centum’ which mean ‘by the one-hundred’. The prefix ‘cent’ is all around us. For example, a ‘century’ is years, ‘centimeter’ is of a meter, a ‘cent’ is of a dollar. In spanish, ‘cien’ is . You can also think of percent as a unit. per or out of every .

Explain to a young child how to solve the following problems.
  • Chandice has started collecting small toy vehicles and has toy vehicles. Of the vehicles, are race cars, are pick-up trucks, are semi-trucks, and are fire trucks. Without using a calculator, what percent of Chandice’s toy vehicles are race cars? Draw a picture that supports your computations.
  • Kajal wants to give a tip for food delivery but needs to enter the tip amount as a dollar amount. She purchased worth of food. Without using a calculator, what is the amount Kajal should give as tip?

Percent Increase and Decrease

In the problems where percent increases or decreases are calculated, we calculate not only the percent change, but also the “actual value” change as well.

It can be helpful to have terminology to describe the answers to these type of problems.

The absolute change for the bus pass was – the actual dollar amount that the price changed. The relative change for the bus pass was 30% decrease – the percent or proportion that the dollar amount was changed.

The annual number of attendees at the pride parade in a town rose by in 2012 and by in 2013.
  • What was the total percent change in attendees over the two years? Let’s first try this problem with a specific number of attendees to start with. In your group, find the change if there were certain number of attendees in 2011 (choose a 3 digit number that does NOT end with a 0). Then try to think about how to do this problem without knowing a specific number of attendees.
  • It might be tempting to say that the change over the two years was a increase. Why might someone think it is a increase (i.e., how do you come up with from the numbers in the problem?)
  • Why is incorrect?