The following tasks are designed to build on students' prior knowledge about percentages, which they are more familiar with than fractions or decimals.
Beakers have the same widths, but different heights. Clearly show the given percentages on the beakers provided.
1. What would 50 percent look like on both beakers?
2. What would 25 percent look like on both beakers?
3. What would 75 percent look like on both beakers?
4. How much is missing from each beaker above? 50%, 75%, 25%
5. What would 125 percent look like on both beakers?
6. Back to question 1, does one of the beakers have more liquid than the other? Explain.
Yes, the left beaker will have more in these situations because the left beaker is bigger. This question shows that the "referent", or more commonly the "whole" in elementary textbooks, may be different for percentages.
Think like a teacher: What is a pedagogical reason for question #4?
This question shows that 1 whole is the same as 100%.
This is a sidewalk. It is 10 sections in length.
This is Perdy, 
, a PERcent budDY. For each question, Perdy starts at the left edge of the sidewalk then walks to the right.
1. Perdy walks 20 percent of a sidewalk. Where is Perdy now?
Right edge of 2nd section.
2. Perdy walks 40 percent of a sidewalk. Then Perdy walks 50 percent of a sidewalk. Where is Perdy now?
Right edge of 9th section.
3. Perdy walks 40 percent of a sidewalk. Then Perdy walks 50 percent of the amount already walked. Where is Perdy now?
Right edge of 6th section.
4. Be careful with wording. Which of the following are the same (if any)?
Think like a teacher: What are advantages to using "sidewalks" versus beakers?
The horizontal orientation make transfer to number lines easier for students. For teachers, "sidewalks" are called "strip diagrams" and are also useful for teaching algebra.
Try to answer the problems in as many different ways as possible, e.g., visually, percent tables, decimals, equivalent fractions, etc.
1. What is 75 percent of 60 cm?
percent tables: 100% → 60; 50% → 30; 25% → 15; 75% → 45
answer is 45
equivalent fractions: \( \frac{75}{100} = \frac{7.5}{10} = \frac{15}{20} = \frac{45}{60} \)
answer is 45
decimals: \( 0.75 \cdot 60 = 45 \)
2. What is 12 and a half percent of 60 cm?
percent tables: 100% → 60; 50% → 30; 25% → 15; 12.5% → 7.5
answer is 7.5
equivalent fractions: \( \frac{12.5}{100} = \frac{25}{200} = \frac{75}{600} = \frac{7.5}{60} \)
answer is 7.5
decimals: \( 0.125 \cdot 60 = 7.5 \)
3. What is 125 percent of 40 cm?
percent tables: 100% → 40; 50% → 20; 25% → 10; 125% → 50
answer is 50
equivalent fractions: \( \frac{125}{100} = \frac{12.5}{10} = \frac{25}{20} = \frac{50}{40} \)
answer is 50
decimals: \( 1.25 \cdot 40 = 50 \)
4. What is 125 percent more than 40 cm?
percent tables: 100% → 40; 125% → 50; 225% → 90
answer is 90
equivalent fractions: \( \frac{225}{100} = \frac{900}{400} = \frac{90}{40} \)
answer is 90, note: 225 is from 100% + 125%
decimals: \( 40 + 1.25 \cdot 40 = 40 + 50 = 90 \)
Think like a teacher: Why is it important to solve mathematics problems using different methods or representations?
Each method helps to better understand the mathematical concept. The more methods (or representations) we have, the better we understand the concept.
As a teacher, I also do math problems multiple ways to make sure my answers are consistent.