MATH 612 --PP Explorations
Fractions and Proportions: K - 8 Learning and Teacher Practices
(Math task group tag order is D-A-B-C)
PP Math Task D: Special Effects
 The following task should help you think about the connections between percents, fractions
and decimals, as well as how they are all modeled on 10 × 10 percent grids.
 Include all of your work, sketches, answers and written explanations for the math tasks (I)
as well as your answers to the “Question for you” (II) in your write up.
10 × 10 percent grid models (where one 10 × 10 grid is always 100%)
Special Effects Questions (from a sixth grade school text)
I. Answer all of the questions on the page (#1 -6) and illustrate each answer with a 10 × 10
(100%) grids (or tenths grid, if you prefer). Explain!
Question for you (please answer)
II. What connections in these questions are helpful for students? Give at least two examples.
What connections or phrasing in these questions are potentially confusing for students?
Give at least two examples.
MATH 612 --PP Explorations
Fractions and Proportions: K - 8 Learning and Teacher Practices
PP Math Task A: Increasing and Decreasing Percents
 The following task should help you think about the common percent misconceptions
associated with amounts increase and decrease.
 Include all of your work, sketches, answers and written explanations for the math tasks (IIII) as well as your answers to the “Questions for you” (IV-V) in your write up.
10 × 10 percent grid models (where one 10 × 10 grid is always 100%)
Increasing and Decreasing Perc4nts Questions
For each part a & b:
i) Give a specific numerical example (no story, just the numbers) that demonstrates the
situation
ii) Use 10 × 10 (100%) grids to illustrate the solution, explain!
iii) Give a number sentence, such as 4 + 75% of 4 is 4 + .75(4) = 4 + 3 = 7, that checks your
conclusion in part i)
I.
a. If an amount decreases to 1/4 of its original value, by what percent has the amount
decreased?
b. If an amount quadruples in value, by what percent has the amount increased?
c. If the (new) amounts in parts a. and b. increase or decrease back to the original value,
by what percent has the (new) amount increased or decreased? How would you
describe this increase or decrease in words (doubles in value, etc.)?
II.
a. If an amount halves in value, by what percent has the amount decreased?
b. If an amount doubles in value, by what percent has the amount increased?
c. If the (new) amounts in parts a. and b. increase or decrease back to the original value,
by what percent has the (new) amount increased or decreased? How would you
describe this increase or decrease in words (doubles in value, etc.)?
III.
a. If an amount decreases to 1/3 of its value, by what percent has the amount decreased?
b. If an amount triples in value, by what percent has the amount increased?
c. If the (new) amounts in parts a. and b. increase or decrease back to the original value,
by what percent has the (new) amount increased or decreased? How would you
describe this increase or decrease in words (doubles in value, etc.)?
Questions for you (please answer)
IV. If a decrease is “to 1/p of its original value” (p > 0), by what percent has the amount
decreased? Explain and make sure your formula works for your results in I, II and III.
V. If an increase is “to p × the original value” (p > 0), by what percent has the amount
increased? Explain and make sure your formula works for your results in I, II and III.
MATH 612 --PP Explorations
Fractions and Proportions: K - 8 Learning and Teacher Practices
PP Math Task B: Percent and Whole, Whole and Part and Percent and Part
 The following task should help you think about the defining characteristics between the
three main types of percent questions, and how this is reflected in modeling their solutions.
 Include all of your work, sketches, answers and written explanations for the math tasks (IIII) as well as your answers to the “Question for you” (IV) in your write up.
10 × 10 percent grid models (where one 10 × 10 grid is always 100%)
Percent and Whole, Whole and Part and Percent and Part Questions
For each of the following percent questions, do all of the following:
a. Write a simple Story Problem suitable for an elementary school student that results in
the given percent question. Your story problems should make real-world sense.
b. Illustrate the solution to the problem using 10 × 10 (100%) grids, explain!
c. Determine the solution to the problem using a proportion, explain!
d. Describe the type of question (Percent and Whole (1), Whole and Part (2) or Percent
and Part (3) by identifying the components.
I. If 3% of an amount has a value of $45, what is the value of 100% of the amount?
II. If an amount has a value of $45, what is the value of 150% of the amount?
III. If an amount has a value of $45, what percent of the amount has a value of $25?
Question for you (please answer)
IV. What key phrases or ideas would help an elementary school student identify which type of
question is being asked? Percent and Whole (1), Whole and Part (2) or Percent and Part (3)?
Give a variety of examples.
MATH 612 --PP Explorations
Fractions and Proportions: K - 8 Learning and Teacher Practices
PP Math Task C: Tips and Newspapers
 The following task should help you think about
 Include all of your work, sketches, answers and written explanations for the math tasks (I-II)
as well as your answers to the “Question for you” (III) in your write up.
Proportions and ratio tables
a c2

b
10
1 5 10 15 20
3 6 9 12 15
Tips and Newspapers Questions
I. Suppose you have gone out for dinner with four of your friends. The bill for dinner is $36
which includes tax of $2.42. If you will give a tip of 15% on your part of the bill (divided
evenly) before tax, how much is the tip? Explain your solution.
II. Todd sells 3 newspapers every 10 minutes at his newsstand, and Shauna sells 4 newspapers
every 12 minutes at her newsstand. If they work at these rates and combine their sales,
how many newspapers will they sell in 30 minutes? You may use 10 × 10 (100%) grids,
proportional reasoning, equivalent ratio tables, or whatever calculation method makes the
most sense to you. Illustrate and explain your solution.
Question for you (please answer)
III. What key phrases and ideas in these questions would help an elementary or middle school
student make sense of, and solve the problem correctly. Which methods of solution would
make the most sense to such children?