Fractions and Problem Solving

Drew PollyUNC Charlotte

Williamsburg City is dividing up square regions of the city. Each region is 10,000 square yards.

Each region is broken into 3 sections... In the bags! ◦ Region A: 1, 11, 12◦ Region B: 2, 7, 13◦ Region C: 5, 10, 15◦ Region D: 4, 6, 14◦ Region E: 3, 8, 9

How large is each section in terms of square yards and as a fraction of a square region?

Splitting up Williamsburg

Approaches?

Splitting Up Williamsburg

How would your students do?

Ways to differentiate…◦ High-end?◦ Low-end?

Splitting Up Williamsburg

Your grandpa decides to sell his stamp collection. His stamps come in sheets, but they are only partially filled.

What fraction of each sheet is filled? What decimal of each sheet is filled? If each sheet has a value of $100, how

much is each sheet worth?

Selling Stamps

Approaches?

Mathematical Concepts?

Selling Stamps

Let’s look at some students’ work◦ 5th grader…

Selling Stamps

Grid #234/80 is the fraction since there are 80 squares and 34

are shaded34/80 is equal to 0.425 and 42.5%T: “how did you get your answer?”N: “used the calculator and did 34 divided by 80.”T: “anyone do this another way?”Deanna: “multiplying both the numerator and

denominator by 10 means the fraction is equal to 340/800 and then divide both by 8. You get 42.5/100 or 0.425 or 42.5 percent.

What do these strategies show about both students? How have they used the diagram to support their

answer?

Student work- Natalie

Teacher asks: “Is each of the 80 squares going to be more than, less than or equal to 1%?”

Students shout: “all.” Teacher asks for explanation Rashid: “We have 100% to shade across 80 squares, so if each

square gets 1% there is still 20% leftover.” Teacher: “So how much of that 20% does each of the 80 squares

get?” Bonnie: “I think ¼ of a percent since 20 is ¼ of 80.” Teacher: “So how much percent of the whole grid is in one

square?” Bonnie: “1 and ¼ or 1.25% percent.”

What do these students know?How have they connected the diagram to the mathematical

concepts?

Connecting to the decimal grid

100 percent sharing them across 80 squares

It’s “easy” to just make the numerator the decimal even if the numerator isn’t 100

When we teach students to write decimals from decimal grids, a lot of the time we tell them to “just count.”

What is the main dillemma?

Teacher: “So what was Natalie’s answer as a percentage?”

Sam: “42.5 percent.” Teacher: “How much percent is in one square?” Sam: “1.25 percent.” Teacher: “So if 34 boxes are squared what

percent is shaded?” Sam (uses overhead calculator): 34 * 1.25 =

42.5 Sam: “42.5 percent”

Connecting the grid to the answer

How has the teacher “leveraged” student responses during this discussion?

Discuss the mathematical focus in each of the three stages.◦ Was it on computations, representations,

connecting mathematical ideas or a combination?

Examining the Teacher Moves

Draw a 12x8 array on your paper (12 across, 8 tall)

Shade 3/8 of the grid Show it 3 different ways

What is the decimal number and percent for the shaded region?

Shading Grids

Here is Devon’s progression of shading 3/8. What did he do? What concepts does he understand based

on his representations?

Student work- Devon

“I broke the rectangle into four parts. There would be six columns in each half and then three columns in each fourth. “

“Then, I split the fourths into eighths. There are 1 ½ columns in each eighth. Then I shaded in 3 of the 8ths.”

Student work- Devon

“For the percent it’s either 25 or 50 percent. I’m not sure.” T: “Why are you not sure? What do you know about how

you shaded the rectangle?” D: “I broke it into eighths. Fourths are 25% and eighths are

half of that (uses calculator to divide 25 by 2). Each eighth is 12.5% so it would be 12.5+12.5+12.5 which is 37.5 percent.

T: “How would we write that as a decimal?” D: “I know 37.5% is less than one so it’s 0.375.”

Based on Devon’s comments above what mathematics does he know?

How did the teacher moves help Devon connect his strategy to the mathematical concepts?

What would you do next with Devon to make sure he understood the concept?

Student work- Devon

There is a chocolate cake and a vanilla cake in the kitchen. Both are the same size. The chocolate cake has 8 slices and 4 have been eaten. The vanilla cake has 5 slice and 3 have been eaten.

Which cake has more left? Prove your answer in two ways:

◦ A picture◦ Some computational method

Fractions Comparing…

Student work

Comparing Fractions

Draw four squares on your paper that are the same size

Divide two of them into fourths so that each fourth is the same size and the same shape

Divide two squares into fourths so that the fourths are the same size but each square has at least two different-shaped fourths

Where to start

What makes a good fraction task (word problem)?

Characteristics of Tasks

InterMath

Other resources?

Fractions Tasks

Drew Polly Drew.polly@uncc.edu http://elemath.pbworks.com

Questions?