Fractions and DecimalsGrades 2-5

MSP Nov. 16 and 18KCS

Decimal Work

 

Opening Task• Represent each decimal grid as a:

o Fraction (in simplest form)o Decimalo Percent

Approaches- How and why…• Easiest….

• Hardest…

• What did you need to know to be successful?

• 100 percent sharing them across 80 squares

Connecting the grid to the answer• 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”

Examining the Teacher Moves• How has the teacher “leveraged” student

responses during this discussion?

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

connecting mathematical ideas or a combination?

Shading Grids• Shade 3/8 of a 12x8 grid (12 across, 8

tall)• Show it 3 different ways• What is the decimal number, fraction and

percent for the shaded region?

Student work- Devon• 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?

Great White Shark Ratio• A great white shark is 6 meters long (average)• A great white shark’s tooth is 8 cm long1.Find the proportion(ratio) of the great

white shark’s body length to tooth length2.Measure your height3.Using the great white shark ratio above,

find the “expected” length of your tooth4.Compare you and your partner?

Shark• 8 cm/ 600 cm = 1 cm / 75 cm• 1 cm of tooth for every 75 cm of body• 129 cm human body• 129-75 = 54, 1 r 54…. 1.4• 129/75

Share your inner “shark”

How can we organize our data?

Graphing our data

Equivalent Fractions• Find equivalent fractions for • 1/2

• 2/3

• 3/7

Equivalent Fractions• How did you do to find equivalent

fractions?

• How can you prove that they are equivalent? o With manipulatives? o With an area model (circle or rectangle)o With an equation?o With words?

Comparing Fractions• Two pizzas are on the counter. The

pepperoni pizza is cut into 5 slices. The cheese pizza is cut into 8 slices.

• If you take two pieces from the pepperoni pizza and four pieces from the cheese pizza, which pizza has more left?

• How do you know?

Comparing Fractions• What did you to solve the task?• Can you prove your answer with:

o A circular shaped pizzao A rectangular shaped pizzao Individual pieces of pizzao Cross multiplication (butterfly )o Finding equivalent fractions with common

denominators

Comparing Fractions• Let’s look at student work

• What strategy does the student use?• What do they know?• What misconceptions do they have?• Ideas for interventions?

Fractions on a Number Line• Let’s count around the class by 3/8• What number do you think we will land

on?o Why?

• As we count, write down your number.

• Where did we end up?• How close was our estimation?

Fractions on a Number Line• Stand if your fraction can be simplified to a

whole number

• Stand if your fraction can be simplified to ½

• Stand if your fraction can be simplified to ¼ or ¾

• Stand if your fraction cannot be simplified at all

Fractions on a Number Line• Mark the following fractions

• 1/12, 13/12, 9/8, 15/8, 7/4, 9/6, 7/13

Fractions on a Number Line• In Between

• What strategies did you use as you played?

Fractions on a Number Line• In Between

• What mathematical concepts are embedded in the task?

Fractions of a group• In a basket there are 24 apples. 1/3 of

them are red. ½ of them green. The rest are yellow. How many of each color are there?

• Prove your answer using both pictures and an equation

True or false • 2/5 of 50 = 20% of 100

• Prove this using a picture and an equation.

• Explain your answer in words.

True or false • 2/5 of 50 = 20% of 100

• Approaches?

True or false • 2/5 of 50 = 20% of 100

• Explanations?

Comparing Fractions of a Group Tasks

• Look at these two taskso 1) 2/5 of 50 = 20% of 100 o 2) In a basket there are 24 apples. 1/3 of them

are red. ½ of them green. The rest are yellow. How many of each color are there?

• How are they similar?• How are they different? • What are the different issues and struggles

students have?

Proportional Squares• If the smallest square is 1 unit by 1 unit,

what is the area of each square? The entire unit?

Proportions• Make a pattern block picture that includes

at least 10 shapes and is ½ blue.• How can you prove that it is ½ blue?

• Make a pattern block picture that is ¾ blue. How can you prove that it is ¾ blue?

What does this look like in Grades 3-5?

• It takes you 45 minutes to drive 30 miles. How long will it take to travel 120 miles?

• How would we expect students to solve this task?

• Where would they struggle?

Proportional Reasoning in 5th Grade• How much oatmeal and flour for 120

cookies?

Condominium Demographics• In a condominium, 9/17 of the people are

married and 3/5 of the men are single.• How many people are:

o Married men?o Married women?o Single men?o Single women?

• Extension: Find more possible answers (there are many)

Approaches

Solutions