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Fractions and Decimals Grades 2-5. MSP Nov. 16 and 18 KCS. Decimal Work. Opening Task. Represent each decimal grid as a: Fraction (in simplest form) Decimal Percent. Approaches- How and why…. Easiest…. Hardest… What did you need to know to be successful?.

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Presentation Transcript
opening task

Opening Task

  • Represent each decimal grid as a:
    • Fraction (in simplest form)
    • Decimal
    • Percent
approaches how and why

Approaches- How and why…

  • Easiest….
  • Hardest…
  • What did you need to know to be successful?
connecting the grid to the answer

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

Examining the Teacher Moves

  • 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?
shading grids

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

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 devon1

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 devon2

“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

great white shark ratio

Great White Shark Ratio

  • A great white shark is 6 meters long (average)
  • A great white shark’s tooth is 8 cm long
  • Find the proportion(ratio) of the great white shark’s body length to tooth length
  • Measure your height
  • Using the great white shark ratio above, find the “expected” length of your tooth
  • Compare you and your partner?
shark

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
equivalent fractions

Equivalent Fractions

  • Find equivalent fractions for
  • 1/2
  • 2/3
  • 3/7
equivalent fractions1

Equivalent Fractions

  • How did you do to find equivalent fractions?
  • How can you prove that they are equivalent?
    • With manipulatives?
    • With an area model (circle or rectangle)
    • With an equation?
    • With words?
comparing fractions

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 fractions1

Comparing Fractions

  • What did you to solve the task?
  • Can you prove your answer with:
    • A circular shaped pizza
    • A rectangular shaped pizza
    • Individual pieces of pizza
    • Cross multiplication (butterfly  )
    • Finding equivalent fractions with common denominators
comparing fractions2

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

Fractions on a Number Line

  • Let’s count around the class by 3/8
  • What number do you think we will land on?
    • Why?
  • As we count, write down your number.
  • Where did we end up?
  • How close was our estimation?
fractions on a number line1

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 line2

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 line3

Fractions on a Number Line

  • In Between
  • What strategies did you use as you played?
fractions on a number line4

Fractions on a Number Line

  • In Between
  • What mathematical concepts are embedded in the task?
fractions of a group

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

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 false1

True or false

  • 2/5 of 50 = 20% of 100
  • Approaches?
true or false2

True or false

  • 2/5 of 50 = 20% of 100
  • Explanations?
comparing fractions of a group tasks

Comparing Fractions of a Group Tasks

  • Look at these two tasks
    • 1) 2/5 of 50 = 20% of 100
    • 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

Proportional Squares

  • If the smallest square is 1 unit by 1 unit, what is the area of each square? The entire unit?
proportions

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

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 5 th grade

Proportional Reasoning in 5th Grade

  • How much oatmeal and flour for 120 cookies?
condominium demographics

Condominium Demographics

  • In a condominium, 9/17 of the people are married and 3/5 of the men are single.
  • How many people are:
    • Married men?
    • Married women?
    • Single men?
    • Single women?
  • Extension: Find more possible answers (there are many)