# Fractions_ Decimals_ Percents

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```					Fractions, Decimals, Percents
Big Ideas, Concepts,
Relationships, Operations
With a nod to Small (and VdW)
What is a Fraction (anyway)?
• We can represent them in many ways:
Symbolically, Pictorially, Concretely
• They are about Part and Whole relationships
(the foundation of fractions, decimals, and
percents)
• How about a ―Big Idea…‖
Big Ideas…
• Chapter 9 (Small): (Fractions)
• Small: Fractions can represent parts of wholes,
parts of sets, parts of measures, division, or
ratios. Fractions are not meaningful without
knowing what the whole is.
• [VdW]: Fractional parts are equal shares or
equal-sized portions of a whole or unit. A unit,
which is counted as 1, can be an object or a
collection of things. On a number line, the
unit is the distance from 0 to 1.
Whole: Object or Collection of Things?
• They LOOK a lot different, but they‘re both written as
one number over the other (symbolically you can‘t tell
the difference!)
• ―Collections‖ can be especially challenging figuring out
‗part‘ and ‗whole‘ (recall the ‗other‘ fraction PPT)
• Object or Collection…… quick…
• A pizza
• A box of Smarties (different colours)
• A classroom of children
• A chocolate bar
• A carton of eggs
A Bit of Terminology
• What do we mean by ‗denominator‘?
▫ And where is it?
• What do we mean by ‗numerator‘?
▫ And where do we find it?
13
• What if the fraction was    18
▫ What does it mean?
That Grade 6, 7, or 8 Scenario…
• Carl-Donna- ―third‖ of a pizza problem…
• A way to get a sense of students‘ basic
understanding of ―fraction‖…

• Do PLENTY of those kinds of things-hooks,
short queries, etc…. ―authentic‖ situations…
Let’s Stop Here for a Few Minutes…
• And imagine ourselves in a couple of classroom
scenarios.
we consider the concept of ―fraction.‖ We looked at-
▫   Fraction Fourths
▫   Fraction Development
▫   ( These were PowerPoint files from Resources)
▫   DO LOTS!!
• And then in Grade 6, where we look at the number
line and ordering fractions (some not so easy…)
▫ Let‘s do this now…(time permitting)
• Okay, Let‘s Continue Working with Fractions
• Showing Equivalence…
• Using concrete materials, visual images
• But First--
• What do we mean by “Equivalent
Fractions”? Try to explain in your own words.
• Let‘s see what Small says…
Another Fraction Concept Big Idea
(Chapter 9…)
• Equivalent Fractions…
▫ Fractions have more than one name…for example,
1/2 is another name for 2/4
• Time to introduce some manipulatives--
▫ Let‘s pause here for a few minutes… (screen)
▫ Geoboards, pattern blocks, Cuisenaire rods… in
Equivalent Fractions
•   Show that the following are true:
•   First—Use manipulatives—and then explain
•   Then—Create your own algorithm for one
•   And defend it---
•   i) 1/2 = 3/6            ii) 4/6 = 8/12
iii) 1 ¾ = 7/4

• Complete the following:
• i) 3/4 = □ /8          ii) 3/6 = 2/□
Patterns, patterns, patterns….
• Concrete and visual models are marvelous, but
how can we…
• Help students move from the concrete to seeing
that, with equivalent fractions, we can get from
1/2 to 3/6 by multiplying both numerator and
denominator by the same number (3), or from
2/8 to 1/4 by dividing both N & D by the same
number (2)
• Discuss in your groups. Then briefly share with us.
Another Big Idea…
• For addition and subtraction, it is critical to understand
that the numerator tells the number of parts and the
denominator the type of part.

• So, suppose a grade 7 student wrote:
½ + ¼ = 2/6
• And said, happily, ―You just add the numerators
• How do you help him?
• Some additions with manipulatives, then explain… (Just
do some from each).
• Some require more than one set of fraction circles…
• i) ¼ + ¼                ii) 2/3 + 2/3 =
iii) 1/6 + 3/6 =

• iv) ¼ + ½ =              v) ¾ + ⅜ =
vi) 1 ½ + ⅝

• vii) 1/3 + ¼             viii) 4/6 + ¾ =
• And some subtractions (manipulatives) (again,
just do some from each)…

• ix) 2/3 – 1/3            x) 3/6 – 1/6
xi) ¾ - ½

• xii) ¾ - 5/8 =           xiii) 2/3 – ¼
xiv) 1 ½ - 3/8 =
Multiplying Fractions… Pictorially
―What is 1/3 of 1/4 of a whole?‖
1/3 × 1/4 ----- Explain the diagrams below:

Therefore 1/3 x 1/4 = 1/12
Dividing Fractions… Pictorially
• Try ½ ÷ ¼ = ? ―How many ____________
are there in _____________ of a whole?‖
• Explain the diagram:

Therefore 1/2 ÷ 1/4 = 2
Patterns, Patterns, Patterns…
• As for addition and subtraction…
• The same for multiplication and division…
• The keys are (1) do lots; (2) have the students
articulate what they see (patterns, patterns,
patterns) in their own words; (3) draw out
connections through the concrete and visual to
the symbolic.
• (Think about the last division…can you see how
by doing enough problems ―invert and multiply‖
just might have some meaning?)
Contextual Problems…
• Can help give meaning to fraction operations
(algorithms)
• Can provide opportunities for you to ask:
▫ ―Does this answer make sense?‖
• Can provide opportunities for students to ask
themselves:
▫ ―Does this answer seem reasonable?‖
• Chapters 10 & 11 (Decimal and Percents)
▫ Addition and subtraction with decimals are based on
the fundamental concept of adding and subtracting the
numbers in like positional values—a simple extension
from whole numbers.
▫ Multiplication and division of two numbers will
produce the same digits, regardless of the position of
the decimal point. Thus…it is not necessary to develop
new rules for decimal multiplication and division…the
computations can be performed as whole numbers
with the decimal placed afterward by way of
estimation.
Another Big Idea
• Decimal numbers and Percents are simply other
ways of writing Fractions.

F

P          D
Fractions, Decimals, and Percents
• Percents: Simply 100th fractions
• 10 × 10 grids (recall the Fraction Development
PowerPoint) (decimals and percents)
• Text author Small has plenty of great ideas for
developing fractions, decimals, percents 9and
ratios and rates)
• And here‘s more ways to think of percents and
fractions… from the Resources page
▫ Let‘s stop and check them out…

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