Transcript Key strategies for interventions: Fractions

KEY STRATEGIES FOR
INTERVENTIONS:
FRACTIONS
TODAY’S AGENDA
Concepts and procedures
Assessments
Games, activities and simulation
GENERAL
STRATEGIES
1. Use visual representations to explain concepts.
2. Provide tasks that engage students’ thinking.
3. Teach procedures explicitly, verbalizing your thinking.
4. Require substantial practice where students are expected
to explain their thinking. Provide corrective feedback.
5. Connect types of fraction problems to types of whole
number problems (putting together, taking from, repeated
addition, measurement division, etc.)
6. Provide guided practice with corrective feedback and
frequent cumulative review.
TOO MANY
PROCEDURES
𝟒 𝟏
× = ___
𝟓 𝟐
Do I need common denominators?
Do I cross multiply?
A cake recipe makes 36 cupcakes. We only need 24. The
original recipe requires 3½ cups of flour. How many cups of
flour do we need if we want to reduce the recipe to make only
24 cupcakes?
WHAT IS A FRACTION?
Several definitions –
1) parts of a whole
2) parts of a set
3) a number on the number line
4) a way to write division
Introduce fraction circles and fraction bars, then relationship
rods and pattern blocks. Build on 1st/2nd grade cutting and
coloring circles, squares, etc.
Illustrate each one with a drawing and symbols.
Fluency involves identifying the fractional part of a whole (or
part of a set), and drawing an illustration of a fraction as part
of a whole or part of a set.
COMPARING
FRACTIONS
1) same denominator
2) same numerator
3) relative to benchmark fractions (close to 0, close to 1,
greater or less than ½)
This is a conceptual task, not one that calls for procedural
fluency.
SAME DENOMINATOR,
SAME NUMERATOR
Which is larger? Explain your reasoning. Use manipulatives
or drawings.
𝟑 𝟓
𝟖 𝟖
𝟑
𝟒
𝟑
𝟓
BENCHMARK
FRACTIONS
1
Using fraction pieces, find a fraction that is close to but
2
smaller.
1
2
Find a fraction that is close to but larger.
1
2
1
2
Find a fraction that is closer to 0 than .
Find a fraction that is closer to 1 than .
Find a fraction that is close to 1 but larger.
ORDERING
FRACTIONS
Order this set of fractions from smallest to largest:
3 1 5 1 3 14
, , , , ,
4 10 12 4 5 15
Explain how you figured this out.
Test your understanding:
http://phet.colorado.edu/en/simulation/fraction-matcher
MORE REASONING
WITH FRACTIONS
Compare
3
4
and
2
3
. Which is closer to 1? Why?
USING REASON TO
THINK ABOUT ADDITION
Is this reasonable? Use manipulatives to check or show.
NUMBER LINES – A
USEFUL TOOL
What’s hard about this for many students?
ONE APPROACH
1. Draw a line across a page. Using a fraction bar piece for
1
, measure lengths of the bar starting from the left end
4
1 2 3 4
of the line. Mark the end of each length as , , , etc.
4 4 4 4
(Where should you put zero?)
2. Locate and label
5 6 7
, ,
4 4 4
on the number line.
15
3. Locate and label on the number line. How much is this
4
as a mixed number?
This is the introduction to mixed numbers.
NUMBER LINE
ACTIVITIES
Almost! I almost got it exactly. I’m going to turn it over and
try again to see if I can get the paperclip to land right on the
½ mark.
¼? I just moved my clip what I thought was half-way down
the line and then cut that in half. I got pretty close.
From Bridges, 3rd grade CCSS supplement
EQUIVALENT
FRACTIONS
Use manipulatives
to find multiple
4. Equivalent fractions are the same
(the same total amount of the
equivalentsize
fractions
whole) but they are divided into
different numbers of pieces. For
Develop a procedure (scaling up
both the total number of parts and
the number we have, developing
proportional reasoning).
Fluency involves using this
procedure to find equivalent
fractions.
1 1×2 2
=
=
4 4×2 8
COMPARING
FRACTIONS (PT. 2)
Which is larger, which is smaller?
3
4
5
8
3 6
4 10
First practice drawing the two
fractions to see which is
larger/smaller. Then find the
equivalent fractions with common
numerators or denominators.
4.NF.2 Compare two
fractions with different
numerators and different
denominators, e.g., by
creating common
denominators or numerators,
or by comparing to a
benchmark fraction such as
1/2.
COMPARING
FRACTIONS (PT. 2)
Which is larger, which is smaller?
2
6
5
8
Compare each fraction to
See the problem set in the
handouts.
1
.
2
IS IT REASONABLE?
from Operations with Fractions & Decimals Packet
FRACTION ADDITION –
COMMON DENOMINATORS
3 1 1 1 2 1
= + + = +
4 4 4 4 4 4
1
2 7 5 12
2 +1 = + =
=4
3
3 3 3
3
Adding thirds or fifths or fourths is like adding apples or
oranges or pears. 2 thirds plus 2 thirds is like 2 pears plus 2
pears (= 4 pears, or 4 thirds). Model this with fraction
manipulatives.
This is a joining (putting together or adding to) problem. In this
case the “pears” are 1/3-size pieces. This is the KEY learning.
FRACTION ADDITION –
COMMON DENOMINATORS
3
8
After a class party, of one pan of brownies is left over and
2
8
of another pan of brownies is left over. How much is left over
altogether? (Use fractions pieces or drawings, then explain
the procedure.)
3
Britney walks of a mile to school each day. How far does
4
Britney walk to school in one week (5 days)? Put a dot on the
appropriate place on this number line to show your answer.
Be as accurate as you can be.
FRACTION ADDITION –
“FRIENDLY” DENOMINATORS
The procedure of finding a common denominator has to
arise out of extensive work with fraction manipulatives,
asking: How can we add two different things?
Use fraction pieces to find this sum:
𝟑 𝟏
+
𝟖 𝟒
What procedure arises from work with the manipulatives?
C-R-A
Try this addition on a number line:
𝟑 𝟏
+
𝟖 𝟒
1. Concrete (Objects)
2. Representational (Picture)
3. Abstract (Symbols)
𝟑 𝟐
+
𝟖 𝟖
FRACTION ADDITION – NOT
“FRIENDLY” DENOMINATORS
The extension to any denominator is straightforward: Scale
them both up until we find a common denominator.
𝟐 𝟏
+
𝟑 𝟒
Try it.
𝟐 𝟒 𝟖
𝟑 𝟔 𝟏𝟐
𝟏 𝟐 𝟑
𝟒 𝟖 𝟏𝟐
𝟖
𝟑
𝟏𝟏
+
=
𝟏𝟐 𝟏𝟐 𝟏𝟐
A great deal of work needs to be done
1. to connect these kinds of fraction problems to joining and
separating problems with whole numbers
2. to develop the sense of proportionality in scaling up the
fractions, and
3. in eventually moving away from using manipulatives to
using the multiplication procedure.
Fluency comes from practice, where thinking about the
trades gets replaced with the scaling up procedure.
RECOMMENDATIONS FOR
FRACTION INSTRUCTION
• Base early understanding on fair shares
• Use representations of different kinds
• Develop estimation skills for comparing fractions by basing
comparisons on benchmark fractions such as 1/2.
• Develop the meaning of a fraction as a number (a place on
the number line); connect a point on the number line to a
fraction of a whole through the meaning of denominator
and numerator
• Use real-world examples including measurement
• Develop the reasons behind procedures and expect
students to explain their thinking
PRACTICE VS. DRILL
Practice is used to establish a procedure.
Drill is used to get fast with it.
What procedures have we looked at for adding fractions?
Same denominator
One denominator a multiple
Family or no: common denominator
PRACTICE GAMES
Fraction Tracks, also called Fraction Game
What do
students
learn playing
this?
PRACTICE PROBLEMS
1/10 of the M&M’s in a bag are red and 1/5 are blue. What fraction of all
the M&M’s are red and blue? What fraction of the M&M’s are NOT red
or blue?
You give 1/3 of a pan of brownies to Susan and 1/6 of the pan of
brownies to Patrick. How much of the pan of brownies did you give
away? How much do you have left?
You go out for a long walk. You walk 3/4 mile and then sit down to take
a rest. Then you walk 3/8 of a mile. How far did you walk altogether?
Pam walks 7/8 of a mile to school. Paul walks 1/2 of a mile to school.
How much farther does Pam walk than Paul?
A school wants to make a new playground by cleaning up an
abandoned lot that is shaped like a rectangle. They give the job of
planning the playground to a group of students. The students decide
to use 1/4 of the playground for a basketball court and 3/8 of the
playground for a soccer field. How much is left for the swings and play
equipment? Draw a picture to show this.
DRILL PROBLEMS
Once students understand the type of problem and the
procedure they’ll use, then they can do drill problems that
are often just “naked number” problems.
DIAGNOSTIC
ASSESSMENTS
IISD Mathematics wiki
inghamisd.org – WikiSpaces – Mathematics
FRACTION
RESOURCES
IISD Elementary Math Resources wiki, 4th-5th Grade
inghamisd.org – WikiSpaces – Elementary Math Resources
FRACTION
MULTIPLICATION
The learning progression outlined in the CCSS needs
to be followed systematically and explicitly.
1. A whole number times a fraction
(generalizing multiplication as repeated addition)
2. A fraction of a whole number (generalizing
work with geometric figures in 1st and 2nd grades)
3. A fraction times a fraction (generalizing the
area model)
FRACTION
MULTIPLICATION
1. A whole number times a fraction (generalizing
multiplication as repeated addition)
2
5×
3
2
5 hops of 3
The key to fraction multiplication is knowing
how to estimate the size of the answer.
WHOLE NUMBER
TIMES A FRACTION
A game idea: Roll two dice, one with 1-6, one with fractions.
3
Write the multiplication shown by the dice, e.g. 3 × , say the
product out loud,
line.
9
,
4
9
4
4
then write at the correct spot on a number
It becomes obvious after playing the game
awhile that a simple procedure is to multiply
the whole number times the numerator.
Dominoes can be used instead of fraction dice.
Choose which fraction it can represent.
FRACTION OF A
WHOLE NUMBER
1. A whole number times a fraction (generalizing
addition of fractions)
2. A fraction of a whole number (generalizing
work with geometric figures in 1st and 2nd grades)
2
𝑜𝑓 6
3
You have 6 donuts. You want to give 1/3 of them to your
friend Suzi, 1/3 of them to your friend Sam, and keep 1/3 of
them for yourself.
2
𝑜𝑓 6
3
Why is this multiplication? How is it related to division?
1
3
2
3
3
3
C-R-A
R: a visual representations - find 1 third of 6, then take 2 of
that amount:
A: an abstract method – divide the whole number by the
denominator (6 ÷ 3 = 2) then multiply by the numerator (2 x 2
2
3
= 4). So × 6 = 4.
PRACTICE
Create six practice problems, three contextual and three noncontextual, to practice this procedure.
The procedure is: Divide the whole number by the
denominator (to find one group), then multiply by the
numerator (to find the total).
Does it also work to multiply by the numerator first, then
2
divide by the denominator? 3 × 6 = 4 (yes, but no meaning)
What if the denominator is not a factor of the whole number?
3
× 10
4
FRACTION
MULTIPLICATION
1. A whole number times a fraction (generalizing
addition of fractions)
2. A fraction of a whole number (generalizing work
with geometric figures in 1st and 2nd grades)
3. A fraction times a fraction (generalizing the area
model)
TYPE 1: SOLVING
VISUALLY
3
2
A track is of a mile in length. If you run of the track, how
4
3
much of a mile have you run?
https://www.teachingchannel.org/videos/multiplying-fractions-lesson
SIMPLE VISUAL
PROBLEMS
3/4 of a pan of brownies was sitting on the counter. You
decided to eat 1/3 of the brownies in the pan. How much of
the whole pan of brownies did you eat?
1
3
𝑜𝑓
3
4
2
3
3
4
1
3
, which is represented as ×
× =?
3
4
For these problems, we don’t have a procedure other
than using a drawing.
Will this visual approach work with
1
3
𝑜𝑓
2
4
or
3
8
×
4
5
ANOTHER VISUAL
REPRESENTATION
What if the denominator is not a factor of the
whole number? Start with a simple problem
of a fraction times a whole number.
3
× 10
10
4
40 fourths
3
4
× 10 is 30 fourths,
30
4
Let’s generalize the area model from the picture of
1
2
to a picture for ×
3
4
3
4
× 10
C
R
A
SIMPLIFYING
FRACTIONS
You have 2/3 of a pumpkin pie left over from Thanksgiving. You
want to give 1/2 of it to your sister. How much of the whole
pumpkin pie will this be?
2
1
The algorithm results in but the drawing method results in .
6
3
Are they equivalent? Does the form matter?
FRACTION DIVISION
Learning progression:
1. Division of a fraction by a whole number (use
manipulatives or drawings)
2. Division of a whole number by a fraction
(generalize the “measurement division” or
repeated subtraction process using drawings)
3. Division of a fraction by a fraction (continue
using drawings to support conceptual
development; provide alternative algorithms)
GENERALIZING FROM
WHOLE NUMBERS
Partitive (fair shares)
We want to share 12 cookies equally among 4 kids. How
many cookies does each kid get?
How would you solve this with objects?
The number of groups is known; the number in each group is
unknown.
Measurement division (repeated subtraction)
For our bake sale, we have 24 cookies and want to put 4
cookies in each bag. How many bags can we make?
How would you solve this with a picture?
The number in each group is known; the number of groups
is unknown.
How many bags of 4 are in 24?
WHY IS THIS
IMPORTANT?
1
3
A box of Cheerios contains 12 cups. Each serving is cups.
2
4
How many servings are in a box of Cheerios? How much is
left over?
Is this Partitive or Measurement division?
Write 3 additional problems like these. (We’ll solve this later.)
DIVIDING A FRACTION
BY A WHOLE NUMBER
We have ½ of a pizza and want to share it equally among 4
people. How much pizza does each person get?
𝟏
÷𝟒
𝟐
Try this with fraction manipulatives.
Doing these kinds of problems is essential to build number
sense about the size of the expected answer.
Try this:
𝟖
÷𝟒
𝟑
While the approach is different, it’s still “fair shares” or
partitive division.
DIVIDING A WHOLE
NUMBER BY A FRACTION
We have a dozen large cookies and want to give ½ cookie to
each child. How many children can we serve?
1
12 ÷
2
(How many times does ½ go into 12?)
Solve this by making a drawing.
The procedure: Each cookie is divided in half, making two
pieces. There are 12 cookies, so 12 x 2 pieces.
DIVIDING A FRACTION
BY A FRACTION
A serving size is ¼ cup. How many servings are in 5/4 cup?
Solve it. Write it as an equation.
5
4
÷
1
4
How many ¼’s are in 5/4?
This is the same as asking “How many 1’s are in 5?”
Procedure: Get common denominators.
DIVIDING A FRACTION
BY A FRACTION
1
3
A box of Cheerios contains 12 cups. Each serving is
2
4
cups. How many servings are in a box of Cheerios? How
much is left over?
1
2
Translated to symbols 12 ÷
3
4
Procedure? Find common denominators...
50
4
÷
3
4
then ask “how many 3’s in 50?”
Write two more similar problems to solve with common
denominators.
Why does “invert and multiply” work?
𝟕 𝟏
÷
𝟖 𝟖
𝟕 𝟖
𝟏 𝟖
( × )÷( × )
𝟖 𝟏
𝟖 𝟏
See pp. 29-34 in Operations with Fractions for an
instructional approach.
DECIMALS
The development of decimal fractions has several steps:
Place tenths on the number line up through 3 or 4. (Equate
5/10 to 1/2 and 10/10 to 1).
Introduce the new notation for tenths: 1/10 = 0.1, 2/10 = 0.2, 1
4/10 = 1.4, 2 7/10 = 2.7. Be clear that this is just a new name
for tenths, a new way of writing tenths, called decimal
numbers. The “decimal point” separates the ones from the
tenths.
Introduce hundredths through the money system: $4.55 is
read 4 dollars and 55 cents, where each cent is 1/100 of one
dollar (one hundred are needed to make one dollar).
Use the hundreds block as a visual representation.
Expand the place value system to show:
hundreds tens ones . tenths hundredths
WHY DO WE “COUNT
DECIMAL PLACES?”
2.3
Yellow 2 x 1
x 1.8
Orange 0.3 x 1
Blue 2 x 0.8
.2 4
Green 0.3 x 0.8
1.6
1.3
2.3
2
5.1 4
Decimals Forever!
1.8