© TUNING UP FRACTIONS LINDA WEST [email protected] SMARTTRAINING, LLC.
-
Upload
lesley-jones -
Category
Documents
-
view
215 -
download
0
Transcript of © TUNING UP FRACTIONS LINDA WEST [email protected] SMARTTRAINING, LLC.
![Page 2: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/2.jpg)
©
BIG IDEAS IN MATHEMATICS - COMPETENCIES
– Visualization– Making Connections (looking for patterns
in order to generalize)– Communication
Problem Solving Number Sense
• Based on the Theories of:– Jerome Bruner– Zoltan Dienes– Richard Skemp
![Page 3: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/3.jpg)
©
NUMERATOR
•An adjective•The counting number
•Tell how many•The “numberator”
![Page 4: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/4.jpg)
©
DENOMINATOR•A noun•A label•Tells what kind of unit
•The “deNAMEnator”•An Ordinal number
![Page 5: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/5.jpg)
©
What makes these types of numbers so difficult for
students?
![Page 6: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/6.jpg)
©
• Notation confusing• “bigger” numbers indicate “smaller amount”• Same number indicates different amount• Multiplication sometimes yields smaller
result while division sometimes yields a larger one
• The “whole” must always be held in the mind • Fractions, decimals and percents
traditionally taught as disparate topics
![Page 7: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/7.jpg)
©
FRACTIONS
FractionsQuantityProportion
Percentage is exclusively used for ProportionDecimal is exclusively used for Quantity
![Page 8: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/8.jpg)
©
Whole vs. Fractional Number Sense
Whole Numbers Fractional Numbers
Counting units Relationships to
other numbers• More/Less• 5 & 10 benchmarks• Part/Whole relationships
Counting units Relationships to
other numbers• More /Less• 0, ½, 1 Benchmarks• Part /Whole Relationships• Equivalency
![Page 9: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/9.jpg)
©
Whole Numbers Fractional Numbers• Equal units based on
10’s with relationships to each other
• Connect constructs with abstract notation and operations
• Whole universe of “units” that are less than 1, but are related to each other!
• Connect constructs with abstract notation and operations
![Page 10: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/10.jpg)
©
Keys to Understanding
• Concepts and Problem Solving before Rules and Drill
• Connections before Calculations
![Page 11: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/11.jpg)
©
The Dangerous Rush to Rules• None of the rules help students think about
the meaning• Rules give students no means of assessing
whether an answer is reasonable• Surface mastery of rules is quickly lost• Algorithm rules do not immediately apply to
every situation• Incredibly defeating for students
![Page 12: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/12.jpg)
©
2 ÷ = ?Please:
1. Solve.
2. Draw a picture.
3. Write a word problem.
![Page 13: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/13.jpg)
©
12 ÷ 4 = ____
![Page 14: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/14.jpg)
©
Our real question is:
How many halves are contained in 2 ¾?
![Page 15: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/15.jpg)
©
EXAMPLE:A recipe calls for 2cups of flour.
The only size measuring cup available is a cup measure. How many measuring cups are required to complete the recipe?
WOULD ANYONE LIKE TO SHARE THEIR WORD PROBLEM?
Jacqui ran 2 ¾ miles. This was ½ the distance that she runs each day. What is the total distance that Jacqui runs each day?
![Page 16: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/16.jpg)
©
NOW TRY THIS ONE2 ½ ÷ ⅓
• Use some of the pattern blocks on your table to solve this problem.• Verify with algorithm.• Write a word problem that can be solved using this equation.
![Page 17: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/17.jpg)
©
2 ½ ÷ 1/3
2 ½ ÷ 2/3
![Page 18: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/18.jpg)
©
FUN WITH FRACTIONS
+ =
Using digit tiles 1 – 9, once only in each equation, how many equations can you create?
![Page 19: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/19.jpg)
©
HOW DO WE DEVELOP VISUALIZATION SKILLS WITH FRACTIONS?
• Take out the yellow hexagon. Cover it with as many different pattern blocks as you can until the entire hexagon is filled.
• How can we express what we have done concretely in a number sentence?
![Page 20: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/20.jpg)
©
HOW DO WE USE
LogicTo explain
The fraction division algorithm?
![Page 21: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/21.jpg)
©
6 ÷ = ____WHAT DOES THAT MEAN?
Let’s go to an easier problem:6 ÷ 3 = ____ What does that mean?
• How can I read that differently?
• How many threes are in six?
• 6 is 3 of what number?
![Page 22: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/22.jpg)
©
6 ÷ = ____WHAT DOES THAT MEAN?
• Who can read the equation differently?
• How many ½’s are in 6?
• 6 is ½ of what number?
![Page 23: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/23.jpg)
©
DEVELOPING THE CONCEPT
• 6 ÷ 6 = ____
• 6 ÷ 3 = ____
• 6 ÷ 2 = ____ Does anyone see a pattern?
• 6 ÷ 1 = ____
• 6 ÷ = ____
![Page 24: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/24.jpg)
©
DO YOU SEE A PATTERN?
• How many halves are in 1? (2)• How many halves are in 2? (4)• How many halves are in 3? (6)• Let’s focus on thirds:
– How many thirds are in 1? (3)– How many thirds are in 2? (6)– How many thirds are in 3? (9)– How many thirds are in 4? (12)
![Page 25: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/25.jpg)
©
DO YOU SEE A PATTERN?
• LET’S FOCUS ON FOURTHS:– How many fourths are in 1? (4)– How many fourths are in 2? (8)– How many fourths are in 3? (12)– How many fourths are in 4? (16)
![Page 26: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/26.jpg)
©
DO YOU SEE A PATTERN?
• LET’S SWITCH IT UP A LITTLE BIT:– How many halves are in 1?– How many thirds are in 1?– How many fourths are in 1?– How many fifths are in 1?– How many halves are in 2?– How many thirds are in 2? etc., etc.
![Page 27: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/27.jpg)
©
AREA MODEL TO REAL LIFE
We need multiple embodiments. (Zhang 2012)
![Page 28: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/28.jpg)
©
DIVISION BY A FRACTION
• 1 =
0 1
![Page 29: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/29.jpg)
©
USING BAR MODELS WITH FRACTIONSDivide 3 by 2/3
2/3
There are four 2/3’s. and another half of a 2/3 in 3. So there are four-and-a-half 2/3’s in 3.
3 ÷ 2/3 = 3 x 3/2 = ?
![Page 30: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/30.jpg)
©
3 DIVIDED BY
• What is your word problem?
![Page 31: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/31.jpg)
©
A man travelled from Town X to Town Y in three days. He covered of the journey of the first day and of the remainder of the journey on the second day. What fraction of the total journey did he cover on the last day?
4
He travelled of the total journey on the last day.
![Page 32: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/32.jpg)
©
You painted of a room. of the area painted was green. What fraction of the room was green?
![Page 33: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/33.jpg)
©
A PICTURE IS WORTH …
• Find ¼ of 48.
• Find ¾ of 48.
• 1/3 of a number
is 48. Find the number.
• 4/5 of a number
is 48. Find the number.
![Page 34: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/34.jpg)
©
• 0 1 2 3
Send your text message to this Phone Number: 37607
Strongly Disagree
Strongly Agree
Disagree Agree
Speaker was well-prepared and
knowledgeable (0-3)
Speaker was engaging and an effective presenter (0-3)
Session matched title and description in
program book (0-3)
Other comments, suggestions, or
feedback (words)
___ ___ ___ ___________ _______
Example: 38102 323 Inspiring, good content
poll code for this session
(no spaces)
Non-Example: 38102 3 2 3 Inspiring, good content
(1 space) (1 space)
Non-Example: 38102 3-2-3Inspiring, good content
![Page 35: © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC.](https://reader036.fdocuments.us/reader036/viewer/2022062423/5697bfc71a28abf838ca7b88/html5/thumbnails/35.jpg)
©
Teaching Tips, Product Discounts, Workshop Updates & More
SMARTTraining: Singapore Math Training
MathSingapore