Transcript Fractions and Decimals Workshop

Rational Numbers and
Representations
Fractions and Decimals Grades 3 -5 Workshop
Longwood University
Dr. Virginia Lewis
Cathlene Hincker
Miriah Eisenman
Before we get started
› Instructor Introductions
First…Pre-Workshop Content Assessment
› Remember, you are not be graded during this workshop!
› Please answer the questions to the best of your ability.
› At the end of our third day together, you will take a postworkshop assessment to see how this workshop has
impacted your knowledge of Grades 3-5 fractions,
decimals and representations.
3
Community of Learners
› Complete 3 X 5 note card:
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–
–
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Name
Email
Where do you teach?
Number of years teaching & grade levels
Favorite mathematics topic
Why are you here?
What weaknesses/concerns do you have about your own
understanding of fractions and decimals?
› Introductions – Introduce another person in our class to
everyone!
4
What do students need to know about
fractions and decimals?
› What is the essential knowledge?
Goals of the Workshop
1. To know more about rational numbers than you expect your
students to know and learn.
2. An awareness of different models and representations to
enhance thinking about rational numbers.
3. To become familiar with the connections between fractions,
decimals, and place value.
Goals of the Workshop
4. To know what mathematics to emphasize and why in planning
& implementing lessons.
5. To anticipate, recognize, and dispel students’ misconceptions
about fractions and decimals.
6. Build on prior grades’ fraction ideas and know later-grade
connections  vertical alignment.
The Standards
National Council of Teachers of Mathematics Standards
and the Virginia Standards of Learning
What do NCTM’s Principles and Standards for School
Mathematics say?
Understand numbers, ways of representing numbers, relationships
among numbers, and number systems
› Grades 3 -5
– develop understanding of fractions as parts of
unit wholes, as parts of a collection, as locations
on number lines, and as divisions of whole
numbers;
– use models, benchmarks, and equivalent forms to
judge the size of fractions;
– recognize and generate equivalent forms of
commonly used fractions, decimals, and
percents;
National Council of Teachers of Mathematics. 2000.
Mathematics. Reston, VA: NCTM.
Principles and Standards for School
What do NCTM’s Principles and Standards for School
Mathematics say?
Compute fluently and make reasonable estimates
› Grades 3 -5
–develop and use strategies to estimate
computations involving fractions and
decimals in situations relevant to students'
experience;
–use visual models, benchmarks, and
equivalent forms to add and subtract
commonly used fractions and decimals;
National Council of Teachers of Mathematics. 2000.
Mathematics. Reston, VA: NCTM.
Principles and Standards for School
Fractions/Decimals in Virginia SOLs K-2
Kindergarten
› K.5 Identify parts of a set and/or region that represent fractions
for halves and fourths
1st
› 1.3 Identify parts of a set and/or region that represent fractions
for halves, thirds, and fourths and write the fractions
2nd
› 2.3 Identify parts of a set and/or region that represent fractions
for halves, thirds, fourths, sixths, eighths, and tenths (connects to
decimals later)
Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public
Schools. Richmond, VA: Commonwealth of Virginia Board of Education.
What do students currently learn about
fractions/decimals in your grade level?
› Organize into grade level groups
› Without looking at the SOLs brainstorm and record what
you teach in your grade level about fractions and
decimals.
› Look over the SOLs and adjust anything that needs
adjusting on your chart paper.
› Record the SOL number next to each of your big ideas.
Identify on the grade level charts concepts
that build on knowledge from previous
grades.
Building on grades 3-5
In grade 6
– Describe and compare data, using ratios using the appropriate
notations a/b, a to b and a:b.
– Investigate and describe fractions, decimals, and percents as ratios
– Identify a fraction, decimal, or percent from a representation
– Demonstrate equivalent relationships among fractions, decimals,
and percents
– Compare and order fractions, decimals, and percents
Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public
Schools. Richmond, VA: Commonwealth of Virginia Board of Education.
Building on grades 3-5
More grade 6
– Demonstrate multiple representations of multiplication and division
of fractions
– Multiply and divide fractions and mixed numbers
– Estimate solutions and then solve single-step and multistep
practical problems involving addition, subtraction, multiplication and
division of fractions
– Solve single-step and multistep practical problems involving
addition, subtraction, multiplication, and division of decimals
Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia
Public Schools. Richmond, VA: Commonwealth of Virginia Board of Education.
Building on grades 3-5
In grade 7
– Compare and order fractions, decimals, percents, and numbers
written in scientific notation;
– Identify and describe absolute value for rational numbers
– Solve single-step and multistep practical problems, using
proportional reasoning
Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public
Schools. Richmond, VA: Commonwealth of Virginia Board of Education.
Building on grades 3-5
In grade 8
– Simplify numerical expressions involving positive exponents, using
rational numbers, order of operations, and properties of operations
with real numbers; and
– Compare and order decimals, fractions, percents, and numbers
written in scientific notation
– Describe orally and in writing the relationships between the subsets
of the real number system
– Solve practical problems involving rational numbers, percents,
ratios, and proportions
Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public
Schools. Richmond, VA: Commonwealth of Virginia Board of Education.
The Common Core Standards
› You will find a copy of the grades 3-5 standards for
fraction instruction in your packet
› While Virginia is not currently participating in the
Common Core it is interesting to see these standards
too.
› http://www.corestandards.org/math
But there’s more to the standards
› In your group
– Read the Introduction to the Standards of
Learning
– Highlight anything of interest you would like to
discuss
– Read your assigned Process Standard. Be
prepared to summarize for the class what this
standard encompasses.
What representations do you currently use
during fraction and decimal instruction?
Multiple Representations
› How many different ways can you represent ¾?
Fraction Models
› Area Models for ¾.
Fraction Models
› Set Models for ¾.
Fraction Models
› Measurement Models for ¾.
Problems and Models
› In Mrs. Park’s class there are 24 students.
the students play soccer.
One third of
› My dad made me a pan of brownies for my birthday.
ate 5/8 of my pan.
I
› I tried to run from my school to my favorite ice cream
shop. I ran 9/10 of the way before I stopped because I
was tired.
Adapted from page 98 of
McNamara, J. & Shaughnessy, M. 2010.
Beyond Pizzas and Pies. Sausalito, CA: Math Solutions.
Using representations to solve problems
› It takes 18 minutes for John to walk home from school.
He has walked 2/3 of the way at a constant speed. How
many minutes has he been walking?
› There are 18 marbles in Suzanne’s marble collection.
2/3 of the marbles are green. How many green marbles
does Suzanne have?
› Joanna filled 18 bowls with 2/3 cup of flour in each.
How much flour did Joanna use?
Adapted from page 130 of
Van de Walle, J.A., Bay-Williams, J.M., Lovin, L. H., & Karp, K.S. 2014. Teaching Student-Centered
Mathematics Developmentally Appropriate Instruction for Grades 6-8 (2nd ed). Boston, MA:
Pearson Education, Inc.
What are the meanings of fractions?
› Fractions as part of a whole or part of a set
› Fractions as quotients- the result of division
› Fractions as ratios or rates – comparing quantities with like
(ratio) or unlike (rate) units
› Fractions as operators – Stretches or shrinks the magnitude
of another number
› Fractions as measures – Rational number thought of as a
unit fraction to be repeated
Lamon, Susan J. 2007. Rational Numbers and Proportional Reasoning Toward a Theoretical
Framework for Research. In Frank K. Lester, Jr (Ed.), Second handbook of research on
mathematics teaching and learning (pp. 629 – 667). Charlotte, NC: Information Age Publishing,
Fractions as part of a set
Fractions in our Class
› I need 10 volunteers to come to the front of the room.
– What fraction are girls?
– What fraction are wearing jeans?
– What other fraction questions can we ask and answer?
– Why is the denominator for every fraction 10? What does the
denominator represent?
– Why is the numerator different? What does the numerator
represent?
Use Student Data to Write Fractions
› Which of the following statements is true for you?
– My last name is longer than my first name.
– My last name and first name are the same length.
– My last name is shorter than my last name.
First and Last Names
My last name is
longer
My names are the
same
My last name is
shorter
Tally
› What fraction of the class has last names that are longer?
› What fraction of the class has last names that are shorter?
› What fraction of the class has names that are the same
length?
Use Student Data to Write Fractions
› How many syllables are in your first name?
Number of Letters in Our First Names
1
2
3
4
5
Tally
› What fraction of the class has one syllable in their first
name?
› What fraction of the class has fewer than three syllables
in their first names?
› What fraction of the class has more than three syllables
in their first names?
Fractions as a part of a whole
› Use color tiles to build a rectangle that is ½ red, ¼ yellow,
and 1/8 green and 1/8 blue. Make a drawing of your
rectangle.
› Can you find another rectangle that also satisfies the
requirements? Make a drawing of your rectangle
› Use color tiles to build a rectangle that is 1/6 red, 1/2
green, 1/3 blue. Make a drawing of your rectangle.
› Can you find another rectangle that satisfies these
requirements? Make a drawing of your rectangle.
› Why do you think we used the ½, ¼, and 1/8 fractions in
one rectangle and the ½, 1/3, and 1/6 in another
rectangle?
Adapted from pg 280 Burns, M. 2007. About Teaching Mathematics a K-8 Resource (3rd ed).
Sausalito, CA: Math Solutions
Why is the whole important?
› If the whole is a group of 12 color tiles, what is ½?
› If the whole is a group of 10 color tiles, what is ½?
Blocking Out Fractions: An AIMS activity
› This activity can be purchased in PDF form from Aims
Education Foundation at the following web address.
› http://www.aimsedu.org/item/DA6539/Blocking-OutFractions/1.html
Grade 3 Sol Practice Item Made Available by the Virginia
Department of Education (VDOE)
How does
instruction
which
focuses on
the role of
the whole
help
students
interpret this
question?
Unitizing
Is the ability to think of a quantity in different sized chunks
Fold an isosceles triangle in half three different times.
unfold the triangle…
Then
Can you see eighths?
Can you see fourths?
Can you see halves?
Students need to be able to think flexibly about fractions!
What is Partitioning?
› Partitioning is the act of breaking the whole into parts
› Students need experiences where partitioning in different
ways results in equivalent amounts
› For example the following partitions have different
representations but all are equivalent to 1 whole.
Partitioning the Whole
› How can you divide this square into halves? Is
there more than one way to divide your square?
› Write a number sentence to represent how the
halves combine to make a whole.
› How can you divide this square into fourths? Is
there more than one way to divide your square?
› Write a number sentence to represent how the
fourths combine to make a whole.
Adapted from pg 52 of
Schuster, L., and Anderson, N.C. 2005. Good Questions for Math Teaching . Sausalito, CA: Math
Solutions.
Partitioning the Whole
› How can you divide this square into eighths?
› Write a number sentence to represent how the
eighths combine to make a whole.
› How can you divide this square into halves,
fourths, and eighths?
– Label and color-code each fractional part.
– Write a number sentence to represent how the
parts combine to make the whole.
Adapted from pg 52 of
Schuster, L., and Anderson, N.C. 2005. Good Questions for Math Teaching . Sausalito, CA: Math
Solutions.
Egg Carton Fractions
› Can you make halves? Thirds? Fourths? Sixths? Twelfths?
› Can you make 2/3? How many twelfths are the same as
2/3? How many sixths are the same as 2/3?
› Can you make ¾? How many twelfths in ¾?
› Put two cartons together and make:
– 1
– 1
½
5/6
Equivalent names using red and yellow
counters
› You need 18 counters, 6 red and 12 yellow.
› The 24 counters make up the whole.
› Can you group the counters to “see”
–
–
–
–
6/18
12/18
4/6
1/3
Adapted from pp. 153-154 in Van de Walle, J.A. and Lovin, L.H. 2006. Teaching StudentCentered Mathematics Grades 3-5. Boston, MA: Pearson Education, Inc.
Find the fraction name for each piece.
Fractions, Decimals and the Open Number Line
› Draw an open number line like this:
0
›
›
›
›
2
Locate 0.50 on the number line with your finger.
Locate ¼ on the number line with your finger.
Locate 0.75 on the number line with your finger.
Locate 1 ½ on the number line with your finger
› What could this task reveal to a teacher using it for formative
assessment?
Fractions as Operators
Input
Output
Input
Output
2
1
1
4/3
3
1.5
2
8/3
6
3
3
4
8
4
4
16/3
Input
Output
1
2/3
2
4/3
3
6/3
4
8/3
Find the input-output rules for these function machines.
Adapted from pg 83 of Chapin, S. H. and Johnson, A. 2000. Math Matters Grades K-6
Understanding the Math You Teach. Sausalito, CA: Math Solutions Publications.
A focus on Visualization
› Can you see 4/10 of something? What is the whole?
› Can you see 1 ½
of something? What is the whole?
› Can you see 2/3 of 3/5? What fraction would that be?
Adapted from pg 329 of Van de Walle, J. A., Karp, K. S., and Bay-Williams, J. M. 2013.
Elementary and Middle School Mathematics Teaching Developmentally (8th ed.) Boston, MA:
Pearson Education, Inc.
Is this 3/8?
Wrapping Up Wholes and Parts
› Students need opportunities to think about the whole
– Use materials where the size of the whole changes
› Students need to encounter unequally partitioned areas
and number lines
› Students need to design their own strategies for
partitioning areas and number lines
Decimal Basics
Cube activity: What’s special about base 10?
› Grab two handfuls of unifix cubes.
specified group size.
Number of
cubes in each
group
Number of
Groups
Divide the cubes into the
Number of
Leftovers
Total Cubes
7
3
8
10
› Record the number of groups and the number of leftovers in your
chart. For each different sized group do three trials.
› What do you notice when we make groups of size 10? What is so
special about base 10?
Adapted from pp. 47-50. Cohen, S. C., Lester, J. B., and Yaffee, L. 1999. Building a System of
Tens Casebook. Parsippany, NJ: Dale Seymour Publications.
Ways to Build a number with Base Ten Blocks
› Build 156 with your base ten blocks
› With your partner build and record all the possible ways
you were able to make 156.
› How many different ways are there?
› How did you know when you had found them all?
Introduction to Decimals
› The decimal point separates the whole units from the fractional
parts.
› The whole units are to the left of the decimal point.
parts are to the right of the decimal point.
The fractional
› Represent the fraction 3/10 with the Decimal grids in tenths.
use decimal notation to name this number.
Then
› Represent the fraction 3/10 with the Decimal grids in the
hundredths.
› Why is 3/10 = 30/100? How could we write this equation using
decimals?
Representing decimals
› Represent the decimals 0.7 and 0.07
–
–
–
–
On 10 x 10 grids
Coins
Base-ten blocks
Number lines
› Consider the number 7,777.777
Place
Thousands Hundreds
Tens
Ones
.
Tenths
Hundredths
Thousandths
7
Digit
7
7
7
7
.
7
7
Value
7,000
700
70
7
.
7
10
7
100
7
1000
Ways to Build a Number with Base Ten Blocks:
Part 2
› Build 1.56 with your base ten blocks
› With your partner build and record all the possible ways
to make 1.56.
› How was your thinking the same or different from when
you modeled 156 with base ten blocks?
Decimals and Base Ten blocks
› If I use a flat to represent one whole, a long to represent
tenths, and a unit to represent hundredths, what numbers
can I represent using exactly 8 pieces?
› How do you know you have found them all?
› Put the numbers you were able to build in order from
smallest to largest.
Guess my rule
In
Out
3
30
4
40
5
50
6
60
In
Out
.3
3
.4
4
.5
5
.6
6
In
Out
.03
.3
.04
.4
.05
.5
.06
.6
We often “tell” students when you multiply by 10 you “add a zero”. Does this
rule hold true with decimals?
Another rule we “tell” is that when you multiply by 10 you move the decimal
one place to the right and when you divide by 10 you move the decimal one
place to the left. Why does this rule work? What conceptual understanding
do students need in order to “see” why this rule works?
Comparing money and decimal numbers
› How could we show $2.20 using the fewest number of
dollar bills and coins?
› How could we show 2.20 using the fewest number of base
ten blocks?
› How could we show $2.50 using the fewest number of
dollar bills and coins?
› How could we show 2.50 using the fewest number of base
ten blocks?
How much money with 10 coins?
› Find all the different amounts of money we can
represent using exactly 10 coins (only dimes and
pennies).
› List them in order and look for patterns in your list.
Decimal Designs
› This activity can be purchased in PDF form from Aims Education
Foundation at the following web address.
› http://www.aimsedu.org/item/DA10352/Decimal-Designs/1.html
› Teaching Tip: Post the designs on a bulletin board (minus the
decimal information) to make an interactive board or center where
you can post questions about their designs and have students
answer them. Your questions could focus on decimals and/or
fractions. You could also have them to write number sentences on
how the colors add to 1 and focus on addition.
Mrs. Sam’s Quilting Project
› She wants to design a new quilt square that has….
– three-tenths of the square red
– fourteen-hundredths yellow
– two-tenths blue
› Mrs. Sam’s wants the remaining two sections of quilt square to be
orange and green.
› Use a hundredths grid to make a sketch of her quilt square.
› In your design, what portion of the quilt square will be orange?
Green?
› Share your solution with your classmates. Why do some students’
quilt squares look different than others? Are they “correct” given
the requirements? (Discuss as both fractions and decimals).
“It’s natural to read decimals such as 2.7 and
0.34, for example, as ‘two point seven’ and ‘point
thirty four.’ However, students should know that
these decimals can also be read as ‘two and
seven-tenths’ and ‘thirty-four-hundredths,’ and
they should be able to relate decimal fractions
to common fractions. Paying attention to this
difference reinforces the fact that decimals are
fractions written in a different form.”
Burns, M. 2007. About Teaching Mathematics a K-8 Resource (3rd ed).
Sausalito, CA: Math Solutions. p 284.
Dealing with Decimals: An AIMS activity
› This activity can be purchased in PDF form from Aims
Education Foundation at the following web address.
› http://www.aimsedu.org/item/DA7818/Dealing-WithDecimals/1.html
› How can this activity be used as an informal assessment?
› Other ideas
– Use the cards to play decimal war
– Deal two tenths, hundredths, and thousandths cards to each player.
Players have 30 seconds to make the largest (or smallest) number
possible. The player with the largest (or smallest) wins. Each player
discards the cards they used and draws three more cards and play
again.
More Place Value Fun
› Each player should draw the game board as shown.
____ ____ ____ ____ . ____ ____ ____
› Players take turns rolling the ten sided die (numbered 0 - 9)
writing their number in one of the spaces on the game board.
continue play until all blanks are full. The winner is the
player with the largest number.
› Play again: this time the person who makes the smallest
number wins
› How did your strategy change when you were trying to get a
large number vs a small number?
Adapted from Burns, M. 2007. About Teaching Mathematics a K-8 Resource (3rd ed). Sausalito,
CA: Math Solutions. p 290.
Grade 4 Sol Practice Item
Made Available by the VDOE
How does
instruction
which
focuses on
place value
help
students
interpret this
question?
What strategies do you currently teach students
to help them compare and order fractions
and/or decimals?
Equivalence
What do students need to know to
understand equivalent fractions?
Equivalent fractions have the same value
– They identify the same point on the number line
Equivalent fractions have different representations
Equivalent fractions can be generated by multiplying
the numerator and denominator by the same value
Can parts of a set have different names?
› Everyone should have 12 two-color counters.
› Turn your counters over so that only one counter is red.
– What fraction of your counters is red?
› Turn over another counter so that now you have two
counters that are red.
– What fraction of your counters is red?
› Turn your counters over so they all are yellow.
› Divide your counters into six equal groups.
› What fraction of the counters are in each group?
– Did you say 1/6?
How about 2/12?
Adapted from Burns, M. 2007. About Teaching Mathematics a K-8 Resource (3rd ed). Sausalito,
CA: Math Solutions. p 270.
Can parts of a set have different names?
› Put all your counters in a pile yellow side up.
› Divide your counters into four equal groups.
› Turn over the counters in one group so they are red.
– What fraction of your counters are red?
› Did you say 3/12? Or ¼?
› Turn over the counters in another group so now two
groups are red.
– What fraction of your counters are red?
› Did you say 6/12? 2/4? Or ½?
– Even though 6/12 = 3/6 would 3/6 make sense for this
situation?
Adapted from Burns, M. 2007. About Teaching Mathematics a K-8 Resource (3rd ed). Sausalito,
CA: Math Solutions. p 270.
The Fraction Kit
› Make your own fraction kit. You will find the instructions in
– Burns, Marilyn. 2001. Lessons for Introducing Fractions Grades 4-5. Sausalito, CA:
Math Solutions.
› As you are making the kit pose questions like…
› Who can explain why it makes sense to label each part as I did? (Four parts and each
part is one of the 4)
› How many sections do you think this one will have when we unfold it?
› How should I label the parts on each of these parts?
› The top number in this fraction is called the numerator. Does anyone know what we call
the bottom number?
Cover Up
› Play the game Cover Up.
› The directions for the game can be found in
Burns, Marilyn. 2001. Lessons for Introducing Fractions Grades 4-5.
Sausalito, CA: Math Solutions.
Reflecting on Cover Up
› What are the benefits of Steps 3 and 4 in the Cover Up
game?
› What information would the teacher circulating around the
classroom be able to collect?
› What questions could the instructor pose to students as
they are playing the game?
Playing Uncover
› Now play Uncover version 1
› How does this game compare to Cover Up?
› Now play Uncover version 2. How does the addition of
the New Rule affect how you play the game and the
mathematics you are using?
The directions for both versions of Uncover can be found in
Burns, Marilyn. 2001. Lessons for Introducing Fractions
Grades 4-5. Sausalito, CA: Math Solutions.
Another Cover Up
› Play Cover Up …the instructions for the game can be
found in
– De Francisco, C., and Burns, M. 2002. Teaching Arithmetic:
Lessons for Decimals and Percents, Grades 5 – 6. Suasalito, CA:
Math Solutions Publications. p. 166.
Pursuit of Zero
› Play Pursuit of Zero…the instructions for the game can be
found in
› De Francisco, C., and Burns, M. 2002. Teaching Arithmetic:
Lessons for Decimals and Percents, Grades 5 – 6. Suasalito, CA:
Math Solutions Publications. p. 167.
› How are the two versions of Cover Up similar and
different?
› How are the games Uncover and Pursuit of Zero similar
and different?
Same Name Frame
› This activity can be purchased in PDF form from Aims
Education Foundation at the following web address
– http://www.aimsedu.org/item/DA3377/Same-Name-Frame/1.html
› Use the chart to explore patterns with equivalent fractions.
› How could this chart be used to help students to simplify
fractions?
› Why does this chart work?
Fraction and Decimal Equivalents
› Use your hundredths grids to help you find the fraction
equivalents.
Fraction
½
¾
3/5
9/10
1/20
3/25
Equivalent Fraction
with a denominator of
100
Decimal number
Mixed number and Decimal equivalents
How much money are five quarters worth?
Word
Fraction
Money
Five-fourths
5/4 or 1 ¼
$1.25
Drawing
Eleven-tenths
Six-fifths
How does using representations to explore these equivalents
add to students’ conceptual understandings in a way that using
the calculator to divide the numerator by the denominator does
not?
Base Ten Go Fish
› Create a deck of cards for several decimal numbers with
the decimal representation of the number on one card,
the base ten block representation on another card, and
the corresponding money on another card.
› Deal each player several cards (depending on the size of
your deck)
› Turn up the top card on the pile of left-over cards to start
a discard pile.
› Play Go Fish taking turns asking for a particular number.
› The first player to get three matching representations
wins the hand.
Fraction and Decimal Equivalence Display
› Assign each pair of students a decimal.
› Provide time for each partner group to determine the
fraction equivalent for their decimal.
› Students then create a poster that meets the following
criteria
– Include at least one other fraction and one other decimal that
are equivalent to your original pair.
– Use words, pictures and/or numbers to provide an explanation
that proves your fractions and decimals are equivalent.
Grade 5 Sol Practice Item
Made Available by the VDOE
How does
instruction
which
focuses on
equivalence
help
students
interpret this
question?
Grade 4 Sol Practice Item
Made Available by the VDOE
How does
instruction which
focuses on
representations
when learning
fraction and
decimal
equivalents help
students interpret
this question?
Grade 4 Sol Practice Item
Made Available by the VDOE
How does
instruction which
focuses on
representations
when learning
fraction and
decimal
equivalents help
students interpret
this question?
Benchmarks
Thinking about Halves
› Start with a 4 x 2 rectangle on grid paper
– Choose two different color markers
– Shade half of the rectangle one color and half of the rectangle
the other color
– Be creative! You want each grid to look unique!
› Start with another 4 x 2 rectangle on grid paper
– Shade one fourth of the rectangle blue.
– Again, be creative so we can see a variety of “one-fourths.”
› Share the different ways that the students “saw” one-half
and one-fourth. The SOL test questions often give the
students questions where they have to find the fraction
but the parts that are shaded are scattered and not sideby-side.
Grade 3 Sol Practice Item
Made Available by the VDOE
Grade 4 Sol Practice Item
Made Available by the VDOE
Why is the benchmark of ½ the best strategy
for comparing these fractions?
23
50
5
8
3
5
8
18
7
12
4
9
Decide if each scenario is more or less than ½?
› James ate 6 out of 13 brownies.
› Elizabeth worked 5 hours of the 9 hour shift
› Taleke ran 8 miles of his 15 mile goal
How can you tell when a fraction is ½?
› Find four fractions that are equivalent to ½.
› What do you notice about all fractions that are equivalent to
½?
More or Less than 1?
› Are the following sums larger or smaller than 1? Explain
how you know.
› 3/8 + 4/9
› 1/2+ 1/3
Do you need less than or more than $1.00?
› If you want to buy an apple for $0.39 and raisins for
$0.25, how much money do you need?
– Estimate: What do you think more or less than $1.00? Why?
› Use a representation to help you determine exactly how
much money you will need.
Using Benchmarks to compare or order
› For each of the following fractions is it closest to 0, ½ or 1?
1/6
3/8
5/7
› How do benchmarks help you order these fractions?
› For each of the following decimals is it closest to 0, 0.5, or 1?
›
0.4
0.05
0.85
› How do benchmarks help you order these decimals?
In Proper Order
› This activity can be purchased in PDF form from Aims
Education Foundation at the following web address
› http://www.aimsedu.org/item/DA10146/In-ProperOrder/1.html
Ordering Decimals
› 0.04
0.40
0.44
› For each decimal write how you would say it
› Shade a hundredths grid to represent it
› Build it with base ten blocks
› Use your representations to help you order the decimals
greatest to least. Be prepared to explain to the class
how you decided on your ordering.
More or Less Cards
› Write a fraction and a decimal to compare on your first
index card.
› On your second index card write a mixed number and a
decimal to compare
› Swap your cards with another student.
› For each card decide which number is the greatest
› Use a representation to justify which is more and which is
less.
› How does your representation help to determine how
much more or how much less?
Put In Order
› The directions for this activity can be found in
Burns, Marilyn. 2001. Lessons for Introducing Fractions Grades 4-5.
Sausalito, CA: Math Solutions. pp. 105-115.
Open number line assessment
› Draw an open number line from 0 to 2 and place your fraction on the
number line. (use fractions equivalent to well known terminating
decimals).
› Find the equivalent decimal and place it on the number line too.
› Place a fraction on your number line that is smaller than your original
number.
› Place a decimal on your number line that is larger than your original
number.
› What kind of information can we gather about the student’s ability to
compare and order fractions and decimals from this formative assessment?
Individual Put In Order: Another formative assessment
› Put the following numbers in order from smallest to largest and
explain your reasoning in writing
1 15
14 1
, , 0.10, ,1 , 0.50
8 16
10 2
Adapted from Burns, Marilyn. 2001. Lessons for Introducing Fractions Grades 4-5.
Sausalito, CA: Math Solutions. pp. 135.
Dueling decimals: An AIMS Activity
› This activity can be purchased in PDF form from Aims Education
Foundation at the following web address
› http://www.aimsedu.org/item/DA4714/Dueling-Decimals/1.html
› Before doing the activity use the wheels to do the following…….
› Represent 3/10.
– How many hundredths are in 3/10?
– How can you write 3/10 as a decimal?
› Represent 0.25
– How many tenths and how many hundredths in 0.25?
› Represent 0.42
– How many tenths and how many hundredths in 0.42?
› How could you use the circles to show 0.3 + 0.6 = 0.9?
› How could you use the circles to show 0.9 – 0.4 = 0.5?
The Greatest Wins
› When we paid our bill Bob received three dollars and
thirty cents in change. I received three dollars and three
cents in change. Who received more change?
$3.30 > $3.03
› Shante and James are trying out for the track team.
Shante is able to sprint eight-tenths of a mile. James is
able to sprint for sixty-nine hundredths of a mile. Who
was able to sprint the furthest distance?
.8 > .69
The Greatest Wins
› The instructions for the game can be found in
› De Francisco, C., and Burns, M. 2002. Teaching Arithmetic:
Lessons for Decimals and Percents, Grades 5 – 6. Suasalito, CA:
Math Solutions Publications. p. 183.
Grade 3 Sol Practice Item
Made Available by the VDOE
How does
instruction
which
focuses on
benchmarks
help
students
interpret this
question?
Grade 4 Sol Practice Item
Made Available by the VDOE
How does
instruction
which
focuses on
benchmarks
help
students
interpret this
question?
Grade 5 Sol Practice Item
Made Available by the VDOE
How does
instruction
which
focuses on
benchmarks
help
students
interpret this
question?
Grade 4 Sol Practice Item
Made Available by the VDOE
How does
instruction
which
focuses on
benchmarks
help
students
interpret this
question?
Comparing fractions with unlike denominators
› Juan and Christina each had individual pizzas that were
the same size. Juan sliced his pizza into eight slices and
ate six of them.
Christina sliced her pizza into six slices
and ate four of them. Who ate more pizza?
› Use a drawing to represent this problem situation to help
you solve this problem.
Comparing fractions with unlike denominators
› Marsha and Karl each had pizzas that were the same size.
Marsha cut her pizza into six slices and ate two of them. Karl
sliced his pizza into three slices and ate one of them. Who ate
more pizza?
› Use a drawing to represent this problem situation to help you
solve this problem.
› Why is important to say in these problems that the pizzas were the
same size?
› Could we say who ate more pizza if we didn’t know the size of their
pizzas?
› Sort the drawings for these two problems into those who drew
circles and those that drew rectangles. What are the challenges
when using circle drawings to represent fractions?
Strategies for comparing fractions
› Fractions with a numerator of 1 (unit fractions): The
bigger the denominator the smaller the fraction. Why?
› Fractions with the same denominator: The fraction with
the largest numerator is the largest. Why?
› Fractions with the same numerators (other than 1): Use
a representation to help you write a rule for comparing
fractions with the same numerators. Why does the rule
work?
Strategies for comparing fractions
› Fractions with different numerators and denominators
– Benchmarks
– Equivalent Fractions
– Parallel number lines
› Use each of these methods to decide which is closer to 1?
5/6 or 7/10
Addition and Subtraction
Use your fraction circles or another fraction
manipulative to solve these problems
1 1

› What is
4 2
3 1

› What is
4 2
› What is
1
1
2
Solve these problems too!
1 1

› What is 2
2 4
› What is
1 1
1 
4 2
1 1
3


› What is
4 8
More Addition
› What is
1 1
5
1  2
4 2
8
?
› Be prepared to explain your method for finding the
sum.
Egg Carton Fractions Addition
› ½ + 1/3
› ¾ + 1/6
› 7/12 + 5/6
› ¾ + 3/4
Egg Carton Subtraction
› ½ - 1/3
› 5/6 – 2/3
› 1/3 – 1/12
› 1 1/6 – ¾
› 1 1/3 – 3/4
Egg Problems
› Juanita bought a dozen eggs at the store. She used
three of them to make a breakfast cake, used ½ of them
to make scrambled eggs, and used 1/6 of them to make
waffles. How many eggs are left?
› Hannah bought three dozen eggs at the store. Onefourth of the eggs in the first carton broke on the way
home, 1/3 of the eggs in the second carton were broken,
and 1/12 of the eggs were broken in the third carton.
How many unbroken eggs does Hannah have?
Adding and Subtracting with Pattern Blocks
› Devin bought a small cake to celebrate his birthday. He ate
1/3 of his birthday cake with lunch and ate 1/6 of his
birthday cake for dinner. How much birthday cake did Devin
eat?
› Andre baked 4 pizzas for his super bowl party. He ate 1/6 of
one pizza before the party started. At the party 2 ½ pizzas
were eaten. How much pizza is left-over for Andre to take to
work the next day?
› How does working with models help us make sense of the
processes?
Adapted from pg 91 of Chapin, S. H. and Johnson, A. 2000. Math Matters Grades K-6
Understanding the Math You Teach. Sausalito, CA: Math Solutions Publications.
What is the Sum?
› Each pair of partners is given a card with an addition
problem, for example 1/6 + ¼ , or a decimal addition
problem.
› Player 1 uses manipulatives to build the first number and
put it in the sock.
› Player 2 then uses manipulatives to build the second
number and also adds it to the sock.
› Both players guess the sum and then look in the sock to
see if they found the sum correctly.
What is the Difference?
› Each pair of partners is given a card with a subtraction
problem on it, for example ¾ - 3/8, or a decimal
subtraction problem.
› Player 1 builds the first fraction and places it in the sock.
› Player 2 then removes the pieces from the sock needed
to find the difference. (Trades may be needed too).
› Both players find the difference and then look in the sock
to see if they are correct.
Fractions, decimals, and money
› I have eighteen coins in my pocket.
› One-sixth of the coins are quarters.
› One-third of the coins are dimes.
› One-ninth of the coins are nickels.
› The rest of the coins are pennies.
› How much money do I have in my pocket?
Adapted from pg 55 of
Schuster, L., and Anderson, N.C. 2005. Good Questions for Math Teaching . Sausalito, CA: Math
Solutions.
Race to $1.00
› Players take turns rolling the six-sided die.
› Players may take that number of quarters, dimes,
nickels, or pennies. Keep track of the value of the
money by recording the value as a decimal.
› After six rolls record how much money each player
has.
› It’s OK to go over $1.00
› The winner is the player closest to $1.00
› How far is each player from $1.00?
Pigs Will Be Pigs
› Read the book Pigs will be Pigs by Amy Axelrod
› Then answer the following questions…
› How much money did the pigs collect when they searched
their house? Be prepared to share your method for
calculating the total.
› How much do four specials cost?
› After purchasing the specials how much money did the pigs
have left over?
Adapted from De Francisco, C., and Burns, M. 2002. Teaching Arithmetic: Lessons for
Decimals and Percents, Grades 5 – 6. Sausalito, CA: Math Solutions Publications. p. 134.
Addition and Subtraction Concepts
› Jill has 3/5 of a candy bar. If her mother gives
her 1/3 of a candy bar, what portion of a candy
bar will Jill have?
› (Join- result unknown)
› Andre jogged ½ mile before breakfast and then he
jogged some more after dinner. If he jogged 1 ¼
miles that day, how far did he jog after dinner?
› (Join-change unknown)
Addition and Subtraction Concepts
› Tony has 25 ¾ ft of copper wire. If he gives 12 ½ ft to Bill for a project,
how much wire will Tony have left?
› (Separate – result unknown)
› Tony has 25 ¾ ft of copper wire. He gave some to Bill for a project.
Now he has 13 ¼ feet of copper wire left. How much copper wire did
Tony give to Bill?
› (Separate – change unknown)
› Tony has some copper wire. He gave 12 ½ ft of copper wire to Bill for a
project. Now Tony has 13 ¼ feet of copper wire left. How much copper
wire did Tony have to start with?
› (Separate – start unknown)
Addition and Subtraction Concepts
› Susan has 3/8 of a 24 pack of soda. Tim has
4/6 of a 24 pack of soda. Who has more soda?
How much more?
› (Compare)
› Jennifer has 3½ yards of fabric. 1 ¼ yards are
red and the rest of the fabric is blue. How much
blue fabric does Jennifer have?
(Part-part-whole Part unknown)
Are these ½ + 1/3 or ½ - 1/3?
› Starting at her apartment, Sally runs ½ mile down
the road. Then Sally turns around and runs 1/3
mile back towards her apartment. How far has
Sally run since leaving her apartment?
› Starting at her apartment, Sally runs ½ mile down
the road. Then Sally turns around and runs 1/3
mile back towards her apartment. How far down
the road is Sally from her apartment?
Multi-Step Problems
› Justin works part-time and earns $160.00 every two
weeks. She has the following budget:
–
–
–
–
1/5 of his paycheck is for snacks/meals
¼ of his paycheck is for gas
1/4 of his paycheck is for recreation
The rest is for college savings
› How much money does Justin save for college every two
weeks?
Adapted from Lappan, G., Fey, J.T., Fitzgerald, W.M., Friel, S.N., Phillips, E.D. 2004. Bits and
Pieces II Using Rational Numbers. Needham, Massachusetts: Pearson Prentice Hall. p.88.
Can we add or subtract fractions regardless of the
whole?
What would ½ - 3/8 be if
the ½ is ½ of the smaller
circle and the 3/8 is 3/8 of
the larger circle?
Grade 3 Sol Practice Item
Made Available by the VDOE
How does
instruction
which
focuses on
the use of
representati
ons help
students
interpret this
question?
Grade 3 Sol Practice Item
Made Available by the VDOE
Grade 4 Sol Practice Item
Made Available by the VDOE
Grade 4 Sol Practice Item
Made Available by the VDOE
Grade 5 Sol Practice Item
Made Available by the VDOE
Decimals and Remainders
The Doorbell Rang
› Read the story The Doorbell Rang by Pat Hutchins
› Pause for the children to determine how many cookies
each person gets as the new guests arrive.
– Have the children to draw a representation for each problem
› How many cookies would each person get if there were
36 cookies to share among 8 people?
Balloons and Brownies:
Remainders vs Mixed Numbers
› The instructions for this activity can be found in
› Burns, Marilyn. 2003. Lessons for Extending Fractions Grade 5.
Sausalito, CA: Math Solutions. pp. 27 - 38.
Decimals and Remainders
› Divide nine apples between two people.
› How much does each person get?
– Use a drawing to determine how much each person gets and
how much is leftover.
› What does your solution mean?
– Use a drawing to determine how much each person gets as a
fraction and as a mixed number.
› What does the whole number mean in your mixed number? What does
the denominator of the fraction mean in your mixed number? What does
the numerator mean in your mixed number?
› What does the denominator of the fraction mean in your fraction solution?
What does the numerator mean in your fraction solution?
Decimals and Remainders
› Divide nine apples between two people.
each person get?
How much does
– Use base-ten blocks to determine how much each person gets
as a decimal.
› What does the whole number in your solution mean? What does the
decimal portion of your solution mean?
– Use long division to determine how much each person gets.
› How does thinking of division in these ways add to our
understanding of the connectedness whole number,
fraction, and decimal computation?
Decimals and Remainders
› Graph your fraction, mixed number, and decimal solutions
on a number line. How is your thinking different as you
graph each solution?
› Divide nine apples between two people.
each person get?
How much does
9
1
9  2   4  4.5  4 R1
2
2
Decimals and Remainders: Another one
› Share five brownies among four children.
does each child get?
How much
– Use a drawing to determine how much each person gets and
how much is leftover.
› What does your solution mean?
– Use a drawing to determine how much each person gets as a
fraction and as a mixed number.
› What does the whole number mean in your mixed number? What does
the denominator of the fraction mean in your mixed number? What does
the numerator mean in your mixed number?
› What does the denominator of the fraction mean in your fraction solution?
What does the numerator mean in your fraction solution?
Decimals and Remainders: Another one
› Share five brownies among four children.
does each child get?
How much
– Use base-ten blocks to determine how much each person gets
as a decimal.
› What does the whole number in your solution mean? What does the
decimal portion of your solution mean?
– Use long division to determine how much each person gets.
› Which of your solution methods most closely parallels
the work you do when you are calculating using the
standard algorithm?
Decimals and Remainders
› Share five brownies among four children.
does each child get?
How much
5
1
5  4   1  1.25  1R1
4
4
› Graph your fraction, mixed number, and decimal solutions
on a number line. How is your thinking different as you
graph each solution?
Decimals and Remainders: Can you divide a
smaller number by a larger number?
› Share four brownies among five children.
› How much does each child get?
– Use a drawing to determine how much each person gets and
how much is leftover.
› What does your solution mean?
– Use a drawing to determine how much each person gets as a
fraction.
› What does the denominator of the fraction mean in your fraction solution?
What does the numerator mean in your fraction solution?
› Is it possible to write how much each person gets as a
mixed number? Explain.
Decimals and Remainders: Can you divide a
smaller number by a larger number?
› Share four brownies among five children.
does each child get?
How much
– Use base-ten blocks to determine how much each person gets
as a decimal.
› What does the whole number in your solution mean? What does the
decimal portion of your solution mean?
– Use long division to determine how much each person gets.
› How does thinking of division in different ways help the
students to understand why the whole number portion of
their decimal solution is a zero?
Decimals and Remainders
› Divide four brownies among five children.
does each child get?
How much
4 8
45  
 0.8  0 R 4
5 10
› Graph your fraction, mixed number, and decimal solutions
on a number line. How does graphing your solutions
help students to “see” that the solution is less than 1
whole?
Multiplication
Building Rectangles
› This activity can be purchased in PDF form from Aims
Education Foundation at the following web address
› http://www.aimsedu.org/item/DA1924/BuildingRectangles/1.html
Writing Rectangles
› Given the dimensions of a rectangle build the
representation of the product using your base ten blocks
and the array model.
› Examine the partial products algorithm and how it relates
to the array model.
Tens to Tenths Again
› The array model extends to the multiplication of decimal
numbers.
› Build each of the products with your base ten blocks.
› Use the partial products algorithm to determine the
product too!
Where does the decimal go?
› Use the array model to represent the product 12 x 13
› Then use the array model to represent this product
– 1.2 x 1.3
› How are these representations similar? Different?
Adapted from Van de Walle, J.A. and Lovin, L.H. 2006. Teaching Student-Centered Mathematics
Grades 3-5. Boston, MA: Pearson Education, Inc. p. 199.
Which answer makes sense?
1) Solve the problem on page 175 of
De Francisco, C., and Burns, M. 2002. Teaching
Arithmetic: Lessons for Decimals and
Percents, Grades 5 – 6. Sausalito, CA: Math
Solutions Publications.
2) Examine the student work samples. What is
Patrice’s misunderstanding? What rule is being
incorrectly applied by Crystal?
Calculators?
How do you use calculators during fraction
and decimal instruction?
Multiplying and Dividing by Numbers Close to 1
› What can you tell me about these two products?
– 0.8 x 20
– 1.2 x 20
› How does an activity like this contribute to a
student’s ability to be successful when estimating?
Decimal Calculator games
› Play each of the following calculator games
–
–
–
–
Decimal Nim
Target
Race to Zero
Target II
› Be prepared to share any strategies you develop to win the
games.
› How does using calculators in this way contribute to the
students’ understandings of decimals and place value?
Instructions for the games can be found in De Francisco, C., and Burns,
M. 2002. Teaching Arithmetic: Lessons for Decimals and Percents,
Grades 5 – 6. Sausalito, CA: Math Solutions Publications. pp 184-187.
More calculator fun!
› On many calculators by pressing the equal sign over again repeats
the previous action.
› Enter 0.1 in your calculator and then press equal.
› Make a prediction about the next number you will see.
› Enter equals again. Continue this process until your calculator
says 0.9. What do you think the next number will be? Make a
prediction.
› Then press the equals again to check your prediction.
› When you add one-tenth to a number with a nine in the tenths
place, why does the digit in the ones place increase by 1?
More calculator fun!
› If you enter 0.001 in your calculator, how many times will
you need to press the equal sign to get the calculator to
reach 0.01?
› What number do we need to add to two and forty-seven
hundredths so that the display shows 2.473?
Knockout!
› Instructions for this activity can be found in
› De Francisco, C., and Burns, M. 2002. Teaching Arithmetic:
Lessons for Decimals and Percents, Grades 5 – 6. Sausalito, CA:
Math Solutions Publications. P. 164.
“Giving students rules to help them develop facility with
fractions will not help them understand the concepts.
The risk is that when students forget a rule, they’ll have
no way to reason through a process.”
Burns, M. 2007. About Teaching Mathematics a K-8
Resource (3rd ed). Sausalito, CA: Math Solutions. p 268.
Place logo
or logotype here,
otherwise
delete this.
Sources
› All the practice items can be found on the VDOE website
at
http://www.pen.k12.va.us/testing/sol/practice_items/in
dex.shtml#math
› You will also need the following books:
– Axelrod, A. 1994. Pigs Will Be Pigs. New York : Maxwell
Macmillan International.
– Burns, Marilyn. 2003. Lessons for Extending Fractions Grade 5.
Sausalito, CA: Math Solutions.
– Burns, Marilyn. 2001. Lessons for Introducing Fractions Grades
4-5. Sausalito, CA: Math Solutions.
– De Francisco, C., and Burns, M. 2002. Teaching Arithmetic:
Lessons for Decimals and Percents, Grades 5 – 6. Sausalito, CA:
Math Solutions Publications.
– Hutchins, Pat. 1994. The Doorbell Rang. New York : Mulberry
Books.