Download Lesson Objectives

Lesson Topic: Working with Rational Numbers-Day 2
Lesson Objectives
1. Order and compare rational numbers (fractions, decimals, percents, and integers)
2. Apply operations to fractions and integers
3. Visually determine fractional, decimal or percent parts of a whole
4. Use different models for representing operations on fractions and integers
Standards Addressed: PSSM Standards Addressed
Number and Operations Standards for grades 6-8
 Work flexibly with fractions, decimals and percents to solve problems
 Compare and order fractions, decimals and percents efficiently, and find
their approximate locations on a number line
 Develop meaning for integers and represent and compare quantities with
them
 Understand the meaning and effects of arithmetic operations with
fractions, decimals and integers
 Develop and analyze algorithms for computing with fractions, decimals
and integers and develop fluency in their use
Pre-requisite Skills
 Knowing magnitude of fractions, decimals, integers, and %’s and being
able to order them
 Being able to convert from fractions to decimals to %’s
Materials needed
Engager activity: “Stuck on Rational Numbers”
 Set A post-it notes with the following fractions –1/2 ¾ ¼ 1/5 1/3 7/8
8/9 1/8 5/8 1/10 11/12 5/6 2/3 4/5 1/6 4/6 6/8 1/9 5/12 3/10
 Set B post-it notes with the following fractions, decimals, and %’s 2/5
40% 0.4 3/8 37.5% 0.375 11/22 50% 0.5 1/9 11 1/9%
0.111… 1/10 2/3 66 2/3% 0.666… 5/12 0.41666… 41 2/3%
7/12
 Set C post-it notes with the following fractions and decimals –5/9
-9/5
-4/4

9/5
½
-1/2
7/8
-7/8
4/3
-4/3
30/15
-18/9 -30/15 15/10 -6/4 0/12 11/12 -17/12
Adding machine tape—three 6 foot long sections
-4.5/4.5
Activity 1: Fraction Strips
 Six 1” X 11” strips of paper for each student (cut from 8 ½” by 11” paper)
 copies of fraction table -1 per student
 transparency of fraction table
 colored transparency strips for use on the overhead
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Activity 2: Fraction Circles
 Fraction circle pieces 1 set per student pair
6--whole pieces
5 --1/2 pieces 7—1/3 pieces 8—1/4 pieces
12—1/6 pieces
18—1/8 pieces
 2 sets of fraction circle pieces with magnetic strips on back to use on white
board
Activity 3: Coveralls
 Prepare 3 dice per student pair (Die 1: .01 .03 .09 .15 .21 .33)
(Die 2: 2% 4% 8% 16% 24% 30%) (Die 3: 1/25 2/25 1/10 1/5
3/10 1/20). Make each die a different color for ease with distribution.
 Coverall Game Sheet-1 per student pair; 1 transparency of Game Sheet
 1 copy of “Take a Look at Some Numbers” song
Activity 4: The Charge Model for Integers
 1 bag with 20 red beans and 20 white beans per student -larger beans are
easier to handle. (Note: Beans may be a management nightmare,
therefore you could cut forty 1” squares using two different colors to make
twenty of each.)
 red and white transparent squares or circles for the overhead to model
integers
 1 sheet of white paper per student to use as a working mat for problems
 5 blank transparencies and markers for overhead
 Charge model transparency A (blue font) with the following problems and
solutions:
2 -6 = -4
5 – (-5)= 10 -8 – (-4)= -4 -2 – (-7)= 5 -3 – 4=-7
3 – (-4)= 7
-1 – 9= -10
 Charge model transparency B (red font) with the following problems and
solutions:
2 + (-6) = -4 5 + 5= 10
-8 + 4= -4
-2 + 7= 5
-3 + (-4)=-7
3 + 4= 7
-1 + (-9)= -10
 Adding Integers Transparencies
 Subtracting Integers Transparencies
 Multiplying Integers Transparencies
 1 set of 5 x 8 cards (magnetic strips on back) with the following problems:
3 x 5= 15 3 x 4= 12 -2 x -3= -6 -3 x 5= -15 -1 x -5= 5 2 x -3= -6
-4 x -3= 12 2 x 4= 8
Activity 5: Walking the Number Line
 1 set of 3 x 5 cards with the following job descriptions for each group of 4:
walker
checker
reader
recorder
 1 long number line (-6 to 6) per group of 4 students made on masking
tape for putting on the floor
 1 set of question cards per group of 4 (2 + 3 -2 + 5 -3 + -1
5–2
-2 – 3
3 - -2
-1 - -4
-6 - -2 -2 –3 -1 + -2 3 + -5 -4 + 1)
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Activity 6: Integer CONTIG
 Prepare 3 dice per student pair (Die 1: -1, 2, 3, -4, -5, -6)
(Die 2: 1, -2, 3, 4, -5, 6) (Die 3: 1, -2, -3, -4, -5, 6). Make each die a
different color for ease with distribution
 Transparency of Integer CONTIG game sheet
 Integer CONTIG game sheet –one per student pair
 2 different colored markers per student pair
Activity 7: Fresno
 3 ten-sided dice numbered 0 to 9 for the teacher
 Fresno overhead transparency
 Fresno student game transparency
 1 Fresno student game sheet for each student
 Chocolate coins for prizes (optional)
ENGAGER - Number line activity “Stuck on Rational Numbers”
Teacher Action
Engager Set A post-it
notes:
Place all three adding
machine tape number lines
on the board or wall.
(Note: Adding machine
number lines include:
Place one above the other.
*0 at one end and 1 at the
other end
*0 at one end and 1 at the
other end
Place this one in a different
place than the other two.
*-2, -1, 0, 1, and 2 evenly
spaced)
Teacher Talk
Give each student a number Discuss with the others at
from the Set A post-it notes. your table where you think
your number should be
placed on the number line.
Send one table at a time to
Place your fraction on the
place their number on the
appropriate place on the
number line. Discuss each
number line.
group’s placement of
numbers.
Student Response
Students discuss the
placement of their numbers.
Students place their
numbers on the number
line.
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Teacher Action
Teacher Talk
After all numbers are placed What helped you in
discuss their placements.
deciding where to place
your number?
Are any fractions in the
same place?
Discuss with the others at
your table where you think
your number should be
placed on the number line.
Place your number on the
appropriate place on the
number line.
Engager Set B post-it
notes:
Give each student a number
from the Set B post-it notes.
Send one table at a time to
place their number on the
number line. Discuss each
group’s placement of
numbers.
After all numbers are placed What helped you in
discuss their placements.
deciding where to place
your number?
Engager Set C post-it
notes:
Give each student a number
from the Set C post-it notes.
Send one table at a time to
place their number on the
number line. Discuss each
group’s placement of
numbers.
After all numbers are placed
discuss their placements.
Are any numbers in the
same place?
Discuss with the others at
your table where you think
your number should be
placed on the number line.
Student Response
Benchmarks numbers such
as ½, ¼, and ¾.
When the numerator and
denominator are just 1 away
from each other it is close to
one, etc.
¾ and 6/8; 2/3 and 4/6
Explanations will vary.
Students discuss the
placement of their numbers.
Students place their
numbers on the number
line.
Benchmarks numbers such
as 50% and 0.5.
I changed mine from
fractions to decimals or
decimals to %’s.
50% and 0.5, 11/22, etc.
Students discuss the
placement of their numbers.
Place your number on the
appropriate place on the
number line.
Students place their
numbers on the number
line.
What helped you in
deciding where to place
your number?
Are any numbers in the
same place?
The numbers that were
already on the number line.
Teacher Talk
What do you notice about
the fraction table?
Student Response
All the rows are the same
length.
-18/9 and -30/15 are both 2, etc.
Activity 1: Fraction Strips
Teacher Action
Using the Fraction Table
to Change Terms of
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Teacher Action
Teacher Talk
Fractions.
Place fraction table
transparency on overhead
and hand out a copy of the
fraction table to each
student.
Distribute one 1” paper strip Take your paper strip and
to each student.
mark ¾, fold, and tear off.
Use your ¾ strip to identify
other fractions on the
fraction table that have the
same length.
What are they?
Use the table to explain a
procedure for changing a
fraction to an equivalent
fraction that uses more
pieces.
.
Are there any other
fractions on the table that
name the same length?
Check to see if your
procedure works on
fractions you’ve identified
as equivalent.
Use the table to explain
how to change a fraction to
an equivalent fraction that
uses fewer pieces.
Student Response
The rows are divided up
differently.
Some of the fractions line
up with the fractions below.
The larger the denominator,
the smaller the piece, etc.
6/8, 9/12, 12/16, and 18/24.
The 12th’s strip is divided
into 3 times the number of
pieces that the 4th’s strip is
divided into. So you are
multiplying the number of
total pieces as well as the
number of parts you
actually have both by 3.
Answers may vary:
1/3 = 2/6, 3/9, etc.
Answers will vary.
For example 2/3 is
equivalent to 8/12. The
12th’s strip has 12 pieces
and the 3rd’s strip has 3
pieces. It takes 4 of the
12th’s to equal 1 of the
3rd’s. If I divide the
number of 12th’s by 4, it
tells me the number of
3rd’s. This means that
when I go from 12th’s to
3rd’s I have to divide both
numbers by 4.
This is called simplifying to
lowest terms because we
used the fewest number of
pieces to name our fraction.
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Teacher Action
Addition of FractionsPutting Together:
Write the problem 1/6 + ¾
on the board or overhead.
The teacher will now
address this common
misconception.
Distribute five 1” strips to
each student.
Lead students to connect
their prior knowledge
regarding changing terms of
fractions and how the strip
has helped them find a
common denominator.
Teacher Talk
When we do operations
using fractions, we make
sure our final answers are
simplified to lowest terms.
We can use this fraction
table to model addition of
fractions.
Student Response
Let’s look at the problem
1/6 + ¾. Locate these
pieces on your fraction
table.
What do you think the
answer might be for this
problem?
(Some students may say
4/10. This is a wrong
answer typically given by
middle school students.
However, if students don’t
give any answer to this
because they don’t have any
idea what to do, the teacher
could suggest just adding
the two numerators together
and the two denominators
together.
.
Let’s see if we can prove
that 4/10 is the right answer.
Take one of the strips of
paper, line the strip up next
to 1/6 and mark it. Move
this mark to the zero on the
fourths line and mark ¾.
What does the marked area
of strip represent?
Now using the fraction
table. How long is this
strip?
Allow students to discuss
why the answer 11/12 and
not 4/10.
If you don’t just add the
numbers together in this
fraction, how do you get
11/12?
The strip represents adding
1/6 and ¾.
Students may notice that the
strip is equal to the fraction
11/12. If they do not notice
this, prompt with a
demonstration.
Students should discuss
with a partner.
If you look at the table, you
can see that 1/6=2/12 and
¾=9/12. You can add 2/12
and 9/12 since now they
have the same denominator.
(Note: This is an excellent
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Teacher Action
Teacher Talk
example to clarify the
misconception of adding
numerator plus numerator
and denominator plus
denominator.
Example: 1/6 + ¾ does not
equal 4/10.)
Let’s use a new strip to try
the following problem: 2/3
+ 1/8.
Student Response
What did 2/3 + 1/8 equal?
Explain how you know this
is correct.
19/24
2/3 = 16/24 and 1/8 = 3/24
or 19/24
Allow students time to work
together to mark 2/3 on a
new strip and 1/8.
Repeat discussion for
solving the problems:
2/9 + 1/6
1/6 + 3/8
Allow students to work
together to find their answer
and prepare an explanation.
Discuss questions and
comments about addition of
fractions.
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Teacher Action
Subtracting Fractions
Write the problem 11/12 –
5/8 on the board or
overhead.
Subtracting Fractions
Write the problem 11/12 –
5/8 on the board or
overhead.
Teacher Talk
First mark a strip which is
11/12 long and tear off at
this mark. We want to take
5/8 away from this, so put
the right end of the strip at
the 5/8 and mark the 0 point
on your strip. Place a large
“X” on the section that
represents the 5/8. Find the
length of the remaining part
on the fraction.
First mark a strip which is
11/12 long and tear off at
this point. We want to take
5/8 away from this, so put
the right end of the strip at
the 5/8 and mark the 0 point
on your strip. Place a large
“X” on the section that
represents the 5/8. Find the
length of the remaining part
on the table.
Notice that 11/12= 22/24
and the length of the strip is
5/8=15/24. The difference
is (22-15)/24. Use this to
develop the algorithm.
Student Response
Students use the fraction
strips and table to model the
problem.
Students use the fraction
strips and table to model the
problem.
Do the following problems
similarly: 8/9 – 5/18
5/6 – ¼
7/8 – 5/6
Do the following problems
similarly:
8/9 – 5/18
5/6 – ¼
7/8 – 5/6
Extension: Students make up an addition problem and a subtraction problem that will
work using the table, and trade with a partner so they can work each other’s problem.
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Teacher Action
Multiplying Fractions
Students will continue to
use 1” strips and fraction
table.
Teacher Talk
Remember that one model
of multiplication would
have us read 3 x 4 as 3
groups of 4.
Write 3 x 4 on the board or
overhead.
Write ½ x 5/6 on the board
or overhead.
When we see ½ x 5/6 that
means find half of 5/6.
Student Response
We will use the paper strips
and fraction table to model
multiplying fractions on this
problem.
Teacher will demonstrate as
students work with the
strips.
Mark a length equal to 5/6
on your paper strip and tear
this amount off (very
important to tear off).
Then fold it in half.
Students begin marking and
following teacher’s
directions. Provide
assistance to students as
needed or assign a partner
student to those having
difficulty following.
Use your fraction table to
help you find the length of
this piece.
What did you find?
Write 2/3 x 7/8 on the board Let’s try another fraction.
or overhead. Demonstrate
2/3 x 7/8 means take 2/3 of
with another strip.
7/8.
5/12
Mark a length equal to 7/8
on your paper strip and tear
this amount off.
Then fold this length in
thirds and open out 2 of
them.
Compare these pieces to
your fraction table.
How long is this piece?
Students may say 15/24 or
5/6. Discuss why both
answers are correct.
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Teacher Action
Display the following
problems:
¾x½
2/3 x 5/6
½ x 5/9
5/6 x ¾
Teacher Talk
Student Response
Allow students to work
individually or in pairs to
use the fraction strips to
solve.
Monitor students’
discussions as they work.
Many students may be
ready to make a conjecture
about multiplying fractions
mathematically.
For those who think they
see a pattern, ask them to
explain.
To multiply fractions you
multiply the numerator and
denominator. You might
need to take the fraction to
lowest terms.
For those students not ready
to summarize, continue
allowing use of the strips
Journal:
Students can be asked to create a problem and record a drawing of it in their journal with
an explanation.
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Teacher Action
Dividing Fractions
Write the problem ¾ 1/4
on the board or overhead.
Students will need paper
strips and the fraction table.
Teacher Talk
We will be using the
measurement model for
division. 20  5 means I
have 20 and want to know
how many groups I can
make with 5 in each group
Student Response
Students use the fraction
strips and table to model the
problem.
Three ¼’s will fit so the
answer is 3.
Teacher will demonstrate as
students work.
¼
¼
¼
Write the problem 5/6 ÷
5/12 on the board or
overhead.
1/6 1/6 1/6 1/6 1/6 1/6
Mark and tear off a strip
equal to the 3/4.
Use your fraction strips to
see how many lengths of
1/4 will fit in this.
Use a new paper strip, mark
and tear off 5/6.
How many times will 5/12
fit in this?
Two 5/12’s will fit so the
answer is 2.
Yes there are two groups of
5/12.
 - - 5/12 - - -  - - - 5/12 - 
Write the problem 2/3 ÷1/6
on the board or overhead.
1/3
1/3
Mark and tear off 2/3 and
see how many times 1/6
will fit in this.
Four 1/6’s will fit so the
answer is 4.
Now try 7/8 divided by ¼.
Three ¼’s will fit but there
is ½ of the ¼ left so there
are 3 ½ one fourths in 7/8.
 1/6  1/6   1/6  1/6 
Write the problem 7/8 ÷ ¼
on the board or overhead.
Mark and tear off 7/8. See
how many lengths of 1/4
will fit.
Continue this procedure
with 5/12 2/12 .
Mark 5/12 and tear off.
See how many groups of
2/12 will fit in this length
There are 2 complete
groups, and 1 of the 2 parts
needed to make an
additional group of 2/12.
So the answer is 2 ½.
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Teacher Action
Display the problem 2/3
4/9
Teacher Talk
Mark 2/3 and cut off. See
how many groups of 4/9 are
in this length.
Student Response
There is one complete
group, and the remaining
part has 2 of the 4 segments
needed to make another
group of 4/9. So the answer
is 1 2/4, which simplifies to
1 ½.
More examples to work are:
5/8  3/16
5/6  7/24
7/8  5/24
¾  5/12
Option: You may want to illustrate an algorithm for division by converting both fractions
to equivalent fractions with a common denominator and then divide the numerators. For
example, 2/3 ÷1/6 equals 4/6 ÷1/6 equals 4÷1 equals 4.
Alternatively, no attempt will be made to develop the artificial algorithm of invert and
multiply.
Extension: Students make up multiplication and division problems that can be done
using the table, and have them trade with a partner to work.
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0
halves
0
thirds
½
1/3
0 fourths
0 sixths
1/6
2/9
2/12
3/8
4/12
4/16
0 eighteenths2/18
4/18
4/24
6/24
6/18
8/24
5/8
4/9
6/16
5/9
6/12
7/12
8/16
8/18
10/24
4/4
4/6
4/8
5/12
3/3
¾
3/6
3/9
3/12
0 sixteenths 2/16
2/24
2/4
2/6
2/8
0 ninths 1/9
0
2/3
¼
0 eighths 1/8
0 twelfths 1/12
2/2
12/24
6/8
6/9
8/12
10/16
10/18
14/24
5/6
12/18
16/24
6/6
7/ 8
7/9
9/12
8/9
10/12
12/16
9/9
11/12
14/16
14/18
18/24
8/8
16/18
20/24
22/24
12/12
16/16
18/18
24/24
twenty-fourths
Fraction Table
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Activity 2: Fraction Circles
Teacher Action
Changing mixed numbers
to fractions and vice
versa:
Teacher will demonstrate as
students work with the
circles to demonstrate
addition and subtraction of
mixed numbers.
Teacher Talk
Student Response
Put out 1 circle and five 1/8
pieces. What does this
represent?
Students use the fraction
circles to model the
problem.
1 5/8
Trade the whole circle for
8ths pieces- how many are
there?
13/8
So we can say 1 5/8 = 13/8
Continue using the fraction
circles to change mixed
numbers to fractions:
1¾
1 3/6
1 3/8
Now put out seven ¼
pieces. What does this
represent?
Rearrange them to form a
circle(s) with pieces left
over. How many circles do
we have?
Lay out five ½ pieces.
What does this represent?
Rearrange them to form a
circle(s) with pieces left
over. How many circles do
we have?
7/4
1¾
5/2
2 1/2
Continue using the fraction
circles to change fractions
to mixed numbers:
8/3
10/4
11/6
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Teacher Action
Adding mixed numbers
Display the problem
1 5/8 + 2 7/8
Teacher will demonstrate as
students work with the
circles.
Display the problem using a
conventional algorithm and
show how this compares to
the concrete model.
Display the problem:
2 ¾ + 1 5/8 + 1 1/2
Teacher Talk
Student Response
Lay out circles to represent Students use the fraction
1 5/8. Now lay out circles circles to model the
to represent 2 7/8. Combine problem.
the similar pieces.
What do we get?
Can we change this for a
mixed number where the
fractional part is not larger
than 1.
Many people record what
we just did by writing:
1 5/8
+2 7/8
3 12/8 = 4 4/8 = 4 1/2
Lay out the appropriate
pieces to represent each
mixed number.
You get 3 12/8.
Trade in 8/8 for a whole
circle and then trade the
remaining 4/8 for larger
pieces.
What do we have to do Make some trades so they
before we can add these have the same denominator.
numbers?
Right, first trade the 8ths,
4ths and half for pieces of
the same size (all eighths).
Write the algorithm on the
board or overhead.
Now we have 2 6/8 + 1 5/8 4 15/8
+ 1 4/8.
What do we get when we
combine
these
mixed
numbers?
How can we simplify this
When we replace 8/8 with a
mixed number?
whole circle, we have 5 7/8.
Again, show how to do the
problem using a
conventional algorithm and
show how this compares to
the concrete model.
Do the following problems
using the fraction circlesfirst trading to a common
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Teacher Action
Teacher Talk
shape of fraction, then
trading for whole circle
when possible, and then
simplifying for larger
fraction pieces when
possible.
Student Response
2 2/3 + 1 5/6 + 1 ½
2 1/6 + 2 ½ + 1 2/3
Teacher Action
Subtracting mixed
numbers
Write the problem 3 ¾ 1 ¼ on the board or
overhead.
Teacher will demonstrate as
students work with the
circles.
Display the problem
3–1¾
Teacher Talk
The object is to start with
3 ¾ and take 1 ¼ away
from it.
Lay out 3 ¾ fraction circles
and remove 1 ¼.
Student Response
Students use the fraction
circles to model the
problem.
What did you get for an
answer?
This time we put out 3
circles and try to take 1 ¾
away from them.
There are no fourths to take
the ¾ from so what could
we do?
1 2/4 or 1 1/2
Lay out 6 circles and ½.
We want to take away 2 ¾.
Students use the fraction
circles to model the
problem.
Since there are no fourths,
what can we do?
Trade the ½ for 2/4
Trade a circle in for 4/4 and
then take away the 1 ¾,
getting 1 ¼.
Now show the problem
using the traditional
algorithm and show how
this concrete model matches
the method.
Display the problem
6 ½ - 2 ¾.
Again show the problem
worked with the traditional
algorithm and show how
this relates to the concrete
model with trading.
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Teacher Action
Teacher Talk
There are still not enough
fourths to take away ¾ so
we can trade a whole circle
in for 4/4. Now we have 5
6/4 and can easily take
away 2 ¾.
What is left?
Student Response
3¾
Do the following problems
using fraction circles:
5 1/3 – 1 5/6
5 ¼ - 2 7/8
6 ½ - 3 5/8
Continue comparing the
circle fraction model to the
traditional algorithm.
Teacher Action
Multiplication cannot be
modeled conveniently
with fraction circles, so
will not be done.
Teacher Talk
Division of Fractions
Display the problem 2  ½
2  ½ means how many
groups of ½ are in 2?
Display the problem 1 ½
 ¾.
This would be easier if we had
halves.
Trade the 2 circles in for half
pieces. How many ½ pieces
are there?
1 ½  ¾ means how many
groups of ¾ are in 1 ½?
Lay out 1 and ½. Begin this
problem by trading the 1 ½ for
fourth pieces- getting 6/4.
How many groups with 3/4 in
each can you make?
Student Response
Students use the fraction
circles to model the
problem.
4
2
Day 2 – Page 17
Missouri Mathematics Academy
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Teacher Action
Display the problem 2 
¾.
Teacher Talk
2  ¾ means how many
groups of ¾ are in 2. First
trade the 2 wholes in for
fourths, since we’re making
groups of ¾.
How many ¾’s can you find?
Student Response
There are 2 and some
extra.
That’s right.
Each group requires 3 1/4th
pieces, and after taking out 2
groups of ¾, you have 2 of the
3 pieces needed to make
another group. So the answer
is 2 2/3.
While doing this problem with
the table, also write it on the
board as:
3 8 3 8/3 8
2
2   
 2
4 4 4
1
3
3
Display the problem
1 5
1 
2 8
1 ½ 5/8 means how many
groups of 5/8 are in 1 ½.
What trade would help us get
started?
Now, we’re trying to find how
many groups of 5/8 are in the
12/8. How many groups of
5/8 can you find?
Trade the 1 ½ for 12/8
2 complete groups and 2
extra of the 5 that I need
for another group
While doing this, write the
problem on the board:
1 5 12 5 12
2
1  
 
2
2 8 8 8 5
5
The algorithm we use for
division could be to find a
common denominator and
divide across. This is similar
to the way we multiply across
for multiplication.
So the answer is 2 2/5
Day 2 – Page 18
Missouri Mathematics Academy
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Teacher Action
Display the problem 1 1/3
 5/6.
Teacher Talk
1 1/3 5/6 means how many
groups of 5/6 are in 1 1/3.
What trade would help us get
started?
How many 5/6 can you find?
Student Response
Trade the
1 1/3for 8/6
There is 1 group of 5/6
with a remaining 3 of the
5 pieces needed to make
another group - so 1 3/5.
Do the following problems
using the fraction circles
1 1/2  7/8
1 3/4  3/4
1 2/3
Day 2 – Page 19
Missouri Mathematics Academy
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Activity 3: Coveralls
Directions:
Play the game in pairs. Give each pair a game sheet and 3 dice as described in the materials.
A player rolls all 3 dice and shades in the value of ONE of the die on one of the 100 squares. (They
choose which die they would like to use.) If all amounts are too large, no part may be shaded. The
winner is the first to EXACTLY fill the square.
Purpose: Recognize the relationship between fractions, decimals, and percents. Compare fractions,
decimals, and percents.
Three dice marked:
.01
2%
1/25
.03
4%
2/25
.09
8%
1/10
.15
16%
1/5
.21
24%
3/10
.33
30%
1/20
After the game is completed, initiate a discussion of what mathematics is involved in the game. Include
in this the strategies students used to determine which of the 3 die to use for their turn. Also discuss
other options for the die—what would be some considerations in determining a particular fraction, such
as 1/3, should be on a die.
Day 2 – Page 20
Missouri Mathematics Academy
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Take a Look at Some Numbers
(Written by Cindy Bryant & sung to the tune of “Take Me Out to the
Ballgame”)
Take a look at some numbers
There are some things you’ll see
Those that name the same amount
Are known as equivalencies
Some name parts of whole numbers
Some may name more than one
So it’s fractions, decimals, and percents
For us to learn
Day 2 – Page 21
Missouri Mathematics Academy
Revised July 2003
Activity 3: Coveralls
PLAYER A
PLAYER B
ROUND 1
ROUND 2
Directions:
Play the game in pairs. Give each pair a game sheet and 3 dice as described in the materials.
A player rolls all 3 dice and shades in the value of ONE of the die on one of the 100 squares.
(They choose which die they would like to use.) If all amounts are too large, no part may be
shaded. The winner is the first to EXACTLY fill the square.
Three dice marked:
.01 .03 .09 .15 .21 .33
2% 4% 8% 16% 24% 30%
1/25 2/25 1/10 1/5 3/10 1/20
Activity 4: Charge Model for Integers
Day 2 – Page 22
Missouri Mathematics Academy
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Teacher Action
Give each person a blank
sheet of paper and a
collection of both kinds of
beans- at least 20 of each.
White beans will be positive
and red beans will be
negative. Put beans on the
paper ONLY IF they are
involved in the problem
being worked. Then the
teacher in front can easily
tell if the person is doing
the problem correctly.
Teacher Talk
Student Response
White beans will be positive
and red beans will be
negative.
Put out 4 white beans –
what is the value?
Students use beans to model
teacher directions.
4
How about 3 red beans-3
what is the value?
Now put out 5 white and 3
red beans. What is the
value?
2
A red and a white bean will
cancel each other out, just
like a positive and a
negative charge do. We call
these “zero pairs”.
Put out 3 white and 7 red
beans. What is the value?
Using the charge model to
add, subtract, and
multiply.
Now make –4 using fewer
beans
now using more beans…
Display ( –2) using 8 beans
Display 3 using 7 beans
Display 5 using 13 beans
Now we’re ready to add,
subtract and multiply using
the charge model.
Addition: Record the
problems and solutions on
the white board or
chalkboard.
2+3
2 + 3 Addition is a putting
together operation. Put 2
white beans and then 3
white beans on the paper.
What is the value?
-4
Responses will vary
Responses will vary
3 white and 5 red
5 white 2 red
9 white 4 red
5
Day 2 – Page 23
Missouri Mathematics Academy
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Teacher Action
-2 + -5
-3 + 5
Do the following problem
similarly: 4 + 5
(See Adding Integers
Transparency)
-2 + -6
(See Adding Integers
Transparency)
-3 + 7
(See Adding Integers
Transparency)
Do the following problems
similarly:
-5 + 2
-10 + 4
3+7
-2 + -1
5 + -5
-3 + 6
2 + -5
Teacher Talk
- 2 + -5. Put on the paper 2
red beans and then 5 red
beans.
What is the value?
Ask for an explanation.
-3 + 5 Put on the paper 3
red beans and 5 white
beans. What is the value?
Who can explain this
answer?
Student Response
-7
2
3 pairs of red and white
beans will equal zero and
can be taken out.
This time we will draw a
model to represent what we
are doing. Negatives
should be drawn as shaded
circles and positives drawn
as open circles.
Draw a model to solve the
first four problems and use
the beans to represent the
rest of the problems.
Day 2 – Page 24
Missouri Mathematics Academy
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Teacher Action
Teacher Talk
Look at the worked
examples on the board (the
ones we’ve just done with
the beans). We’d like to
determine an algorithm
through inductive reasoning
(seeing lots of examples
where we’ve gotten the
answer somehow, and
looking for generalizations
to make). First we’ll just
look at the size of the
number without the sign.
For the first problem (2 +
3) what do we do with 2
and 3 to get 5?
Now the 3rd problem (-3 +
5) what do we do with 3
and 5 to get 2?
Go through the list writing
either add or subtract
depending on which would
have been required to get
the answer (ignoring the
sign for now).
What generalization can
you make about the ones
that required adding?
And then what tells you the
sign for the answer?
Look at the problems where
subtraction was required.
Student Response
Add.
We subtract.
Students mark the list.
Both numbers had the same
sign.
Whatever the sign of the
numbers was.
What do you notice about
the two numbers if the
answer was zero?
They had the same number
of positives as negatives.
What do you notice about
the two numbers if the
answer is positive?
When there are more
positives than negatives (or
when the positive number
has the larger absolute
value).
Day 2 – Page 25
Missouri Mathematics Academy
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Teacher Action
Teacher Talk
What do you notice about
the two numbers if the
answer is negative?
Student Response
When there are more
negatives than positives (or
when the negative number
has the larger absolute
value).
What procedure could you
use for adding a positive
and negative number
together instead of using the
charge particle method?
To get the number part of
the answer, you subtract.
To get the sign, you
determine which has the
larger absolute value.
Day 2 – Page 26
Missouri Mathematics Academy
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Adding Integers Transparency
Activity 4: Charge Particle Method
Source: Focus on Pre-Algebra by Margaret A. Smart
4
+
5
=
9
-2
+
-6
=
-8
-3
+
7
=
4
Day 2 – Page 27
Missouri Mathematics Academy
Revised July 2003
Teacher Action
Subtraction:
Again, record the problems
and solutions on the board.
5–2
(See Subtracting Integers
Transparency)
-4 – (-2)
(See Subtracting Integers
Transparency)
-3 – 5
(See Subtracting Integers
Transparency)
Teacher Talk
We’ll use the “take away”
model for subtraction,
where we put the first
number on the paper and
take the 2nd number away
from it.
Student Response
(Read 5 – 2 as “5 take away
2”.)
To solve 5 – 2 put 5 white
beans on the paper. To
solve the problem we must
take 2 positives away from
a value of 5. Do the 5 white
bean have a value of 5?
Then we can easily take
away two positives.
What value does that leave?
-4 – (-2) means to take
away two negatives from
something that has a value
of -4. How would you
represent that with the
beans?
What value does that leave?
-3 – 5 means to take away
five positives from
something that has a value
of -3.
How is this problem
different from the other
problems we’ve done?
Remember we must begin
with a value of -3 and to
solve the problem we must
take 5 positives away from
something that has a value
of -3.
What do you do in order to
have 5 positives to take
away?
3
Put 4 red beans on the paper
and take away 2 red.
-2
You don’t have any white
beans to take away.
(Some students may think
you could put in five
positives and then take them
away.)
Day 2 – Page 28
Missouri Mathematics Academy
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Teacher Action
To clarify this common
misconception, model the
solution the student has
suggested by placing 5
positives in with the 3
negatives.
4 – (-2)
-3 – (-5)
Teacher Talk
If we put the 5 positives in,
what is the value of our set
before we take the 5
positives away?
Remember we must take 5
positives from something
that has a value of -3 and
when we put in the 5
positives we got a value of
2 which isn’t the same.
How could we modify our
set to include 5 positives
and not change the value of
-3?
Model the problem with
your beans. What is the
solution?
Explain how you would
model this problem with the
beans.
Explain how you would
model this problem with the
beans.
Student Response
2
We can put in as many
“zero pairs” as we want but
we must put in enough to
finish the problem. So 5
zero pairs is enough.
8 red beans, or –8.
4 – (-2) means put out 4
white beans and take away
2 red beans.
2 zero pairs will need to be
added in order to take away
the 2 red beans.
This will leave you with 6
white beans, or 6.
-3 – (-5) means put out 3
red beans and take away 5
red beans.
This time you only need to
add 2 zero pairs of beans.
After removing 5 red, you’ll
have 2 white beans left.
Place the Activity 4:
Transparency A on
overhead and cover
solutions.
Solve these problems using
the beans and record your
solutions
Day 2 – Page 29
Missouri Mathematics Academy
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Teacher Action
Place the Activity 4:
Transparency B on
overhead and cover
solutions while giving
students time to solve the
problems.
Put Activity 4:
Transparency A on top of
transparency B.
Teacher Talk
What are the solutions to
the problems?
Look at these problems.
Can you generate any rules
for subtracting integers?
(Students will generate
numerous rules that have
limited applications.)
It appears that our rules for
subtraction aren’t as
obvious as the addition
rules. So let’s go back and
review some addition
problems.
What are your answers?
Compare red and blue
problems 1 and their
solutions.
Student Response
2 – 6= -4
5 – (-5)= 10
-8 – (-4)= -4
-2 – (-7)= 5
-3 – 4= -7
3 – (-4)= 7
-1 – 9 =-10
Answers may vary.
2 + (-6)= -4
5 + 5= 10
-8 + 4= -4
-2 + 7= 5
-3 + (-4)= -7
3 + 4= 7
-1 + (-9) =-10
One is addition and one is
subtraction but we got the
same answers.
What do you notice?
Continue comparing the
problems and solutions on
the remainder of the page.
Can you develop a rule for
subtraction?
We changed subtraction to
adding the opposite. So –4 –
(-3) becomes –4 + 3 and 2 –
7 becomes 2 + (-7). Then
we use our addition rules.
Day 2 – Page 30
Missouri Mathematics Academy
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Subtracting Integers
Activity 4: Charge Particle Method
Source: Focus on Pre-Algebra by Margaret A. Smart
5
-
2
=
3
-4
-
-2
=
-2
-3
-
5
=
8
Day 2 – Page 31
Missouri Mathematics Academy
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Activity 4: Transparency A
1.
2 – 6 = -4
2.
5 - (-5) = 10
3.
-8 - (-4) = -4
4.
-2 - (-7) = 5
5.
-3 - 4 = -7
6.
3 - (-4) = 7
7.
-1 - 9 = -10
Day 2 – Page 32
Missouri Mathematics Academy
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Activity 4: Transparency A
1.
2 + (-6) = -4
2.
5 + 5 = 10
3.
-8 + 4 = -4
4.
-2 + 7 = 5
5.
-3 + (-4) = -7
6.
3 + 4 = 7
7.
-1 + (-9) = -10
Day 2 – Page 33
Missouri Mathematics Academy
Revised July 2003
Teacher Action
Multiplication
Record the worked
examples on the board or
overhead.
3X5
(See Multiplying Integers
Transparency)
3 X –4
(See Multiplying Integers
Transparency)
-2 X 3
(See Multiplying Integers
Transparency)
Teacher Talk
3 X 5. We’ll use the sets
model for multiplication.
The first number tells how
many sets to put in or take
out, and the 2nd number tells
how many are in each set.
We always start with a
value of zero on the paper
(not necessarily an empty
sheet.) Since the 3 is
positive, 3 X 5 means PUT
IN 3 groups of 5 on the
sheet, or put in 3 groups
with 5 white beans in each
group. You’ll then have 15
white beans, or 15.
3 X –4 means PUT IN 3
groups of –4 each, or 3
groups with 4 red beans in
each group.
How many beans?
How is this problem
different than the other
multiplication problems
we’ve worked?
-2 X 3 means TAKE OUT
two groups with three in
each group because the first
number is negative.
How can you create a set
that has a value of zero
where you can take away
two groups of three?
If you put in more than two
groups of three zero pairs
will it change your answer?
Student Response
Students will follow the
teacher’s modeling.
You’ll have 12 red beans, or
–12.
You have to put in at least
two groups of three zero
pairs.
No, because the zero pairs
you don’t use will cancel
each other out.
Day 2 – Page 34
Missouri Mathematics Academy
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Teacher Action
-2 X -3
(See Multiplying Integers
Transparency)
Teacher places index cards
(magnetic strips on back)
with multiplication
problems written on them
on the board after the
students have solved the
problems.
Teacher Talk
Explain how you would
solve this problem using the
beans. Remember to start
with a set of beans that has
a value of zero.
How would you work this
problem? Remember to
start with a set of beans that
has a value of zero.
Do the following problems
using beans:
2X4
-3 X 5
-1 X -5
2 X –3
-4 X –3
Now look at the worked
problems and sort them in
to groups.
(Note: Students may group
problems in different ways
than the intended groups of
positive times positive,
negative times negative,
positive times negative, and
negative times positive.
The teacher may need to
lead the class in this
direction.)
Student Response
In order to do this, we’ll
need to start with some
beans on the sheet (actually
6 zero pairs of beans in
order to take out 2 groups
with 3 white beans in each).
So begin with 6 ZERO
PAIRS of beans on the
sheet;
this is still equal to zero.
Now you can take out 2
groups with 3 white beans
in each.
In order to do this, we’ll
need to start with some
beans on the sheet (actually
6 zero pairs of beans in
order to take out 2 groups
with 3 red beans in each).
So begin with 6 ZERO
PAIRS of beans on the
sheet; this is still equal to
zero. Now you can take out
2 groups with 3 red beans in
each.
Students go to board and
arrange cards into grouping
and explain their basis for
grouping the cards.
Day 2 – Page 35
Missouri Mathematics Academy
Revised July 2003
Teacher Action
Teacher Talk
Lead a discussion to
develop rules for
multiplication that will
work for all problems.
Student Response
Day 2 – Page 36
Missouri Mathematics Academy
Revised July 2003
Multiplying Integers Transparency
Activity 4: Charged Particles
Source: Focus on Pre-Algebra by Margaret A. Smart
Example: 3 x 5
Example: 3 x -4
Example: -2 x 3
Example: -2 x -3
Put in three groups of 5.
Put in three groups of -4.
3 x 5 = 15
3 x -4 = -12
Put in 2 sets of 3 zero pairs. Take out 2 sets of 3.
-2 x 3 = -6
Put in 2 sets of 3 zero pairs. Take out 2 sets of -3.
-2 x -3 = 6
Day 2 – Page 37
Missouri Mathematics Academy
Revised July 2003
Activity 5: Walking the Number Line for Integers
Teacher Action
In advance, place number
lines on the floor with
numbers -6 to 6.
Draw a number line on the
board with the numbers -6
to 6.
Prepare Job Cards with job
descriptions for each group.
Select three students to
demonstrate walker,
recorder, and checker while
the teacher serves as the
reader.
Teacher Talk
We’re going to model the
way your group is to work
each of these problems.
Student Response
Walker will start at zero
facing the positive
direction.
If a number is positive,
walk forward, if negative,
walk backward, and
subtraction means turn
around.
Our first problem is 5 + -2.
Recorder, write the problem
down.
Walker, start at zero facing
positive direction.
Walk forward 5.
We’re adding -2, so walker
should walk backwards 2.
Where is the walker now?
Checker, is this correct?
Recorder, please record the
answer.
Students trade job cards.
Our second problem is
3 - 4.
Recorder, write the problem
down.
Walker, start at zero facing
positive direction.
Walk forward 3.
Since it’s subtraction, the
walker will turn around and
walk forward 4 because 4 is
a positive number.
Where is the walker now?
Checker, is this correct?
Walker is standing at zero.
Walker walks forward 5.
Walker walks backward 2.
3
Yes. The answer is 3.
Walker is standing at zero.
Walker walks forward 3.
Walker turns around and
walks forward 4.
-1
Yes. The answer is -1.
Day 2 – Page 38
Missouri Mathematics Academy
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Teacher Action
Teacher Talk
Students trade job cards.
Our second problem is
-4 – (-3). Recorder, write
the problem down.
Walker, start at zero facing
positive direction.
Walk backward 4.
Since we are subtracting,
the walker will turn around
and move backward 3 since
3 is a negative number.
Where is the walker now?
Checker, is this correct?
Distribute a set of Job Cards Each group will need pencil
and a set of Question Cards and paper to record their
to each group of four
work.
students.
Student Response
Walker is standing at zero.
Walker walks backward 4.
Walker turns around and
walks backward 3.
-1
Yes. The answer is -1.
Students go to number lines
and work problems in their
groups making sure to
change job roles after every
two problems.
Day 2 – Page 39
Missouri Mathematics Academy
Revised July 2003
Activity 6: Integer Contig
Teacher Action
Distribute game sheets and
colored markers and set of
three colored dice to each
student pair.
Teacher Talk
We’re going to play a game
that will give you an
opportunity to practice and
review integer operations. A
player rolls 3 dice and
decides how to combine the
3 numbers (add, subtract,
multiply, divide, or a
combination of operations)
to determine one of the
numbers on the game board.
This number becomes a
shaded square as a player
colors it with his marker.
The player’s score is the
number of shaded squares
that his new square is
touching, regardless of the
color of the square’s
shading. It may either
touch by a side or touch by
a corner. Play continues
until time is up, and the
player with the larger score
wins. (Note: The first
player will not score on his
first turn.)
Student Response
Day 2 – Page 40
Missouri Mathematics Academy
Revised July 2003
Activity 6: Integer Contig
Game Board
-180
-144
-125
-108
-100
-90
-75
-60
-54
-48
-42
-40
-39
-38
-36
-35
-34
-33
-32
-31
-30
-29
-28
-27
-26
-25
-24
-23
-22
-21
-20
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
40
41
42
45
50
64
72
80
96
120
150
216




2 players.
A player rolls 3 dice and decides how to combine the 3 numbers (add, subtract, multiply, divide,
or a combination of operations) to determine one of the numbers on the game board. This
number becomes a shaded square as the player colors it with his marker. The player’s score is the
number of shaded squares that his new square is touching, regardless of the color of the square’s
shading. It may either touch by a side or touch by a corner.
Play continues until time is up, and the player with the larger score wins. (Note: The first player
will not score on his first turn.)
Use 3 dice with the following integer values: (-1,2, 3,-4,-5, -6)
(1, -2, 3, 4, -5, 6)
(1, -2, -3, -4, -5, 6)
Day 2 – Page 41
Missouri Mathematics Academy
Revised July 2003
Activity 7: Fresno
Teacher Action
Distribute Fresno game
sheet.
Place Fresno transparency
on overhead and cover up.
Have three 10-sided dice
ready to roll.
Uncover transparency to
show solution.
Teacher Talk
Student Response
This game will give you the
opportunity to work with
rational numbers.
I have three 10-sided dice
with the numbers 0 – 9 on
each die. I’m going to roll
and tell you what the three
numbers are.
You will write them in the
first three boxes on your
game sheet.
Suppose we have rolled 5,
3, and 2. The goal is to use
these three numbers and any
mathematical operations to
generate the largest single
digit possible.
You must use each number
rolled, and it can only be
used once. You can use any
combination of symbols and
operations.
An explanation of how they
Obviously, the largest
got they got 9.
single digit is 9.
Can you find a way to make
9 with these 3 digits?
Here’s how I got 9.
Before we roll again, you
must place your 9 in the box
you choose below the first
game.
Your goal is to generate the
largest 5-digit number
possible.
What is the largest 5-digit
99,999
number you can make?
Why would you want to put I might not be able to get
your 9 in the first box?
another 9.
This 9 must be placed
before I roll again.
Now, we’ll start the actual
Day 2 – Page 42
Missouri Mathematics Academy
Revised July 2003
Teacher Action
Roll the dice for each of the
5 rows, recording their
calculations, and entering
the resulting digit in their
chosen location.
Teacher Talk
game with me rolling the
dice during the game.
We’ve completed our first
round. Who got the largest
number?
(Note: You may want to
give the winner of round a
chocolate coin or Hershey
Nugget.)
Before moving to round 2
we have one more thing you
have to do. You must select
3 of your digits from your
round 1 number to record in
the final number boxes at
the bottom of your sheet.
Remember, your goal is to
generate the largest 9-digit
number possible.
(Note: Continue play until
the game is over.)
Student Response
Students should report and
give proof of their answer.
(Whoever has the highest
score must give proof.)
Students select and place
their digits in the final box.
Day 2 – Page 43
Missouri Mathematics Academy
Revised July 2003
(Diggin’ for the largest digits)
Activity 7: Overhead Transparency
Name: _________________
5
3
2
(5 -2) x 3
9
9
Day 2 – Page 44
Missouri Mathematics Academy
Revised July 2003
(Diggin’ for the largest digits)
Name: _________________
The Final Number is …
Day 2 – Page 45
Missouri Mathematics Academy
Revised July 2003
Diverse Learners
These lessons employ a variety of activities that would meet the needs of a diverse
population of students. Kinesthetic learners would be provided the opportunity to learn while
moving around during the engager and while walking the number line. Visual learners would be
given the opportunity to use visual processing skills while using the fraction table, fraction
circles and beans for integers.
Students who prefer linear models should find “Stuck on Rational Numbers”, Fraction
Strips, and Walking the Number Line engaging. Conversely, students who are more comfortable
with area models will like the Fraction Circles, Coveralls, Integer Contig, and visual
representations for the Charge Model for Integers.
Use of cooperative learning groups aids students who learn through oral communication.
Cooperative groups also help relieve the level of anxiety in learning and allow for better
acceptance of all students as they participate in various roles. Using job descriptions that change
throughout the activities provides a variety of experiences and challenges while increasing
student involvement and enhancing opportunities to make mathematical connections. As students
are encouraged to verbalize an answer during group work they will have to internalize the
concept. Explaining it to someone else will afford them the opportunity to refine their thoughts.
Assessment
During the engager activity teachers would be able to observe whether students were able
to correctly order the rational values. During use of the fraction table and fraction circles,
various students could be asked to explain how they arrived at an answer using the
manipulatives.
After playing Coveralls, students could be given a possible game scenario where 3
numbers were provided. The student could decide which choice to make and how many squares
to shade.
Day 2 – Page 46
Missouri Mathematics Academy
Revised July 2003