CAMMP 2015
College of Education
Reading and Elementary Education
Names:
Jonathan Lin
Cary Dufresne
Connie Ortiz
Integrated, Thematic Problem Solving Unit: Gifts from Ancient Greek Culture
Grade Level: 6
Arithmetic Topic (operation): Pre-Algebra
Day 1
Hierarchy Step #1
Understanding variables and constants.
Theme Problem #1
There are 10 baklavas in the market. Some of them were stolen by birds when Mercury, the
patron god of commerce, was not looking. How many were left?
Summary Table
Action Language: 10 takeaway what is 6?
Open Number Sentence: 10 – x = 6
Manipulative: Cuisenaire Rods
Quantitative Solution: 4
Referential Meaning: baklavas stolen
Concrete Manipulative
Cuisenaire Rods
Theme Problem #2
Socrates went to the market in Athens and bought 3 equally priced scrolls for 12
drachma, the ancient Greek currency, in total. How much did each scroll cost?
Summary Table
Action Language: N/A
Open Number Sentence: 3x = 12
Manipulative: Hands-On Equations
Quantitative Solution: 4
Referential Meaning: drachma
Concrete Manipulative
Hands-On Equations
Student Problems
Determine the variables and constants in the following.
1. x = 5
2. 10 = 2x
3. 15 = 3x
4. 2 + 3 + w = x
5. y + 3 = 293,819,305 + z
Student Problems Answer Key
1. Constant: 5
Variable: x
2. Constant: x
Variable: x
3. Constant: 15
Variable: x
4. Constant: 2 & 3
Variables: w & x
5. Constant: 3 & 293,819,305
Variables: y & z
Hierarchy Step #2
Modeling balanced equations (including those with variables on both sides).
Theme Problem #1
Zeus has 3 bags of gold coins. Each bag had the same number of gold coins in it.
Poseidon had a bag with 10 gold coins and another bag with the same amount as each of
Zeus’s bags. If both gods have the same number of gold coins, how many gold coins were in
each bag?
Summary Table
Open Number Sentence: 3x = 10 + x
Manipulative: Hands-On Equations
Quantitative Solution: 5
Referential Meaning: gold coins in each bag
Concrete Manipulative
Hands-On Equations
Representational Manipulative
Open Number Sentence
Student Problems
1. Help Themis balance the scales of Justice:
4x = 2x + 6
2. Aristotle was visiting Rome and he bought 6 gladiator tickets and one $1 bunch of
grapes for the same price as 3 gladiator tickets and ten $1 bunches of grapes. How
much does each gladiator ticket cost?
3. Help Themis balance the scales of Justice:
3x + 5 = 4x
4. Plato had several students. His friend asked him how many students he had and Plato
responded, “I have many students. Find 4 times a number, increased by 2, which is
the same as three times a number increased by 9. That number is how many students
I have.”
Student Problems Answer Key
1.
2.
3.
4.
Hierarchy Step #3
Manipulating models to obtain a solution.
Theme Problem #1
Zeus has 3 bags of gold coins. Each bag had the same number of gold coins in it. His
brother, Poseidon, had a bag with 10 gold coins and another bag with the same amount as each
of Zeus’s bags. If both gods have the same number of gold coins, how many gold coins were
in each bag?
Summary Table
Open Number Sentence: 3x = 10 + x
Manipulative: Hands-On Equations
Quantitative Solution: 5
Referential Meaning: gold coins in each bag
Concrete Manipulative
Hands-On Equations
Representational Manipulative
Open Number Sentence
Student Problems
1. Help Themis balance the scales of Justice by finding x:
4x = 2x + 6
2. Aristotle was visiting Rome and he bought 6 gladiator tickets and one $1 bunch of
grapes for the same price as 3 gladiator tickets and ten $1 bunches of grapes. How
much does each gladiator ticket cost?
3. Help Themis balance the scales of Justice by finding x:
3x + 5 = 4x
4. Plato had several students. His friend asked him how many students he had and Plato
responded, “I have many students. Find 4 times a number, increased by 2, which is
the same as three times a number increased by 9. That number is how many students
I have.”
Student Problems Answer Key
1. Quantitative Solution: 3
Referential Meaning: N/A
2. Open Number Sentence: 6x + 1 = 3x + 10
Quantitative Solution: 3
Referential Meaning: cost per gladiator ticket
3. Quantitative Solution: 5
Referential Meaning: N/A
4. Open Number Sentence: 4x + 2 = 3x + 9
Quantitative Solution: 7
Referential Meaning: students
Transitional Activities
Hands-On Equations Class Worksheets Lesson #1
Hands-On Equations Class Worksheets Lesson #2
Day 2
Hierarchy Step #4
Check answers with modeled equations by substituting variables with values.
Theme Problem #1
One hen laid twice as many golden eggs as the younger hen. This gave the farmer just
enough golden eggs to keep 9 and sell 9. How many golden eggs did each hen lay?
Summary Table
Open Number Sentence: 2x + x = 18
Manipulative: Hands-On Equations
Quantitative Solution: 6
Referential Meaning: golden eggs
Concrete Manipulative
Hands-On Equations
Representational Manipulative
Open Number Sentence
–
Teachers demonstrated substitution of x with 6
and work through the problem so students
understand how to check their answers.
Student Problems
1. Help Themis balance the scales of Justice by finding x:
2(4x + 2x) = x + x + 20
2. Athena put 3 boxes of snakes in a cage with 10 other snakes. She also put 5 boxes of
scorpions in the next cage. If all the boxes have the same number of critters and the
cages now contain the same number of critters, how many are each in box?
3. Help Themis balance the scales of Justice by finding x:
3x = x + 20
4. Demosthenes asked, “If Socrates is 3 years older than twice the age of his little
brother, Question Mark, and their ages add up to 15, how old are each of them?”
Student Problems Answer Key
1. Quantitative Solution: 2
Referential Meaning: N/A
2. Open Number Sentence: 3x + 10 = 5x
Quantitative Solution: 5
Referential Meaning: critters per box
3. Quantitative Solution: 10
Referential Meaning: N/A
4. Open Number Sentence: 2x + 3 + x = 15
Quantitative Solution: 4 & 11
Referential Meaning: Question Mark’s age & Socrates’s age
Transitional Activity
Hands-On Equations Class Worksheets Lesson #3
Hands-On Equations Class Worksheets Lesson #4
Hierarchy Step #5
Manipulating whole numbers on both sides of a balanced equation.
Theme Problem #1
Aristotle bought one toga and received 4 drachma in change. His student bought two
togas and received 3 drachma in change. If they each gave the merchant the same amount of
drachmas for the purchase, how much did each toga cost? How much did each give the
merchant?
Summary Table
Open Number Sentence: x + 4 = 2x + 3
Manipulative: Hands-On Equations
Quantitative Solution: 1 & 5
Referential Meaning: cost per toga & drachma given to merchant
Concrete Manipulative
Hands-On Equations
Representational Manipulative
Open Number Sentence
Student Problems
1. Help Themis balance the scales of Justice by finding x:
x + 2 + 2x = x + 10
2. Medusa guessed a number. Heracles guessed a number that was double the sum of 3
and the number guessed by Medusa. If the sum of numbers guessed by Medusa and
Heracles is 18, what are the numbers?
3. Help Themis balance the scales of Justice by finding x:
5x + x + 5 = x + 25
4. Odysseus found a riddle that will help him get home. It says, ‘If you can find these
three numbers, you will be 18 days closer to being home to your wife. Three
consecutive numbers add up to 18. Find these numbers.”
Student Problems Answer Key
1. Quantitative Solution: 2
Referential Meaning: N/A
2. Open Number Sentence: 2(3 + x) + x = 18
Quantitative Solution: 4 & 14
Referential Meaning: number guessed by Medusa & number guessed by Heracles
3. Quantitative Solution: 4
Referential Meaning: N/A
4. Open Number Sentence: x + (x + 1) + (x + 2) = 18
Quantitative Solution: 5 & 6 & 7
Referential Meaning: consecutive numbers
Transitional Activity
Hands-On Equations Class Worksheets Lesson #5
Hands-On Equations Class Worksheets Lesson #6
Codebreakers
Day 3
Hierarchy Step #6
Understanding “opposite” numbers & variables.
Theme Problem #1
Ares moved the football to the + 30 yard line. On the next play, the quarterback was
“sacked” for a loss of 20 yards. Where is the football now?
Summary Table
Open Number Sentence: +30 - 20 =
Quantitative Solution = + 10
Referential Meaning: 50 yard line
Concrete Manipulative
Football Field
Theme Problem #2
A bunch of Greek gods and goddesses decided to time travel to the time American
football was invented and played.
Hermes started at the 40 yard line with the football. He was then penalized 15 yards
for cheating with his winged boots, gained 5 yards on a running play, and then lost 10 yards on
a running play. After these three plays, what is the position of the football?
Summary Table
Open Number Sentence: 40 – 15 + 5 – 10 =
Quantitative Solution = + 20
Referential Meaning: 50 yard line
Concrete Manipulative
Football Field
Transitional Activities
Football Game (Constructing Number Sense pg. 490)
Materials Required: Position Markers, Football Field
Rules: The player with the football draws three cards which are played one at a time to
earn a first down (+ 10 yards). If a first down is gained, three more cards are
drawn, and the pattern is repeated. If a first down is not gained, the player draws
the next available number card and multiplies its value by 4 to determine the
distance traveled with a “punt” to the other player.
Card Values: red = negative face value
black = positive face value
any king = 40 yard pass completion
any queen = intercepted pass
any jack = fumble recovered by other team
all face cards are ignored on fourth “down” (draw)
number cards = yards gained or lost (red or black)
Day 4
Hierarchy Step #7
Introducing ★ and “negative” numbers to modeling balanced equations.
Theme Problem #1
When a number is diminished by its opposite, the result is 12. Find the number.
Summary Table
Open Number Sentence: x – ★ = 12
Manipulative: Hands-On Equations
Quantitative Solution: 6
Referential Meaning: N/A
Concrete Manipulative
Hands-On Equations
Representational Manipulative
Number Line
Student Problems
1. If ★ = 3, evaluate: ★ + 2x + 2★ + 7
2. If x = -5, evaluate: 2x + 4★ - ★ + 1
3. Find x and ★: 3★ = x + ★ + 9
4. Find x and ★: 2★ + 3★ + 1 = ★ + 11 + 3★ - ★
Student Problems Answer Key
1. Quantitative Solution: 10
Referential Meaning: N/A
2. Quantitative Solution: 6
Referential Meaning: N/A
3. Quantitative Solution: x = -3, ★ = 3
Referential Meaning: N/A
4. Quantitative Solution: x = -5, ★ = 5
Referential Meaning: N/A
Transitional Activities
Hands-On Equations Class Worksheets Lesson #10
Hands-On Equations Class Worksheets Lesson #11
Codebreakers
Day 5
Hierarchy Step #8
Applying knowledge and skills to advanced problems.
Theme Problem #1
Themis is thinking about a question while she wonders whether it is balanced on her
golden scales of Justice. There are three consecutive numbers on her scale. Five times the
opposite of the first number, increased by 20, will result in the third number. What were the
numbers that Themis was trying to balance?
Summary Table
Open Number Sentence: -5x + 20 = x + 2
Manipulative: Hands-On Equations
Quantitative Solution: 3 & 4 & 5
Referential Meaning: numbers Themis was trying to balance
Concrete Manipulative
Hands-On Equations
Student Problems
Themis is trying to figure out a bunch of numbers on her scales of Justice.
1. The sum of two consecutive integers is 15. Find the numbers.
5. The sum of two numbers is 18. Twice the smaller number decreased by 3 equals the
larger number. What are the two numbers?
6. The sum of three consecutive even numbers is 48. What is the smallest of these
numbers?
7. The sum of five consecutive even numbers is 50. What are the numbers?
Student Problems Answer Key
1. Open Number Sentence: x + x + 1 = 15
Quantitative Solution: 7 & 8
Referential Meaning: numbers on scale
2. Open Number Sentence: x + 2x - 3 = 18
Quantitative Solution: 7 & 11
Referential Meaning: numbers on scale
3. Open Number Sentence: x + x + 2 + x + 4 = 48
Quantitative Solution: 14
Referential Meaning: smallest number on scale
4. Open Number Sentence: x + x + 2 + x + 4 + x + 6 + x + 8 = 50
Quantitative Solution: 6 & 8 & 10 & 12 & 14
Referential Meaning: numbers on scale
Transitional Activities
Codebreakers
Day 6
Hierarchy Step #9
Finding values with simple simultaneous equations.
Theme Problem #1
Plato thinks that you can put equations in equations and find values for everything. He
writes down:
2x = y + 9
y=x+3
Is he correct? If so, can you find a value for x and y?
Summary Table
Open Number Sentence: 2x = x + 3 + 9
Manipulative: Hands-On Equations
Quantitative Solution: 12 & 15
Referential Meaning: numbers Plato was thinking about
Concrete Manipulative
Hands-On Equations
Student Problems
Aristotle does not agree with Plato. Prove that Plato correct by finding a value for x and y.
1. 3x = y + 2
y=x
2. 4x = 18 + y
y=9+x
3. y = 9x + 3
x=2
4. 11y = 10x + 28
x=2+y
Student Problems Answer Key
1. Open Number Sentence: 3x = x + 2
Quantitative Solution: 1 & 1
Referential Meaning: x & y
2. Open Number Sentence: 4x = 18 + 9 + x
Quantitative Solution: 9 & 18
Referential Meaning: x & y
3. Open Number Sentence: y = 18 + 3
Quantitative Solution: 2 & 21
Referential Meaning: x & y
4. Open Number Sentence: 11y = 10(2 + y) + 28
Quantitative Solution: 50 & 48
Referential Meaning: x & y
Transitional Activities
Codebreakers