Poly needs help learning more about addition and subtraction. Will you help Poly represent the following situations?

Carlos has 3 pencils.

Jaz gives him 4 more.

How many pencils does Carlos have now?

Jaz has 6 erasers.

How many erasers does Carlos need to give her so she will have 11?

Jaz has 12 pencils.

She gave Carlos 4.

How many does she have now?

Carlos had some erasers.

He gave 5 to Jaz. Now he has 8 left.

How many erasers did Carlos start with?

Tyrone has 3 red flowers and 8 blue flowers.

How many flowers does he have?

Kyle has 15 flowers.

7 are blue and the rest are red.

How many of Kyle’s flowers are red?

Tyrone has 16 flowers.

Kyle has 9 flowers.

How many more flowers does Tyrone has than Kyle?

Kyle has 3 flowers.

Tyrone has 8 more than Kyle.

How many flowers does Tyrone have?

Can you help Poly learn to divide?

Can you add 5’s to make 25?

How many 5’s did you use?

Can you take away 5’s from 25 and end with 0?

How many 5’s did you take away?

Can you subtract 15’s from 60 and reach 0?

How many 15’s did you subtract?

Can you subtract 6’s from 25 and reach 0?

Do you reach 0? If not how close did you get?

Can you add 5’s and reach 26?

Do you reach 0? If not how close did you get?

How many 5’s did you add? What is the gap?

Can you add 5’s and make 27?

Did you reach 27? If not how close did you get?

How many 5’s did you need? What is the remainder?

Use the right symbol:

less than <, or, greater than >

...and make the chip True!

Add maximum number of 7s and make the sum less than 29.

Please note how many 7s are needed!

Subtract 3s from 10 as much as you can and keep the remainder positive and less than 3.

Please note how many 3s can you use and what is the remainder.

Subtract 13s from 66 as much as possible and the remainder be positive and less than 13.

Please note how many 13s can you use.

What is the remainder?

Subtract 3s from 10 as much as possible and put suitable remainder on the stack 2 such that chip will be True!

Add as much as possible 3s on the stack 2 and add remainder on the stack 3 such that make the chip True!

Please note how many 3s you have used and what is the remainder?

Can you help Poly connect money and percentages?

One poly-buck is 100 poly-cents.

There are four kinds of coins

• one-poly-cent coin

• five-poly-cents coin

• ten-poly-cent coin

• twenty-five-poly-cents coin

Poly wants to know what coins Poly could use to make 25% of 2 poly-bucks?

Help Poly figure out how much 25% of 2 poly-bucks is.

Poly wants to know what coins Poly could use to make 25% of 2 poly-bucks?

Find one way of making 25% of 2 poly-bucks with coins.

Poly wants to know what coins Poly could use to make 25% of 2 poly-bucks?

Find another way of making 25% of 2 poly-bucks with coins.

Poly has been trying to save some money. Poly has decided to save 15% of what Poly makes. Poly helped a neighbor mod their computer and got 24 poly-bucks for it.

Help Poly figure out the amount of money Poly wants to put in his savings jar.

Poly has been trying to save some money. Poly has decided to save 15% of what Poly makes. Poly helped a neighbor mod their computer and got 24 poly-bucks for it.

What coins and bills could Poly put in his savings jar?

Find one way of making that amount of money with bills and coins.

One poly-buck is 100 poly-cents.

There are four kinds of coins

• one-poly-cent coin

• five-poly-cents coin

• ten-poly-cent coin

• twenty-five-poly-cents coin

Imagine you have stacks of one-poly-cent coins in these groupings:

3
5
6
7
9

• Each grouping has some one-poly-cent coins in it.

• The stacks of one-poly-cent coins inside each unique grouping are equal.

• Each grouping can have different size stacks of one-poly-cent coins than the other groupings

What would be the size of the stack of one-poly-cent coins inside each grouping if the sum of all the one-poly-cent coins is 1 poly-buck?

Each of the chip 1 and chip 2 in the machine is an attempt by Poly to answer the question above, but he didn’t finish them.

Your task is to go finish the machines so they are good representations of the problem. In each of the machines, find a different solution to the number of coins in each stack.

How many different solutions can you find?

How many different solutions are there?

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The circumference of a circle is slightly more than three times as long as its diameter. This exact ratio is called π, or Pi.

C/d = π = 3.14159265358...

If a circular pie has a diameter of 10 centimeters, what is its area?

If a circular pie has a perimeter of X centimeters, what is its area?

I have a cool circular pie cutter that has a radius of 5 cm. My pie has a radius of 10 cm. If I cut the pie using the pie cutter and give the piece to my friend, how much pie will I have left?

I cut my favorite apple pie into X sweet pieces and Y sour pieces. If I randomly choose a piece to eat, what is the probability it is sweet?

An apple pie has a radius of 10 centimeters. I take a knife and cut it like below, with 3 cuts that form an equilateral triangle. What is the area of the pie outside the cuts?

Sara bought 3 boxes of pencils and ended up with 18 pencils.

How many pencils would she have purchased if she had bought 5 boxes of pencils?

How many pencils would she have purchased if she had bought 5 boxes of pencils?

Sara bought 3 boxes of pencils and ended up with 18 pencils.

How many boxes did she buy if she bought 42 pencils?

How many boxes did she buy if she bought 42 pencils?

44 students are studying in the 3rd grade at Pacifica elementary school.

If there are 3 calculators for every 4 students, how many calculators are there in the whole class?

If there are 3 calculators for every 4 students, how many calculators are there in the whole class?

On a map, ⅓ inches represents 5 miles.

If San Jose and Santa Barbara are 18 inches apart on the map, what is the actual distance between them?

If San Jose and Santa Barbara are 18 inches apart on the map, what is the actual distance between them?

If Hana drives with a speed of 45 mph, it will take 2 hours to reach home.

What speed should she go to cover the same distance in 1.5 hours?

What speed should she go to cover the same distance in 1.5 hours?

A car spends 4.5 Gallon of gas per 100 miles inside the city and 3.3 Gallons per 100 miles in highway.

How much total gas is needed if the car travels 54 miles inside the city and 180 miles in highway?

How much total gas is needed if the car travels 54 miles inside the city and 180 miles in highway?

Last year 50 students tried out for the basketball team, this year 70 students came to the tryouts.

By what percentage did the number of students trying out increase?

The number of students changed 20 (70-50) out of 50, so how much is that out of 100?

By what percentage did the number of students trying out increase?

The number of students changed 20 (70-50) out of 50, so how much is that out of 100?

The temperature in Sunnyvale fell from 75 to 60 degrees on Sunday.

What is the percentage change in temperature?

What is the percentage change in temperature?

In her first month of soccer training, Sara made 80 shots. The following month she made 90.

By what percentage did she improve?

By what percentage did she improve?

The Apple Store in Palo Alto had 200 customers yesterday but 240 customers today.

What is the percentage increase in customers?

What is the percentage increase in customers?

Last week, the Apple Store in Palo Alto had 240 customers but 200 customers on the next day.

What is the percentage of change?

Is your answer different than that for the previous chip?

Which answer was greater?

Why?

What is the percentage of change?

Is your answer different than that for the previous chip?

Which answer was greater?

Why?

The Google stock started today at 1091.44 at Nasdaq and ended at 1099.90

What is the percentage of change?

What is the percentage of change?

Make a calculator for NASDAQ that takes the stock price of any company in opening and closing of the market in a day, and calculates the percentage of change from the beginning to end of the day.

How does it show if the stock went up or down?

How does it show if the stock went up or down?

Hana bought a coat on sale with a 20% discount. The original price of the coat was $85.

How much did the coat cost after the discount?

How much did the coat cost after the discount?

Hana’s coat was $85. She received a 20% discount and then paid 8% tax on the discounted price.

How much did she pay in total?

How much did she pay in total?

Hana noticed that the discount was applied before calculating the tax. She wondered what would happen if it had been done the other way around.

In the previous chip we found that if we add tax after the discount Hana should pay $73.44.

Now find out how much she would pay if we added the 8% tax before taking the 20% discount.

In the previous chip we found that if we add tax after the discount Hana should pay $73.44.

Now find out how much she would pay if we added the 8% tax before taking the 20% discount.

Lia also bought a winter coat. The original price was $90 but she only paid $72, she was telling Hana that she has had a better deal than the 20% discount Hana got earlier. But Hana didn’t believe her.

Can you determine what was the percentage discount Lia got?

Can you determine what was the percentage discount Lia got?

Hana also bought her mother a coat. Her mother’s coat had a 25% discount and she paid $105 for it.

Can you find what the original price was?

Can you find what the original price was?

Hana went for shopping to a mall. She spent 80% of her money in the first store, 80% of remaining money in the second store, and she had $12 left.

How much money did she have when she entered the first store?

How much money did she have when she entered the first store?

On this chip show that:

x% of y = y% of x, for any x and y.

Test your chip for x = 8 and y = 50.

Internet Shopping Bill

Input:

Input:

x original cost

y shipping charge

z tax

y shipping charge

z tax

Both the shipping cost and the tax are percentages of the original cost.

Make a machine to calculate the total bill with these 3 inputs.

Test with x=56, y=8%, z=15%

Make a machine to calculate the total bill with these 3 inputs.

Test with x=56, y=8%, z=15%

Positive and Negative Numbers

When you run poly, a number will fall on top of your stack.

Create a machine that will turn that number into its opposite.

5 <--(opposite number)--> -5

Create a machine that will turn that number into its opposite.

5 <--(opposite number)--> -5

Sara receives some number of polybucks (which will fall on your stack twice when you run poly) and has to pay a bill of that same number.

How much will she have left?

How much will she have left?

An elevator starts at level 2, goes up 4 levels, and down 4 levels.

Where does it end?

What does this tell you about how 4 and -4 relate to each other?

Where does it end?

What does this tell you about how 4 and -4 relate to each other?

A robot started at 50 meters from the door. It moved 5 more meters away from the door, then 50 meters towards the door, and finally 9 more meters away from the door.

How far from the door did it end up?

What does this tell you about how 50 and -50 relate to each other?

How far from the door did it end up?

What does this tell you about how 50 and -50 relate to each other?

TESLA stock started at X, went down by Y, and went down again by Z.

Where did it end up?

Test for X=243, Y=11 and Z= 58

Where did it end up?

Test for X=243, Y=11 and Z= 58

Temperature Problems

On a cold day, the temperature at 6:00 am was -5°C, and at 10:00 am the thermometer was showing 0°C.

How much did the temperature increase?

How much did the temperature increase?

The next day at 6:00 am it was -4°C and at 8:00 am it was -1°C.

How much did the temperature rise?

How much did the temperature rise?

On the same day, the temperature went up by 5°C in the next two hours.

What was the temperature at 10:00 am?

(Reminder: It had been -4°C at 6:00 am and -1°C at 8:00 am)

Can you draw a graph of the temperature that day between 6:00 am and 10:00 am?

What do you predict the temperature will be at 12:00 pm?

What was the temperature at 10:00 am?

(Reminder: It had been -4°C at 6:00 am and -1°C at 8:00 am)

Can you draw a graph of the temperature that day between 6:00 am and 10:00 am?

What do you predict the temperature will be at 12:00 pm?

Kai wanted to keep track of the temperature change. The thermometer showed -13°C in the morning. Next time Kai looked at it the temperature increased by 5°C and then the following time it had increased by 1°C. When Kai looked at it for the third time, it had decreased by 3°C and then decreased by 2°C more before they went to bed.

What was the temperature the last time Kai looked at the thermometer?

Do you have enough information to graph Kai’s temperature measurements throughout the day?

What time do you think his highest and lowest measurements occurred?

What was the temperature the last time Kai looked at the thermometer?

Do you have enough information to graph Kai’s temperature measurements throughout the day?

What time do you think his highest and lowest measurements occurred?

The next day Kai decided to track the temperature change again. This time they did it every 4 hours starting at 6:00 am. At that time the thermometer showed -12 degrees Celsius. For their next measurement the temperature had risen by 4 degrees and on the one after that it had increased by 2 more degrees. For this fourth measurement, the temperature had dropped by 5 degrees and on their last one it had fallen 3 additional degrees.

What was the temperature on Kai’s last measurement?

Do you have enough information to graph Kai’s temperature measurements throughout the day?

Do you see a pattern in both days’ measurements?

What was the temperature on Kai’s last measurement?

Do you have enough information to graph Kai’s temperature measurements throughout the day?

Do you see a pattern in both days’ measurements?

Can you calculate the percentage of temperature decrease if it changed from -4 °C to -6 °C?

Students play the game Target 20, in which players roll four dice and use operations to combine the results to get as close to 20 as they can.

Students are challenged to think flexibility about using operations and how choosing different operations can led to different outcomes.

Students are challenged to think flexibility about using operations and how choosing different operations can led to different outcomes.

Die Roll: 1 4 1 3, make 20.

Die Roll: 6 4 1 1, make 20.

Die Roll: 5 6 1 2, make 20.

Die Roll: 5 3 2 4, make 20.

Die Roll: 6 3 1 1, make 20.

Die Roll: 2 4 3 6, make 20.

Students are challenged to think flexibility about using operations and how choosing different operations can led to different outcomes.

Die Roll: 5 3 4 2, make 20

Die Roll: 6 2 2 3, make 20.

Die Roll: 3 3 2 5, make 20.

Die Roll: 2 4 6 1, make 20.

Die Roll: 6 3 4 3, make 20.

Die Roll: 5 2 5 6, make 20.

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For any polyhedron (3D polygon with straight edges), we can define:

a = The number of vertices in the polyhedron.

b = The number of edges.

c = The number of faces (sides)

The Euler Characteristic is given by a-b+c

Find the Euler Characteristic for a Tetrahedron, shown here. Set the values for a, b, and c in the first stack by putting the appropriate value on top of each variable "set" block. Then, use the second stack to form the expression of a, b, and c that gives the Euler Characteristic.

What is the Euler Characteristic for a cube? Hint: You might have to combine numbers to make the values you need!

What is the Euler Characteristic of an octahedron?

What is the Euler Characteristic of an dodecahedron?

What is the Euler Characteristic of an icosahedron?

What pattern do you notice about all of these Euler Characteristics?
Do you think this pattern is always true?

What happens to the Euler Characteristic of a cube if we add an edge on a face, as below?
Determine the new Euler Characteristic of the cube.

What happens if we add an extra point on the cube in the middle of an edge?
Determine the new Euler Characteristic of the cube.

Hint: Drag from the bottom of the second stack to create the orange diamond block that runs the second stack!

What is the Euler Characteristic of a cube with a hole in the middle, shown here?

What do you notice now about the Euler Characteristic?

If h is the number of holes in the object, can you make a conjecture about the Euler Characteristic in terms of h? Try objects with more holes to confirm your suspicions!

— Problems like this are extremely important to mathematicians, since they can give intuitions about objects we can't see in our universe.

— Maryam Mirzakhani was one the great mathematicians who worked on the study of geometry of mathematical objects. She was the first woman who received the Fields medal (the world's most prestigious math award) in 2014 for her work on these topics.

— Maryam passed away in 2017 due to cancer at the age of 40 in 2017. Her birthday, May 12, is recognized as Women in Mathematics Day.

Maryam Mirzakhani