When you give $\unicode{x00a9}$ two numbers, it gives you the third number in the addition/subtraction pattern. For example, $\unicode{x00a9}\left\{ {21,23} \right\} = 25$ (going up by $2$’s), and $©\left\{ {95,90} \right\} = 85$ (going down by $5$’s).

Find $\unicode{x00a9} \left\{ {6,11} \right\}$ and $\unicode{x00a9} \left\{ {11,6} \right\}$.

Find $\unicode{x00a9}\left\{ {11,\unicode{x00a9}\left\{ {6,11} \right\}} \right\}$.

Write an equation for $\unicode{x00a9}\left\{ {m,n} \right\}$.

When you give $\unicode{x24}$ two numbers, it gives you the third number in the multiplication/ division pattern. For example, $\unicode{x24}\{3,15\} = 75$, and $\unicode{x24}\{48,24\} = 12.$

Find $\unicode{x24}\{2,3\}$, $\unicode{x24}\{12,13\}$, and $\unicode{x24}\{102,103\}.$

What is $\unicode{x24}\{2009,1\}$?

If $\unicode{x24}\{x,2\} = \unicode{x24}\{3,1\}$, then what is $x$?

The symbol $¤$ takes a single number, squares it, and then subtracts 4.

What is $¤(6)?$

Can you ever get a negative answer for $¤(x)?$ Why or why not?

Find an $x$ so that $¤(x)$ is divisible by 5.

Let the symbol $\unicode{xa5}$ mean: Add up the two numbers, then take that answer and subtract it from the product of the two numbers. What is $5\unicode{xa5}8$? $8 \unicode{xa5} 5$?

Look back at the problem above. Do you think the same thing would happen for any pair of numbers, if you used the same symbol? Explain your answer.

Fergie claims that each of the following symbols has the commutative property. Examine each of his claims.

$x \Downarrow y$ means add 1 to $y$, multiply that answer by $x$, and then subtract $x$.

Take two whole numbers $x$ and $y$. To do $x \% y$, you divide $x$ by 2, round down if it’s not a whole number, and then multiply by $y$.

To calculate $x\unicode{xa3} y$, imagine that you walk $x$ miles east and then $y$ miles northeast. $x\unicode{xa3}y$ is how far away you end up from your starting point.

$\unicode{xbf}$ works by adding up the two numbers, multiplying that by the first number, and then adding the square of the second number.

$\unicode{xa2}$ takes two numbers. You reverse the first number (for instance, 513 becomes 315); one-digit numbers stay the same), then add the reversed number to the second number, and finally add up the digits of your answer.

Let $x \unicode{x2605} y = x^2-y^2$. For example, $4 \unicode{x2605} 3 = 16-9 = 7$. True or false: $x \unicode{x2605} y$ always equals the sum of the two numbers — for example, $4 \unicode{x2605} 3 = 7$ which equals $4 + 3$. If it’s true, justify your claim. If it’s false, try to find out what kinds of numbers do make the claim work.

When you give $\heartsuit$ two numbers, it finds the sum of the two numbers, then multiplies the result by the first number. Finally it subtracts the square of the first number. Flinch claims that $\heartsuit$ is commutative. Is he correct?

Here’s how you might prove Flinch’s claim in problem 19 for $any$ two starting numbers.

Let’s call your first number $m$ and your second number $n$. Write down and simplify as much as you can the expression for $m \heartsuit n$.

Write down and simplify as much as you can the expression for $n \heartsuit m$.

Is the rule below commutative?

The rule $\unicode{x223c}$ adds up the two numbers, doubles the answer, multiplies the answer by the first number, then adds the square of the second number and subtracts the square of the first number.

Let $x \clubsuit y = \frac{1}{{1,000,000,000}} \thinspace {x^y}$.

Is there a value of $y$ such that $10 \clubsuit y > 1$?

Is there a value of $y$ such that $1.001 \clubsuit y > 1?$

The symbol “&”, applied to $any$ two numbers (not only whole numbers), means that you take the first number and then add the product of the numbers.

When you calculate $x$&$y$, you get 120. What could $x$ and $y$ be? Give several different answers.

When you calculate $x$&$y$, you get 120. Write an equation that expresses this fact, then solve for $y$ in terms of $x$ (meaning, write an equation $y = $…. with only $x$’s in the equation).

The command “CircleArea” is a rule that finds the area of the circle with the given radius. For example, ${\rm{CircleArea(3)}} = 9\pi $.

Find CircleArea(4).

Write the equation for CircleArea($x$).

Can you find CircleArea(-4)? Why or why not?

It’s January 1st, and you are counting the days until your birthday. Let $m$ be the month (as a number between 1 and 12) and $d$ the day (between 1 and 31) of your birthday, and pretend that there are exactly 31 days in each month of the year.

How many days are there until January 25? Until April 10?

For January 25th, $m = 1$ and $d = 25$, and for April 10th, $m = 4$ and $d = 10$. By looking at what you did in part a, explain how you can use the numbers $m$ and $d$ to count the days from January first to until any day of the year.

Let the symbol $♥$ represent this count. Write an algebraic rule for calculating $m♥d.$ Test your rule with an example.

Now, count how many days are from your birthday to New Year’s Eve (December 31st). Represent this with the symbol $\unicode{x2648}$. Again, pretend there are 31 days in each month.

Write an equation for in terms of $m$ and $d$, and use an example to show that your equation works. (You might want to try explaining it in words or in an example first.)

“max($a$,$b$)” takes any two numbers and gives you the larger of the two. The symbol $\partial $ is defined by $\partial (a,b)$= max($a$,$b$) – min($a$,$b$). Is $\partial $ commutative?

We say that the counting numbers (i.e., 1, 2, 3, ...) are “closed under addition” because any time you add two counting numbers, you get another counting number. Decide whether or not the counting numbers are closed under each of the following operations. In each case where the answer is no, try to find a group of numbers that $is$ closed under that operation.

* (multiplication)

- (subtraction)

/ (division)

The symbol ∂, from problem 27

The rule $\unicode{x2190}$ adds twelve to a number and divides the sum by four. What number $x$ can you input into $\unicode{x2190}(x)$ to get an answer of 5? An answer of -3?

To do the rule $\forall$, add 5 to the first number and add 1 to the second number, then multiply those two answers.

What’s $3 \forall 1$?

What’s $x \forall y$?

Your friend tells you that she needs to find numbers $x$ and $y$ so that $x \forall y$ gets her an answer of $A$— a whole number that she does not reveal. In terms of $A$, tell her what to plug in for $x$ and $y$. Make sure that your strategy would always get her the answer she wants.

What values of $x$ and $y$ give you an odd answer? Prove that your description is true and complete. (Make sure you know what it means to prove it’s complete!)

What values of $x$ and $y$ would give you an answer of zero?

Create 3 different rules that give an answer of 21 when you plug in 2 and 8.

Let $f$ be a rule that acts on a single number.

Create a rule for $f$ so that $f(2)=12$ and $f(11)=75$.

Now, create a new rule $g$ such that $g(5)=26$ and $g(10)=46$.

Let $x \triangle y = x^2+2xy$. ($x$ and $y$ have to be integers.)

If you plug in 6 for $x$, find a number you could plug in for $y$ to get an answer of zero.

Using part a as an example, describe a general strategy for choosing $x$ and $y$ to get an answer of zero, without making $x$ zero. Explain why your strategy works.

Suppose you use the same number for $x$ and $y$— call this number $N$. (So, you’re doing $N \triangle N$.) What is your answer, in terms of $N$? Simplify as much as possible.

Suppose $x \triangle y = 20$. Solve for $y$ in terms of $x$.

Describe a strategy to get any odd number that you want. Give an example, and also show why your strategy will always work (either give a thorough explanation, or use algebra to prove that it works).

Tinker to find a rule for $x \# y$ that gets the following answer: $5 \# 1 = 24$. Then try to write a rule that gives $5 \# 1 = 24$ and $4 \# 2 = 14$.

Create a rule $\alpha$ that works with two numbers, so that $\alpha(1,1) = 5$ and $\alpha(2,3) = 10$. Give your answer as an equation in terms of $a$ and $b$. $\alpha (a,b) = $ __________________________.

Let $\otimes \{a,b\}$ be the two digit number where the tens digit equals $a$ and the units digit equals $b$. For example, $\otimes \{9,3\} = 93$.

In terms of $f$, what do you get when you do $\otimes \{5,f\}$? Write your answer as an equation. (Remember, writing $5f$ doesn’t work, because it means $5 \cdot f$).

What do you get when you do $\otimes \{a,4\}$?

When you calculate $\otimes \{a,4\} - \otimes \{4,a\}$, what do you get in terms of $a$? Simplify as much as possible.

$\otimes \{6,x\} - \otimes \{x,3\} = 12$. Find $x$. Show your work algebraically.

Now, let’s redefine $\otimes \{a,b\}$. Let $\otimes \{a,b\}$ be the three-digit number where the hundreds and units digits are $a$, and the tens digit is $b.$

In terms of $b$, what is $\otimes \{b,2\}?$ (Again, $b2b$ won’t work).

In terms of $a$ and $b$, what do you get when you calculate $\otimes \{a,b\} - \otimes \{b,a\}$? Simplify as much as possible and check your answer with an example.

For any two positive numbers $a$ and $b$, let $a \bot b$ be the perimeter of the rectangle with length $a$ and width $b$.

Is $\bot$ commutative?

Is $\bot$ associative? In other words, does $(a\bot b) \bot c = a\bot(b\bot c)?$

A positive whole number is called a “staircase number” if the digits of the number go up from left to right. For example, 1389 works but not 1549.

The rule $x \nabla y$ works by gluing $x$ and $y$ together. For example, $63 \nabla 998$ gives an answer of 63998. If $x$ and $y$ are both staircase numbers, and $x$ is bigger than $y$, is it always true that $x \nabla y$ is bigger than $y \nabla x$ ? If you think it’s always true, explain why. If not, explain what kind of input would make it true.

Suppose that, for any two positive numbers $a$ and $b$, $a$■$b$ represents the area (ignoring units) of the rectangle with length $a$ and width $b$.

Is ■ associative?

In problem 23, you were asked to find some pairs of numbers $x$ and $y$ so that $x$&$y = 120$. (Recall that $x$&$y$ takes the first number and adds the product of the numbers).

Do you think you could find numbers $x$ and $y$ to get any number you wanted? Try a few possibilities and explain what you find.

Develop a rule for choosing $x$ and $y$ to get the number that you want, which we will call $N$. (Hint: Try starting out by picking a number for $x$, and then figuring out what $y$ would have to be. You might have to try different numbers for $x$ to find a solid strategy.)

Don’t use a calculator for this problem.

Find $\frac{3}{4} + \frac{5}{6}$

Find $2 \div \frac{1}{2}$

Simplify $3x - 2(x + 1)$

Write $.\overline 6 $ as a fraction.

Find $2 \frac{1}{4} +5 \frac{7}{8}$

Let $x \square y = xy - x - y$. Develop a strategy to pick $x$ and $y$ so that you can get any number you want.

For a number $x$, $f(x)$ subtracts twice $x$ from 100.

What is $f$(10)?

What is $f$(-4)?

Write an expression for $f(x)$.

Write an expression for $f$(3$x$).

Let Let $x \square y = xy - x - y$. Prove or disprove: if $x$ and $y$ are both larger than $2$, then $x \square y$ gives a positive answer. Make sure your explanation is thorough.

Write an equation for a rule $a \lozenge b$ , so that the answer is odd only when both $a$ and $b$ are even.

Let $a{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{\% }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b = {a^2}b - {b^2}a$. What would have to be true about $a$ and $b$ for $a{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{\% }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b$ to be positive?

Let $a \ ☺ \ b = ab + b^2$. What would have to be true about $a$ and $b$ for $a \ ☺ \ b$ to be negative?

Let $x \ominus y = xy + y/2$ , where $x$ and $y$ are whole numbers.

Describe what values of $x$ and $y$ would get you a negative answer.

Describe what values of $x$ and $y$ would get you an even whole number answer. (Careful! Your answer won’t be just in terms of odds and evens. You’ll have to tinker to see what type of numbers work. Think through it step by step and explain your reasoning).

What could $x$ and $y$ be to get 21? Give at least 3 different options.

Find a strategy to get any whole number you want, called $N$. You can explain your strategy in algebraic terms (“To get an answer of $N$ let $x$ equal … and let $y$ equal …”) or in words. Show an example to demonstrate that your strategy works.

For a number $x$ that might not be a whole number, $\Phi \left( x \right)$ represents the integral part of $x$— for example $\Phi \left( {6.51} \right)$ is 6 and $\Phi \left( {10} \right)$ is 10. Using the $\Phi $ notation, write a rule @ that gives an answer of 0 if $x$ is a whole number, and gives a non-zero answer if $x$ is not a whole number. For example @(20) should equal 0 and @(20.3) should give an answer not equal to zero.

The rule $\lfloor b \rfloor$ is called the “greatest integer function” — it outputs the largest integer that is not above $b$. For example, $\lfloor 9.21 \rfloor = 9$, $\lfloor {12} \rfloor = 12$, $\lfloor .92 \rfloor = 0$, and so on.

Write an equation for the rule $\left\{ b \right\}$, which rounds $b$ down to the hundreds — for example $\left\{ {302} \right\} = 300$, $\left\{ {599} \right\} = 500$, and $\left\{ 2 \right\} = 0$. For your equation to calculate $\left\{ b \right\}$, you can use any standard operations, and you should use the operation $\lfloor b \rfloor$.

To understand the rule behind the symbol $\diamondsuit $, you need to draw a picture (graph paper will help). To draw the picture for $5\diamondsuit 3$, pick a point to start and then draw a line stretching 5 units to the right. From there, draw a line stretching 3 units up. Then draw a line stretching 3 units to the right, and then a line stretching 5 units up. Finally, draw a line back to your starting point.

Let $5\diamondsuit 3$ be the area of the shape you just drew. Calculate this number exactly.

Pick 2 new numbers (the 1st number should still be bigger). Draw the picture and calculate the answer that $\diamondsuit $ would give for your numbers.

Draw a diagram to help you find an algebraic rule for $x {\kern 1pt} \diamondsuit {\kern 1pt} y$ (again, you can assume that $x$ is a bigger number than $y$). Check your rule by making sure it would give you the right answer for $5\diamondsuit 3$ and for your numbers in part b.

Will your rule still work if the two numbers are the same, such as $4\diamondsuit 4$? Explain why it will or will not always work.

Will your rule still work if the first number is smaller, such as $3\diamondsuit 7$? Explain why it will or will not always work.