Activity A1: What’s My Rule? (OLD)
Activity The Game
This activity is played between one person and others. In this case, we will say the instructor is playing with their students, but a group of two or more students can play as well.
The instructor thinks of a rule, and when the rule is applied to the input number, the result is the output number. The instructor should give at least two pairs of numbers that fit the rule and should write the ordered pairs out so students can see the guesses. Ask the students to give you another ordered pair (input, output) that the students think may fit the rule and record the solution. (Give them more than two ordered pairs if they cannot give you other pairs of numbers that fit the rule.)
Important Guidelines:
- It is very important that students not tell the rule or yell it out before it is asked for.
- The instructor must tell students if their pair of numbers fit the rule. Say: "Yes that fits my rule" or "No, that doesn't fit my rule."
- Give the majority of the class an opportunity to figure out the rule.
- Write down the rule in words first (e.g., "multiply by three").
- Next, talk about how you might shorten or simplify a rule using mathematical symbols (e.g., $3 \times n$).
- This should be a mental math activity—encourage students not to use paper or pencil.
Suggested Rules:
Select a hidden rule and type a number to see the output. Can you guess the rule?
Concept PS1: Combining Like Terms
In the mathematical expressions '$5x$, $4a$, and $16rt$', the letters '$x, a, r, t$' used to represent numbers are variables, and the numbers '$5, 4, 16$' are coefficients. The expression $7a$ means 7 times $a$.
In $2^3$, the 3 is called the exponent, and the 2 is called the base. The exponent indicates how many times the number or expression is multiplied. For example, $2^3 = 2 \cdot 2 \cdot 2 = 8$.
In mathematics, we combine like, sometimes called similar, terms, which have the same variables and exponents. They do not have to have the same coefficients.
Can Combine (Like Terms)
- $2b + 3b = 5b$
- $49a + a = 50a$ (implied 1a)
- $7c - 9c = -2c$
- $5t + 5t = 10t$
- $3xy + 7xy = 10xy$
- $5x^2 + 9x^2 = 14x^2$
- $23x^4y^7z^2 + 5x^4y^7z^2 = 28x^4y^7z^2$
Cannot Combine (Unlike Terms)
- $7xy$ and $3x$
- $4x^2$ and $9x$ (different exponents)
- $2ab^2$ and $5a^2b$
Two ways of thinking
To combine like terms, we can change $2b$ to $2 \times b$ which also means $b+b$, and the $3b$ to $3 \times b$ which also means $b+b+b$. Then,
$2b + 3b = (b+b) + (b+b+b) = 5b$
The problem can also be solved by the distributive property. In the expression $2b+3b$, the factor $b$ is common to both terms and can be factored out:
$(2+3)b = 5b$
In conclusion, you can either:
- Break the problem down into addition and/or subtraction.
- Apply the distributive property.
- Just add the coefficients when you have like terms.
Radicals
Next, let’s consider combining expressions with radicals. This will be useful later. We can combine the following:
$7\sqrt{x} + 3\sqrt{x} = 10\sqrt{x}$
As long as the radicals are exactly the same, we can treat them as like terms and simplify when adding or subtracting.
$5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$
You cannot simplify: $3\sqrt{2} + 2\sqrt{3}$ (radicals are different!)
Thought Questions:
Why is $9\sqrt{7} - \sqrt{7} = 8\sqrt{7}$?
Why is $3xy + 2xy = 5xy$?
Multiplication Note: When you multiply radicals, you multiply coefficient times coefficient and radical times radical:
$3\sqrt{2} \cdot (5\sqrt{7}) = 15\sqrt{14}$
Problem Set 1
Some problems will require a written explanation.
a) Draw the next two figures in the pattern. (Mental exercise)
b) Complete the table for the number of squares in each figure.
| Figure # | Squares | Figure # | Squares |
|---|---|---|---|
| 1 | 1 | 6 | |
| 2 | 4 | 7 | |
| 3 | 9 | 8 | |
| 4 | 16 | 9 | |
| 5 | 25 | $n$ | $n^2$ |
| 10 | 25 | ||
| 100 |
Solve without a calculator:
Find $\Box$ in the equation: $9 \times 8 = \Box - 25$
Explain which equations are true for all values of $N$:
Add the numbers: $\frac{7}{10}$, $\frac{7}{100}$, and $\frac{7}{1000}$. Write as a decimal.
Looking in my backyard one day I saw some boys and some dogs. I counted 40 heads and 100 feet. How many boys and how many dogs were in my backyard?
Is the following statement Always, Never, or Sometimes true? Explain your reasoning.
$M + N = M + P$
If I double a number and subtract 5, the answer is 25. What number did I start with?
This is 1/8 of the figure. What does 3/4 of the figure look like?
(Draw on your own paper or mental visualization)
Find the area of each shape without using formulas.
The third figure is a trapezoid. The formula for the area of a trapezoid is $A=\frac{1}{2}(b_1+b_2)h$.
Given: $b_1=4$, $b_2=2$, $h=2$. Show the formula gives the same area as counting squares.
Alex needs exactly 8 cups of flour to make 14 donuts. How many donuts can he make with 12 cups of flour?
Explain which equations are true and which are not true:
Each statement below is only true some of the time. Tell which values of $a$ make it true all the time.
Graph the inequality on a number line: $x \ge -3$ and $x < 2$.
Simplify by combining like terms.
$3x^2 + 7x - 4 + 9x^2 - 2x + 8$
$3xy^2 - 9xy + 8xy^2 + 5xy$
Explain why $5n + 4n = 9n$.
Hint: Use the distributive property: (5+4)n
Give the coordinates of point U which will make a rectangle with R(1,2), S(6,2), and T(1,5).
a) $17x - 9x$
b) $14\sqrt{3} + 5\sqrt{3}$
c) $18y - y$
d) $23xyz + 48xyz$
e) $5\sqrt{6} + 4\sqrt{3}(\sqrt{2})$
f) $16xyz - 29xyz$
