Chapter 01: What’s My Rule?
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., ).
- This should be a mental math activity—encourage students not to use paper or pencil.
Instructor Notes A1 🔐
One of the most common problems in teaching algebra at a higher level is that students do not have an understanding of the concept of variable. They think of a variable as a specific number and have difficulty reasoning abstractly about variables.
When using algebraic thinking (or working on algebra problems), “letters” are used as place holders. Place holders can represent unknowns (a solution with a fixed value) or variables (a solution that is a relationship). In algebra, we often exchange the word variable and unknown when describing the place holders. In “What’s My Rule?” N is a variable because it changes. In comparison, when solving the equation X + 4 = 7, X does not change. It is just a number that students may or may not be able to find, but once they do find it, X is fixed and is not a variable.
Examples of simple equations, like x + 4 = 7, are often counterproductive in teaching transformations to solve equations. Students do not need to subtract 4 from both sides to find the value of x. A student might say, “I know that 3 + 4 is 7, so x = 3!” They do not need transformations to solve this equation!
Write the rule in words and using algebraic notation. Write down the rule or rules in words. Once the rule is written in words ask, “Can anyone express the rule algebraically?”
Talk about how you might write down the rule using mathematical symbols.
Two examples of different rules are:
• Double the first number and subtract one.
• Add the first number, and the number 1 less than the first number.
The Power of Algebra When the rules are expressed in words, they are different, especially the first and the last one. The power of algebra is showing that these rules represent different thinking but are mathematically equivalent.
2N – 1 = N + (N – 1).
Suggested Rules:
2n + 1
5n – 1
n² – 1
n² + n + 1
3n – 1
5n + 1
n² + 1
n² + n – 1
3n + 1
5n – 2
n² – 2
n(n + 1)
2n + 3
5n + 2
n² + 2
3n – 10
2n – 5
5n – 6
n² – 4
3n + 10
3n – 5
5n + 7
n² – 5
6n – 5
2n – 7
5n – 7
n² + 5
6n – 7
2n + 9
5n – 9
4n – 1
6n + 7
Instructor Notes A1 🔐
The intent of this section is to help students develop a conceptual understanding of combining like terms.
Combining like terms is structural. In some ways , you are thinking of two objects as one object. You are transforming two objects or two processes into one!
Examples are important, and this introduction provides several examples.
Non-examples, examples of what does not work, are also important!
This section is designed to help students see the reasoning behind combining like terms. Mathematics should not be magic and it should be a sense making activity!
PS1: Combining Like Terms
In the mathematical expressions ‘5x, 4a, and 16rt,’ the letters ‘x, a, r, and t’ used to represent numbers are variables, and the numbers ‘5, 4, and 16’ are coefficients. The expression 7a means 7 times a.
In 23, 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.
The following examples illustrate like terms and when added or subtracted can be combined.
2b and 3b 2b + 3b = 5b
49a and a 49a + a = 50a
7c and -9c 7c – 9c = 2c
5t and 5t 5t + 5t = 10t
3xy and 7xy 3xy + 7xy = 10xy
5x2 and 9x2 5x2 + 9x2 = 14x2
23x4y7z2 and 5x4y7z2 23x4y7z2 + 5x4y7z2 = 28 x4y7z2
The following are examples of unlike terms and cannot be combined.
7xy and 3x 4x2 and 9x 2ab2 and 5a2b
Two ways of thinking
To combine like terms, we can change 2b to 2b × b which also means b + b and the 3b to 3 × b which also means b + b + b. Then, 2b + 3b = 2b × b + 3 × b = b + b + b + b + b = 5b.
The problem can also be solved by the distributive property. In the expression 2b + 3b, the factor is common to both terms and can be factored out to ( 2 + 3 )b = 5b.
In conclusion, you can either 1) break the problem down into addition and/or subtraction, 2) apply the distributive property, or 3) just add the coefficients when you have like terms.
In the second example, 49a + a, there is an implied 1 in front of the a or 1a. Therefore,
\(
\begin{array}{rcll}
(1) & 49a + a & = & 50a \\
(2) & 23x – x & = & 22x \\
(3) & x^2 + x^2 & = & 2x^2
\end{array}
\)
Other examples
\(
\begin{array}{ll}
(1)\; 7c – 9c = -2c \\
(2)\; 5t + 5t = 10t \\
(3)\; 3xy + 7xy = 10xy \\
(4)\; 5x^2 + 9x^2 = 14x^2 \\
(5)\; 23x^4y^7z^2 + 5x^4y^7z^2 = 28x^4y^7z^2
\end{array}
\)
Instructor Notes PS1 🔐
This page was recently added to TWA. Our analysis has shown that students struggled with working with radicals on course assessments. TWA takes a mixed-review or distributed-practice approach. Our intent is to introduce working with radicals early in the curriculum and provide students with multiple experiences with radicals. Part of the problem is students do not have good symbol sense with radicals. They may not understand that:
5√2 + 3√2 = 8√2 or
5√2 x 3√2 = 15√4 = 15 x 2 = 30.
Next, let’s consider combining expressions with radicals. This will be useful later. We can combine the following:
\(7\sqrt{x} + 3\sqrt{x} = 10\sqrt{x}\)
The notation \(7\sqrt{x}\) means ‘7 times .\(\sqrt{x}.\)’ Since multiplication is repeated addition, you are adding \(\sqrt{x} + \sqrt{x} + \sqrt{x} + \sqrt{x} + \sqrt{x} + \sqrt{x} + \sqrt{x}\). Likewise, \(3\sqrt{x}.\) means \(\sqrt{x} + \sqrt{x} + \sqrt{x}.\) Adding the two expressions together is \(\sqrt{x}.\)
As long as the radicals are exactly the same, we can treat them as like terms and simplify when adding or subtracting. We can also treat the radical of a number like a variable. For example,
\(5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}\)
You cannot simplify or combine like terms for the following:
\(3\sqrt{2} + 2\sqrt{3}\)— the radicals have to be exactly the same!
Why is ? \(9\sqrt{7} – \sqrt{7} = 8\sqrt{7}?\)
Why is the following true? \(3\sqrt{xy} + 2\sqrt{xy} = 5\sqrt{xy}\)
For future reference, when you multiply radicals, you multiply the coefficient times the coefficient and the radical times the radical. For example,
\(3\sqrt{2} \,(5\sqrt{7}) = 15\sqrt{14}\)
Instructor Notes PS1 🔐
1. This problem lays the groundwork for future problems. Students are utilizing their geometry skills and concepts. The table is important. Tables can help students focus on and find both the recursive pattern—down the table in the ‘Number of Squares’ column—and the explicit pattern—across the table (i.e., the relationship between the figure # and the number of squares in the figure.)
Try to connect the rule to the figures. For the rule 7n + 5, how are the constant and the coefficient of the variable represented in the figures?
One of the most difficult aspects of these problems is that students quickly find the recursive pattern. The pattern increases by 7 each time, or +7. However, many, inevitably, think the explicit pattern, the pattern connecting the figure number with the squares in the figure, is ‘n + 7’. I ask them, “What are you doing each time?” and the students typically respond, “Adding 7.” Therefore, they are repeatedly adding 7. Sometimes I ask what repeated addition is, and most know that it is multiplication. Therefore, the rule is not +7 but rather involves 7 times the number or 7n. Of course, this is not the final rule. Students can pick any value in the table and I ask how you can go across the table using the 7n. This is the explicit pattern and algebra!
Structurally, 7n + 5 is a generalization using symbols of two operations representing one rule.
A common solution to this problem is 7n + 5. Another rule could be 5(n + 1) + 2n. These solutions are equivalent but represent a different way of looking at the figures. It can be challenging to help students to generalize without telling them the answer, but it is worth it. Students must construct the mathematics themselves. They cannot intuit it from our telling!
Problem Set – PS1
Some problems will require a written explanation.
1. a) Draw the next two figures in the pattern.
b) Complete the table for the number of squares in each figure.
| Figure # | Number of Squares in the Figure |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| 25 | |
| 100 | |
| n |
Instructor Notes PS1🔐
2a. Some students may line up the numbers in a column or use a calculator.
Ask if there are any other ways to do the problem or if there is an easier way to find the solution without a calculator.
Problem 2a encourages students to develop number sense and can be done readily with a good understanding of place value.
2b. One intent of this problem is to provide an incentive for students to use the associative property of multiplication. Many students do this problem the long way. This problem is designed to encourage them to look at the structure of the number sentence and use the associate property to make the calculation easier. They do not have to name the property.
Ask: “Can you always move the parenthesis if you have all multiplication?”
This problem may lead into a discussion of the order of operations and if problems must be solved from left to right.
Properties are part of the framework of algebra.
2. Solve each of the following without using a calculator:
a) 1,000,ooo – 5 = ________
b) ( 418 × 25 ) × 4 = ________
Instructor Notes PS1🔐
3. This problem is designed to help students come to a better understanding of the concept of equivalence and the equal sign. Some students may put 72 in the box, some may put 47, and some may put 72 in the box and then add an equal sign and 47 (i.e., they will write “= 47” to the right of the 25).
We will have several of these types of questions in the Problem Sets, so they do not have to get it the first time. For many students, the equal sign means “and the answer is.” Equality is a concept that students need to understand in algebra, especially when solving equations.
Ask: “Are both sides equal if they have 72 (a number other than 97 in the box)?”
The property of equivalence allows us to represent structural aspects of algebra.
3. Find ⌂
9 × 8 = ⌂ – 25
Instructor Notes PS1🔐
4. This problem addresses combining like terms. Emphasize that both 4b and 4c represent 5N. 4a and 4d represent common errors that students make. If students are making these selections, talk about these errors. We will have several of these types of questions in the Problem Sets, so they do not have to get it the first time. For many students, the equal sign means “and the answer is.” Equality is a concept that students need to understand in algebra, especially when solving equations.
Ask: “Why is 4a false? Why are 4b and 4c true?”
Consider how 5N =N + N + N + N + N is five ‘objects’ thought of as one ‘object’.
5. Encourage students to make the connection between fractions and decimals.
Ask: “What is easier; adding the decimals or the fractions?”
6. In order to succeed in algebra, students must be able to problem solve! Students are more likely to draw pictures or use trial and error to solve this problem. A pictorial solution that students may use is to draw 40 circles representing the 40 heads, then put 2 feet on each head, which is 80 feet, and then distribute the remaining 20 feet (100 – 80 = 20) 2 at a time until they are used up. Make the connection to algebra. If students do not come up with an algebraic representation, show them one: b + d = 40 and 2b + 4d = 100.
Ask: “How did you solve the equation?”
4. Explain which equations are true for all values of N :
a. 5N = 5 + N
b. 5N = 5 × N
c. 5N = N + N + N + N + N
d. 5N = 7 + 7 + 7 + 7 + 7 = 35
5. Add the numbers; \(\frac{7}{10}, \; \frac{7}{100},\) and (\frac{7}{1,000}.\) Write the sum as a fraction and a decimal.
6. 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?
Instructor Notes PS1🔐
7. Students may erroneously reason that since N and P are different letters or variables, they cannot be the same number. This problem is trying to help students develop a deeper understanding of the concept of variable.
Ask: “Can N = P? Why or why not?”
8. Translating words to algebraic expressions is an important algebraic skill. Students may solve this problem by trial and error, which is a viable solution. Hopefully, some students will express the problem algebraically. If not, you might show them this representation; 2N - 5 = 25.
Ask: “How do you solve this equation?”
9. Fractions can be problematic for students at all levels. Notice how these types of problems are different from traditional problems where students are given the whole and then asked to divide it into fractions. Our intent is to help students develop a strong conceptual understanding of fractions before they begin operations with fractions.
Ask: “How did you solve the problem? Did anyone do it in a different way?”
7. Is the following statement Always, Never, or Sometimes true? Explain your reasoning.
M + N = M + P
8. If I double a number and subtract 5, the answer is 25. What number did I start with?
9. This is 1/8 of the figure. What does ¾ of the figure look like?
Instructor Notes PS1🔐
10. An important idea here is that area is measured in square units, that is, the number of squares that it takes to cover the figure. A rectangle can be composed of several squares. Students can mentally take two triangles and put them together to make a square. This idea or concept for finding area is conservation—the whole is equal to the sum of its parts.
10a. The application of formulas is likewise important in algebra. Students may not understand the subscripts for the variables. They need to learn to substitute values into formulas.
11. This problem involves proportional reasoning. There are multiple ways of solving the problem. Students may find a unit rate—1 cup makes 1 and ¾ donuts—or a composite unit rate— multiplying the ratio “8 cups make 14 donuts” by 1 and ½ or reducing 8 cups make 14 donuts to 4 cups make 7 donuts and then multiplying each number by 3. Another method is the building up method of repeatedly adding—4 cups make 7 donuts; therefore, 8 cups make 14; therefore, 12 cups make 21 donuts. Another method is cross multiplication— solving 8/14 = 12/n by cross multiplication (i.e., 8n= 168 and n = 21).
12. This problem provides practice combining like terms, illustrating some common errors in a, b and d. If students are making these selections talk about these errors.
13. Ask: “Can anyone give an example where 13b and 13c are false?”
13. This problem connects the concept of variable and negative numbers. Many students may believe that –a is less than 0 because it is negative. However, as a variable, a can represent any number. Therefore, the value of ‘–a’ cannot be determined. This problem gets at the dual meaning of variables and negatives.
10. Find the area of each shape without using formulas.
10a. The third figure is a trapezoid. The formula for the area of a trapezoid is \(A = \left( \frac{1}{2} b_1 + b_2 \right) h,\) where b1 is the length of one base and b2 is the length of the other base. In the problem above, b1 = 4 units, and b2 = 2 units. The height, h, is the length between the two bases. In the problem above, the height is 2 units. Show the formula, \(A = \left( \frac{1}{2} b_1 + b_2 \right) h,\) gives the same area as counting the squares.
11. Alex needs exactly 8 cups of flour to make 14 donuts. How many donuts can he make with 12 cups of flour?
12. Explain which equations are true and which are not true:
a. 2X + 3Y = 5XY
b. 2X + 3Y = 6XY
c. 2X + 3Y = X + X + Y + Y + Y
d. 2X + 3Y = 6XY
13. Each statement below is only true some of the time, tell which values of a make it true all the time.
a. – a = 0
b. – a < 0
c. – a > 0
Instructor Notes PS1🔐
14. This is a precursor to the section on graphing linear inequalities. This may give an indication whether some students already know the material.
15. Examining the similarities and contrasts between two representations, an algebraic expression, and a graph helps students develop structural understandings.
16. This is asking students to reiterate the explanations provided in the introduction.
Combining like terms is combining two objects or the process of performing an operation on two objects into one. The process becomes an object.
Do students know what 5n + 4n means or are they just performing a mindless symbol manipulation?
17. Ask students if they can name the quadrants. Examining the similarities and contrasts between two representations, an algebraic expression, and a graph, helps students develop structural understandings.
14. Graph the inequality on a number line:
x ≥ –3 and x < 2
15. Simplify by combining like terms.
a. 3x2 + 7x + 4 + 9x2 – 2x + 8
b. 3xy2 + 9xy + 8xy2 + 5xy
16. Explain why . 5n + 4n = 9n
17. Give the coordinates of each point: R, S and T. Give the coordinates of a fourth point U which will make a rectangle when all the points are connected. Locate point U on the graph.
18. Simplify:
a. \(17\sqrt{x} – 9\sqrt{x} = 8\sqrt{x}=\)
b. \(14\sqrt{3} + 5\sqrt{3} =\)
c. \(18\sqrt{y} – \sqrt{y}\)
d. \(23\sqrt{xyz} + 48\sqrt{xyz}\)
e. \(5\sqrt{6} + 4\sqrt{3}(\sqrt{2}) = 9\sqrt{6}\)
f. \(16\sqrt{xyz} – 29\sqrt{xyz}\)
