Expressions, Equations, and Functions
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C HAPTER
Chapter 1. Expressions, Equations, and Functions
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Expressions, Equations, and Functions
Variable Expressions One of the first obstacles for students as they work with expressions is the use of math verbs. Be sure that the students understand the math verbs that are used and the many different words that can be used to mean addition, subtraction, multiplication and division. In addition, the words “less than” are often mixed up. Look at this example. 14 less than the number 25 When students see this, they often will write.
14 − 25 They don’t read the problem accurately. Instead they see the word “less” and just assume that it is subtraction and that the values are written in the same order as they are in the problem. To help clarify this, you can point out to students right away that certain words will require them to move the values around. The words “less than” are one of these examples. Here is the correct answer.
25 − 14 This will be relevant whether the students are working with integers as in the example or with variables and integers. The next issue can be with notation. Teachers are advised not to allow notation to become an obstacle for their students from the beginning. Writing should be in-line and organized and maintain a logical flow. Students may misread their work if their writing is illegible. Zs can look like. 2s, +s like lower-case ts, tiny negatives disappear into the paper, other stray marks become minus signs, etc. Students often compress their answers on paper, especially when they are used to writing across instead of in a downward fashion. 3x − 1 = 8 = 3x = 9 = x = 3. The equals sign is being used in at least two ways: equality between quantities, and equivalence (or implication) between equations. When evaluating algebraic expressions, students can easily lose track of negatives. Evaluate 1 − x, where x = −1. Evaluate 1 − x2 , where x = −1. Evaluate 1 − x−3 −3 , where x = −1. The first example involves a variable with a negative “out in front” and the value being substituted is negative. The second involves the vulnerable combination of negatives and exponents. The third involves negatives and fractions. As a check, ask your students if they can tell the difference between −12 and (−1)2 1
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Order of Operations One of the first errors can be found when we write the order of operations. P E MD AS The students will think that multiplication always comes before division and that addition always comes before subtraction. Be sure to include the words “ In order from left to right” when you share this step with the students. You can say, “We complete multiplication and division in order from left to right” and “We complete addition and subtraction in order from left to right.” This helps the students to keep this straight especially if they learned it incorrectly in an earlier grade. The next thing to pay attention to is the P. P does not only stand for parentheses, but grouping symbols. The students may not have been introduced to other grouping symbols besides parentheses, so it is a good idea to include this and not assume that they know. After all, bracket does not begin with P. Students should be reminded occasionally that subtracting negatives is equivalent to adding positives.
Patterns and Expressions When working with students, be sure to clarify if they make a mistake, why they have made the mistake that they have made. Part of the learning process consists of learning from mistakes. If students are told that their answers are wrong without good reasons or explanations, they will simply lose out on what could have been a learning experience. Consider the following subtle example. Students were asked to simplify the expression:
(−x)(−x)(−x) One student gave the answer:
(−x)3 The exponent is odd, (−x)3 so it can be further simplified to −x3 . The two expressions can be interpreted differently if order of operations is the focus: (−x)3 means to cube the opposite of x −x3 means to take the opposite of x cubed You can also require students to check their work. One way to do this is to ask the students to work with a partner and explain their solution and how they got it. Create a check list for the students when checking a solution. Here is an example of a useful checklist. • Does the solution make sense? If not, why not? If so, why? • Is the sign of the number correct? 2
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Chapter 1. Expressions, Equations, and Functions
• Is the answer written in the correct form? Always remind students of the skills used in the previous lesson. Many will forget that they will need to use the order of operations to evaluate expressions.
Equations and Inequalities Be sure that the students understand the definition of an equation and an inequality. You can use the words “quantity on one side of the equals balances with the quantity on the other side,” for an equation. This will help the students to think in terms of quantity or values and not just numbers. Explain that an inequality can sometimes be equal and sometimes not. Review the inequality symbols and what they mean. Do not assume that the students have learned them correctly. In fact, many students will not have learned them correctly. You can tell students that the symbol always “eats” the larger value. ≤ ≥ > < Practice writing inequalities to describe different situations. This will help the students to apply the symbols in the correct way. Many times the students will have a conceptual understanding of which symbol needs to be used, but have a difficult time executing the use of the symbol. Have students write out and identify the givens in problems. If variables other than x are used students need to be careful that Zs and 2s are not confused. Students have trouble labeling the variables correctly. Have them locate the key question in the exercise such as: What? How much? When? How long? Where? How far? Students should be cautioned to avoid the common mistake when fractions are involved; failing to divide the entire numerator will lead to incorrect results. For example: 2x + 10 6= x + 10 2
Functions as Rules and Tables Function notation can be an obstacle for many students, primarily because they do not understand its purpose. Some students will try to multiply the f and the x. To avoid this, begin by helping the students to understand the word “function.” In order for the students to be able to apply the concept of one thing being a function of another, we have to use real-world examples so that they can make connections. For example, we could say that the number of batches of cookies that we can make is a function of the number of eggs we have. We can’t make 10 batches of cookies if we don’t have enough eggs. Therefore, one variable depends upon the other variable. The eggs are independent of the cookies, however, the cookies are not independent of the eggs. We must have the eggs to make the cookies. A common error can be for students to mix up the input and the output. You want them to understand that the input is the x value and the output is the y value. Then once they have that straight, you can show them how function notation can be substituted into the function. 3
www.ck12.org A rule means the same as a pattern. It is just written in another way. Be sure that the students understand that the rule can be expressed in several different ways. It can be written as an expression or in words. These are interchangeable. When working with a table, give the students a lot of practice substituting values into a given expression to find an output. This is going to be key for generating ordered pairs when graphing. If the students don’t understand how to generate their own inputs/outputs, then creating ordered pairs will be challenging when it comes to graphing. Now that the students are using new concepts in their work, sometimes they will forget things that they have already learned. Review multiplying by a fraction and also the order of operations.
Functions as Graphs Begin this lesson by reviewing tables and how to create a table of values. You can give the students an example such as f (x) = 2x + 1. Remind them that the function notation is the same as the output or y. Many times students will keep getting this mixed up. You can show them the following tables to illustrate the point.
TABLE 1.1: x 0 1 2 3
y 1 3 5 7
AND
TABLE 1.2: x 0 1 2 3
f (x) 1 3 5 7
This will help the students to see that the results are the same. Once this is clear, you can help them to notice how they can create ordered pairs from a table. The values form ordered pairs. Remind the students that the output is always the y value of an ordered pair whether it says y or is written in the function notation f (x). When plotting points on the coordinate plane, have students place arrow heads (↔, l) above the points in an ordered pair to help them remember which direction to move when plotting the point. Functions cannot have more than one output per input, but two inputs can have the same input. Examples that clarify this will help the student understand the definition of a function.
A Problem-Solving Plan Begin by telling the students that a word problem is a story that involves mathematics. This will help to take the pressure off of students when they first approach a word problem. Then you can tell them that there is a certain method for solving a problem. The steps of a problem solving plan will help them. Be sure that the students write the steps of the problem solving plan down in their notebooks. 4
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Chapter 1. Expressions, Equations, and Functions
Then remind students to read the problem through and underline math verbs. The following list includes possible sources of mistakes for students while solving an applied problem: • • • • • •
Not listing the given information Not restating the question being asked in your own words Not selecting a variable to represent the unknown quantity Not clearly stating what the variable represents Not looking for possible patterns Not looking up a definition or formula
A common mistake students make is checking their answers in the equations that they constructed instead of in the original wording of the problems.
Problem-Solving Strategies: Make a Table; Look for a Pattern When teaching students problem solving strategies, you want them to see them as strategies. A common problem is that the students assume that they already know the strategy and understand it completely. Because of this, they aren’t able to incorporate the strategy into their memory and often fail to remember it when solving a problem. To help with this, be sure that the students take notes on each strategy and write down good times to use each strategy. When a problem has data that needs to be organized, a table is a highly effective problem-solving strategy. A table is also helpful when the problem asks you to record a large amount of information. Patterns and numerical relationships are easier to see when data are organized in a table. When values repeat or are changed in the same way from step to step, then it is a good time to look for a pattern for solving the problem. The following list includes possible sources of mistakes for students while solving an applied problem: • • • • • •
Not listing the given information Not restating the question being asked in your own words Not selecting a variable to represent the unknown quantity Not clearly stating what the variable represents Not looking for possible patterns Not looking up a definition or formula
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