Solving Simple Equations

When solving a simple equation, think of the equation as a balance, with the equals sign (=) being the fulcrum or center. Thus, if you do something to one side of the equation, you must do the same thing to the other side. Doing the same thing to both sides of the equation (say, adding 3 to each side) keeps the equation balanced.

Solving an equation is the process of getting what you're looking for, or solving for, on one side of the equals sign and everything else on the other side. You're really sorting information. If you're solving for x, you must get x on one side by itself.

Addition and subtraction equations

Some equations involve only addition and/or subtraction.

Example 1

Solve for x.

x + 8 = 12

To solve the equation x + 8 = 12, you must get x by itself on one side. Therefore, subtract 8 from both sides.

To check your answer, simply plug your answer into the equation:

 

Example 2

Solve for y.

y – 9 = 25

To solve this equation, you must get y by itself on one side. Therefore, add 9 to both sides.

To check, simply replace y with 34:

 

Example 3

Solve for x.

x + 15 = 6

To solve, subtract 15 from both sides.

To check, simply replace x with –9 :

 

Notice that in each case above, opposite operations are used; that is, if the equation has addition, you subtract from each side.

Multiplication and division equations

✕Powered by ExploreadsSome equations involve only multiplication or division. This is typically when the variable is already on one side of the equation, but there is either more than one of the variable, such as 2 x, or a fraction of the variable, such as

 or 

In the same manner as when you add or subtract, you can multiply or divide both sides of an equation by the same number, as long as it is not zero, and the equation will not change.

Example 4

Solve for x.

3 x = 9

Divide each side of the equation by 3.

To check, replace x with 3:

 

Example 5

Solve for y.

To solve, multiply each side by 5.

To check, replace y with 35:

 

Example 6

Solve for x.

To solve, multiply each side by .

Or, without canceling,

 

Notice that on the left you would normally not write  because it would always cancel to 1 x, or x.

Combinations of operations

Sometimes you have to use more than one step to solve the equation. In most cases, do the addition or subtraction step first. Then, after you've sorted the variables to one side and the numbers to the other, multiply or divide to get only one of the variables (that is, a variable with no number, or 1, in front of it: x, not 2 x).

Example 7

Solve for x.

2 x + 4 = 10

Subtract 4 from both sides to get 2 x by itself on one side.

Then divide both sides by 2 to get x.

To check, substitute your answer into the original equation:

 

Example 8

Solve for x.

5x – 11 = 29

Add 11 to both sides.

Divide each side by 5.

To check, replace x with 8:

 

Example 9

Solve for x.

Subtract 6 from each side.

Multiply each side by .

To check, replace x with 9: 

Example 10

Solve for y.

Add 8 to both sides.

Multiply each side by .

To check, replace y with –25: 

Example 11

Solve for x.

3 x + 2 = x + 4

Subtract 2 from both sides (which is the same as adding –2).

Subtract x from both sides.

Note that 3 x – x is the same as 3 x – 1 x.

Divide both sides by 2.

To check, replace x with 1:

 

Example 12

Solve for y.

5 y + 3 = 2 y + 9

Subtract 3 from both sides.

Subtract 2 y from both sides.

Divide both sides by 3.

To check, replace y with 2:

 

Sometimes you need to simplify each side (combine like terms) before actually starting the sorting process.

Example 13

 Solve for x.     

3 x + 4 + 2 = 12 + 3

First, simplify each side.

Subtract 6 from both sides.

Divide both sides by 3.

To check, replace x with 3: 

Example 14

Solve for x.

4 x + 2 x + 4 = 5 x + 3 + 11

Simplify each side.

6 x + 4 = 5 x + 14

Subtract 4 from both sides.

Subtract 5 x from both sides.

To check, replace x with 10: 

Probability

Probability is the numerical measure of the chance of an outcome or event occurring. When all outcomes are equally likely to occur, the probability of the occurrence of a given outcome can be found by using the following formula:

Example 1

Using the spinner shown in Figure 1, what is the probability of spinning a 6 in one spin?

Since there is only one 6 on the spinner out of ten numbers and all the numbers are equally spaced, the probability is .

Figure 1. Spinner with equally divided sections.

 

Example 2

Again using the spinner shown in Figure 1, what is the probability of spinning either a 3 or a 5 in one spin?

Since there are two favorable outcomes out of ten possible outcomes, the probability is  or .

When two events are independent of each other, you need to multiply to find the favorable and/or possible outcomes.

Example 3

What is the probability that both of the spinners shown in Figure 2 will stop on a 3 on the first spin?

Since the probability that the first spinner will stop on the number 3 is  and the probability that the second spinner will stop on the number 3 is , and because each event is independent of the other, simply multiply.

Figure 2. Spinners with equally divided sections.

 

Example 4

What is the probability that on two consecutive rolls of a die the numbers will be 2 and then 3? (A die has 6 sides numbered 1–6.)

Since the probability of getting a 2 on the first roll is  and the probability of getting a 3 on the second roll is , and since the rolls are independent of each other, simply multiply.

Example 5

What is the probability of tossing heads three consecutive times with a two‐sided fair coin?

Because each toss is independent and the probability is  for each toss, the probability is

Example 6

What is the probability of rolling two dice in one toss so that they total 5?

Since there are six possible outcomes on each die, the total possible outcomes for two dice is

6 × 6 = 36

The favorable outcomes are (1 + 4), (4 +1), (2 + 3), and (3 + 2). These are all the ways of tossing a total of 5 on two dice. Thus, there are four favorable outcomes, which give the probability of throwing a total of five as

Example 7

Three green marbles, two blue marbles, and five yellow marbles are placed in a jar. What is the probability of selecting at random a green marble on the first draw?

Since there are ten marbles (total possible outcomes) and three green marbles (favorable outcomes), the probability is .

Example 8

In a regular deck of 52 cards, what is the probability of drawing a heart on the first draw? (There are 13 hearts in a deck.)

Since there are 13 favorable outcomes out of 52 possible outcomes, the probability is  or .

Arrangements

If there are a number of successive choices to make and the choices are independent of each other (order makes no difference), the total number of possible choices is the product of each of the choices at each stage.

Example 9

How many possible combinations of shirts and ties are there if there are five different color shirts and three different color ties?

To find the total number of possible combinations, simply multiply the number of shirts times the number of ties.

5 × 3 = 15

Example 10

 A combination lock has three settings, each of which contains numbers from 0 to 9. How many different possible combinations exist on the lock?

Note that each setting is independent of the others; thus, because each has ten possible settings,

10 × 10 × 10 = 1,000

There are 1,000 possible combinations.

Permutations

If there are a number of successive choices to make and the choices are affected by the previous choice or choices (dependent upon order), then permutations are involved.

Example 11

How many ways can you arrange the letters S, T, O, P in a row?

The product 4 × 3 × 2 × 1 can be written 4! (read 4 factorial or factorial 4). Thus, there are 24 different ways to arrange four different letters.

Example 12

How many different ways are there to arrange three jars in a row on a shelf?

Because the order of the items is affected by the previous choice(s), the number of different ways equals 3! or

3 × 2 × 1 = 6

There are six different ways to arrange the three jars.

Example 13

If, from among five people, three executives are to be selected, how many possible combinations of executives are there?

This is a more difficult type of arrangement involving permutations. Notice here that the order of selection makes no difference. The symbol used to denote this situation is

C( n, r), which is read the number of combinations of n things taken r at a time. The formula used is

Because n = 5 and r = 3 (five people taken three at a time), then the solution is as follows: 

Now solve.

If the problem involves very few possibilities, you may want to actually list the possible combinations.

Example 14

A coach is selecting a starting lineup for her basketball team. She must select from among nine players to get her starting lineup of five. How many possible starting lineups could she have?

Because n = 9 and r = 5 (nine players taken five at a time), the solution is as follows.

Example 15

How many possible combinations of a, b, c, and d, taken two at a time, are there?

Since n = 4 and r = 2 (four letters taken two at a time), the solution is as follows: 

You may simply have listed the possible combinations as ab, ac, ad, bc, bd, and cd.

How to Teach So Students Remember

Introduction

1. It is suggested in the Introduction that each of us follows the same process when learning and mastering something new. Review these stages and identify something you learned in this way. Share your experience with your colleagues.
2. Some updates and changes to the book are listed. Which of these do you find most intriguing? Can you see where applying this information will benefit your students?

Chapter 1. Step 1: Reach and Teach

1. Discuss the attention and motivation strategies that you use. What works best for your grade level or content area?
2. Using emotion in the classroom may be more difficult in some content areas. Where would it be the easiest? Brainstorm ideas for adding emotion in your content area(s).
3. As educators we are all familiar with Maslow's hierarchy of needs. How do you feel about Matthew Lieberman's thoughts about a hierarchy with social needs represented first instead of physical needs? How would you adapt this idea to your classroom?
4. There is a strong research base for using advance organizers. Share examples of these with your colleagues. Create an agree/disagree chart for this chapter.
5. Consider the common ground you have with your students. Brainstorm areas that could be used to reach students in relation to your content area.
6. In your own words, explain to a colleague the importance of Step 1.

Chapter 2. Step 2: Reflect

1. We are often asked to reflect at the end of a lesson or unit. This is important, however, the reflection step in this chapter comes after the introduction of a lesson or concept. Do you currently give your students this opportunity to make connections with prior knowledge? Can you see a difference in the retention of information when you allow your students to reflect?
2. What habits of reflection do you incorporate into your teaching?
3. Questioning is one way to get students to think about their thinking. Make a list of essential questions for this chapter.
4. There are several types of wait time mentioned in this chapter. Which do you use?
5. Model for your students how you reflect as you read. Choose a difficult selection and read and reflect aloud as you encounter difficulty with meaning or as you are making connections.
6. In your own words, write the important concepts in this chapter.

Chapter 3. Step 3: Recode

1. Recode the concepts in this chapter as though you are explaining the concepts to a colleague.
   Writing helps information "stick." How would you explain this to students?
2. Examine the seven cognitive processes described in this chapter. How do you use these in your classroom? How could you use them more?
3. Pair up and write on sticky notes each idea that comes to mind when you think of the word "memory." Put the ideas in categories and create titles for each category. Stick these on chart paper, and compare categories and ideas. Discuss how you might adapt this activity for your classroom.
4. Ask your students to draw pictures of the concepts that you are studying. Then have them write about the concepts. Determine if recoding in writing was easier after drawing. Compare results with your colleagues.

Chapter 4. Step 4: Reinforce

1. How do your students respond to your feedback? Does the feedback change the way they are learning?
2. Assessment is feedback. Think about the difference between formative and summative assessment. Then look at your assessment techniques. Do you have a balance between the two? More formative assessment provides you with what kind of information?
3. Create a graphic organizer for the three types of feedback. Alone or with colleagues, fill in examples of motivational feedback, informational feedback, and developmental feedback that you use. Which do you use the most? Is there one type that you do not have in your repertoire? How could you add it to your current practice?
4. What are you learning from an episode of feedback? Feedback is like a dance in which you lead your partner (the student). If you are leading well, your partner follows. If not, you must change your technique.
5. Review the memory processes involved in reinforcement.

Chapter 5. Step 5: Rehearse

1. Discuss how rehearsals can take students from lower levels of thinking to higher levels of thinking.
2. On page 109, there is a list of ideas for cross-curricular teaching. The author suggests that, "Activities such as the ones above should be preceded by reflection time and followed by reflection and reinforcement after they are completed." Think about this practice. Do you agree with it? Do you use it? Write about a time that you "stepped back and then stepped forward" in this way.
3. Choose a concept that you will soon be teaching. Examine Figure 5.2. Create rehearsals that will place information in the various memory pathways.
4. Field trips create episodic memories if they are meaningful. Establish characteristics of a field trip that will make it memorable. Plan the field trip using the five memory "lanes."
5. Students need to know that sleep affects memory. Discuss with your students their sleep habits and have each keep a chart representing the amount of sleep they get each night.
6. Which mnemonic devices do you use? Try the peg system activity with your students, which will show them how powerful their memories can be. In your content area, find a way to use the peg system to encourage further use.
7. Consider your own practices in using rehearsal strategies. Do you vary your rehearsals to keep students engaged and to reach different learners? Discuss these with your colleagues and share ideas.

Chapter 6. Step 6: Review

1. Instead of saying, "We are going to have a pop quiz today!" try something like, "Let's have a retrieval practice!" That may be less threatening to your students. Practice testing raises test scores. See if this strategy works with your students.
2. How often do you review? Set up several reviews in your lesson plans for a unit. See if these added reviews make a difference.
3. Examine the review practices for your summative assessments. Determine if they support permanent memory, if you need to add more reviews, and if increasing your reviews affects student memory and achievement.
4. How can you orchestrate your reviews to encourage transfer?
5. With colleagues, set up a review for this chapter.

Chapter 7. Step 7: Retrieve

1. Have you found that students rely on their surroundings (episodic memory) for memory triggers? Notice which areas (you, the whiteboard, bulletin board, etc.) provide more cues. How can you use this information for your lessons?
2. How much test anxiety do your students experience? How do you help students through this? Talk with colleagues and share strategies.
3. If you begin with the end in mind, your assessment is created before your lesson plans. Therefore, your instructional strategies should match your assessment. Bring in assessments and unit plans to evaluate how closely aligned they are.
4. Vocabulary is coming to the forefront with its effects on student achievement. Susan did not match her test vocabulary to the vocabulary she used instructionally. Discuss why the vocabulary during rehearsal and review must also be used on the assessment.
5. Examine your summative assessment strategies. Which type of assessment do you use more, recognition or recall? Do you have a balance? Compare each student's success on the various assessments. Could you help a student by increasing the number of this type of assessment?
6. The suggestion of a blank sheet of paper at the end of a test for students to write down what they know is intriguing to the author. Try this strategy and see if you discover more information about how your students learn and remember.

Chapter 8. Realization

1. The seven steps all begin with R. Can you name them? Did this mnemonic device help you?
2. Now take the seven steps and see if you are able to identify the forms of memory used for each step.
3. Many teachers use these steps in creating daily lesson plans. Suppose that you have been asked to present information on memory to the parents of your students, perhaps at a PTO meeting. Decide what would be valuable and helpful for parents to know. Create an outline using the seven steps for your presentation.
4. Discuss with your colleagues the type of professional development you have had to increase student learning and memory. What are next steps for you and professional learning?

Appendix A. Brain Briefing

1. Review the lobes of the brain and their functions. Draw your own diagram.
2. Discuss the difference between explicit and implicit memory.
3. Survival, novelty, and choice are suggested as reasons for remembering. How do these fit into your classroom strategies? How do you approach these areas in order to help your students remember?

Appendix B. Graphic Organizer

1. Consider the graphic organizers in this appendix. As a group, classify these organizers according to how you could use them in the context of your content.
2. Share other organizers that you have found useful. Organizers lose their novelty if they are overused. Add these new organizers to the list you created in question one.
3. If you teach at a low-performing school, you may want to choose a few organizers that will be used at all levels. This continuity will help students remember how they are used and what information they call for.