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2.9 – Chapter Review

I haven’t gotten here yet, but I’ve made a review sheet to put here.

Algebra 02-9 Review

2.1 – Proportions

• Rename fractions as decimal numbers
• Write ratios and proportions that express relationships in data
• Solve proportions by multiplying to undo division
• Solve proportions by inverting both ratios
• Solve problems using proportions
• Review skills in working with percents

2.2 – Capture-Recapture

• Work with the idea of sample
• Become familiar with representative samples
• Understand the capture-recapture method

2.3 – Proportions and Measurement Systems

• Review the English measurement system and the metric system
• Convert measurement units using conversion factors
• Convert measurement units using dimensional analysis
• Learn and use the term rate

2.4 – Direct Variation

• Learn the properties of a direct variation equation
• Graph a direct variation equation
• Read a direct variation graph to find missing values in the corresponding table
• Use a direct variatnion equatoin to extrapolate values from a given data set
• Develop an intuitive understanding of the concepts slope and linear equation

2.7 – Evaluating Expressions

• Review or learn the rules governing order of operations
• Use calculator list operations to investigate the concepts variables, terms, and expressions
• Rewrite algebraic expressions that include multiplication and division, or addition and subtraction, by the same number

2.8 – Undoing Operations

• Build toward symbolic equation solving by working backward and undoing operations
• Translate situations into equations and solve by undoing.

2.8 – Undoing Operations

Vocab:

• equation
• solving equations
• solution
• undo

Objectives:

• Build toward symbolic equation solving by working backward and undoing operations
• Translate situations into equations and solve by undoing

Day 1:

To begin the lesson, I’ll talk a bit about what it really means to solve an equation. I’ll include examples where there is more than one solution as well such as x² = 9. I’ve certainly been working on the “undoing” parts over the last two weeks, so that should be part of their vocabulary by now.

I’m going to do the investigation on the board because in order for the students to understand questions 7 and 8 on the homework, they’ll need to see the “Sequence” column which isn’t part of the blank tables that are given in the book for step 4 of the investigation or example B in the lesson.

Then I will do example A with them on the blank tables that I will hand out to them – from there I will let them do Example B on their own.

I’ll make the kids do question #1 on their homework without the use of the calculator. It certainly shouldn’t be needed. I’ll be sure to tell them that if the work is shown correctly, with n numbers in the problem it should take n lines to do, including the problem itself. I’m finding that I’m a real stickler when it comes to showing work correctly.

Undoing Operations Blank Worksheet

My PowerPoint demonstration is too large in the old Office 2003 format, and the Office 2007 format is apparently a security risk here on WordPress so I’m not going to put it here. If you would like it, give me a jingle.

Day 2:

I gave questions 2 – 6 for homework yesterday, so that didn’t include question #1’s order of operations question. I made a practice worksheet for the students to do in class with me so that they can see the proper way of writing down the evaluation of numeric expressions.  The second part of the worksheet was three “undo tables”. I didn’t collect these so that they could use them for help with the homework tonight (#1, 7, 10).

2.8 Practice Worksheet (Order of Operations and “Undoing Equations” Tables)

2.7 – Evaluating Expressions

The order of operations seems to not be stressed during the lesson, so I spent a bit more time with it. Instead of having the students do the investigation, I led them through it by having them take notes that were turned in at the end of the period. It might be advisable to put question #1 from section 2.8 on the homework for this section instead of the next one.

I want the be able to make sure that students will be able to solve these equations later on, so their understanding of the order in which statements are applied to an algebraic expression is really important. Without an intuitive understanding of writing the rules of expressions in the correct order, they will have difficulty with the next section.

I was going to spend two days on this section, but at this point, I think one will be enough.

Slide number four’s problem has a solution of 4 no matter what number you begin with. So if you want the kids to do the “There are no grey elephants in Denmark” bit, it should work.

2.7 – Powerpoint Presentation

2.4R – Worksheet

I had to have a substitute so I gave my students the 2.3 and 2.4 Review Worksheets to do. 2.3 was turned in and was graded…they were quite poor. Section 2.4 seems to be easier for them. Upon return I had a shortened class so I read the answers to both sections and did many of them on the board paying particular attention to how the work should be shown:

• units need to be shown in the work and given in the answer
• direct variation problems need three steps:
  1. writing the equations
  2. showing the substitution step
  3. solving
• the differences between changing of units (using multiplication) and scaling (using proportions)

Because of a field trip, the last day of the quarter I gave a quiz on sections 2.3 and 2.4.

2-4 Quiz (version 1)

2-4 Quiz (version 2)

2.4 – Direct Variation

Day 1:

Vocab:

• Constant
• directly proportional
• direct variation
• constant of variation

The investigation works rather well. Since we’ve already learned how to put data into two lists and operate on them, finding the constant of variation was rather quick. Then we graphed the function and looked at the table. Having TI-Smartview on the screen is very helpful. Also, since the hand-drawn graph is the same function as what is needed for homework questions 1-3, a graphing calculator isn’t absolutely necessary for the homework.

I spent a great deal of time with the equation y = 1.6x, explaining what each part meant, and how knowing the value of one of the variables will give us the value of the other, just like the graph does. And now we have the opposite of how we’ve been “undoing” division when solving proportions, so solving for x isn’t a big stretch.

I also went over the Y=, WINDOW, TRACE, ZOOM, GRAPH, TABLE, and TblSet buttons on the calculator to get them more accustomed to using it.

Day 2:

After spending so much time with the calculator and how to set up equations the day before, (the investigation is very long), I made a worksheet for them to do with me in class to get used to solving these equations. Then I gave them the problems from the book for homework.

2.4 Worksheet

2.3R – Review

Once the students began working on the homework from section 2.3, which came after MEA break, it was apparent that they could not remember how to solve proportions. Coupled with the fact that solving proportions and doing dimensional analysis looked so similar on paper but were fundamentally different, I thought it wise to take a step back and have a review day.

The book’s objectives below are just about all met with the worksheet although some of them are concepts that will need to be discussed in class. My plan is to go over the homework in class paying particular attention to the differences between how and why proportions are used compared to dimensional analysis while paying particular attention to the mathematical vocabulary.

2.1 – Proportions

• Rename fractions as decimal numbers
• Write ratios and proportions that express relationships in data
• Solve proportions by multiplying to undo division
• Solve proportions by inverting both ratios
• Solve problems using proportions
• Review skills in working with percents

2.2 – Capture-Recapture

• Work with the idea of sample
• Become familiar with representative samples
• Understand the capture-recapture method

2.3 – Proportions and Measurement Systems

• Review the English measurement system and the metric system
• Convert measurement units using conversion factors
• Learn and use the term rate

2.1 to 2.3 Review

Quiz 2.3 (version 1)

Quiz 2.3 (version 2)

2.3 – Proportions and Measurement Systems

I thought that my students would have spent too much time measuring objects with a ruler, so I did this part a little differently. I put three segments up on the board, and asked the first three kids that entered the room to measure one of them in both centimeters and inches. I then showed the students that 39.5 inches was about a meter, and then asked how many centimeters the meter stick must be (to bring up the “cents” part once again). Finally, I measured the length of a desk, the length of a book, and the height of a student in both units. Using that data, I had the kids put numbers into their calculators in order to find the conversion. Luckily in all the classes they all said 2.54 cm to an inch.

I then used that to change the length of a girl’s head from inches to cm and the height of a student’s head from cm to inches.

For the next example, I found that I could eat 21 goldfish crackers in 47 seconds. I then talked about both unit rates (goldfish/sec) and (sec/goldfish). And then we found how many goldfish I could eat in an hour.

Lastly, the students noticed that I had grown a beard over the MEA break and I looked to find that the average beard grows at 1/2 inch per month. So we decided to find out how fast my beard must be growing in miles per hour. It gave me an opportunity to talk about scientific notation on the calculator as well – 1.096 * 10^(-8) miles per hour.

The students had trouble with question 1a to begin with, so I will have to have a review day to get them back up to speed with proportions.

2.2 – Capture-Recapture

Day 1:

To begin the lesson, I talked about how environmentalists often need to estimate how many fish are in a lake and asked the kids how they thought that it might be done to which they gave me a couple of ways. I then told them about the most common way – the capture/recapture method.

I didn’t do the investigation as it was set up. I first drew a lake on the board and drew in a bunch of “x”s to represent the number of fish. I then “captured” ten of them, tagged them, and threw them back into the lake denoting them with a different color “x”.

I then told the kids that I would wait for awhile for the fish to move around so that all my tagged fish weren’t in one area. Then I showed them that I could recapture a sample, and use that information to write a proportion to solve for the number of fish in the entire lake.

Next, I showed the kids the computer Exploration on page 105 and did the experiment four times. We discussed why the answers weren’t all the same (they ranged from just over 100 to more than 300) and what we could do to with those results to get a better estimate to the true number of fish in the lake.

Then I handed out the beans and told them to come up with their own simulation of what happened on the board and on the computer. After ten minutes, I had two groups come up and show what they had done.

Day 2:

The second day was spent going over all the homework and then doing questions 3 and 4 from the Extra Practice 2.2 worksheet that was put onto the screen. I put the problem up on the board and the students were given about two minutes for each to write a proportion that would represent the problem. I’ve found that it is useful to ask a student for “the ratio where you know both numbers” and then from there, label each part of the ratio, then ask the class what the second ratio must be. This way the kids realize that there is more than one way to write a proportion but that once one ratio is done, there is only one way to write the second.

2.2 Three Extra Practice Problems for Class

At the end of day 2, I gave the following quiz.

Algebra Sections 2.1 and 2.2 Quiz (version 1)

Algebra Sections 2.1 and 2.2 Quiz (version 2)

2.1 – Proportions

Vocabulary:

• ratio
• rational number
• terminating decimal
• repeating decimal
• proportion
• variable
• reciprocal

Lesson Objectives:

1. Rename fractions as decimal numbers
2. Write ratios and proportions that express relationships in data
3. Solve proportions by multiplying to undo division
4. Solve proportions by inverting both ratios
5. Solve problems using proportions
6. Review skills in working with percents

Lesson:

I began with a discussion about ratios – they can be expressed a few different ways. I divided 7/9, and 7/10 on the board to show what repeating and terminating decimals were, but also did mention non-repeating, not-terminating decimals as well.

Then I gave the definition of a proportion and talked awhile about equivalent fractions. Then I introduced a variable and we worked on solving proportions with the variables in the numerator and in the denominator. (Example A)

I then put some word problems on the board for the kids to write proportions for. I used Example B first, and then wrote some more and solved each of them.

Finally, I did Example C from the text book as well as two other problems involving percentages.

I assigned questions 1 – 6, 8, 9.

02-1 Word Problems to Proportions