Additional Notes on
Algebra: Themes, Tools, Concepts

Many if not most lessons in this book should be started with the book closed. The explorations that initiate many of the lessons offer an opening that should make it much easier for students to understand what the questions addressed in the lesson are. After working on that, you can determine how much of the lesson students should do in the book.

Like many textbooks, ATTC contains a lot more material than can be covered in a single year. You need to be judicious in selecting which lessons to do, and which parts of each lesson.

While non-traditional, the sequencing of the book makes sense. Chapters 1-4 are essentially pre-algebra. Ideally, students would have already seen much of this. Chapters 5-9 are the core of an introduction to algebra. The final chapters offer additional topics to choose among, many of which I prefer to teach in later courses. However, the starts of chapters 8 and 13 should happen earlier than is suggested by their placement in the book.

A Word from the Authors

We listed all the math educators whose ideas influenced us in the writing of the book. Would that all math curriculum authors listed their inspirations!

Letters

You might create your own version of the 'Dear Student' letter.

You can ask your students to write a 'Dear Teacher' letter, in response to the 'Dear Student' letter. Ask them to include information about their math background and history, their math fears, their math hopes. This can be the start of a useful conversation.

Later, you can follow up by asking students to have their parents read 'Dear Parents'. Students can interview their parents about their algebra (or other math) memories. Ask students to compare their algebra experience with their parents'. (Perhaps in the form of a letter to you, or as a newspaper article about changes in math education.)

Lesson 1.1

For #17, one student came up with the generalization: 'subtract the shortest perimeter from the longest; divide by 2; add 1'

Lesson 1.2

#17: change to 'for an area of 20'

Lesson 1.4

Repeat the instructions from problem 1 in #2-4.

#5: Suggest writing algebraically before evaluating. Figure seems to consist of three parts.

1.A

Suitable for homework: 7

The table is difficult to read. It needs to be broken up into four separate tables, and an illustrated example is needed.

Lesson 1.5

#11: You must use more than one block.

Lesson 1.6

Suitable for homework: 1, 6, 10-22

Lesson 1.7

The subtitle and teacher notes for the last three problems correspond to #20-21 in 1.11.

#29-31 should be moved to be before #21, and retitled More Perimeter Puzzles.

Lesson 1.8

The Teachers' Guide says: 'It's preferable you give no help'. Actually if needed, use the following hints:

• start by analyzing one-pane windows
• don't forget that window panes include both glass and wood -- how much of each material may help determine the price

The windows should look (graphically) like the ones in Ch 7.2, and they should each be labeled, a, b, c, etc. to facilitate discussion.

1.B

Make clear that no window should be left uncovered.

#4: reword so that report is a letter to Ms Tall. (In fact the strategy of suggesting a specific audience for the reports should be used whenever possible.)

#3, 4 solution: 5 undecided windows can get long drapes, $17 left over

Lesson 1.11

In the instructions between #7 and #8, and beyond, use the word 'change' instead of the word 'step'.

#16-21 should be below the line, anywhere in the chapter.

#17 sols are missing. #19 sols need to be illustrated. We need a better explanation of why 27 is impossible.

#24: the reference is to the previous problem, not to problem 11.

Lesson 1.12

Suitable for homework: (using dot paper) 4-6, 11-17

Useful for assessment: 17

#1: 'at least three', rather than 'as many as you can'

#15a and c: Note that there are only four right and acute triangles that satisfy the given conditions. Accept solutions that are mirror images of each other, or seize the opportunity to show that there are obtuse triangles that work.

1. Essential Ideas

Instructions preceding #2-3 are unclear. Should be rewritten as a, b, c, d:

a. write what the blocks show in terms of x and y

b. use substitution to evaluate if x = 0 and y = 2

c. etc

Nine Function Diagrams is a useful activity to do at some point while working in this chapter.

Lesson 2.1

#6d: one of the five-blocks is supposed to be upstairs.

Lesson 2.3

#2: Do not use 3-D blocks.

#5 is not an exploration.

Lesson 2.4

#20: the question should end after the word 'multiplication'. This problem cannot be done with the blocks.

Lesson 2.6

Suitable for homework: add 1-5

Lesson 2.7

#7: Write them in the form y = a function of x

#22: Not a 'Generalization'

Lesson 2.8

If possible, the last diagram on p 64 should be at the top of p 65, with #5.

#6 needs a 'key'.

#4 solution: Change 'Using the same scale...' to end of paragraph to: 'In general, the speed is the distance traveled per unit of time.'

Lesson 2.10

#1c. Explain the pattern in terms of the figure. Give examples.

The table should say '# of blocks' instead of 'Figure #'

#2-6: The following are made of 5-blocks. (The figures need to make that clear, with double thickness between the blocks.)

Lesson 2.11

Answer to #8 is very incomplete.

Lesson 2.12

The answer to #9 should be 22.5.
The answer to #10b should be 27 for all the triangles.

#12: not negative (instead of nonnegative)

#17: Change the hint to: You may use the square (1,3), (1,4), (2,4), (2,3)

If your students have a good understanding of this material, they may feel this lesson gets tedious. It is OK to skip many problems here, or to break it up into smaller chunks.

Essential Ideas

#2: Do not use parentheses in your answer.

Before #5, change 5-8 to 5 and 6.

#9: do not use 1 as an exponent

#11, 12: perhaps more points are needed

Lesson 3.1

#8: patterns (instead of pattern)

Lesson 3.2

#11: Change to 'When x is negative, y is greater than 5. What does this say about two negatives?'

Delete #13b, which is very tedious.

Lesson 3.3

#18. Do not use a calculator.

Lesson 3.4

#1 is not an exploration. If this cannot be changed, at least the end-of-exploration graphic should be inserted after #1.

4a. 'numbers' instead of 'answers'

5, step (2): Multiply by 4.

A

Remove the thick horizontal line from the tables at the top of the page. (OK in the one at the bottom.)

Lesson 3.5

#23: change 'when' to 'whenever'

Lesson 3.6

#10: …in problem 9, if there is no remainder, write the related…

We need more problems of the type in #11. Not necessarily here.

#19: put a 'y2' to the left of the bottom row of the table

#18-19: no sols given!

Lesson 3.7

#4: put 'different' in italics

#12: not a report, but a key

#13: Hint: what is 1/82 as a decimal?

#17: not an exploration

Lesson 3.8

#6: 'approximately' should be in italics

#5: suggest a scale for the axes, either in the TG, or right in the problem. Perhaps a figure.

Add a problem after #15: redo #8, using the formula and trial and error

Lesson 3.9

#14 solution: 'beddy-bye' is in the Random House dictionaries.

Lesson 3.10

#1 should not be there. Kids resent making a diagram that's already in the book. (As a rule, problems where they are asked to make tables, diagrams, or graphs that are already in the book should be removed, and replaced with questions involving reading the table, diagram, or graph.)

Lesson 3.12

#3a. Change 'bottom right' to 'bottom left'

Lesson 4.3

You will need: graphing calculators (optional)

You can use the graphing calculators for #12-16.

#3, 8: Where it says to use eight values, suggest instead 7 (or 9) values, including 0.

Lesson 4.4

8b. should be 'to problem 7.'

Instructions preceding 10-12: a. Don't make a table of values if you are using a graphing calculator.

14-17: Replace 'function' with 'polynomial function'

A

1, 2b answers: Sally probably got a ride with Neil in the final stretch, with her bike on the rack.

Lesson 4.5

Add to the instructions preceding #21: 'It is possible to encounter three-digit numbers while carrying out this procedure.' (Or change the example so that it does.)

Lesson 4.6

#7 should be screened.

#19. Break it up as a, b, with a at the beginning, and b starting with: 'Do all your answers agree?' Then add part c:

c. How could one get the real density from Gabe's data?

#23: …in all the data. (See Problems 5, 6, 15, 16.)

Lesson 4.7

#12: …to be acceptable? Decide how many degrees off is OK, and give the range within which his method works. Explain.

#13 would be a lot more interesting if the words were removed from the second column. (Leaving the first as an example, perhaps: 'shorts and T-shirts'.)

Change #13: Report: Write a letter to Mr. and Mrs. Gral about estimating temperatures. Include easy methods for Fahrenheit/Celsius conversion, and a clothing reference table like this:

Lesson 4.8

#11. Remember that the ticks are supposed to mark off equal amounts of liquid.

Lesson 4.9

#15 answers: pastry, not loaf

B

#1-3 should be screened, and labeled as non-core in the Teachers' Edition.

#4: add a question before a and b (changing them to b and c):

a. Is y = 2x an example of direct variation?

#7: put 'area function' in italics

#8: insert the question: 'Is y = 3 an example of direct variation?'

Lesson 4.10

#6: …20 mph, etc. up to 80 mph.)

#13: replace the key with a lightbulb.

#13 is difficult to understand.

Answers: Add the following to #13:

'However, the validity of the rule should be judged in the worst case scenario, and that would be when it is used by someone who uses the image of a small car.'

Answers: take out the words 'to the nearest cent' from #15b.

Add a problem at the end:

#17. Is the sales tax you found in #16 the same as the one you found in #15? If not, find a sales tax that works for both problems.

Lesson 4.11

Make #11 a Summary, rather than a Report.

Lesson 4.12

#15. Find five functions of x whose value…

Essential Ideas

#18: make graphs start at the origin, or on one of the axes.

Lesson 5.1

#2: 'the two right-hand dials' or 'the two dials on the right', instead of 'the last two dials'.

Lesson 5.2

#12: 'two positive values for P' and 'two negative values for P', instead of 'several'.

Lesson 5.3

'Division and the Distributive Law': the example needs to be explained in much more detail.

#11: …the above division.

A

#10. 'at least one curve' rather than three

Lesson 5.6

Move #1 to the end of the lesson, screen it, and insert: '…whole numbers other than 1, is it possible…'

Lesson 5.7

#14-16: 'Multiply', instead of 'Use the Lab Gear'.

Answers: both function diagrams need to be symmetric. (And in-out lines should be added.)

Lesson 5.8

#2 answer should be: All multiples of five except 5 and 15 are possible. No other numbers are possible.

The answers to #26-28 are missing.

B

#8 answer: 2xy cannot equal 4xy if x and y are not zero.

Lesson 5.9

#7a. 'cubes' instead of 'tiles'

#3 answer: in the table, for 15, it should be 1+2+3+4+5, not 12+3+4+5.

Lesson 5.12

Before #2: 'move' should be in quotes, not in italics.

Answers, #8: show table grid lines

Practice

#1a. Open parentheses before 6s

Lesson 6.1

Make clear that 'free miles' are per day, not for the whole duration of the rental.

#2. 'Which company do you think offers the best deal?' (Instead of 'Which car...')

Clarify that #1-5 refer only to one-day rentals.

Lesson 6.2

#2: change the y's to x's.

Lesson 6.4

#28: change 12 to 12x

A

Before #5: remove the words 'every two weeks'.

#4: change '$K' to '$x'

#7 answer: Lara will run out of money after 17 and a half months. (Not 20).

Lesson 6.5

Use graphing calculators for #10-20.

Lesson 6.6

Before #13: Add: 'Assume she receives no interest.' Or remove the references to a savings account.

Lesson 6.7

#4 answers: the diagonal part of the graph and its description should be removed

Lesson 7.1

#1 answer is incomplete.

#11 answer: shading is incorrect in figure. See #12 and #13 for the correct way to do it.

Lesson 7.6

#6: should be '…exactly two solutions?'

#17 is probably too hard. Cut it?

The beginning of this chapter should be earlier in the book.

Lesson 8.5

#11: very awkward sentence

TG: 'A doubling population' The crossreference to 7.11 is wrong.

Lesson 8.9

At start of lesson, add: 'Do not use the exponent 1.'

#10: change key to lightbulb

#12: Add: 'Do not use the exponent 1. One is impossible.'

Lesson 8.10

Before #17: should be 4x8

Lesson 8.11

#21 solution: change 'subtraction' to 'addition'

Lesson 9.3

#9: change first rectangle to by 2.

#24: 'are of length 5 and start at the origin, what are the possibilities…'

Lesson 9.10

#9, 12 sols: show grid

#10 sol: show sol

Lesson 10.1

Can part of the lesson be made optional? It is very long and gets tedious.

Lesson 10.3

Mystery Containers, Teachers' Edition: Delete the last sentence.

#1c answer is incomplete. There are three cases.

Lesson 11.3

#4 sol: 7/11, not 7.11

Lesson 11.5

I learned Game 1 and Game 2 (in the Exploration) from a book by Marilyn Burns.

Lesson 11.7

#25 should be broken up into two problems.

A

In the figure, L/2, L/3, etc represent the length of the segments, and should be next to a representative segment, not centered on each figure.

Support Masters

1.2 graph axes: should be 'square'