Chapter 20: Pre-Algebra

1. Chapter 20: Pre-Algebra By Elisabeth, Joanne, and Tim

2. Agenda • Chapter 20 Overview: New Skills • Pre-skills and Pre-Assessment • Example Lesson: Intro to Variables and Tables • Chapter 20 Application Items

3. New Skills (see overview) • Using the coordinate system to visually represent values of variables (example lesson to come) • Finding a missing value in a ratio equation and other ratio-related problem solving • Using prime numbers to reduce fractions • Simplifying expressions that involve exponents • Operating on integers (using rules to work with negative numbers, see next slide)

4. New Skills (cont’d) • Rules for combining integers (Format 20.3) • If the signs of the numbers are the same, you add • If the signs of the numbers are different, you subtract • When you subtract, you start with the number that is further from zero on the number line and subtract the other two • The sign in the answer is always the sign of the number that was further from zero (Another option is to use a number line to teach the actual concepts involved)

5. New Skills (cont’d) • The traditional four rules for multiplying: • Positive x Positive = Positive • Negative x Negative = Positive • Negative x Positive = Negative • Positive x Negative = Negative • A one-rule alternative is described (p.451)

6. Pre-Skills & Pre-Assessment Revise any statements that uses its underlined word incorrectly. • The specific unit measured by a ruler is how much the numbers stand for (for example inches and centimeters are examples of specific units) • The type of unit measured by a tape measure is distance • When a problem involves an unknown amount, you can use a symbol like x (which is called a variable) to stand for the unknown value. • In algebra, a function is a way to relate two variables to each other • The ratio of a dirt bike ramp's height to its width might be 1 to 2  • 2, 3, 5, 7, and 11 are prime numbers Use the number line for these ones. • How far is 4 from -3? • Draw a vertical axis at 0 • Label the x axis and y axis on your drawing • Plot a point at (3,5) Also think about including some of these on your pre-assessment: • Solving problems involving proportion or percent, multiplying two powers of the same base number, reducing fractions, and operating on negative numbers

7. Coordinate System Concepts

8. (Smart Art can be distracting)

9. Example Lesson • This is the opener lesson for a unit that culminates in the use of Graphical and Algebraic models • Students first need a good understanding of what variables are and how to represent them numerically (in a table)

10. Opener: Small Groups (4 or fewer) • Hand out envelopes • In these envelopes are cards, each labeled with the name of a measurable aspect of a person’s life • Group task (3 min): • each person take one of the cards • on scratch paper, list 3 reasonable values for the aspect labeled on your card (your own current value for that aspect is just one that you can use – think of some for other people too) • Switch cards and do it again until you have listed 3 values for each of the four aspects

11. New ideas: Whole Class • The four cards are examples of aspects about a person that can have different values. • What should we call things like this that can vary so much in their value? • Choose a fitting one-letter symbol for each of the four aspects we looked at

12. New ideas: Whole Class (cont’d) • In algebra, when you have a bunch of values for a variable, there is more than one way to display those values • You can display it using a table. The symbol for that variable goes at the top of the column. Values for the variable go (in ascending order) below it. • You can also display the values for a variable using a number line. Here, the number line (or axis) itself is labeled with the symbol for the variable and dots are made along it where the values are located. • Individual task: • Display the numbers from your four lists using the four tables and four number lines on the handout

13. Discussion: Whole Class • We’ve seen how to display the values for a variable when it is alone • Variables can also be related to each other Which of the four aspects make good pairs?

14. New Ideas: Whole Class • If we transfer the values for one of the (cause) variables in a pair to a 2-column table, we can think about the resulting values for the other variable in the pair • On your own: • Choose one of your pairs of related variables • Put the values for the causing partner into the first column of a 2-column table (on back side of sheet) • Put reasonable accompanying values for the other variable in the second column

15. Next Time • A number line is only able to display values for one variable • (There is only one axis and it can stand for only one type of value) How can we display with dots when there are two variables involved and we have to show the value of one based on the value of the other?

16. Chapter 20 Application Items • 1. Complete the following function table.

17. Application items (cont’d) • 2. Construct a set of problems for use in Format 20.3 for combining integers. • Make a poster board titled: Steps to Combining Integers • Step 1: If number signs (negative, negative) and (positive, positive) are the same:add numbers. • Step 2: If number signs (positive, negative) and (negative, positive) are different: subtract numbers. • Subtract from the number farthest from the zero on number line. • Step 3: Solve the problem. • Step 3: Always, look and take the number sign from the higher number closest on the number line from zero and attach it to your answer. • For example: -20 + 8 = ____ • The number signs are different, so subtract 8 from 20. (negative, positive) • Result: -12 • Twenty is the larger number and it has a negative sign in front of it, put a negative sign in front of 12. Final answer is -12 • Another example: 15 – 8 = ____ • The number signs are different, so subtract 8 from 15. (positive, negative) • Fifteen is the larger number and it is positive, so put a plus sign or leave it blank in front of 7. Final answer is 7. • One more example: 4 + 5 = ____ • The number signs are the same, so add them both together. • Five is the larger number and it is positive, so put a plus sign or leave it blank in front of 9. Final answer is 9.

18. Application items (cont’d) • 3. Construct a set of examples and nonexamples for students to use in detemining if a number is a prime number. • Rule: All prime numbers have only two different factors: 1 and itself, that’s it. • Non-example 1 x 1 = 1 all the same factors • Example 1 x 2 = 2, 1 and 2 are the 1 and itself factors • Non-example with 4, 1 x 4 = 4, 2 x 2 = 4, 3 different factors, 1, 2, and 4, does not follow rule • Example: 1 x 5 = 5, 1 and 5 are the 1 and itself factors • Non-example with 10, 1 x 10 = 10, 2 x 5 = 10, 4 different factors, 1, 2, 5 and 10 does not follow rule

19. Application items (cont’d) 4. Explain why initial teaching and practice of the concept of absolute value is best accomplished using the phrase farther from zero rather than absolute value. Integers are made up of positive and negative numbers. Conceptually it can be more challenging, some students may find it difficult to understand negative numbers, therefore, Stein recommends referring to a number line for visual representation with the number zero as the reference point for the necessary steps, such as left of line as negative and right of line as positive. Also for number sign to final answer by looking for the greater number from the number zero. Unlike the combining integers strategy, teaching the conventional notation (teaching absolute value, first), the student needs to understand the first number with the opposite number and determine correct number sign, for example, -4 – 5 =, the student would have to understand that conceptually the problem is 4 – 5 = because the absolute value of -4 is 4. More complicated problems, could be even more difficult to understand, such as 4 – (-5) = 4 – 5 =, and so forth. Another concept to grasp, when the problem has a plus sign for computation, whether it is (positive, positive, or negative, negative) the final answer will always be positive. 4 + 3 = 7, -4 + (-3) = 7. Furthermore, the absolute value rules are different for multiplication and division.

20. Application items (cont’d) • 5. Outline the steps in teaching students to find the prime factors of a number. • Let’s take the numbers 18 and 32. We can also make this a fraction 18/32. • Work out all the factors. • 18 divided by 2 = 9 • 9 divided by 3 = 3 • Now the prime number rule: two different factors, 1 and itself. • Take all the prime factors: 2 x 3 x 3 = 18 • 32 divided by 2 = 16 2 x 3 x 3 = 18 3 x 3 = 9 • 16 divided by 2 = 8 2 x 2 x 2 x 2 x 2 = 32 can reduce, 2 x 2 x 2 x 2 = 16 • 8 divided by 2 = 4 • 4 divided by 2 = 2 • 2 x 2 x 2 x 2 x2 = 32

21. Application Items (cont’d) • 6. Write a classification problem with fractions to be solved with ratio table, and show the solution. • Three-fourth of the fifth grade class are boys. There are 10 girls in the fifth grade class. There are a total of 40 fifth grade students. How many boys are there?

22. Application Items (cont’d) • 7. Show the solution process to the following comparison problem with percentages, using the ratio table shown. • Show the solution process to the following comparison problem with percentages, using the ratio table shown. • Ben is 35% taller than his mom. If Ben is 21 inches taller than his mom, how tall is Ben and how tall is his mom? (Hint: Ben is being compared to his mom, so who is equal to one or 100%) • Solution process: 21=.35X.