Chapter 20: Pre-Algebra

By Elisabeth, Joanne, and Tim

Chapter 20 Overview: New Skills

Pre-skills and Pre-Assessment

Example Lesson: Intro to Variables and Tables

Chapter 20 Application Items

Agenda

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)

New Skills (see overview)

Rules for combining integers (Format 20.3)1. If the signs of the numbers are the same, you add2. If the signs of the numbers are different, you

subtract3. When you subtract, you start with the number that

is further from zero on the number line and subtract the other two

4. 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)

New Skills (cont’d)

The traditional four rules for multiplying:1. Positive x Positive = Positive2. Negative x Negative = Positive3. Negative x Positive = Negative4. Positive x Negative = Negative

A one-rule alternative is described (p.451)

New Skills (cont’d)

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

Pre-Skills & Pre-Assessment

Coordinate System Concepts

1Variable

• Units• Table

2Number Line

• Horizontal• Axis (1-dim.)• Scale• Ray vs. Line• Operations

3Coordinate Plane

• Vertical• Axes (2-dim.)• Coordinate• Ordered Pair• Function

(Smart Art can be distracting)

1) What is a Variable? • Unit

• Table

2) How do you use a Number Line?

• Horizontal• Axis (1-dim.)• Scale• Ray vs. Line• Operations

3) What is a

Coordinate

Plane?

• Axes (2-dim.)• Vertical• Coordinate• Ordered Pair• Function

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)

Example LessonLesson 1: Pages 443 to 445

Introduction to VARIABLES

1. Given a single variable, students will generate and consider reasonable values

2. Given a table with values of one variable, students will plot points on a number line

3. Given two related variables, students will generate and consider reasonable pairings of values and use a table to display their pairs

Small group (4 or fewer) activityMake lists of values for units

Present next ideas to large groupDecide on symbols to use for unitsMake a table of values for a unitMark values on a number line

Individual practice Turn lists into tables, mark values

Present next ideas to large groupAdd a column to a tableGenerate reasonable pairsUse statements to check pairs

Small group practiceAdd columns, generate pairs, check, save tables for later

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

Opener: Small Groups (4 or fewer)

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

New ideas: Whole Class

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

New ideas: Whole Class (cont’d)

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?

Discussion: 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

New Ideas: Whole Class

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?

Next Time

1. Complete the following function table.

Chapter 20 Application Items

x Functionx times 4

Answery

0 0 x 4 0

1 1 x 4 4

2 2 x 4 8

3 3 x 4 12

4 4 x 4 16

5 5 x 4 20

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.

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

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.

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

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?

Application Items (cont’d)

Fraction Ratio Quantity

5th grade

boys

3/4 3 30

5th grade

girls

1/4 1 10

Total fifth

graders

4/4 = 1 4 40

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.

Application Items (cont’d)