Algebraic Concepts and Expressions

ASE MA 01:

Algebraic Concepts and Expressions

Steve Schmidt

abspd.appstate.edu

NotableQuote

“Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time.”

- Thomas Edison

You Can Write on this Packet!

You can find everything from this workshop by Googling abspd. Look under: Teaching Resources, Adult Secondary Resources, Math, ASE 01: Algebraic Concepts and Expressions

Agenda

8:30 – 10:00 Teaching Math Reasoning

10:00 – 10:15 Break

10:15 – 11:45Algebra: Concrete to Abstract

11:45 – 12:45 Lunch

12:45 – 2:00 Active Math Learning

2:00 – 2:15 Break

2:15 – 4:00Battle for Hearts and Minds: Growth Mindset

Workshop Objectives

- Learn to teach algebra in context

- Understand the benefits of using an active approach for learning algebra

- Learn how to generate increased algebra understanding by using manipulatives and teaching from

concrete to abstract

- Understand ways to help ease math anxiety and strengthen adult secondary learners' content

knowledgethrough understanding the growth mindset

Winning the Battle for the Hearts and Minds

______is the most failed high school class. Some of our students have failed it more than 5 times before they reach our classrooms!

Carol Dweck – Mindset

Fixed Mindset / Growth Mindset
Math Belief / Math is about your ability
Challenges / Avoids challenges
Effort / Effort is fruitless
Feedback / Ignores feedback
Success / Threatened by others’ success
Result / Failure

Salman Khan (Khan Academy) on the Growth Mindset

My 5-year-­old son has just started reading. Every night, we lie on his bed and he reads a short book to me. Inevitably, he’ll hit a word that he has trouble with: last night the word was “gratefully.” He eventually got it after a fairly painful minute. He then said, “Dad, aren’t you glad how I struggled with that word? I think I could feel my brain growing.” I smiled: my son was now verbalizing the tell­-tale signs of a “growth­ mindset.” But this wasn’t by accident. Recently, I put into practice research I had been reading about for the past few years: I decided to praise my son not when he succeeded at things he was already good at, but when he persevered with things that he found difficult. I stressed to him that by struggling, your brain grows. Between the deep body of research on the field of learning mindsets and this personal experience with my son, I am more convinced than ever that mindsets toward learning could matter more than anything else we teach.

Researchers have known for some time that the brain is like a muscle; that the more you use it, the more it grows. They’ve found that neural connections form and deepen most when we make mistakes doing difficult tasks rather than repeatedly having success with easy ones.

What this means is that our intelligence is not fixed, and the best way that we can grow our intelligence is to embrace tasks where we might struggle and fail.

However, not everyone realizes this. Dr. Carol Dweck of Stanford University has been studying people’s mindsets towards learning for decades. She has found that most people adhere to one of two mindsets: fixed or growth. Fixed mindsets mistakenly believe that people are either smart or not, that intelligence is fixed by genes. People with growth mindsets correctly believe that capability and intelligence can be grown through effort, struggle and failure. Dweck found that those with a fixed mindset tended to focus their effort on tasks where they had a high likelihood of success and avoided tasks where they may have had to struggle, which limited their learning. People with a growth mindset, however, embraced challenges, and understood that tenacity and effort could change their learning outcomes. As you can imagine, this correlated with the latter group more actively pushing themselves and growing intellectually.

The good news is that mindsets can be taught; they’re malleable. What’s really fascinating is that Dweck and others have developed techniques that they call “growth mindset interventions,” which have shown that even small changes in communication or seemingly innocuous comments can have fairly long­-lasting implications for a person’s mindset. For instance, praising someone’s process (“I really like how you struggled with that problem”) versus praising an innate trait or talent (“You’re so clever!”) is one way to reinforce a growth ­mindset with someone. Process­ praise acknowledges the effort; talent­ praise reinforces the notion that one only succeeds (or doesn’t) based on a fixed trait. And we’ve seen this on Khan Academy as well: students are spending more time learning on Khan Academy after being exposed to messages that praise their tenacity and grit and that underscore that the brain is like a muscle.

How do we develop the growth mindset in our students?

1. Share with students the research about malleable intelligence! (See handout)

2. We can also:

Teaching Activities

The handouts for the following teaching activities are all on the ABSPD web site. Look under: Teaching Resources, Adult Secondary Resources, Math, ASE 01: Algebraic Concepts and Expressions

Clothesline Algebra

Set up a clothesline somewhere in your classroom. Stringing the clothesline between two chairs on a table works well. (You can also make a “virtual” clothesline by drawing a line on a whiteboard.) Put the number 0 in the middle of the clothesline and also place some other numbers like 1, -1, 2, -2, etc. Give all students a card with an unknown on it. Tell them that they will be told a value for their unknown (like n = 1) and that they now have to put their card on the clothesline.

When all students have placed their cards on the line, gather around the clothesline and see if the cards were placed correctly. When we come across cards placed incorrectly, ask students to talk with a partner about where the card should go. This activity can be extended by changing the unknown value.

Human Coordinate Plane

In this activity, students become points on a coordinate plane! Before this activity, use masking tape to create a coordinate plane in your classroom or hallway. Tile floors with 1 foot squares work best. The activity has three parts:

1. People in a group are given coordinates and place themselves on the grid

2. The group creates a letter like T, Z, or I on their graph and writes down the points where each member is standing

3. Groups trade cards and try to guess what the other groups’ letter is!

Creating Rules: The S Pattern Task

As we teach algebra, developing students’ algebraic thinking skills should be a key goal. “Algebraic thinking includes recognizing and analyzing patterns, studying and representing relationships, making generalizations, and analyzing how things change” (Seeley, 2004). The S Pattern Task asks students to first visualize, and then fill in a chart describing a pattern. They will then make a generalization as they develop a rule that describes the changes they see in the pattern.

Algebra Tiles

Sharma (2005) describes how math should be taught from concrete to abstract. Instead of diving into an abstract algebra concept like or 3x, students should begin with a more concrete representation. Algebra tiles use squares and rectangles to show , x, and whole numbers. There are plenty of great videos on YouTube that show how to use algebra tiles. As Algebra 1 is the most failed class in high school, teaching students abstract concepts in the same way they did not understand them before is not a best practice.

Algebra Card Games

Algebra card games are another way to teach algebra from concrete to abstract. These four games use playing cards to represent integers. Red cards show negative numbers and black cards show positive numbers. These games can be easily adapted and help students become more fluent in adding, subtracting, multiplying, and dividing integers. Understanding integers is a crucial building block for algebra success.

Phone Plans

This lesson plan is from the EMPower math series book Seeking Patterns, Building Rules. I love the EMPower series because it teaches matematical thinking instead of just rote memorization of concepts. There are eight books in the series and they cover topics from basic math up through algebra and geometry.

The phone plan is a wonderful activity since it is contextual (almost all students have a cell phone). This activity is not on the ABSPD web site since it comes from a copyrighted book. Information about the EMPower series is available at:

Discovering Exponents

This activity does double duty as it helps build algebraic thinking while teaching students to use the official GED calculator, the TI 30XS. Students first guess about which of two exponents is bigger and then use their calculator to find out for sure. Then, students are asked to make generalizations about their findings.

Use Other Variables than X

For some students, using another variable than x to show an unknown unlocks the door to algebra. Use n for number, use t for t-shirts, use f for feet etc. Sometimes little things mean a lot when teaching a subject students have not been successful in before.

Teach the Equal Sign

From their study of arithmetic, many students think the equal sign means, “the answer comes after this” (3 + 4 = ___ ). Help them understand that in algebra, the equal sign means that whatever is on one side must balance with whatever is on the other side of an equation: 2n + 3 = n - 1.

UPS ✔Problem Solving Method

1. Understandthe problem

What are you asked to do?

Will a picture or diagram help you understand the problem?

Can you rewrite the problem in your own words?

1. Create a plan

Use a problem solving strategy:

Guess and check Solve an easier problem

Make a list Experiment

Draw a picture or diagram Act it out

Look for a pattern Work backwards

Make a table Change your viewpoint

Use a variable

1. Solve

Be patient

Be persistent

Try different strategies

1. Check

Does your answer make sense?

Are all the questions answered?

What other ways are there to solve this problem?

What did you learn from solving this problem?

Source: Polya, How to Solve It

Understand
Plan
Solve
Check

Practice Problems

1. Two boys, Shawn and Curtis, went for a walk. Shawn began walking 20 seconds earlier than

Curtis.

• Shawn walked at a speed of 5 feet per second.

• Curtis walked at a speed of 6 feet per second.

For how many seconds had Shawn been walking at the moment when the two boys had walked

exactly the same distance?

2. The math club sells candy bars and drinks during football games.

• 60 candy bars and 110 drinks will sell for $265.

• 120 candy bars and 90 drinks will sell for $270.

How much does each candy bar sell for? (Note: Put the answer in dollars and cents.)

3. Katie and Jennifer are playing a game.

• Katie and Jennifer each started with 100 points.

• At the end of each turn, Katie’s points doubled.

• At the end of each turn, Jennifer’s points increased by 200.

At the start of which turn will Katie first have more points than Jennifer?

4. Alex walked 1 mile in 15 minutes. Sally walked 3,520 yards in 24 minutes. In miles per hour, how

much faster did Sally walk than Alex? (Note: 1 mile = 1,760 yards)

5. On an algebra test, the highest grade was 42 points higher than the lowest grade. The sum of the two grades was 138. Find the lowest grade.

6. Dominic earns $285 per week plus an 8% commission rate on all his sales. If Dominic sells

$4,213 worth of merchandise in one week, how much will his total earnings for the week be?

Sources: NC Algebra 1 End of Course Test, GED® Testing Service

Understanding Quadratic Equations

A quadratic equation in standard form looks like a + bx + c = 0 With numbers, it’s: 4 + 3x + 2 = 0

We’ll look at two methods for solving quadratic equations: factoring and the quadratic formula.

Factoring

Common Factor

With this type of problem: + 15x = 0 we’ll first search for a common factor.

Since there is an x in both terms, we’ll factor it out:

+ 15x = 0

x (x + 15) = 0

Then we’ll set both factors equal to 0 and solve

x = 0 x + 15 = 0

- 15 -15

x = - 15

We can check both solutions by plugging them into the original equation:

+ 15x = 0
+ 15(0) = 0
0 + 0 = 0 / + 15x = 0
+ 15(-15) = 0
225 + (-225) = 0
0 = 0

Product of Two Binomials

Since this equation + 7x + 12 = 0 does not have any common factors, we’ll look to factor it into two parenthesis: (x + or - number) (x + or - number)

When the is by itself, we’ll look for two numbers that when multiplied equal the last term and when added equal the middle term. For + 7x + 12 = 0 we’ll look for two numbers that multiply to 12 and add up to 7:

Multiply / Add
1 (12) = 12 / 1 + 12 = 13
2 (6) = 12 / 2 + 6 = 8
3 (4) = 12 / 3 + 4 = 7

Hint: Many times the factors that are the closest together on the number line are the correct factors

Now we’ll create two parentheses using this guide to help with the signs:

Signs in Problem / Factored Signs
(+) (+) / (+) (+)
(+) ( - ) / (+) Larger Number
( - ) Smaller Number
( - ) (+) / ( - ) ( - )
( - ) ( - ) / ( - ) Larger Number
( + ) Smaller Number

Since the signs in our problem are both positive, our factors will be too: (x + 4) (x + 3) = 0

We’ll set each factor equal to zero and solve:

(x + 4) = 0 (x + 3) = 0

x = - 4 x = - 3

We can go back to the original equation and plug in each solution to see that it works:

+ 7x + 12 = 0
+ 7(-4) + 12 = 0
16 + (-28) + 12 = 0
0 = 0 / + 7x + 12 = 0
+ 7(-3) + 12 = 0
9 + -21 + 12 = 0

0 = 0

Quadratic Formula

In standard form, a quadratic equation looks like: a + bx + c = 0 Using the quadratic formula, we’ll first identify a, b, and c and plug them into the formula.

+ 7x + 10 = 0

a (number with ) is 1

b (number with x term) is 7

c (whole number) is 10

Now we’ll plug these numbers into the quadratic formula:

If students are having trouble factoring, the quadratic formula may be the best way for them to solve a quadratic equation. The quadratic equation always works!

Let’s Practice

1. - 11x = 0 / 2. + 20x = 0 / 3. – 9x + 8 = 0 / 4. + 16x + 48 = 0
5. - 5x - 14 = 0 / 6. - 5x - 36 = 0 / 7. + 3x - 28 = 0 / 8. - 3x - 1 = 0
9. - 15x + 54 = 0 / 10. + 2x - 80 = 0 / 11. + 5x - 36 = 0 / 12. - 8x + 12 = 0

Resources

Professional Development

Google: nc college and career readiness training calendar

The 2015 - 16 training calendar shows all the professional development opportunities available

across the state. Professional development will be held in each region.

AHS Algebra 1 Course

Search for AHS Algebra 1 under continuing education courses

This is a complete Algebra 1 course that can be used for Adult High School

Algebra Resources

Google: algebra with pizzazz

This classic work which has Algebra 1 puzzles students can do to gain extra practice in

various concepts.

Google: get the math

Get the Math show how algebra is used in the real world in music, sports,fashion, video

games, restaurants, and special effects.

Manipulatives

Google: national library of virtual manipulatives

This site has algebra manipulatives for instruction across grade levels

Algebra Tiles

Google: algebra 4 all learnport

This site provides an easy to use set of virtual algebra tiles.

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