ASE MA 01:Algebraic Concepts and Expressions

Steve Schmidtschmidtsj@appstate.edu

abspd.appstate.edu

Notable Quote“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

Agenda8:30 – 10:00 Problem Solving to Build Math Reasoning

10:00 – 10:15 Break

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

11:45 – 12:45 Lunch

12:45 – 2:00 Active Math Learning

2:00 – 2:15 Break

2:15 – 4:00 Achieving the Growth Mindset

Workshop ObjectivesPage 1 | ASE MA 1: schmidtsj@appstate.edu

This course is funded by:

- Learn to teach algebra using a research based approach

- 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 knowledge through understanding the growth mindset

A Trip Back in Time and What Research SaysHistorically in adult basic skills, students started studying math and were immediately sent to “fraction land.” They learned how to add, subtract, multiply, and divide fraction problems by hand. This took students several weeks (or months), and they did hundreds of problems in workbooks. If they stayed in the program and mastered fraction land, their reward was moving on to “decimal land” where a similar process took place. Many students stopped out or dropped out along the way before even reaching “percent land.”

Times have changed. The new generation of high school equivalency tests (GED, TASC, HiSET) focus on math reasoning. To help students, we need to focus on research-based approaches instead of “we’ve always done it that way before.” What does research say are best practices for teaching math? Here are some:

- View computation as a tool for problem solving, not an end in itself

- Provide opportunities for group work

- Situate problem-solving tasks within familiar, meaningful, realistic contents in order to facilitate transfer of learning

- Address and evaluate attitudes and beliefs regarding both learning math and using math

- Develop understanding by providing opportunities to explore mathematical ideas with concrete or visual representations and hands-on activities.

(Ginsburg and Gal in Nonesuch, 2006)

UPS ✔ Problem Solving MethodPage 2 | ASE MA 1: schmidtsj@appstate.edu

1. Understand the 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?

2. Create a plan

Use a problem solving strategy:

Guess and check Solve an easier problemMake a list ExperimentDraw a picture or diagram Act it outLook for a pattern Work backwardsMake a table Change your viewpointUse a variable

3. Solve

Be patient

Be persistent

Try different strategies

4. 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

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Understand

Plan

Solve

Check

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Practice Problems1. At the end of January, Carson Real Estate had a total net worth of - $15,382. At the end of

February, the company’s net worth was - $13,011. At the end of March, the company’s worth was - $14,217. Which shows the correct increase or decrease in net worth for February and March?

February MarchA. - $2,371 - $2,658B. - $1,974 $2,658C. - $1,974 $1,206D. $2,371 - $1,206

2. An auditorium has a seating capacity of 1,500. For an upcoming community theater play, tickets cost $4 for adults and $2 for children. At last year’s play, twice as many children attended as did adults. If the attendance at this year’s play has the same ratio, which of the following expressions represents the best estimate of the total amount of money from ticket sales when the auditorium is filled to capacity?

A. 1,500 × $4 B. 1,500 × $2 C. (750 × $4) + (750 × $2) D. (1,000 × $4) + (500 × $2) E. (500 × $4) + (1,000 × $2)

3. Tamara works in the lab at Biogen Industries. She studies bacteria growth. In experiment #234-17, she is working with a bacteria that doubles every day. She starts with 2 bacteria cells and checks back after 5 days. She notices there are 32 bacteria. She checks back after 10 days and notices there are 1024 bacteria. Which expression below show the rate of bacteria growth?

A. x2

B. 2xC. 2x2

D. 2x

4. Burt is trying to find out the best price to sell his custom metalwork. He will double his prices every week until his sales start to fall. If the price of a metalwork candle holder is m and he doubles the price for 7 weeks, how much will it cost?

5. The Main Street Deli started the day with 36 cookies in inventory. They sold 21 during lunch and more during the afternoon. They had 4 cookies left at the end of the day. Write an equation that shows how many cookies (c) they sold during the afternoon.

6. 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?

Sources: NC Algebra 1 End of Course Test, Essential Math Skills, Essential Ed

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Teaching ActivitiesThe 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 may be extended by changing the unknown value.

Human Coordinate PlaneIn 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 TaskAs 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.

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Algebra Tiles Sharma (2005) describes how math should be taught from concrete to abstract. Instead of diving into an abstract algebra concept like x2 or 3x, students should begin with a more concrete representation. Algebra tiles use squares and rectangles to show x2, 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 PlansThis lesson plan is from the EMPower math series book Seeking Patterns, Building Rules. I love the EMPower series because it teaches mathematical 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: http://empower.terc.edu/

Discovering ExponentsThis 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 XFor 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 SignFrom 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.

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Winning the Battle for the Hearts and MindsAlgebra 1 is the most failed high school class. Some of our students have failed it more than 5 times before they reach our classrooms! What can we do to help them?

Carol Dweck – Mindset

Fixed Mindset Growth Mindset

Math Belief Math is about your ability Math can be _____________

Challenges Avoids challenges ________________ challenges

Effort Effort is fruitless Effort leads to _______________

Feedback Ignores feedback ________________ feedback

Success Threatened by others’ success ____________ by others’ success

Result Failure Success!

“Researchers have known for some time that the brain is like a muscle; that the more you use it, the more it grows. They have 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.” (Salman Khan of Khan Academy on the growth mindset)

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Viewing Guide: Carol Dweck - The Power of Believing That You Can Improve

1. What is the difference between a growth and fixed mindset?

2. How do we praise wisely?

3. For our students, what is the power of yet?

4. What would developing your students’ growth mindset look like in your classroom?

https://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve?language=en Page 9 | ASE MA 1: schmidtsj@appstate.edu

Teaching with Questions1. What does the instructor mean when he says, “Sometimes the best thing I can do is to walk away?”

2. How can we be too helpful as instructors?

3. What does the instructor mean when he says, “Wrong answers provide the best opportunity for learning?

4. What does the instructor do to instill curiosity in his students?

5. An instructor once told me that her students had “learned helplessness.” What does this mean?

Source: https://youtu.be/hv7MBSEcpW8

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Understanding Quadratic EquationsA quadratic equation in standard form looks like: 4n2 + 3n + 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: n2 + 15n = 0 we’ll first search for a common factor.

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

n2 + 15n = 0

n (n + 15) = 0

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

n = 0 n + 15 = 0 - 15 -15

n = - 15

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

n2 + 15n = 0

02 + 15(0) = 0

0 + 0 = 0

n2 + 15n = 0

−152 + 15(-15) = 0

225 + (-225) = 0

0 = 0

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Both solutions work!

Product of Two Binomials Since this equation n2 + 7n + 12 = 0 does not have any common factors, we’ll look to factor it into two parenthesis: (n + or - number) (n + or - number)

When the n2 is by itself, we’ll look for two numbers that when multiplied equal the last term and when added equal the middle term. For n2 + 7n + 12 = 0 we’ll look for two numbers that multiply to 12 and add up to 7:

Multiply Add1 (12) = 12 1 + 12 = 132 (6) = 12 2 + 6 = 83 (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: (n + 4) (n + 3) = 0

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

(n + 4) = 0 (n + 3) = 0 n = - 4 n = - 3

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

n2 + 7n + 12 = 0

(−4)2 + 7(-4) + 12 = 0

16 + (-28) + 12 = 0

0 = 0

n2 + 7n + 12 = 0

(−3)2 + 7(-3) + 12 = 0

9 + -21 + 12 = 0

0 = 0

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3 and 4 fit!

Both solutions work!

Quadratic Formula

In standard form, a quadratic equation looks like: an2 + bn + c = 0 Using the quadratic formula, we’ll first identify a, b, and c and plug them into the formula.

n2 + 7n + 10 = 0

a (number with n2 ) is 1

b (number with n term) is 7

c (whole number) is 10

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

n=−b±√b2−4 ac2a

n=−7±√72−4 (1)(10)2(1)

n ¿ −7±√49−402

n=−7±√92

n=−7±32

n=−7+32

n=−42

n=−2

n=−7−32

n=−102

n=−5

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!

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Let’s Practice1. n2 - 11n = 0 2. n2 + 20n = 0 3. n2 – 9n + 8 = 0 4. n2 + 16n + 48 = 0

5. n2 - 5n - 14 = 0 6. n2 - 5n - 36 = 0 7. n2 + 3n - 28 = 0 8. n2 - 3n - 1 = 0

9. n2 - 15n + 54 = 0 10. n2 + 2n - 80 = 0 11. n2 + 5n - 36 = 0 12. n2 - 8n + 12 = 0

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ResourcesReal World Algebra

Google: get the math http://www.thirteen.org/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 and Algebra Tiles

Google: national library of virtual manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html

This site has algebra manipulatives for instruction across grade levels

Google: algebra 4 all learnport http://a4a.learnport.org/page/algebra-tiles

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

Research BaseDweck, C. S. (2006). Mindset: The new psychology of success. New York: Random House.

National Institute for Literacy. (2010). Algebraic thinking in adult education. Washington, DC: Author.

Nonesuch, K. (2006). Changing the way we teach math: a manual for teaching basic math to adults.

Duncan, B.C., Canada: Malaspina University-College.

Polya, G. (2014). How to solve it: a new aspect of mathematical method. Princeton NJ: Princeton

Science Library.

Saliga, L.M., Daviso, A., Stuart, D., Pachnowski, L. (2015). Steps forward and back in adult

numeracy teacher professional development: a reflection on a teacher workshop experience.

Journal of Research and Practice for Adult Literacy. 4(1). 48-54.

Seeley, C.L., (2004). A journey in algebraic thinking. NCTM News Bulletin. September 2004.

Sharma, M., & Travaglini, L. (Eds.). (1988). Levels of knowing mathematics. Framingham, MA:

Center for Teaching/Learning of Mathematics.

U.S. Department of Education, Office of Career, Technical, and Adult Education. (2014).Page 15 | ASE MA 1: schmidtsj@appstate.edu

TEAL math works! guide. Washington, DC: Author.

Answer KeyPage 5

1. DFebruary -15,382 + x = -13,011 +15,382 +15,382 x = 2,371

March -13,011 + x = -14,217 +13,011 +13,011 x = -1,206

2. “Twice as many children attended as adults,” so for each group of 3 tickets:

2 Children + 1 Adult = 3 Tickets

1500 divided by 3 is 500. So there would be 1000 children’s and 500 adult tickets. Each child ticket costs $2 and each adult ticket costs $4 so the answer would be: E. (500 x $4) + (1000 x $2)

3. D. 2x After 5 days 25 = 32. After 10 days 210 = 1024.

4. m7

5. 36 – 21 – c = 4

6.

60 Candy Bars and 110 Drinks = $265

120 Candy Bars and 90 Drinks = $270

60C + 110D = 265120C + 90D = 270

- 2(60C + 110D = 265)

-120C - 220D = - 530 + 120C + 90D = 270 -130 D = - 260

D = 2

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60C + 110D = 265

60C + 110(2) = 265

60C + 220 = 265 - 220 - 220

60C = 45 60 60

C = .75

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Math can be learned

Embraces challenges

Effort leads to mastery

Learns from feedback

Inspired by others’ success

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1. 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.

2. Praise effort and process instead of intelligence. “I love the way you worked so hard to get the right answer on that problem,” versus “you’re so smart.”

3. The power of yet means that hard work and effort will lead to eventual success. We might not reach our goal today but will sometime down the road.

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1. Walking away allows the students to continue to focus on a problem instead of being told the answer by the teacher.

2. Immediately supplying an answer robs students of the chance to struggle through and get a solution on their own.

3. Making mistakes is a great way to see what does not work.

4. He presented a question and had students work to find the answer. Our brains want to solve mysteries, and we will work hard until we find the answer.

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5. When students are always told the answer, they do little on their own.

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1. n2 - 11n = 0

n(n – 11) = 0

n = 0 n – 11 = 0

n = 11

2. n2 + 20n = 0

n (n + 20) = 0

n = 0 n + 20 = 0

n = - 20

3. n2 – 9n + 8 = 0

(n – 8) (n – 1) = 0

n – 8 = 0 n – 1 = 0

n = 8 n = 1

4. n2 + 16n + 48 = 0

(n + 4)(n + 12) = 0

n + 4 = 0 n + 12 = 0

n = - 4 n = -12

5. n2 - 5n - 14 = 0

(n – 7) (n + 2) = 0

n – 7 = 0 n + 2 = 0

n = 7 n = - 2

6. n2 - 5n - 36 = 0

(n – 9) (n + 4) = 0

n – 9 = 0 n + 4 = 0

n = 9 n = - 4

7. n2 + 3n - 28 = 0

(n + 7) (n – 4) = 0

n + 7 = 0 n – 4 = 0

n = - 7 n = 4

8. n2 - 3n - 1 = 0

¿ 3±√132

9. n2 - 15n + 54 = 0

(n – 9) (n – 6) = 0

n – 9 = 0 n – 6 = 0

n = 9 n = 6

10. n2 + 2n - 80 = 0

(n + 10) (n – 8) = 0

n + 10 = 0 n – 8 = 0

n = - 10 n = 8

11. n2 + 5n - 36 = 0

(n + 9) (n – 4) = 0

n + 9 = 0 n – 4 = 0

n = - 9 n = 4

12. n2 - 8n + 12 = 0

(n – 6) (n – 2) = 0

n – 6 = 0 n – 2 = 0

n = 6 n = 2

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