Short Range Lesson Plan - Wikispaces-+Unit+Lesson...Short Range Lesson Plan Solving One-Step Equations using Addition & Subtraction Mastery Algebra 9th Grade Joy DiMuzio Objectives:

Short Range Lesson Plan

Solving One-Step Equations using Addition & Subtraction

Mastery Algebra

9th

Grade

Joy DiMuzio

Objectives:

• Student will solve for a given variable in one step by using addition or subtraction.

• Student will verbally explain their steps after solving a one-step equation.

• Student will write a mathematical equation in one variable when given an application with

missing information.

• Student will follow steps to determine whether or not their solution is correct.

• Student will demonstrate an understanding of the meaning of equality.

SCSDE Curriculum Standards Addressed:

• 7-3: the student will demonstrate through the mathematical processes an understanding of

proportional relationships

o 7-3.4 – Use inverse operations to solve two-step equations and two-step inequalities.

(NOTE: the work in this lesson is actually somewhere in between the 6th

grade standard and the

7th

grade standard. The 6th

grade standard is for applying inverse operations to solve an equation

in one step – however, it is for whole number coefficients and solutions only. In this lesson, we

will be solving equations that have fraction solutions)

• EA-1: The student will understand and utilize the mathematical processes of problem solving,

reasoning and proof, communication, connections, and representation

o EA-1.3 – Apply algebraic methods to solve problems in real-world contexts.

NCTM National Curriculum Standards Addressed:

• Algebra

o Represent and analyze mathematical situations and structures using algebraic symbols

� Use symbolic algebra to represent situations and to solve problems, especially

those that involve linear relationships

� Recognize and generate equivalent forms for simple algebraic expressions and

solve linear equations

• Communication

o Communicate their mathematical thinking coherently and clearly to peers, teachers,

and others

Prerequisites:

• Students must:

o Be able to translate sentences into algebraic equations

o Be able to express situations symbolically

o Be able to simplify algebraic expressions using distribution, combining like terms, and

finding common denominators

o Be proficient in multiplication and division, including problems that involve decimals,

negative numbers, and fractions, with or without a calculator

Materials/Preparation:

• Smartboard and Smartboard notebook software

• Prepared slides for lecture

• TI-84 or equivalent calculator

• 1 group activity sheet for each group of 3

• Paper & Pencils

Procedures:

• Introduction/Pre-learning (15 minutes)

o Slide 1 – Recall/Review – Writing algebraic expressions

� Have 3 students come to the board to share and explain their answers. I will ask

the class if they agree with the answers or if anyone got a different answer.

� Uncover answers and discuss different ways of writing the expressions.

o Slide 2 – Recall/Review – Evaluating expressions

� Have 3 students come to the board to share and explain their answers. I will ask

the class if they agree with the answers or if anyone got a different answer.

• Guided Learning/Class Discussion (25 minutes)

o Slide 3 – 123 =−m

� No Answer, Please – without giving the answer, have students explain what

might be happening with these numbers. Write down student responses.

o Slide 4 – 123 =−m

� Ask students to come up with scenarios that this equation might describe. Write

them on the board. Lead students in a discussion about taking away vs. giving,

how “giving” and “taking” are related to each other.

� Transition the discussion into how “giving” and “taking” are related to addition

and subtraction…

� Discussion about how addition and subtraction are opposites.

o Slide 5 - Balance

� Ask students what the equals sign might mean in this context.

o Slide 6 – Balance

� Ask students if this is a true statement.

� Practice adding and subtracting different numbers to both sides and show that

it still remains a true statement

� Discuss the importance of keeping the equation balanced by doing operations to

both sides

� Ask students to work with a partner to come up with a conjecture in their own

words about what they have just seen about adding and subtracting from both

sides of the equation.

� Ask pairs to share their conjectures. Ask the class if they agree and if anyone

came up with a different conjecture.

o Slide 7 – Addition Property of Equality

� Don’t spend a lot of time here – this is to confirm that their conjectures are true

o Slide 8 – Subtraction Property of Equality

� Don’t spend a lot of time here – this is to confirm that their conjectures are true

o Slide 9 – 123 =−m

� Ask students to use their conjectures to find out what value of m would make

this a true statement.

� CHECK solutions by substituting the variable with the solution.

o Slide 10 – 97142 =+ d

� No Answer, Please

� Prompt students to mention that you are adding something to 142, but the

result is smaller – what must this mean about the variable?

o Slide 11 – 8

1

4

3 −=+g

� No Answer, please

� Discuss fractions, negatives – make sure all aspects are discussed

� Ask students for suggestions on how to solve – there are many different ways

o Slide 12 – Write and solve an equation - A number increased by 5 is equal to 42. Find the

number.

� Do this together as a class. Ask if there is more than one way to write the

equation, more than one way to solve.

� CHECK the solution

o Slide 13 – Write and solve an equation

� In the fourteenth century, the part of the Great Wall of China that was built

during Qui She Huangdi’s time was repaired and 1000 miles was added to the

length of the wall. When the wall was completed, it was 2500 miles long. How

much of the wall was added during the 1300’s?

� Ask students to find the information that is necessary for solving this problem

� Ask what information is not known? What do we want to know? What can we

represent that value with? Can we put this into words? (The variable is the

additional length – could be called a. The original length plus the added length

equals 2500 miles)

� How can we write an equation that might represent the information we are

given in this problem? (1000 + a = 2500)

� How can we solve? Check?

o Slide 14 – Write and solve an equation

� A midsize car with a 4-cylinder engine goes 10 miles more on a gallon of gasoline

than a luxury car with an 8-cylinder engine. A midsize car consumes one gallon

of gas for every 34 miles driven. How many miles does a luxury car travel on a

gallon of gasoline?

� Have students try this on their own. Circulate around the class to see how they

are progressing.

� After a while, ask students to share their solutions with a partner.

� Ask a volunteer to come to the board and share their answer

� Ask the class if everyone agrees? If someone solved it differently?

� Solution: words – the distance a luxury car can travel plus 10 equals the

amount a midsize car can travel, variable – the distance a luxury car can travel

(d), equation – d + 10 = 34

o Slide 15 – Try these – circulate around the room as they work, answer questions. If there

are common questions, have students stop working and explain what needs to be

explained.

� 5413 =+t , t = 41

� 1821 −=+ q , q = -39

� 2715 =−s , s = 42

• Reflection Activity – Group Work (30 minutes)

o Have students get into math groups (pre-determined heterogeneous groups of 3)

o Give each group a set of 6 problems (see handout)

o Circulate as groups work together – encourage them to seek help from group members

before asking for the teacher’s help

o While circulating, assess each student’s input during collaboration, make notes for

participation grades.

• Closing/Wrap-up (20 minutes)

o Collect group worksheets – BE SURE ALL GROUP MEMBERS’ NAMES ARE ON ALL

PAPERS!!

o Assign homework

Assessment:

• Informal assessment of skills during pre-learning recall/review activities. This assessment is

based on student’s responses and communication during discussion and is meant to gauge the

overall understanding of the class.

• Formal assessment of group papers and collection of homework the next day. Group papers will

be graded for accuracy, but will be graded leniently. Random homework problems will be

graded for accuracy and combined with a completion grade.

Adaptations:

• Class discussion will be methodic and simplified in order to make sure that students of differing

ability levels will be able to follow the discussion and take notes properly. I will not move ahead

until I can assess that all students are ready to do so.

• For the group work, if there is a group of two, I will work with them a little more closely than I

do the other groups. If there is a group of four, I will split them into two groups of two and will

work with both groups.

Follow-up Lessons/Activities

• Follow-up lessons will include review of one-step equations using addition & subtraction and

instruction on using multiplication and division to solve one-step equations. Subsequently, two-

step equations will be introduced.

Slides_Lesson1_Solving_Equations

1

November 12, 2009

Nov 11-3:35 PM Nov 11-3:35 PM

Nov 11-4:01 PM Nov 11-6:44 PM

Nov 11-6:38 PM Nov 11-6:38 PM


2

November 12, 2009

Nov 11-7:00 PM Nov 11-7:00 PM

Nov 11-6:44 PM Nov 11-6:44 PM

Nov 11-6:44 PM Nov 11-7:14 PM


3

November 12, 2009

Nov 11-7:14 PM Nov 11-7:14 PM

Nov 11-7:38 PM

Group Activity Sheet

You will work in groups to solve the following equations. If you have a question, please discuss it with your

group members BEFORE asking for help.

Write your answers on a separate sheet of paper and show your work to . Make sure your papers have ALL

group members’ names on them. They will be collected at the end of class.

1. Write three equations that are equivalent to 2714 =+n .

2. Compare and contrast the Addition Property of Equality and the Subtraction Property of

equality.

3. Solve the following problem in as many different ways as you can. Show your work and

check your solution.

6

5

2

1 =−h

4. Determine whether or not xxx =+ is sometimes, always, or never true. Explain your

reasoning.

5. Determine whether or not xx =+ 0 is sometimes, always, or never true. Explain your

reasoning.

6. A) Look at the diagram below. Write an equation you could use to solve for x and then

solve for x .

B) Now write an equation you could use to solve for y and solve for y .

78cm

(y – 17) cm 24cm

(x + 55) cm

Group Activity Sheet

You will work in groups to solve the following equations. If you have a question, please discuss it with your group

members BEFORE asking for help.

Write your answers on a separate sheet of paper and show ALL of your work (when appropriate). Make sure your

papers have ALL group members’ names on them. They will be collected at the end of class.

1. Write three equations that are equivalent to 2714 =+n .

Examples: n – 3 = 10, n + 23 = 36 – any equation with the solution n=13.

2. Compare and contrast the Addition Property of Equality and the Subtraction Property of equality.

The Addition Property of Equality and the Subtraction Property of Equality can both be used to solve

equations. The Addition Property of Equality says you can add the same number to each side of an equation.

The Subtraction Property of Equality says you can subtract the same number from each side of an equation.

3. Solve the following problem in as many different ways as you can. Show your work and check your

solution. 6

5

2

1 =−h

This can be solved by adding ½ to both sides first, by finding a common denominator first, etc. I will accept all

correct ways as long as the solution is correct (h = 4/3)

4. Determine whether or not xxx =+ is sometimes, always, or never true. Explain your reasoning.

Sometimes – if x = 0, x + x = x is true. Students may try to subtract x from both sides, leaving x = 0.

5. Determine whether or not xx =+ 0 is sometimes, always, or never true. Explain your reasoning.

Always – any number plus 0 is always the number. Students may try to subtract 0 from both sides…

6. A) Look at the diagram below. Write an equation you could use to solve for x and then solve for x .

x + 55 = 78, x = 23

B) Now write an equation you could use to solve for y and solve for y .

y – 17 = 24, y = 41

78cm

(y – 17) cm 24cm

(x + 55) cm


Solving One-Step Equations using Multiplication & Division

Mastery Algebra

9th

Grade

Joy DiMuzio

Objectives:

• Student will solve for a given variable in one step by using multiplication or division.









(NOTE: the work in this lesson is actually somewhere in between the 6th

grade standard and the

7th

grade standard. The 6th

grade standard is for applying inverse operations to solve an equation

in one step – however, it is for whole number coefficients and solutions only. In this lesson, we

will be solving equations that have fraction solutions)




• EA-3: The student will demonstrate through the mathematical processes an understanding of

relationships and functions.

o EA-3.7 – Carry out a procedure to solve literal equations for a specified variable.


• Algebra


� Use symbolic algebra to represent situations and to solve problems, especially

those that involve linear relationships

� Recognize and generate equivalent forms for simple algebraic expressions and

solve linear equations

• Communication


and others

Prerequisites:

• Students must:










• Stop watch, meter stick, roll of masking tape and worksheet for each group of 3 students

• Paper & pencils

• Arrange with teachers/principal for space outside (or in a hallway if weather doesn’t permit

outside work)

• Mark off a long distance with flags, chalk or masking tape (either a long sidewalk, grassy area, or

hallway if inside)

Procedures:

• Introduction/Pre-learning(15 minutes)

o Go over questions students have on their homework assignment

� Work out problems on the board

o Collect homework for grading

o Slide 1 – Review – Solving 1-step equations with addition & subtraction

� Have students work 3 problems on their own

� Have 3 volunteers come to the board to work them out and explain methods

� 149 =−v , v = 23 4418 =+ z , z = 26 15)65( =−−b , b = -50

o Slide 2 – Review – equivalent equations, properties of equality

� Ask students which equations are equivalent to the original.

� Why are they equivalent?


o Slide 3 –

� No answer please – without giving the answer, have students explain what

might be happening with these numbers. Write down students’ responses.

o Slide 4 –

� Ask students to come up with a scenario that this equation might describe.

Write them on the board.

� Lead students in a discussion about what “3t” might mean besides just

multiplication. Talk about groups and what is the opposite of putting items into

groups.

� Discussion about how multiplication and division are opposites.

o Slide 5 – Balance

� Remind students of how we keep the equation balanced by doing operations to

both sides of the equals sign.

� Practice multiplying and dividing both sides by the same numbers and show that

it still remains a true statement.

� Remind students of the conjecture they came up with for addition & subtraction

� Ask students if they believe the same is true for multiplication & division

� Find out who agrees & disagrees – ask for a conjecture.

o Slide 6 – Multiplication Property of Equality

� BRIEFLY show students their conjecture is correct

o Slide 7 – Division Property of Equality

� Same as above

o Slide 8 –

� Ask students what is happening in this equation

� Ask students if they can apply their conjectures in order to solve this problem

� Are there different ways to solve? How would you start? Would anyone start a

different way?

� How would you check to see if your solution was correct?

o Slide 9 – Try these – - circulate around the room as they work

� 10

7

30=t

� 336

1 =w

� 123 =− x

� Ask for volunteers to come to the board to show and explain their solutions.

� Ask if students found the correct solutions in a different way.

o Slide 10 – Find the error

o Slide 11 – sample student papers

� Ask students which one of these is wrong?

� Why is it wrong?

o Slide 12 - rtd =

� Explain what this equation represents

� Ask students what’s different about this equation and the ones they’ve been

looking at (no numbers)

� Explain the term “literal equation” and what it means (quantities are expressed

either wholly or in part by letters)

� Ask students how they might use this equation to find an unknown?

� What could the unknown be?

� How many pieces of information have to be given in order to find out one that is

unknown?

� This equation has distance on one side. What if you wanted to know rate? How

might you solve this equation for rate?

� What pieces of information would you have to know if you wanted to find out

the value for time?

� Ask students how would rewrite the equation to solve for different variables.

Write these on the board.

o Slide 13 –River Discharge

� Have students write a literal equation for river discharge (D=wds)

� Ask students how many pieces of information must you have in order to find

one unknown.

o Slide 14 – Find depth of Mississippi River

� Ask students which variable they are solving for

� Have them try on their own to solve

� Ask a student to share their solution with the class. Ask whether or not

everyone came up with the same answer. Address any wrong solutions to find

where a mistake was made.

o Introduce group activity

• Reflection Activity – Group Work (30 minutes)

o Explain that they will be filling out a “peer critique form” for their group members

o Explain directions for group activity (will be on their handouts as well)

� Have students get into math groups (pre-determined groups of 3)

� Hand out data collection sheets, meter sticks, and stopwatches to each group

� Each group will need a calculator

o Visit each group as they are working to see if they have questions


o Go over data collection

� Compare solutions for the unknown distances

� “What factors may have contributed to the error in your calculations for your

rate and subsequently for your unknown distances?

� “How could you conduct your trials with more accuracy?”

� “Is it possible to calculate a rate knowing only the distance traveled?”

� “What information is necessary to calculate a rate? Distance? Time?

o Collect group data sheets – BE SURE ALL GROUP MEMBERS’ NAMES ARE ON ALL

PAPERS!!

o Hand out “peer critique form” – have students fill out and rate the work done by their

group members – have them turn in as they leave.

o Assign homework.

Assessment:

• Informal assessment of skills during warm-up and review activities. This assessment is based on

students’ responses and communication during discussion and is meant to gauge the overall

understanding of the class. This is not meant to determine understanding on an individual basis.

• Informal assessment during class discussion and example problems of one-step equations.

• Formal assessment of group papers and collection of homework the next day. The group papers

will be graded for accuracy. The homework will be a combination of a completion grade and a

correctness grade for random problems.

• Participation assessment from peer critique forms – these will be recorded as participation

grades

Adaptations:




• For the group work, if there is a group of two, they will share the work of recorder. If there is a

group of four, I will split them into two groups of two.


• Follow-up lessons will include review of one-step equations using multiplication & division and

instruction in solving two-step equations. There will be an enrichment activity assigned with the

homework that will challenge students to think mathematically. Multi-step equations will be

introduced after two-step equations.

Reflection

I taught this lesson to my peers. As I was teaching, I found several problems with my plan that I had

not anticipated. The first was the fact that I didn’t include opportunities for students to show

different ways of solving problems that involve multiplication and division. I revised this plan and

included multiple discussions on how some problems can be solved in more than one way.

When I did my peer lesson teaching, I was hurrying through the lesson so that I could hit the

important parts and still stay within my time frame. I know that my performance in class did not

reflect the amount of time I spent planning and preparing this lesson. I normally would spend more

than twice the amount of time teaching this in a real classroom than I did in my peer classroom so

that I could spend an acceptable amount of time making sure the students were gaining

understanding about the material.

I do believe the d-r-t activity is a very good activity. Because of time and space, I couldn’t have

everyone in the class participate. In a real classroom, there would be multiple sets of data so the

end discussion would have been much more interesting. This activity has a lot of potential.

I was also disappointed that I wasn’t able to get to the river discharge problem at the end. When I

revised my lesson plan, I put this problem before the group activity. It’s a very interesting problem

that I believe students will find fun and challenging.

Slides - Lesson 2 Solving One-Step Equations

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November 12, 2009

Oct 18-5:46 PM Oct 18-5:46 PM

Oct 18-6:13 PM Oct 18-6:13 PM

Oct 18-6:13 PM Oct 18-6:13 PM


2

November 12, 2009

Oct 18-6:13 PM Oct 18-6:13 PM

Oct 23-3:23 PM Oct 23-3:23 PM

Oct 24-11:24 AM Oct 23-3:23 PM


3

November 12, 2009

Oct 23-3:23 PM Oct 23-3:23 PM

Solving One-Step Equations – Group Activity Sheet

Group Members: ___________________________________

___________________________________

___________________________________

Step 1: On the ground, mark off a distance of 4 meters with masking tape.

Step 2: Have the “walker” walk the distance using a normal, steady walking pace. The “timer” should time each of the 5

trials. The “recorder” will record the information in the table below. Repeat this process four more times for a

total of 5 trials. Be sure to remind the walker to try to keep a steady pace.

Distance = Rate X Time

D = rt

Distance (meters) Time (seconds) Rate (meters/second)

Step 3: Using the distance and time information you collected, calculate the rate for each trial and enter it in the table.

This is the walker’s speed.

Step 4: Calculate the average rate of speed for the walker.

Average rate = ________________________

Step 5: Now, walk the length of the hallway as directed by your teacher at the same pace. Time how long it takes to

complete the distance. Record this information in the table below.

Distance (meters) Time (seconds) Average Rate (meters/sec)

Step 6: Using the average rate from step 4 and the time recorded in step 5, determine the length of the hallway.

Step 7: Measure the hallway. How close was your calculated value? What could account for your error? How could you

minimize your error?

Peer Contribution CritiquePeer Contribution CritiquePeer Contribution CritiquePeer Contribution Critique

Your Name _______________________________________________________

Your role in the group (circle one): walker timer recorder

Your group members ____________________________________________________________________

Rate your group members’ contribution to the group on a scale from 1 to 4, with 1 being “did

absolutely nothing” and 4 being “really contributed and completely did their job”.

Circle the rating that is most appropriate:

Walker 1 2 3 4

Timer 1 2 3 4

Recorder 1 2 3 4

Peer Contribution CritiquePeer Contribution CritiquePeer Contribution CritiquePeer Contribution Critique

Your Name _______________________________________________________

Your role in the group (circle one): walker timer recorder

Your group members ____________________________________________________________________

Rate your group members’ contribution to the group on a scale from 1 to 4, with 1 being “did

absolutely nothing” and 4 being “really contributed and completely did their job”.

Circle the rating that is most appropriate:

Walker 1 2 3 4

Timer 1 2 3 4

Recorder 1 2 3 4


Solving Two-Step Equations

Mastery Algebra

9th

Grade

Joy DiMuzio/Ashley Burgess

Objectives:

• Student will solve for a given variable in two steps by using a combination of addition or

subtraction and multiplication or division.













• Represent and analyze mathematical situations and structures using algebraic symbols

o Use symbolic algebra to represent situations and to solve problems, especially those

that involve linear relationships

o Recognize and generate equivalent forms for simple algebraic expressions and solve

linear equations

Prerequisites:

• Students must:

o Know what “opposite operations” are and how to use them

o Be proficient in solving one-step equations using addition/subtraction or

multiplication/division


o Be proficient in basic arithmetic operations either with or without a calculator





• One- and two-step equation strips with solutions

• Possible answer sheets (1 per student)

• BINGO cards (1 per student)

• Bowl

• Chocolate (suckers for students with allergies)

• Paper & pencils

Procedures:

• Introduction/Pre-learning (15 minutes)

o Class discussion/quick review of previous two lessons on solving one-step equations

� Review “opposite operations” & conjectures/properties of equality

o Slide 1 – Recall/Review –One-step equations

� Have students solve and check on their own

� Have 3 volunteers come to the board to share and explain solutions

� Ask if anyone solved a different way but got the correct solution

o Slide 2 – Recall/Review – contextual problem

� Have students solve and check on their own

� Class discussion about solution and ways to solve


o Slide 3 – 42102 =+p

� No Answer please – without giving the answer, have students explain what

might be happening with these numbers. Write down students’ responses

o Slide 4 – 42102 =+p

� Ask students to think of a scenario that these numbers might represent

� Discuss opposite operations

� Ask students to lead through finding the solution to this equation

� Check the solution

o Slide 5 – 694

=+x

� Ask students what is happening in this equation (what operations are being

done)

� Ask students how they might solve it

� Make sure all the ways are discussed

o Slide 6 – Beach trip problem

� Have students write and solve an equation

o Slide 7 – Try these

� Circulate as students work these on their own

� Ask for volunteers to share and explain their solutions on the board

� There are multiple ways to solve these – allow students to share different ways

� 53

4 =−d

� 61143

2 =−a

• Reflection Activity – Equation BINGO (30 minutes)

o Distribute 1 BINGO card and 1 sheet of possible answers to each student

o Explain that a one- or two-step equation will be drawn from the bowl and students will

solve and check the problem on a piece of paper that will be collected at the end of

class. If the answer they find is one that they filled in on their BINGO card, they may

cross off that space. If a student completes a BINGO (row, column, diagonal), they

receive a piece of chocolate.

o Some answers will be fractions. Be sure you simplify them if you aren’t using a calculator

o Pull equations out and write them on the board – mark off on the equation sheet as

they are chosen so BINGO winners’ cards can be checked against the equation sheet.

• Closing/Wrap-up (20minutes)

o Collect students’ solutions and checks sheets

o Assign homework

Assessment:

• Informal assessment of understanding during warm-up of one-step equations

• Informal assessment during class discussion and example problems of two-step equations

• Formal assessment of BINGO solutions and checks that are collected and graded

Adaptations:




• BINGO equations will include a wide range of difficulty levels from very basic to challenging. I

will give students sufficient time to finish the solutions and checks before I draw another

equation.


• Possible enrichment activities might include a worksheet with word problems that the students

must write equations for.

• Follow-up lessons will include review of two-step equations and instruction in solving multi-step

equations that involve variables on both sides of the equation, distribution, and combining like

terms.

Reflection

I put a lot of thought into this lesson. While I was planning, I kept in mind that I wanted the students to

experience something different from what they were used to with Miss Burgess. I wanted them to be

part of class discussion, share their answers and methods with their classmates, and have fun while

learning. I tried twice the previous week to practice with the Smart Board in the computer lab, but each

time there was a technical issue that kept me from practicing. This caused me a little bit of anxiety since

I had never used a Smart Board before. I taught this lesson to two classes and I will reflect on them

individually.

• Second period mastery algebra – I was extremely nervous. I was nervous about the Smart Board

and about teaching in general. I was prepared, but I still didn’t know what to expect. I began

with the warm-up review and had three students come to the board to share their answers. I

was surprised because even though they had come up with the correct answer (mostly by doing

the math in their heads) they couldn’t properly explain their steps. We moved quickly into the

new material because I wanted to get to the game at the end. As I wanted to do, I let the

students dictate to me how we should solve the problems. When there was a disagreement

among them about proper steps, I stopped and took all suggestions into account. They were

able to see that there were several ways to get the correct answer, but that there was usually

one way that took less steps and looked a little less messy. There were even a couple of

instances where I was able to show them how their suggestion would NOT work because it

would produce a wrong solution. I don’t think Miss Burgess was impressed by that. She likes to

teach them one way and have them mimic her.

I was rushing through the slides. After the lecture part, I gave them the game and we had to

rush through it because time was running out. I found out at the end that the majority of the

students had not really learned what I wanted them to learn. Many of their solutions on the

BINGO game were wrong. Nobody won. I felt like I let them down and that Miss Burgess was

going to have to come behind me and do some re-explaining.

• Sixth period mastery algebra – I was no longer nervous. I was comfortable with the Smart Board

and I knew that I had to make changes to my lesson plan. I eliminated the game and just spent

class time explaining, modeling, and listening. I took my time when they wanted to see other

possible ways of solving a particular equation and made sure after each example that everyone

understood before we moved on. At the end, I gave them an 8-problem formal assessment and

collected them to grade. I found that the majority of the students had indeed achieved the

objectives I wanted them to learn.

The thing that is most disappointing is the fact that I knew the second class wasn’t having any

fun. I wanted so badly to throw in something different for them. If I had it to do over, I would

ask my mentor teacher if I could have two consecutive days to teach this lesson. I would spend

one class period on lecture and discussion and then the next on review and an activity to

reinforce the learning. 45 minutes just wasn’t enough for both.

This lesson was written to fit into the constraints of Mrs. Burgess’s class period and to make sure I

covered everything she wanted covered (in particular, the specific examples she wanted me to do).

However, if I was teaching this lesson to a class of my own, I would make several changes to the basic

structure of the lesson. Instead of beginning the lecture portion of the lesson with an example like

42102 =+p , I would start with an example with smaller numbers that are easier to work with

mathematically. I would still use all even numbers so that we could get into a class discussion about

dividing by 2 before subtracting 10. I would still have a class discussion about what is happening in the

equation and try to lead students to explain the mathematics in their own words. I would have them

come up with a scenario to match the equation and then lead them through the steps to solve it – using

more than one way to do it. I could show them that they could divide through by 2, subtract 10, or even

add a (-10). These are all things that I didn’t have time to do when teaching Mrs. Burgess’s class because

I was limited to the 45 minutes and the content she wanted covered.

I would take each of the examples I included in the lesson (each was a representation of the types of

problems the students would be expected to be able to solve) and I would preface them with a similar

problem that was a little more simple. This wouldn’t add too much time to the lecture and would easily

fit in to an hour & a half class. If I wind up having a 45 minute class period in the future, I will make sure I

am covering the basics of this concept through daily discussion in my classroom. That way the students

would be familiar with the content before we actually covered it. I understand that there are

opportunities to throw in “future learning” into daily activities & discussions.

Slides - Lesson 2 Solving Two-Step Equations

1

November 12, 2009

Oct 3-10:01 PM Oct 3-10:01 PM

Oct 3-10:01 PM Oct 3-10:01 PM

Oct 3-10:01 PM Oct 3-10:01 PM

Beach Trip

Slides - Lesson 2 Solving Two-Step Equations

2

November 12, 2009

Oct 3-10:01 PM

Try these:

1. 1296 =−x x= 7/2

2. 29113

=+t t= 54

3. 4

3212

x−= x= -46/3

4. 212

13 =+b b= 41/3

5. 2154 =+a a= 4

6. 1354 =− x x= -9/5

7. 33310 =+x x= 3

8. 50347 −=− x x= 1

9. 613116 =+y y=15/8

10. 10235 +=− p p= -45/2

11. 5.235

=+−m

m= 2.5

12. 143

5 −= x x= 57

13. 3517 =−x x= 52

14. 14210 =+x x= 132

15. 183 =x x= 6

16. 215

=x x= 105

17. 568 −=x x= -7

18. 6214 =−b b= 76

19. 162

=b b= 32

20. 15 =+x x= -4

21. 92

3 =−z z= 21

22. 1462

−=m m= -292

23. 14103 =−x x= 8

24. 109 =−z z= 19

25. 418 += y y= 14

26. 28212 −= x x= 20

27. 123

96 =+n n= 9/2

28. 172

4 =− p p= -26

29. 667 =+b b= 0

30. 524 =+n n= 3/4

31. 356 =−r r= 4/3

32. 83 =−− x x= -11

33. 1343 =+− m m= -3

34. 124

=x x= 48

35. 512 −=−t t= -2

36. 83 =− x x= -5

37. 117 −=x x= -11/7

38. 2110011 =+− q q= 79/11

39. 8

1

4

3 −=+g g= -7/8

40. 592

=−k k= 28

1. 1296 =−x x= 7/2

2.

29113

=+t

t= 54

3. 4

3212

x−=

x= -46/3

4.

212

13 =+b

b= 41/3

5. 2154 =+a

a= 4

6. 1354 =− x

x= -9/5

7. 33310 =+x

x= 3

8. 50347 −=− x

x= 1

9. 613116 =+y

y=15/8

10. 10235 +=− p

p= -45/2

11. 5.23

5=+

−m

m= 2.5

12.

143

5 −= x

x= 57

13. 3517 =−x

x= 52

14. 14210 =+x

x= 132

15. 183 =x

x= 6

16.

215

=x

x= 105

17. 568 −=x

x= -7

18. 6214 =−b

b= 76

19.

162

=b

b= 32

20. 15 =+x

x= -4

21.

92

3 =−z

z= 21

22. 146

2−=m

m= -292

23. 14103 =−x

x= 8

24. 109 =−z

z= 19

25. 418 += y

y= 14

26. 28212 −= x

x= 20

27.

123

96 =+n

n= 9/2

28. 17

24 =− p

p= -26

29. 667 =+b

b= 0

30. 524 =+n

n= 3/4

31. 356 =−r

r= 4/3

32. 83 =−− x

x= -11

33. 1343 =+− m

m= -3

34.

124

=x

x= 48

35. 512 −=−t

t= -2

36. 83 =− x

x= -5

37. 117 −=x

x= -11/7

38. 2110011 =+− q

q= 79/11

39. 8

1

4

3 −=+g

g= -7/8

40.

592

=−k

k= 28

32 -9/5 8 -45/2

-46/3 52 79/11 -7/8

105 14 1 2.5

3 -26 0 -11

15/8 19 9/2 20

48 -2 -5 28

-11/7 132 6 41/3

76 7/2 -4 21

-292 -7 54 4/3

-3 57 4 3/4

B I N G O

Name______________________________________ Class Period__________


Solving Multi-Step Equations

Mastery Algebra

9th

Grade

Joy DiMuzio

Objectives:

• Student will solve for a given variable in an equation that involves addition, subtraction,

multiplication, division, distribution, or combining like terms.

• Student will verbally explain their steps after solving a multi-step equation.





• 8-3: The student will demonstrate through the mathematical processes an understanding of

equations, inequalities, and linear functions.

o 8-3.2 – Represent algebraic relationships with equations and inequalities

o 8-3.3 – Use commutative, associative, and distributive properties to examine the

equivalence of a variety of algebraic expressions

o 8-3.4 – Apply procedures to solve multistep equations





• Algebra


� Understand the meaning of equivalent forms of expressions, equations,

inequalities, and relations

� Use symbolic algebra to represent and explain mathematical relationships

• Communication


and others

Prerequisites:

• Students must:






o Understand the meaning of equivalent expressions and how to keep equations balanced





Procedures:

• Introduction/Pre-Learning (15 minutes)




o Slide 1 – Review – Solving 2-step equations



� 465 −=+n , n=-2 104

17 −=− s, s=57 316

5=+

−m

, m=-125


o Slide 2 – Using an equation to determine if two populations are equal

� In 1995, there were 18 million Internet users in North America. Of this total, 12

million were male and 6 million were female. During the next five years, the

number of male Internet users on average increased by 7.6 million per year, and

the number of female Internet users increased by 8 million per year. If this trend

continues, these two expressions represent the number of male and female

Internet users in x years after 1995.

� Ask students if we want to know when the number of male Internet users will

be the same as the number of female Internet users, what might we do? (set

the expressions equal to each other)

o Slide 3 – Equation

� What do you notice about this equation that is different from the ones we’ve

been solving? – (There is a variable on both sides)

� Ask students how we might begin to solve this equation. Is there more than one

way?

� X = 15 years – so when will the two populations be the same? – 2010

� In the year 2010, what will the two populations be? – 126 million male & 126

million female – a total of 256 million users

� The latest 2009 data shows that there are approximately 251.7 million Internet

users in North America. Is this what you would expect? Why or why not?

o Today we’re going to solve equations that require multiple steps to solve. There will be

some with variables on both sides of the equation and some that may require

distribution. You will use what you have already learned about solving equations and

simplifying expressions. We will just be putting a lot of that information together today.

o You will also see for the first time today that there can be a situation where there is no

solution to an equation! Let’s look at one of those now:

o Slide 4 – 3252 −=+ mm

� Ask students if they can tell why there might not be a solution to this equation

� You can see that there is no value for m that will make this statement true. This

is a false statement and it has NO SOLUTION.

o Slide 5 – 235)1(3 −=−+ dd

� This one is going to require some distribution

� Have students work in pairs to solve

� Have pairs share and explain their solutions

� You can see here that no matter what value you put in for d, the statement is

going to be true.

� This is what we call an identity and its solution is ALL NUMBERS.

o So you can see that there are 3 possible outcomes for equations –

� there is one solution,

� the solution results in a false statement so there is no solution, or

� the solution results in an identity and the solution is all numbers

o Slide 6 – 13876 −=+ nn

� We’re going to solve this together, but with each step, we’re going to write the

reason we can do what we’ve done

� Let students lead through steps to follow – justifications will be simplification,

properties of equality

o Slide 7 – Try this!! 10

7

5

23 =−m

� Circulate as students work – informally assess understanding

� Have a volunteer come to the board to share solution

� If you observe that there have been multiple ways of solving, ask those students

to share their methods as well.

o Slide 8 – Try these: Remember to check your solutions!

� Circulate as students are working

� 6798 +=+ ss , s = -3

� 92)1(27 +=++ ww , all numbers

� )53(48 += c , c = -1

� )2(437 rrr +−=− , no solution

o Slide 9 – Geometry

� The rectangle and the square have the same perimeter. Find the dimensions of

each figure

� Students may solve in different ways – example:

• 2(3x+1) + 2x = 4(3x) – x = ½, dimensions = 2 ½ X ½ and 1 ½ X 1 ½

• 3x + 1 + 3x + 1 + x + x = 3x + 3x + 3x + 3x

o Slide 10 – Helpful Tips for Solving Word Problems

� Explain to students that these are just some of the ways to approach word

problems.

o Slide 11 – Roller Coaster problem

o Slide 12 – Step 1 – Draw a picture & assign variables

� Important – assign one unknown a variable and name the other unknowns in

terms of that variable

o Slide 13 – Step 2 – Make a table

� x, 70, 70x / 8x, 10, 80x

o Slide 14 – Step 3 – Write an equation and solve.

o Introduce group activity

• Activity – Group Work (30 minutes)

o Explain directions for group activity (will be on their handouts as well)

� Slide 15 – Map of auto race

� Students get into groups of three – one group of four is okay

� Provide each group with the “Auto Race” problem sheet

� Each group will need paper, pencils, & a calculator

� After 10 minutes give hint – make two separate tables (draw on the board)

� Walk around to each group and answer questions


o Go over results

o Look at map – based on the picture, is that what you would expect to find?

o Collect group papers – Make sure each group member’s name is on the papers.

o Homework assigned from book.

Assessment:

• Informal assessment of skills during warm-up and review activities

• Informal assessment during class discussion and example problems of multi-step equations.

• Informal assessment of individual understanding by walking around to check students’ papers

during individual work.

• Formal assessment of group papers and collection of homework the next day.

Adaptations:





• Follow-up lessons will include review of solving equations using all methods learned so far. A

good enrichment activity is “Solving Equations using M&M’s”.

Lesson 4 - Solving Multi-Step Equations

1

November 12, 2009

Oct 25-9:51 AM Oct 25-9:52 AM

Oct 25-9:52 AM Oct 25-9:52 AM

Oct 25-9:52 AM Oct 25-9:52 AM


2

November 12, 2009

Oct 25-9:52 AM Oct 25-9:52 AM

Oct 25-9:52 AM Oct 25-9:52 AM

Oct 25-9:52 AM Oct 25-9:52 AM


3

November 12, 2009

Oct 25-9:52 AM

downhill

Oct 25-9:52 AM

Oct 25-9:52 AM

Group Activity – Auto Race

Instructions: Below is a description of a cross-country auto race between two teams of drivers. The race was

completed in relay form, with each driver picking up where the one before him left off.

Use the strategies for solving word problems and the formula (D=rt) to find out which team won the race.

Do all your work and calculations on a separate sheet of paper and be sure to include all group members’

names on your solution page(s).

Ten drivers participated in an auto race from Orlando, FL to Seattle, WA. The two five-person relay teams each

selected the route they thought would be fastest.

Team 1

Ade left Orlando on the Florida Turnpike and took I-75 to I-10. He stayed on I-10 through Mobile, AL and New

Orleans, LA until he came to Baton Rouge. Gabi continued on I-10 to Houston, TX, took I-45 into Dallas, and

then I-35 up to Oklahoma City. Rico continued the drive to Wichita, KIS where he used I-135 to connect to I-70.

When he reached Denver, CO, he caught I-25 up to Cheyenne, WY. Mac left Cheyenne on I-80 to Twin Falls, ID.

Pam continued on I-84 but switched to I-82 just past Pendleton, OR. I-82 joins I-90 to go into Seattle.

Team 2

Meanwhile, Felix was taking I-75 up to Chattanooga, TN. There he got on I-24 for Nashville. Keshia continued

via I-24, I-57, and I-64 to St. Louis, where she got on I-70 for Kansas City, MO. Daisy drove up I-29 to Sioux

Falls, SD, and then turned west on I-90 for Rapid City. Bree continued on I-90 all the way to Butte, MT. Jacob

drove the last leg of I-90 into Seattle.

Including gas stops, Mac, Bree, and Pam were able to average 60 mph in the wide open west. Four drivers

could average only 55 because the highways passed through the metropolitan areas of Denver, Houston,

Atlanta, and Omaha. Dense fog on the river crossing from Kentucky to Illinois and accidents around Biloxi, MS

and Spokane, WA caused three drivers to average only 50mph.

Felix, Mac, Keshia, and Jacob all drove for the same amount of time. Daisy, Ade, and Gabi each drove one hour

longer than Mac. Rico drove one hour less than Mac, whereas Bree drove two hours less than Mac. Pam drove

an hour and a half less than Mac. If the total distance driven by the ten drivers was 6495 miles, which team

won?

Group Activity – Auto Race

• Instructions: Below is a description of a cross-country auto race between two teams of drivers. The

race was completed in relay form, with each driver picking up where the one before him left off.

• Use the strategies for solving word problems and the formula (D=rt) to find out which team won the

race.

• Do all your work and calculations on a separate sheet of paper and be sure to include all group

members’ names on your solution page(s).

Ten drivers participated in an auto race from Orlando, FL to Seattle, WA. The two five-man relay teams each

selected the route they thought would be fastest.

Team 1

Ade left Orlando on the Florida Turnpike and took I-75 to I-10. He stayed on I-10 through Mobile, AL and New

Orleans, LA until he came to Baton Rouge. Gabi continued on I-10 to Houston, TX, took I-45 into Dallas, and

then I-35 up to Oklahoma City. Rico continued the drive to Wichita, KIS where she used I-135 to connect to I-

70. When he reached Denver, CO, he caught I-25 up to Cheyenne, WY. Mac left Cheyenne on I-80 to Twin

Falls, ID. Pam continued on I-84 but switched to I-82 just past Pendleton, OR. I-82 joins I-90 to go into Seattle.

Team 2

Meanwhile, Felix was taking I-75 up to Chattanooga, TN. There he got on I-24 for Nashville. Keshia continued

via I-24, I-57, and I-64 to St. Louis, where he got on I-70 for Kansas City, MO. Daisy drove up I-29 to Sioux Falls,

SD, and then turned west on I-90 for Rapid City. Bree continued on I-90 all the way to Butte, MT. Jacob drove

the last leg of I-90 into Seattle.

Including gas stops, Mac, Bree, and Pam were able to average 60 mph in the wide open west. Four drivers

could average only 55 because the highways passed through the metropolitan areas of Denver, Houston,

Atlanta, and Omaha. Dense fog on the river crossing from Kentucky to Illinois and accidents around Biloxi, MS

and Spokane, WA caused three drivers to average only 50mph.

Felix, Mac, Keshia, and Jacob all drove for the same amount of time. Daisy, Ade, and Gabi each drove one hour

longer than Mac. Rico drove one hour less than Mac, whereas Bree drove two hours less than Mac. Pam drove

an hour and a half less than Mac. If the total distance driven by the ten drivers was 6495 miles, which team

won?

Sum of distances = 6495

550x – 105 = 6495

X = 12 hours

Team 2 (northern route) won

Team 1

r t d time

Ade 50 x + 1 50 (x + 1) 13

Gabi 55 x + 1 55 (x + 1) 13

Rico 55 x - 1 55 (x -1) 11

Mac 60 x 60x 12

Pam 60 x - 1.5 60 (x - 1.5) 10.5

59.5 hrs.

Team 2

r t d time

Felix 55 x 55x 12

Keshia 50 x 50x 12

Daisy 55 x + 1 55 (x + 1) 13

Bree 60 x - 2 60 (x - 2) 10

Jacob 50 x 50x 12

59 hrs.


Solving Equations and Formulas

Mastery Algebra

9th

Grade

Joy DiMuzio

Objectives:

• Student will solve equations for a given variable.

• Student will solve literal equations and formulas for a given variable.

• Student will use formulas to solve problems


• 8-3: The student will demonstrate through the mathematical processes an understanding of

equations, inequalities, and linear functions.

o 8-3.2 – Represent algebraic relationships with equations and inequalities

o 8-3.3 – Use commutative, associative, and distributive properties to examine the

equivalence of a variety of algebraic expressions

o 8-3.4 – Apply procedures to solve multistep equations




• EA-3: The student will demonstrate through the mathematical processes an understanding of

relationships and functions

o EA-3.7 – Carry out a procedure to solve literal equations for a specified variable


• Algebra


� Understand the meaning of equivalent forms of expressions, equations,

inequalities, and relations

� Use symbolic algebra to represent and explain mathematical relationships

• Communication


and others

Prerequisites:

• Students must:



o Understand the meaning of equivalent expressions and how to keep equations balanced

o Know how to interpret word problems into mathematical expressions and equations





• 1 baggie of M&M’s per student

• 1 Student M&M Questionnaire per student

• 1 large baggie of M&M’s for teacher

Procedures:

• Introduction/Pre-Learning (15 minutes)




o Slide 1 – Review – Solving multi-step equations



� 5379 +=+ xx , x = -1/3, 148)173(22 =++ ww , w = 57/4,

yyy2

14

2

3 +=− no solution


o Slide 2 – Using an equation to design a roller coaster

� Ron Toomer designs roller coasters, including the Magnum XL-200. This roller

coaster starts with a vertical drop of 195 feet and then ascends a second shorter

hill. Suppose when designing this coaster, Mr. Toomer decided he wanted to

adjust the height of the second hill so that the coaster would have a speed of 49

feet per second when it reached its top. If we ignore friction, the equation

2

2

1)195( vhg =− can be used to find the height of the second hill. In this

equation, g represents gravity (32 feet per second squared), h is the height of

the second hill, and v is the velocity of the coaster when it reaches the top of

the second hill.

� Explain that equations can have more than one variable. You may need to solve

for one of those variables, based on information that you are given.

� Ask students what information would we need in order to find the height of the

second hill. (gravity and velocity)

o Slide 3 – 743 =− yx

� Ask the students what they think may be happening in this equation, what

might the two variables represent?

� Ask students to come up with a scenario where there might be two pieces of

information that are unknown

� Ask students why they may want to solve for one of these variables

� Ask students what they might need to do in order to solve this equation for y.

� Allow students to lead through the steps to solve for y

o Slide 4 – 743 =− yx

� Try this! – have students get into pairs and solve this equation for x

� Have pairs share their solution and methods.

� Ask if anyone solved it a different way

o Slide 5 – 52 +=− smtm

� Try this! - Solve for m

� Ask for student to share their answers

� Ask students if they can see a problem with the denominator

� Explain – If you wind up with a solution that is a fraction with a variable in the

denominator, then you aren’t through. Remember – you cannot have a zero in

the denominator because division by zero is undefined. So you have to “throw

out” any number that would make the denominator zero. In this case, s cannot

equal 2. You must include that in your solution.

o Slide 6 – Example 3 - rC π2=

� Formula for the circumference of a circle – solve for r.

� Go through steps to solve for r

� Ask – “why might we want to solve this equation for r?” – if we knew the

circumference and not the radius, we could use the circumference to find the

radius.

o Slide 7 – use the formula to find the size of the hailstones in Kansas City in 1898.

� On May 14, 1898, a severe hailstorm hit Kansas City. The largest hailstones were

9.5 inches in circumference. Windows were broken in nearly every house in the

area. Use the formula to find the radius of the largest hailstones. (r = 1.5 inches)

� Ask students if the radius was 1.5, what was the diameter? (3 inches)

o Slide 8 – 2

2

1ats =

� This formula represents the distance (s) that a free-falling object will fall near a

planet or the moon in a given time (t). In the formula, (a) represents the

acceleration due to gravity.

� Solve the formula for a.

o Slide 9 – Use the formula to find the value of gravity on the moon.

� A free-falling object near the moon drops 20.5 meters in 5 seconds. What is the

value of a for the moon? – (a = 1.64 m/s2 )

o Slide 10 – Teacher’s M&M candy data

� Introduce class activity

• Reflection/Activity – (30 minutes)

o Each student needs paper and pencil

o Hand out M&M baggies

o Tell students to record the teacher’s M&M data from the board, then count their own

M&M’s and make a similar table for their data. THEY ARE NOT TO SHARE THEIR DATA!

o Hand out Questionnaires

o See “Building Equations Using M&M’s” for instructions


o Collect papers & questionnaires.

o Assign homework.

Assessment:

• Informal assessment of skills during warm-up and review activities

• Informal assessment during class discussion and example problems of multi-step equations.

• Informal assessment of individual understanding by walking around to check students’ papers

during individual work.

• Formal assessment of students’ equation sheets & questionnaire and collection of homework

the next day. The equation sheets will be graded for completion only. The homework will be a

combination of a completion grade plus an accuracy grade for random problems.

Adaptations:





• Follow-up lessons will include review of solving equations using all methods learned so far.

Slides - Lesson 5 - Solving Formulas and Equaitons

1

November 12, 2009

Oct 25-5:33 PM Oct 25-5:46 PM

Oct 25-5:50 PM Oct 25-5:50 PM

Oct 25-6:09 PM Oct 25-6:30 PM

Slides - Lesson 5 - Solving Formulas and Equaitons

2

November 12, 2009

Oct 25-6:30 PM Oct 25-6:44 PM

Oct 25-6:54 PM Oct 25-6:58 PM

M&M Questionnaire

1. I would have to add (or eat) _______ red candies to have the same number of red candies as the

teacher. How many red candies do I have?

2. If I doubled the number of blue candies I have, then I would have _______ blue candies. How many

blue candies do I have?

3. If I tripled the number of yellow candies I have, I would have _______ more yellow candies than the

teacher. How many yellow candies do I have?

4. If I added 15 brown candies to my bag, the teacher would have to add ______ brown candies to her

bag in order for us to have the same number of brown candies. How many brown candies do I have?

5. If I ate 3 of my orange candies, then put my orange candies together with the teacher’s orange

candies, we would have _______ orange candies all together. How many orange candies did I start with

originally in my bag?

6. Suppose another student had a bag of M&M’s exactly like mine. So we each started with the same

number of each color candy. If we combined our candy, then I ate 5 of our red candies, we would have

_______ red candies left. How many red candies did I start with originally in my bag?

7. My brown, yellow, and green candies total ______. I have _____ more (or fewer) brown candies than

yellow candies. I have _______ fewer (or more) green candies than yellow candies. How many brown

candies do I have? How many yellow? How many green?

8. I have a total of _______ candies in my bag. I have ______ more (or fewer) brown candies than

orange candies. If I eat all my brown and orange candies, I will have ______ candies left. How many

brown candies did I eat? How many orange candies did I eat?

Documents

Short Range Lesson Plan - Wikispaces-+Unit+Lesson...Short Range Lesson Plan Solving One-Step Equations using Addition & Subtraction Mastery Algebra 9th Grade Joy DiMuzio Objectives: