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Week 4: Overview

NEW Week 4 - Exponents and Order of Operations

Lesson Content and Issueo Making the shift from computation to problem-solving strategies -

teaching students how to apply what they've learned to new situations.

Teaching Strategy - Authentic Instruction, specifically Problem-based instruction

Math Concept Leveling Integration Resources Assignments

o App Cornero Readings

o Apply Teaching Strategy

Week 4: Lesson Content

Introduction

Our students tend to be great at memorizing facts, figures and formulas in order to help them solve problems. However, there is an ongoing issue in today's students where when we present them with a problem, they have difficulty in transferring this knowledge to actually solving problems. The issue then becomes being able to shift student knowledge from simple computation to actual problem-solving strategies. It is important that we present the students with not only the what, but also the why to solving problems so that they can truly apply what they are learning to real-world problems. Let's take a look at problem-solving as a process.

Problem solving begins with the thinking process. When students use reasoning skills, decision making, and critical and creative thinking, they are problem-solving. When designing problems for students to solve, you can do this in several ways. First, you can create a problem with a

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specific goal in mind, one that is clear and concise. For example, you may give the students a set of numbers and ask them to round each of these numbers to the nearest tens, hundreds, and thousands or you could give a worksheet of multiplication problems. Finding the correct answer would be the specific goal or end result. Likewise, you can design a problem with no set goal or clear instruction, where students need to use critical thinking and come up with a plan on their own. For example, you could give the students a scenario and have them develop a word problem using the facts. For example, "Jerry had 5 flashlights, Tom had 7 lanterns, and Sarah had 3 pillows. Design a story using this information." While the first type of problem just requires mathematical reasoning, the scenario requires a deeper level of critical thinking.. Just think of it this way: Clear and specific problems are frequently found in a classroom setting, while less structured problems are much more likely to occur in life. We need to help prepare our students for solving both types of problems. (Mayer and Wittrock, 2009).

Selecting from a wide variety of Instructional techniques can help enrich the learning experience for students. These could include:

Explicit instruction Drills Research Class Discussion Group Work Presentations Role Playing Modeling Using manipulatives

When engaging students in problem-solving in math, you should select the type of strategy that would work best for this purpose. For example, although problem-solving may start with lecture, it is important that the students are given the opportunity to "dig in" and really use their higher order thinking skills to solve the problem. One strategy that works well in this scenario would be group work, where the students collaborate to come up with answers to the problem being presented.

Also, by aligning the math curriculum to state standards, giving our students the time to practice problem-solving skills, using frequent and ongoing assessment strategies to determine learning outcomes, and providing help and remedial activities for students who need to review a concept again, will enable our students to learn and use problem-solving skills successfully. Improving Math Performance , outlines teaching strategies and ideas that have worked for schools around the nation to help implement problem-solving skills in their students.

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Remember, when you expect your students to simply memorize by rote, retention will most likely occur, but students could eventually have issues with application or transfer of this knowledge. But by incorporating strategies where students are expected to demonstrate a mastery of the math concept itself, the process becomes meaningful, there is better retention, and transfer of knowledge is a success. This should be our ultimate goal as teachers; emphasizing the problem-solving process, teaching this process in the confines of the subject area (in this case, math), and making the process as authentic as possible are key in making problem-solving skills the norm in your classroom. In the next section, we will discuss problem-based instruction, an authentic teaching strategy.

Week 4: Learning Strategy

Remember that our goal in instruction is to ensure student success and to choose teaching strategies that will meet their individual needs. This week, we will examine problem-based learning as an effective and authentic instructional strategy.

Strategy: Authentic Instruction and Problem-based Instruction

This week, we will focus on one type of authentic learning – problem-based learning. As the name implies, students explore new concepts by using critical thinking skills to solve a problem. Would you rather have a worksheet containing 20 questions on multiplication or work with a team to determine how many _____________________________________________?

Watch this short introductory video about inquiry and problem-based learning and how it changes the feel of learning and using math.

Remember, as we teach our students how to solve problems, it is just as critical that they understand why we solve a math problem as it is to understand how we solve it. Problem-based instruction involves the teacher presenting a problem or scenario to the students, who then work together in groups cooperatively, researching, inquiring and discovering an answer to the problem. In problem-based instruction, the teacher has a big part to play in the direction of the process, as they ask probing questions to increase creative thought and collaboration among the group members.

In this short video, the teacher engages the students in a "new way to learn." It introduces a problem-solving strategy called I2 (I-squared). The teacher serves as the facilitator and first

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introduces a challenge. The students then get in groups and brainstorm about possible solutions. They use hands-on activities to help solve the problem. Then the group presents their solution to the class. Finally, the teacher conducts a wrap-up.

In problem-based instruction, the students are given a real-life problem that they can relate to and are personally interested in. The problem should be one where there is no obvious immediate answer, but should require the students to ask questions and investigate in order to arrive at the solution. The students should then be given the opportunity to discuss the process in a community of learners where each voices personal ideas for finding a solution. Lastly, the students need to be able to inquire, observe, and direct themselves as they come up with a solution to the problem. For example, the unit of study is equilateral triangles. You place your students in groups of four and give each group six toothpicks. Then you ask them to create one equilateral triangle using only these 24 toothpicks. Then you ask them to create two equilateral triangles using six toothpicks. Try to build three...now build four, again, only using six toothpicks. Ask them if it is possible to make all four equilateral triangles. Why or why not? This type of problem gives the students a hands-on, collaborative environment where can they work together, direct themselves, and solve the problem.

When teaching our students how to problem-solve, there are some practical strategies that can help them with the process. Consider the following four steps and how they could assist students in solving problems. They are summarized from a lengthy document, "Classroom Cognitive and Meta-Cognitive Strategies for Teachers", found here.

Step 1: Understanding the problem - This is a step that is sometimes ignored in the problem-solving process - where students should begin getting a handle on what the problem involves. During this step, it is important to verify that the students understand the vocabulary used in the initial problem. A good check for this is to have the students retell the problem in their own words.

One strategy to incorporate here is called SQR (Survey, Questions, Read). In other words, students will read the problem, paraphrase the problem, determine what needs to be solved, and rephrase the question to eliminate information that they won't need to use. Other strategies to use are the Frayer Vocabulary Model, mnemonic devices, graphic organizers, paraphrasing, and cooperative learning groups. The article at the above link goes into much more detail about these strategies.

The key step for students is understanding how to determine what it is they are being asked to solve. There are several different strategies you can teach them to help them with this. First, they can mentally think through what it is they are looking for. They could also begin picturing or diagramming the problem on paper. This is particularly useful when introducing a new concept. They should also verify that they have enough information to begin the problem-solving process. Working in groups can also be useful, as the team discusses and reaches a consensus on how to proceed.

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Step 2: Devising a plan to solve the problem - This is the step where the students will work together to estimate, share, and explain their strategies with one another. Students begin to hypothesize and estimate, reading and paraphrasing the problem, and coming up with guesses, while starting to create a list of possibilities. They will also use reasoning to decide what stays and what goes. They will draw pictures and diagrams to help them visualize their plan. Once they put this plan together, they can then start explaining their plan of action to solve the problem.

Students should ask themselves if the problem looks familiar to something they have worked on in the past. Some strategies that should be emphasized during this step are encouraging the students to guess and check, to make lists, to get rid of unnecessary information, to use reasoning skills, while looking for patterns. If the students are given a simpler problem to solve initially, it may help them to better see how to solve the problem at hand.

Step 3: Implementing a solution plan - This is the step where the students experiment with the different plans they came up with in step 2. It is important that the students are allowed to make mistakes. They need to be encouraged to show all their work along the way and discuss with their group members if the process is working. They can make multiple attempts at solving the problem to see which solution works best.

During this step, students should be encouraged to double check each step, possibly trying a different plan if this one isn't working. Mistakes are okay, at this point. Lastly, when they come up with a solution, they will check their answer to make sure it is correct.

Step 4: Reflecting on the problem - looking back - This is the final step where the students reflect on the plan after the answer is discovered. They will check that their answer is correct, then come up with other strategies that could have worked just as well or even more efficiently. Then they should be encouraged to take the process and their answer and extend what they learned to other problems to increase their knowledge and problem-solving capabilities.

Reflection includes making sure that the result can be verified and that the answer makes sense. It also involves distinguishing which methods of problem-solving worked and what didn't. In this step, students realize what they learned through the process. And it should be determined if they can apply what they learned to solve a similar problem in the future.

Take some time to think about how you can use problem-based instruction in your math classroom. What problems would you present? What strategies would you use? How would you implement this type of instruction for your students?

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Next week, you will learn about another aspect to authentic learning - project-based instruction. But for now, we will dive right into the math topics included in this week's lesson - exponents and order of operations.

Week 4: Math Concepts

Teaching students the concepts of exponents and order of operations will enable them to grow in their math skills, introducing them to skills necessary in algebra. Once students grasp the concepts of exponents and order of operations using simple numerals, they will be better able to move from the concrete (using just numerals) to the abstract where variables are introduced.

The concepts of exponents and order of operations may be difficult for some students initially, however, when they are taught in a systematic and consistent fashion, step-by-step, the students will see a pattern develop and will be able to apply this pattern to a multitude of problems.

As you continue with this class, Core Math Concepts for Elementary, we hope that you increase your knowledge, comfort and love of math. Please remember to use the knowledge and resources that you gain in the lesson so that you can impact your new generation of students by passing on your passion for the topic.

Exponents

When first exposed to exponents, it could be a little difficult for students to understand the terminology. Saying “3 to the fifth power” can sound foreign to some students. But, once teachers break it down into steps for students, exponents end up being a simple, useful and fun task! In fact, learning to use exponents can prove to be a time-saver in the long-run.

We already know that multiplication is simply repeated addition. Well, exponents are simply repeated multiplication! If your students know the why behind multiplication already, then teaching them exponents should prove to be an easy transitional task for them. Watch this video which provides an in-depth, but clear explanation of the key aspects of exponents. There are other subsequent lessons available at YouTube that you can use for more advanced or older students.

Here are a few suggestions:

Elementary Algebra Lesson 1.2: Exponents Part 2 Elementary Algebra Lesson 2.1: Addition and Subtraction of Like Terms, Part 1

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Elementary Algebra Lesson 12: Multiplying Binomials

Teaching your students the new terminology that accompanies exponents is very important. In other words, instead of saying "math problems," use the term expressions. When you say base number, your students should recognize this as "the number that is being multiplied." The word exponent (or power) means "how many of that base number that will be multiplied." Writing in exponential form will include a base number and an exponent, and will look something like this: 24 . This same number written in word form will look like this: "two to the fourth power." Expanded or factor form will be written like this: "2 X 2 X 2 X 2." And standard form will be the number in simplest form, which, in this case is 16. Here is an overview of what you have just read:

Base number 24

Exponent or power 24

Exponential form 24

Word form Two to the fourth powerExpanded/Factor form 2 X 2 X 2 X 2Standard form 16

When you first start teaching exponents to your students, a proven method to use is a factor tree. Factor trees can be great jumping off point for teaching exponents. What is a factor tree? "A factor tree is a diagram used to break down a number by dividing it by its factors until all the numbers left are prime." (Sebastian). Remember, a prime number is a number that is only divisible by one and itself. All the numbers left in red in the factor tree (the prime numbers) are numbers that you will use when writing your exponents. See the example below:

Once you have determined your prime numbers at the bottom of the factor tree, in this case 2, 2, 2, 2, 2, and 3, you can begin writing these using exponential form. First, count how many twos that you have and then how many threes that you have. You have 5 twos and 1 three. Using exponents, you would write this as 25 X 31. When the exponent happens to be a "1", it is

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not necessary to actually write the one next to the base number because, in this case, you are not multiplying the three by another three. Therefore, the above expression can also be written as 25 X 3.

See the chart below for an overview of what the above factor tree produced:

Exponential form 25 X 31 or 25 X 3Word form Two to the fifth power times three to the first powerExpanded/Factor form 2 X 2 X 2 X 2 X 2 X 3Standard form 96

Once you introduce exponents to your students, give them the opportunity to see patterns and to develop their own rules for exponents. Of course, these rules are pre-established, but allowing the students to develop these rules in their own language based on their own experiences will make this concept more meaningful and will demonstrate the problem-based authentic learning theory.

Khan Academy has a mini-unit on exponents that would be good for you to view, as well as show to your students and their parents. Take a moment to watch it now.

Have your students engage in cooperative group work to facilitate discussions about exponent rules and exceptions. Tell them to place various exponents behind a number and solve them. Then have them make the exponent a "one." Ask them what they notice about the result. Ask them what happens when the exponent is a "zero." What would the result be? Is there a pattern that develops? Is there an exception? And present them with a whole line of the same base being multiplied. Ask them if there is a shorter way to come to the solution more quickly. Here are some basic rules of exponents that you will want your students to arrive at through this cooperative discussion and group work:

Any number raised to the first power equals itself. ( For example, 31 = 3) Any number raised to the zero power is always 1. (For example, 60 = 1). The exception

to this is when you solve 00. 00 equals 0. When multiplying exponents with the same base, just add the exponents. (For example,

23 X 26= (2 X 2 X 2) X (2 X 2 X 2 X 2 X 2 X 2), or 29. Remember, just add the exponents and keep the same base).

Be sure and remind them that this is for multiplication only and not for addition!

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Order of Operations

Just suppose that two classes at your school are attending a field trip to the park. In one class, there are 24 students. In the other class there are 22 students. The teachers know it's hot outside, so they want to make sure the students stay hydrated by providing them with 2 bottles of water each. How many water bottles should the teachers bring with them to the park? In mathematical form, the equation would look like this: 2 X (24 + 22), the 2 representing the number of water bottles each, and the 24 and 22 representing to total number of students. The answer would equal the total number of water bottles they should bring for the kids. This is but one example as to why using order of operations in solving problems is so important. Let's try it both ways:

Order of operations tells us to compute what is in parentheses first:

2 X (24 + 22) = 2 X 46 students, which equals 92 total water bottles.

However, when using standard form (from left to right), here is what happens:

2 X (24 + 22) = 2 X 24 first, which equals 48. Then add this number to 22, which equals 70.

So, do you see the dilemma here? If we don't use order of operations to solve a problem, we come up with a different solution, in this case, not enough water for your kids.

Order of operations is a mathematical concept that some students may have difficulty understanding at first. It is the conventional way of solving problems using a specific order of computation. How do we break this difficult mathematical concept down in a way that students can understand? The most popular method has been to use the PEMDAS method. However, when students use PEMDAS, to solve problems using order of operations, they often make mistakes when they get past the "P" and the "E". You may ask: "Then, why use it?" Let's first look at what PEMDAS means and you will gain better understanding of why we should present our students with an alternative.

PEMDAS is the acronym that many people use to remember Order of Operations. Most of us learned it as, “Please Excuse My Dear Aunt Sally.” Of course, each letter stands for one of these operations.

ParenthesisExponentsMultiplication and DivisionAddition and Subtraction

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What this seems to mean is that when given an equation to solve, you are to solve it in the order of the above hierarchy - solving what is in parentheses first, then exponents, then multiplication, division, addition and subtraction. However, there is an inherent problem with this method. When students solve problems using PEMDAS, they often believe that they are to solve the equation by multiplying first, then dividing. Likewise, they assume that addition must always be solved first, before subtraction. This is not the case. In reality, multiplication is just as important as division, so sometimes division comes first. Addition and subtraction are equally important, so sometimes subtraction comes first. This is probably the number one misconception with order of operations. So, teach them this alternative form of PEMDAS:

M A

P E or or

D SSo when solving an equation using order of operations, this is how you'll do it:

Solve what is in parentheses first. Solve all exponents next. Solve multiplication and/or division next (whichever comes first from left to right). Then solve addition and/or subtraction last (whichever comes first from left to right).

Now, let's put this new knowledge to the test in your classroom!

Start by writing or displaying a simple expression on the board.

4 + 6 X 5

Have the students evaluate this expression. Some students will say the answer is 50. Other students will say the answer is 34. Which is it? Facilitate a discussion as to which of these answers is correct. Start this discussion by asking if students can write a word problem that works with this expression. Take a look at this example:

Stacy worked for six hours today babysitting. She charged $5/hour. She did such a great job that she was given an additional $4 as a tip. How much money did she make today?

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Draw a diagram on the board showing hours and money. Visual learners always do better when they can actually see what it is that you mean. For example you can draw this:

Hour 1 = $5Hour 2 = $5Hour 3 = $5Hour 4 = $5Hour 5 = $5Hour 6 = $5

Extra pay = $4

When the kids see it in this way, they should say 6 hours X $5 = $30 + $4 for extra pay = $34. Is that the same as 4 + 6 X 5? Shouldn’t it be?

That’s when you explain that mathematicians decided collectively on the order of operations so that we would all evaluate expressions in the same way no matter the order in which the numbers are written. We cannot simply solve all equations from left to right. Ask your students why this is. They should establish that the order of operations was put in place so that we would all arrive at the same solution for math expressions. Then introduce them to the PEMDAS alternative that we showed you earlier.

Remember that a misconception by students is that all multiplication should happen before all division because the multiplication comes before division in the acronym. Since multiplication and division have the same precedence, they should be evaluated as they appear from left to right. Start with 12 divided by 3, which gives you 4. Then multiply by 4, for a final answer of 16.

 

Incorrect

 

Correct

Similarly, although addition comes before subtraction in the acronym, remember they have the same precedence. Consider the problem 4 + 10 - 5 + 8. To solve, you will add 4 and 10, which is 14, and then subtract 5, which gives you 9. Finally, add 9 to 8 for a final answer of 17.

 

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Incorrect Correct

As another whole group discussion, give students the example below. What is the correct order in which to evaluate this expression?

Remember to use the PEMDAS alternative. First, there are no parenthesis, so move on to exponents. Again, there are no exponents, so you will multiply or divide, depending which comes first left to right. Divide 9 by 3, which is 3. Multiply 6 and 5, which is 30. It is important to emphasize to your students that as they compute using order of operations, that they use this step-by-step process, and simply bring down the other numbers and symbols to the next line (as you can see in line 2). The danger is that students tend to want to skip steps, but unfortunately, unless they're lucky, they end up with the wrong answer. It is important that they know that it is okay to take the time to write out the steps.

Take a look where we left off in line 2. What operations do you still have left? I see a division symbol at the end, so this will be the next operation you will compute , 30 divided by 2 is 15. Bring that down to line 3 (along with the other numbers and operational symbols). What do you have left? Addition and subtraction. Remember the alternative: add and subtract in order from left to right. Therefore, 2 added to 3 is 5. Subtract 5 and get 0. Add 15. Your final answer is 15.

It is a good habit to practice a few expressions with just the four basic operations as we did above, using addition, subtraction, multiplication and division. This will keep your students from being overwhelmed with the new process. Then add in exponents gradually, then finally, parentheses.

After the initial introduction to the concept of order of operations as a whole class, you will want the students to work in small groups to solve additional problems. Try this one.

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Encourage the students to insert parentheses to make this expression true. Tell them that it's okay to try solving the problem using different orders. Once the answer they get is 50, have them share with the whole group how they came up with the right answer.

Problem: 4 x 6 + 2 + 3 x 6 = 50

Answer: 4 x (6 + 2) + 3 x 6 = 50

*Make sure that students know that the X is a multiplication sign and is not representing an unknown number. You may have some students with some algebra experience who may bring that to your attention. We’ll talk more about finding the value of X in Week 5. For now, these are still multiplication signs.

Now let's try a few problems using parentheses and exponents. Using the PEMDAS alternative, you will solve what is in parentheses first, (3 x 6) = 9. Bring that down, along with the other numerals and symbols. Then you will solve the exponents next, 52 = 25. Bring that down, along with the other numerals and symbols. Do you see any multiplication and division? Yes, you do, 2 x 25 = 50. Bring everything down. Lastly, solve addition and subtraction in order from left to right. Your final answer is 34.

75 - 2 x 52 + (3 x 6) =

75 - 2 x 52 + (9) =

75 - 2 x 25 + 9 =

75 - 50 + 9 =

25 + 9 = 34

Let's try another one using all of the components of PEMDAS. This one is a little tricky. We have found the parentheses, but within the parentheses is an exponent. Now what do I do? Until you solve the exponent, you cannot move on. So, you have to pretend the parentheses are not there for a moment. Solve the exponent first, 33 = 27. Bring that and everything else down. NOW, it looks more familiar. Let's move on. Refer back to the PEMDAS alternative and solve the parentheses now, 27 - 7 = 20. Bring that and everything else down. Next, solve the exponent, 42 = 16. Bring that and everything else down. Next comes multiplication and division. But notice that the D comes before the M this time (division before multiplication). This is correct according to the PEMDAS alternative. Solve 16 ÷ 16, which is 0. Now multiply 20 x 6, which equals 120. Bring that and everything else down. If you got 120 as an answer, you are correct!

16 ÷ 42 + (33 - 7) x 6 =

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16 ÷ 42 + (27 - 7) x 6 =

16 ÷ 42 + 20 x 6 =

16 ÷ 16 + 20 x 6 =

0 + 20 x 6 =

0 + 120 = 120

Khan Academy – Order of Operations – There is an entire unit on Order of Operations in Khan Academy. This would be a great link to share with students and parents for home as well.

You and your students are now ready to take on Algebra! Please see the resources section in this module for a number of resources available to help supplement this week's content.

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Leveling

Introduction

Because your students will possess various levels of understanding in the subject of math, you will need to consider differentiating instruction to give each of your students a fair opportunity to learn the concepts successfully. Below, we’ll explore some differentiated lesson ideas in the areas of exponents and Order of Operations as they pertain to fifth grade math Common Core Standards.

Common Core Standard:

MACC.5.OA.1.1Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. - This strategy will help students to group and visualize each step of the alternate PEMDAS method, enabling them to see the process in a more digestible manner.

MACC.5.OA.1.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

Low-level & Students with Learning Disabilities (SLD)

Students with learning disabilities tend to become overwhelmed very quickly when it comes to math expressions with multiple steps involved in evaluating them. This is where we can bring the alternative PEMDAS strategy back in a different way. These students would benefit from a checklist when evaluating equations (and later as well with more complicated algebra equations). Create a checklist on an index card for these students to use to check off each operation as they complete it.

Another idea would be to encourage students to rewrite the expression after completing each step. Give students a highlighter to highlight the step they will complete. Have them rewrite the expression each time operation is evaluated. By doing this, both the teacher and the student can easily retrace any missteps and correct the problem.

Here’s an example:

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(5 2 + 6) - 1 + 5 X 7 X 3(25 + 6 ) – 1 + 5 X 7 X 3

31 – 1 + 5 X 7 X 331 – 1 + 35 X 3

31 – 1 + 10530 + 105

135

English Language Learners (ELL)

You will also find that some of your students are English Language Learners (ELL students). These students will need additional assistance with understanding the vocabulary, especially with a new foreign mathematical vocabulary. YouTube is a great resource for supplemental instruction in this area. It may be beneficial for a student to watch a video in Spanish to help you get this mathematical concept across to a Spanish-speaking student.

Watch this Order of Operations in Spanish video as a sampling of what your ELL students can use to help them better understand the concept.

ITunes apps are another great way to enhance and supplement instruction for an ELL student. Order of Operations by Brainingcamp is one example of an app that allows the student to demonstrate their understanding of this concept with very simple instructions to follow.

On Level

There are a number of games and resources available online for additional practice with Order of Operations. This one is something that groups of students can play together.

Order of Operations Millionaire Game is a game best played using a SmartBoard or a Mimio device and a projector. That way, your students will feel like they are playing on a real game show, which is great for engaging your whole class. Students should be divided into two groups and given slates. A team captain for each team is chosen. Students must work each problem through on the slate and decide as a team which answer they will choose. The team captain gets to click on the answer on the whiteboard. It’s a fun way for them to gain extra practice in this skill

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Above Level & Gifted

Because you will also have students in your class who completely understand the concepts, the above-level kids will need a task beyond simply evaluating expressions using the order of operations. They will need to see the why behind what they are learning. Accelerating gifted and above-level students’ curriculum is a very common way to keep these students challenged.

Above level and gifted students can begin tackling additional topics related to exponents, such as scientific notation and powers of ten. Allow the gifted students to begin work through these challenges. Using the powers of ten, these students can determine how to balance equations, in the following activities: Exponents: Scientific Notation and Exponents and Powers of Ten.

If these students need an additional challenge, introduce them to the previously uncharted world of a Base 5 system. Using a place value chart that they create, students will be asked to evaluate numbers in base ten to their equivalent number in base five.

Once they understand the base five system, have them work in different bases with various operations. Subtraction in Different Bases works with subtraction in bases 2, 4, 8, 10 and 16! There’s an online abacus to help them visually see the patterns.

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Week 4: Integration Corner

Introduction

When first introducing order of operations to your students, they may want to know why it exists - why solving phrases in a certain order is so important. In this section, you will explore some activities that will help you explain the why behind order of operations to your students. Let’s look at some real-world applications for order of operations and exponents.

Reading & Language Arts

Math concepts can lend themselves well to writing assignments. Why not have students write a story about how order of operations came to be? What chaos ensued before it was determined that order of operations would rule over our mathematical calculations? The stories that they write should include examples and non-examples of the PEMDAS alternative. Have students share their stories. Can they find any expressions embedded within the stories? Can they determine whether or not they follow the law of order of operations?

Integrating math and language arts offers a way for the teacher to use books to teach math concepts. Also, You're Not In Math Class Anymore: Integrating Math Across the Curriculum gives some good insight on how to use math across other curriculums.

Science

Why do we need exponents? Well, when you work with exceptionally large numbers, it’s so much easier to write them using exponents. Scientists use exponents all the time – and this is probably why we call this numerical writing system scientific notation! Think about the distance from the earth to the sun in miles. We could write 93,000,000 miles or, in Scientific Notation, 9.310 miles. Have students find other ways that we can use exponents or scientific notation to write larger numbers and share them with the class.

Exploring Math and Science is a webquest about the numerous connections between science and math concepts that will give you some great ideas integrating these two subjects in your classroom. This Math Song is also a nice way of introducing scientific notation to your students when teaching them about exponents.

Social Studies

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Why not take your students on a trip down memory lane to the world of the Eniac and Atari? Have they ever seen a 5 ½” floppy disk? Have the ever heard of magnetic tape? Take students on an exploration of the computer history. Any time you can go back in time to create a timeline of events is a great way to show your students how easy it is to integrate social studies and math. Integrating Math into Social Studies is a great website for giving you other ideas about how to integrate math and social studies in practical ways. Math and Social Studies also discusses the integration of economics and geography into your math curriculum.

Once you have your students hooked integrating basic math and social studies concepts, ask them, “How do computers communicate?” What language do they speak? That’s when you can introduce them to the binary numbering system. This is a base two system as opposed to the base ten system that they’ve come to know and love. Try Your “Hand” at Binary is a “hands-on” lesson that allows students to decode and write code using nothing more than a pen and their hand.

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Week 4: App Exploration

In this section, we will provide you with a number of applications, or apps, that you can download and use in teaching exponents and order of operations to your students. Of course, this list is not exhaustive, and new apps are being created every day. Take a look at some of the apps below that are available for use in your classroom.

* Order of Operations Game – This is s a game that can be played on the *computer or on an iPad!

5 Dice: Order of Operations – A math game intended for upper elementary and middle school students that will help students enjoy practicing order of operations. The game encourages students to use higher order thinking to solve the "target" number by working backwards given the answer but not the equation. One great feature about this simple math game is that teachers are able to receive immediate feedback of their students’ progress through email.

32HD – This app is based on a simple concept. Your child is given more number tiles than they need and must arrange them within the mathematical sequence displayed in order to reach the total of 32. $

AKW BEDMAS – BEDMAS is a free educational application for mathematics. It is directed towards curriculum from grade 1 to 9. This app is also free from advertising and completely functional without in-app purchases.

Exponents Exponents - This app is designed to allow students to learn about the rules of exponents. The purpose is that this app will be a resource in the learning of exponent rules.

Everyday Mathematics – Name that Number – Players use five number tiles to build number sentences that equal a target number. Players can name the target number by adding, subtracting, multiplying, and/or dividing the numbers using as many tiles as possible. The more tiles the player uses, the more points he or she earns. There are a total of 5 rounds in a game. Visual and audio reinforcement of correct and incorrect answers helps players master math facts and order of operations. $

Math Order of Operations – Students will learn to perform operations in a calculation in the correct order, use the mnemonic PEMDAS to remember the Order of Operations, practice Order of Operations in equations and check answers, create their own problems and solve with a group or partner. $

My Dear Aunt Sally – My Dear Aunt Sally is a new mathematics game that reinforces number sense, number operations, problem solving, and order of operations. Students build

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automaticity with the order of operations by playing at over 300 difficulty levels and enjoy funny, colorful animations of Aunt Sally and her animal friends. $

Order of Operations by Brainingcamp – Everything you need for teaching and learning Order of Operations: narrated lessons, practice questions, virtual manipulatives in a challenging game. $

This = That – Drag numbers and choose operations to build creative equivalencies while practicing math facts.

Zombies vs. Exponents - With firing cannons, sinking ships, and a zombie pirate theme, Zombies vs. Exponents helps students grades 5-8 develop confidence using scientific notation in an approachable game-like environment. $

$ = Cost to App = Free

Week 4: Assignments

1. App-tivity Create & Share

Review the apps presented in this week’s [App Exploration] section. Then, search education-based app websites and/or the iTunes store to review apps that align to this week’s math focus, Order of Operations and exponents. Select one app (found in App Exploration or through your own research) and create an app-tivity.

To share your app-tivity with fellow classmates, click or tap the “reply all” command in the email titled, ‘Week 3 App-tivity Share’ sent by your Instructor. Then, copy and paste your app-tivity into the body of that e-mail message.

2. Read & Reflect

Part A: Write an informal 100-150 word reflection that summarizes your exploration of this week’s content. In your summary, include new insights you have gained, integration ideas you might adopt, and any best practices that have been reinforced via the Lesson Content, Strategies for All, and Integration Corner sections.

Part B: In 100-150 words, review and reflect on a minimum of two resources provided in this week’s [Readings and Resources].

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Note: Please do not summarize what you explored; instead, discuss your personal reactions, new insights, and ideas for integration; you might also discuss how the readings and resources can be used to support your professional growth in facilitating Mathematics instruction.

Part C: Review a minimum of two apps from this week’s [App Exploration] and write a 25 word reflection on each app that you review. In your reflection, you might choose to describe the benefits of using this app, it’s alignment to common core objectives, any classroom management considerations associated with using this app, and/or ideas for incorporating this app in or outside your classroom.

3. Submission Instructions

Type your assignment (items 1 and 2 above) single-spaced with an extra line between paragraphs. Create your assignment in a word processor and save often. Complete both assignments in the same document. Include the appropriate heading at the beginning of each section (i.e. App-tivity Create & Share, Read & Reflect Part A, Part B, Part C). Then, copy and paste your assignment into the body of an e-mail message. Do not send your assignment as an attachment. Enter Week 3 Assignment in the Subject line of your e-mail message. Submit your assignment on or before midnight of the activity’s due date. Review the Schedule under Course Home for due dates.

Important Note: If you desire graduate credit, [click or tap here ] for more information. You must complete graduate credit application requirements by the end of this week. After Week 1, an application for graduate credit will not be accepted due to accreditation requirements. In addition, please let your instructor know when submitting this week’s assignment.

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Week 4: Readings & Resources

Readings & Resources

Each week you will find a variety of resources that can help you further investigate the various topics that covered this week. Below is information on and links to this week’s readings. Please click on the links to interact with each article or video.

Problem with PEMDAS points out the problem with teaching PEMDAS as a hierarchy and includes a diagram to clarify understanding.

PEMDAS Alternative is a blog post that provides an alternative to the hierarchal PEMDAS and an explanation of how you can use it to teach your students.

Learning the Order of Operations is a site that focuses on how to solve multi-step problems, providing practice in which is a large part of learning the Order of Operations.

Learning and Teaching the Order of Operations explains in depth how to go about introducing the Order of Operations to your students and helping them grasp its importance.

Helping Your Children with Order of Operations is the follow up of the article Learning and Teaching Order of Operations. This article gives tips about how to help students at various levels and provides examples of problems to use with students.

Teaching Order of Operations – Third Grade Lesson Ideas is a site that is more specific to younger students whereas Teaching Order of Operations – Fifth Grade Lesson Ideas has tips on how to teach Order of Operations specifically to fifth grade students.

Tips on Teaching Exponents with Examples is an article that gives tips on introducing exponents to students, vocabulary you should use, and common misconceptions that happen when teaching exponents.

CoolMath: Order of Operations Lessons – This Web site summarizes PEMDAS in a way that is clear and understandable. It’s a bit colorful and has ads, but it’s another free resource for teachers and students.

Exponents PowerPoint – This is a PowerPoint presentation introducing the concept of exponents. Teachers can download this PowerPoint and adjust it to meet their own needs.

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The Math Page - Exponents – This page could serve as a self-guided lesson for students on the meaning of exponents, the rules of exponents and negative bases.

Welcome to Problem-based Learning - This is a web resource about problem-based learning.

Problem-solving in Mathematics - This PDF is 75 pages, but a great resource if you want to know more about problem solving in mathematics.

6 Routines to Support Mathematical Thinking - This is a great article that discusses organizational strategies that will encourage effective problem-solving.

Welcome to Problem-based Learning - This article is a good introduction to the topic of problem-based learning.

Why Have an Order of Operations Anyway? - This resource gives reasons for order of operations and gives examples.

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References

Ask Dr. Math. (2013). Why order of operations? [Web resource]. Retrieved from

http://mathforum.org/library/drmath/view/57319.html

CLS Consulting (2012). What is PBL? [Web resource]. Retrieved from

http://www.pblearning.com/what-is-problem-based-learning.html

Ed.gov. (n.d.). Improving math performance. [Web resource]. Retrieved from

http://www2.ed.gov/programs/nclbbrs/math.pdf

FLDOE (2010). Classroom cognitive and meta-cognitive strategies for teachers: Research-

based strategies for problem-solving in mathematics K-12. [Web PDF]. Retrieved from

http://floridarti.usf.edu/resources/format/pdf/Classroom%20Cognitive%20and

%20Metacognitive%20Strategies%20for%20Teachers_Revised_SR_09.08.10.pdf

"Illuminations: Order of Operations Bingo." Illuminations: Welcome to Illuminations. N.p., n.d.

Web. 12 Oct. 2013. <http://illuminations.nctm.org/LessonDetail.aspx?id=L730>.

Mayer, R. & Wittrock, M. (2009). Problem solving. [Web resource]. Retrieved from

http://www.education.com/reference/article/problem-solving1/

"order of operations idea bank." Math Cats -- fun math for kids. N.p., n.d. Web. 12 Oct. 2013.

<http://www.mathcats.com/grownupcats/ideabankorderofoperations.html>.

Perkins, D., Laur, D., Kubik, T., & Malefyt, T. (2013). What is authentic learning? [Web

video]. Retrieved from http://youtu.be/UNP7hv0d0Rk

Scott, Stephanie T.. "Tips on Teaching Exponents With Examples | eHow." eHow | How to

Videos, Articles & More - Discover the expert in you.. N.p., n.d. Web. 12 Oct. 2013.

<http://www.ehow.com/way_5200232_tips-teaching-exponents-examples.html>.

Sebastian. (2010). What is a factor tree? [Web resource]. Retrieved from

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http://thefactortree.com/2010/10/what-is-a-factor-tree/