# Inca Ancient Civilization Picture Problem 4.NBT.B.4

The image displayed above could be used in an integrated math classroom to help teach 4th grade math students about mathematical practices from ancient civilizations as well as record keeping techniques. Often times, we get caught up in the paper and pencil way of showing one’s work. By integrating a bit of Inca history in our math classroom we can elicit a different form of showing work and have some creative and artistic fun while we’re at it. A rich math task to help students model adding multi-digit whole numbers is to have students learn about Inca culture and create a quipu, which is a knot technique used to add multi-digit whole numbers.

Students would be able to read about Inca life and customs and then create a quipu using numbers of their choice. Once students are taught the fundamentals about what certain knots represent and how they are positioned, they can create their own quipu that displays an algebraic equation. An extension that can be used with this task is to have students trade quipu and determine what the equation represented is and check to make sure the answer represented is correct. Another extension would be to have students divide into stations and go around to each station and use a handout to write down the equations represented on various quipu and move about the room until they have been to each quipu station.

This task is a rich math activity because it can be done by all students of varied skill levels. Advanced students  can create more complex equations, while struggling students can create much simpler equations. This task also provides multiple pathways in the sense that students can creatively represent their equations on their quipu with a variety of colors and string lengths.

The multicultural aspect of this picture activity is that it not only integrates a different form of writing math equations, but it also introduces students to Inca culture and other social studies content. Writing and literature are another integration that can be used with lesson because students read informational text about people of the Inca culture and then get an opportunity to write about how the advancements of mathematics have evolved, Inca history itself, and how students created their own quipu.

Integrated Common Core State Standards and Mathematical Practices Addressed:

CCSS.Math.Content.4.NBT.B.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.

CCSS.ELA-Literacy.RI.4.7
Interpret information presented visually, orally, or quantitatively (e.g., in charts, graphs, diagrams, time lines, animations, or interactive elements on Web pages) and explain how the information contributes to an understanding of the text in which it appears.

CCSS.ELA-Literacy.W.4.2.d
Use precise language and domain-specific vocabulary to inform about or explain the topic.

CCSS.Math.Practice.MP4 Model with mathematics.

CCSS.Math.Practice.MP5 Use appropriate tools strategically

CCSS.Math.Practice.MP6 Attend to precision.

# Fence Fractions – 4.NF.B.3

Students are engaged in real world application of mathematical modeling when creating a fence using fractions that have unlike common denominators to add to make a whole. Tommy is building a fence and has one side length finished (1 whole) and needs help finding the other sides to his fence. The students will be given different fractions to analyze on geogebra–an online tool for students to use to help conceptualize fractions with unlike denominators. This lesson is designed for a 4th grade class modeling addition of fractions.  Students will be working individually to help Tommy create his fence.

Fence Lesson Plan-25objui

http://www.geogebra.org/m/ZrHpgCQv

# Pizza Fractions 5.NF.1

This is a fifth grade mathematics lesson integrated with technology. The focus of this lesson is adding fractions with unlike denominators. This activity would take place when students are beginning to understand how to add fractions with unlike denominators. The pizza problem is a classic example used when it comes to fractions, however this activity is going to have a little bit of a twist.

Normally, math is in the morning but for the day of this lesson, math will be moved to after lunch. During lunch students would be able to eat pizza that would be provided. After lunch, students will need to add up how many pieces of pizza are left, the top of the boxes will say how many slices each pizza was cut into. This should be fairly easy because the pizzas would most likely be cut into the same amount of pieces.

After the students have added up how much pizza is left, they will figure out how much pizza the class ate as a whole. Again, should be fairly simple for fifth grade students.

This is when the lesson will take advantage of the technology provided. You can either do this activity as a class or, depending on how many computers you have, you can have them work individually or in small groups.

On VisualFractions.com, there is an adding unlike fraction circles activity that students can do. The website will give the students the first addend and they will have to input the fraction that is shaded. Then, they will be provided with the second addend, again, having to provide a fraction for the shaded amount. Finally, they will have to add the two unlike fractions by finding equivalent fractions. If students get the answer wrong, it will tell them if the solution is greater than or less than the answer they gave.

Education Technology EALR 1.1.2 Use models and simulations to explore systems, identify trends and forecast possibilities.

CCSS-Math 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

# Painted Cube F.LE.A

The image can be use to teach students to create exponential functions, arithmetic series, description of the relationship of the different colors. Students make a list of the patterns in the cubes to help them create an equation that represent each of the cubes; to expand their thinking, they can create an equation that would give how many cubes of a given color depending on the area of the cube or the number of cubes used. To make the activity more engaging, providing students with wooden squares to recreate this image, would help students deepen their understanding of the problem. The image can be to teach ratios/proportions between objects.

HSF.LE.A.2

# The power of Books 5.OA.B.3

This 3-Act Math Task is called “Nana’s Lemonade” by Dan Meyer. In this task there are three acts. Act one is a video that shows a small glass of water with only one lemon wedge, towards the end of the video a second glass comes in which looks like it can be about three to four times the size of the first glass. Along with act one is a questions asking students “How many lemon wedges should we use to make it taste the same?” it asks for the students’ guess as well as a guess that us too high and one that is too low. In act two there a couple questions asking “what information would be useful to know here?” and “guess the volume of the larger cup” in this act there are also two images showing the volume of the cups. With the information given in act two the students are able to get to act three which is a video with the answer.

This task is aligned to the Common Core State Standards:

CCSS.MATH.CONTENT.6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

CCSS.MATH.PRACTICE.MP4 Model with Mathematics.

Students are also given an opportunity to challenge themselves with the sequel questions which range in difficulty. Since we are in the topic of numbers and operations I would maybe have the students answer “how many ounces of water are in each container?” The students already know how many cups of water each container has but now they will have to convert to ounces.

# 5.NF-FrAcTiOn fUn – Add and subtract fractions with unlike denominators

FuN wItH fRaCtIoNs  by Mike Prelesnik & John Broin

CCSS:  5.NF.A.1   Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

CCSS:  5.NF.A.2   Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. for example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Mathematical Practices:

Make sense of problems and persevere in solving them–This is achieved when the project is started with the explanation of what the whole is and how a fraction is breaking the whole into “pieces”  and then adding  “pieces” together from different wholes in order to get an equivalent sum.

Reason abstractly and quantitatively–When  introducing  fractions with different denominators and show the students that they still equal the same whole, the students then start to look more abstractly at the task and start putting together that 3/3 is the same as 2/2  and also finding it incorrect to add 1/3  and 1/2 unless the pieces are converted into the same size piece and they see this visually.

Model with mathematics–This is accomplished throughout the entire project starting with the teacher modeling right up front about what our whole number is and continues through the students modeling the actual addition of fractions using the tools provided (which also covers the Practice of  “Use appropriate tools strategically”)

Attend to precision–This practice comes in different forms for this project.  Each strip is broken down into smaller and smaller pieces (sizes) and it requires extreme precision to make sure that 5/6 and 7/8 aren’t mistaken as the same size.

Materials and Equipment:

There is no technology needed for this modeling activity.  All that is needed is precut strips of paper of equal size and shape, colored pencils, white board, and the worksheet.  You can also purchase plastic folding strips from a teachers supply store if budget permits.

Modeling Activity:

This activity helps teach the mathematical concept of adding fractions by using strips of paper to represent the whole and then folding the strips to help represent how many pieces the whole is split up into.  This is a hands on activity that will give the students a visual representation of the value of each fractions

In reaching the CCSS of 5.NF.A.1 (which is replacing given fractions with equivalent fractions in order to produce an equivalent sum)  the students will each have several strips of paper, all of equal shape and size.  Each student will then be asked to represent the value of 3/5 by folding the strip into 5 equal parts and then shading 3 of these parts with a colored pencil.  The students will then use a different strip to represent the value ¾ using the same procedure.   The student will then use these 2 strips and discuss in groups of 4 how they can add the pieces together.   The teacher will walk around the classroom during this part of the activity and ask leading questions to help the students discover, on their own, the conclusion that if you break each piece into the size of piece from the other strip, that they will become equivalent size pieces on both strips, and then they can count the shaded parts and add those together.   Once the teacher is satisfied that the groups have a clear understanding, a representative from each group should stand up at the white board and show, graphically, how they added their 2 fractions.

See video for visual reference:  http://www.youtube.com/watch?v=lidrNnp2ga0

Addressing CCSS: 5.NF.A.2  is done through the worksheet that is sent home with the students for a more summative assessment.  This worksheet will have the Title picture above to give the student a visual representation of a whole broken down into different size pieces.  Then,  a set of story problems for each student to work out on their own will involve real world application like using recipes and having  to adjust them to meet different scenarios.  I.E.

1. You give 1/3 of a pan of brownies to Susan and 1/6 of the pan of brownies to Patrick. How much of the pan of brownies did you give away?

2. You go out for a long walk. You walk 3/4 mile and then sit down to take a rest. Then you walk 3/8 of a mile. How far did you walk altogether?

For students who are having a hard time understanding the procedure of breaking each piece of one strip into the size of piece from the other strip, the teacher can simplify the problems by using larger fractions like ½ and 1/3.  Also, you could draw the activity on a full size sheet of paper representing each fraction as a bar that is broken into its respective pieces and then have the students write down what they see on the paper.

Discussion Questions:

1.   What’s the top number of a fraction represent?

2.  What’s the bottom number of a fraction represent?

3.  Why can you add the top numbers if they are different but you can’t add the bottom number if they are different?

4.  Can the top number be larger than the bottom number?  If so, what does this represent?

5.  Is it possible to have 2 different denominators and have the fractions be equivalent?

6.  Where would you use these skills in the world outside of the classroom?

# 4.OA – Cards Up … Seven Up

Concept: Operations

Procedures: Addition, Subtraction, Multiplication, and Division

Common Core State Standard Targeted:

CCSS.MATH.CONTENT.4.OA.A.2

Use the four operations with whole numbers to solve problems: Multiply or divide to solve word problems involving multiplicative comparison

While playing this game, students are using all four operations, with an emphasis on multiplication and division, to solve a real world problem. By definition, a real world problem is something that is concrete, not abstract, and uses a concept in a real setting or application. To students, a game qualifies as such. Therefore, this game targets and helps students to meet this standard. The game could be used to teach and practice the standard or assess the mastery of it.

Mathematical Practices Used:

• Make sense of problems and persevere in solving them: The students are given a situation where they are asked to problem solve as there is no one “right” way to solve the problem. The students are also asked to work through situations in which a pre-described method or set of steps is not provided to them. They must utilize strategy when playing against themselves or a partner, and must make adjustments to that strategy, in order to use the most cards and ultimately win the game.
• Model with mathematics: The students are modeling the situation as they create number sentences (with the cards) that represent the given situation (product number), which exemplifies the use of modeling with mathematics. By modeling, the students are taking an abstract concept of operations and are turning it into a concrete and tangible representation. This transfer from ideas on a page to the tangible real world, in the form of a game, can solidify knowledge in students. It can also give the students a reason why it is important to learn, adding motivation for the students to master the concept. Lastly, modeling can make the concept come alive, leading to better retention, higher ability to transfer the knowledge, and allowing for higher thinking skills to occur in regards to the concept.
• Look for and make use of structure: The students are taking a general idea of operations and using it in a specific situation, using the structure of the operations as their guide to solve the problems.
• Construct viable arguments and critique the reasoning of others: The students will engage in dialogue about their mathematical reasoning as they explain to their partner how they arrived at their target number. The partners will have discussion about whether or not they agree as well as what modifications would need to be made to make the equation true.

Technology

A use of technology that could be included in this activity would be the use of a calculator. However, the purpose behind the activity was to practice mental math and quick access of math facts. This purpose would not be served if the use of a calculator was permitted.

Mathematical Modeling Aspects Present in the Activity

•  Realizing when revisions need to be made: Students must realize when their answers are incorrect and make the necessary changes in order to make their equation equal their targeted number.
•  Make improvements on their model or strategy: Students will choose a strategy that gives them the most cards. However, as the game is continued and more time is spent playing the game, students may change their strategy in order to receive more cards. Strategy may also change when playing with a partner versus themselves. This idea of strategy and how it changed is discussed during the wrap up period.
•  Interpret their mathematical results in the context of the situation and reflect on whether the results make sense: Students will evaluate their answers through self-check and peer discussion to establish of their results makes sense and are correct.

Objective: Students will be able to use multiple operations (addition, subtraction, multiplication, and division) to create equations that equal specific answers.

Time: 25 – 30 minutes (with an additional discussion and wrap-up for an extension, if applicable or time permitting)

Players: One or two

Materials: Deck of cards (with jokers and face cards removed), two dice, one tally sheet* per partner (if playing in partners), one board* per partner (optional)

How to Play – Individual:

1. If not already done, remove the jokers and face cards from the deck of cards.
2. Shuffle the cards.
3. Roll the two dice and multiply the numbers together. This is your target number for this turn.
4. Take the first seven cards of the deck and flip them face up, placing them in a row (on the board spots if using one).
5. Using the seven up facing cards, add, subtract, multiply, and\or divide the numbers to achieve the target number. The object of the game is to use as many cards as possible of the seven facing up. (Note: Ace cards are worth 1)
6. After the target number is achieved, place the used cards in a pile to the side (or on the board where labeled) and leave the remaining, unused cards in the row.
7. Replace the used cards so seven cards are facing up again.
8. Roll the dice again for a new target number and complete steps 3-7 until the deck is gone.

How to Play – Partners:

1. If not already done, remove the jokers and face cards from the deck of cards.
2. Shuffle the cards.
3. Determine which partner will go first.
4. Distribute seven cards to player one. Player one will take the seven cards and flip them face up, placing them in a row (on the board if using one).
5. Player one will roll the two dice and multiply the numbers together. This is their target number for this turn.
6. Using their seven up facing cards, player one will add, subtract, multiply, and\or divide the numbers to achieve the target number. The object of the game is to use as many cards as possible of the seven facing up. (Note: Ace cards are worth 1)
7. After the target number is achieved, player one will explain to player two what operations and steps they used to get to their target number.
8. Player two has the opportunity to challenge any flaws they see in player one’s explanation at this point.
9. Once the two players have agreed on the equation and the target number was achieved, player one will place the used cards in a used card pile on the table (or board), tally how many cards they used, and leave the remaining, unused cards in the row.
10. Now it is player two’s turn. Player two will place the next seven cards in the deck on their board.
11. Player two will follow the same steps listed in direction number 5-9.
12. At the start of each players next turn, they will need to replace the cards so that seven cards are facing up at the start of each turn.
13. The players will alternate turns, following directions 5-9, until the deck is gone. Once the deck is gone, a winner will be determined based on which partner has the most tallies.

For students who need a change of some kind, rules can be set that changes what operations the students can use. The rules can limit, expand, or include all operations to meet the student’s needs.  Rules can also be made that set a minimum in the number of cards that must be use in order to challenge the student with how many operations they use in order to use that many cards and still achieve the target number.

For example, students who are below the fourth grade level, the game can be adapted to use only addition and subtraction for both the target number operation, and the operations used to achieve that target number.

An example for students who are above the fourth grade level would be to set a minimum of four cards to be used each turn so that the student must challenge their thinking in order to use more cards.

Discussion Questions

1. What was a strategy that got you the most cards?
2. Was the strategy that you used different when you played against yourself than when you played against a partner? If so, how was it different?
3. Did you and your partner have any disagreements about their reasoning or mathematical equations? If so, what were they? How were they resolved?
4. Which operation did you use the most? Why do you think that was?
5. Which operation did you use the least? Why do you think that was?
6. How are these skills used in real life?
7. Why are these skills important to learn?

Wrap Up

By allowing the students to engage in math talk about the game that they just played, mathematical connections can be drawn and the students can really dive deep into the embedded concepts as well as the mathematical practices that are included in the game. This discussion can serve as a reflection and a time for further and deeper mathematical learning to occur. Of course, these questions can be used as a quick debrief or a more extensive conversation starting point.

Currah, J., Felling, J., & MacDonald, C. (1992). All hands on deck, math games using cards and dice. (Vol. 2). Alberta, Canada: Box Cars & One-Eyed Jacks

# Using iPad to Teach Long Division

In today’s classroom, every student from Kindergarten on knows what an iPad is and how to work it to some degree. Incorporating the use of iPads, when available, can really enhance the learning environment. In addition to this, teachers can motivate their students to create a personal need to achieve the learning outcome, because students will not actually realize they are actively learning, but rather just playing games. Using an iPad is fun! Students will want to keep using it for as long as possible. This is practice, and practice leads to the outcome being learned.

An example of incorporating the use of iPads into the classroom is using the app called “Long Division Touch – Classroom Edition.” This app will help you   teach the math standards:

CCSS.Math.Content.6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm.

CCSS.Math.Content.6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

The app contains an introduction, which explains to the student how to use the app. They must go through the motions with the app to complete it. In addition, the app contains five “lessons,” which include “Zero Up Top,” “Remainders,” “Decimals,” “Repeating Decimals,” and “Decimal in Divisor.” Each of these lessons contains an “Explain” button that will show the student how to solve the problem using the standard algorithm. They also have a “Practice” button, which gives the student random problems to practice.

The app also allows for teachers to create “Activities,” which are assignments that the student must complete. In this section, the teacher can name the activity or assignment, choose the total number of problems and what kinds of problems from seven categories. These include one digit divisor, zeroes in answer, decimals, decimal in divisor, two-digit divisor, remainders, and repeating decimals. A great thing about this is that a teacher is not limited to just one category, but can choose as many of the seven categories as desired to complement the class.

You might be asking yourself how can I actually see that each student is completing the activity/assignment that I made without physically monitoring each and every student in the class? Well this is an easy solution! This app also has the feature of Kodiak reporting. Kodiak reporting is a free educational tool that lets teachers monitor their students’ progress and usage on educational apps. Kodiak can be accessed anywhere with an Internet connection, and the work that students do on the “Long Division Touch” app will automatically be sent to Kodiak. Another great feature of Kodiak is the “Dashboard,” which allows the teacher to see every iPad screen in the classroom from his/her own device. This makes it easy to monitor every student as they work from the app, which gives you the added bonus of being able to see who is struggling so that you can offer help.

The app is available through the App Store and is only available for the iPad. Unfortunately there is a cost for this particular app – \$1.99. However, there is in fact a free version of the app but the practice problems given are very limited and Kodiak reporting is not a feature. I suggest purchasing the app if possible in order to provide each student with adequate practice to achieve the learning outcome and allow you to monitor progress.