BIG IDEA 7
Using Symbols to Describe the World
Algebra is a very important part of mathematics for expressing relationships, but students’ approach to algebra and their chance of enjoying and appreciating algebra are often destroyed by the ways it is taught in traditional textbooks. Algebra is a language—a way of describing relationships—but textbooks introduce it as a set of methods and rules to memorize. When students are introduced to variables, they should learn that they are used to represent unknowns and can stand for different numbers, as illustrated in the following pattern.
The shape has a constant value of 5, shown by the green cross in the middle. But the tails of the cross grow each time, and the growth is always 2x, two times the case number. In case 1 there are 2 squares; in case 2 there are 4. For this reason, the linear expression that describes the growth is 2x + 5. One reason that we know this growth is linear is that a graph of the pattern growth, showing case number against number of squares, would be a straight line.
In the expression illustrated here, the value of x changes depending on the case number. In case 1, x = 1, and in case 2, x = 2. This shows the meaning of the word variable: the value of x varies.
The problem comes in US classrooms when students are taught, at the beginning, to solve for x. They do repeated exercises solving for x. They get the idea that x has to be one number that does not vary. This is not a good representation of a variable. In the artificial situation set up by the textbook question, there is only one number that satisfies x, but in most uses of algebra, x is a variable representing something that varies. When students have solved for x for enough hours, they become resistant to seeing algebra as something that can describe patterns and growth, with variables that vary, when this is arguably the most important part of algebra.
In the Visualize and Investigate activities in this big idea, we encourage students as they investigate relationships to use variables as symbols whose value can vary. In the Play activity, students will meet a situation where the variable is one number. These are the different uses of variables. Students should know that variables can vary and that in some situations like the one presented in the mobile activity each of the variables represents one number. In the Visualize and Play activities in this big idea, we encourage students to think about equivalency in interesting puzzles and patterns. We ask them to build, draw, write, and discuss the relationships they see. We stress this type of work so that they can engage themselves in deep thinking about balance and what the equal sign means. We are focusing on flexibility with numbers as students work to create multiple representations of equivalent expressions.
In our Visualize activity, we bring in Cuisenaire rods, one of my favorite manipulatives, which shows numbers visually and physically. In this activity, students use two rods to define a whole and then choose rods to represent half of the unit. There are different possibilities, as students may choose one rod or a combination of rods to represent the half. The goal of this activity is for students to use multiple representations for what they are defining as a fraction (
) or mixed number (1
). We also ask them to make 1 in another way, which will require different rods. They may say that if light green + yellow = 1, then dark green + red = 1. This can lead to interesting questions, such as: “If purple + brown = 1, then what is ?” Students are asked to sketch the Cuisenaire rod proof they build, describe the relationship with words, and then describe the relationship with symbols. We anticipate that students will develop their own symbols because they are tired of writing the words for the rod colors over and over again.
In the Play activity, there is an opportunity to focus on equality. It is always good to remind students that the equal sign (=) means that whatever appears on one side of it is equal to whatever appears on the other. Some students get the mistaken idea that the equal sign means, “Do something; perform a calculation!” In this activity, students will get the chance to think of equality through balance in visual puzzles. There are multiple different ways that students can come up with for finding values of shapes that create balance. We think this will lead to some good classroom discussions.
In the Investigate activity, students will see a growth pattern using pattern blocks. We chose hexagons, trapezoids, squares, and rhombuses to make the growing pattern. Students are given the first three cases and asked if they can see how the pattern is growing and whether they can predict how many of each tile there would be in the different cases. This is an opportunity for students to use variables to represent the changing number of shapes and to describe growth. This is an important use of algebra: a way of describing patterns and growth. It is also an opportunity for students to see algebra visually and create brain connections when both visuals and symbols are used.
Cuisenaire Rod Equivalents
Snapshot
By developing symbols to represent the relationships between Cuisenaire rods, students explore the ways that symbols can be used to efficiently and clearly communicate relationships.
Agenda
| Activity | Time | Description/Prompt | Materials |
| Launch | 5–10 min | Introduce Cuisenaire rods if students have not seen them before. Show students the purple and brown Cuisenaire rods and tell them that we are going to consider these two rods together to add to 1 unit. | Purple and brown Cuisenaire rods, to display |
| Explore | 10 min | Partners explore the question, If purple + brown = 1, then what is ? Ask students to develop ways to communicate in writing the relationships they find. | Cuisenaire rods, one set per partnership |
| Discuss | 10–15 min | Discuss the ways the class can use words and symbols to record the relationships they found. Discuss which ways are most effective and clear. Make sure students focus not just on communicating the answer but on the relationship between rods. |
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| Explore | 20–30 min | Partners explore the relationships between rods on the Cuisenaire Rod Equivalents sheets, with a focus on recording these relationships in pictures, words, and symbols. |
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| Discuss | 15 min | Discuss the ways that students developed to communicate the relationships they found and the symbols they used. Discuss why symbols were useful. Tell students that this is what mathematicians do to develop ways of communicating clearly with one another. | Chart and markers |
| Extend | 20+ min | Students create their own Cuisenaire rod relationship questions, which they record and solve on the sheet provided. |
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To the Teacher
The intent of this activity is somewhat different than many of our others. Students are asked to do many tasks rather than one long one. We want these tasks to be challenging, but the recording should become a bit tedious so that students have a reason to invent variables out of their own need to make writing equations more efficient. When students invent the variables, the variables all clearly represent something students understand. Students will typically use the first letter of the colors to stand for that block, but this can create additional challenges, such as what to do with two different greens or the blue, brown, and black rods, which all begin with b. These challenges are themselves motivations for having a common system for naming the rods that can help students communicate with each other. This is how mathematical conventions of all kinds arise, and it is worth highlighting to students in the closing discussion that they have engaged in the authentic work of mathematicians as they invented a common language for communicating mathematical ideas.
We use fractions in this activity to give students a chance to think with fractions, which are critical to developing algebraic thinking but are not well represented in sixth grade overall. Note that there are multiple solutions to the tasks we’ve posed. When students can provide evidence and communicate it clearly, celebrate these diverse ways of thinking about the tasks and use these multiple correct solutions to promote thinking about equivalence.
Activity
Launch
Launch this activity by telling students that today they are going to explore the relationship between different blocks, or Cuisenaire rods. If students have never seen Cuisenaire rods, you may want to show them the entire set and ask them what they notice about the structure of these blocks.
Show on the document camera the purple and brown rods side by side, and tell students that we are going to consider these two rods added together to be 1 unit. Tell students that recording relationships clearly is important to our work. Write out this relationship on the board or chart: purple + brown = 1. Ask students, If purple + brown = 1, then what is ?
Explore
Provide partners with a set of Cuisenaire rods to explore the question, If purple + brown = 1, then what is ? Ask students to focus on developing ways to record the relationship between these rods clearly. How could you do it with words? How could symbols help you be clear? Students will likely need only a brief time for this part of the exploration.
Discuss
Once students have developed some ideas, come together to discuss briefly what students found and how they recorded the relationship. Ask students, If purple + brown = 1, then what is ? Invite students to share their answers and come to agreement that the dark green rod is . They may have also found other smaller rods that can be summed to the same length. The focus of this discussion, however, is on how to record this relationship clearly and precisely. Ask students, How did you record what you found? Record horizontally on the board or a chart all the different ways students came up with so that they can see them side by side.
Discuss the following questions as a class:
- Which ways are the clearest? Why?
- Which ways are precise? Which are not? Why?
Draw attention to the ways of recording that are the most accurate and take into account not just what dark green represents as a number () but what it is of, such as dark green = of purple + brown.
Explore
Provide partners with the Cuisenaire Rod Equivalents sheets and ask them to continue to develop both verbal and symbolic ways of recording. Students should build, sketch, and record all the relationships in each task. Students will likely begin to substitute letters for the color words simply to make recording easier. Be sure to push students toward precision in the use of these symbols. For instance, there are two different greens. How will they indicate each? There are multiple colors that begin with the letter b. How will they handle this?
Discuss
Gather students together to discuss the following questions:
- What ways to communicate the relationships did you develop?
- Which ways do you think are clearest? Which are the most precise?
- Why did we use symbols to communicate? How do the symbols help us?
Record the ways students developed for communicating that they think are the clearest and most precise. Tell students that mathematicians use symbols to help them communicate relationships clearly and precisely, just as students did today. You might ask students when they have seen similar equations with letters to represent quantities. They may name formulas for area, volume, or other measurements they have seen in the past. Students may think of these as abbreviations. Name for students that in mathematics, we consider these variables, since we don’t always know the value of things or that values may vary.
Extend
Invite students to pose their own Cuisenaire rod relationship questions and to record what they find using words, numbers, and symbols. Provide students with the Design Your Own Cuisenaire Rod Equivalents sheets to record the questions they create and solutions they find. Students will need access to the full set of Cuisenaire rods and colors.
Look‐Fors
- Are students communicating the full relationship, or just the answer? Students may be tempted to make recording simpler by only writing the answer, such as green = . However, this does not capture the full relationship, in that green only represents when brown and purple are 1 unit. Hold students accountable to recording this full relationship in words and using symbols to make the recording more efficient. After students have recorded a precise relationship in words, ask them, How could you show this same idea using some symbols? What would be an efficient way of recording the relationship?
- Are students writing the relationships in words first? As students move down the Cuisenaire Rod Equivalents sheets, you may notice that students are skipping the writing column. This is not inappropriate as long as students are capturing the relationship in symbols accurately and completely. In fact, the tedium of recording in words is intended to motivate the use of symbols as a more efficient representation of the relationship. If you see students skipping the words, ask them why and encourage them to articulate why symbols seem more useful.
- How do students’ symbols relate to their words and sketches of the relationship? One challenge of representing an idea in multiple forms, as we have asked students to do here, is that these representations can end up not being equivalent. If students discover that one form does not match another, they may also be confused about which one is accurate. Be sure to probe students to explain how their pictures connect to the words they have written, and how both of these are reflected in the symbols they have used. You might point to parts of the equation and ask, What does this represent in your picture? What words do these symbols represent? If students do encounter errors, ask them, How will you decide which makes sense? How can you decided which relationship you want to represent in the rods?
- Are students inventing or using variables? The goal of this activity is to support students in developing symbolic representations, including variables, for relationships. Students may do this in ways that are not conventional for mathematics, and this is appropriate at this point. For instance, students may deal with the challenge of having a light green and a dark green by abbreviating them as “lg” and “dg.” In mathematics, we would not conventionally use two symbols to stand for a single quantity, but this decision makes sense for students who are inventing variables. If in the discussion it comes up from students themselves that the use of two letters could be confusing, spend time discussing why. This can support students in refining the conventions they are developing.
Reflect
How do symbols help you communicate mathematical ideas?
Cuisenaire Rod Equivalents
Design Your Own Cuisenaire Rod Equivalents
Math Mobiles
Snapshot
Students play with math mobiles, which show relationships of balance and equivalence, to determine the value of each shape in the mobile. Students explore how they might represent these relationships with symbols.
Agenda
| Activity | Time | Description/Prompt | Materials |
| Launch | 10 min | Show students the Math Mobile Puzzle sheet and ask students what they notice. Use words to annotate features that students notice. | Math Mobile Puzzle sheet, to display |
| Play | 20+ min | Partners work with the Math Mobile Puzzle sheet to determine the value of each shape. Students explore what they notice that helps them find these values and how they might represent these relationships symbolically. | Math Mobile Puzzle sheet, one per partnership |
| Discuss | 15+ min | Discuss what students found in the Math Mobile Puzzle sheet and how they used what they noticed to find the values of the shapes. Focus on the ways students represented relationships and how they used symbols. | Math Mobile Puzzle sheet, to display |
| Play | 30+ min | Partners choose from Mobile Puzzles 1–4 to explore the relationships in the mobiles, how they might represent these symbolically, and the value of each shape. | Mobile Puzzle 1–4 sheets, multiple copies for partnerships to choose from |
| Discuss | 15–20 min | Discuss the observations students made about the relationships in the puzzles, how they represented those relationships, and how they used these to figure out the value of shapes. Celebrate students’ observations, questions, challenges, and surprises. | Mobile Puzzle 1–4 sheets, to display |
| Extend | 30+ min | Partners design their own math mobile puzzles, providing their audience with enough information to solve the puzzle. Students can swap puzzle drafts with another group to test them and get feedback. |
To the Teacher
The mobiles in this activity are based on puzzles designed by Lou Kroner (1997) in his book In the Balance for grades 4–6, which, sadly, is out of print. If you do find a copy of either of his books in this series, we highly recommend these puzzles for algebraic thinking.
When looking at these math mobiles, we are reminded of mobiles created by sculptor and painter Alexander Calder. If possible, we recommend finding some images of his work to share with students and asking them what they know about mobiles. The central idea that students need to understand and draw on in this activity is balance. The components of a mobile must be in balance so that it does not tip over or collapse. The mobiles in these puzzles all balance, and this balance creates the opportunity to use symbols and equations to express relationships between the different pieces within the mobile.
For the first puzzle, the entire structure is based on balance and halving. The “weight” of the mobile, or the sum of the shapes, is 16. Sixteen is a particularly flexible and advantageous number for halving, and we hope this task will give students an entry point to finding that the shapes are worth 4, 2, and 1, each a result of halving repeatedly. We have not provided answer keys for the puzzles students explore in the second half of the activity. We highly recommend that you sit down, perhaps with colleagues, and try to solve these yourselves to see how students might do so. One last note: Mobile Puzzle 4 has multiple solutions. Be sure to push students to consider what the shapes might represent and to figure out different ways to solve this puzzle.
Activity
Launch
Launch the activity by showing students the Math Mobile Puzzle sheet on the document camera. Tell students that this is a mobile and ask them what they know about mobiles. Use this as an opportunity to ensure that students understand that mobiles must balance to work. Tell students that this mobile is perfectly balanced and has a total weight of 16 units.
Can you find the weight of each shape to balance this math mobile?
Ask students, What do you notice about the mobile? What observations can you make? Invite students to turn and talk to a partner about their observations. Collect and record some observations from the class and ask students to justify what they see. Record these observations using words, as you recorded relationships in the Visualize activity. Pose the question that students will explore today: What do you think is the value of each of these shapes? Ask students to think about how they could represent these statements using symbols and how symbols could help them solve the puzzle.
Play
Partners work together using the Math Mobile Puzzle sheet to try to find the value of each of the shapes in the mobile, remembering that the mobile must balance and have a total weight of 16. As students work, they explore the following questions:
- What relationships do you notice in the mobile?
- How could you represent these relationships using symbols?
- How could these equations help you solve the puzzle?
Discuss
Gather students together to discuss the following questions:
- What was the first thing you noticed that helped you figure out the value of one of the shapes? How did it help you?
- What strategies did you use to find the values of the shapes?
- What equations did you write to describe the relationships in the mobile? How did these help you?
Be sure to highlight the ways students are thinking about the mathematical relationships of equivalence in this puzzle and how those might be represented with symbols and any examples of where these symbolic representations were equations. Representing relationships in some way provides evidence for the values students find and can help students remember their chain of reasoning.
Play
Partners choose from Mobile Puzzles 1–4. Some groups may work through several puzzles, while others may spend their time focused on one puzzle. Expect struggle and be sure to celebrate every observation students make about the mobiles, as each observation moves students along on their journey. Partners work on one or more puzzles, recording their thinking on the puzzle sheets and exploring the following questions:
- What relationships do you notice in the mobile?
- How could you represent these relationships using symbols?
- How could these equations help you solve the puzzle?
- What is the value of each of the shapes in the mobile? How do you know?
Discuss
Gather the class to discuss the following questions:
- What were the most useful observations you made about the puzzle you tried? Why?
- How did you use symbols to represent the relationships you found? How did that help you?
- How did you use the different clues, observations, or relationships together to reason about the value of the shapes?
- What was the hardest part of the puzzles you tried? Why? How did you tackle that challenge?
- What mistakes did you make? What did you learn from them?
Be sure to discuss any interesting strategies, observations, or moments you noticed as you observed students at work. Invite students to share their realizations, questions, and surprises.
Extend
Invite students to create their own math mobiles. These are challenging to make because students will need to think about how to balance the mobile and how to make sure that there is enough information in the puzzle so that others can figure it out. When students have a draft of a puzzle, have them ask another group to try it and give feedback. Students will likely make many mistakes in designing these mobiles, but the opportunity to create a puzzle engages students in kinds of reasoning that are different from those used in solving puzzles. When students have created and tested their puzzles, they can swap with other groups or create a puzzle bulletin board or display for others to engage in.
Look‐Fors
- Are students using equivalence? The central idea in this lesson is using equivalence to determine relationships and find the missing values. As you watch students working, look for students decomposing the mobiles into parts where equivalence is visible and useful. They may isolate one branch of the mobile and locate the relationship inside—for instance, that two triangles are the same as one rectangle. Students may not know yet what to do with these individual observations, but making them is the first step. Encourage students to start by simply noticing the relationships they see to help them locate an entry point into the task. Students may be overwhelmed by the full mobile. You might encourage decomposition by saying something like, “Is there a part of the mobile where you can see a relationship? Choose a section and let’s talk about what you see.”
- How are students representing relationships? After students notice relationships, they may not know how to coordinate them in order to find the values of the shapes. But we don’t want students to lose these observations. This is one reason that recording and representing the relationships students see is crucial. If you hear students discussing equivalence or other relationships, ask, How could you record that relationship so you don’t forget it or so you can use it later? How might you represent it? What symbols would help you capture what you just said?
- Are students using variables? It is not necessary that students use variables to represent the relationships they see, but we expect that some students, following the Visualize activity, will choose to represent the shapes with variables. It makes sense that students may want to designate triangles as t and squares as s to make it easier to record how they are related. If so, look for students making a key of some kind to track what the letters stand for. You might ask, How will you remember what each letter stands for? Or, How would someone else know what each letter stands for? However, you may also see students simply using the shapes themselves, drawing triangles to represent triangles. This is entirely appropriate, and it makes sense to ask students to discuss these decisions and their reasoning behind them in the whole‐class discussions.
Reflect
What strategies were most helpful in finding the values of the shapes in the mobiles? Why?
Reference
- Kroner, L. (1997). In the balance: Algebra logic puzzles grades 4–6. New York, NY: McGraw‐Hill.
Math Mobile Puzzle
Mobile Puzzle 1
Mobile Puzzle 2
Mobile Puzzle 3
Mobile Puzzle 4
