Truth and consequences: talking about solving equations

The language we use when we talk about solving equations can be a bit of a minefield. It seems obvious to talk about an equation such as $3x + 2 = x + 5$ as saying that $3x+2$ is equal to $x + 5$, and that’s probably a good place to start. But there is a hidden assumption in there that the equation is true. In the Illustrative Mathematics middle school curriculum coming out this month we start students out with hanger diagrams to represent such equations:

The fact that the hanger is balanced embodies the hidden assumption that the equation is true. It is helpful for explaining why you have to perform the same operation on each side when solving equations; if you take two triangles from the left side you have to take two triangles from the right side as well in order to preserve the balance. This leads to a discussion of how performing the same operation on each side of an equation preserves the truth of the equal sign.

But what happens with an equation like $3x + 2 = 3x + 5$? In this case, the hanger diagram is a physical impossibility: the right hand side will always be heavier than the left hand side. I can imagine that students who have an idea of an equation as “the left hand side is equal to the right hand side” might be confused by this situation, and think this is not a proper equation. Especially when they reduce it to $2 = 5$. Students learn to say that this means there are no solutions, but it’s hard to make sense of that response rule without understanding what’s really going on with equations.

The fact is, an equation with a variable in it is neither true nor false, because it is merely a phrase in a longer sentence, such as “If $3x + 2 = x + 5$ then $x = \frac32$.” This sentence is true, but the phrases within it are not sentences and have no inherent truth or falsity. When we perform the same operation on each side of an equation, we are not only preserving the truth of the equal sign but also preserving the consequences of the equal sign. If we use if-then language when talking about equations, then we can make sense of equations with no solutions. A sentence like “If $x$ is a number satisfying $3x + 2 = 3x + 5$ then $2 = 5$” makes perfect sense. It’s the mathematical equivalent of “If the moon is green cheese, then I’m a monkey’s uncle.” It’s a way of saying the moon is not green cheese . . . or that there is no solution to the equation.

The middle schooler’s version of if-then language might not always use the words “if” and “then.” You might say “Imagine there is a number $x$ such that $3x + 2 = x + 5$. What can you say about $x$?” Just as you say “Imagine this hanger is balanced and the green triangles weigh one gram. How much do the blue squares weigh?” I think it’s a useful approach with students to remember that equations are a matter not just of truth, but of truth and consequences.

Why is a negative times a negative a positive?

OK, I can hear the groans already. There are many contexts for answering this question and they are dubious in varying degrees because the real answer is “because I said so.” That is to say, the rule for multiplying negatives is a convention; adopted for good reasons, but a convention nonetheless. Those good reasons are mathematical: we want to make sure that when we extend multiplication and addition to negative numbers the properties of operations still apply. In particular, we want the distributive property to apply. Meditate on this:
$$3\cdot(5 + (-5)) = 3\cdot5 + 3 \cdot (-5).$$
The left side is really $3 \cdot 0$, so it had better be zero. So the right side had better be zero as well. The first term on the right side is 15, so the other term had better be $-15$. So $3 \cdot (-5) = -15$. We want the commutative law to hold, so we had better say $(-5)\cdot 3 = -15$ as well. Now meditate on
$$(-5)\cdot(3 + (-3)) = (-5)\cdot 3 + (-5)\cdot(-3).$$
The same reasoning tells us that $(-5)\cdot(-3) = 15$.

Trouble is, all this is really hard to explain to middle schoolers, so people invent contexts. One context I’ve seen has something to do with sending out bills. If you receive 5 bills for 3 dollars then you have $5 \cdot (-3) = -15$ dollars. Sending out is the opposite of receiving, so if you send out 5 bills for 3 dollars, you have $(-5)(-3)$ dollars. But once you receive payment, you have \$15. So$(-5)(-3) = 15$. One problem with this is that you have to buy more conventions to believe it: the convention about negative amounts of money representing debt, the convention about negative receiving being kinda sorta like sending out. That’s a lot of conventions to prove something that is, as I said, a convention itself. Another problem is that all this context really shows is that$-(-3) = 3$, five times. The multiplication in this context is really just repeated addition; it doesn’t work for numbers that are not integers. You can’t send out 5.6 bills. There is one context that I think does a better job here, and that is$\mbox{distance} = \mbox{speed} \times \mbox{time}$. This does work with non-integers, and you can make sense of all of the quantities involved as negative numbers. Let’s assume that an object is moving along the number line, and that you measure its position at different times, setting your stop watch to 0 when it passes through the origin. Negative distance is distance to the left; negative speed is speed from right to left; and negative time is time before you started measuring. (Later we use the terms displacement and velocity, but there’s no need to introduce them right away.) So if the object is moving at$-5$m/sec, where is it at time$-3$seconds? Well, it’s moving from right to left and it has 3 seconds before it hits the origin, so it is 15 m to the right of the origin. So$(-5)(-3) = 15$. Was I cheating there? Is this context subject to the same objections I made about the money context? Didn’t I just make up a whole bunch of conventions about negative distance, time, and speed? I think these conventions pass the cognitive sniff test better. They don’t seem as artificial to me. You can really make quantitative sense of negative distance, speed, and time. It feels more like the real world and less like an accountant’s convention. (No offense to accountants intended.) In a way, we have replaced the mathematician’s desire to have the properties of operations continue to hold with the physicist’s desire to have the laws of physics continue to hold. So where is the distributive property in all of this? I think it is built into our physical intuition about this context. If I travel for 3 hours, and then for another 2 hours, I can figure out how far I have gone by just adding the times and multiplying by my speed, or I can add the distances traveled in each time period. That’s the distributive property. If you dig into the reasoning I gave for the object moving at$-5$m/sec in the light of this common sense, questioning each claim, you end up with something not too far from the mathematical reasoning I gave earlier. By the way, this is the approach we take in the Illustrative Mathematics middle school curriculum. Finding contexts for mathematical ideas that are faithful to the mathematics is difficult and requires real sensitivity to both the mathematics and the way students think. Our brilliant curriculum writing team is up to that challenge. Talking about fractions, decimals, and numbers When students first learn about fractions, we want them to learn that they are just numbers; new numbers, but numbers nonetheless, that fit into the same system as the whole numbers they are familiar with. The number line can help with this, with whole numbers and fractions sitting together, and located in essentially the same way; choose a unit (1, 1/3, 1/10) and then count off a number of those units. It also helps students understand that equivalent fractions are just different ways of writing the same number. When (finite) decimals come along, they get added to the list of representations. The Common Core emphasizes this unity by treating decimals as just a different way of writing fractions, e.g. in 4.NF.C: “Understand decimal notation for fractions, and compare decimal fractions.” In this view, 0.3 is not a new sort of number, just a different way of writing the number 3/10. This leads to some difficulties in the use of language, because at some points in the curriculum you do want to distinguish between decimals and fractions, for example when you ask a student to write 4/5 as a decimal or to write 0.125 as a fraction. (“You told me it’s already a fraction!” the smart student might reply.) The IM curriculum writing team was talking about these difficulties the other day and Cathy Kessel had a useful comment: There’s a developmental issue. When fractions are introduced, the distinction between number represented and representation is blurred, and similarly for decimals (finite, then repeating). But, when the two types of representations are seen as representing the same thing, then the thing and its representations start to separate more. Because we want students to develop a conception of the number behind the representation, we start out saying decimals are also fractions. Later we build a negative addition to the number line and add the opposites of fractions. Once we have a robust conception of the number line, inhabited by rational numbers, we want to talk about different ways of expressing those numbers: fractions, decimals, infinite decimals, expressions involving square root symbols and exponents. So we start to distinguish between fractions and decimals, not as numbers, but as forms for expressing numbers. We initially suppress their role as forms in order to gain a robust conception of number; once they are firmly attached to that conception we can distinguish between them. They only way to do this without giving multiple meanings to the same words would be to invent new words and be consistent in their use. This harks back to the distinction between “numeral” and “number” in the New Math, which didn’t take hold. Illustrative Mathematics Session at the Joint Mathematics Meeting Illustrative Mathematics organized a special session at the Joint Mathematics Meeting on January 7, 2016 in Seattle, WA called Essential Mathematical Structures and Practices in K-12 Mathematics. Here is a description of the session: The mathematics curriculum in the US has been shaped by myriad forces over the years, including the competition for market share among publishing companies, economic realities of school districts’ purchasing power, the ease with which teachers can deliver the material, traditional expectations of what mathematics classroom work should look like, and so on. Surprisingly absent from these forces is the nature of the discipline of mathematics itself. The focus of this special session was on identifying and describing the essential mathematical structures of the K-12 curriculum, as well as the key mathematical practices in the work of mathematicians that should be mirrored in the work of students in K-12 classrooms. New Illustrative Mathematics website, with K–5 blueprints Illustrative Mathematics has a new look today. There’s a video explaining some of the new features on the Illustrative Mathematics Facebook page. One big new feature is the course blueprints. At the moment we just have K–5 blueprints. We’ll be adding more content to those and also adding high school and middle school blueprints over the next few months. I’ve made a forum here for people to comment and ask questions about them. Fall Virtual Lecture Series from Illustrative Mathematics Welcome back to school! This fall Illustrative Mathematics will be offering our second Virtual Lecture Series, this one focuses on the theme: Working with Number in the Elementary Classroom The following lectures are scheduled in the series on Thursday nights from 7-8pm Eastern on Adobe Connect. Watch them live with the ability to ask questions, or watch the recordings at any time: September 25, 2014 Linda Gojak, Immediate Past President, National Council of Teachers of Mathematics, Director, The Center for Mathematics Education, Teaching, and Technology, John Carroll University “Using Representations to Introduce Early Number and Fraction Concepts” October 23, 2014 Dona Apple, Mathematics Learning Community Project, Regional Science Resource Center, University of Massachusetts Medical School “Supporting students’ conceptual understanding about number through reasoning, explaining and evidence in both their oral and written work” November 20, 2014 Brad Findell, The Ohio State University December 11, 2014 Francis (Skip) Fennell, Professor of Education McDaniel College, Past President NCTM “Fractions Sense – It’s all about understanding fractions as numbers (and this includes those special fractions – decimals!) – use of representations, equivalence, comparing/ordering and connections” January 22, 2015 Susan Jo Russell, TERC: Mathematics and Science Education and Deborah Schifter, Education Development Center (EDC) “Operations and Algebraic Thinking in the Elementary Grades” Sign-up here! This school year we will offer two series. In the fall we are featuring “Working with Number in the Elementary Classroom” and this spring we will offer “Incorporating Mathematical Practices into the Middle and High School Classroom.” The intended audience for these series is classroom teachers, district and state mathematics specialists, and mathematics coaches. The five hour long sessions will include 40 minutes of presentation from national experts on Adobe Connect, followed by 20 minutes of Q&A. The sessions will also be recorded for participants that are not able to join in person. The cost to virtually attend each series is$150.

Here is a flyer to circulate among friends that might be interested or to post in the staff room!  Hope to see you there.

K–5 Elaborations of the Practice Standards

Illustrative Mathematics, with the assistance of Mary Knuck, Deborah Schifter, and Susan Jo Russell, has been working on developing grade band elaborations of the Standards for Mathematical Practice. Here is a draft of the K–5 document.

As usual, please comment by starting a new thread in the forums. I’ve created a new forum for the practice standards there.

Virtual Lecture Series! (and we’re back from a rocky end of year)

Are you interested in engaging with national experts around mathematics education without the travel, hassle, and costs associated with attending a conference? Introducing Virtual Lecture Series, brought to you by Illustrative Mathematics. Virtual Lecture Series bring together top speakers from around the country for a series of talks, as well as time for questions and answers, giving you a chance to learn and interact with experts without leaving your classroom or office. Illustrative Mathematics will be offering a variety of Virtual Lecture Series on different themes.

Our first Virtual Lecture Series will meet around the theme: Preparing and Facilitating Engaging Professional Development for Teachers around the Common Core, on the last Wednesday of the month at 7pm Eastern/4pm Pacific from January through May. The intended audience for this series is district and state mathematics specialists as well as teacher leaders. The five hour long sessions will include 40 minutes of presentation from national experts on Adobe Connect, followed by 20 minutes of Q&A. The cost to virtually attend the entire series is \$150 which includes access to the following presentations:

January 29th: Diane Briars, President Elect of NCTM, Topic: Effective Instructional Practices to ensure all your students are “Common Core Ready”
February 26th: Bill McCallum, Lead writer of the CCSSM, Topic: Preparing K-12 Teachers for the Pathway to Algebra
March 26th: Mary Knuck, Arizona Department of Education Retired, Topic: Math Talks
April 30th: Ashli Black, NBCT and Cal Armstrong, Math Teacher Leader, Topic: Involving Teacher Leaders in Preparing and Facilitating Professional Development
May 28th: James Tanton, Mathematician and Author of Thinking Mathematically! Topic: Instilling a Love of Mathematics

Also the blog is back from a rough time over the new year.  Sorry if you had trouble with any of the posts or forums, we were not as quick as we could be in renewing the domain.  Let us know if you continue to have trouble accessing anything.