No Series: Concept First, Notation Last


Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP4:  Model with mathematics.

    Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Download Common Core State Standards (PDF 1.2 MB)


Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP6:  Attend to precision.

    Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Download Common Core State Standards (PDF 1.2 MB)


Common core State Standards

  • Math:  Math
  • 7:  Grade 7
  • EE:  Expressions & Equations
  • B:  Solve real-life and mathematical problems using numerical and algebraic expressions and equations
  • 4b: 
    Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

    a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

    b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Download Common Core State Standards (PDF 1.2 MB)

Concept First, Notation Last

Lesson Objective: Develop a conceptual understanding of inequalities
Grade 7 / Math / Inequalities
11 MIN
Math.Practice.MP4 | Math.Practice.MP6 | Math.7.EE.B.4b


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Discussion and Supporting Materials

Thought starters

  1. How does the dotting of solutions help students build conceptual understanding?
  2. How does Ms. Alcala build off of her students' prior knowledge?
  3. What might you do after this lesson to move from conceptual understanding to procedural skill and application?


  • Private message to Harriette Huang

I love Ms. Alcala's approach.  Every problem, she asks students to give a few numbers to test and try if they work.  Then she asks Ss to plot and find the trend.  At last, she guides Ss to use mathematical notations to describe the solution concisely and beautifully.

When students learn math this way, they are not learning for scores or for colleges.  They are learning problem-solving that they can apply to their life the same day they go back home.  How much sugar do we want in this cake? Try it and taste it.  Find all the solutions that work and write down as a recipe.

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  • Private message to Kathleen Taylor

The title of this lesson is what caught my attention first- conceptual understanding is paramount, and something many teachers don’t take the time to develop. TIME is the key here, and though I feel like there is never enough of it, I am committed to taking the time necessary to develop conceptual understanding. Thank you for this!!!

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  • Private message to Carolynn Molleur

Thank you for this video! I work with a team of Algebra teachers who recently watched this video together, and it allowed us to teach inequalities in a way that engaged and made sense to students!

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  • Private message to Melissa Fenner
Do you have a place to subscribe to YOU?!?!?!
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  • Private message to Bion Shelden
What an excellent lesson! (Question: Did you REALLY accomplish all this in one class period?). Minutes before coming to Leah's excellent video I watched another video on teaching the exact same concept by a teacher whom I'm sure has a reputation for excellence, but which left me depressed. THAT teacher was extremely effective at drilling her students into successful memorization of the "rules" (like flipping when multiplying by a negative) but I saw no evidence that UNDERSTANDING was being achieved. Superb teaching, Leah.
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  • Concept First, Notation Last Transcript

    Speaker 1: Beautiful. Start putting solutions on your number line.
    I always want every math problem

    Concept First, Notation Last Transcript

    Speaker 1: Beautiful. Start putting solutions on your number line.
    I always want every math problem to make sense. How do I show that -4.9 also works? My kids, they're not robots. They're not machines. They're not looking at a problem and applying this technique that I taught them to do. They're looking at a problem, understanding what it means, and coming up with a solution that makes sense to them.
    Does -5 work?
    Today you're going to see my lesson on linear inequalities. I like to start all of my lessons with concept first, notation last.
    Is 0 bigger than -3?
    Male: Yes. Wait. Yes.
    Speaker 1: Should we put a dot there?
    Male: Yes.
    Speaker 1: Okay.
    We have never done any inequality work at all in this class. We've only done equation work, so that extends to inequalities.
    Who can give me a number that works?
    There's definitely a standard that students need to solve linear inequalities. You have a number, it gets multiplied by another number, and added to another number. It's less than or greater than, or greater than or equal to some other number. We have not fully developed that whole standard in just this lesson. We definitely have more work to do.
    Why is 5 a solution?
    This was very much a basic lesson on understanding what it means to solve an inequality and how to read the notation of inequality problems.
    Is that going to be less than 10? Let's put ...
    Many of my kids already know how to multiply, add, and subject with positive and negative numbers. If I can make that inequality problem just that, just thinking about number. Then lay on top of what they already know, some new information. It has something to stick onto.
    I'm going to hand out a piece of lined paper.
    I started by just giving them a basic inequality. Give me a number that is bigger than -3.
    Male: -2.
    Speaker 1: Is -2 bigger than -3?
    Male: Yes.
    Speaker 1: What does it mean to be bigger? Nicolas?
    Nicolas: Further to the right on the number line.
    Speaker 1: Further to the right on the number line.
    The kids came up with numbers that represented solutions. [Keyona 00:02:23]
    Keyona: 1,000,000,000
    Speaker 1: 1,000,000,000
    Keyona: All the positive numbers.
    Speaker 1: All the positive numbers.
    You can't actually list all the solutions. I actually had a student in sixth grade who tried. It seems ridiculous to try and write them all out. We know it's really going to be all numbers bigger than -3.
    We're going to try and describe these solutions in a different way. Please rotate your paper 90 degrees clockwise. We decided to graph them in order to be able to list more than we could by listing them out in a number. We're going to start by putting a dot on -2. Why am I going to put a dot on -2?
    Male: Because it works.
    Speaker 1: Because it works. How do I get a 1,000,000,000 on here?
    Male: A arrow.
    Speaker 1: An arrow. Which uses Nicolas's rule to the right of -3. How do I put dots on all positive numbers? Let me see what you would do. Then I was looking when they were doing the graphing. Did they really understand the infinite-ness of this? I love hearing the dotting. What about 2 and a half?
    Did they just put dots on the solutions they had already listed? Were those dots allowing them to see other solutions that they hadn't thought of before? Did you get all the positive numbers? Look at some more. Beautiful.
    To not start with the ray, but to rather start with the dots and say these dots represents solutions. This ray is just an infinite number of solutions smooshed so close together, allows them to see the utility of a graph.
    What we are going to do is circle the spot around -3, and show the world that there is no dot there. It's called the boundary point. If you fill in all these dots, you will get this graph.
    After we graphed them, I tried to add the notation on top of that. What do we call numbers that we don't really know at the moment?
    Male: X.
    Speaker 1: X. How do we write bigger than -3?
    Female: Sign [thing 00:04:37].
    Speaker 1: The little sign thing. How does it look?
    Female: Crocodile mouth.
    Speaker 1: It looks like a crocodile mouth, which way does it point? It points towards the solutions. Which side of the boundary point has my solutions?
    Male: To the right.
    Speaker 1: Point it to the right. This is the greater than symbol that I want to guys to learn today. You read inequalities left to right. I want to show you that this is like a little mini arrow. Notice that it matches up with this arrow.
    We're modeling the situation of a number greater than -3. We can even model more realistic situations as they get more sophisticated. Apply a real world situation to this very abstract idea, and create answers that make sense.
    New problem, on the back of your paper. A number multiplied by -2 produces a number less than positive 10. I would say almost every adult would get that problem wrong, if they just tried to do it remembering their high school or middle school math. Even though the problem starts out by being a less than problem, the solution is all numbers greater than a particular number. That switch of the inequality symbol, divide both sides by -2 and flip the sign is what is very confusing.
    You can start by just giving me a single number that works. That is a good place to start.
    Female: -9. [crosstalk 00:06:10].
    Male: 2.
    Speaker 1: What is -2 times 2?
    Male: -2 times 2 is 4.
    Speaker 1: -4.
    Male: That's less than 10.
    Speaker 1: Write that 1 as well.
    Male: In the positive number box, it tells what's positive and what's negative.
    Male: 6 times -2 is -12. That's also 10.
    Speaker 1: Many of my students were able to do that. I definitely saw some that were not. We're going to have to go back with those kids, and continue to work on multiplication and addition. It doesn't mean it's hopeless for you. It became obvious to me that they were struggling.
    Now what we're going to do, is try and describe with a sentence. All the numbers that do work. [Paloma 00:06:58].
    Paloma: Any number higher than ... no. Any number greater than 95 is a solution to this problem.
    Speaker 1: I love everything you just said. This is what we're trying to do today. You're trying to identify, not just a few of the solutions, but all of them.
    How do we notate that -5 does not work?
    Beautiful. Notice what he did here with the open dot.
    Female: Okay.
    Speaker 1: That allows me to see that it goes right up to -5 and stops.
    Male: It can be anything close to -5, but it can't be -5.
    Female: It's an open [inaudible 00:07:36] if it does not include the -5.
    Male: Yeah.
    Speaker 1: Okay you guys, let's look at a few examples. This idea allows us to identify that everything right up to here works. Why would an arrow help in this situation? [Dasia 00:07:55]
    Dasia: You would know that any numbers beyond that point were [inaudible 00:07:58].
    Speaker 1: Beautiful.
    Now we're going to write this solution in mathematical notation. What could we call any number?
    Male: X.
    Speaker 1: I need to point which way the numbers go. This arrow all numbers to this side of -5. This is 2 ways to show the solution to the problem.
    Write this 1 down. I want the solutions to that problem. You have 3 options. One is just to list every number that works. Two is to make a graph of every number that works. Three is to write in notation all the numbers that work.
    Just try a number. Give me a number.
    Male: 10.
    Speaker 1: 10. What's 4 - 10?
    Male: That's 6.
    Speaker 1: You're on the right track. Look up here. 4 - 10 is that really 6?
    Male: Wait. [inaudible 00:09:06] negative.
    Speaker 1: Yeah. Because when you subtract 10 from 4, you have to go a long way.
    Male: Okay.
    Speaker 1: You would get to -6. Is -6 less than 12?
    Male: Yeah.
    Speaker 1: 10 works. Your first guess was right, keep going.
    Male: 4.
    Speaker 1: What's 4 - 4?
    Male: 0.
    Speaker 1: Is that less than 12?
    Male: Yes.
    Speaker 1: You got one.
    The whole lesson was to get them away from treating it as a procedure, and towards getting them to understand that I am looking for numbers that make this true. If that's what you think about every time you see a problem. This is a problem about numbers. I know a lot about numbers. I can apply my previous understanding of numbers to this new situation and get the right answer. I'm giving the kids the power to remember that when it comes up again later.
    Male: 8 is 12. [crosstalk 00:09:57]
    Male: Anything bigger than -8.
    Female: To the right of -8.
    Speaker 1: 0 works.
    Female: All positives.
    Speaker 1: All positive numbers, because 4 - and ... yes. -8 is the boundary point. Can you pass your cards to your card collector? Let's look at what we got.
    This 1 shows the math going in. 4 minus something is less than 12. Negative 3 works, so -3 is the solution. This is a graph of the solutions. Negative 8 doesn't work, because that makes a 12. Everything to the right of it. Finally, to along with that, all numbers pointing to the right of -8. Tonight on your homework, I'm going to have you do all 3 parts.
    One of the things that we're dealing with at this school is I have a huge diversity of mathematical understanding at this point. I feel the big thing for me, I say this to my kids a lot, is everybody likes to learn. Not everybody likes school. As long as I'm allowing every kids access to the content, that allows them to be engaged. The idea that they could learn something is a very appealing idea.

School Details

Martin Luther King Middle School
1781 Rose Street
Berkeley CA 94703
Population: 989

Data Provided By:



Leah Alcala
Math / 7 8 / Teacher


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