Lesson 1: Evaluation Blocks and Arithmetic Expressions
Students will begin using Evaluation Blocks to explore the concept of math as a language, and more specifically, a programming language. By composing arithmetic expressions with Evaluation Blocks, students will be able to visualize how expressions follow the order of operations.
In this lesson students get their first taste of the programming language and environment that they’ll use throughout this course. While the environment and block-based language we’re using here look a lot like things students may have seen in Scratch or the Hour of Code, it’s important to understand that the kind of programming we’re doing here is a bit different. In order to better align with the rules of Algebra we are following a programming paradigm known as Functional Programming. This difference may not matter to your students, but for those who have some existing programming experience you may ask them to research this different paradigm and report back on the similarities and differences.
Activity: Evaluation Blocks
Common Core Math Standards
- A.SSE.1 - Interpret expressions that represent a quantity in terms of its context.
Students will be able to:
- Convert arithmetic expressions to and from code.
- Use Evaluation Blocks to reflect the proper order of operations for an expression.
For the Students
- Evaluate - Perform the computation in an expression, producing an answer.
- Evaluation Block - A block of code that represents the structure of an expression
- Expression - A computation written in the rules of some language (such as arithmetic, code, or an Evaluation Block).
- Function - A mathematical object that takes in some inputs and produces an output.
- Value - A specific piece of data, like 5 or "hello".
Activity: Evaluation Blocks
The programming language you are going to learn uses Evaluation Blocks to visually represent mathematical functions. Each block of code is either a Function, or a Value - head to CS in Algebra Stage 2 in Code Studio to get started programming.
A mathematical expression is like a sentence: it’s an instruction for doing something. The expression 4 + 5 tells us to add 4 and 5. To evaluate an expression, we follow the instructions in the expression. The expression 4 + 5 evaluates to 9.
Sometimes, we need multiple expressions to accomplish a task. If you were to write instructions for making a sandwich, it could matter very much which came first: melting the cheese, slicing the bread, spreading the mustard, etc. The order of functions matters in mathematics, too. If someone says “four minus two plus one,” they could mean several things:
- Subtract two from four, then add one: (4 - 2) + 1
- Add two and one, and subtract the result from four: 4 - (2 + 1)
Depending on which way you read the expression, you might have very different results! This is a problem, because we often use math to share calculations between people. For example, you and your cell phone company should agree upfront on how much you will pay for sending text messages and making calls. Different results might mean that your bill looks wrong. We avoid problems by agreeing on the order in which to use the different operations in an expression. There are two ways to do this:
- We can all agree on an order to use
- We can add detail to expressions that indicate the order
Mathematicians didn’t always agree on the order of operations, but now we have a common set of rules for how to evaluate expressions. When evaluating an expression, we begin by applying the operations written at the top of the pyramid (multiplication and division). Only after we have completed all of those operations can we move down to the lower level. If both operations are present (as in 4 - 2 + 1), we read the expression from left to right, applying the operations in the order in which they appear.
Evaluation Blocks provide a visual way to indicate the order of operations in an expression.
All Evaluation Blocks follow three rules:
- Rule 1: Each block must have one function, which is displayed at the top of the block.
- Rule 2: The values for that function are placed below, in order from left to right.
- Rule 3: If a block contains another block as a value, that inner block must be evaluated before the outer block.
Before students get started on the computers, you can have them work through the Evaluation Blocks Worksheet in the student workbook.
Common Core Math Standards
EE - Expressions And Equations
- 6.EE.2 - Write, read, and evaluate expressions in which letters stand for numbers.
MP - Math Practices
- MP.1 - Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
- MP.2 - Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
- MP.3 - Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
- MP.4 - 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.
- MP.5 - Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
- MP.6 - 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.
- MP.7 - Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
- MP.8 - Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
NS - The Number System
- 6.NS.5 - Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
- 6.NS.6 - Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
- 6.NS.8 - Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
OA - Operations And Algebraic Thinking
- 5.OA.1 - Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
- 5.OA.2 - Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Q - Quantities
- N.Q.1 - Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- N.Q.2 - Define appropriate quantities for the purpose of descriptive modeling.
REI - Reasoning With Equations And Inequalities
- A.REI.1 - Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
SSE - Seeing Structure In Expressions
- A.SSE.1 - Interpret expressions that represent a quantity in terms of its context.
- A.SSE.2 - Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
- A.SSE.4 - Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★