1. Math Basics & Diagrams Foundations of Algebra Unit 1 Lesson 1

2. Lesson Objectives: AF 1.3 Have students understand BASIC rules of mathematics through the use of simple diagrams such as number lines, basics shapes and plus & minus signs. Have students reason and make logical sense of simple diagrams that model basic mathematical rules. Have students use diagrams that represent their way of thinking (how they see the problem), as well as to understand how other students think (how others see the problem).

3. Introduction How would you show the math expression 3+2 using basic shapes? Depending on how your brain is wired, here are two possible ways of seeing this problem using circles: + + and Both diagrams show the sum of 3 and 2.

4. Terminology DIAGRAM A diagram is a pictorial representation of some kind of idea, concept, or math problem. The diagram describes the math problem. EXPRESSION An expression is a math problem that is written using numbers, operations, letters and symbols. There are two kinds of expressions: Numeric (numerical) expression: A math problem that contains only numbers and operations. Algebraic expression: A math problem that contains numbers, operations, letters and/or symbols.

5. Terminology (Continued) TERM A term is a value in a math problem separated by an operation. For example, in 3+2, the number 3 is a term and the number 2 is also a term. INTEGERS Integers are all numbers & their opposites. The number –6, for example, is an integer because it has an opposite value, 6.

6. The Four Operations & Diagrams ADDITION To add means to combine terms that are alike. For example 6 + 5 means we are going to combine 6 of something with 5 of the same kind of something: + Is the answer different if the numbers were –6 + –5? The answer would be negative, but the diagram would be the same because we are combining two like-terms, only negative.

7. The Four Operations & Diagrams SUBTRACTION To subtract means to combine OPPOSITE terms. For example, 5 – 3 can be shown like this: + – + One blue circle is the opposite of one red circle, so when you combine them, they cancel each other out. After all opposites have canceled out, the answer remains. The answer to 5 – 3 is 2.

8. The Four Operations & Diagrams MORE ON SUBTRACTION You can rewrite a subtraction problem using a plus sign. For example, 5 – 3 can be rewritten as 5 + – 3. Make sure that the negative sign travels with its number, though! Notice in both diagrams the opposite values cancel each other out even though we have two different diagrams. The idea here is to remember that opposite values cancel each other out and 5 pluses added to 3 negatives equals 2. +– – +– +

9. Double Signs It’s not a good to have consecutive signs, or double signs, because they can confuse you. For example, find the value of 4 – – 5. If you are lost, then use a diagram: +– – – The answer you get is –1, but THAT IS NOT CORRECT because you didn’t remove the double negatives. Consecutive negatives cancel each other to become a positive. This means that the red circles actually become blue and they are added to the other blue circles. The final answer is 9, NOT –1. + +

10. ▬▬  To remove double signs, use the Alien Face (two negatives turn into a positive). The Alien Face is also used when multiplying or dividing numbers with different signs. Remove Double Signs

11. The Four Operations & Diagrams MULTIPLICATION To multiply means to take a value and add it to itself as many times as the other number indicates. Let’s make a diagram that shows the expression 3 x 2. This can be shown two ways: x =+ or + + 2 groups of 3 3 groups of 2 The answer is 6

12. The Four Operations & Diagrams DIVISION To multiply means to take a value and make a specified number of groups containing the indicated amount of values. Let’s make a diagram that shows the expression 12 ÷ 4. In other words, how many groups of 4 can we make with 12? = According to our diagram, we can make three groups of 4 if we have 12.

13. Making Diagrams To make a mathematical diagram, you can use circles, stars, or any shape you wish. The purpose of this unit is to present you with a familiar topic in a way that may help you make better sense of it. Example: Create a simple diagram showing 3 + 5:    +      or      +    Remember, we are combining like-terms, so the shapes are the same.