Algebra Equations：代数方程.doc

Algebra EquationsLevel 1: Solving Simple Linear EquationsI. IntroductionA. Prerequsite: This work can be started as soon as the child can do basic integer arithmetic with the four arithmetic operations. Equations with subtraction in them require an introduction to negative numbers. Equations with division in them can be created so as not to need fractions, but more typically, will require an understanding of the fraction form of integer division (as opposed to the integer formi.e., quotient and remainderof division.)In addition, the equations that involve parenthesis typically require a refresher on the use of the distributive law (such as introduced in the earliest work with the trinomial cube.) This refresher should show multiplication as a large rectangle that contains two smaller rectangles signifying two multiplications. Hence a(b+c), the one rectangle equals ab+ac, the two rectangular parts of the large rectangle (the whole is the sum of the parts.)B. Description:C. Material: Something to represent units. Gram-centimeter cubes work well. Also something to represent X. It is possible to use cups that represent an unknown number of unit cubes. Rods also work well, especially if they resemble your units in cross section. I use repainted cuisenaire rods that were originally made for use with the decanomial square presentation.D. You also need two platforms to represent the two sides of the equation. In a pinch, two pieces of plain paper will do. Place these with an equal sign between them.II. Plan of the Work:A. 1-Step Equations1. Addition: e.g. X+6=202. Multiplication: e.g. 5X=303. Subtraction: e.g. X-4=114. Division: e.g. X / 3 = 2B. 2-Step Equations with multiplication and addition: e.g. 2X+5=17C. 3-Step Equations requiring combining like terms: e.g. 3X+4+2X=19D. 3-Step Equations with unknowns on both sides: e.g. 4X+2=6+2XE. 3-Step Equations involving parenthesis: e.g. 2(3X+5)=28III. Presentation:Book 1Lay out two sheets of paper with an equals sign in between (see illustration.) Construct the equation X + 2 = 8. Say, “These are the same number.” Point to the side with the 8 and say, “This is 8.” Point to the side with the X + 2 and say, “This is also 8. They are both 8.” Now simultaneously take two units away from both sides and say, “Now they are both 6. So X (point to the X) is 6 (point to the 6.)” Now construct X + 3 = 7. Ask “How much is this?” (point to the 7 units.) Ask, “And how much is this?” (point to the X + 3.) If the student doesnt answer 7 to both questions, review what “=” means through a representation or otherwise coach the student. Now ask, “What does X have to be?” (fall back to discussion like, “if you add 3 to it, you get 7” or “what can you add 3 to to get 7?” as needed.) note: at this level, students relate better to seeing what X has to be and will later develop the idea that you subtract 3 from both sides to undo adding 3 to X. Look for the opportunity to bring out the latter fact, but be patient. The student has to see this for himself. Now build other types of 1-step equations. For multiplication, use multiple Xs on one side. Describe it as 2X or 3X etc. when talking about it. Use questions like, “What do you have to multiply X by to get ” For subtraction, you will have to use something to represent a negative (equate it to a hole that can be filled with a unitso when weve taken away a unit, we are left with this hole.) For division, you can use a fraction circle to represent X, then the quarter circle will represent X 4. Use questions like, “How much must X be if one quarter of it is ”Book 2Lay out two sheets of paper with an equal sign between as in the diagram above. Build the equation 3X + 5 = 17. Say, “These are the same number. What are they?” ans: 17. Say, “If I take 5 units from both of these 17s, what will they both be?” Direct the student to do it. ans: 12. Say, “Record what you get.” student writes 3X = 12. Ask, “3 times what is 12?” ans: 4. Separate the Xs vertically on one side of the equation. Distribute the 12 units into three groups vertically on the other side of the equation. Say, “Each X gets 4.” Remove all but one X on one side while removing all but 4 on the other. Say, “Record what you get.” student writes X = 4. Work more problems if needed until student can work them on his own.Book 3Lay out two sheets of paper with an equal sign between as in the diagram above. Build the equation 2X + 4 + 3X = 14. Ask “What do I get when I add 2X (point to the 2X) and 3X?” (point to the 3X) ans: 5X Move the 3X next to the 2X and ask the student to write what he sees. 5X + 4 = 14. Work through the rest of the problem as in book 2.Book 4Lay out two sheets of paper with an equal sign between as in the diagram above. Build the equation 6X + 7 = 2X + 19. Point to the 6X + 7 and the 2X + 19 and say, “These are the same number.” Point to 2X on both sides of the equation and say, “These are the same number.” As you pick them up and take them away. “And we can take the same number from both sides of an equation.” Say, “Record what you see.” Student writes: 4X + 7 = 19. Work through the rest of the problem as in book 2.Book 5Lay out two sheets of paper with an equal sign between as in the diagram above. Review the distributive law from the cover of the binomial cube materials. Build 3(2 + X) = 18. Stack the left side so that it is a 3 rows of (2 + x), one on top of another. Point to the left side and ask, “How much do we have here?” ans: 6 + 3X. Say, “Record what you see.” Student writes: 6 + 3X = 18. Work through the rest of the problem as in book 2.IV. Follow Up Activities: Have students do a variety of exercises from the Algebra Equation Level 1 books. Have students practice “chunking” problems. This is where they cover a part of the equation and visualize it as a single number. For instance, in the equation 3X + 5 = 17, cover the 3X and ask, “What plus 5 is 17?” ans: 12. Now uncover the 3X and say, “So 3X must be 12.” (later, you can put this as a question, “So what must 3X be?”) Have students work with “unravelling” the left hand side. For example in 3X + 5 = 17, block out the 17 and ask, “If I am given a value for X, how do I calculate 3X + 5?” ans: multiply by 3 and add 5. So how do I UNDO this? ans: subtract 5 and divide by 3. This may take an example or two. You can prepare an equilateral triangle with vertices A, B, and C labeled on both sides. Now rotate the triangle and flip it vertically. Ask how to undo this. Student will find that he must do the opposite of each operation in the opposite order. The triangle allows a variety of complicated examples. A Rubic Cube can also illustrate this very well. How do you undo a front clockwise twist followed by a side clockwise twist? Back to the equation: it can help to list the operations vertically. In the example above, Multiply by 3 Add 5Then you can draw a line (think like it is a special mirror.) and write Subtract 5 Divide by 3Let students practice doing this on problems from book 2. Create more difficult “unravelling problems. For example, 3(2 + 3X) 5 = 10 mult X by 3, add 2, mult result by 3, subtract 5, you get 10. To unravel, take 10, add 5, divide by 3, subtract 2, divide by 3, the result is X. “Chunk” these more difficult unravelling problems. cover the 3(2 + 3X) and ask, “What must I have to subtract 5 and get 10?” Answer: 15. Have the student write 3(2 + 3X) = 15. Continue by now covering the (2 + 3X) and ask, “What must I have to multiply by 3 and get 15?” Answer, 5. Have the student write what he gets