Investigating Expressions with Algebra Tiles
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1 Investigating Expressions with Algebra Tiles Work with your partner to investigate the tiles you have in your packet. You and your partner will be sharing these tiles. How many different tiles are in the packet? What do you notice about the color of the tiles? Using at least one of each of the different tiles, build a rectangle that has no gaps or spaces. You may use more than one of each tile but you must use all the different types. When you have completed your rectangle, share it with your partner for their approval. Then call the teacher over to check out what you have built. When your rectangle has been approved by your teacher, sketch a detailed picture of it in the space below. Use the space below to trace each tile and label its dimensions as your teacher leads this part of the investigation. You may be asked to do this part in your math notebook. Use the area names of the different Algebra Tiles to write an expression for the rectangle you created earlier.
2 USING AN EXPRESSION MAT As a reminder, the shaded tiles are positive, and the un-shaded tiles are negative. Below are several tiles with their associated values. 1. An Expression Mat is an organizing tool that is used to represent expressions. An Expression Mat has an addition region at the top and a subtraction region at the bottom. You could, for example, represent 5x 2y on an Expression Mat as shown above. W orkin g with a partner, write algebraic expressions for each representation below. Start by building each problem with your algebra tiles. Do not simplify your expressions.
3 2. How could you represent parentheses on an Expression Mat? Discuss with your partner as you build the expression 2x 2 ( 4) (3 y) on an Expression Mat. 3. On your Expression Mat create each of the following expressions using algebra tiles. Share your different representations with your partner. 1. 3x (y 2) 2. 0 xy + (x 2 3) 4. 5 (3 2x) Imagine the following situations: a) Ramona babysits her neighbor s baby and stuffs the $15 she earns into her purse. When she gets home, the $15 is nowhere to be found. It must have fallen out of her purse. b) The Burton Pumas football team completes a pass and gains 12 yards. But on the very next play, the quarterback holds onto the ball too long and gets sacked, losing 12 yards. c) You are at the beach. You dig a hole in the sand and place the sand you remove in a pile next to your hole. Someone comes along and pushes the pile back into the hole. What do these situations have in common? How can you use math symbols and numbers to represent each of them?
4 5. Gretchen used seven algebra tiles to build the expression shown at right. 1. Build this collection of tiles on your own Expression Mat and write its value without simplifying. 2. Without changing the value of the expression, use a different number of tiles to represent Gretchen s expression three different ways. Be ready to share your representations with the class. Hint: Think about the previous problem and how you might represent zero with tiles on an Expression Mat. 3. Build each expression below so that your representation does not match that of your partner. Once your partner is convinced that together you have each found different and valid representations, sketch your representation on your paper and be ready to share your answer with the class. i. 3x y ii. (3 4x) 6. Wade created an Expression Mat for the espression 1 + 2x 2. Explain what Wade did in each step below. Is he allowed to simplify his Expression Mat in this way? Why or why not?
5 7. Write the algebraic expression shown on each Expression Mat below. Build the model and then simplify the expression by removing as many tiles as you can without changing the value of the expression. Finally, write the simplified algebraic expression. COMPARING EXPRESSIONS Two expressions can be represented at the same time using an Expression Comparison Mat. The Expression Comparison Mat puts two Expression Mats side-by-side so you can compare them and see which is greater. 8. Build the Expression Comparison Mat shown above. Write an expression without simplifying that represents each side of the Expression Mat. a. Simplify each of the expressions so that fewer tiles are used. Develop an organized method to simplify both sides of the Expression Comparison Mat. Be prepared to justify your method to the class. b. Which side of the Expression Comparison Mat do you think is greater (has the largest value)? Agree on an answer as a team. Make sure each person on your team is ready to justify your conclusion to the class.
6 9. As Karl simplified some algebraic expressions, he recorded his work on the diagrams below. a. Explain what he did to each Expression Comparison Mat on the left to get the Expression Comparison Mat on the right. b. If necessary, simplify further to determine which Expression Mat is greater. How can you tell if your final answer is correct?
7 10. Use Karl s legal moves to determine which side of each Expression Comparison Mat is greater. Record each of your legal moves on the Resource Page provided by drawing on it the way Karl did in problem 9. After each expression is simplified, state which side is greater (has the largest value). Use the resource page provided. 11. WHICH IS GREATER? Write an algebraic expression for each side of the Expression Comparison Mats given below. Use the legal moves you worked with in this lesson to determine which expression on the Expression Comparison Mat is greater.
8 JUSTIFYING AND RECORDING YOUR WORK Although using algebra tiles can make some situations easier because you can see and touch the math, it can be difficult to remember what you did to solve a problem unless you take good notes. 12. Use the simplification strategies you have learned to determine which expression on the Expression Comparison Mat above is greater. Record each step as instructed by your teacher. Also record the simplified expression that remains after each move. This will be a written record of how you solved this problem. Discuss with your team the best way to record your moves.
9 13. While Athena was comparing the expressions shown below, she was called out of the classroom. When her teammates needed help, they looked at her paper and saw the work shown below. Unfortunately, she had forgotten to explain her simplification steps. Can you help them figure out what Athena did to get each new set of expressions? starting expression 3x + 4 x ( 2) + x 2? 1 + x 2 + 4x (4 + 2x) 3x ( x) x 2? 1 + x 2 + 4x + ( 4) + ( 2x) 3x ( x) + 2? 1 + 4x + ( 4) + ( 2x) 2x + 6? 2x 5 6? 5 6 > 5, so the left side is greater
10 14. For each pair of expressions below, determine which is greater. Carefully record your steps as you go. If you cannot tell which expression is greater, write Not enough information. Make sure that you record the result after each type of simplification. For example, if you flip all of the tiles from the region to the + region, record the resulting expression and use words or symbols to indicate what you did. Be ready to share your work with your partner. a. b. c. 5 (2y 4) 2 or y (1 + y) + 4 d. 3xy + 9 4x 7 + x or 2x + 3xy (x 2)
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