(insert your name ‘s) Answer Sheet for WebQuest:
Modeling and Graphing Linear Equations
Part 1: The variable as a closed brown bag
In the first picture you see a bag and three blocks. What equation is represented by this setup, do you think?
If the scales are balanced, how many blocks are in the bag?
In the next configuration you see three bags and a block on the left side, and two bags and 7 blocks on the right hand side.
- If x is represented by a bag, and the number 1 by a block, what equation is represented by the configurationon the scale? Write down the equation.
- When you have completed steps 2a-c, how many blocks must be in the bag? How do you know?
- Solve the equation you wrote up in 1. here, the way you have done earlier in math class. Then compare your answer with what you found in the previous bullet point.
Click on the button B below the image. You will get a new configuration
- If x is represented by a bag, and the number 1 by a block, what equation is represented by the configurationon the scale? Write down the equation.
- When you have completed steps 2a-c, how many blocks must be in the bag? How do you know?
- Solve the equation you wrote up in 1. here, the way you have done earlier in math class. Then compare your answer with what you found in the previous bullet point.
Click on the button labeled C, and again repeat steps 1 through 3.
- If x is represented by a bag, and the number 1 by a block, what equation is represented by the configurationon the scale? Write down the equation.
- When you have completed steps 2a-c, how many blocks must be in the bag? How do you know?
- Solve the equation you wrote up in 1. here, the way you have done earlier in math class. Then compare your answer with what you found in the previous bullet point.
Reflection:Is there more than one way of solving (more than one possible list of steps for solving) the equations?
Reflection:What kinds of solutions can not simply be represented by number of blocks on a scale? (Hint: How would you represent x=-3?)
Solving 2x+5=9.
- What happens to the seesaw when you place the last block in place?
- What happens to the blocks and bricks when you click to subtract five from both sides?
- What happens to the blocks and bricks when you click to divide both sides by two?
- What is the solution of the equation?
Enter the same equation,2x+5=9, over again. This time, you are following a different set of steps.
- (“Step 1”) What happens when you subtract 1 from both sides?
- (“Step 2”) What happens when you divide both sides by two?
- What happens when you subtract 2 from both sides?
- What is the solution of the equation?
- How does this answer compare to the one you found above, when you solved by first subtracted five from both sides?
- Why would switching the order of steps 1 and 2 above not work in this case? (Try it out with your model if you are unsure.)
Solving 3x+5=2x+6
Write down the list of steps that you use to solve this equation (copy from the window where each operation appears after you choose it).
Part 2B: Thevariable as a brick of unknown weight or as a balloon of unknown buoyancy
What happens when you drag and drop a red –x balloon to the seesaw?
What happens when you drop an x-brick on the same side?
Why can we use balloons to represent negative numbers?
Solving 3x-5=-x+3
Write down the steps you chose to solve this equation, and also the answer. Copy from the window in which every operation appears as you choose them.
Solving -4x+4=-x-2
Write down the steps you chose to solve this equation, and also the answer. Copy from the window in which every operation appears as you choose them.
Part 3: Solving equations graphically
What expression was the word "graph" originally a short version of?
Write down the etymology of the word “graphic”.
On the basis of what you have read about the word "graph", how could you restate the phrase "the graph of an equation" in different words?
Solving 2x-5=9
What is the point of intersection (x,y) that you find when you have completed steps 1-8?
What is the x-value for which the two expressions 2x-5 and 9 are equal?
Compare your answer to what you found in Part 2 when solving 2x-5=9 using the seesaw.
Solving3x+5=2x+6
What is the point of intersection (x,y) of the graphs?
What is the solution of the equation?
Solving3x-5=-x+3
What is the point of intersection (x,y) of the graphs, and what is the solution of the equation?
Solving-4x+4=-x-2
What is the point of intersection (x,y) of the graphs, and what is the solution of the equation?
Reflection: You have solved the same equations algebraically and graphically. Which method do you prefer, and why?