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Solving Linear Equations in Two Variables
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  • Improve Individual Responses to Notebooks and Pens (10 minutes)
    • Have students do this task at the beginning of the next lesson if you do not have time during the lesson itself. Some teachers like to set this task for homework.

      • Return to the students their original assessment task, Notebooks and Pens along with a second blank copy of the task sheet.
        • Look at your original responses, and think about what you have learned in this lesson.
        • Using what you have learned, try to improve your work.

      If you have not written questions on individual pieces of work, then write your list of questions on the board. Students are to select from this list only those questions appropriate to their own work.

  • Solutions: Notebooks and Pens
    • Dan is incorrect:

      Dan has misinterpreted n to mean, "notebooks sold" rather than "the number of notebooks sold."

      So he has read the equation "4n = p" as "there are four notebooks sold for every pen sold."

      The equation actually means, "4 times the number of notebooks sold equals the number of pens sold," or "the store sells four time more pens than notebooks."

      Emma is incorrect:

      Emma has also misinterpreted n to mean "notebooks" rather than "the number of notebooks."

      In the second statement, 5n does not mean, "there are 5 notebooks." It means "5 times the number of notebooks."

      Since each notebook costs $5, 5n gives you the amount of money taken from selling notebooks, and since each pen costs $2, 2p gives you the amount of money taken from selling pens. So 5n + 2p = 39 means that $39 was taken altogether from selling notebooks and pens at these prices. However, the equation does not, in isolation, tell you how many notebooks or pens were sold.

      Using the first equation to substitute 4n for p in the second equation gives n = 3 and p = 12.

      3 notebooks and 12 pens were sold.

  • Solutions: Cash Registers
      1. The number of dollar bills is three times the number of quarters.

        Four times the number of quarters plus the number of dollar bills totals 70.

      2. Possible values: (4, 12) or (8, 24).
      3. Possible values: (12, 22) or (7, 42).
      4. x = 10, y = 30
  • Comments on Sample Student Work
    • Ava used "guess and check" with both equations.

      Strengths: Her work is systematic and easy to follow.

      Weaknesses: Her method is inefficient and, although it is systematic, she has not reflected on each answer to determine the next set of values to check.

      Her lack of progress leads to her abandoning the task.

      Ava could add an explanation about her solution method.

      Ethan used an elimination method.

      Strengths: This method can work if equations are manipulated carefully.

      Weaknesses: Ethan makes a mistake when rearranging the first equation. Consequently, when the two equations are added together, a variable is not eliminated, but instead Ethan has created an equation with two variables.

      Ethan briefly used guess and check. This gives many solutions. Ethan has simply opted to figure out two solutions. Both answers are incorrect. Ethan has not explained his working or why he was happy with the second set of values.

      If the first equation had been 3x + y = 0, what would still be wrong with Ethan's method?

      Would this method ever obtain just one solution?

      Joe used a substitution method.

      Strengths: This is an efficient method.

      Weaknesses: Joe failed to multiply all the terms on the left-hand side of the equation by three, so he obtained an incorrect answer.

      If Joe had substituted 3x for y into the second equation the solution would have been very straightforward.

      Mia used a graphical approach.

      Strengths: This method can work.

      Weaknesses: In this case a graphical approach is not a very efficient strategy.

      Mia has made an error in her second table: y = 66 not 56.

      Mia could have used the co-ordinates (20, −10) to help plot the second line. There are no labels on either axis. The scale of Mia's graph means that the lines are not plotted accurately.

      Was Mia right to abandon (20, −10) as a point to be used to plot the second line?

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