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Modeling Situations With Linear Equations
During the
  • Interactive Whole-Class Introduction (30 minutes)
      • Give the class copies of the situation Making and Selling Candles and the mini-whiteboards.
      • Work through this sheet with the class, using episodes of whole-class discussion interspersed with short paired or individual work. Keep all students interested by asking them to show their answers using the mini-whiteboards. When a student offers an answer, ask other students to comment or explain, rather than evaluating the answer yourself.

      Students' spoken and whiteboard responses will give you information about what they are finding difficult.

      • Students work together in small groups on those parts of the task so as to produce a joint response.
      • Use the joint responses as the basis for further class discussion.

      Questions 1 and 2

      For example, you could work on Questions 1 and 2 orally.

        • What numbers should I write in for k, n, and s [Question 1]?
        • How could we figure out the total profit made, p, from the other numbers [Question 2]?
        • Use your mini-whiteboards to show me the equation you need to solve.
      • Allow students a few minutes to discuss their ideas in pairs, and then ask them to present their equations. Ask students to justify every step. Keep linking the context and the representation. For example:
        • You have to multiply 4 by 60 [or n by s].
        • What does that tell you? [The amount of money he makes from selling the candles].
        • What do you have to do next? [Subtract 50 (or k)].
        • Why subtract? [Because the cost of buying the kit will reduce his profit].
        • How can I write this equation using the values of k, n, s?

      Students will probably find more than one way of writing the equation.

      • When asking students to show equivalence, ask them to explain what the equation means in words, not just rearrange the equation.
        • How else can I write this equation using the values of k, n, s?
        • Anthony, you wrote the equation a different way. How did you write it?
        • Are these two equations the same? How do you know?
        • Kay, do you agree?
      • Ask students to explore different values of one of the variables:
        • What if you charged $3 for each candle? $7?
        • What other values could n take?
        • What else could I change?
        • What method do I use to calculate the value of p?
        • Does the calculation method change when I change the value of n? s? k?

      In this way, show that p = 60 × 4 − 50 and that in general, p = ns − k.

      This generality is not a trivial step. By calculating using different values for each variable, students should become aware that the same method works whatever values are given to those variables. You may need to spend some time establishing this point.

      • Give students two or three minutes to record their solutions to Question 1 and Question 2, then ask them to read Question 3.
      • Now look together at Question 3. Insert the answer for p as 190 and erase the selling price, s.

      Question 3

        • Suppose we know the cost of the kit, the number of candles that may be made from the kit, and the profit we want to make. How could we figure out the correct selling price?
        • Write your method on your mini-whiteboard.
        • I see p + k and p − k. Which do you think is correct? Why? [You first add the cost of buying the kit to the profit, 50 + 190 or p + k].
        • What does that tell you? [The total amount of money you need to make from selling the candles].
        • Then what do you do? [You divide the answer by the number of candles to find the selling price per candle].

      Students should reason that

      s =    190 + 50
      and that
      s =    p + k

      To focus all students on understanding the relationship between the context and the algebra, it can be useful to get a number of different students to explain the same answer.

        • Alicia—please explain why Alex is correct, in your own words.
        • Kay—you explain it too, in your own words, please.
      • Ask students to explain whether the same method works for different values.
        • What method do you use to calculate the value of s?
        • Does the calculation method change when I change the value of n? p? k?
      • If students produce different equations, get them to link their descriptions of the situation with the different numerical and algebraic representations:
        • Are these two ways of writing the same numeric equation? How do you know?
        • Are these two ways of writing the algebraic equations equivalent? How do you know?
      • Finally, focus students' attention on what they have been doing when they know three values and solve to find the fourth:
        • You've figured out how to solve for p, and how to solve for s. What else could I change?
      • Now turn to Question 4. Erase two numbers, n and p:

        • What numbers could n and p be?
        • Is this the only solution? Give me another solution...and another...and another...
        • This time, there are two numbers I don't know, n and p. What is different from not knowing one number?
        • Write a table of possibilities, and draw a graph to show how p depends on n.
  • Working in Pairs (10 minutes)
      • Hand out graph paper, and allow time for students to work on the problem in pairs. As they do this, go round and prompt them to try different pairs of numbers using the structure of the situation:
        • If n were 20, what would this equation mean?
        • How could you calculate the profit?
        • How can you write this method in words or symbols?
      n 20 30 40 50
      p 30 70 110 150
        • What equation fits this table?
        • What would the graph look like?
        • What does the graph show?
      • During the paired work, you have two tasks: to note student difficulties and to support them in their thinking about the graphing activity.

      Note student difficulties.

      Students have not been given a table to complete, nor is there a set of axes or scales on the axes. This allows you to find out whether students are able to use their knowledge in context.

      • Is students' use of algebraic notation accurate? Conventional?
      • Do they choose a sensible range for the number of students when drawing the graph, including values of n from 0 to 60?
      • Do students calculate an appropriate scale for each axis to fit all sensible values of n and p?
      • Do they label axes accurately and use equal increments?
      • Do they know how to draw and complete an accurate table of values?
      • Do they plot the points accurately?
      • Do they make a point plot or join the points to make a linear graph?
      • Do students use features, such as intercepts and linearity, to check the accuracy of their plots?

      Support students' thinking.

      • If students are stuck or making errors, try to support their thinking rather than solve the problems for them. If a pair of students is stuck on a problem, suggest that they seek help from another pair who have dealt with that problem successfully.
        • You have chosen values of n from x to y. Why did you choose those values?
        • How do you know that your plot is accurate?
        • What does the x-intercept show?
      • Encourage students to continue to link their mathematical representations with the context.
        • What happens to the profit if the kit only makes five candles?
        • At this point, n = 3.5. Does that make sense?
      • If students complete this quickly, set one of the following questions:
        • Is the graph best drawn as a point graph or as a linear graph? Explain your answer.
        • Explain the relevance of the x- and y- intercepts.
        • Erase two different variables [s and p or n and s], and express the relationship between them if all other variables are kept constant.
  • Whole-Class Discussion (10 minutes)
      • Ask students to try to answer Question 5 in pairs. Although students have already done part of this question, they need to recognize this for themselves.
        • Q5: Write a general formula for showing the relationships between the variables. What do you think you are being asked to do?
        • Suppose you know s, k, and n, and you want to calculate p. What method are you going to use?
        • Does the method change if you change the value of n?
        • What's the method for calculating p, whatever the values of s, k, and n?
      • If students are really stuck, refer them back to their earlier work:
        • Look back at Q2. What did you do to calculate s when you knew p, k, n?
        • What would happen to the calculation method if n were any other value?
        • Tell me in words how to calculate the total amount of money you collect. How would you write that using algebra? How do you calculate the selling price from that?
      • After a few minutes, ask each pair to show you the four general formulas using their mini-whiteboards:
      • p= ns − k k = ns − p
        n = p + k
        s = p + k
      • If there is disagreement about formulas, write different versions on the board. Ask students to show the equivalence of different formulas.
      • Ask students to say how they figured out their answers. Hopefully, some will have focused on writing relationships from the situation, generalizing numerical versions of the equation, and others will have used algebraic manipulation.
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