If a linear programming problem has a solution, then it must occur at an interior point of the feasible set \( S \) .Ĥ. A linear programming problem consists of a linear objective function to be maximized or minimized subject to certain constraints in the form of linear equations or inequalities.ģ. The solution set of a system of linear inequalities is unbounded if it can be enclosed by a circle.Ģ. Determine whether the following statements are true or false.ġ. The transition to the tape diagram is beautiful! If you’d like to see it in action, check out Topic E of Grade 1 - Module 4.True or False. Prior to that module, students are provided with the foundational skills that lead to an understanding of the part/whole relationship and what the tape diagram represents. ![]() Students are first introduced to the tape diagram in Lesson 19 of Module 4, Grade 1. The tape diagram serves as an access point for the 5th grader, whereas the 9th grader’s fluency with this reasoning allows for instantaneous articulation of the same concept. Both draw the conclusion that 4 loads of bricks and 4 loads of sticks is equivalent to 12 loads of sticks and pursue the answer from there. Note that the essential reasoning by both the 9th grader and 5th grader are the same. ![]() ![]() S: Yes, because the weight of 1 load of bricks and 3 loads of sticks is a little less than half the total weight. 1 load of bricks and 3 loads of sticks weigh 321.25 kilograms. Now draw another tape to represent the weight of the bricks. Draw a tape to represent the weight of the sticks. T: Let’s show that relationship using a tape diagram. T: What do we know about the bricks and the sticks? S: We know that there are bricks and that there are sticks. What is the total weight of 1 load of bricks and 3 loads of sticks? The total weight of 4 loads of bricks and 4 loads of sticks is 771 kilograms. This example shows not only the direct power of the tape diagram, but it also shows the prior learning evident in the modules that brings students to the point where they can solve this problem with success.Ī load of bricks is twice as heavy as a load of sticks. Through the RDW process, the student would read and reread the problem, draw a tape diagram to help make sense of the information in the problem, solve the problem mathematically, write an answer statement, and then revisit the original problem to determine if his/her answer makes sense.Ĭonsider the following dialogue as a possible interaction with a 5th grade student who is working to solve the problem. ![]() So, how would a 5th grader solve the problem? One way would be to follow the RDW process and to draw a tape diagram. “But,” she concluded, “I have no idea how a 5th grader would do that!” When asked why she did it that way, she explained (in one very long run-on sentence) that “since the bricks were twice the sticks and since there are 4 loads of each, that’s really 12, so I divided by 12, and then I multiplied by 3 and then by 2 (since the bricks are twice the sticks) and I added my answers together”. She then added the two products together to get her answer. To solve this Grade 5 problem, my 9th grade daughter first divided 771 by 12 and then multiplied the quotient by 3 and by 2.
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