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Example Test Answer Key Grades 6-7

Name: _____________________

1. a. Mark the right fraction on each card on the clothing line.
   b. Below each card, hang at least one extra card with an equivalent fraction.
   

   Solution Description Goal: Order fractions on a number-line. Special instructions: Compare this problem to “Fractions and clothing lines.2,” which is its counterpart. Comments: Some teachers used this problem for a classroom activity. They actually stretch a rope through the classroom and show the unit by hanging 0 and 1 in their places. Students were asked to hang fraction cards (with a thin cord on them) in their proper place with clothes-pegs. This activity reveals common misconceptions about fractions. Criteria Grades 6 - 7, Number, Level 1, Single Answer, Essential, Less than two minutes




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The second problem is also on number content, but it is definitely more challenging. This problem offers students an opportunity to show different strategies, and it offers teachers the opportunity to use these strategies in their feedback, thereby enhancing the teaching/learning process.



2. There are 3 ways to express 15 as the sum of consecutive whole numbers:
   15 = 1 + 2 + 3 + 4 + 5
   15 = 4 + 5 + 6
   15 = 7 + 8
   
   17 can only be expressed in 1 way:
   17 = 8 + 9
   
   16 can not be done at all.
   
   a. Investigate other numbers.
   b. Make a report of your findings, arranging them in a logical way. Show what you did to find the numbers.



Solution 1 + 2 = 3 2 + 3 = 5 3 + 4 = 7 1 + 2 + 3 = 6 2 + 3 + 4 = 9 3 + 4 + 5 = 12 1 + 2 + 3 + 4 = 10 2 + 3 + 4 + 5 = 14 3 + 4 + 5 + 6 = 18 Description Goal: using whole number operations and computations in order to investigate whole number concepts Special instructions: Preferably students should work in small groups Criteria Grades 5 - 7, Number, Level 2, 3, Extended Response, No Context, About 30 minutes





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The Eskimo problem is geometrical in nature, but not in the traditional sense. This problem focuses on how we see things and why. Although one can expect students to draw all actual vision lines before answering the questions, there is also the possibility that they will 'see' immediately that Es is in the 'best' position. The reasoning then becomes an interesting aspect of this problem.



3. The eskimos Es, Ki, and Mo are sitting in their igloo. They look through the door and see a number of polar bears. In the figure below you see a top view of this situation:
   a. Which of the three eskimos has the largest angle of vision?
   b. Who sees the most polar bears?
   c. Mo's angle of vision is larger than Ki's angle of vision. Still, Mo sees fewer polar bears than Ki. Give an explanation.
   
   

   Solution a. Es b. Es c. Mo is closer to the side of the door, but Ki is more in line with the door's opening. If I draw a straight line between Mo and each of the polar bears (like his line of vision) and I do the same with Ki, more lines connect with Ki. Ki has a better line of vision. Description Goal: Illustrate lines of vision. Criteria Grades 5 - 7, Geometry, Level 2, 3, Multiple Question, Essential, About 5 minutes




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4. In an advertisement for videotapes it says; “Buy 5 tapes, pay for only 4.”
   Normally these tapes cost $6.25 each.
   What is the discount percentage if you buy 5 tapes?
   
   

   Solution The discount is 20%. Description Goal: Determine a discount percentage by reasoning or by calculating with rational numbers. Special Instructions: A calculator may be used. Student Strategies Students may calculate: normally paying for 5 tapes: 5 x 6.25 = $31.25. With the discount, one pays 4 x $6.25 = $25.00. The difference is $6.25, which is 1/5 of $31.25 or 20%. It is much easier not to look at the prices to pay. Buy 5 and pay only for 4, so 1/5 is the discount. Criteria Grades 5 - 7, Number, Level 2, Single Answer, Essential, Less than two minutes




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The last problem on the test is one that can be called a pre-algebra problem. Many teachers will be tempted to translate the problem into 2x + 2y = 44 and x + 3y = 30 and then use their algebraic tools and techniques. In this case, our students are around 12 years of age and do not know much, if any, algebra. However, given the opportunity, they can be excellent problem solvers and engage in some surprising reasoning. And that’s all that is needed in this problem. Of course, it does help if students have seen more mathematics than simple basic computational skills. But even then, one might be surprised to see the different strategies that students use in this case. Note the example strategy shown below: If two of each cost $44, then one of each costs $22. That means that, looking at the second picture, two cokes must be $8. Or, some may use a more geometrical solution, using patterns: First picture is 2,2 and $44. And second picture is 1,3 and $30. So make the third picture 0,4 and $16. So four cokes must be $16. These types of problems can be very interesting to solve. There are many like these that are centuries old but often neglected in modern textbooks. And they have a lot of potential for informing the teaching/learning process.



5. How much does a T-shirt cost?
   And how much is a soda?
   Explain how you got your answer.


Solution A t-shirt costs $18.00 and a soda $4.00. Description Goal: Solve a problem with two unknowns. Student Strategies Students can use a variety of informal strategies to solve this problem (combination chart, exchanging, matrix notation, reasoning, trial and error or guess and check), and can also write two equations with 2 unknowns. By exchanging 1 T for 1 S the price decreases by $14.00, so the next combination would be no T-shirts and 4 sodas for $16.00, which means a soda costs $4.00. Criteria Grades 5 - 7, Algebra, Level 3, Open Question, Essential, About 15 minutes