One of the issues I have with some of the Khan Academy videos is how he seems to select numbers without any planning. To be fair, he is not alone in this approach of using numbers in examples without considering the impact these numbers will have on learning. Many of the novice teachers I work with struggle with the same view that any number will do. Fortunately, reading The Teaching Gap and exploring Lesson Study introduce our teachers to the importance of being thoughtful about the numbers they use throughout a lesson.
But what happens when the numbers come from a textbook? Can these numbers be trusted? During an observation, a teacher used the following problem from Holt McDougal Mathematics Course 1.
When the teacher asked students how they tried to answer the questions, they provided three different approaches.
Many of the students originally did the same thing in both cases - divided by four. One student suggested that because the area was given for the first square that they only needed to divide 81 by two. This proved to be a turning point in the discussion as many of the students backtracked on their first instinct and said they needed to find a number that multiplied by itself resulted in 81. Perhaps they had missed "an area" the first time.
Thinking that he had addressed the issue, the teacher asked, "Which figure has the longest side?" The student he called on responded that the second square had the longer side. Then the teacher asked the student, "How much longer?" to which he got the response of 0.75 feet. Clearly, this student had missed part of the discussion, but that is not what I am concerned with here.
Why had the textbook authors used 84 feet for the perimeter of the second square? Most middle school teachers could tell you that 20.25 feet would be a common mistake among their students. So why have a number (84) that makes the first part of the answer correct even when the work is done incorrectly? I would have used a perimeter of 80 feet so that both parts would have been wrong if the anticipated error had been made. Am I wrong in this choice? This is a genuine question to which I would appreciate your feedback.