Lily is working through 4th and 5th grade level math on Khan Academy’s Mission section. Her comprehension is at about 50% – 60% without much involvement on my part. When a question is presented that she has no comprehension of, she can either choose to indicate that she’s not learned “this” yet, or she can ask for a hint, and/or watch a video that explains the concept. There are a few that I choose “not learned yet” because even with a video/hint, it would be confusing for her, but mostly, she chooses which route she prefers.
Below is a question that asks her to plot 4/3 on the number line, and choose which line represents this plotting. I reviewed this question prior to Lily seeing it, and began the video as her introduction.
After watching the associated video, and working through a few examples together on paper (images below), including reinforcing 3/3 = 1, while 4/3 = 1+1/3 (and other examples of fractions that are equivalent to 1 whole, she was shown the question in the first image below, and immediately chose option B. Her comprehension was obvious, and though we may revisit this concept in a day or two in order to verify, I’m seeing a solid foundation in basic understanding of fractions in this more abstract form, whereas before, and for the last two years, her demonstration of comprehension focused on a more concrete use of portioning. For instance, in baking, in divvying up items among herself and friends, in choosing how much food or what proportion of her plate/bowl/cup she wanted fill with contents for consumption, etc..
Fractions on a number line: Graph and locate fractions on a number line.
Her written numbers have improved significantly since this spring, when she was still struggling with forward/backward and even proportion. Every once in a while, she still writes a number backward, but it’s quite infrequent and most often, self corrected.
In a lot of our diagrams and working through of functions, problems, and depictions, we use connecting lines (other similar objects to show connection) to demonstrated associated flow, and to illustrate how we’re thinking through something. Previously, I typically had the pen in hand and drew the connections, but in the last month or two, she has begun to share this part of the process about 1/2 the time.