Educating Risa

I saw these videos about using unifix cubes to show how to get the average (mean) of three numbers, and figured they would mesh well with a problem of the day. The kernel for my version of this problem was Day 16 here. Here it is:

Last week, the high temperature of the day was 8C on Monday, 10C on Tuesday, 13C on Wednesday, 8C on Thursday, and 11C on Friday. What is (1) the average (“mean”) temperature; (2) the median temperature? (3) the most common temperature (the “mode”)?

We started off by making a chart of our data:

Daegan was able to see that the answer to (3)—the most common temperature, or mode, was 8C, as it was the only temperature to occur more than once. We then moved on to the median, by making the temperature values in block form:

  

and then putting them in order from smallest to largest to find the middle term:

Daegan was able to see that the middle term is 10. There are two temperatures colder than (less than) 10, and two warmer than 10. 10 is the median value. We then got into a discussion about why knowing the mean, median and mode mattered (house prices are one common application, but that means little to a seven-year old, so I stuck to discussing dinosaur sizes), and how they are not always the same number. At this point I realized the mean of the values I have given Daegan would also work out to be 10, so I changed the question slightly, in a way that affected only the mean (not the median or mode). I changed the 13C on Wednesday to 18C.

Daegan again got out the blocks and modeled the question:

We talked about how to obtain the average, or mean value, by redistributing the blocks equally—like in the video. (I modelled this with the much simpler idea of doling out cookies). In order to do this, we needed to change the solid 8-rod (brown) of the 18C value to 8 ones:

Daegan then broke the ones off and redistributed them among the columns, yielding the answer: 11.

Now it was Daegan’s turn to create a question for me. He asked me to find the average size of the species Triceratops, given the two subspecies Triceratops horridus (size 30 ft.), and Triceratops maximus (size 38 ft.), which he drew:

I solved it numerically, and then pictorally, so that Daegan could follow along more easily:

I drew tens (the long rectangles) and ones (the small squares)—think Base 10 blocks—to show Daegan how to solve this problem pictorally. (My 7 year old cannot yet solve 68/2 numerically).

I thought we were done for the day, but Daegan immediately asked another question: What number doubled is 38? So I seized the teachable moment. I began by drawing 3 tens and 8 ones on the board, and asking him what we needed to do. He saw that we needed to divide the group in two to figure out the number than need be doubled to yield 38. If your child does not see this, change it to a simpler, more intuitive question, like “what number doubled is 10?” :

Now, our job was to divide up those tens and ones. Daegan had no problem giving each group 1 ten, and 4 ones. But that left 1 ten rod left to be divided between the groups. Daegan said “Oh! So you give each 5!” and I said showed him how we could do that by changing our ten for 10 ones, and giving 5 ones to each group. He could then see from the pictures that 19 doubled yields 38. 

Totally enjoying this Problem of the Day approach right now. Daegan has asked me more math questions out of the blue in the past week (how long is a decade? a century? a generation? What fraction of your life have you lived in Calgary, Mom?, etc.) than he has over the past several months.