INVESTIGATION 1: Visual Patterns
We spend one lesson investigating patterns we choose from this website:
It is best to start with patterns like #2, #15, #18, that add a constant number of objects at each step.
Concrete/pictorial students will spend the time they need to build or draw the next few steps in a pattern, and count the new totals at each step. Watch for leaps of intuition when a student gradually begins to see a pattern emerging, and takes the risk of predicting the 10th or the 43rd term.
This student (who is a fairly visual learner) saw the following pattern in this set of triangles – She saw it as a cascading set of diagonal triangles: the 3rd figure has 3 on the left diagonal, then 2, then 1. Therefore, the 7th figure would have 7 on the far left side, then 6, then 4, etc. Neither of us had SEEN that relationship until she did!
Other students (abstract thinkers) don’t need to build or draw, and easily jump to the representation of any pattern in a chart. They can see the pattern emerge by looking at the numbers only. Remember, however, that this is probably not a majority of your class! Give them enough patterns, and harder patterns, so that they’re engaged and challenged, too.
INVESTIGATION 2: Happy numbers
The process for investigating happy numbers is given in the attached classwork. Beware: it is quite easy for some students (the algorithmic, abstract thinkers) to follow the steps, and quite difficult for the concrete thinkers to follow, since the whole process is abstract. Communicate the importance of working at your own best speed. This investigation pushes them into a zone of discomfort – they’ll say it makes little sense – but we do want to gently push them towards the abstract, and they usually do figure out the pattern and enjoy it by the end of the lesson.
Here is a video of a teacher’s approach to her own understanding and teaching of this pattern:
Try a couple numbers together as a class, then pass out the worksheet:
Because this lesson is difficult for the most concrete learners, we usually draw the first 9 squares on the whiteboard for referral. The repeated referral to the squares’ areas builds the students’ visualization.
Here are 2 HW’s that follow up on this investigation:
INVESTIGATION 3: GAUSS
This power point: cw 10_20Gaus includes a link to a youtube video of the famous episode of Karl Friedrich Gauss at about the same age as our fifth graders, and his addition of all the numbers from 1 to 100.
We follow it up with this classwork:
The difficult part of this is NOT the consecutive connecting of the two ends of each series. It is the counting of the NUMBER of pairs included. The series with an ellipse (…) are off-putting for some students.
Those students benefit from writing out the series. Eventually, they will devise their own shortcuts to figuring out how many numbers there are, and therefore how many PAIRS. Give them the time they need.
You will notice that HW is still reviewing fraction drawing at Level 1, and this will continue for weeks, since it is so difficult for some students and so important.