“Reversed” Ages

In August, at a summer board meeting of the Adult Numeracy Network, the fabulous Sarah Lonberg-Lew (@MathSarahLL) shared a problem. Well, it wasn’t really a problem, more like something she noticed. In the meeting, she asked what we noticed and what questions we might ask.

In this CAMI meeting, I basically did the same thing. I started by sharing the two sentences above, followed by the request:

Consider the situation, then pose some questions.

I then asked pairs to share their questions and come up with a few more.

Then I gave strips of paper to each pair and asked them to write a few of their favorite questions to place on the wall (a la Usha Kotelawala):

I then asked everyone to choose a question they were interested in and spend some time individually to work towards answering it. We worked independently for about 10 minutes, then small groups shared what they had discovered. The small groups worked together for about 30 minutes and shared some of their work on chart paper.

The first group laid out Sarah and her mom’s ages over time on a 100-grid and noticed a few things:

• The “reversible” ages are in a diagonal on the chart
• The “reversible” ages happen every 11 years for Sarah and her mother
• The ages alternate even and odd
• The digits sum to an odd number. These sums increase by 2 in each pair of ages. For example: 3 + 0=3, 4 + 1 = 5, 5 + 2 = 7, 6 + 3 = 9, etc.

Ramon and Greg noticed other things:

• Greg’s mom (68 years old) and Greg (34 years old) don’t seem to have any “flipped” ages.
• The difference between “flipped” or “reversed” numbers is a multiple of 9. For example, 81 – 18 = 63 and 84 – 48 = 36. Based on this, they made a conjecture that in order to be “reversible,” two ages have to have a difference that is a multiple of 9. The difference in ages between Greg and his mom is 34, which is not a multiple of 9, so they would never have reversible ages.

A few more questions can be posed based on the work these groups did:

• Why do “reversible ages” happen only if the difference is a multiple of 9?
• Why is there a gap of 11 years before ages are “reversed” again?

I want to recognize a few people who helped me prepare. Before the meeting, I shared the problem on Twitter and the following people responded to post questions and make suggestions. Thank you all! @MathSarahLL, @PatriciaHelmuth, @Rivera_Con, @mtrushkowsky, & @benjamindickman

Hey @MathSarahLL @PatriciaHelmuth @v00121 @Rivera_Con, I’m sharing Sarah’s age problem with our math teachers’ circle next week. Want to help me develop some #supportpush cards? #noticewonder #writesomequestions #ANNmath (shh. don’t tell #nyccami) pic.twitter.com/qtggtYO3QN

— Eric Appleton (@eappleton) September 13, 2019

I shared a draft version of the problem with support/push questions with the #nyccami group. Please share comments and suggestions.

In Attendance: Avril, Eric, Greg, Joneil, Maya, Ramon

Programs Represented: BMCC ALC, CUNY Adult Literacy PD team, CUNY Start