Looking for challenging problems?

 Check out Reflections and Tangents postings of "old" calendar problems! Search the name of the month in the search box on her blog. Here are two samples ... 

YouTube Channel worth exploring ...

 

Looking for challenge problems that address SAT, ACT, AMC and a whole host of other ABCs??

I like this Youtube Channel - I think you'll find interesting problems and complete solutions!  

How might you use this in class?  

1) What if you used it as a bonus for each unit? Or an extension for students who master the required content early? 

2) How about asking students if they can find a different way to solve - working on flexibility?

3) What about asking paired students to choose one problem (from your curated selection) to present to the class?  Build in cooperative skills, presentation skills, as well as working on math skills?

4) Only half of your class present due to a field trip or other event?  Don't want to teach "new" material?  Use one of these challenge problems!

How else might you use them?

Here's a sample:

Instant Replay #3: Adding "richness" using strategies

Planning for rich problems is tough!  It can be tricky and varies depending on the set up in the school.  In my last placement I experienced a well-organized PLC program, a detailed curriculum plan, and a calendar!  The calendar was the more difficult piece to adjust to ... the 5, 6, or 7 teachers teaching algebra 2 were expected to test on the same day. At first I found this annoying - I wanted more freedom!  But in the end I learned that it was a blessing.  I explained the calendar to students and parents.  Admittedly sometimes it was hectic and I felt like we could use more time to delve deeper or practice more.  But the calendar enabled me to hold students to high expectations and insist on their participation in their own learning.

Our PLC planned a unit at a time.  We started with the assessment, then the activities, and mapped out the lessons.  We had options so that teachers on the team could vary activities based on the needs of their students and their own teaching preferences.  As I looked at the whole unit, I looked for ways to infuse critical thinking ... through a "rich problem" or through the strategies we used in class.

And that leads me to this thought expressed in previous online discussions ... sometimes we add "richness" to the work in class, not by a specific problem, but by the strategies we use.  For example, others have mentioned using error analysis as a way to add richness.

So what strategies add "richness" to the work in class?  "I do, we do, you do" quite possibly stifles the richness opportunities and trains students to expect you to guide their every move.  It slows down the training of perseverance! At least that has been my experience.  Obviously we have to give notes, demonstrate procedures some of the time.

So what strategies add "richness" during the practice of skills/concepts?  Here are a few I have found to be particularly useful ...

Desmos Activity Builders - especially when designed well.  Check out Meyer's post on Desmos building guide!

Card sorts ... as students work through cards they have to analyze the details given to make matches.  The conversations are often rich around those details.

WODB ... the analysis challenges students to look for similarities, differences, comparisons, and to find as many different varied responses as possible.

Error Analysis Tasks ... these can vary from "Find the Mistakes" Gallery Walk, to stations or even having students work problems incorrectly to swap with a partner.  Becoming aware of common errors in math helps students to think more deeply about procedures.  Asking students to create Two Truths and a Lie questions ... swapping with partners, posting online, creating a collaborative slide show are ways to implement the activity.

I mentioned other strategies in a previous post and questioning strategies as well.

How do you build "richness" into the ordinary lessons?  What strategies have you found to promote deep thinking?

Instant Replay: Rich Tasks #2

So in Part 1 in exploring the topic of Rich Problems, I outlined typical characteristics of Rich Problems. They are non-routine in that they are not typical practice of a skill or algorithm; they have little scaffolding; there are multiple entry points; and students of varying skill levels can benefit from Rich Problems. Rich Problems have natural extensions - one question leads to another. They are difficult and interesting at the same time. Rich Problems are revealing! They give insight to teachers about students' understanding. They help students experience the essence of mathematics. 

So if you are like me ... you want to see some examples of Rich Problems.  Examples help us think about our own work and how we might tweak a question to make it more open-ended, more accessible to students and yet keep it challenging!

I did a Google Search and found a few problems I think would fit nicely in Algebra 2, First Semester curriculum, and could potentially be worthy of discussion!  Let me know what you think of these!  

Systems of Equations ... From Illustrative Math ... Cash Box

Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said,

I wonder whether the dollar belongs inside the cash box or not.

The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found$200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not?

Systems of Equations ... from Riverpoint Advanced Mathematics Partnership-Algebra Project ... Systems from Sequences

Make up an arithmetic sequence of six numbers: a0, a1, a2, a3, a4, and a5. Then write and solve the system of equations (feel free to use technology to help you solve these).

Compare your answer to others who used a different sequence. Come up with conjectures as to both what is happening and why.  Be ready to justify your conjectures and to share your work.

Quadratic Functions ... from Underground Mathematics

Given that two of the parabolas have the equations, y=x2−12x+27  and y=−x2+12x−36, can you find the equations of the other parabolas?  Can you recreate the diagram on Desmos?

Quadratic Functions ... from Underground Mathematics

When are coefficients of a quadratic equal to its roots? 

The equation 𝑥^2 + 𝑎𝑥 + 𝑏 = 0, where 𝑎 and 𝑏 are different, has solutions 𝑥 = 𝑎 and   𝑥 = 𝑏.  How many such equations are there?

Non-Linear Systems of Equations ... from Riverpoint Advanced Mathematics Partnership-Algebra Project ... Intersections

Part 1: A line with slope 5 passes through the vertex of the parabola above. Does it intersect the parabola in another point (other than the vertex)? If so, find the point of intersection. If not, explain why.

Part 2: Think of all possible lines that pass through the vertex of the parabola shown above. Which lines intersect the parabola again at another point and which ones do not? Explain.

Part 3: Think of all possible lines with a slope of 5. Which of these lines intersect the parabola shown above? How many times? Explain.

There are others!  Certainly many of the Desmos Activity Builders are rich, worthwhile problems - as are 3-Act Math Tasks.  

Do you have a favorite "Rich Problem?"  Do you have a favorite source for problems??  If so, please share in the comments!

Resources for problems include ... 

Riverpoint Advanced Mathematics Partnership-Algebra Project 

Underground Mathematics

Open Middle Problems

NRich Enriching Mathematics

Mathematics Assessment Project

UT Dana Center Benchmark Tasks

Instant Replay: What are Rich Problems

I love @Druin's blog post today about rich problems!  If you haven't read it yet ... definitely trip over there now!

I'm going to repost a few ideas I wrote a year or two ago!  Hopefully there is something here to interest and inspire!

First Week Idea - Chessboard Squares Extension

I love the YouCubed Week of Inspirational Math ideas!  I love the idea of starting at the very beginning of the year inviting our students to explore patterns, create generalizations, and communicate math!!

With that in mind, I love to stumble on ideas ... some olds ones that need revamping ... like the one I'm sharing today!

I imagine many of us have challenged students with the question - "How many squares are on a Chess board?"  And of course, the first response is often 64, which allows us the opportunity for a sly smile, raised eyebrows, and the comment, "Really?  Are you sure?"

One of the fun things I am doing in retirement is sharing Usborne Books & More with folks ... yes, I'm not just a math lady ... I'm also a book lady!  So I was pleasantly surprised the other day when I was looking through Anna Weltman's, This is Not Another Math Book**, I found a great extension of the question about squares on a Chess board!

Obviously in doing this activity, you want to start with the basic challenge!  And have you ever noticed the pattern??

1    8x8 square

4    7x7 squares

9    6x6 squares

16  5x5 squares ... YES!

The number of subsequent numbers of squares are all square numbers!!  WooHoo!

The extension in Weltman's book above asks students to consider a rectangular board.  How many squares are on a 5 x 8 board or a 4 x 6 board?  Can they not only answer those questions ... but can they generalize the rule??!  Can you make a rule for how many square there are in ANY SIZE rectangular chessboard!

**If you are interested in knowing more about the book I've referenced, let me know.  It is currently on special.  Publisher's Weekly says this about the book:  This companion to THIS IS NOT A MATH BOOK continues to explore the connections between mathematics and art via more than two dozen step-by-step projects. Most of the projects require only the paper templates provided, along with writing utensils, a ruler, tape, and scissors. They include drawing kaleidoscopic patterns (as part of an exploration of symmetry), making a forest of fractals trees, and assembling five-square pentomino shapes (they resemble Tetris pieces) into pictures. There isn’t much discussion of how artists use math (Mondrian is the only fine artist mentioned), but Weltman’s projects ably demonstrate how creative mathematics can be.  

#BLAugustChallenge2018

#MTBOS

#ITeachMath

#UsborneBooks&More

#Alg2Chat Rich Problems Part 3

Last night we had a great Twitter chat about Rich Problems.  We noted that they are hard to find -
especially for Algebra 2, and it is challenging to find time to fit them in our curriculum.  But we also noted that they are worth the effort!  Watch for the Storify link ...

A couple of discussions stuck with me.  One was about planning for  Rich Problems.  Planning is tricky and varies depending on the set up in the school.  In my last placement I experienced a well-organized PLC program, a detailed curriculum plan, and a calendar!  The calendar was the more difficult piece to adjust to ... the 5, 6, or 7 teachers teaching algebra 2 were expected to test on the same day. At first I found this annoying - I wanted more freedom!  But in the end I learned that it was a blessing.  I explained the calendar to students and parents.  Admittedly sometimes it was hectic and I felt like we could use more time to delve deeper or practice more.  But the calendar enabled me to hold students to high expectations and insist on their participation in their own learning.

Our PLC planned a unit at a time.  We started with the assessment, then the activities, and mapped out the lessons.  We had options so that teachers on the team could vary activities based on the needs of their students and their own teaching preferences.  As I looked at the whole unit, I looked for ways to infuse critical thinking ... through a "rich problem" or through the strategies we used in class.

And that leads me to this thought expressed last night ... sometimes we add "richness" to the work in class, not by a specific problem, but by the strategies we use.  Last night others mentioned using error analysis as a way to add richness.

So what strategies add "richness" to the work in class?  "I do, we do, you do" quite possibly stifles the richness opportunities and trains students to expect you to guide their every move.  It slows down the training of perseverance! At least that has been my experience.  Obviously we have to give notes, demonstrate procedures some of the time.

So what strategies add "richness" during the practice of skills/concepts?  Here are a few I have found to be particularly useful ...

Desmos Activity Builders - especially when designed well.  Check out Meyer's post on Desmos building guide!

Card sorts ... as students work through cards they have to analyze the details given to make matches.  The conversations are often rich around those details.

WODB ... the analysis challenges students to look for similarities, differences, comparisons, and to find as many different varied responses as possible.

Error Analysis Tasks ... these can vary from "Find the Mistakes" Gallery Walk, to stations or even having students work problems incorrectly to swap with a partner.  Becoming aware of common errors in math helps students to think more deeply about procedures.  Asking students to create Two Truths and a Lie questions ... swapping with partners, posting online, creating a collaborative slide show are ways to implement the activity.

I mentioned other strategies in a previous post and questioning strategies as well.

How do you build "richness" into the ordinary lessons?  What strategies have you found to promote deep thinking?

#Alg2Chat Rich Problems Part 2

So in Part 1 in exploring the topic of Rich Problems, I outlined typical characteristics of Rich Problems. They are non-routine in that they are not typical practice of a skill or algorithm; they have little scaffolding; there are multiple entry points; and students of varying skill levels can benefit from Rich Problems. Rich Problems have natural extensions - one question leads to another. They are difficult and interesting at the same time. Rich Problems are revealing! They give insight to teachers about students' understanding. They help students experience the essence of mathematics. 

So if you are like me ... you want to see some examples of Rich Problems.  Examples help us think about our own work and how we might tweak a question to make it more open-ended, more accessible to students and yet keep it challenging!

I did a Google Search and found a few problems I think would fit nicely in Algebra 2, First Semester curriculum, and could potentially be worthy of discussion!  Let me know what you think of these!  

Systems of Equations ... From Illustrative Math ... Cash Box

Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said,

I wonder whether the dollar belongs inside the cash box or not.

The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found$200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not?

Systems of Equations ... from Riverpoint Advanced Mathematics Partnership-Algebra Project ... Systems from Sequences

Make up an arithmetic sequence of six numbers: a0, a1, a2, a3, a4, and a5. Then write and solve the system of equations (feel free to use technology to help you solve these).

Compare your answer to others who used a different sequence. Come up with conjectures as to both what is happening and why.  Be ready to justify your conjectures and to share your work.

Quadratic Functions ... from Underground Mathematics

Given that two of the parabolas have the equations, y=x2−12x+27      and y=−x2+12x−36, can you find the equations of the other parabolas?  Can you recreate the diagram on Desmos?

Quadratic Functions ... from Underground Mathematics

When are coefficients of a quadratic equal to its roots? 

The equation 𝑥^2 + 𝑎𝑥 + 𝑏 = 0, where 𝑎 and 𝑏 are different, has solutions 𝑥 = 𝑎 and   𝑥 = 𝑏.  How many such equations are there?

Non-Linear Systems of Equations ... from Riverpoint Advanced Mathematics Partnership-Algebra Project ... Intersections

Part 1: A line with slope 5 passes through the vertex of the parabola above. Does it intersect the parabola in another point (other than the vertex)? If so, find the point of intersection. If not, explain why.

Part 2: Think of all possible lines that pass through the vertex of the parabola shown above. Which lines intersect the parabola again at another point and which ones do not? Explain.

Part 3: Think of all possible lines with a slope of 5. Which of these lines intersect the parabola shown above? How many times? Explain.

There are others!  Certainly many of the Desmos Activity Builders are rich, worthwhile problems - as are 3-Act Math Tasks.  

Do you have a favorite "Rich Problem?"  Do you have a favorite source for problems??  If so, please share in the comments!

Resources for problems include ... 

Riverpoint Advanced Mathematics Partnership-Algebra Project 

Underground Mathematics

Open Middle Problems

NRich Enriching Mathematics

Mathematics Assessment Project

UT Dana Center Benchmark Tasks