My guess is many of you had math classes similar to mine in middle & high school. Each day I would be assigned a set of problems from our text book. As the teacher wrote the assignment on the board, “Please do problems 2-40….” and we would all groan as she wrote one final word, “even!”. We groaned because we knew that we could not use the answer key found in the back of the text book. On days that a problem set included the word ‘odd’ we would inwardly cheer, thinking this would allow us to cheat somehow. I, like many of you, bookmarked these pages. I used them religiously.

It was not until I was a teacher that I realized that using the answers in the back of the book was not cheating It was a great study skill. I was very good at working backwards to figure out a problem. If my answers were wrong it forced me to go back and find my mistakes and fix them. As a teacher giving students the solutions to a homework set is something I want to do. The problem with my 2017 classroom is that I don’t use a textbook. The phrase “check your answers in the back of the book” means nothing to my students.

I find it valuable for students to have solutions in 2017 so I often include them on my homework. Including answers takes away from the rush to get an answer and puts the focus on students providing the work that supports the solutions given. I don’t though always take the time to add the answers. My real goal with students is to empower them to know how to check their answers for themselves.

This tweet from Michael Fenton caught my attention.

Michael’s tweet resulted in over 100 responses. You should check out the thread. Lots of good stuff in there. I saw this tweet when I was brainstorming how to get my students to use Desmos and thought, I hope my students would have checked their work in Desmos and found their mistake on their own before I told them there was a mistake. So how do I create this culture in my classroom.

In the last few years I’ve started writing the phrase, “**Check your solutions in Desmos.**” on the bottom of any assignment that Desmos would benefit students (which is many of them). If it would only benefit students on a few problems, I name which problems to use Desmos on.

I’ve had success with some students doing this, but not all. I have found that students who do use my suggestion to use Desmos to check their work are more likely to use it on their own to check work or as a tool for exploring & learning. This last year I had a new mission as a teacher: **How do I get every, yes every student to use Desmos as a tool to check work, explore and learn on their own without my prompting**.

I know better. I know I should not assume that students know how to use Desmos, even if I’ve told them to use it and helped them sign up for an account. I know that I must always model any tool I want students to use. I have done this on my smart board, but I needed to do something more. Here is my new plan for modeling the student use of Desmos to check work and more. I call this ‘**DESMOS IS THE NEW BACK OF THE BOOK**“. I’ve shortened this to #DITNBOTB (ha. not this acronym is short)

I started working on this idea in San Francisco in November at Desmos headquarters along with other Desmos Fellows. I worked for an hour with the amazing Stephanie Blair trying to figure out how to put my idea into Activity Builder. We started but did not finish. I’ve been meaning to finish all school year. This summer I picked this up again so I can use what I create this fall.

I decided to abandon Desmos’ ‘Activity Builder’ as the vehicle for lots of reasons. First, students will almost solely use either the online graphing calculator or the Desmos app on their phone when working at home. There are a few things that look different on Desmos and Activity Builder and I did not like this. Second, I know if I am going to get all students to use Desmos on their own I need to model it using the tool they will use most often and activity builder is not that tool. I know activity builder is the sexy tool for all you Desmos geeks, but I find myself often creating things in the online graphing calculator to use with students. (though I also love and use AB).

So here is my plan for this year. Sometime in week 1 I plan to give a homework assignment to explore using Desmos to check work. I have not decided which of the following I will do yet. I am going to give you a few options. I’d love to hear what you think. These are still a work in progress. (I will update this post after I use these with student).

#### OPTION #1:

I teach Advanced Algebra. Have students do 7 review algebra problems for homework. The next day in class I will have them check their work on Desmos. This may be teacher led with me or I may use a version of what I’ve made in option #2 to do with a partner. I selected these 7 problems because they are things I want my students to know how to do on Desmos at the start of Adv Alg. It also gives me a chance to see the algebra skills they have or don’t have at the start of the year. Here is OPTION #1’s homework assignment: Advanced Algebra Start of the year Algebra Review You will see these exact same problems in option #2.

#### OPTION #2:

I give the following (DITNBOTB homework) to students for homework and ask them to use Desmos to check their work. The same 7 algebra review problems are included. The next day in class they work with a partner to fill in any holes they could not figure out in class. As a class we will discuss some together. I am leaning towards this option. I like the idea of them using Desmos to figure out which of the 7 problems are incorrect (note: one of the 4 that is incorrect is because I am being picky on notation).

#1 is one of the reasons I left Activity Builder. On the online graphing calculator you can see the solution to any 1-variable equation you type in. This does not show up in AB & I really want my students to use this to check solutions.

Here are some of the Desmos graphs I may use with students to go over each/some of the problems. These are all simple, but they represent what I used to make each problem.

- Problems 1-4
- Problem #5
- Problem #6
- Problem #7
- jpegs for each of the 7 problems: DITNBOTB homework with jpeg solutions

#### What’s Missing?

I love the assignment I’ve created to start the year. I am confident that modeling using desmos will increase the number of students who will use it. That said, there is some things I want students to know missing from this assignment. I plan to do a few things more with my students throughout quarter 1. Here is what I still need to develop.

One big thing missing is how great Desmos is amazing for showing **Equivalency**. I am thinking of doing related to solving equations in this tweet from the pride of Minnesota, Megan Schmidt and a reply from Max Ray.

So I graphed the 3 expressions in Megan’s tweet & they do not line up (not equivalent).

I will be putting something together to assure students will use Desmos to look for equivalency. Perhaps with solving equations and putting each step in Desmos OR perhaps for looking at forms of linear equations and which are equivalent. When I do (or if you want to develop something for me) I will post it here.

In addition to ‘equivalency’ missing from the assignment I created, what else is missing? I’d love to hear your ideas.

#### Closing thoughts.

The graphing calculator has been the great differential tool of the last 25 years in our mathematics classroom. It has allowed students with fragile numeracy skills to engage in grade level topics. It has given our top students a tool to explore on their own. Desmos is the great differentiation tool for the next 25 years. Desmos is good for our students at all levels. If it is such a great tool, then it is my responsibility as my students teacher to make sure they know how to use it and are using it daily. It is my opinion that all of us should be making space to use it the first several weeks of school.

A word of caution. Don’t make the ‘field of dreams’ mistake in your classroom. Just because Desmos exists does not mean your students will use it. Eli (and his team) built it but it is the job of us teachers to market desmos to our students. It is our job to model it. I would also argue it is our job to model its use in the mode our students will use it the most – on their cellphones. If you want students to check work in Desmos on their own – have them check their work on their phones in class at least 2-5 times during the first few weeks of school. Model it and they will do it. Don’t assume just because it exists students will use it.

I really do believe ‘**Desmos is the new back of the book**‘ for our students. It is my go to when I am working on mathematics and I am on a mission to create this culture in my students. #ditnbotb

As always, I’d love to hear from you. Email me at saravdw@gmail.com, tweet me @saravdwerf or comment below.

Sara: Love this! I don’t know about older, advanced algebra students, but my 8th grade algebra students would need modeling of how to use Desmos to check their work vs how to use Desmos to get an answer without having done the work. I like your idea of starting with option #2. Instead of starting the year with something that looks like every other post-summer review, you’re starting with a cool new tool used in a new way. I’m curious why you want to tell them how many errors there are. Is it b/c one is tricky to find as it’s only a formatting error? I’m not sure you need to tell them there are 4. Your next assignment could then be more like option #1 (with different review problems), where they do them, check them, but you also give them permission to use Desmos to find the answer if they can’t find it themselves, but expect them then to work backwards and try to solve it (as you mention that you, and we all did, when we used the back of the book answers). Excited to see how this evolves.

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Thanks for the feedback. My gut is to tell them how many errors since it is a start of the year activity. My goal is to have assignments that are safe to engage in without fear of failure (yet). I’ve found that telling how many errors there are does not take away from students engaging in all.

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One of my co-workers many years ago called the back of the book “BOB” which would shorten your acronym a bit.

Love the ideas. We have just gone 1 to 1 with chromebooks and I would love to be able to use Desmos 100% of the time, but we have to use calculators on our state test. Additionally, there is the challenge of PSAT/SAT/ACT and the calculator/no calculator sections.

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This is great stuff Sara. What I love is not just the cumulative review to support prerequisite skills and understandings and the critical role that Desmos can play in this, but the way this approach builds ownership of both the mathematics and the learning of math – in direct opposition to the typical and mind-numbing 4 weeks of telling students what they are supposed to have remembered and somehow never do. I would only encourage you to balance the skills (solve, simplify, factor, find) on your worksheets that Desmos can help check with some explain, justify, convince yourself, create a picture, that build the foundation for all of the new learning for the year.

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