# Making Thinking Visible with a Card Sort

I am a firm believer that students need to practice math concepts again and again for them to become proficient.  Some students may need more practice than others, but everyone benefits from the review.  Worksheets are good when there is not a lot of time, but they can be quite boring!  However, give a student a pair of scissors and some glue and now they are engaged.  That is what I like about Jacquie’s Expanding and Simplifying Card Sort.  There are six problems for students to simplify using the distributive property and combining like terms.  The problems are similar enough that you must do the work to figure out the answers.  I wanted the pieces to be bigger so I retyped the problems in Keynote and put a border around each expression.  (Don’t worry,  I first purchased Jacquie’s set and I am not looking to sell my version.)

I thought this would be a great activity on a day that I wouldn’t be at school.  I gave instructions for the substitute teacher to read to the students, along  with all of the materials.  Unfortunately, students seemed to glue pieces in random order.  By just looking at the papers and not knowing what went on while I was away, I’m not sure if the students did not fully understand the directions or they just didn’t want to do the work.  I do not fault the substitute teacher or the students.  Rather it got me to thinking, what could I do better to keep all students accountable for actually doing the problems and for me to see the students thinking?

So, I re-ordered the cards on the handout so that the problems were in the first two rows, the middle steps were in the next two rows, and the final expressions were in the last two rows.

Instead of giving students all of the pieces at the start, they would only be given the initial problems to cut out and glue on their paper.  Then they would have to work out the distributing and show me their work before getting the next set of cards.  Students would be able to check their answers when they cut and glued these cards underneath their work.  Finally, they would combine like terms and show me their final expressions before getting the last of the cards.

In the end, students still get to use the scissors and the glue.  However, I can quickly glance at each student’s work to see who understands the concepts and if not, where they are making mistakes.

# Introduction to Combining Like Terms

Last week my 7th grade students looked at combining like terms.  As a visual introduction, I handed each student one of three different colored  “tickets” to a Giants game.  I presented the students with this scenario: “An anonymous donor bought tickets for the entire class to go to a Giants game.  Unfortunately, not all the seats are in the same area.”  I put a map of MetLife stadium on the board showing the different seating areas. The different colored tickets showed the seat number, level, and the price of the ticket.   Green tickets were seats in the upper level, yellow tickets were seats in the mezzanine level, and orange tickets were seats in the lower level.  After getting in groups of four, I had the the students find out how much the donor paid for all of the tickets.  Each group had to write their answer on a small whiteboard.  After about 4 minutes, I had each group put their whiteboard on the front ledge so that we could discuss it as a whole class.

The conversation went something like this:

Me:  What did the donor pay for all of the tickets?

Student: \$4200.

Me:  How did you come up with that amount?

Student:  The oranges tickets cost \$2400, the yellow tickets cost \$1200, and the green tickets cost \$600.

Me:  What do you mean?

Student:  There are 6 orange tickets, 4 yellow tickets, and 6 green tickets.  Orange tickets cost \$400, so \$400 times 6 is \$2400.  Yellow tickets cost \$300, so \$300 times 4 is \$1200.  Green tickets cost \$100, so \$100 times 6 is \$600.

Me:  Why didn’t you just add the total number of tickets and multiply that number by \$400? or 300? or \$100?

Student:  Because the different colored tickets are worth different amounts.

Me:  So, why can you only combine the yellow tickets with other yellow tickets?

Student:  Because they are all worth the same amount.

Me:  And why can you only combine the green tickets with other green tickets?

Student:  Because they are all have the same value.

Me:  And what about the orange tickets?

Student: You can add the orange tickets together because they have the same value.

At this point I introduced combining like terms with variables and constants.  Each time a student tried to add something like 3x + 5 + 2x to get 10x, I reminded them of the different colored tickets.  For instance, the 3x and the 2x are like the green tickets and the 5 is like the yellow ticket.

After we worked on a few problems in our interactive notebook, students worked with a partner to play a Combining Like Terms Dice Activity.  I put stickers on a pair of wooden dice.  On one die were different coefficients (positive and negative)  with the variable x and on the other die were different coefficients (positive and negative) with the variable y.  Students had to roll the pair of dice twice and write down the terms in the squares on the handout and then combine like terms.  After completing four problems like this, they had to combine the answers of #1 and #2 together, and the answers for #3 and #4.

Partners would work on the problems independently and then check answers with each other.  If they got different answers and could not agree, they could raise their hands to ask me for help.

After these two activities, I felt students were ready to tackle similar problems for homework.  I wish I could incorporate visuals and games with every new concept!

I would love to hear about anyone else’s way to introduce combining like terms.  Please share!

# Expanding and Simplifying Algebraic Expressions

This week, my 8th grade Algebra class will be having their first test.  Two topics that I want to review with them again before the test are the distributive property and combining like terms.  Jacquie of Mathematters has a great cutout activity that I use with my 8th grade pre-algebra class.  However, I wanted something that would take less time but still have the students engaged.  Also, I wanted the activities to make their thinking visible.  So I made a handout using the same idea, but students have to draw arrows to the correct corresponding step instead of cutting and glueing the pieces in the proper order.

Here is an example of a student’s work:

I was able to go around the room to see that students were showing their work with the distributive property and then circling and boxing in like terms to get the final answer.

After that, students worked in groups to figure out the errors in this free handout from Secondary Math Solutions.

Students had to locate the mistake, as well as explain how to correct the mistake using complete sentences.

In both activities, students needed to either show their steps or write out the process to complete the problems.  Both activities allowed me to see who understood the problems and who needed more direction.  Now for the test…