Function Machines and Arithmetic Racks

Using the arithmetic racks 3 times in a week?  *gasp*  I was so excited, so you know the kids were amped!  Today we used the racks as function machines.  You put one number in the machine, the machine changes it by adding or subtracting, and a new number comes out.  This is always one of the most difficult (and frustrating) concepts of the year!  I have to admit, using the rack was a last minute brainstorm on my part, but I think it may have worked.  I'll have to see tomorrow when we revisit the concept!

Personally, I think the written function machine is very overwhelming to students the first time they see it, so I just got out our enormous arithmetic rack and starting posing situations to them:

I put 1 bead in the function machine.  2 beads came out.
I put 2 beads in the function machine.  3 beads came out.
I put 3 beads in the function machine.  4 beads came out.

By now, the students are starting to notice that I'm ending with 2, 3, 4, and can predict that next 5 will come out, then 6, then 7... so we back up. I went over each situation again and asked how the machine changed the number.  They discovered that the machine added 1.

We did this with a few other rules before moving on to the written version:

 

I apologize for part of the chart being covered up... I found the hooks for the rack after we'd already started charting and was excited not to have to hold it up any more!  This picture is showing 18 beads coming out of the machine.

By writing the in and out numbers down, it also allowed me to extend the activity and connect it to missing addends.  When doing function machines, the first several pairs of in and out numbers are given.  The students have to find the rule, which in this case is a missing addend.  We see that the out numbers are bigger than the in numbers, so the machine is adding beads.  Each time I wrote the in and out numbers in blue and once we discovered how the machine altered the in number, we filled in the missing addend into the number sentence (in green).  Missing addends are really tough, too, so maybe using the rack as a visual aid will help!

Next, we did some more problems at the rug, but the students had their own racks as well.  After this, students went to their seats to try to work more independently.  Instead of doing problems as a class, I let them try with partners.  Then we went over them together.  Here is an example from the document camera:

We are working on #1 "Your turn."

First, we got 10 beads ready to go into the machine.

 Next, we slid the 10 beads into the machine (sliding them to the right- acting it out helped some).

 The machine takes 3 beads away and the rest come out.  We can see that 7 beads come out.

Note: There are stars next to 9 and 11 because I asked the students if there was any way to predict what the answer would be before we started using those two numbers. 


Onto #2.

To find the rule, I presented the problem as a missing addend.  We had 5 beads and when we put them in the machine, 10 beads came out.  How many beads did the machine add?  We started with 5 beads and started adding beads until we had 10 total.  The color of the beads was helpful on this one!  Students could easily see that 5 beads had been added.

We then double-checked the rule with the other two pairs of numbers before finding the out number for 16 and then coming up with our own.

This is the 4th year that I have taught function machines, and I have to say that it was not at all frustrating.  Now, that is not to say that every students "got it," but every student participated and came away with some new level of understanding.  I love when I can file something away that has been successful!

Some high points of the lesson:
One student asked to make arithmetic racks again like we did earlier in the year (see my very first post)- I was so happy that it stuck out in their minds!

Another student was disappointed that there were only 2 rows on the arithmetic racks.  Never fear!  Soon we will be able to project our iPads and I have an arithmetic rack app that has more than 2 rows.  Score!

 

Ta-da!  It's the Number Rack app and it's FREE!!!