I wanted students to exercise some mental math sense today so I played a working backwards game called "What's my Number." We are working on Math Practice Standard 1 and 2, by solving a problem and thinking it through.It's a great time and kids love it. It is really simple for most students if I keep the numbers reasonably small.
It goes like this:
I said: I am thinking of a number, when I add 6 and divide it by 3, my answer is 4. What number am I?
Students thought for a minute. I saw people get pencil and paper out and I encouraged them to remember how we solved these in the past, using a start, change, change result strategy. I encouraged students to do it mentally without pencil and paper.
One student raised his hand and said "12". I asked him to explain his answer. He explained that he started with 4 ( his start) then he reversed the operations ( change, change) So he "timesed" 4 by 3 and then subtracted 6 and got 12...then he said "Whoops, I meant 6."
We tried another one: I am thinking of a number, when I add 17 and subtract 30, I get three. The double digits seemed to challenge them a little bit. And having to subtract 17 from 33 was mentally challenging. Fewer people raised their hands this time. One student answered 14.
I stepped them through it using a Start, Change, Change Result strategy.
Start: n + 17 -30 = 3. So, 3 + 30 - 17 = n. I talked it through out loud, "33 -17 is what?"
Soon one boy piped up and said 16.
I plan on doing this game more frequently using subtraction because mental subtraction is difficut for them.
I wanted to head off any questions about how to create a numberline with fractions and be sure that students understood what the denominator means. I drew a numberline on the board and labeled it 1/2. I asked them how many parts the number line would be divided into equally? They quickly answered "2", and I think I heard someone say "duh". So, I got a little more complicated and didn't bother to mark the numberline with 1/2. I moved on to 7/8 to be a little more challenging.
I asked the same question and everyone said "8". I asked how I could go about dividing up this blank numberline in 8 parts. No one knew. I wanted to reinforce Math Practice Standard 6 as we work to attend to precision through understanding the common mistake.
I took them through the division process by referring them back to their understanding of benchmark fractions. I did this because the standard expects students to know where benchmark fractions are and therefore connect new concepts about fractions to that.
So, I told them that our benchmark fraction is 1/2. I drew the 1/2 in this time by drawing a line right in the middle of the horizontal line. I asked how we should divide the numberline again, and they told me that we could divide the 1/2 in half. So, I did. I handed the pen to another student and asked them to draw 8ths. He did perfectly. I handed the pen to another student and said, now shade the line to 7 of the eight parts. I turned to my students as he did this and asked, what does the 7 represent?
Right away, I could tell they understood because they were anxious to tell me that the numerator told us that was that many parts that we had.
So, I concluded, the denominator is...A student finished my sentence: " How many parts the whole is divided up into."
And the numerator shows: " How many we got!" was another's reply.
I told them that the common mistake was that student didn't divide up the parts equally and showed them on another drawing how 2/3 would look if each part was not divided equally. I told them that as we learn to compare fractions on a numberline, accuracy in their drawing was essential.
This SB Example of Numberline Lesson shows how I used the SB while students had grid paper to draw on during the whole class instruction. I simply brought up the SB and searched grid paper and then slid it over to the pages. It makes a great practice board. I drew two lines on top of one another. We worked through the concept of lining the lines right on top of one another so it was easier to compare. As we plotted out fractions together on the lines, I really stressed how we were counting spaces and not lines.
It was time to independently practice. I turned to the last page of the SB file. Using interactive dice to create fractions, we rolled twice and started to compare the fractions we created. I told them that the numerator had to be less than the denominator. I told them that if they were comfortable with plotting improper fractions, they could do that.
I roved the classroom as they worked, looking at work. I saw that they all were drawing the same size whole and working with the graph paper. I stopped at one student's desk to notice that his parts were not equal. He redrew the pieces and I emphasized the accuracy. It is hard to divide up the line for some, so I showed him the tricks of using the benchmark fraction of 1/2 once again. He was proud that he figured out that if the denominator was even, 1/2 was the "go to" fraction to help us divide it up. I was pleased to see his thinking, but couldn't resist getting him to think farther. Common Core expects that we push kids to think as we satisfy not only the standard, but the Math Practice standards as well.
I asked: What if the denominator is odd? I showed him how to divide thirds up (again) by finding 1/2 and then "eyeballing" the division, checking to see that I wasn't too far from 1/2 and that all pieces were even. Then we could divide those in half to make 6ths. He was happy! Explaining his understanding shows the enthusiam and understanding about he just learned. This was a student who grasped the fractional model method 100%, but struggled with the number line!
I could see that after about 20 minutes students were able to create their number lines without a problem.
I assigned 5 problems for them to show using a number line. I gave them the choice of paper. One student asked if he could use technology using Educreations and the grid paper. I told him, sure! I was excited that he thought of it!
1. 4/8 O 3/4 2. 7/9 O 4/10 3. 3/6 O 610 4. 8/10O 3/12 5. 6/8 O 9/10
I think that if they cannot show their understanding to me in 5 problems, I will need to remediate.
I told them that they could work 20 minutes on IXL.com Level F Q.6 if they would like more practice using different models. I like IXL because I have a membership and I can monitor progress through the records I am provided with on the site. It is internationally used, so CCSS are only a part of it and you have to set it to your state and then Common Core. OR they could work "Oh No! Fractions" and use the models to prove their answer.