This is a math series follows our unit for introduction to fractions in grade 3. This is one of several, and I’m combining several days into one post. This part of the unit all involves number line.

## Day 3

My intention for our fraction work was to continue with our circles, but it was so beautiful outside today that we took our math outside to work on our fractions on a number line. We first drew our number line out two feet between each whole number using a ruler in one color and marked our whole numbers with a foot long mark out to the side. Then I asked her to use another color to half each one finding our half mark. These marks were 6 inches long, half of the foot mark. With another color, I then asked her to half each half finding our ¼ marks, which were delineated by a 3 inch tic on the line. In yet another different color we found the half of each ¼ finding our 1/8 marks which were just a short tics. These were as low as we went on this number line.

After we had all of our marks, we began counting by fractions, first by halves- ½, 2/2 (by the whole number “1”), 3/2, 4/2…and so on. Then we counted by ¼’s—1/4, 2/4, ¾, 4/4, and so forth. Finally we counted by 1/8’s in the same way. After we had counted out all of our marks, we noticed that some numbers had several numbers—equivalent fractions. We discussed this.

When we finished with this line, we began to write the 2^{nd} line. This time we divided the whole numbers into thirds on the number line and then divided the thirds into half, just like we did previously with our circles. (See this Post). Just like the previous line we started counting first by 1/3’s and then by 1/6’s and talked about the equivalent fractions on the number line. This mark was the same size, but could be expressed in different fractions or whole numbers.

## Day 7

### Review of Counting Fractions

Day 7 found us back outside. When we explored fractions with our circles, we explored first dividing them into fractions with the understanding that each fraction was part of a whole. We also went through a certain sequence of exploration. First we sorted and compared. Then we sequenced them, or counted by that fraction, i.e. 1/2, 2/2, 3/2… From here we explored equivalent fractions, and then we began our four processes first adding and subtracting, then multiplying and dividing. That sequence is the same exploration that I plan for the number line. It goes slightly faster as she is already familiar now with fractions from our circle explorations.

Today we began as we did with our first exploration of a number line on Day 3 above. We drew out a number line from 0 to 7. Then I had her half each of the numbers, and we counted as we marked each of the fractions: 1/2, 2/2, 3/2…

### Exploring the Four Processes- Addition, Subtraction, Multiplication, and Division

From here we began exploring the four processes. We just made up some expressions, using the number line she figured out what the solution was. We used the counting up method here. So for 3/2 + 7/2, she would choose 7/2 to stand on (we’ve used the counting on mental math method, so she knows to begin with the larger number,) and then count an addition 3/2 up coming to 10/2, which is easy to see from the number line that it was also 10. We did maybe 5 to 6 of these expressions.

After exploring addition we moved on to subtractions. The method is the same, but moving in the opposite direction when subtracting, and of course order matters here, and the first number is the number she started on.

After subtraction we worked on multiplication. This was done by “skip counting” the number of 1/2 we were multiplying by. For instance if you multiply 7 by 1/2, you would count 1/2 – 7 times to come up with 3 1/2 or 7/2. After doing several of these we explored division. This was accomplished by asking how many 1/2 fit inside a certain number. For 4 ÷ 1/2, she would count the number of 1/2 marks until she came to the number 4, ending up at 8. Doing this several times, she realized that the number would just be doubled.

When finished with halves, I had her 1/2 her 1/2’s and come up with 1/4 marks for the number line. We then counted – 1/4, 2/4, 3/4…- our fourths up the number line. We did a few addition and subtraction problems before I began to really loose her interest, so we stopped there for the day.

## Day 8

Moving our number line inside, we put Addition and Subtraction of fractions with like denominators into her notebook. We drew several number lines in the same way that we did outside, halving and halving again, counting the sequence for each fraction. Using our number lines, we began with 1/4’s and added a couple of expressions, beginning at the first fraction and then jumping the number of increments in whatever fraction was asked for in second number of the expression. For instance in the first expression pictured below, 1/2 + 1/2, she placed her colored pencil on the first 1/2 on the number line, and “jumped” another 1/2 to the 2/2 mark. In the third expression, 3/4 + 2/4, she began with her pencil on the 3/4 and jumped an addition 2 increments of 1/4 to the 5/4 or 1 ¼ mark. We color coded her expressions by coloring a box around the expression and using that same color to “jump” on her number line. From there we moved onto 1/4’s with a couple of addition problems. After a couple of problems with 1/4, we paused to see patterns, using this pattern to find the solution to the final expression using 1/8’s which was not marked our number line.

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