We continued our work with understanding our place value system and how each place to the left is ten times bigger than the space on the right. This week we counted a huge pile of beans.
We started by putting ten beans in a small cup. When we had ten small cups filled, we poured them into a larger cup to hold 100 beans. Students worked together through this process over and over again. When we ran out of large cups we dumped them back into the original container and kept track of how many hundreds of beans we had with tally marks. We had so many beans that we got to see that ten 100’s is one thousand. I don’t remember how many beans it was exactly, but it was over 11,000 - they worked hard!
This may seem like a tedious exercise to adults, but kids need lots of experiences with
equating ten 10’s to one hundred, and ten 100’s to one thousand. They just need lots of
experiences of working with large numbers in general too. So while they had fun working
together, there was some important math behind what we were doing.
We reviewed our subtraction strategies of: removing BY a friendly number, removing TO a
friendly number, find the difference (the distance between the minuend and subtrahend),
and constant difference (shift the problem to nicer numbers). Then we added the standard
algorithm to our repertoire of subtraction strategies. The standard algorithm works with the
ones place first, works toward the left, and can involve regrouping (borrowing).
I offer the standard algorithm as one of many strategies to have, but I want kids to have all
the strategies - not just one favorite that they use all the time. We worked in our student
books solving each problem with two different strategies. While I don’t emphasize the
standard algorithm because it is not a sense-making, transparent process, it is quite efficient
when the problem does not require regrouping.
Once again, we used the base 10 blocks to build the concept of regrouping. The difference
being that in addition we regroup ten small units into one larger unit, and in subtraction we
break up a large unit into ten smaller pieces. We used the base ten pieces for developing a
conceptual understanding and then moved to the abstract, numerical written model.
We started with an individual assessment in class, but I quickly saw that we needed more guided and independent practice with finding common denominators. The assessment served its purpose, and we shifted gears to build proficiency with finding common denominators.
We reviewed using ratio tables to find common denominators.
We then used skip counting to find the lowest common denominator. These are similar
processes, but the ratio table allows for the flexibility to find multiples out of order.
Using the ratio table model not only helps us identify a common denominator, but also
produces the equivalent fraction. With the skip counting method, we still have to use the
Identity Property of Multiplication to create an equivalent fraction.
Continuing with the Ratios and Rates unit, we used ratios like 4 to 3 to find a single number
ratio (per 1 unit) of 1.33 (see section A). We looked at absolute comparisons and relative
comparisons. See page 13 for an example.
We also used a guitar to explore the ratios of string length to change pitches. This is based on the work of the Greek mathematician Pythagoras. We used a digital instrument tuner to “see” the different pitches. It’s all part of working with ratios.