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Do Your Kids Need a Dawning Division Moment?
After sitting in a week’s worth of meetings about Common Core instruction and best math practices, I feel as if my way of instructing students is gradually morphing into a new beast, which I am not sure I fully can describe yet. In the midst of all of this, I went back to the fifth graders I have been working with on the day after these meetings. I decided to try out some of the teacher practices I had been learning. Sidenote: This set of fifth graders isn’t yet proficient at long division or ladder division for that matter. I posed a long division problem to the students, but instead of giving it to them in “naked numbers” alone, I gave it to them in word problem form as well. In reflecting, I should have started out with the word problem alone and let the students figure out that they were supposed to divide. After watching the students struggle with solving using ladder division or the traditional long division algorithm, I encouraged the students make equal groups to see if that helped them. One student in particular that I worked closely with (whose work is pictured below) managed to find a correct answer using equal grouping, however he could not find his mistake when he used ladder division. Another student…one I might add who typically doesn’t have much perseverance to solve problems…related completely to the contextual situation and persevered through to solve the problem to the end. To get to my MAJOR DIVISION REVELATION, let’s look at this student work below:
One student's incorrect ladder division and solving with equal grouping correctly.
The same student's equal grouping and the relationship to the correct ladder division which I showed him. Notice how 20, 30, 10, and 5 are the same in the circles and to the side of the division problem.
So, you ask, what is the big AH-HA? If you notice that the numbers in the student’s circles (equal groups) are the same numbers to the side of the division problem (or the ladder). The numbers to the side of the ladder are actually the chunks that students would naturally break off if they were naturally dividing cookies among people etc.! Now I know according to my PD classes that I shouldn’t have just told and shown the student this relationship, but I should have allowed him to figure it out. I was just a little too excited to hold in my own personal discovery I suppose.
Another discovery I had in experimenting with having students discover their own strategies, I learned that students actually do better and persevere more when the problem is in a context familiar to them and it isn’t just a naked numbers problem. Having a contextual situation actually gives the students an entry point into the problem, and so they don’t give up as quickly. I have always thought backwards. Students MUST UNDERSTAND THE NAKED NUMBERS FIRST and THEN they can SOLVE HARD WORD PROBLEMS. But in reality after my experiment with these fifth graders, they were much better at solving this division problem in their own way when they had a context. I have had a complete paradigm shift, and you were all here to see it above– that is.
My New Singapore Math Manipulatives!
I get the luxury of ordering new math materials for my school. To prepare for all of the common core instruction in place value, fractions, and decimals, I ordered lots of fraction circles, some more unifix cubes, and some place value strips. When my order came in this week, it was like Christmas! That is always the feeling I have anytime I rip open boxes of new math materials! I am most excited about the place value strips which are pictured below. As you can see in the pictures, they come apart so that students actually see the value of the number. These are one of the Singapore Math manipulatives. Each color matches the Singapore math discs if you decide to get those as well which I use with 1st, 2nd, and 3rd grade. If you want to have a smaller size, there is also a copy, cut and laminate version in one of Sandra Chen’s books, The Parent Connection for Singapore Math.
Place Value Strips
Place Value Strips Showing the Value of the Digit
Decimal Strips
Decimal Strips Showing the Value of the Digit
Free Test Taking Rubric or Checklist
I showed the fifth graders that I have been teaching for the past few weeks this page before they tested. I let them know that I was going to be looking for these actions or test taking strategies while they tested. Our principal gave the students extra recess time at the end of the day if they worked hard on the test all morning long. I wanted a way to measure “working hard on the test”, so I used this checklist/rubric. If students did 4 of the 6 actions or testing strategies listed on the sheet, then they were able to have extra recess. Across the top of the page the categories read:
- Underlined Key Words
- Brain Dumped– Writing important information down on the math reference sheet that they may forget
- Eliminated Wrong Answers (on multiple choice)
- Used P.E.C.E (an acronym that stands for using a picture, equation, complete sentence, and elaboration to solve an open response)
- Persevered When Problem Solving
- Checked Work or Used the Entire Time to Work
If you would like to use this form, you can download it for free here. I am posting it in Word format so that you can open it and change the wording to suit your needs.
Testing Rubric Checklist
Do You Need a Fun Way for Students to Experience Perimeter, Area, or Volume?
Students built the following houses out of food items and then calculated the perimeter of them. We did this project as a relaxing activity after testing. I allowed students to build houses out of graham crackers, frosting, red hots, marshmallows, and Smarties. Then they used a measurement tool to calculate the perimeter in centimeters. If I had really wanted to use this activity to stimulate mathematical thinking, I would have had the students calculate the surface area and the volume using large marshmallows. Since I just wanted the kids to take a fun break after testing, I didn’t have them calculate anything except the perimeter. If I had to pick their favorite after testing activity from this week, this would have to be it!
One of the most creative houses the kids built. If you look on the napkin, you can see the perimeter calculations.
One student discovered an array with her Smartie shingles!
Measuring perimeter of the graham cracker house with a base ten rod. (each unit is a cm)
Yummy, Fun, Math with Circles!
I did this today to reward students who had put forth their best effort on our state tests. Originally, I had just planned on bringing cookies, but we had just been studying the different parts of a circle. Cookies being round just naturally lent themselves to be used for this spontaneous activity. I bought some Twizzlers and told students to use those to separate the strands to use for the chord, radius, and diameter. A spoonful of heaping icing was distributed to each child to help all of the parts stick. Students got red hots to use for the center of the circle. According to the common core standards, unfortunately, the parts of a circle will no longer be in the curriculum for fifth graders after I came up with this nifty activity 
. Maybe a teacher somewhere out there that teaches middle school students or older could benefit from this in the future. Below are pictured the circles with their Twizzler radius, diameter, and chord. In reflection, if I were to do this over I wouldn’t have bought Twizzlers. I would have bought those individual licorice colors that are single strands. I saw a package at Walgreens after I had already bought the Twizzlers. I think those would work better because they are already single strands AND I could have differentiated what I wanted students to use for each color. For example, make your chord red, your radius blue, and your diameter yellow licorice. If students did this, then I would really know that they could distinguish between the parts. All in all, the students ATE THIS UP–literally!
A student who followed directions and separated the strands of licorice.
And a student who didn't quite follow directions with the Twizzler strands still stuck together, but recognized the parts of the circle nonetheless. These cookies were a little crumbly from their trip to school. Hence, the imperfect circle shape.
How Can You Use Pattern Blocks to Simplify Fractions?
Use pattern blocks to help students find equivalent fractions. Students simply take the blocks and trade them in for larger and larger blocks until they can not use any larger blocks to make the same shape. See below for some examples. The first row shows the blocks as fractions of 1 hexagon.
To make using hexagons for fractions more engaging, call them cookies since they are yellow and the size of a cookie. There is a TERC math investigation lesson called “Hexagon Cookies” which is in the Fair Shares book (for 3rd grade). Hexagon Cookies makes a great lesson to teach previous to simplifying fractions with the pattern blocks.
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