5.OA.A.1 Visualizing the Order of Operations with Snap Cubes

 My students struggled with order of operations, so I wanted find a way to help them visualize why the order of each operation was important. I had seen students use the pneumonic of PEMDAS—“Please excuse my dear Aunt Sally” or “Parentheses, Exponents, Multiply or Divide in order from left to right, and finally Add or Subtract in order from left to right,” but this made little sense to my 4^th grade students. “Who cares?” they asked. “Why is that so important?” It didn’t matter to them that the answers were different!

While reflecting on the day’s lesson that had not been as convincing to my students as I had hoped, I started playing with snap cubes and realized that I could show them each step. Maybe this would help them clarify their thinking about why multiplication and division had to be done before addition and subtraction.

I wrote problems on a sheet of paper and projected them on the screen. “How can I build 4?” I asked. “Show me on your desk.” The students responded and each connected 4 snap cubes on his or her table. “Now show me 3 times 3,” I continued. “And what does 2 times 2 mean? Then what if I divide the 2 times 2 in half? Now what does it look like?” The class followed along with me, constructing each step on their desks. 

“Alright,” I paused. “Now what if we combine each of our steps? What do we get when we add the 4 plus 3 times 3 plus 2 times 2 divided by 2?” The children carefully combined each step by adding  4 + 9 + 2 to get 15. Then I went back to the order of operations rule. “If we wanted to force the addition of 4 + 3 before multiplying by three, then we could use the parentheses,” I said and built it on the overhead for them to see. Suddenly, the students burst forth with an understanding, “Ohhhh!” I chuckled. “Okay, now you try one on your own."

I gave them 16 ÷ 2 + 3 x 2—the parentheses were added later when we debriefed the problem.

Here Rigoberto is finding 2 x 4 + 2 + 3 x 3. "That's 8 plus 2 plus 9," he stated triumphantly. "That's 19."

Several were starting to get it! I know I will need to review these concepts to keep them alive for the children, but at least we have a starting place from which to build.

4.MD.A.2 Pablo's Pizza Problem

4.MD.A.2 

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals....




“Proportional reasoning is one of the most fundamental topics in middle grades mathematics. Students’ ability to reason proportionally affects their understanding of fractions and measurement in elementary school, and it supports their understanding of functions and algebra in middle school and beyond. Given the importance of ratio and proportion, it is typical to see extensive class time devoted to the topic in upper elementary and middle school grades.” –NCTM, posted 3-11-2013


Alejandra tried to solve the pizza problem with 28 cubes. She was able to successfully divide it into two halves, but then was not able to cut the remaining 14 cubes into thirds for Cristal to get a 1/3 of the remaining pizza. 





Jason helped her discover that the problem could easily be solved with 24 cubes. “First, just divide it in half,” Jason said. “That’s 12 and 12. Pablo eats half and then gives 1/3 of the remaining pizza to Cristal. That’s 1/3 of 12, so it’s 4. And 3 times 4 is 12, so it works.”

I let them present their solution to the class and then asked the children, “What other numbers work for this pizza problem? And what would be a reasonable size for the pizza that Pablo and Alexis bought?” 

Galilea tried 36 cubes. She said that Pablo ate 18/36 of the pizza and gave 6/36 to Cristal, leaving 12/36 for Alexis. “The whole pizza has 36 pieces. That’s 36/36 or one whole,” she smiled.

 Alejandra, Patricio,and Rigoberto also tried 6 pieces of pizza and 24 pieces to represent the whole pizzas. They decided that 6 pieces was too small to share with 3 people--especially since Pablo ate half!

Cesar made two pizzas. The first one had 6 pieces. “Pablo ate ½ and gave one piece to Cristal,” he said. “That leaves 2/6 for Alexis and 
2/6 = 1/3.” I showed him that 1/3 of ½ = 1/6 and he added that to his paper.

Cesar’s second pizza had 24 pieces. After dividing the cubes in half and then cutting the remainder into 3 equal parts, he saw that Pablo must have given Cristal 1/6 or 4 pieces of pizza. At this point, several of the children realized that the total amount of pizza pieces had to be an even number and a multiple of 3. 

As a result, Eligio tried 66 cubes. He said that Pablo ate half and wrote ½ = 33/66. He divided up the remaining pizza into 3 equal parts and wrote that Cristal ate 11/66 or 1/6 of the pizza. I was fascinated with the numbers Eligio chose, because it was easy to see the same pattern that kept appearing around the classroom for several others:  ½ + 1/6 + 1/3 = 1 entire pizza. In Eligio’s case, though, you didn’t even need to factor out the elevens to see the same pattern of 3/6 + 1/6 + 2/6 = 6/6 = 1. 

We made a table to summarize our data.

 Doing all of this math made Pablo really hungry!  It was almost lunchtime at school, so he decided that the pizza in his story problem must have had at least 12 pieces—maybe even 18, since he had to give ½ of it away.  Alexis said that 4 pieces of pizza were plenty for him, so he decided that 12 pieces was a reasonable size of pizza to buy. “Besides,” he laughed, “it won’t cost so much.”

4.NF.A.2 The Best Candy Deal

For our Problem of the Day, I asked the children, “Which is a better deal? Two pieces of candy for 3 cents, or 3 pieces of candy for a nickel?”







Motivated by the idea of buying candy, the children eagerly jumped into the problem. I showed them how to organize their information on a table. The first row was labeled, “Assorted candy” and the second row was called, “Coins.” I helped them get started by showing them how to draw 2 candies for 3 pennies in the top chart, and 3 candies for a nickel on the bottom table.

Kimberly saw the fractions and had correct ratios, but did not know how to compare numerators to see that 6 candies for 9 cents was a better deal than 6 candies for 10 cents. 

The children enjoyed pretending that the pattern blocks were their assorted candies and several used protractors to help them trace circles that represented the coins.

Since some were struggling with correctly building the ratios, I told the students to pretend that the candies were sold in bags of 2 candies for 3 cents or bags of 3 candies for a nickel. This helped them make sense of the problem and better organize their ratios. 

Jessi helped his classmates see that 2 bags of 2 candies each (or 4 candies) would cost "3 cents + 3 cents." This helped some who were confused about how to use the table for organizing their information.

Finally, Ediverto was the only one who correctly label his work with the right answer. He wrote, "I can prove 9¢ for 6 candies is a better deal." 

Pleased with his efforts, I had him come up to the front of the class and share his discoveries. Ediverto beamed with great pleasure and did a little "victory dance" to show his delight.

He is a child with autism and never speaks a word, but he  is well accepted by the class and a brilliant mathematician.

Here Ediverto is showing the other students that "6 candies for 9 cents" is a better deal than paying "10 cents for 6 candies." The class clapped and cheered for him. We were all very proud of Ediverto!!

Below you can see the back side of his candy chart.

4.NF.A.1 The next day I followed up with a related activity and showed the children why 2/3 and 4/6 and 6/9 and 12/18 were equivalent fractions. Here are two different student samples. In each case, we multiplied by the big "1" and proved that multiplying a fraction by n/n may change the name of the fraction and the size of its parts--but not the total area.  2/3 times 2/2 = 4/6 and 4/6 is an equivalent fraction of 2/3.

  

4.NF.B.4c Hand Puppet Problem

In order to solve this hand puppet problem, I gave each of my 4th graders a sheet of 2-cm graph paper and 18 snap cubes. 

The cubes were arranged in sets of 6 and included 3 different colors to help each child connect with the fabric colors in the story problem.

 Immediately, the discussion began with “How much is 5/6^th of a yard?” After Eligio determined that he could represent this with 5 of the 6 snap cubes, then Fatima struggled with “How much is 3 times 5/6^th of a yard?”

Patricio said that  3 x 5/6  was 15/18, but Rigoberto disagreed. Rigo showed the class that “three times a number” was the same as adding it 3 times. Therefore,  
5/6 + 5/6 + 5/6 = 15/6.

Martha saw that each cube (1/6) was worth $2, because there were 6 cubes (6/6) in a yard and each yard cost $12.

Esmeralda wrote, “The girls need to buy 2 ½ yards of felt. It will cost $30. I know this for sure, because 1/6 = $2 and $2 x 5/6 = $10. So $10 x 3 yd. = $30 and that’s how I know.”  She confused the 3 times 5/6 with 3 yards, but was definitely making progress in her understanding. 

Patricio showed the class that he could move 2 cubes from the third 5/6 of a yard to make 2 complete yards, plus ½ a yard. “That equals 2 ½ yards of felt,” he beamed. “Each block costs $2 and 15 x 2 = 30. I got the 15 from each block.” It was obvious that he had clarified his earlier thinking that multiplying 5/6 by 3 was not the same as multiplying 5/6 by 3/3. He also realized that ½ a yard was $6 and not ½ a dollar or 50 cents.

The children agreed that problem solving was messy work, but they enjoyed the social aspect of working together and helping each other be successful in finding the solution. 

MP3 Math Disagreements

After reading the NCTM article, “3 Strategies for PromotingMath Disagreements” written by Angela Barlow and Michael McCrory, I decided to watch for opportunities to try these strategies in my third grade classroom.  The article encourages teachers to watch for differing opinions during classroom discussions and use this time as an opportunity for students to debate, prove, and deepen their thinking about a mathematical concept.  This is the perfect atmosphere for MP3 Construct viable arguments and critique the reasoning of others.

A few days later, my students were discussing why fractions have equal parts. I quickly realized that several were confused about how to turn our “Crusty the Clown” pattern block model into equal parts. The textbook and student workbook pages always neatly did this for the children, so they never had to think about forming equal parts of an irregular shape on their own.

As stated in the NCTM article, I asked students to choose a side of the classroom that represented their viewpoint. Immediately, two of my sharpest students went to the wrong side, declaring that Crusty had 5 equal parts. Jeffrey and Jorge were correct that part of Crusty was made up of 5 equal blocks, but they were not looking at the whole clown shape. Others who were unsure or undecided trusted the boys and stayed with them—even after much proof and evidence was offered by students on the other side of the room that Crusty had 14 equal parts. The children were delighted to debate and state their views with a microphone in hand. Emotions surged and intensified on both sides of the discussion; however, I stayed neutral and continued to encourage them to listen to each other and clearly state their opinions.

 After exchanging pattern blocks several times in an attempt to get equal parts, the students hit a deadlock.  I told them to take a breather and read library books for ten minutes to clear their thoughts. When we came back together, some children were more open to actually hear what others were saying, and at last cleared up the misconception about “What does it mean to have equal parts?” And "Where is the whole?"

Students realized that the entire shape had to be equally divided and that if you add area to the original shape you have just changed the whole. In the end, I told the both sides that I was extremely proud of them for standing their ground and methodically proving their ideas. The smaller group consisted primarily of soft-spoken Latinas. Although they were outnumbered by aggressive, dominant boys who knew they were right, I was delighted to watch the girls become more confident and show their evidence of why the green pattern blocks had to be used to evenly divide up the entire shape without redefining the whole.

14 equal parts of Crusty the Clown 

The next day, we used snap cubes to show “5 of 6 equal parts of 18,” and I noticed that the children were immediately engaged and knew how to do it. Several enjoyed making the connection back to dividing our pattern block Crusty into equal parts as well. Success!  I now have them beginning to listen to each other and starting to support their views with concrete evidence and simple proofs.

Source of the Math Disagreements article: Teaching Children Mathematics, v17 n9 p530-539 May 2011, NCTM

4 equal parts--but is the whole complete? 

6 equal parts--what happened to Crusty's hat? 

5 equal parts--but is it still Crusty the Clown?