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Modeling Multiplication and Division with Watermelon Plates

I live for classroom hacks! I enjoy discovering new ways to deliver math instruction. Recently, I found a fantastic package of watermelon plates at Wal-mart. I was attracted to the seeds in the middle of each plate. The seeds are great for counting in primary grades, but kids can also use them for lessons highlighting the importance of multiplying and dividing.

Items Needed:

Watermelon plates
Counting Cards
Sharpies
Paper or Journal
Pencil

The Hook:

How can watermelons help us learn about multiplication and division?

After passing out plates to students, tell them you need to know how many seeds are "in the watermelons." Next, ask students "How can we find the total number of seeds?" Students can answer this question in journals or discuss possible responses with partners, small groups, or in a whole group format. Based on responses, extend the conversation by asking them to choose the fastest method.

Watermelon Work:

Distribute "counting cards." The cards should indicate the number of seeds students should group together. For example: "Circle groups of 3 seeds." Another option is the "A, B, C, D, E." Assign students groups based on the letter they call in a whole group format,

Recommended groupings: two seeds through seven seeds.

Student asked to group seeds by five.

After students group seeds, ask them how the groupings can help them count faster. Students should be able to respond that they can count the number of groups and multiply by the number of seeds in each group. There may be seeds left over. Students can add the remaining seeds to the product. Make sure students write the multiplication expressions in their journals or worksheets. Extend this part of the lesson by allowing students to discuss how and why different groupings resulted in various multiplication expressions.

Wide Divide:

After students determine the number of watermelon seeds on each plate, discuss how the groupings also represent the inverse of multiplication (division). First, ask students how the groupings model division. Seek input and responses from students. The goal: students realize that the total number of seeds divided by the number of seeds in each group results in the quotient (the number of groupings created with a possible remainder).

Beyond the Plates:

Watermelon is healthy, nutritious, and TASTY! If possible, offer watermelon cubes or slices to bring a little bit of the "real-world" into this delicious summer math lesson.

Happy Hacking!

Error Analysis in Mathematics

It will happen. Students will make mistakes when solving math inquiries. In the past, these errors were met with red ink and low grades, leaving generations of students feeling as if math was a bottomless pit they could never escape. Now, errors can serve as learning opportunities students can use to master mathematical concepts.

Why error analysis?

Error analysis is the process of reviewing solutions to math inquiries, identifying mistakes, explaining why they were made, and making corrections. For example, if a student completes 20 problems focused on multiplying rational numbers, and two of the answers are incorrect, did the student make calculation errors or does the student have an issue understanding the proper algorithm? Would your response change if the student answers 15 out of 20 incorrectly? 

While it is important for educators to understand the types of mistakes students make, there is a crucial element missing. Students also need to learn how to identify and correct mistakes. The most effective types of error analyses focus on 5 common mathematical errors: conceptual, operations, reading comprehension, procedural, and silly (EA CORPS).

Conceptual errors

Conceptual errors arise from the failure to recognize, understand, or connect math ideas and principles. Indicators of conceptual errors include errors based on definitions of math concepts or repeated procedural or calculation errors. A student who identifies a four-sided curved figure as a quadrilateral may not recognize that quadrilaterals are formed by four straight line segments. Conceptual errors are the hardest to address and require planning and consistent practice to overcome. 

Operations errors

Operations errors - also called calculation errors - can be a major roadblock to students trying to master mathematics. Addition, subtraction, multiplication, and division are at the core of the subject, and mistakes here can cause major frustrations. Many times, operations errors are minor, and students can prevent them by checking their work and taking additional time when solving problems. Repeated operations errors may indicate issues with conceptual or procedural understanding.

Reading comprehension errors

Some would argue that math has become "reading comprehension with numbers." Indeed, mathematics is a language. Therefore, it should come as no surprise that reading comprehension is a major issue in error analysis. This type of error occurs when a student misreads or misunderstands the text. It can stem from a lack of focus, rushing, or the inability to analyze text. Students understand the basic concept and apply correct procedures. There are also no calculation errors. Review the example below:

There are 5 cars and 3 trucks in the parking lot. Write the ratio of trucks to cars in three different ways.  

Jon's response: 5 to 3, 5 : 3, and 5 / 3 

Jon's response shows that he understands the concepts of ratios and how to write them. Jon's mistake stems from failing to order the objects being compared based on the prompt. Students can avoid reading comprehension pitfalls by highlight key words, close reading, using context clues, and using graphic organizers. 

Procedural errors

Procedural errors are mistakes in processes when solving math problems. Errors include incorrect operations, incorrect algorithms, and incorrect or missing steps. This type of error can manifest when students work with the order of operations, for example. Some students will perform multiplication before division and addition before subtraction despite the order of the operations (left to right). Procedural errors also arise when students learn "shortcuts" before understanding concepts. They rely on shortcuts and fail to develop the conceptual or procedural fluency needed to master concepts.

Silly errors

Silly errors are not really silly, but they are the easiest types of error to fix. Many silly error excuses are easy to identify:

"I ran out of time."
"I thought the answer was right."
"I changed my answer."
"I didn't pay attention."

There are two important ways students can combat silly errors: focus and confidence. Focusing on problems, one at a time, can alleviate problems caused by failing to pay attention. Practice builds confidence, which is needed when nervous guts tell students to change the answers, but clear minds know the first answer is correct.

Join the EA CORP!

We're looking for a multitude of great teachers and students who are ready to improve classroom environments by embracing mistakes and not condemning students for making them. There's nothing to sign and no phone calls to make. Just commit to using error analysis as a tool for growth in math classrooms. Want a way to get started? Download the guide students can use to analyze errors in class. It's perfect for back-to-school and useful all year round.

Click here for student error analysis guide. 


Join the conversation on Twitter using #eaCORP.

Further reading:

Simple Activities can Help Reveal Students' Issues with Math Concepts

Is Reading Comprehension the Real Problem with Math?