## Challenging Word Problems

# Advanced Model Drawing Virtual Class Homework #6

Try a bar model and post in the comment section below.

Try a bar model and post in the comment section below.

Try a bar model and post in the comment section below.

Try a model and share in the comment section below.

Dear Sarah,

I am dying with my 5^{th} grade…they are struggling with fractions (the chapter is hard) and with not a strong Singapore foundation from last year…well…we are on our 4^{th} week! Should I do the 4^{th} grade fractions chapter?? I honestly am at a loss.

Thanks

Math facts are simply the basics: addition, subtraction; multiplication; division. They are basic number combinations and calculations we do every day. So why is it that learning math facts creates such huge problems for teachers and students in our classrooms?

Is it that some of us are just naturally better at math than others? Perhaps. The good news is you as a teacher can help anyone improve their math skills. You might be wondering how. Well, rote memorization alone won’t get us there.

Often we teach the meaning of equality as the equal sign indicating “put the answer.” To prepare students for future mathematics, we need to rethink the way we communicate this representation to students. The equal sign is too important to attach such a limited meaning, especially when students are moving to abstraction. When moving to abstraction, it is important to use this symbol as a “reader” versus a “speller.” What do I mean by that? Let me explain. First, a speller sounds foreign because the student is trying to make sense of abstract symbols that they don’t fully understand. A reader connects meaning that they can comprehend while connecting the story to the symbol.

For example, when teachers record the following equation,

A speller would say, “Four plus five equals nine”.

A reader would say, “Four apples and five apples is the same as saying ‘I have nine apples’” .

In the journal prompt below, we want students to see the statement as 5 and 4 is** the same as** 2 less than 11. If we focus too much as a speller, students see symbols as performing actions rather than relating ideas, which leads to misconceptions of the equal sign.

Less is more. Hesitate in thinking math becomes harder by creating larger numbers, rather provide depth early on using children’s intuitive understanding of basic number operations. A simple journal entry can help assess what students understand about operations and the meaning of equality. Once you find that a student understands the meaning of addition and subtraction, it is time to create tasks that dive deeper in understanding equality. These task can be as simple as asking questions such as, 4 + 5 = ____ + 6. Additional equality tasks can be found in the Algebraic Thinking and Reasoning with Numbers books (Groundwork Series) by Greenes, Carole, et al., (https://www.mheducation.com/prek-12/program/MKTSP-O8302M06.html).

In the book, “Thinking Mathematically,” Carpenter, Thomas P., et al. suggest some benchmarks to keep in mind while moving towards the conceptual understanding of equality.

- Establish a starting point. What are students’ initial ideas regarding the equal sign?
- Mix it up. Don’t always write equations in the form
*a + b = c*, rather*c = b + a*. - See the equal sign as a relational symbol. Emphasize students proving that each side is the same as the other.
- Compare sides without calculations. Encourage students to look for relationships without performing calculations.

Listen to students and be aware of where they are in the process. Encourage them to question if it is true. If so, why is it true? By listening and looking we may be pleasantly surprised at what we find out.

Below is student work demonstrating various levels of understanding within the concept of equality.

*Spiky says that 5 + 4 = 11 – 2 is false.*

*Curly says that it is true.*

*Who do you agree with and why?*

Many teachers ask how to help students record the renaming process in the addition algorithm. What language should I use?

I learned to “make trades” or “borrow” when adding and subtracting that require renaming. Is this still correct?

Teachers do not have time to prepare multiple tasks and lessons to meet the needs of each student in the classroom. The key is giving an anchor task that all students can enter and knowing how to adjust the task based on our observations.

In this class, students were guided to use “Katell’s” method (aka-left to right strategy) to solve the problem 65 – 12. Notice how the student recorded their thinking process.

During the guided structure component of the lesson, teachers are observing and questioning students. This student demonstrates and can explain the left to right strategy. What comes next for this student? During our planning phase, we need to ask ourselves, “What can I do for the student that already knows the answer”? Looking at this student’s work, how can we deepen the mathematical understanding?

After listening to the student explain his thinking, the teacher commented, I can understand your verbal explanation, but your written work is confusing to me. I wonder why I am confused? The teacher then walked away, and the student grappled with the idea of equality. Differentiation is meeting the student where they are at and extending the learning.

Do not try to create more problems for students to practice. Plan your task thinking of what mistakes students typically make and how you can help extend the thinking. Use your observations to go deeper and differentiate!

What are other ways you could take the task 65 – 12 and extending the task? Share your thought below.

Too often when we pose a problem and students shout out an answer. We need to ask ourselves do students understand the concept or are they obtaining the correct answer by fitting the symbols and numbers into a structure they know?

In this example, the anchor task was to subtract 34 from 87. The teacher wanted to screen the children first wondering, *Do students know what it means to subtract one number from another?* To find out more the teacher posted the following problem, removing the numbers.

Students were asked to set up the expression that could represent the situation. The student A on the far right was the only student in the class that seemed to understand. When asked to explain his thinking, many observing teachers felt he understood the concept and that he gave us a platform to generate a discussion.

Following this analysis, the numbers were inserted into the problem. Subtract 34 from 87. It was interesting to see student A’s work. Much to our surprise, Student A who seemed to understand the concept had a hard time transferring that knowledge to another situation. Notice his equation is not correct but he gets the final answer.

**Less is more.** Spend more time on conceptual understanding and listening and watching students versus solving more problems. The answer does not help assess student reasoning or how we can extend or guide the learning process.

Participants were asked at the 2017 Fall Mathodology Institute if it was possible to use the digits 0-6 only once to make a true statement?

Is it possible? Participants worked and worked and did not find a way to solve the problem with the given conditions. The teacher allowed the task to move on without closure for a purpose. When students work to become confident problem solvers we want them stating, “this is not possible, I have exhausted every situation.” Listen to their reasoning, and look for a systematic way to prove it is not possible. All too often we focus on finding the solution, and we can deepen the process and allow for practice when we alter the task.

Try giving a task where it is not possible and see how your students handle it. Follow-up by asking, “if it is not possible, can we change one of the symbols to make it a true statement?”

When we do not provide immediate closure, we allow students to continue to explore. Many participants emailed after the institute because they would not quit until they figured it out. Below are a few of the pictures they sent.

Great perseverance by participants who refused to give up?

This is what we want from students…next question for Shelly…can I change any symbol to make it a true statement? If so is there only one way?

Ok…..I’ve worked literally for hours and I’ve come to the conclusion that this problem is NOT possible. If the lesson was to teach us to let kids struggle so they work harder like a 🦋 I get it. I’ve done at least 100 fraction problems over the weekend. If you would’ve given us the answer I would’ve never thought about this problem again……but you’ve let us struggle and now it’s just painful. Please tell me it’s not possible my sanity is at stake.

-Shelley

Students using the think!Mathematics program reinforce the count on strategy. Games/activities can be used during the guided practice part of a lesson. Students are encouraged to count on from the larger number.

Use of number line to show rounding. In each case what place is the student rounding to? How do you know?

My students are really struggling with multiplying and dividing with powers of 10 (especially division).

Do you guys have any suggestions for reinforcement activities/anchor tasks?

My students are really struggling with multiplying and dividing with powers of 10 (especially division).

Do you guys have any suggestions for reinforcement activities/anchor tasks?

A journal entry from a grade 1 student. This student can draw it, write an equation, and give a number bond for the given task.

How might you further assess this student?

Give two questions you can ask this student to extend the thinking that is recorded here?

Can you see the CPA progression? Modeling with actual items in making the lesson come to life first. Then move to pictorial representation with pictures of the objects. Last student model with cubes and record in a number bond.

Modeling with actual items in making the lesson come to life first. Then move to pictorial representation with pictures of the objects. Last student model with cubes and record in a number bond.

Then move to pictorial representation with pictures of the objects. Last student model with cubes and record in a number bond.

Lastly, student model with cubes and record in a number bond.

Hi Sarah,

Your name came up during today’s fourth grade math planning meeting, as all three of the teachers attended the Mathodology conference, this past summer. We were discussing math apps to practice the multiplication table, and they said you recommended Reflex, which I’m not familiar with. Would you be able to help answer questions for an individual use?

Can you picture spending several classroom periods on the number 5? Specifically, breaking it down into parts, and putting it back together again? Sounds like a lot of time, right? But, spending this amount of classroom time to decompose numbers using number bonds, allows students to gain a deeper, more flexible understanding of numbers. You might be wondering what those periods would actually look like, how you’d keep your students engaged and learning for that amount of time. In this blog, we will explore exactly that.

Simply put, number bonds are the different ways we can break apart numbers. Number bonds are all about the relationship between numbers and quantities. The relationships of parts to a whole. Building the foundation for all mathematical operations. Building mental images of number relationships. While it sounds like a simple concept, it can be difficult for students to learn. So, it is important to dedicate time for them to learn the concept.

Who has the largest number? Students use cards to compare different representations of a number.

Number cards from mathodology coming soon!