I hope you are all healthy, happy, loved, and well-rested.

I have been fortunate these last few weeks to work with Ms. Morales and Mrs. Aguirre. They have shown me how to slow down and be more explicit when explaining my thinking and teaching process to others. What seems so clear in my mind has not always been made as clear to others. Thank you ladies for your questions, feedback, and tips!

This week, I want to share my thoughts on using small numbers to teach big ideas. I am very aware of the standards that students must meet to be proficient and advanced proficient in mathematics. Therefore, I need a minimum of 10 lessons on one topic to help them grow from beginning to advanced strategies. We begin by using very small numbers, then we advanced to more challenging numbers every day.

Setting High Expectations

I have high expectations for students. I set high expectations because I know students will reach a little higher to meet them and develop their self-confidence and skill set along the way.

On day one of any new lesson or when I demonstrate in a classroom, I always begin by posing a problem using small and simple numbers. Using small numbers in a problem allows me to assess students’ needs:

• 1) do the students understand the basic concept of the problem
• 2) what mathematical practices do students utilize when solving problems,
• 3) what mathematical connections to teach,
• 4) what tools do I need to introduce,
• 5) what interventions do students need,
• 6) what concepts need revisiting.

In my opinion, supplying an answer to a problem or writing an equation is not enough. I expect students to apply the Standards for Mathematical Practice to their work every day. By teaching students to use the Standards for Mathematical Practice daily, I can ensure that all students will have an in-depth understanding of the mathematical concepts, develop confidence, explain their thinking, develop and use reasoning, and be prepared for state assessments.
I can only teach these big ideas when I begin with small numbers.

Standard for Mathematical Practice #6: Attend to precision.

Array or Disarray?

I know that disorganization leads to silly mistakes. Organization is one of the first skills I teach students to utilize when they begin problem-solving. I ask students to look at the two options I have written on the board:
Option A and Option B. I ask them which way of setting up their tools or drawings will lead to more accurate results. Option A- disarray or Option B- an array?

Students always choose Option B. I then ask them to look at their tools or drawings. Are they in an array or in disarray? Students easily refine their drawings and tools because they only have a small amount to reorganize. Refining or realigning 9 objects is far easier than 90 objects. By the time students begin using large numbers, they will have developed the habit of organizing their tools in arrays or organized lines.

Standard for Mathematical Practice # 8: Look for and express regularity and repeated reasoning.

Organized mathematical models can help students see and use repeated reasoning to solve problems that involve large numbers.
If you look at the student’s work below, you can see they have chosen a large number to solve and, based on the existing model, it may require a lot of time and effort to solve the problem, and errors may occur. However, their organization can help them solve more accurately and efficiently.

I would ask this student, “How much is in the first column going down?”
(If all of the pieces are together, it totals 2 1/4).
“If the first column is 21/4, what will the second column equal, the third column, and the fourth?”
Organization is a lifelong skill. A student’s ability and willingness to organize their work will serve them far better in the long run than quickly calculating an answer.

Standard for Mathematical Practice #7: Look for and make use of structure.

Relational Thinking

Relationships are easier to see with small numbers. One skill that students must learn is the relationship between addition and subtraction. Teachers often tell me they give students fact family problems to solve. While I see students solve fact families, I rarely see them utilize fact families to solve problems.

Imagine asking students to solve the problem: 11 – 8 = __ on a number line. Young and older children may begin at the number 11 and go back and subtract “8” landing on number 3. Now imagine asking them to find the difference or space between 8 and 11 on a number line. They might begin with 8 and jump 3 times to get to 11. The answer again is 3. We can test out the relationship between counting down to subtract and counting on to subtract several times if we use small numbers. Students can test the relationship out for themselves, confidently arriving at the same answer when we introduce the concept with small numbers.

Then… we can ask students if the same approach will work with large numbers.

642 – 597 = __

Stand back and watch logic and reasoning develop.

Standard for Mathematical Practice #4: Model with mathematics.

Mathematical Models, Equations, Number Lines, and Ratio Tables

Mathematicians verify their results before submitting them.
“How did you verify your work is accurate?”
Is a mantra I am repeating in grades K – 6th this year. Do not submit your work unless you have verified your results with another model, tool, or labels.

The expectations are clearly stated, repeated, and modeled. (Yes, I model expectations!)
It is now the student’s responsibility to meet those standards because I consistently modeled over the course of ten lessons, how to apply big ideas using small numbers.

You know I love hearing from you!