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How I Lay Out A Lesson

As always, I hope this email finds you happy, healthy, and loved.

I was listening to an interior designer explain how he designed and decorated homes the other day and I got a little frustrated because he described general ideas as opposed to specifics. I felt like he left out a lot of little, yet important details someone would need to create similar results. I think that when people do something on the everyday basis for years, they can take for granted what they do, but it can leave the rest of us wondering what next. I have been planning standards-based mathematics lessons without the use of a textbook for almost 10 years now. It is something I enjoy and a process I probably take for granted. I thought it would be a good idea for me to slow down and unpack the process I use multiple times a day to plan for multiple grade levels.

Always start with your standards.

Honestly, the best part about the Common Core standards is that they are actually very informative and specific. In addition, a word problem can address multiple standards depending on the wording and the units.

I looked through the second grade standards and selected 3-4 standards that can be addressed with one word problem.

(Table 2: Equal Groups Problem)
James is sorting the coins that he found in a jar. So far he found ____ nickels. How much money does he have?

Represent and solve problems involving addition and subtraction.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

Count within 1000; skip-count by 5s, 10s, and 100s.

Students can solve this problem using tally marks, repeated addition, or multiplication. The students can draw a model of the problem i.e., circles representing the coins or tally marks. They can write equations that match how they counted to find the total i.e., 5 + 5 + 5 or 3 x 5.

To differentiate nickels can be changed to pennies, dimes, quarters or 100 dollar bills.
To increase the rigor a second step can be added to the problem.

James is sorting the coins he found in the jar. So far he found ___ nickels.
__ of the nickels are damaged and cannot be used. How much money does James have now?

To increase the rigor even more, differentiate the instruction, and connect multiple mathematical representations, students can be asked to use an alternative model to represent the problem(s).

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

James is sorting the coins that he found in a jar. So far he found ____ nickels. How much money does he have?

Represent how James might have counted the amount of money he found using a number.


Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems1 using information presented in a bar graph.

James found ___ nickels in a jar. His sister Meghan found ___ nickels. Create a graph to represent how much they each found. Write an equation to compare the difference.

In the above problem students would have the opportunity to skip count by 5’s, practice creating graphs, compare quantities, and write equations using missing addends (0.15 + ___ = 0.30)  or subtraction (0.30 – 0.15 = ___).

I will stay with this line of problems for multiple days so that students have multiple opportunities two work with a number of different monetary units, practice getting better at number lines, equations, and graph. I expect their to be mistakes that can  be corrected with practice,  familiarity with money will improve, the accuracy of graphs will get better with time, and flexibility with equations will improve.

Using language to pose problems that enable students to clearly envision the situation will allow them to make meaning and make reasonable decisions when problem solving.

(Multiplicative Comparison Problem TypeAlex lives in a one story house that is 20 feet tall. John lives in a an apartment building that is 10 times as tall. How tall is John’s apartment building?

(15, 12) (21, 17) 

Depending on the numbers you choose and your follow-up questions you can address multiple additional standards:
Round your answer to the nearest (hundred, thousand, etc.) 
Create a graph to represent the difference in the heights between Alex and John’s homes. 
Architects and contractors often represent the height of buildings in inches and feet. Create an input/output table or ratio table to convert inches to feet. Represent the height of their homes in inches and feet.

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1

Generalize place value understanding for multi-digit whole numbers.

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Use place value understanding to round multi-digit whole numbers to any place.

Solve problems involving measurement and conversion of measurements.

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …

Generate and analyze patterns.

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

The above problems are just a couple of examples of how I use the standards to extend students’ thinking. The number choices can be altered to address a variety of student needs and/or standards.

The number choices can be altered to address a variety of student needs and/or standards. Through your follow-up questions and in the ways students represent the solutions to the problems multiple standards can be addressed.

The students needs are addressed through the fact that they can relate to real-life scenarios, they can solve using the numbers that are best suited to their needs. I find that some second grade students count by fives inaccurately, they often skip over 15. In addition, many fourth graders think the difference between 40, 400, and 4,000 is that a zero has been added. They do not really understand that is the power of 10, not zero. Students are given multiple opportunities to use multiple mathematical representations. They are not forced to solve in one way, but can make connections to multiple ways equations, drawings, graphs, and number lines.

Using a CGI approach looks like and sounds like…

looking for evidence of what is going well. I ask students to explain their solutions, I ask students to try representing their thinking using a different mathematical model. I tell students, “I don’t care if you look at someones else’s paper, we are here to learn from one another. We are here to practice becoming better mathematicians, this is not a test. I tell students I don’t care if you get it wrong, I only care that you try.” Then, I praise them for their efforts and explicitly name what they did. “I noticed how you did not give up. I noticed that you gave “John” constructive feedback about his strategy. I noticed that you were willing to try a new strategy. I noticed that you crossed off when you counted again. I noticed that you asked for help when you got stuck.”

What does your CGI approach look like and sound like? Who do you ask for help when you get stuck?

You know I love hearing from you. Send your comments, questions, and wonderings to

If you are interested class is still open. Enroll in the University of San Diego’s online CGI math course, today!

Take up to 6 months to complete the course, earn up to 3 graduate level units, and learn ore about your students’ mathematical thinking.

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