Thought for the day: Sometimes, you can use a series of Creative Math Prompts one at a time in order to scaffold a concept.

Concepts: meanings of fractions, decimals, percentages; equivalent fractions; proportional reasoning; translating between fractions, decimals, and percentages (including estimation)

(T), (M), and (B) represent the Top, Middle, and Bottom pair of lines respectively.

Beginning

I notice one number line on top of another with the zeros lined up on the left. (T)
I notice that 7 is a little closer to 13 than to 0 both as a number and in the picture. (T)
I wonder where the dotted line crosses the bottom number line. (T)
I notice that the picture appears to be about fractions (or decimals). (T)
I notice that the bottom line ends in 100. (T)
I notice that the diagram reminds me of equivalent fractions. (T)
I notice that the top line could represent the fraction 7/13.
I notice that the picture could be about percentages. (T)
I notice that the dotted line must hit the bottom line at a number a little greater than 50. (T)
Students may try to guess the number (say to the nearest 10).
I notice that the dots make it easier to estimate the place where the dotted crosses. (M)
I notice that the dots represent multiples 10% (M).
I notice that 7/13 is between 50% and 60%, but closer to 50%. (M)
I notice that the double-number line illustrates 1/13. (B)

Exploring

I wonder how I could calculate the exact place where 7 crosses the bottom line.
I predict that it is easier to figure out where 1 crosses than where 7 crosses. (M) and (B)
I notice that knowing where 1 crosses can help me figure out where 7 crosses. (M) and (B)
I wonder where the dots on the lower (0 to 100) line will hit the top (0 to 13) line. (M)
I wonder if knowing this could help me calculate the place there 7 crosses. (M)
I wonder if i can think of stories to fit these double number lines.

Creating

As they notice and wonder, students may create their own estimates, strategies, diagrams, or stories. Their creations may be inspired by things that they have wondered about. For example, they may

Create estimates for various pairs of locations on the number lines.
Create multiple strategies for calculating the place where the dotted line for 7 crosses the lower line.
Create double number line diagrams for other fractions and percentages.
Create stories or contexts for the double number lines.
Create different ways (other than double number lines) to represent the percentage for 7/13.

Reflecting and Extending

I notice that double number lines help me visualize and understand percentages.
I notice that there is more than one way to calculate a percentage for a fraction.
I notice that a percentage is a special kind of proportional relationship.
I wonder what would it would look like if I drew magnified pictures of different parts of the double number line.
I wonder if it could ever be useful to have a triple number line.
I wonder if I could represent my different calculation strategies as algebraic formulas.

Notes

In these notes, I will share what happened when I used this particular set of images with a sixth grade advanced math class.

To set the stage: This lesson was part of a professional development activity for using concept-based lessons and activities with advanced learners. All of the students understood percentage as “parts of 100” and were familiar with fraction/percentage relationships for halves, fourths, fifths, and tenths. Most had been taught methods for writing fractions as decimals but were not yet fluent with these procedures. A few had been taught how to set up proportions and use cross products to solve them (probably by parents or previous teachers who wanted to show them “advanced” methods), but they did not understand why these procedures worked.

I began by showing students the top image and asking them what they noticed and wondered. They saw a connection to fractions (and decimals) right away. Interestingly, it took a while for them to focus on the number 100 and make a connection to percentages. They were immediately curious about where the dotted line through 7 crossed the lower line and soon realized that this question was equivalent to finding a percentage for the fraction 7/13. They saw that it must be at a number greater than 50 (both visually and because 7 is greater than half of 13). However, most of them estimated the number to be closer to 60 than 50. At this point, we took some time to talk about stories or real-world contexts that the number lines might represent.

As the discussion developed, I showed the second picture in order to get the students thinking about multiples of 10% and to assist with the estimate. This image also helped everyone begin thinking about concrete strategies for calculating a more precise value of the percentage of 7/13 (though some students had already begun developing their own approaches based on previous learning about converting fractions to decimals or using cross products).

The final picture suggested a specific strategy, especially for students who had not yet found one. In the end, three strategies emerged:

(1) 100 ÷ 13 • 7 (Divide 100 by 13 to find the percentage for 1/13. Then multiply by 7 to find the percentage for 7/13.)
(2) 7 ÷ 13 • 100 (Divide 7 by 13 to write 7/13 in decimal form. Then multiply by 100 to write the decimal as a percentage.)
(3) 100 • 7 ÷ 13 (Write a proportion and use cross products.)

Comparing the three strategies led to a discussion of properties of multiplication and division. Students noticed that the same three numbers appeared in each calculation but in different orders. Eventually, they saw that they had divided by 13 in every case and either started with or multiplied by 100 and 7 every time.

In summary, by the end of the lesson, students had connected concepts around fractions, decimals, and percentages; made estimates and refined them; created stories suggested by the number lines; and developed and compared three strategies for calculating a percentage for 7/13 . Near the end of the class, a few students had spontaneously begun trying to create algebraic expressions to represent their strategies.