The following visual math talk is a string of related problems that are crafted in a way to foster discussion and the construction of viable arguments. As such, they are open ended and have more than one possible response.
As the facilitator of the following visual math talk prompts, be prepared to notice and name the thinking students are using (i.e.: additive thinking or multiplicative thinking).
While you could simply share the first image with the prompt for students, introducing via the silent solution animations provided may be more helpful. Although each visual math talk prompt video models a possible strategy, we encourage you to represent student thinking by drawing on the board or on the screen rather than simply playing the video and ignoring the thinking of your students.
Consider utilizing the following visual math talk prompt and be prepared to pause where indicated:
Prompt:
What do you notice? What do you wonder?
After students share some of what they notice and wonder about the shapes that have ‘popped’ on the screen, continue playing the video until the next prompt appears:
Describe as many relationships as you can between the circles and the squares.
Some possible relationships students might share include:
- There are 10 more squares than circles
- There are 3 times as many squares as there are circles
- For every circle there are 3 triangles
- \(\frac{3}{4}\) the shapes are squares (\(\frac{1}{4}\) of the shapes are circles)
- 75% of the shapes are squares
It is worthwhile to note the observations that students will share, as a way of determining if students are thinking in absolute or relative terms, and whether they are able to consider both part:whole and part:part relationships.
As you are making note of students’ observations, consider modeling the use of ratio language and notation, including:
- The ratio of ____ to ____ is _____ : ______. For example, “the ratio of squares to circles is 3:1 or the ratio of circles to squares is 1:3”.
- For every ______, there are ______.
Some students may make use of rate language such as:
- There is a rate of _____ per _____
- For example, “there is a rate of 3 squares per circle” or “there is a rate of 1 third circles per square”.
Of course, these are just some possible observations and many more ideas may be highlighted by your students.
Consider encouraging students to find other equivalent ratios by scaling in tandem (ratio reasoning) or by using the constant of proportionality (rate reasoning) and record their thinking in a ratio table.
Consider utilizing the following visual math talk prompt and be prepared to pause when indicated:
The following prompt will appear on the screen:
Describe as many relationships as you can between the circles and the triangles.
Some possible relationships students might share include:
- There are 9 more triangles than circles
- There are 4 times as many triangles as there are circles
- For every circle there are 4 triangles
- \(\frac{4}{5}\) of the shapes are triangles (\(\frac{1}{5}\) of the shapes are circles)
- For every white triangle there are 2 green triangles
Again, consider the language students are using. Is it in absolute terms (i.e.: additive thinking) or in relative terms (i.e.: multiplicative thinking)?
If students are using multiplicative thinking, are they using ratio reasoning (i.e.: comparing squares and triangles as separate quantities) or rate reasoning (i.e.: summarizing the relationship via a single quantity with a compound unit such as 4 squares per circle”)?
After celebrating the different relationships that students may have shared using various types of reasoning, be sure to highlight again the ratio relationship via a ratio table (or double number line) and ask students to share some equivalent ratios that we can reveal from this relationship. You might also consider highlighting the constant of proportionality (i.e.: finding the rate) by using partitive division to divide the number of circles with the corresponding number of squares or visa versa from multiple rows in the ratio relationship.
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