As students become familiar working with growing patterns and explaining them using algebraic reasoning, additional contexts and problem situations can be introduced. Below are some contexts I have used in the past. Each context is described on this page. Click the image or the link below the image to learn more about each context.
Contexts
Two of Everything
This is a great way to connect literature and mathematics. This children’s book by Lily Hoy Tong is a story about Mr. and Mrs. Haktak. They discover a magic pot that doubles whatever goes into it, including Mr. Haktak. Begin by exploring the doubling pattern in the story. Students should use tables labeled with input (what goes into the pot) and output (what comes out of the pot). Students should work to generalize and write an equation to describe the pattern.
Questions to consider as students engage in this work:
- What if 300 pens were dropped into he pot?
- What if 40 tomatoes were pulled out of the pot? How many were put into the pot?
The next day, explain that the magic pot is not doubling anymore. It is using a different rule. Provide students with partially completed tables (minimum of 3 rows) that represent different rules such as x + 5, 4x, .5x, etc. For each, ask students to add examples to the table, then explain the rule in words and using an equation.
These activities work well for ELL students because the situation can be acted out with concrete objects or illustrated, completing the tables does not require a great deal of vocabulary, and there are a number of opportunities for students to communicate (practice speaking and listening).
These activities also work well for students with disabilities because using an image of a pot and manipulatives along with a table of values can assist students as they reason about the input-output relationship.
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The Crow and the Pitcher
Another great connection to literature is uses Aesop’s fable, The Crow and the Pitcher. This fable can be used to engage students to determine how many stones it might take to raise the water enough for a bird to drink it. Present the fable to the students. You can begin by either reading this fable from a book of Aesop’s Fables, or use a video like the one below.
After sharing the story and discussing the moral, ask students if there is anything they are curious about in the fable. Students will likely ask questions such as:
- Does this really happen?
- Are crows the only birds that do stuff like this?
- Are other animals clever like this?
- How many stones would it take to raise the water?
Materials:
- A graduated cylinder or beaker (one per small group)
- A tall, clear, plastic cylinder such as a jar or vase and a ruler may also be used
- Small stones or marbles (approximately 100 per group)
If you don’t have access to a graduated cylinder or beaker, use a clear, plastic, cylindrical vase or jar. the best size, I think, is about 8-10 inches tall with a diameter of about 2-3 inches. Students can tape a ruler to measure the change in water height as “stones” are added.
Students should determine to what height (or capacity, if using a beaker) the water should get to on the jar for a bird to successfully drink it. To do this, they can research the length of a crow’s beak, and cut a straw to that length, then tape it to the top of the rim, extending into the jar.
To begin, students record the height of the water before any marbles are added (zero marbles). Students then add marbles until they can measure a new height/capacity for the water easily. Students should add several data points to a table, until they feel comfortable using the data to make a prediction. Desmos can also be used to plot points from their table of values to help make a prediction
Ask students to share their reasoning for their predictions and facilitate a discussion that focuses on the rules developed and the reasoning used. Highlight student strategies that are unique and/or those strategies that relate well. If groups of students use different “stones,” discuss the similarities and differences in the “stones” predicted and the size of the “stones.”
After students have shared, ask them if they would like to see the results using their “stones.” This is a live version of the third act of a 3-act task. Students will very likely get excited as you approach their predicted number of “stones.”
After the “reveal,” ask students about any discrepancies between their predictions and the actual number of “stones.” They can think-pair-share, then add their thoughts to their journals.
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Number Tricks
There are endless “magic” tricks showcasing your ability to “read the minds” of your students. A few can be found below. In addition, I have a whole page dedicated to the Mathematics of Magic with even more ideas to engage students in mathematical and algebraic reasoning.
Tricks like these should be explored with small bags or boxes, or even post-it notes, to represent the unknowns and counters or cubes to represent numbers used in the operations. As students model tricks like these, they will begin to connect the algebra to these models.
The figure below shows how the trick above might be modeled using materials like these.
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Smashing Play-doh
In this investigation, students use a small can of play-doh to determine if there is a pattern to how the diameter changes based on the amount of weight set on it and to predict the diameter of the play-doh after the large weight is placed on it.
Investigating this can be challenging only because balancing the weight on top of the play-doh, can be a bit tricky.
Materials:
- play-doh – one small can per small group or pair
- Weights – Any group of weights will work from 0.5 lb to around 10 lbs. (at least three different weights per small group or pair)
- One large weight – for the reveal – approximately 20-25 lbs.
- 10″ to 12″ cake rounds – one per small group or pair
The ideal weights for this investigation are “plate” weights like you find in a gym to add to barbells. These will fit nicely on the cake rounds and make it less cumbersome to balance on the play-doh.
First, introduce the problem. Bring out the play-doh and the large weight. Ask students to turn and talk about what will happen to the play-doh when the large weight is placed on top? Ask students to predict what will happen and be precise with their language as they share. Facilitate a discussion and ask them to predict: How large will the diameter of the play-doh be after this large weight is placed on it?
Share estimates to collect class data, highlighting different ranges where estimates seem to cluster.
Introduce the materials and discuss the procedures for smashing play-doh:
- create a cylinder of play-doh with a circular face that has a diameter of 5 cm.
- Carefully, place the weight on the cake-round.
- Gently, place the cake-round with the weight on top of the play-doh cylinder.
- Wait about 1 or 2 minutes until the play-doh is no longer being smashed by the weight.
- Carefully remove the cake-round and the weight
- Measure and record the weight used and the diameter of the play-doh after it was smashed.
- Repeat these steps for the next weight.
- Use the data you collect to predict the diameter of the play-doh for the large weight.
- You may want to graph the ordered pairs from the data you collect in Desmos.
Ask students to share their reasoning for their predictions and facilitate a discussion that focuses on the rules developed and the reasoning used. Highlight student strategies that are unique and/or those strategies that relate well. If groups of students use different weights, discuss the similarities and differences in the diameters predicted and the size of the weights used.
After students have shared, ask them if they would like to see the play-doh smashed with the large weight? This is a live version of the third act of a 3-act task. Students will very likely get excited as the play-doh gets smashed and as you measure the diameter.
After the “reveal,” ask students about any discrepancies between their estimates, predictions and the actual diameter. They can turn and talk, then add their thoughts to their journals.
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Contextual Algebraic Thinking Lessons
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