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## Supporting Probability Understanding through Area Models

Moving beyond memorization of probability rules, the area model can be useful in making some significant ideas in probability more apparent to students. In particular, area models can help students understand when and why they multiply probabilities and when and why they add probabilities.

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## GPS: Spring Frogs

April 2020's GPS department provides tasks for each grade band that invite students to reason with age-appropriate number theoretic concepts.

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## Is the Last Banana Game Fair?

Is the “Last Banana” game fair? Engaging in this exploration provides students with the mathematical power to answer the question and the mathematical opportunity to explore important statistical ideas. Students engage in simulations to calculate experimental probabilities and confirm those results by examining theoretical probabilities

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## Problems

A monthly set of problems targets a variety of ability levels.

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## Threatened to extinction

Hundreds of species of animals around the world are losing their habitats and food supplies, are facing extinction, or have been hunted or otherwise negatively influenced by humans. Students learn about some of these animals and explore multiple solution strategies as they solve this month's problems. Math by the Month features collections of short activities focused on a monthly theme. These articles aim for an inquiry or problem-solving orientation that includes four activities each for grade bands K–2, 3–4, and 5–6.

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## Problems

A monthly set of problems is aimed at a variety of ability levels.

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## Problems

A monthly set of problems is aimed at a variety of ability levels.

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## Using PhET Simulations in the Mathematics Classroom

Imagine that you and your language arts colleagues are teaching Edgar Allan Poe's short story, “The Pit and the Pendulum.” This thrilling story takes us to the Inquisition during which a prisoner is surrounded by hungry rats and bound to a table while a large pendulum slowly descends. The prisoner believes that the pendulum is 30-40 feet long and estimates that it should take about 10-12 swings before he is hit, leaving him with about a minute or a minute and a half to escape. Are his estimations correct? If so, will he make it out in time?

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## Problems

A monthly set of problems is aimed at a variety of ability levels.

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## Relationships Between Opportunity to Learn Mathematics in Teacher Preparation and Graduates' Knowledge for Teaching Mathematics

We investigated whether the mathematics studied in 2 content courses of an elementary teacher preparation program was retained and used by graduates when completing tasks measuring knowledge for teaching mathematics. Using a longitudinal design, we followed 2 cohorts of prospective teachers for 3 to 4 years after graduation. We assessed participants' knowledge by asking them to identify mathematics concepts underlying standard procedures, generate multiple solution strategies, and evaluate students' mathematical work. We administered parallel tasks for 3 mathematics topics studied in the program and one mathematics topic not studied in the program. When significant differences were found, participants always performed better on mathematics topics developed in the program than on the topic not addressed in the program. We discuss implications of these findings for mathematics teacher preparation.