Math DEEP groups

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding."

- William Paul Thurston

Third Grade DEEP Math Highlights

Lines, Angles, and Geometric Relationships

During the second term, third grade DEEP math students explored the geometry of lines and angles, focusing on how angles are created and how they relate to one another. Students investigated the idea that every angle forms a second angle, examining how angles interact when lines intersect and how these relationships connect to the structure of a circle.

Through hands-on exploration, students discovered that angles formed by intersecting lines from a single point always add up to 360 degrees. Rather than being told this rule, students tested it by measuring, recording, and analyzing multiple examples, allowing them to uncover the pattern through evidence and reasoning.

Students learned and practiced essential geometry skills, including:

Identifying and classifying angles
Measuring angles accurately using a protractor
Drawing circles and constructing angles using a compass
Understanding angle relationships and total degrees around a point

Critical thinking was emphasized as students made predictions, verified results, and explained why their measurements made sense. They compared different angle arrangements, corrected errors in measurement, and refined their technique with mathematical tools.

Several Icons of Depth and Complexity were intentionally integrated:

Patterns: Students identified consistent angle relationships and totals across different configurations.
Rules: Students explored geometric rules governing angle measures and tested whether they always held true.
Relationships: Students examined how angles connect to one another and to the full circle.
Big Idea: The understanding that geometry is built on predictable, connected structures guided student thinking.

Through these investigations, students developed strong measurement skills, spatial reasoning, and logical analysis, while gaining confidence using mathematical tools and thinking like mathematicians who ask questions, test ideas, and justify conclusions.

Third Grade First Term

Ancient number systems

This term, students explored the history of the number system, discovering that numbers can be represented in many different ways across cultures and time periods. As they learned about Egyptian numbers, students strengthened their understanding of place value by translating symbols into expanded form and then into standard form. This required careful attention to quantity, grouping, and the structure of numbers.

Students then studied Roman numerals, applying their number sense as they converted Roman numerals into standard numbers. To deepen engagement and reasoning, students used those numbers in a substitution cipher to decode a hidden message. This activity reinforced number conversion, logical reasoning, and accuracy, while also encouraging perseverance and problem solving.

Next, students compared our base ten number system to the Mayan base twenty system, analyzing how different cultures organize and represent quantities. Students practiced addition and subtraction in base twenty, strengthening their understanding of place value while also gaining exposure to multiplication by 20 and 400 as they worked within the Mayan system.

Students also tackled the Broken Calculator Problem, where they were challenged to create equations that equaled the numbers 1–20 using only the digits 3 and 5 and the operations addition and subtraction. This task required strategic thinking, flexibility, and trial and error, as students explored multiple solution pathways and justified their reasoning.

Throughout this unit, several Icons of Depth and Complexity were emphasized:

Change Over Time: Students examined how number systems have evolved across civilizations.
Multiple Perspectives: Students compared how different cultures represent and use numbers.
Rules: Students worked within the constraints of different number systems and mathematical limitations.
Big Idea: The concept that numbers are a human invention used to represent quantity in many ways guided all learning.

Fourth Grade DEEP Math Highlights – Equivalent & Simplified Fractions Through Fraction Puzzles

This term, fourth grade DEEP math groups developed a deep and flexible understanding of equivalent fractions and simplifying fractions through a series of rich, problem-solving tasks. Students worked with area-based fraction puzzles, breaking larger shapes into smaller pieces and using the shaded portions to represent unit fractions and combinations of fractions. This visual and hands-on approach helped students connect the idea of area to fraction values, making abstract fraction concepts concrete and meaningful. youcubed.org

Students practiced generating equivalent fractions, recognizing that different numerical expressions can represent the same amount of a whole, and applied these relationships to solve puzzles. They also learned strategies to simplify fractions, reducing numbers to their lowest terms to make comparisons and solutions more efficient. These activities required students to be systematic and strategic, carefully decomposing shapes and reorganizing parts to find solutions that fit the puzzle requirements. youcubed.org

Throughout the unit, students strengthened essential math skills including:

Number sense with fractions
Identifying and creating equivalent fractions
Simplifying fractions to their lowest terms
Using visual models to justify numerical reasoning

Students also practiced logical thinking, pattern recognition, and strategic problem solving as they approached each puzzle. They analyzed relationships between fractions, tested multiple pathways to a solution, and explained their reasoning to peers, building both confidence and competence in fraction work.

Several Icons of Depth and Complexity were intentionally embedded in the learning:

Patterns Students noticed consistent equivalences across different fraction representations.
Rules Students applied the mathematical rule that multiplying or dividing the numerator and denominator by the same number creates equivalent fractions.
Multiple Perspectives Students explored more than one approach to making and simplifying fractions.
Big Idea The understanding that fractions can look different but be equal in value guided student thinking.

Fourth Grade

One Number, Many Ways: Understanding Fractions, Decimals, and Percentages

How can we determine the percentage we earned on a test if we answered 5 out of 6 questions correctly? That question launched students into a rich exploration of how fractions, decimals, and percentages are connected.

During the first term, students built a strong foundation by examining how fractions, decimals, and percentages all represent parts of a whole. They began by identifying how each form can represent one whole, then practiced converting between forms: fractions to decimals, decimals to percentages, and percentages to fractions. This work strengthened number sense and helped students see these representations as different ways of describing the same value.

Students then applied their learning creatively by designing pixel art, using their artwork to calculate what fraction, decimal, and percent of the whole each color represented. This activity required careful reasoning, precision, and the ability to justify mathematical thinking.

To deepen understanding, students tackled challenge problems that asked them to shade portions of a hundreds cube based on a given fraction, decimal, or percentage. These tasks reinforced proportional reasoning and encouraged students to visualize quantities in multiple ways.

The unit concluded with a real-world application, allowing students to use their skills in a meaningful context and demonstrating how fractions, decimals, and percentages are used beyond the classroom. Through these experiences, students developed strong computational skills, conceptual understanding, and the ability to think flexibly about numbers.

Fifth Grade Second Term Highlights

Exploring Interest and Financial Math

This term, fifth grade DEEP math students explored the real-world applications of interest rates, learning how they impact everyday financial decisions. Students examined how interest is calculated on credit cards and practiced using math to understand the long-term effects of minimum payments.

As a hands-on application, students calculated how long it would take to pay off a guitar if only the minimum payment was made each month. This task required them to apply multiplication, division, and percentage calculations, as well as to analyze cumulative effects over time. Students developed critical thinking skills as they compared different payment strategies, predicted outcomes, and reflected on how financial decisions affect debt and savings.

Several Icons of Depth and Complexity were highlighted:

Rules: Students applied the mathematical rules governing interest calculation.
Big Idea: Understanding how interest influences financial decisions guided the unit.
Multiple Perspectives: Students explored different payment strategies and their consequences.
Change Over Time: Students analyzed how debt grows over time when only minimum payments are made.

Through this unit, students strengthened both their numerical and problem-solving skills while connecting mathematics to meaningful, real-world financial contexts, preparing them to make informed decisions in the future.

fifth Grade

From Miles to Money: Applying Decimals to Real-World Problems

How much would it cost to drive from West Point, Utah to Washington, D.C.? That real-world question set the stage for a deep mathematical investigation. DEEP students were given the miles-per-gallon ratings of three different vehicles and asked to calculate the total cost of the road trip using the current price of fuel. These calculations provided meaningful opportunities to apply multiplication with decimals, estimate reasonableness, and interpret results in context.

Students then explored how changing variables impacts outcomes by calculating how the total cost would change if the vehicle used premium gasoline instead of regular. Many students were surprised by how dramatically the price fluctuated when fuel type changed, reinforcing the importance of understanding how small differences can lead to large effects.

To extend their reasoning, students calculated how many times they would need to stop for gas based on each vehicle’s fuel tank capacity, combining division, multiplication, and problem-solving strategies. This required students to plan, justify their thinking, and check for accuracy.

After completing their calculations, students reflected on the results and discussed whether they believed gasoline was expensive. Most concluded that it was, supporting their opinions with mathematical evidence. This activity connected math to real-life decision-making and strengthened students’ ability to use numbers to analyze, compare, and evaluate everyday situations.

Sixth Grade DEEP Math & Science Highlights

Measuring the Power of Tsunamis

This term, sixth grade DEEP students explored the immense power of tsunamis by integrating math and science in a real-world context. Using the Across Disciplines icon of depth and complexity, students applied their knowledge of energy transfer, distance, speed, and measurement to understand the scale of natural forces. They also explored the Big Idea of how energy impacts environments and communities, and used the Change Over Time icon to consider how tsunamis can affect areas over extended periods.

The Challenge – Students calculated the total energy produced by small, medium, and large tsunamis, analyzing what that energy could accomplish in real-world terms. For example, they discovered that a single tsunami could generate enough energy to power the Christmas tree in Rockefeller Center for two continuous years.

Math in Action – To understand the real-world impact of tsunamis, students applied a variety of math skills:

Speed, distance, and time calculations to determine how long it would take a tsunami to reach specific locations around the world
Large-number multiplication and estimation to quantify total energy output
Unit conversions and comparisons to relate tsunami energy to familiar applications

By connecting abstract calculations to tangible examples, students gained a deeper appreciation for the scale of natural phenomena and the importance of energy understanding.

Icons of Depth and Complexity highlighted during this work:

Patterns: Students recognized consistent relationships between wave size, energy, and impact.
Rules: Students applied formulas for energy, speed, and distance to solve authentic problems.
Multiple Perspectives: Students compared the energy of tsunamis to everyday energy uses, making abstract numbers meaningful.
Big Idea: Students explored how energy shapes both natural systems and human considerations in coastal regions.

Through this unit, students strengthened mathematical reasoning, measurement, proportional thinking, and applied problem-solving skills, while developing a concrete understanding of the immense power of natural events like tsunamis.

Sixth Grade DEEP Math Highlights – Exploring Algebra and Variables

“How do you solve a problem when some information is missing?” That was the driving question for sixth grade DEEP math students this term. Students began by learning how variables can represent unknown information, then explored different strategies for solving for these unknowns. A key focus was balancing equations, an essential skill for understanding relationships between quantities and solving algebraic problems.

Students applied their learning to a real-world scenario involving pizza costs. Given the price of a medium two-topping pizza and a medium four-topping pizza, students calculated the cost of a cheese pizza and the cost of each topping. They then wrote an equation to represent the cost of a pizza with an unknown number of toppings. Using this equation, students created ordered pairs and learned to graph them, visualizing the relationship between the number of toppings and total cost.

The unit extended to small and large pizzas, where students graphed the cost per topping and analyzed their results to determine which pizza offered the most value. This process required students to use critical thinking, proportional reasoning, and problem-solving skills while connecting algebraic thinking to meaningful, everyday decisions.

Icons of Depth and Complexity were embedded throughout the unit:

Patterns: Students identified consistent relationships between the number of toppings and total cost.
Rules: Students applied algebraic rules to solve equations and graph relationships.
Multiple Perspectives: Students compared different pizza sizes and costs, analyzing efficiency and value.
Big Idea: Understanding how variables represent unknown quantities and how equations model real-world scenarios guided learning.

Through these activities, students strengthened their algebraic reasoning, equation-solving, graphing, and analytical skills, while seeing how mathematics can be applied to practical decision-making.

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