Adding Rigor in Math Class by Getting Students to Think Backward
Rigorous math lessons are important for pushing student thinking and understanding. Today I’m going to share one change that can make a problem more rigorous. This technique can be use to create a problem solving task that really makes your middle school or high school math students think!
A quick note about rigor…
Adding rigor does not mean making a lesson more difficult for no reason. With a thoughtfully created rigorous lesson, the goal is to get students thinking deeper about a concept. It may mean connecting current learning to previous topics, but it doesn’t have to. We want students thinking deeper, creating stronger neural connections as they do.
Recently, I was reading the book Fostering Algebraic Thinking by Mark Driscoll. One of the ways mentioned to foster algebraic thinking was the idea of “reversibility” or “doing-undoing.” When students can solve a problem and then think about how to work backward, we are preparing them for success in algebra. It’s also a great practice for adding rigor to a lesson.
I decided to brainstorm some ways to put this idea into practice in different grade levels. Read on for inspiration.
5th Grade Math
After students have learned to round to certain decimal places, ask them to create their own numbers that would round to a given number. For example: Name 5 numbers that round to 32.04 when rounded to the nearest hundredth.
After students have learned to create common denominators for adding and subtracting fractions, ask them to think when a certain denominator might be used. For example: Martin was adding 1/6 to another fraction. He decided to use 24 as a common denominator. What might the other fraction be?
After students have learned to find volume of rectangular prisms, give them a volume and ask for possible dimensions. For example: A rectangular prism has a volume of 48 cubic units. What might its length, width, and height be?
6th Grade Math
After students learn to simplify expressions by combining like terms, ask them to generate their own expressions that would simplify to a given expression. For example: Write an expression that can be simplified to 5x once like terms are combined. (Bonus points for creativity!)
Once students learn to find the area of a triangle, ask them to create triangles with given areas. For example: Draw a triangle whose area is 4.5 square inches.
After students learn to calculate mean and median, ask them to generate a list of numbers that have a given mean and median. For example: Create a list of 5 numbers whose mean is 25 and median is 30.
7th Grade Math
After students have learned to calculate percentages and discounts, ask them to find an original price given a discounted price. For example: A jacket is $32 after a 20% discount. What was the original price?
After students have to learned to add and subtract rational numbers, ask them to create their own problems with given sums and differences. For example: What are two numbers whose sum is -8? Two negative numbers whose difference is 7? Two numbers whose sum is -1/2?
After students learn to solve equations, ask them create equations with a given solution. For example: Write an equation with at least 2 operations where x = 12.
8th Grade Math
After students have learned exponents rules, ask them to generate their own expressions with exponents with a given solution. For example: Give 2 numbers (written as a base and exponent) that have a product of 8^12.
After students have learn to work with linear functions, ask them to write an equation for a line that passes through a given point. For example: Create as many equations of lines as you can that pass through the point (3, 7).
Once students have learned to used the Pythagorean Theorem to calculate distance on a coordinate plane, ask them to give 2 points with a given distance. For example: Find 2 points whose distance is 15 units apart and do not share an x- or y-coordinate.
Algebra 1
Once students have learned to find the zeros of a quadratic function, ask them to write a quadratic function with given zeros. For example: Write a function with x-intercepts at (5, 0) and (10, 0). Can you find others with the same x-intercepts?
Once students learn to solve absolute value inequalities, ask them to write an inequality with a given solution. For example: Create an absolute value inequality who solution is x < -3 or x > 5.
After students learn to graph scatter plots and calculate the line of best fit and correlation coefficients, ask them to create a scatter plot that has given line of best fit and correlation coefficient using technology. For example: Create a scatter plot with at least 10 points whose line of best fit is close to y = 2x + 5 and whose correlation coefficient is between 0.8 and 0.9.
Geometry:
Once students learn how to use arc and angle relationships to determine a circumscribed angle measure, ask them to sketch a circle with a given circumscribed angle and determine the arc measures. For example: Sketch a circle with a circumscribed angle of 125 degrees. Determine the arc measures of the major and minor arc formed.
Once students learn to find sector area, ask for a circle’s area when given a sector area and its arc measure. For example: A sector whose arc measure is 40 degrees has an area of 5 square meters. What is the total circle’s area?
Once students understand how to find the volume of a sphere, ask them to find the diameter when given the volume. For example: If a sphere has a volume of 50 cubic centimeters, what is its diameter?
These types of question may allow creativity or challenge students’ understanding of a concept.
Many math standards already encourage reversibility. When we ask students to make connections between adding and subtracting or multiplying and dividing, we are are asking them to do and undo. When we ask students to use the Pythagorean Theorem to find a hypotenuse and then later to find a leg, we again are showing the reversibility of mathematical logic.
Reversibility is a powerful thinking ability. It can be rigorous and create the opportunity to think about a concept in a different way.
With your next math topic, think about if you can challenge students to start from the end and work backward. I’d love to hear how you might use this idea in your classroom in the comments.
If you like this idea, check out this free download: 3 Ways to Spice Up Math Problems!
