One of the main reasons so many middle and high school students struggle to master grade-level math concepts is because they lack a strong foundation in numeracy skills. The incremental nature of mathematics requires students to develop their numeracy skills as they progress through the grade levels, but students can seem to be doing OK in the early grades even if they don’t fully understand some of the concepts.
Middle and high school teachers don’t always have time within the instructional block to do a formal review lesson on basic numeracy skills with students. However, there are enrichment activities that we can incorporate into our instructional blocks to help students strengthen their skills. Here are a few that I’ve had success with.
The Hundred Challenge
The Hundred Challenge is a math activity that asks students to apply their basic math skills to create equations that equal each of the digits between 1 and 100—but they can only use the four digits the teacher supplies, like 1,4,7, and 9. The rules are as follows:
- All four digits must be used in each equation.
- Each digit can only be used once in each equation.
- Any operation—addition, subtraction, division, multiplication—or symbol—parentheses, radicals, exponents, etc.—can be used to create equations.
- Digits can be combined to form new digits (e.g., 1 and 4 can be combined to form 14 or 41), fractions (e.g., 1 and 7 can be combined to form 1/7 or 7/1), and decimals (e.g., 4 and 9 can be combined to form 4.9 or .49).
Some numbers will have only one possible equation, given the four digits allowed, while others will have several, so this challenge reveals the differentiated math strategies and learning experiences of the students.
For instance, my high-intervention students tend to create equations for the digits between 1 and 50. They create basic equations by simply inserting operation signs between the required digits, like 49 – 17 = 32, or 1 + 4 + 7 – 9 = 3.
My advanced students usually take more computational risks when creating their equations, and they attempt to create equations for the digits on the higher end of the chart (typically between 75 and 100). The types of equations that these students create typically involve the use of exponents and parentheses to demonstrate their knowledge of order of operations.
I usually use the Hundred Challenge as a collaborative learning activity, but it can function as a living learning center in the classroom for students who finish their independent work early and need more challenge problems to compute. Each student has a sheet with the numbers from 1 to 100 listed, and they fill it up with equations over time.
Do Now: Equation or Expression of the Day
Each and every day, I write a mathematical expression or equation in place of the day of that specific date. When students enter the classroom, they solve the mathematical expression or equation—they know the date, usually, so they’re able to check their work right away. For instance, instead of writing “October 28, 2020,” I may write, “October 3(x – 23) = 15, 2020.” For this particular equation, students would have to solve the equation 3(x – 23) = 15 in order to uncover the day number.
Once students figure out the solution, they then have to create at least two mathematical expressions or equations that also equal the solution. For example, a student may create the expression 4! + (2 x 2) = 28. I give students no more than five minutes to solve the equation and then review the problem with them. This is a great opportunity to have some students come to the board to share the expressions they created.
In my experience, students get really excited about the equations or expressions they create and love to compare them with those of their peers. I’ve even had some students volunteer to create the equations or expressions for future dates.
To make the exercise more challenging, you can set specific conditions for the equations or expressions your students create, which ensures that students don’t give you pedestrian expressions or equations like 28 + 0 or 28 + x = 28.
The ultimate goal of these Do Now exercises is to provide students with opportunities to apply and synthesize their mathematical background knowledge skills. They help students strengthen their abilities to solve and create multistep algebraic equations and expressions. And these exercises are differentiated: Students can access them at their developmentally appropriate level and take academic risks with the knowledge that they’ll receive instructional support from me and their peers.
In a cryptarithmetic puzzle, the digits are replaced by letters of the alphabet. The goal is for students to uncover the puzzle by determining the digit for each letter. The rules for uncovering the puzzle are as follows:
- Each letter represents a digit between 0 and 9. A letter cannot represent multiple digits, and a digit cannot be represented by multiple letters.
- Numbers must not begin with a zero.
- There is only one solution to the puzzle.
For example, the puzzle SEVEN + SEVEN + SEVEN + NINE = THIRTY translates to 49,793 + 49,793 + 49,793 + 3,239 = 152,618 (S = 4, E = 9, V = 7, N = 3, I = 2, T = 1, H = 5, R = 6, and Y = 8). Cryptarithmetic puzzles can also be expressed as multidigit subtraction, multiplication, and division problems. (You can find more cryptarithmetic puzzles here.)
As an extension activity, teachers can challenge students to create their own cryptarithmetic puzzles and have their classmates try to solve them. I typically use these puzzles as a learning center or enrichment activity that students can complete if they finish their independent work early.