If you are looking to increase your students’ fact fluency, prioritizing the skill of missing numbers in equations is one of the keys to unlocking that skill in your student. If your students have struggled with this skill in the past, you’re not alone! Solving for missing numbers in equations can be a difficult skill to learn but the payoff is significant!
A few years back I sat with a first-grade teacher who had been working on missing numbers in equations. The day had ended and she was sitting at the kidney table, head resting on her hands out of pure exhaustion from the day. She had called me down to meet because she had followed the math program to a T and yet more than a few of her students were still struggling to find missing numbers in addition and subtraction equations.
I could so clearly feel her frustration. I had been in her position a year prior when I was trying to support my intervention students… every attempt I made fell flat on its’ face. My students consistently would add and subtract arbitrary numbers to find an answer. Any answer. Regardless of whether or not the answer made sense.
I felt like saying is this even worth it? Can we just move on from this skill?
Why Prioritize Missing Numbers in Equations?
If your students were to memorize every single addition and subtraction fact they would need to commit over 200 individual facts to memory. This is an unreasonable task! When your students can quickly and easily relate facts, automaticity and retrieval are a simple byproduct!
Being able to solve for missing numbers in equations means that your students have a clear recognition of the relationship between addition and subtraction.
A student who knows that 3 + 5 = 8 and recognizes the relationship between addition and subtraction can solve for the missing number in the equation 8 – ___ = 5 because they recognize that these facts are related.
Additionally, solving for missing numbers in equations provides a notation for students who are fluent in composing and decomposing numbers.
Your primary students spend a great deal of time decomposing numbers. For example, students spend time working with partners of ten and know that 1 and 9, 2 and 8, 3 and 7, 4 and 6, and 5 and 5 are all ways to make ten. Students don’t always make the connection between these decompositions and equations on their own!
Asking your students to solve an equation that says 9 + ____ = 10 allows your students to practice connecting the math they already know (a 9 and a 1 make a 10) with notations and equations.
How to Teach Missing Numbers in Equations
Strategy #1: Make the Lessons Hands-On
When we teach math skills and concepts, we want to give consideration to CRA. Your students will understand more clearly when they first experience at the concrete level. If your curriculum or lessons jump straight to equations without hands-on support, you are being set up for failure!
When a number is missing from an equation we are either missing the start the change or the result.
Giving the students two of these numbers and allowing them to experiment with manipulatives will help them to see why future counting strategies work.
In this example, students are given the context “[#]birds are sitting in a tree. More birds fly over! Now there are a total of [#] birds.”
Students use a spinner to generate the initial number of birds in the tree and the resulting number of birds in the tree. They then use manipulatives to solve for the number of birds that flew over to the tree.
Students connect this hands-on experience to an equation as they record their work.
As a teacher you can also model using counting strategies to solve for a missing number using these materials. “Oh! I see you started with 5 birds in the tree. Then you added birds number 6, 7, 8, and 9. So we started at 5 and counted 5 more to get to 9. 5 and 4 more made 9. I can see that in your equation!”
Strategy #2: Be Aware of Moving to Representative Models too Quickly!
This was a mistake I made in a big way. I was sure that if my students understood fact-families and the part-part-whole relationship that they should easily be able to find the missing number in an equation. “Just pop the numbers into a number bond and from there you can easily find the missing number!”
I was asking them to use one abstract concept (part/part/whole) to support another abstract concept (missing number).
Number bonds will eventually be a great strategy to use with your students, however, they are not necessarily the place to start! Remember that we are striving to build a web of understanding for our students. After your students are showing some confidence with hands-on tools, ask them to use those tools alongside a representative model such as a number bond.
Strategy #3: Context is King!
The teacher I was working with was using an abundance of story problems as they were written in the curriculum. The problem was that the contexts provided weren’t necessarily understandable to our students. Worse yet, the context kept changing!
When you first introduce this skill, pick a context and let the students play with it over and over and over so that the context is supporting their ability to make meaning rather than adding another hurdle to understand.
Pair these simple contexts with hands-on tools and ask your students to record the results as equations. It will take time and practice but your students will soon gain the skills and confidence they need.
In the photo you can see students playing with a context around pirate treasure. A treasure chest has a given amount of treasure inside. Students are then told that the pirates either added to or lost some of their treasure. Finally, students are given the resulting amount of treasure in the chest.
Their goal is to use hands-on materials to find the number of pieces of gold that were either lost or found.
Adding this concept rather than asking students to work with naked numbers brings meaning, and therefore greater understanding, to the equations they are writing.