Imagine a special box, a bit like a function machine, but this one loves multiplication! You put numbers in, and it multiplies each one by a secret number hidden inside. Let’s explore this with a fun math puzzle.
In our first puzzle, four numbers went into the box, and four different numbers popped out. Can you figure out what multiplication magic happened inside the box?
We started with whole numbers, and we got whole numbers out. Your mission: discover the multiplication that took place inside the box to get these results. And for an extra challenge, what’s the biggest number that could have been used for multiplication in this first box?
Let’s look at a second box puzzle. This time, imagine we have a new box, and it’s multiplying by a different secret number. Here are the numbers that came out of this second box:
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What number is this second box multiplying by?
How are you working out these puzzles? Think about different methods you could use, and maybe discuss your ideas with someone else. There are often different ways to solve these kinds of problems!
Why is this problem interesting?
This “What’s in the Box?” problem is a fun way to think about function machines and how they work. It’s more engaging than just a typical worksheet and can be adapted to challenge learners of different levels. It’s also a great way to introduce the idea of common factors and encourage students to record their thinking in different ways.
Possible Ways to Approach the Problem
If students are new to this, start simple. Begin with just one number going into the box and one number coming out. This helps everyone understand the process. Then, gradually increase the number of inputs until you get to four, like in the puzzles above. You can adjust how tricky the multiplication is depending on the students’ level.
Once students have had some time to work on the first puzzle, get them to share their methods. Highlight students who have good reasons for choosing particular strategies. This is a good time to introduce vocabulary like “common factors” if it fits with what you’re teaching.
You can also talk about different ways of recording their work. Some might draw pictures, others might write out calculations, and some might do a mix of both.
Key Questions to Guide Thinking
- What multiplication could have happened in the box to get this output number from this input number?
- Would that same multiplication work for the other input and output numbers as well?
Extending the Challenge
Challenge students to explain how they know they’ve found the largest possible number that the inputs have been multiplied by. Some students could even create their own “What’s in the Box?” problems for classmates to solve, maybe even using division instead of multiplication inside the box!
Support for Learners
Some students might find it helpful to start with a simpler version of this task using addition and subtraction. There’s a version called “What Was in the Box?” that uses addition and subtraction which can be a good starting point.
To introduce the multiplication box, you could use physical objects like blocks. Represent a number going into the box with a set of blocks. Cover them up, and secretly add more sets of blocks (representing multiplication) before revealing the new, larger set. Ask questions like: “How many blocks are there now? What happened under the cover?” Encourage them to think about how the number of sets grew, rather than just counting the extra blocks.
When working on the main problem, having a multiplication square or calculator available can be useful for some learners.
