# Math Puzzle Monday!

It’s time for our weekly puzzle! My apologies for not posting last week﹣we were on the road.

This week’s puzzle, like the hexagon puzzle I posted a while back, can easily be done via counting on, but this time, there’s a twist: consecutive squares on the path need to be connected by the move of a chess knight instead of being straightforwardly adjacent! This makes for an appealingly tricky puzzle apart from the arithmetic.

Click on the puzzle to open a copy of the puzzle as a pdf.

And here’s the solution to the puzzle as a pdf, if you want to check your student’s work:

# Math Puzzle Monday!

I thought it’d be fun to start a tradition of posting a math puzzle every Monday! I’ve been coding puzzle generators for a while, which means I should be able to create an unlimited supply of math puzzles for anyone interested. I’ll try to make puzzles that are relevant to recent blog posts. Correspondingly, the puzzle below is good practice for counting on, provided that a child is fluent with counting on without concrete objects and can rote count to 40:

Click on the puzzle to open a copy of the puzzle as a pdf.

And here’s the solution to the puzzle as a pdf, if you want to check your student’s work!

# Lesson Plan: Finishing Counting On

Required skill: fluent addition by counting the first group of objects quickly without pointing at every object, as described in my last post, fluent rote counting starting at any number (not just at 1.)

In my last post, I talked about how you can build fluency with the skill of adding by counting the first group of objects quickly. Since this skill builds on a child’s mental models of addition and quantity, it tends to be retained well with practice. You’ll know that the skill has become second nature if a child can reliably use it to put numbers together without having to think; occasionally forgetting and starting from 1 is not a problem, but a child should be able to quickly change course when reminded. This stage can be reached in anywhere from a few days to a few months.

Once this stage is reached, getting to “traditional” counting on is usually straightforward. Here’s how it goes, using the example of adding 6 and 1. This can be done with or without counters, depending on a child’s preference. (If the below dialogue doesn’t work without counters, definitely try it with counters.)

Parent: Can you tell me what 6 and 1 make together? Remember, you can count the first group quickly.

Child: 1, 2, 3, 4, 5, 6. Then there’s one more, so that’s 7. The total is 7!

Parent: Now I’m going to teach you an even faster way to do it! You counted the first group quickly, and you said 1, 2, 3, 4, 5, 6. So you said the number “6” last.

Wait a minute… we didn’t even need to count that group! We already know that when we count it, we’ll say the number 6 last, because that’s the total number of items in the group. So really, we can just say that number and keep counting from there. That means that we could have said 6 for the first group, then counted a 7 to add on the last 1.

Here’s another example. Say we’re putting together 7 and 2. When you count the first group, what’s the last number you’ll say?

Child: I’ll be counting 7 things… so the last number I’ll say is 7.

Parent: Now keep going! Remember, we want to count the 2 we’re putting together with the 7.

Child: I already got to 7, so now I say 8, 9. The total is 9!

Usually, once a child is fluent in the previous stage of counting on, this stage is not particularly tricky. One possible issue can be a child who understands the idea but is not sufficiently familiar with the counting sequence to starting counting at a number other than 1. In that case, you should continue practicing counting on via fast counting until the counting sequence is absorbed.

To practice this skill, you can use any of the ideas from my previous post, except that this time you will tell the child not to count the first group at all and will instead ask them what the last number they’d say if they had counted it fast would be.

In my experience, this stage transitions smoothly to kids being able to quickly add 1 or 2 to any number. Counting on also opens a lot of doors to mental manipulations of numbers, as we’ll see in later posts!

This is the fourth and final entry in a series of posts about counting on. Here are the first, second, and third entries in the series. These lay out precisely how I teach counting on, as well as my motivation for teaching it like this.

# Lesson Plan: Fluent Counting On

Required skill: understanding how to add two groups by counting the first group quickly, as described in my last post.

So far, we’ve talked about why “counting on”﹣ that is, the method of addition that involves counting up from the first number plus 1﹣is not actually obvious, and about how you can get started on teaching it. If you followed the last lesson, your child can now add 6 and 2 without counting every single item in the group of 6: rather, they can be reminded to “count quickly” and to say 1, 2, 3, 4, 5, 6 without actually pointing at each object.

What should you do next? After all, we don’t want your child to say “1, 2, 3, 4, 5, 6, 7, 8, 9” when adding 8 and 1 indefinitely, no matter how quickly they rattle off the first eight numbers. Eventually, we want to them to simply say that 8＋1 is 9!

The next step in building this skill is to work on fluency. Just because a child has understood an idea doesn’t mean that they have fully internalized it or made it their own. The best way to allow them to do so is to provide concrete situations where the skill is needed, and this is easiest to do using either games or concrete objects. Here are some suggestions:

1. Play Addition War. This game works just like War, except that on each turn, a player puts down two cards instead of one. The winner is the person with the highest total. (And if there’s a tie, that means a war, which is always very exciting for the kids!)

Since cards have the numbers written on them, it tends to be fairly easy for kids to do “fast counting”: they can already see the numeral on the card, and therefore they already know how many dots are on the card and how high to count. Plus, there’s a new addition problem on every turn, allowing your child to get lots of practice.

You can also play other card games involving addition. For example, the game of blackjack was a surprising favorite in many of my classes.

2. Play games in which you add dice. You can take almost any game standardly played with a single die and instead play it with two dice, where the total of the dice is considered to be the roll for that turn. In my homeschool classes, I usually used Chutes and Ladders, but lots of games would work.

This has the advantage of allowing your child to work on subitizing (rapid judgment of quantity) at the same time as counting on. Since you’ll be encouraging them not to count every single dot on the first die, they will need to learn to quickly figure out how many dots are on a die without counting.

In the same vein, any Tiny Polka Dot games involving addition can be used to both practice subitizing and counting on. In my classes, I used to set up games of Concentration in which the object was to flip over two cards that added to 10, but there are lots of other possibilities!

3. Concrete situations which require adding. This can really be anything. “Look, I am showing 5 fingers on this hand and 3 on the other! How many is that in total?” “This bowl has 6 M&Ms and this one has 2. How many M&Ms do we have all together?” If a child has access to counters or is proficient with using their fingers, any addition question can be made into this form: if when asked to do 4＋3, the child shows 4 fingers and 3 fingers and counts them, this can be a good opportunity to remind them how to count on.

Note that this skill is best practiced when the first number is known and the second number is small. If the first number is not obvious, then counting from 1 may very well be the correct strategy, foiling your attempt to teach this shortcut. (If you have a bowl with some unknown number of M&Ms between 10 and 20, and a second bowl with 2 M&Ms, then your child will not be wrong to count from 1!) And I don’t tend to like “counting on” as a strategy when the second number is above 4 or 5, since it gets tedious and ineffective. When the second number is larger, there are better ways to add, as we’ll learn later on!

Another thing I want to emphasize is that this skill is best practiced in situations where there’s a concrete model to fall back on. If a child is having an off day or forgets the trick, they should not be made to feel like they no longer have a successful method for solving the problem: they should still be able to do the problem by simply counting from 1. After they do so, you can remind them how to do this quicker and ask them to do it again, but they should not feel like they did it “wrong.” After all, they didn’t! They were simply slightly less efficient than they may have been.

After a certain amount of practice, you will find that your child almost always uses the “fast counting” strategy instead of counting from 1. How long this takes depends on a lot of factors, such as how often you practice, the age of your child, and their overall level of mathematical aptitude. I recommend practicing this as often as possible to achieve fluency quickly. At that point, you will be ready for the last stage of “counting on,” in which a student can quickly and intuitively answer questions like 9＋1, while still understanding that they are doing the same thing as they do when they count all the items starting from 1. I’ll explain how to get there in my next post.

By the way, the feeling that what they are doing matches their intuition and mental model is precisely the thing that gives kids mathematical confidence. They aren’t learning something new: they are learning shortcuts for things they can already do! As we’ll see over the coming months, this feeling is key.

This is the third entry in a series of posts about counting on. Here are the first, second, and fourth entries in the series. These lay out precisely how I teach counting on, as well as my motivation for teaching it like this.

# Lesson Plan: Starting to Count On

Required Skills: a mental model of quantity, fluent rote counting up to at least 10, a mental model of addition.

In my last post, I discussed the fact that “counting on” is not in fact obvious to most kids. So how does one teach a kid to count on without interfering with their natural number sense but also without waiting for them to discover it themselves?

The key idea is to use a kid’s natural mental models of quantity and addition. What are these models? Most young kids figure out quantities by counting. Through their real-world experience, they also tend to have a robust understanding of what it means to “put numbers together.” That means that kids are, in fact, able to answer questions like “What do we get if we put together 2 cookies and 1 cookie?” They will probably answer it by actually showing the numbers using counters or fingers (or even actual cookies!), but they are already able to solve the problem. As you’ll see, that’s a crucial feature of working with mental models: when a child already has the requisite mental model, they will already be able to solve the question you’re working on. The issue will be that they will not yet be using an efficient method.

All right. Say that we have a child who can figure out a question like “What do 8 and 1 make when put together?” by counting, and say that we want them to do so without the tedious counting of 1, 2, 3, 4, 5, 6, 7, 8, 9 every single time. In other words, we want them to count on. Where do we start?

Here’s our crucial observation: instead of teaching them to count on like it’s separate and different from what they were doing before, we work in the context of the child’s original mental model. We take what they were doing already (which is counting from 1), we work with their perfectly valid process, and we point out that there’s a faster way to do it! Here’s how it works:

Parent: Hey, can you figure out what 5 and 2 make when put together? You can use these counters.

Child: *gets out counters, makes a group of 5 and a group of 2, counts 1, 2, 3, 4, 5, 6, 7.* It makes 7!

Parent: Great job! Now I’m going to teach you a faster way to do this. Let’s arrange your counters in the two groups of 5 and 2 again. Remember, we want to see how many this makes, so we’re going to count all of them together. Let’s start counting with the group of 5: 1, 2, 3, 4, 5…

Hey, wait a minute! We got up to 5. That’s not surprising, is it? That’s a group of 5, so we’d expect to get to 5 if we counted this pile. So really, we could count these quickly, without even looking at them. Let’s try it. Can you count these quickly, without even pointing to them?

Child: 1, 2, 3, 4, 5.

Parent: Very good! Now remember, we’re trying to count all of them. And we’ve already counted the first group! Keep going…

Child: …6, 7.

Parent: Nice work! Look at that… you added the numbers without having to actually to point at any of the counters in the first group. Let’s try it again! Let’s add 7 and 1. *makes a group of 7 and a group of 1.* Now, let’s count how many we have all together. Can you start by counting this group of 7? Try to do it quickly, without actually pointing to each counter.

Child: I can’t!

Parent: Remember, there are 7 counters in this group. We’re putting together 7 and 1. Think about what happens when we count 7 things.

Child: Ohh, right! 1, 2, 3, 4, 5, 6, 7.

Parent: And we need to count all of them, so keep going.

Child: …8. So there are 8 in total! 7 and 1 makes 8.

Parent: Very good! So remember, when we’re putting together two groups, you can actually count the first group quickly, without having to point at each item. That’s a good trick to make putting together numbers faster.

At this point, the child has reached the first stage of “counting on”: they can figure out the total number of items in two groups put together by counting the first group quickly, without actually having to point at every single counter in the group. This will make it faster for them to add.

Note that even though the child now knows this shortcut, they may very well forget it. The good news is that you can remind them within the context of their own mental model and their own methods. “Are you putting together 6 and 2? Yes, you can figure this out by counting all of them if you like. But remember, this group has 6 items. What numbers are you going to say if you count them first?” The child will not feel like they were doing something wrong by counting from 1: they will merely be reminded that there’s a faster way to do it, and that perhaps they should take advantage of it.

This example illustrates a recurring theme of teaching via mental models: a new method will not seek to replace the child’s original way of figuring things out. Rather, it will build on top of it. If a child forgets a new shortcut, they will still be able to figure out the question, and it will be possible to remind them without invalidating the old method. This is because both methods are built on the same mental model. We will see many examples of this in the coming months.

Note that if a child is not yet fluent with rote counting, this may not work, as they may get stuck after saying “1, 2, 3, 4, 5” and not be able to continue the sequence, instead starting at 1 and saying “1, 2.” If that happens, practice rote counting and adding by counting from 1 for a while and come back to counting on when the sequence is fully internalized.

In my next post, I’ll describe how to work on transitioning from the “fast counting” version of counting on to the traditional version.

This is the second in a series of posts about counting on. Here are the first, third, and fourth entries in the series. These lay out precisely how I teach counting on, as well as my motivation for teaching it like this.