Solving the Algebra Puzzle Gr. 1 - 3
Working with equations can be a huge puzzle for students in later elementary and secondary school. Part of the reason for this is that algebra involves a lot more than moving numbers around. There are key concepts underlying algebra. When students understand these key concepts, algebra makes a lot more sense. In this blog, I share information and activities about how to help your child build a strong foundation for later algebra.
What makes algebra so tricky?
Research dating from 1975 to the present shows that many young children have serious misconceptions about the equals sign.
In one study, a significant number of elementary aged children were not able to correctly answer this question:
The most common incorrect answers were 12 and 17.
Students who answered 12 just ignored the 5.
Students who answered 17 added all the numbers.
When children give these answers it's usually because they think the equals sign means “the answer is coming”. They see it as a signal to operate (Van de Walle et al, 2015).
When asked whether these statements are true or false, many children are
only sure that the first equation
(4 + 5 = 9) is true (Carpenter et al, 2002).
Children need to understand that all of these equations are true because in each, one side is 'the same as' the other side. Children need to learn that the equals sign shows a relationship.
This is not an idea that we can just tell children and expect them to understand. If that were the case, these research results would be much better!
This puzzle shows some of the key ideas that children need to piece together to create a a more fulsome, early understanding of equality:
As students develop this more fulsome understanding of equality they will begin to develop strategies for solving problems like 8 + 4 = _ + 5. The strategies they use will evolve over time as they develop a stronger understanding of relationships. The image below shows some of the milestone strategies students might use in grades 2-3. Click on the image to make it bigger.
Activities for Grades K - 3
Many of the games from Easy Games to Grow Big Ideas about Number help your young child build understanding that numbers have smaller numbers nested inside them. The activities described below show you how to continue to build this idea along with the other key ideas related to equality.
Number Race (Grades 1-2)
Roll a pair of dice and find the sum. Move the game token on the corresponding number path forward one space. Cheer for your numbers to come up and race to the finish!
Download this game board or draw one like it on your driveway with sidewalk chalk.
You will also need:
a pair of dice
3 objects for each player to use as game tokens. Choose unique objects that you can tell apart. Coins work well.
Before playing, players need to pick their numbers to cheer for in the race.
Each player rolls one dice. The player with the highest roll gets to pick a number first. That player selects a number and places one game token at the start of that number’s path. Take turns choosing paths until there is a token at the start of each path.
In the game shown above, player one (yellow) had first pick and chose 6. Then player 2 (red) chose 5, and so on.
Playing the game:
On your turn, roll the dice and figure out the sum of the two dice. The sum tells you which token gets to move on the game board. In the game shown above, the sum of 5 has been rolled twice, so the token on the 5 pathway has moved 2 spaces. The sums 2 & 4 have not been rolled at all so those tokens are not yet on a path. If you roll a sum greater than 7, keep rolling until you get a roll that works.
Take turns rolling the dice and moving the tokens forward. When a token reaches the finish line, the game is over. Loud cheering for your numbers is encouraged during the game!
What is my child learning by playing this game?
Mystery Coins (Grade 2-3)
Figure out which coins your partner has hidden in their hand, using as few questions as possible.
Before playing this game, your child needs to know the values of coins up to 25 cents and be able to add these coin values.
You will need a collection of nickels, dimes and quarters.
How to Play:
Secretly select a combination of coins with a total value of 25 cents. Hide the coins in your hand.
Your child will ask questions to try to figure out which coins are in your hand. Only questions with a yes or no answer can be asked.
Are any of the coins dimes?
Are the coins all the same?
Keep track of how many questions your child asks. The goal is to guess the mystery coin combination using as few questions as possible. Play several times with the goal of decreasing the number of questions asked.
Each time you finish a game, record the
combination of coins as an equation, as in the example in the image.
Once you have two different combinations that make 25, write an equation that shows that the combinations are equal to one another.
You can write the equation with or without making them look like coins.
Give your child a turn hiding the mystery coins too.
Extending the learning: Change the total of the coins to fifty cents.
What is my child learning by playing this game?
Partitioning Problems (Gr. 1 - 2)
Here we will take an in-depth look at one example of a partitioning problem:
Jo has 7 pet mice. They live in two cages that are connected by a tunnel so they can go back and forth between the two cages. Show all the ways that the mice can be in two cages.
In solving this problem, your child will be figuring out all the combinations of two numbers that make seven.
You can adapt this kind of problem to use with any number. In the Ontario curriculum, by the end of grade 2, children are expected to partition numbers to 18. You can also change up the context. I’ve suggested some additional context ideas later in this post.
The important thing you want your child to notice with these problems is that no matter how the items are arranged in two groups, the total always remains the same.
Here’s an example of a conversation in which the parent is helping the child notice this important idea:
You'll need to decide whether or not to show your child how to write each combination as an equation. This depends on your child’s prior experience. If they have prior experience with writing addition equations, they are likely ready. If not, wait until you have done this kind of problem a few times. It's important to develop the idea of equality before using abstract symbols like the equals sign to represent the idea.
Your child may come up with four more possibilities for arranging the mice in cages. These will be the opposite arrangements of those shown above.
The order of the numbers in an addition equation doesn't matter: e.g. 1 + 6 = 6 + 1. However, children don't know this at first. Problems like this help develop this important property of addition. Write the equations to match the situation in the cages for now as that will likely make the most sense to your child.
If your child doesn’t find any of these reverse combinations, try a prompt something like this:
Wow, you found 4 different ways the mice could be in the cages. I think I have one more way. Could there be 1 mouse in the red cage and 6 in the blue cage?
When your child has had some experience with using equations, you can introduce using the equals sign to show that two of the combinations have the same value, like this:
6 + 1 = 4 + 3
Tell them that "mathematicians use the equals sign to show that two amounts are the same. This shows that 6 mice plus one mouse is the same as 4 mice plus 3 mice."
If we introduce children to a variety of equation types when they are first learning about equations, children will not develop the narrow understanding of the equal sign that sometimes occurs.
What is my child learning about algebra by solving partitioning problems?
Here are some examples of other contexts for this type of problem:
Eighteen children are at a pool party. Some children are in the pool and some are out of the pool. Show all the ways the children could be in and out of the pool.
Students are making cupcakes at school. Each child is allowed to put 10 candies on their cupcake. There are red candies and green candies. Show all the combinations of red and green candies that could be on a cupcake..
Your child will be most engaged if the problem is about a topic they are interested in. Be creative!
Prompting your child to think about and use relationships between numbers:
When you think your child is ready, explore what happens to the numbers in the equations when you solve a partitioning problem systematically. Here's an example using the pool party problem:
You told me you started with 10 people in the pool and 8 people out of the pool, like this:
10 + 8 = 18
Then you imagined that one person got out of the pool. That made 9 people in the pool and 9 people out of the pool, like this: 9 + 9 = 18.
And you kept doing that. So what would the next situation be if one more person got out?
How would we write that as an equation?
8 + 10 = 18
And the next?
As you are talking, write the four equations as a list and then ask:
Do you notice anything interesting about the numbers?
Why do you think that happens?
There are two things we want students to notice:
One - the combinations all equal 18. Second - as we go from one equation to the next, one number increases by one while the other number decreases by one. And this is why the total remains at 18.
This is an important idea called compensation. Eventually we want students to realize this idea holds true for any number. For example, if we decrease the number of people in the pool by 2, the number of people out of the pool has to increase by 2 to keep the total number at 18.
Your child may not notice anything interesting about the list of equations if they are not yet used to looking for patterns and relationships. If this is the case, point out what you see happening to the numbers on the left. This might prompt them to notice what is happening to the second set of addends. If they don’t notice with that prompt, they probably need more time doing similar problems with concrete materials.
Comparing Cost Problems (gr. 2-6)
Which is more expensive, a toy bear or toy puppy? How much more expensive?
Make a new combination of toy bears and puppies. What is the cost of the combination?
Make a group of only bears or only puppies. Then find the price.
What is the price of one bear? One puppy?
Here's the same problem with larger numbers:
Which is more expensive, an umbrella or a ball cap? How much more expensive?
Make a new combination of umbrellas and caps. What is the cost of the combination?
Make a group of only caps or only umbrellas. Then find the price.
What is the price of one umbrella? One cap?
What is my child learning by doing these problems?
reasoning with unknown quantities
Your child will be developing algebraic thinking skills!
Your child's early math experiences matter!
I encourage you to try out some of these activities with your children.
If you have questions or comments, please post them.
Fosnot, C., & Jacob, B. (2010). Young Mathematicians at Work: Constructing Algebra. Heinemann.
Van de Walle, J., Karp, K., Bay-Williams, J., McGarvey, L., & Folk, S. (2015). Elementary and Middle School Mathematics: Teaching Developmentally, Fourth Canadian Edition. Pearson.