June 2023
Welcome to EFM's June Newsletter!
EFM: Supporting families to play, explore, and love math
News
5 Stages – New Layout for Activities for Families – We have a new look for our Activities for Families in English. The two-column, two-toned look is gone. We now call them “Stages” instead of “Chapters,” and many descriptions have been expanded. Download the new colorful version and have even more fun doing math together!
Playing Cards – The final version of the deck of 52 math puzzles for K-3 is done and the order sent to the manufacturer. They should be available in about two months. Please let us know if you're interested in them. You can check them out at our Deck K-3 page.
Donations – We received a $5000 and a $200 donation this month! Donations make a big difference in what we are able to do. These help make possible trial runs of playing cards, paying for our website and software tools, and perhaps someday paying for physical copies of our annotated storybooks. As a 501(c)(3), all donations to EFM are tax deductible to the extent allowed by law.
Splat! and Quantities
This month I would like to look at some of the free math education ideas of Steve Wyborney from his website: https://stevewyborney.com/. I’d like to discuss his Splat! and quantities material.
Not Just an Answer
As pleasurable as finding an answer is, the guidelines for these activities emphasize the importance of discussing reasoning and fresh thinking. After presenting an answer, a child is asked how the answer was arrived at, and what are some other ways of finding the answer. Discovering additional ways to find an answer is an intriguing puzzle of its own. It emphasizes that finding one answer is not the sole purpose of the journey, and can be the start of further fun.
The discussions are valuable in their own right. Learning how to discuss logical reasoning is an important skill for a child to develop. It also helps reinforce that math should be a social activity that is shared with others in a spirit of fun and adventure.
These activities emphasize quantities and develop subitizing and estimation skills. The method of understanding each quantity through multiple representations and approaches adds significant depth to the learning.
Splat!
Steve’s Splat! idea intrigues me because it starts out with the same ideas as two EFM activities – What’s Missing? (Stage 2) and Shape Sums (Stage 3). That Splat! works so well and is so popular reminds me that an important ingredient of good teaching is storytelling and presentation. Adding a “splat” and a physical aspect makes this much more engaging.
Beginning Splat! starts out the same as What’s Missing?, and then it has many variations that develop into deeper levels. It starts with a number of small dots on a page. The number of dots is indicated on the page or is counted by the child and recorded. Then the “splat” happens – an irregular shape (called the “splat”) is placed over some of the dots.
The typical steps for Splat! are:
Count the number of uncovered dots you see and compare to the total.
Find one way to figure out the number of dots covered by each Splat.
Find other ways to figure out the number of dots covered by each Splat
Verify your answer by removing the Splats.
Example:
In this example, nine dots are shown and counted. A grey splat then covers some of the dots. Four dots are left uncovered, so there must be 9 – 4 = 5 dots under the splat. The 5 can be discovered by counting on, counting down, or whatever methods the child has for finding differences.
What other ways are there to find the 5? If we start with the 4 uncovered and then add 4 more, we are 1 away, so there must be 4 + 1 = 5 dots that are covered. Perhaps your child knows that 9 is three 3’s. Then, counting by 3’s shows that the splat is covering 3 + 3 – 1 = 5 dots. Encourage discovering all sorts of novel and unusual approaches – that is the best part of doing Splat! At the end, take the splat off and check your answer.
Multiple Splats:
One way to add complexity and interest to Splat! is similar to what is done in EFM’s Shape Sums. If more than one Splat is used, then these two rules apply:
Splats with the same color must have the same number of dots
Splats with different colors must have different numbers of dots
Example:
In this example, 17 dots are shown and counted. Then two grey splats and one blue splat are used to cover some of the dots.
There are four visible dots, so 13 dots are covered. What are the possibilities? One answer is that grays have 4 dots and the blue has 5 dots. You can see that by counting by 4’s – 4, 8, 12, and 1 more. How else might this work or be thought about? Perhaps two 3’s and one 7? Perhaps two 1’s and one 11? Why does the blue splat always have an odd number of dots for this puzzle?
Dots With Different Values:
Another layer of complexity is to have the dots have a value other than 1. In beginning Splat!, all the dots have value 1 and you can count them to see their total value. In this version, the dots have values written on them. The values might be drawn from a few whole numbers, such as using 2’s and 5’s, or some of the values may be a basic fraction such as ¼.
Example:
Every dot is either worth 2 or 5. These puzzles are great for skip counting practice. These dots add up to 23. The splat covers up all but 11, so it must cover 12. The puzzle is how to get 12 with 2’s and 5’s. Once 5 + 5 + 2 = 12 is discovered, the next question is whether there are other ways. Perhaps six 2’s? Why won't one 5 work?
Yet another variation is to specify a relationship between different colors of splats. For example, red splats are 3 bigger than black ones, or blue splats are twice as big as green ones. Splats give you the tools to pose many kinds of interesting puzzles that prompt great discussions.
Estimating with Number Clues
Esti-Mysteries use a transparent container containing some colorful objects. The child is asked to estimate how many objects are in the container. There are a small number of numerical clues to help the child narrow down their estimate to find out the exact number. Because this starts with an estimate, and keeps the visual in mind, the numerical clues have more depth because they are grounded in the quantity.
Example:
Estimate the number of toys in this jar. Here are some clues, to be revealed one at a time:
1. The amount is a two-digit number, and its tens digit is less than 4.
2. The amount is an even number.
3. The ones digit is less than 5.
4. There are the same number of toys for every color.
5. The amount is only divisible by one prime.
At this point you know there are 4 x 8 = 32 toys in the jar.
3-Container Estimation
Have 3 containers that have various amounts of some interesting objects. Working with these 3 containers uses the following six steps, with plenty of discussion during each step:
1. Estimate the total number for all three containers
2. Reveal the total number
3. Estimate the number in a chosen container
4. Reveal the amount in that container
5. Estimate the number in the two remaining containers
6. Reveal the amount for each of the containers
Example:
After looking at this picture, estimate the total number of toys. Then reveal that there are 24 total toys. Next pick a container, say the left one, and estimate its quantity knowing that there are a total of 24 toys and it looks like that container has more than either of the other two. After estimating its number and discussing the reasoning, reveal that it has 10 toys. Finally, estimate the final amounts for the remaining two containers knowing that they have 24 – 10 = 14 toys. They appear to have similar amounts, so maybe 7 + 7 or 6 + 8. After discussing everyone’s thinking, reveal that they each have 7 toys.
Subitizing
What do you see, and how else can you see it? Those are the central questions used to develop the ability to see and understand quantities. This is similar to Open Middle’s Dot Card Counting activity.
With the first group on the left, some might see this as three horizontal groups of 5 plus two more. Some might see it as four vertical groups of 3 plus one more vertical group of 5 in the middle.
For the second group, it could be seen as two horizontal 3’s plus one horizontal 5. Alternatively, two of the dots on the top row could be slid down to the gap in the bottom row to form two groups of 5 plus 1 more. It could also be seen as four vertical groups of 2 plus 3 more in the vertical column in the middle.
The third group could be seen as three “+” groups of 5. Or, it could be seen as two diagonal groups of 6 plus another diagonal group of 3.
There is no one right answer for any group of dots. What is important is to develop a visual feel for small quantities and to grow a sense of how numbers relate by fitting the numbers together. Have fun with this, and challenge each other to find fresh ways of looking at each diagram.
If you have any questions or comments, please send them my way. I would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, I would love to talk with you about ways we can work together!
Chris Wright
June 18, 2023
Chris@EarlyFamilyMath.org
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