Yesterday was not one of our more educationally productive days. Still, we did our best to practice what we’ve learned and apply that learning to new situations.
Olive decided early in the day that she wanted to make deviled eggs. She was very eager to explain to me how to make the hardboiled eggs and the materials we would need for making them. We gathered up what we needed and then I asked the all important question, “How many deviled eggs do you want?” Olive’s answer was, ten. I took advantage of Olive’s answer by asking her how many eggs we would need. She, of course, answered with ten eggs. Welcome to the world of ratios (and pre-algebra)! Think back to the days of, “One egg is to two deviled eggs as ____ (fill in the blank) is to ten deviled eggs.” Olive and I used a carton of eggs to look for a pattern. She knew and understood that one egg would yield two deviled eggs. We continued by counting up (by twos) and when we got to four eggs Olive had figured out that we would have eight deviled eggs. When I suggested we add one more egg she told me that we would have nine deviled eggs. I had to repeat the pattern of counting by twos in order for Olive to finally determine that we would get ten deviled eggs from just five eggs. I could tell that Olive was still unsure as to how she arrived at her answer. I suggested that she draw and label a picture of our set up. She agreed and the drawing above is what she came up with. Olive divided her egg drawings in half (and used self talk while drawing the lines) to show that each egg makes two deviled eggs. We added the labels later. By the way, Olive explained to me that I had her drawing upside down. Olive quickly lost interest in our cooking/math lesson and moved onto a new project. She did help peel the eggs, which was not a success. She ate a few and then moved on to playing with her toys. I’m never sure as to handle these situations. That is, when a child or student loses interest in a lesson or project what should I do? In my opinion, there’s not much one can do. Cooking offers up a number of learning ideas, and it relates to math in so many ways! Just remember to avoid math terms and, instead, stick to the practical explanations. Have kids draw and label pictures, look for patterns, and talk about their thinking. Feel free to share your ideas and thoughts on this matter!
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Comparing how much we have of something is a skill we use all the time. Similar to classifying, we use this skill without giving it any thought. I’m sure early humans were quick to realize who had more of just about anything imaginable. But, how did they do this without an understanding of numbers? Quantifying is an abstract concept.
Using Olive’s toy creatures, we played the following game to help us compare who has more and who has less. Materials: A die Twenty or more small toys such as Legos, cars, stuffed animals, etc. Room to place small toys in a row for comparison How to Play: 1. Place all the toys in a pile between players. 2. Take turns rolling the die. 3. The number that shows up is the number of toys taken from the pile. 4. Keep playing until all the toys from the pile have been removed. 5. The winner is the player with the most toys. Okay, there is a LOT of learning going on here and it’s not all mathematics. Olive and I talked about taking turns. We talked about encouraging one another to do our best. I explained to Olive that this is really a game of chance, which she didn’t understand. We talked about which number is the best number to roll. (Olive and I agreed that the number six was the best roll. Six would get us the most toys.) I asked Olive if I could roll a seven and why not when she told me no. Olive had a tough time when I took one of her favorite toys. I had to explain repeatedly that I would give it back. Some times I even went so far as to trade the one I took for a different one. Once all the toys were removed from the pile, we each counted how many we had in our own piles. Olive counted her toys correctly, but she had a difficult time comparing my quantity to her own quantity. It was as if holding those two numbers in her head and then comparing them was just too much to do at once. Olive and I decided it would be easier to compare our quantities by lining them up in two long rows. (See the photo above.) She matched up one of her toys for every one of my toys. As soon as she realized she had more to lay out in her row she knew she had more than me. We practiced counting them again and again talking about which number was more. In the end, however, I suspect Olive needed the visual representation to determine that she had more than me. Variations: Olive and I carefully divided up her small toys (so that we each had the same number), and placed them in a line. That is, Olive’s row of toys were across from my row of toys. We each rolled the die and subtracted that number from our row of toys. The player with a toy or toys still remaining was considered the winner. Conclusion: I talked to Olive about even and odd numbers. She didn’t understand this concept. That’s okay. It will take using her toys and other objects to make sense of even and odd. Olive does understand fairness and can apply this to the times I’ve given her half of my sandwhich, half of a candy bar, or half of a cookie. She also uses the concept of fairness to divide up her toys (evenly) when playing with Mom or me. I’m not always a huge fan of competition when playing these types of games. That is, I’m careful when describing one player as the winner and the other as having lost. For some individuals competition can be a true motivator and for others it can cause a lot of discouragement. That is, learning can get lost in the spirit of competition. What are your thoughts on this matter? How do you keep your kids involved in a game in spite of not always coming out as the winner? Please share your ideas for similar games in which children can compare quantities. I am certain there are games parents can make at home using items around the house. What about money, for example? Just remember, quantifying is an abstract concept and one that must be practiced using tangible objects. Stay tuned! In my next post I’ll write about number sense and ways that Olive and I are practicing this important skill. Look carefully. Can you determine what property Olive used to sort her toy animals? It’s a bit tricky, but it will make sense in a bit.
Identifying properties is a useful skill. We do it all the time, and probably without giving it much thought. It’s how we make sense of the world. Olive used the property of legs to sort these creatures. The animals on the left have legs and the animals on the right don’t have legs. It’s a simple dichotomous key. Remember sorting and identifying leaves in high-school biology? That’s a dichotomous key. I’m sure you’re thinking about some of those creatures on the right. Like me, you’re probably wondering about the starfish. Doesn’t it have legs? Perhaps it does. Olive and I talked about what are legs. At one point, we even compared her starfish to Patrick Starfish from Spongebob. After all, he walks and talks. Patrick has arms as well. (Olive tried walking on her arms.) It was, for a five-year old, an interesting conversation. Does it really matter where that starfish is placed? Probably not. Think back to your biology class. Did the leaves you were sorting fit neatly and nicely into a category? But, guess what? We never did come to an understanding about what constitutes legs. In Olive’s developing brain that starfish did NOT have legs. In fact, upon closer inspection there are several other creatures in her NO LEGS pile that do have legs. What about the octopus? Even Olive knows an octopus has eight legs, but she still put it in the No Legs group. Her answer? “I don’t know. I just did.” Again, does it matter? Again, probably not. So, the next time you and your children are playing with stuffed animals, Legos, beads, dolls, or whatever, ask them for different ways to group their toys. It will make for an interesting conversation. Just remember, there are no wrong ideas! Feel free to share their ideas and what your children come up with. In tomorrow’s post, I’ll talk about how Olive and I came up with a counting game using these same toy creatures and dice. I have finally completed all four video tutorials about how to use the multiplication table for more than just multiplication. Here's the breakdown along with a link to the YouTube videos.
Part I: Using a Multiplication Table for Multiplication Part II: Using a Multiplication Table for Division Part III: Using a Multiplication Table to Find the Least Common Multiple of Two or More Numbers Part IV: Using a Multiplication Table to Find Equivalent Fractions and Missing Quantities in a Ratio Table Please do not forget to subscribe to my YouTube page. Also, please like the videos as well as leave me a comment. Let me know if there are additional video tutorials you would like for me to create and upload. Finally, go to my resource page and download your very own copy of the multiplication table I use in my videos. It's free! Number Sense:
What is number sense? It’s our ability to think about and use numbers flexibly. Number sense is part intuitive and part learned. Ask any five-year old to guess an adult’s age and they will give a range of ages from 10 to 100. Show a child a jar full of M&Ms and ask how many there are and they will answer anywhere from 5 to one million. Number sense is something we develop over time and it comes with our experience with numbers. It happens as a result of playing around with quantities and putting objects together to compare them, to count them, and to manipulate them. Number sense is especially important to understanding concepts such as addition and subtraction. What’s 199 + 201? Without much thought, we know it’s 400. These numbers are very close to one another, but to a young person these numbers are very different. Physically, they look different. After all, there’s a one followed by two nines. Two-hundred one has small numbers, a two followed by a zero and a one. This is all new to young children. Tens Frame: Using the Tens Frame is one way to help children learn about number sense. In addition to the link provided, a complete set of Tens Frames can be found under the Resources tab. I’ve printed off several sets, laminated them, and keep them nearby. I will occasionally pull a few out and have Olive practice identifying the quantity or adding together two Frames. Ways to Use the Tens Frame: The video above shows Olive (and her friends) adding together two Tens Frames. I like to start with the ten frame and one of the other frames, for example the two frame. I hold the ten frame in my right hand and the two frame in my left hand. Olive looks at them and then tells me the total. In this example she would answer with, “Twelve”. Now, and this is where it gets interesting, as the Tens Frames increase in quantity Olive will stop and count on. Watch the video again and listen carefully to when Olive is shown a ten frame and a nine frame. Notice how she pauses and then starting with ten she counts on to arrive at a total of 19. Counting on from a number is a big deal! Olive’s kindergarten teacher calls it Zip Counting. Don’t be surprised if some children count all ten dots and then continue counting the other nine dots. That’s okay. It’s not uncommon for young children to have a difficult time telling an adult what number comes after ten. Before Olive became proficient at counting on she would start at one and count up to tell me what comes after ten. We still practice counting on, especially when it comes to dice. Often times, Olive will count every dot on the dice instead of using the counting on strategy. Conclusion: There are a number of other activities children can do using the Tens Frames. For example, with two sets they could play Concentration. In a future post, I’ll share how Olive and I used two sets to play the old card game called War. What are some other ideas you might have for using the Tens Frames? Feel free to share here or on social media. Parents, I’m going old school in today’s post.
It’s all about the multiplication table. Back in my day . . . before smart phones, tablets, and calculators . . . we had to rely on a multiplication table to help us with those pesky math facts. Oh, how I dreaded the day when we were asked to stand before the class and recite the times tables. (Luckily, our teacher spared us the 11s and 12s tables.) Unfortunately, I’m finding that some of my students are struggling with how to use a multiplication table. For whatever reason, using a multiplication table is taboo. I understand we want our students to learn their math facts, but I don’t want students struggling with inefficient strategies to solve a fact such as 9 x 7. I strongly encourage them to use a multiplication table to help move their learning along. We can work on math facts while at the same time still learn math. With the school year about to start, I have uploaded Part 1 of four video tutorials about the multiplication table. In the first video I review how to use a multiplication table to solve multiplication facts. In the remaining tutorials I will cover how to divide, find the least common multiple of two or more numbers, how to find equivalent fractions, and how to find the missing part in a ratio table. The multiplication table is more than multiplying! To view the video, click here or head on over to the Tutoring page and look for the video titled, Multiplication Table Tutorial, Multiplication. I will continue to upload videos over the next several days so check back often! Don’t have a multiplication table? You’re in luck! Click here, or go to my resource page and download and print a free copy of the multiplication table used in my video. In fact, print out a bunch of multiplication tables to keep around the house, in the car, in book bags, and at school. They’re free! Give some to friends and family. Please let me know if this has helped by leaving a comment below. Better yet, please share this post with friends and family. Thanks!
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