Use formative assessment question prompts in order to encourage independent student thinking in the classroom! Includes question prompts for introduction to regrouping.
Download HandoutK.NBT.A.1 | Understand, explore place value within 11-19. |
1.NBT.B.2 | Understand, relate place value as groups of ten. |
Students identify the value of three digit numbers using a visual tool, the base-ten manipulative Digi-Block.
Download Worksheet and Answer Key2.NBT.A.1 | Understand 3 digit Numbers |
2.NBT.A.1.A | 100 is equal to a bundle of 10 tens |
If you want to change up the pace in the classroom, try switching around the math problems
Forty of the 100 1-blocks we see are green. The other sixty are teal.
We can check this fact visually by combining these blocks into a block-of-100.
So in the clear block-of-100, we can clearly see that 40 of the 100 unit blocks are green, or that 40 percent of the larger block is green.
What about these 100-blocks? Are they 40 percent green?
What we can tell from this is that where the green units are in the 100-block doesn’t matter –– the only thing that matters is the total number of greens.
So all of the images we see are all have 40% green blocks –– even the images where we don’t have a block-of-100!
This can be used as a good activity for students –- give four groups of students the four blocks above, with green blocks mixed throughout. Have your students guess if all four blocks are the same or not! After giving them time to discuss, have the students reorganize their blocks to look like Figure 2!
Download Handout6.RP.A.3.C | Understand percentage as a fraction of 100 |
Use formative assessment question prompts in order to encourage independent student thinking in the classroom! Includes question prompts for introduction to counting.
Download HandoutK.CC.B.4 | Understand, relate numbers and quantities. |
K.CC.B.5 | Understand how to count sets of objects |
A fun way to enrich a student’s understanding of place value is to play games that explore how place value works! Here’s a game that’s easy to set up and should help assess students’ understanding of place value.
All you need is:
Each time you play, you throw the dice, and transfer those numbers to flip cards to help keep track.
Now, you want to explore creating numbers using each die as the place values!
What’s the largest number you can make?
What’s the smallest?
2.NBT.A.1 | Understand 3 digit Numbers |
2.NBT.A.1.A | 100 is equal to a bundle of 10 tens |
2.NBT.A.1.B | 100 is equal to a bundle of 10 tens |
Watch Molly (a 4-year-old!) independently derive the division algorithm. With Digi-Block, long division becomes intuitive. Molly discovers this arithmetic operation without someone teaching her the steps or telling her what to do.
With Digi-Block, division is tackled in 3 intuitive steps. Click on the step to jump to that point in the video.
Step 1: Build the number in blocks that you want to share.
Step 2: Share the blocks equally into the number of groups you’ve determined.
Step 3: Count how many blocks each group got.
Notation, syntax and instruction get in the way of the learning process. All Molly needs to derive the division algorithm is a simple question from her instructor. Digi-Block creates a hands-on learning experience in which children discover place value and the four arithmetic operations.
Other base 10 blocks require teachers to explicitly outline every step of the process. As a result, these children perform memorized steps rather than thinking critically about how to manipulate quantities. With Digi-Block, you’re challenge as a teacher is not telling your students what to do, but rather, it’s how to tell them as little as possible.
4.NBT.6
]]>As children grow, they start trying to understand the relative size of things that they see. A toddler may think that a person and a house are about the same size. As children get older, they may understand that some things are bigger than others, but not have a good understanding of the relative size. An exercise that can be informative to children with both understanding of numbers and understanding of sizes is to use blocks for size estimation. Click here to see the interactive lesson demo. Pull the slider to see how tall a pencil, a child, a house and even the empire state building is in blocks!
For this activity, you will want a few items which have sizes that are easily understood by children. Some nearby examples may be a pencil, a book, and a friend. For each item, ask each child to estimate the height of the item in blocks and write down their estimates. The children may not yet have a sense of how to estimate, so a little guidance on how to estimate will likely be needed. One of the easiest ways to understand estimation is to work the children through multiplication and division– without telling them that’s what they’re doing! It can also help with understanding of base 10. For the smaller items that they will measure, get them to hold a block next to the item and ask them to write down how many blocks they think it would take to be as long as the item. Then, have them estimate how tall the larger items are compared to the small item.
Then have them count the blocks that the estimated for a pencil, and have them put the pencil blocks into holders. When they have finished filling the tens holders, tell them that if the last tens holder is more than half-way full to fill it up and if it is less than half-way full to put put it back in the pile (rounding!). Then have them put the groups of tens blocks together for books and gather as many tens blocks for the book that they estimated is the height of the friend in books and count them (multiplication!).
Next have the children get in groups of two to four so that they can share materials and have a friend to measure. After they measure each item, have them write down the height of the item in blocks so that they can compare their estimates to their measurements.
After that, you can share with them more items from this webpage to get a better understanding of how tall many different items are. Ask the children to reflect on why multiplication is such a useful operation when estimating incrementally larger objects. See all the items we measured in blocks!
Leave a comment with what objects you used in your lessons!
]]>Students set the correct digits for each place value mat. This free 1st grade worksheet also asks students to compare the place value mats to find the greatest number. Includes re-grouping. Comes with worksheet and answer key.
Download Worksheet and Answer Key1.NBT.B.2 | Understand 2 digit Numbers |
1.NBT.B.2.A | 10 is equal to a bundle of 10 ones |
1.NBT.B.3 | Compare 2-digit numbers |
1.NBT.B.2.C | Understand the positional system (e.g. 90 = 9 tens) |
Photo By: multiplication / Paul Goyette / CC BY 2.0
]]>Photo By: multiplication / Paul Goyette / CC BY 2.0
We have all witnessed a major shift in how educators approach the subject of mathematics in US schools. The focus is now on developing a “deep understanding” rather than teaching procedures. The common core math standards ask teachers to delve deeper into fewer topics. Instead of learning one standard algorithm for each arithmetic operation, students are exposed to multiple approaches and asked to explain why each algorithm works. By both positioning the teacher as guide rather than an instructor and empowering students to explore math topics for themselves, math is no longer a series of facts to be memorized but a continual process of reasoning and discovery. Even with this major conceptual shift sweeping the country, the term “math fact” is still alive and well.
In my opinion, the very idea of a “math fact” undermines the entire movement. In George Orwell’s book “1984”, Orwell uses the “math fact” 2+2=4 as the ultimate truth. While Big Brother has the omnipotent power to rewrite history and reduce truth to a intricate web of lies, He has no control over this one, last, dogmatic truth. Orwell chose 2+2=4 as the ultimate truth because this mathematical expression transcends cultural barriers and is seemingly beyond reproach. The very acceptance of 2+2=4 as the archetypal “math fact” illustrates how deeply ingrained a procedural approach to math is in our society. It is as if school-taught math is the true Big Brother.
What if I told you that I could easily make 2+2 equal something other than 4? I don’t need to use torture or a long mathematical proof. I won’t even have to lie to you. 2+2 = 11 in base 3. For many people this statement is utterly jarring. In America, the concept of a base is hardly touched upon on in grade school. Children are taught to unquestionably accept and memorize that we count to 10 by saying: “zero, one, two, three, four, …”. This rhythmic and almost mindless sequence is seared into our brains from the very beginning. But why does 10 come after 9? How can we blame children for approaching math as a set of facts when their first introduction to the subject is in the form of a hard and fast rule?
The English Language Arts also begins in a similar fashion with the alphabet song. However children don’t view the alphabet as a supreme truth. Very early on, children are exposed to the idea of other languages. The alphabet doesn’t necessarily have to begin with “a” and end with “z”. What is important to understand is that letters, or symbolic characters, can be strung together to create words. These words can then be used to represent or describe ideas. Young children soon learn that while the letters and sounds change, this underlying pattern does not. We all freely accept that a “house” is also a “casa”. Just as international languages are variations on a fundamental concept, our base 10 number system is one of many possible numerical schemes following the fundamental pattern of a base.
The purpose of a base is to allow us to encode any quantity with a set number of digits. Without the invention of a base, we would have to invent a new character for each increasingly larger quantity. While we use language to convey ideas and feelings, the numerical system gives us the ability to represent any quantity quickly and efficiently. In base 10, the base we commonly use, an infinite number of quantities can be sufficiently described using only ten digits. In base 2, the base that governs computers, an infinite number of quantities can be described using only two digits! Just like how you can greet people by saying “Hello” or “Hola”, you can represent a set of nineteen things using the numerical representation “19” (base10) or “1011” (base2). When you understand the pattern behind our number system, these possibilities feel natural. When you only know the facts, this alternate language feels viscerally wrong.
The term “Math Fact” is most prevalent when children are learning multiplication. As a child, I could recall my multiplication facts faster than any other student in the class. However, I never understood the idea of partial products. Unless I could solve a multiplication equation using the written algorithm or from memory, there was no way I was going to find the answer. I did not know that the written algorithm for multiplication was based on one of many ways to manipulate quantities. I only knew how to bring down a zero and carry a one.
Even during this time of change in mathematics, I still hear about teachers who ask students to memorize their multiplication facts before they have even introduced the idea of joining equal groups. A popular twitter post ridiculed a school for enacting a strict fact memorization policy. The elementary school defended the policy by stating: “in order for our students to be successful with higher level mathematical concepts, we first need to ensure that they can perform basic calculations fluently.” This statement shows a complete lack of understanding of what “higher-level” mathematics entails. Advanced mathematics is not the ability to reduce and solve longer equations. It is a more advanced form of pattern manipulation and critical thinking. In fact, Keith Devlin, a professor of mathematics at Stanford, describes all of mathematics as the science of patterns.
Schools focus on “fact fluency” because it’s easy to mandate and easier still to assess. Before the age of computers, it was a useful job skill to be able to quickly compute sums in one’s head. Today, companies need critical thinkers who can solve multifaceted and open-ended problems. Just like how children will never learn to read by memorizing word after word, they will never understand math by memorizing equation after equation. Children have to sound out each letter, hear the new sound they made and slowly start to recognize and remember those patterns. Think twice next time someone tells you 2+2=4.
]]>Photo By: Graphics From the Pond / Lita Lita
]]>The hens in the Chicken Coop need help figuring out how many eggs they can lay in a week. In this free 3rd grade worksheet your students will use addition, one digit multiplication, and critical thinking to figure out who lays the most eggs. Comes with worksheet and answer key.
Download Worksheet and Answer Key3.NBT.A.3 | Multiply one-digit by 10 |
3.OA.A.3 | Multiplication under 100 |
3.OA.A.4 | Relating multiplication and division |
3.OA.B.5 | Commutative, associative and distributive properties |
3.OA.A.1 | Interpret products of whole numbers |
Photo By: Krista Wallden
]]>Students will conduct addition under 20 on number lines and complete critical thinking questions. Comes with worksheet and answer key.
Download Worksheet and Answer KeyK.NBT.A.1 | Numbers under 20 |
K.CC.B.5 | "How Many?" under 20 |
1.NBT.C.5 | Find 10 more |
1.NBT.B.2 | place value w/ 2-digit numbers |
Photo By: The Enlightened Elephant
]]>
Give the students a division problem with an answer with a repeating decimal. For instance: 658 ÷ 3 (the answer will be 219.333333333 or 219.3 ̅)
The students start with 658 blocks and 3 paper plates.
When they divvy up the blocks, each plate gets 219 blocks, and we have 1 leftover.
The leftover single block “unpacks” to 10 tenths, and each plate gets 3 of the tenths, with 1 tenth leftover. (Not familiar with decimal blocks? checkout our introduction to decimal lesson plan)
The leftover tenth block “unpacks” to 10 hundredths, and each plate gets 3 of the hundredths with 1 hundredth leftover.
The students can explain what will happen each time we open the leftover block. They also understand that this pattern will continue infinitely. They explain to me what a repeating decimal is!
Similarly, 100 ÷ 3 is a fun problem. Some advanced students are able first to explain/draw/write about what will happen with the blocks. Then they can physically model the problem with blocks to check their answer.
]]>
When you use Digi-Block, the process of regrouping becomes demystified.The Digi-Block model let’s you visualize and conceptualize the act of regrouping. If you want to take 8 away from 24, you’ll quickly realize there aren’t enough singles (or ones) available to take away 8 ones. So you open up one of the tens You now have one less blocks-of-10, this is the “crossing out” of the 2. When you open that block-of-10 you’re revealing 10 singles. That’s “carrying the one.” In essence, you’re adding 10 ones to whatever amount of ones you started with.
Want to see a different problem modeled out? Let us know in the comments!
]]>Watch to see how the same subtraction problems are done with Digi-Block and how they’re done with Base-10 blocks. If you’ve ever seen your students make mistakes while trading (with base-10 blocks), you’ll immediately love that there is no trading with Digi-Block!
The design of the manipulative should never get in the way of your child getting the answer. After you watch the 43 - 15 video, be sure to check out the 430 - 150 video. With Digi-Block, you see how these two problems are essentially the same, just shifted a power of ten (it’s like zooming in and out with a camera). With base-10 blocks, this connection is lost and it seems like an entirely new problem.
Here’s the 43 - 15 video:
Here’s the 430 - 150 video:
]]>Many students struggle with decimals – whether performing basic arithmetic operations, ordering and comparing decimal numbers, rounding with decimals, etc. One of the greatest misconceptions is that the decimal number system is unique. Too often, decimals are introduced as a new (i.e., different) and daunting topic, which sets up students to view decimals numbers as a separate system from the whole number system.
Using Digi-Block decimals right from the start, the students see that decimals are an extension of the number system they already know. They see for themselves that the same patterns hold true: each block opens to reveal ten of the next smaller block. This, of course, is why a solid mastery of operations with whole numbers is essential before the students see the decimal blocks. The decimals come to life when students recognize (dare I say, discover) these same patterns for themselves.
So just promise you won’t introduce the decimals prematurely, okay? Great. Thanks!
Alright, on to the crux of this post! How exactly do I introduce the decimal blocks? There are many ways, but these are my two favorite ways to start the lesson:
I set up what I call the “Progression” of blocks like this:
I point to the block-of-1000 and ask my students, “What happens when I unpack this block?” If they need prompting, I ask, “What’s the first thing I’d find if I open this block?”
The answer, of course, is 10 blocks-of-100. I provide the visual proof:
Then I ask, “What would I find if I unpacked a block-of-100?” The answer, of course, if is 10 blocks-of-10.
Next I ask, “What would happen if I unpacked a block-of-10?” The answer, of course, is 10 single blocks.
I ask my students to explain the pattern explicitly: each time they unpack a block, they find 10 of the next smallest inside.
Now we turn our focus to the single block. I ask, “What would happen if I opened a single block?”
This can either be a class discussion or an individual exercise. You can use this lesson plan and worksheet set as an aid, 1.3 Tenths and Hundredths
You’ll be amazed by answers you get. Here are some samples of student work that never fail to make me smile:
Aren’t these fabulous? From here, I either go to “Option 2,” or directly to “The Unveiling” (see below).
I give my students a division problem that I know will result in a decimal answer. Once again, it’s vitally important that the students have previous experience modeling division with whole numbers before jumping in with decimals. Even for students who are proficient in long division (and especially for those only proficient in the long division algorithm), modeling what happens during the process is very enlightening. Don’t skip these foundational lessons! Make sure your students have had practice modeling division with just whole numbers.
Here, we’ll model the problem 497 ÷ 4. I give them a word problem to ground this problem in reality. Each team of students starts with 497 blocks and 4 paper plates and goes about dividing the blocks evenly between the 4 plates.
When they’ve finished making their groups, they’re left with a rather unsatisfying answer: Each group has 124 blocks, but there’s one extra leftover block, a remainder of 1.
I play up this tension. “Oh! One leftover?! How are you going to share that one evenly?”
Now we’re ready to unveil the decimals!
The Unveiling: Behold the Decimals!
I let some uncomfortable tension build. I hold up a single block and say something like, “Boy, don’t you wish you could break this thing open?” and, “It certainly seems like the single block should be able to unpack.”
Then I ask an important question: “If I unpacked this block, how many smaller blocks do you think there would be?”
The students know the pattern. They answer ten.
Here, I start to ham it up. I say, “What if I told you I have a magic electric Exacto knife that gets this block unpacked?” Some students are skeptical, some are hopeful. With a big act, I take a single block, turn around and pretend to perform surgery on the block, all the while making over-the-top “magic electric Exacto knife” sounds. When I turn around, I’m holding 10 tenth blocks. The students go wild.
Very soon thereafter, something incredible happens. The students intuitively ask, “Do they get even smaller?” Or, “Can we open up one of those?” I love it – this always makes my teacher heart skip a beat. “What do you think?” I ask them. All around, there will be nods. “If I could open it up, how many pieces would there be?” It’s so intuitive to them that, of course, the answer is ten. I repeat the procedure converting a tenth block to 10 hundredths and voila! The kids are hooked.
If you taught with Option 2, let your students resolve their division problem with decimal blocks. Each group gives me their single remainder block, I “dissect” it for them, and hand them 10 tenths. When they’ve shared their tenths evenly, they will have two leftover tenths.
They give me these, and I give them 20 hundredths, which they divide evenly. The answer is much more satisfying now! Each plate has 124.25.
Some astute students will make the connection that each plate received ¼ of the original remaining block, which is represented by the decimal 0.25. To avoid cramming too much into one lesson, I usually don’t explicitly point out this equivalency (we’ll have more lessons to teach this concept later), but I love that the physical model makes the concept so visually clear.
Okay, cool, but are the decimal blocks really equivalent?
For a really powerful visual, I tape 10 tenths together like this and pass it around the room. If you’re dexterous, you can also tape together 10 hundredths. There are usually a few students who take on the challenge of constructing these models for me after school.
The students can compare a group of 10 tenths to a single block……… and a group of 10 hundredths to a tenth.
And of course, it won’t take long for students to ask if they can open the hundredth block. I ask students to draw what they think it would look like. And then to explain what they think would keep happening with the system.
At the end of the lesson, I have my students take an oath that they won’t tell their younger siblings about the decimal blocks. This adds to the special aura of the lesson… the students feel like they’ve been initiated into an elite club. How empowering to realize they can extend their place value knowledge to infinitely small numbers! I balance this carefully, though: The key is that while the decimals feel special, they are part of the same familiar base-10 system, and the same patterns will hold true with the decimal blocks. In the coming year, the students will have dozens of lessons to prove this for themselves. Stay tuned!
For your next decimal lesson checkout the amazing introduction to repeating decimals
Take 10 x 23 = 230. Here is one group of 23 blocks (2 blocks-of-10 and 3 single blocks).
One group of 23 blocks
10 groups of 23 blocks
And here we have our 10 groups of 23, all lined up on the classroom floor.
There are several ways we could pack up the blocks to determine how many blocks there are all together. Today, I want to be strategic about how our class packs up the blocks so we can really examine what happens when we multiply by 10.
So… I help them see we have something convenient right before our eyes. There’s a special, efficient way we could pack them up:
Sometimes, all it takes is setting out some empty holders above our project. The students can envision each column of ten blocks sliding straight up to make a bigger block.
Each distinct column of 10 blocks is packed together…
Once packed, we can verify our answer: 230 (2 block-of-100 and 3 blocks-of-10).
But wait! Doesn’t this look awfully familiar?
I like to pause here so my students can really grapple with this. They realize that what they’re discovering is actually something they’ve known for a long time – that by definition 10 groups ARE 1 of the next sized unit! It’s almost too obvious. But, as with most things in mathematics, the elegance is in the simplicity.
Your students may not have the language to say this quite so specifically, but think of what incredible discoveries they’re making!
Now that’s powerful learning.
Sometimes, for a special twist, at the beginning of the lesson, I split a classroom into three groups. Each group models a slightly different problem, such as:
Group 1: 9 x 23 Group 2: 10 x 23 Group 3: 11 x 23
When each group has laid out the repeated rows of 23 blocks, we pause and briefly discuss the visual similarities to each group’s set-up.
Then, each group packs up their blocks and shows the class their answer:
Group 1: 207 Group 2: 230 Group 3: 253
This is a great way to prompt the observation that there’s something memorable to the look of Group 2’s blocks. In fact, by seeing the three problems side-by-side, there’s a distinct and fun “a-ha” moment: there really is something uniquely special about multiplying by 10.
Want to learn more? Check out Operations with Whole Numbers and Decimals.
A quick side note here. Perhaps your students will initially want to pack up the blocks like this:
Of course, this wouldn’t be wrong. In fact, it is vital that students be given plenty of opportunity to pack the blocks in whatever manner they like, to explore different possible packing patterns, and to reach the stage where they’re organically hungry to hear about why one method might be more strategic than another in certain scenarios. If they haven’t had enough free exploration time, forcing this method may feel overly rigid (or, even worse, it may convey the message that there is a single formulaic way to arrive at a correct answer).
]]>Photo By: Mommy, You’re Going To Be My Wife One Day, Right? / Lotus Carroll / CC BY 2.0
What is the biggest number? How big is infinity? What’s bigger than infinity? In order to explore the mathematical concept of infinity, parents and teachers can apply a number of practical tools.
To introduce the concept to students in grades 6-12, teachers can play an integer-naming game with the students. Players should take turns naming integers, where the winner is the player who names the largest integer. Quick-learning students will soon discover that the strategy to winning this game is simply going last (a billion plus one, for example).
After the game, the teacher should discuss with the class the role of infinity in the game. In the game, the fact that the player going last can always win is a consequence of the ‘infinitude’ of the sets of numbers. A teacher should use this game to get students thinking about the concept of infinity: something to which there is effectively no cap or limit.
Digi-Block blocks can also be used as a concrete model that helps children visualize infinity. Due to the nesting property of the blocks, many children can quickly conceptualize larger and larger blocks. Below is a picture of the blocks. First, ten singles fit into a holder to create a block-of-10. Next, ten blocks-of-10 create a block-of-100. Then ten blocks-of-100 nest to create a block-of-1000.
While Digi-Block only manufactures blocks up to 1000, your students can imagine what they could create if they had ten blocks-of-1000. What if we had ten more blocks of that block?! Take some time to make up names for these bigger blocks. A few of my favorite are ginormahumongous or ‘HUGE! Digi’. If your students are familiar with creating and joining equal groups (aka muliplication), this might be a good time to reflect on what we are actually doing when we create the next larger block (i.e., multiplying by ten).
Now that you have discussed infinitely large whole numbers, ask your students to think about how many times we can unpack the blocks. What if you could unpack the single blocks?
Want to explore the concept through a literary lesson? Try The Cat in Numberland, by Ivar Ekeland. It’s a story of the Hotel Infinity being observed by a cat.
At Mr. and Mrs. Hilbert’s Hotel Infinity, the resident cat is puzzled. The hotel is fully booked — the rooms are full of Numbers, both Odds and Evens — yet guests continue to arrive...
Have the students make lists of quantities that may or may not be infinite and share them with their classmates. Some examples that encourage students thinking about infinity include the number of planets in the universe, the number of humans who have ever lived on Earth, the number of words in the English language and so on.
Bring up the following questions, which frequently double as puzzlers and philosophical queries:
Divide students into small groups and have them further research the concept of infinity and its role throughout the history of mathematics. Have each group choose a particular topic and prepare a short presentation from their research on their use of infinity. After each presentation, have a whole-class discussion about the topic.
The Common Core Mathematics Standards for grades 6-12 include the following standards relating to infinity:
- Apply and extend previous understanding of numbers to the system of rational numbers
- Know that there are numbers that are not rational, and approximate them by rational numbers
- Use properties of rational and irrational numbers
The Common Core Mathematics Standards Practices include:
]]>
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Look for and express regularity in repeated reasoning
Seriously, think of a guess! OK, now I’ll pack the blocks…
Much easier, right?
You could figure out that there are 123 blocks in the first image by individually counting each block. This isn’t a wrong approach, but it’s quite slow and tedious. Viewing each block as an individual unit makes it difficult to efficiently think about large quantities. When you conceptualize numbers with a base ten mindset, as in the second photo, you can see how ones units compose tens units, tens units compose hundreds units, and so on. Understanding this makes it much easier to visualize quantities. With the blocks packed up, allowing us to see this quantity in base ten, it takes just a quick glance to see that the quantity is 123 – composed of 1 block-of-100, 2 blocks-of-10, and 3 ones.
]]>Older elementary students should be using mental math regularly. These students use math to keep track of their grades in their classes, how much of their allowance they have spent, how much money they need to save to purchase the latest video game, etc. Kids do all of this math in their heads, but they probably don’t realize that what they are doing is called ‘mental math.’
Before students can be successful in using mental math, they must first be proficient in their basic math facts. They should be able to automatically add and subtract numbers 1-20, know their multiplication facts up to the 12s, and the basic elements of division. Memorization is not enough; it must be accompanied by understanding.
Younger elementary students need to have experience working math problems using manipulatives, identifying patterns, and practicing reasoning skills. If a student doesn’t know the answer to 4 X 6, the teacher should encourage him or her to think about an answer to a related problem such as 4 X 3 or 2 X 6. That information could be used to build the child up to solving 4 X 6. The answer should not merely be “it’s 24.” It should be reasoned out thoughtfully, with each step explained.
Students need to know when to use mental math. Estimates are frequently not acceptable answers in the classroom or real-world situations (See how far “I have roughly 2 kids” will get you). Sometimes the exact answer is needed. For example, accuracy is important when paying for items at a store. Estimates are fine when modifying recipes or determining how much material to buy for a large project. But to practice the skill of determining when to use mental math and when to find accurate answers, have your child make a list of all the ways his or her family uses math in the real world. Next, have your child divide the list into categories of when mental math is used and when accurate answers are required.
Provide opportunities for students to use mental math. Real-life lends itself to these mental math problems constantly. For example, make a grocery list with prices of the items. Have students figure out the smallest dollar unit they could use to pay for a particular basket of items. Will these 6 things require you to give the cashier a $10 bill, or will you need to hand over $15?
Ask your child also about how much change you should get back when you give the cashier a $20 bill on a $15.75 purchase? Make this sort of double-checking second nature. Another example would be for students to figure out how many loaves of bread they would need to make sandwiches for school lunches for a week.
Some students may ask to use a calculator for these mental math activities. When numbers get too large, this is fine. But conceptual reasoning is always preferred.
]]>When we’re first teaching kids numbers, we’re only concerned with whole numbers (1, 2, 3, 4, etc.) so we can simply say that the number, or digit, that’s furthest to the right corresponds to the ones, which with Digi-Block is the smallest block. But what if we want to be able to express numbers that aren’t whole numbers, such as 20.75? Or 0.3? I can introduce the decimal blocks, but now there is no longer a “smallest” block, because the blocks could in theory always get smaller and smaller, so we can no longer use the simple rule that “the smallest block is the number at the right end.” We need something new!
The way I teach explores these ideas and shows the need for a new device for writing numbers. This engages and excites students. They might come up with a solution that works. Even if they don’t, they’ll now appreciate the solution and have a deeper understanding of why we’ve introduced something new.
First, you need to explore the idea that there’s stuff out there smaller than 1, but bigger than 0. When I teach with Digi-Block blocks, I ask kids, what do I find if I open up a block-of-10? They quickly respond, “10 ones!” I then hold up a one, or single, and ask, what do you think I’d find if I could open this block up too? They think about it and say, “10 smaller blocks inside?” And I say, “Yes, you’re right!” And then I show them the tenths blocks.
Start by writing a multi digit number down, such has “572”. Ask your student to read you the number. Ask him “How do you know which digit is the hundreds? Which digit is the tens? Which digit is the ones?” This question might leave him staring blankly back at you or you might have an interesting discussion about place value.
Regardless, the next step is to open up your student’s thinking. You’ll use different color ink for each power of ten. The idea here is that we can use color, not position, to determine the value of each digit. Tell him “I’m going to use purple for hundreds, blue for tens, and green for ones.” Now write the same digits on a piece of paper, but use green for the “2,” purple for the “5,” and blue for the “7.” Now ask your student to read you this number. If he deciphers the code correctly, he should realize it’s the same number as before: 572! Try a few more examples until he gets the hang of it.
Now ask him if he can think of a different way, maybe without color. I had one student who wrote numbers in different sizes to represent powers of ten. It looked like this:
As he tried to write more and more numbers, and longer numbers that required more than 3 digits, he decided this wasn’t very practical.
I then presented him with our problem: we have things smaller than 1. I bring the tenths blocks back out to show him what I mean. (It’s not important to name these yet, you’re just using them to show a unit smaller than the singles.)
I put out several blocks (a few hundreds, tens, ones, and tenths) and said, how do I write this down? I then took them away and said, or if you write a number down for me, a number that should use those little blocks (the tenths blocks), how will I know what to build?
After some time to think about it. I came back to the “572” we had written before. I asked which way do the digits mean bigger blocks? Which way do they mean smaller blocks? He quickly saw that the digits to the left were for larger blocks and the digits to the right were for smaller blocks. He jumped up and said that the digit for the smaller block must to the right of the one!
He wrote a 4 digit number down, “4823,” but realized it looked like four thousand eight hundred twenty three.
I asked him, what can we do to let me know which digit is for which size block? (Like how we used color before…)
He thought about it and then showed me his idea: he would circle the digit in the ones place. I thought this was fantastic! After some more thinking, I think this is actually better than the decimal point.
Circling the digit in the one’s place shows the relationship between the digits to the left and to the right of the one’s place much more clearly. Ten is one power of ten greater than 1, and one tenth is one power of ten less than 1. 100 is two powers of ten greater than 1 and one hundredth is two powers of ten less than 1, and so on. The decimal point obscures this and makes it look like the relationship between 1 and one tenth, 10 and one hundredth, and so on, should be emphasized…
Back to my student! I told him I thought his idea was great, but a long time ago, when people had to solve this problem, they came up with something a little different: the decimal point. I showed him how to write a number with a decimal point and then asked him to build it in blocks.
]]>These insights spill over into other disciplines, from chemistry to cooking. One way to help your young child learn about measurement is to build a house out of any type of building blocks. Make sure to use terms such as “bigger,” “smaller,” “wider,” “longer,” and “taller.” Your child will be learning math vocabulary and getting a solid foundation—literally—in measurement skills.
Then, why not transition to liquid measurement? Start with two glasses of the same size. Ask your child which has a greater “volume” of water, one glass or the other. Then, pour the contents of one glass into a bigger or smaller glass. Explain that the volume of water hasn’t changed; just the size of the container.
Another activity to help reinforce measurement skills is to go on a measurement hunt. Younger children may not be ready to start measuring with a ruler or measuring tape, but even the youngest child can measure with a string. In math, this is referred to as a “non-standard measuring tool.” Any length of string will do, but around a foot long is ideal because it is easy for a young child to manipulate.
To begin this activity, have a conversation with your child about how things come in different sizes. Point to an object such as toy. Ask your child to find a toy that is larger. Ask your child to find a toy that is smaller and then ask your child to find an object about the same length as the string. Point to other objects. Ask your child if each object is longer or shorter than the string. If your child is handling this activity fairly easily, challenge him or her by asking to choose three objects and put them in order from smallest to largest.
What’s more fun than having a watermelon spitting contest in the summertime? Your child will love this activity and learn about measurement at the same time. All you need is a watermelon and a measuring tape. Each member of your family should take turns spitting watermelon seeds one at a time. Have your child measure from where the person is standing to where the seed lands. Help your child make a chart to record the data. After the contest, compare the data. Ask your children who spit the seed their longest distance. Ask who spit their seed the shortest distance.
Remember, you are your child’s best teacher. You are more qualified than you think to help your child be successful in math. Every day life lends itself to a multitude of math problems. Plus, most kids think these at-home math activities are fun and feel much more like a game than a chore!
]]>Before beginning, make sure your child has some basic background information such as the names and order of the planets in our solar system. Showing your child a model or illustration would be helpful. Explain to your child that most illustrations show the planets in relative size. Discuss with your child the meaning of relative size, showing that the pictures depict how big the planets are when compared to each other and the sun. Ask your child to identify the smallest planet and the largest planet.
Next, help your child gather data about planet sizes and compare the findings. Have your child make a chart and record each planet’s diameter. After their chart is completed, ask your child what they notice about the size of the planets and how they think the planets compare to one another. Ask your child if they think it would be easy or difficult to model the planet sizes. Your child should realize that this would be very difficult due to their great differences in size.
If your child understands the sizes and ratios of planets fairly easily, challenge them a little bit more by explaining the astronomical unit (AU). (An AU is a simplified number used to describe a planet’s distance from the sun. An AU is equal to the average distance from Earth to the sun, approximately 149,600,000 kilometers.) Help your child draw the following conclusions:
Teachers can emphasize the distance of the planets from the sun by taking a class outside to a large area. Different students could each represent a planet by taking the following number of steps away from an object representing the sun:
After portraying the model, the teacher should ask students to describe what they observed about the distances of the planets from the sun and from each other. The teacher should point out the math connection of this activity, and thus explain how closely related math, science, and outer space actually are.
]]>It all depends on your child. If he or she expresses an interest in learning about money, here are some tips and activities to help your child understand the concept of money.
Ages 6-10 For children ages 6-10, here are some basic concepts about money that you can reinforce at home.
Ages 11-13 As your child gets older, his or her understanding of money should increase. Here are some tips for children ages 11-13.
Ages 14-18 By the time your child reaches this age group, he or she should have a good understanding of the value of money. It is still very important to reinforce money concepts even at this age because it will not be long before your child is out in the real world on his or her own. Instilling responsible spending is critical. Here are some tips to help children ages 14-18 learn more about the value of money.
These tips for teaching the different age groups are not set in stone. Your child may be ready to learn these concepts at an earlier age, or may not be ready as soon as you think. Talk to your child and find out what he or she already knows about money and move ahead accordingly.
]]>This math game is similar to ‘I Spy,’ except numbers are used. Tell your child to think of a number between 1-100. You will try to guess your child’s number by asking questions such as, “Is it greater than 20?” “Is it an odd number?” Your child can only answer “yes” or “no.” After you guess your child’s number, you pick a number and let your child try to guess by asking similar questions.
On a piece of paper, have your child make a chart listing numbers 1-50. Like a Bingo game of sorts, whenever they spot a number on a road sign, license plate, billboard, etc., have your child make an X on that number. The first to find all 50 numbers is the winner!
Before getting into the car, your child should write down your license plate number on a piece of paper. Tell them not to write the letters – just the numbers. Every time a car is in front of you, have your child write down its license plate number either under the heading “Less Than” or “Greater Than” depending on if the license plate number is less than or greater than yours. For really long road trips, have them write all of the license plate numbers in order from least to greatest. Your child can also record the names of the different states on the different license plates he or she sees. Your child can tally up which state’s license plate he or she saw the most and which state’s license plate he or she saw the least.
License Plate Add-Up is a game that will allow children to practice mental math skills. Each time a new car is in front of you, call out its license plate number. Have your child add the digits together to get a total. Encourage your child to explain what strategies he or she used used to add the numbers. Then, try different problems with the numbers from the license plate. For example, if the numbers on a license plate were 643, ask “Using the numbers from the license plate, can you use two numbers to make a problem with an answer of 2?” (Yes, 6-4=2) Ask, “Can you add two numbers and get an answer that is odd?” (Yes, 6+3=9)
There are many other games involving math that you can make up while you’re on the road. If you keep your kids busy doing math, you won’t hear “Are we there yet?” nearly as often.
]]>To teach your young elementary school children to identify numbers, have them read through a newspaper and circle or highlight any number they see between 1 and 100. Have your child say the numbers aloud as he or she finds them. Then, make a list of all the numbers that you’ve found together on a page, and order them from smallest to largest.
This is also a great intro to having your children engage with age-appropriate current events.
You and your child can make your own counting book using numbers and pictures found in a newspaper. Have your child cut out pictures from a newspaper and then compile a picture book. For example, for page one of your book, your child will need to find a picture from the newspaper with just one thing in it. Page two of your book should have a picture from the newspaper with two similar things in it, and so on.
At the bottom of each page of the book, have your child write the number of items in the picture and the name of the items. Once the book is complete, have your child read the book to you and other friends and family.
For this activity, you and your child will need a newspaper/magazine, a pair of scissors, paper, and glue. Search the pictures for examples of cylinders, cubes, and other geometric shapes. Cut out pictures and glue them on a piece of paper to make a book of geometric shapes. Group the like shapes together and have your child write the name of the shape at the bottom of each page.
]]>But don’t worry: word problems can be painless. During the summer months, help your child learn to solve word problems so he or she won’t be stressed out when school starts back up in the fall.
Here is a sample word problem and a script you can use to help your child solve it.
Beau had 80 Pokemon cards. He gave 35 of them to his friend Sawyer. How many cards does Beau have left?
Step 1: Before your child even picks up a pencil, have him or her read the entire problem aloud.
Step 2: Help your child identify the question being asked. Have your child underline the question. In the problem above, the question is How many cards does Beau have left? The question usually can be found at the end of the word problem.
Step 3: Have your child identify all of the information in the problem. Information consists of numbers and labels. For example, in the problem above the information is 80 Pokemon cards, 35 of them. (Explain to your child that them refers to Pokemon cards.) Have your child circle the information in the problem. Then, look at what is circled and determined if it is important in solving the problem. Cross out the unnecessary information. (Many problems will include unnecessary information to trick you.)
Step 4: Decide which operation much be used to solve the problem. Teach your child the key words to determine this. The key word left tells us that the above problem is a subtraction problem. (See below for more key words for addition and subtraction word problems.)
Step 5: Solve the problem. Make sure your child labels his or her answer. For example, instead of just writing 45, your child should write 45 cards. (This is very important on standardized tests.)
80 cards
45 cards
Step 6: Have your child check his or her answer to see if it makes sense. (45+35 = 80).
Here are some other key words to help your child determine when to add or subtract.
Addition Key Words:
Subtraction Key Words:
First there is the concept of allowance, designed for recreational spending (candy, ice cream, toys, etc.) Help your child understand that if he or she buys a lollipop, that is fine, but that’s less money to buy a toy car or stuffed animal. This introduces the integral concept of saving.
Playing games with coins will help your child learn about values. These games will also reinforce counting, addition, subtraction, and multiplication skills—all of which we do virtually every time we make a cash transaction. Grocery store coupons found free in newspapers and junk mail can help you teach your child about managing money, addition, subtraction, and even finding percentages.
For this first activity, all you need is some grocery store coupons and a handful of loose change.
For younger children in kindergarten or 1^{st} grade, you first need to make sure they know the values and names of different coins. Observe the different coins and ask your child to name them. Point out how the coins are alike and what makes them different. Once your child knows the names of each coin, their values and relative values—that five pennies equals one nickel, for example—you are ready to move ahead.
Next, gather some change in your hand without letting your child see what you picked up. Tell your child the total value of change in your hand and ask him or her to tell you possible combinations of coins you might have. For example, you might tell your child that you are holding 18 cents. Your child might guess that you have a dime, a nickel, and 3 pennies. Explain that there are different ways of arriving at the same number; 18 pennies could also work.
Next, look at grocery store coupons. Read each coupon and see how much money will be saved. For example, if a coupon says 25 cents off, point out to your child that it means one quarter will be saved, or two dimes and one nickel. Look at different coupons and practice identifying different combinations of coins to equal the savings.
Ask your child what he or she could purchase with his or her savings. A toy car? A book? A pencil? A notepad of paper? Your child needs to have an idea of how much items cost. This will teach your child about number sense and the value of money. Spending money in one place means not spending money in another.
For older children, the coupon activity can be adapted by asking your child to find the percentage of the original price each coupon is worth. Ask your child what could be purchased with his or her savings.
Parents are their children’s best teachers. They want your attention, and what could be better than having fun with your child while reinforcing learning?
]]>