Cuisenaire Rods Lesson 1: Introduce and Explore

Developing Part-Part-Whole Relationships Part 2/3

Developing Part-Part-Whole Relationships Part1/3

Developing 10-Anchor: 10-Facts with a 10-Frame

Part 2: 1,2 More Than and Less Than activities

Part 1: 1,2 More Than and Less Than activities

Children should develop the idea that numbers have relationships to one another. Working with elementary and middle school age children has reinforced that notion. I've noticed that as kids get older and not develop numeric awareness and other relationships, the more likely the child will resort to memorizing facts, require longer time or not fully develop later math concepts. This has nothing to do with capability or intelligence, rather, the child has not developed key number sense areas and this slows, impedes and/or restricts the child's future math growth. Adopting an approach to emphasize these skills early is preferable but these are skills that can be taught to a wide range of ages. Although, I'd stick to basic number recognition and number skills for children 4 and under.

The More Than, Less Than activities help to develop childrens sense of relationships between numbers. Rather than think of one particular number--and only that number--they view it as a range of related numbers. For example, the number 8 is 3 more than 5; 2 less than 10; 1 less than 9; 1 more than 7; 2 more than 6. This is a concept that is important in a wide range of operations; let's look at addition. A child working the problem 8 +13 can solve this problem in a number of ways. A child without a range of strategies or developed number sense will probably have to count out 8 and then add on 13 to arrive at an answer. A child who's developed the 'Counting On' strategy will start with 13 and count on 8 more. A child who's developed an anchor of 10, may think in terms of completing the Ten-Fact for '13'; 7 more completes the Ten-Fact (and gives us 20) and 1 more is 21. Another child who's developed a range of strategies and has some place value sense may know that 10+13 is 23 but 8 is 2 less than 10, so the answer is 21.

It may seem like an overwhelming thought process but children can learn and use a wide range of strategies. Some children may not pick up strategies so easily and need additional activities and modeling to pick up strategies. This is not to say that a child must use one specific strategy. Rather, it's the opposite. It's OK for a child to use a different strategy from another child--or from the adult in the room. I'm more concerned about concepts that a child may be familiar with. If I see a child can not make generalizations about numbers, I work on activities and games that help foster greater number sense.

Developing Math Foundations: Game-Counting On

Developing Math Foundations: Counting On

Counting on is an important strategy that we typically develop in childhood. What is counting on? Counting on is the strategy that if we have two addends, we count on after where the first addend ends.

For example, 17+3. Children who aren't automatic with numbers, haven't developed addition strategies, and haven't developed counting on as a strategy will count out 17 beginning from 1 (using their fingers or a manipulative) and then add 3 more to arrive at the answer, 20.

A child who counts on will start after where the first addend ends. Since the first addend is 17, he/she will count on 3 begining at 18; 18, 19 and 20. A child who develops a count on strategy will have typically developed the strategy after noticing that the sum remains the same using either the shorter, more efficient counting on strategy or the more laborious and time consuming strategy of individually counting out the first addend and then adding the second.

Some children figure it out on their own but others need explicit instruction and modeling to make sense of the strategy. I remember having problems with counting on up until elementary school. My main problem was uncertainty of where I begin counting on from. For example, 17+3. Do I count on beginning with 17 or after it beginning at 18? I eventually figured out the strategy by using logic and easy examples. I would do simple problems for which I knew the answer to--like 5+3. I would then experiment to figure out which was the correct strategy. Since I knew the answer was 8, only one model of the counting on strategy would give me that answer.

This is the kind of experience our kids go through to make sense of numbers and math. We should allow them to explore and make sense of it for themselves but at the same time provide activities, guidance and support for building key concepts.

Using Unifix Cubes Lesson 2: Representing a number in multiple ways

Using Unifix Cubes Lesson 1: Ordering and Sequencing

Another manipulative I use to develop number sense in younger children are Unifix cubes. Unifix cubes are multi-colored blocks that link together to form stacks of ten cubes. It's a great resource to teach counting, compare and contrast quantities, estimation, addition, subtraction, multiplication, measurement and so on.

This activity is an ordering and sequencing activity for children who are still developing early number concepts such as counting and comparing quantities. When children first learn to count they hear number names and soon explore that relationship to quantity. Later they connect those experiences of number names and quantity with number symbols. Working with a manipulative like the Unifix cubes allows children to use their senses (visual and tactile) to make very concrete connections to what is a very abstract concept.

Enjoy the vlog and email me any questions.

using base ten blocks lesson 5: Addition with regrouping

Many students begin learning addition with regrouping--also referred to as carrying over-- in a mechanical and almost robotic fashion. Take 17+3: We start adding up in the ones column. 7+3 is 10; bring down the 0 and carry the 1 to the tens column. 1+1 is 2; bring down the 2. The answer is 20. While they get the answer correct, there is little understanding of the process that they went through and even less understanding of the concept of place value.

Using Base 10 blocks for addition with regrouping helps children feel, touch, manipulate and see addition problems. They then begin to develop connections between numbers, quantities, place value, and operations. A child with experience with base 10 blocks will start to physically take apart and recombine numbers-- using the manipulatives--and later, do so mentally; they begin to internalize what they've learned from their experiences.

Before beginning children/students on addition with regrouping, it's important they have had sufficient time to develop familiarity with the blocks as well as some prerequisites including:

1. Recognizing relationships between units, rods and flats, i.e., 10 units make up a rod, 10 rods make up a flat.

2. Used blocks with base 10 chart to represent values. Children should have experience creating quantities using base 10 blocks on a chart and writing down their values. They should recognize that the number 113 is made up of one flat, one rod and three units. They should also create numbers using base ten blocks and write down its value.

3. Addition (no regrouping).

Optional but helpful:

4. Race to 20 (or 100--depending on what is appropriate for the child). The game helps develop the "trade-in"rule, which is an important concept in regrouping. It familiarizes students with the rule but doesn't overteach it. When students move from the game to addition with regrouping, the trade-in rule is already second nature or something that's very familiar to the student.

Using the tips and techniques listed above, students progressively develop important math ideas and build upon experiences, from prior activities, to successfully transition into new concepts. More intriguing is watching students develop a slew of ideas and concepts that we didn't have to teach them; rather they learned from their own exploration and experiences.

Using Base 10 blocks isn't the panacea for all math challenges and ills but it does help children develop a deeper understanding of numbers and the operations they're engaging in.

Using base ten blocks lesson 4: Addition (no regrouping)

Using variations of the classic card game 'War' to promote addition and subtraction

Another game I like to have kids play to promote number sense, addition and subtraction is the classic card game War. The original version of War is a great game in that it allows kids to think in terms of quantities: They have to think about which number is greater--so they're thinking in terms of quantities. I suggest even teaching younger children to play this but using unifix cube, counters, or even counting the suits on the cards. I prefer younger kids use manipulatives so they can make sense of numbers as well as see and feel larger quantities.

Double War is a twist on the orginal. Instead of one card, players put down two card each. The player with the greater sum wins the round.

Difference War is another twist. For this version you'll need 50 counters. I use counting chips in the video demo but you can use anything: craft gems, small pebbles, paper clips, etc. Each player put down a card. Players take the difference of the two cards in counters. So if player A puts down a 7 and player B puts down a 10; the difference of 10 and 7 is 3. Player B had the larger number and so picks up 3 counters. Players take back their card and place in a discard pile. After the 50 counters have run out, both players count up their counters; player with the most counters wins.

Enjoy watching your child play and learn!

Dot Frames Lesson 2: Five and some more

Dot Frames for Developing Anchors of 5 and 10: Lesson 1

Anchors of 5 and 10 are important ideas for children developing relationships between numbers--specifically, as the name implies, 5 and 10. As children we used the most readily available counter for counting up and down--our fingers. As many of us grew older we internalized the relationships and generalized it for larger numbers. We began to see numbers as some relationship to 5 or 10.

The number 8, for example, is 3 more than 5, or 2 less than 10. The number 32 is simply three 10s and 2 more. A subtraction problem 32-17 might be solved using an anchor of 10. 3 more than 17 is 20, 10 more is 30 and 2 more is 32; 3+10+2 is 15. The process is incremental and children develop it through experience and practice. Some children internalize the process with little to no explanation, while others do not.

Some children don't make those connections as easily and need explicit instruction to develop the concepts. 5-and 10-frames are a great resource in developing 5 and 10 anchors. A 5-frame is a simple 1x5 array and a 10-frame is a 2x5 array. It can be drawn on paper, found as a blackline master in textbooks or online, created on word processing software, etc. I always introduce 5-frames first and move to 10-frames after the student has shown mastery. For some it may be a week or two; for others it may be a day or two.

This lesson is an overview of 5-and 10-anchors, 5-and 10-frames, and a demo of introducing and using a 5-frame.

What's My Number? Board Game