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]]>We’ve also added ** Distance Learning Resources** for each of the lessons, if you’re teaching online.

VISIT: https://singapore-math-blog.com/

The post Website Update first appeared on The Pi Project.

]]>The post Pi Project Website Update first appeared on The Pi Project.

]]>VISIT: https://singapore-math-blog.com/

The post Pi Project Website Update first appeared on The Pi Project.

]]>The post Download Free - A Whole Year of Lesson Plans for 4th, 5th or 6th grade math! first appeared on The Pi Project.

]]>Kathleen and Corrinne are pleased to announce the launch of their new website. You can follow us there on a day-by-day basis – a whole year’s worth of daily lesson plans.

We have currently loaded lessons through September, and promise to stay at least a month ahead of the calendar. Stay tuned!

If you use any lessons, please give us feedback — how did it go? Any problems, suggestions, or improvements?

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]]>The post Teaching Long Division So It Makes Sense first appeared on The Pi Project.

]]>We’ve all heard of the benefits of the curricular approach “CONCRETE > PICTORIAL> ABSTRACT” (C>P>A) — what does that look like in our classrooms, and what does it look like for long division?

We’ve seen this internationally, (the highest ranking countries use it) and we’ve seen it in our own classrooms. We use it because it works.

Singapore developed their C>P>A approach based on the writings of five famous math pedagogues: Piaget, Skemp, Bruner, Dienes, and Vygotsky. Here is an article with visual examples, and here is a short video by Yeap Ban Har (the leading international expert in Singapore Math, in our opinion), describing their contributions to math education.

** The Concrete = the Pictorial = the Algorithm**

The video below shows the integration of the division algorithm across all 3 levels: CONCRETE > PICTORIAL> ABSTRACT. The number bonds and the long division algorithm do nothing more than represent the picture (which uses blocks) in abstract terms. A child who forgets a step can go back and derive the steps all over again.

We all recognize the appeal of this approach, but here’s what we teachers often get WRONG: ** We spend too little time at the concrete/pictorial level.** There are many reasons for this – time pressure from administrators and parents, pressure from standardized testing, poor textbooks, lack of curriculum guides showing how to teach C>P>A, lack of personal experience in our own childhoods. Mainly, though, because of Truth #3:

Children do *not learn like adults. *With our more developed abstract cognitive skills, we can look at a concrete conceptual example once or twice, and integrate it into our existing algorithmic math understanding, *make connections, *and make sense of it. Children, however, need a LOT more time. Some students ( traditionally labeled “good at math”), are quick, abstract thinkers, and move through the 3 levels at an almost adult speed. In fact, they might *prefer* the abstract. A large number of their peers, however, understand the concrete almost exclusively, and a quick transfer to the abstract *DOES NOT WORK* for them. They simply need more time. Where an adult would figure out a concept by laying out blocks 2 or 3 times in a row, a child might need to do it 14 times, or 19, or 23. If they do not get that opportunity, they discard the concrete and try to memorize the algorithm, just to keep up with their class. By 5th or 6th grade, they’ve fallen hopelessly behind, and are in danger of giving up.

For children who learn visually and conceptually, the greatest ally is TALKING. In all our division problems, we write out the problem in WORDS. In the video above, 272 is written as 26 tens and 12 ones. This allows students to approach this problem with blocks – students who otherwise would not know where to begin. They can convert the words (26 tens and 12 ones) to a number by *counting.* Again, we adults often don’t realize the complexity of our own base-10 system, and acknowledge the amount of thinking and building that goes into its mastery.

How do we give students the time needed? We use what we call a “Sandwich Unit”. This means that we introduce the *CONCRETE /* *PICTORIAL* concept using blocks and then drawings, and **stay at the concrete/pictorial level** for at least 2 weeks. Students only have to do one or two of these problems a day (perhaps on homework), as we go on to other topics. Finally, we return to this topic, and slowly develop the abstract representation.

We only actually draw circles and pass out blocks for one day (maybe two) — the FIRST step only in the video above. We give the students 2 or 3 problems on the first day, and have them solve them using blocks. (They have to actually put away their pencils, or this won’t work well.) At this age, they don’t have to be taught to exchange hundreds or tens in order to pass out all the blocks fairly.

Here are 2 problems that might take them some time.

- 13 tens and 5 ones ÷ among 3 children. (This doesn’t lend itself instantaneously to mental math, making it a good problem). The photo shows the answer, after the blocks have been passed out. Note that we use WORDS to pose the problem. This gives an advantage to concrete thinkers, and makes the abstract thinkers more willing to use blocks.

2. Six hundreds, 6 tens and 8 ones ÷ among 5 children. The photo shows the answer, after the blocks have been passed out, although the remainder of 3 is off the screen.

At the 5th grade level, we don’t need to spend much more time using blocks. On the next day, we allow students to use *drawings. *It goes faster, and is just as conceptual.

**Here 2 examples: Notice the lack of number bonds or long division. This is all based on COUNTING what you have drawn (or passed out, for anyone who prefers to use blocks)**

Again, **w****e only do one or two of these a day**, **while we go on to other lessons.** For example, this week we investigated Fibonacci numbers while students did one or two division drawings on their homework. Even children who have already learned long division at home don’t complain about 1 problem a day.

NOTE: Some students will understand and complete the manipulative-based problems very quickly, so any classwork needs to include review and challenge work for them. Students who don’t get to the challenge work do not have to finish it later. This is differentiation by speed. Here is an example of classwork – 2 division problems, one fraction review problem and 2 challenge problems. **CW concrete**

After 2 weeks of intermittent drawing, over 90% of our students show comfort at the pictorial level, and we move on to the recording the number bonds.

Notice that the number bonds strictly reflect the picture drawn.

Step one: Pass out the tens and record the number bond:

Step 2: Pass out the ones and record the number bond. Finally, record the answer by counting the blocks in each circle.

MENTAL MATH. Soon, we give mental math a try. Here is a Power Point with 10 problems, and prompts for the first 5.

Note: Some children find this very hard. They lack background in number sense, and their learning style is visual, which only lends itself to mental math after LONG practice at the concrete/pictorial level.

However, we feel that number sense and mental math are extremely important. More important probably than long division. So we do one of these Power Point warm-ups maybe twice a week, and stop as soon as exhaustion or resignation become visible.

We try to make this fun – students write their answers on a mini-whiteboard and hold them up. Sometimes we have everyone change places after every 3rd problem, just to increase engagement and decrease anxiety. We pretend not to notice incorrect answers, but occasionally ask students to explain their thinking before going on.

After about 2 weeks of intermittent drawing and number bonds, we introduce long division. Again, we only do 1 or 2 problems a day, while we’re working on a different unit altogether.

On the first day, we have students do all THREE steps – blocks, number bonds and long division. They need instruction in how to record the long division. Here is a Power Point that helps engage and build procedural understanding: **DIVISION story **

Here are more examples: In the first photo, the BLUE 120 = Tens passed out.

120 is recorded 3 different ways: (a) concrete – the number of tens rods passed out , (b)mental math – as one of the number bonds and (c) the first entry in the long division problem.

The 15 is (a) the concrete number of ones passed out, (b) one of the number bonds and (c) the second entry in the long division problem.

We do similar problems as homework for 2- 3 days. Then we increase the level of difficulty to include THREE number bonds.

Note: we don’t necessarily expect this level of difficulty as a mental math problem, except maybe as estimation practice.

Gradually, we DROP the Concrete/Pictorial requirement. Students who still need it are encouraged to use the drawing.

Again, we only do ONE or TWO of these a day for the next 3 or 4 weeks. That provides *conceptual *practice, not rote memorization. Visual memory is much stronger than algorithmic memorization, and this procedure can successfully be pulled back up in moments of stress (like a quiz!)

Here is a Word document with long division practice: CW div 2 methods

Important Notes:

1. **This values the thinking, not speed and number size. **You’ll notice we use SMALL numbers for a long time. It’s all about the concept, not the calculation complexity in today’s world. If a child can learn to do 1346 ÷ 11, they can later do bigger numbers if necessary. PLUS – this transfers fairly easily to **decimals, **since it’s so place value based.

2. **This approach teaches place value and number sense. **Even students who prefer the algorithmic approach make place value mistakes, leaving out zeroes in the answer, for example. The more **concept-**based the approach, the less likely this is to happen.

3. **This approach develops estimation and mental math skills. **Even students who enjoy long division make mistakes around answers that have zeroes in the middle.

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]]>Feel free to use or edit as you like. (I fixed the typos!)

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]]>The post More Word Problems first appeared on The Pi Project.

]]>We have seen our 8th graders struggle with algebra problems that involve “the number of bills” and the “value of those bills”. They can write x + y = 27 if there are 27 five and 10 dollar bills altogether, but stumble over the *value* equation: 10x + 5y = 210 when told that the 27 bills add to a value of $210.

So we decided to try to start such distinctions earlier – 5th and 6th grade. Here are 2 Power-Point Warm-Ups that help students begin to make this journey. As always, use manipualtives (we used Cuisenaire rods and Monopoly money, but any blocks will do) and *give them time. *

Photos of the “Money” Problem:

This is a more traditional lesson, since it represents a convention that mathematicians follow to avoid misunderstandings.

Since we are also 8th grade Algebra teachers, we insist on good algebraic form – what we call the “Sacred V”.

Photos of classwork:

Here are all the HW assignments from the last unit, plus their Solution Keys:

HW#1 HW#2 HW#3 HW#4 HW#5 HW#6 HW#7 HW#8 HW#9

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]]>We found on our second* quiz *that many students were still struggling with word problems. (Correct student quiz answers* here.*) So we adapted some of Steve Wyborney’s “Esti-mysteries” to continue reviewing word problems, and students seem to enjoy these ‘mysteries’ and look forward to them.

Important: Take TIME when you show these. We try to slow down the process as much as possible (without totally ruining the tension!), in order to allow more students to spend the time they need thinking through the problems. Number 3 can be used as a “Number Talk’ to see how many ways students can see the problem 7 x 13.

On #1 and #2, students will need a small 100 chart to cross out values as they solve.

Mental Math: On the next warm ups, we don’t allow pencils! We find that this increases student engagement, and they can’t ‘hide behind their pencils’.

Sharky (#6 and #7) is a popular addition to the warm ups!

We humans love patterns. We reproduce them in art, music and literature, we seek them out in nature, in daily life and even in the behavior of others. They allow us to make assumptions, develop strategies and navigate uncertainty. Patterns provide a sense of order – a comforting predictability in an otherwise mysterious universe.

The study of patterns should be central to our learning of mathematics. Luckily, children find them enticing, and enjoy observing and describing predictable changes in growth.

Investigation: We ask students to lay out one tile. Is it a square? What’s its area? Could it be called 1-squared? Now lay out new tiles to show 2-squared, 3-squared and 4 squared. What do you notice?

Hmmm … do you want to change the colors of the tiles to show what you’re noticing?

3. Then students start on their classwork, which increases in difficulty as they go through it. Some students will need the whole time to complete 2 pages, while others will complete the whole 4 pages. This allows for ** differentiation by speed, **which is the only difference we find matters in math. Speed (or lack of it!) is not necessarily an indicator of math ability. Many students who need time to process concepts end up being strong math thinkers in high school and beyond!

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]]>The post Fractions with Legos first appeared on The Pi Project.

]]>We decided that finding fractional *areas* of a rectangle would be a good alternative representation, since we’ve used Cuisenaire rods so often to represent fractions.

The large brown flat has 48 studs (bumps). We asked students to fill the flat completely with smaller pieces, then draw what they’d created on a white board, and label each fractional area. We asked them to repeat if time.

As they labeled the parts, we circulated, asking how they knew what that fractional part was. Almost uniformly, they all answered with a division fact. “Because 8 goes into 48 six times, so it’s a sixth.” We tried rephrasing – “You mean that SIX of those pieces fit on the flat?” Some children actually tested it to see if 6 of the yellow pieces fit on the flat. We feel that this merits reinforcement – a yellow piece IS a sixth because six of them fit on the flat. This is one more step towards tying together the two large concepts of division and fractions.

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]]>The post Fractions: Transferring From Blocks to Word Problems first appeared on The Pi Project.

]]>Our intensive use of Cuisenaire rods came to fruition now as we attempt to transfer fraction visualization to word problems. Here are the 2 lessons we spent on this so far:

CW Frac to WP1 and CW Frac to WP2

We used a format called “Builders and Scribes”.

- Group students in pairs. We then sent each pair to the white boards (we have very long white boards), but they could work at mini-white boards at their desks, too.
- One student starts as the “Builder”, the other as the “Scribe”. The builder uses Cuisenaire rods, laid out on a mini-white board or clipboard. (no pencil/pen for the builder). The scribe uses a white board pen (no blocks for the scribe) and records what the builder has built. The builder stands next to their scribe and they set to work. Of course, there’s a lot of collaboration and role-sharing at both ends of this; that’s absolutely fine
- There are hints for each word problem, but wait as long as possible before showing them. Our first goal is to MAXIMIZE THINKING TIME, not necessarily to get right answers quickly. However, we also want to avoid completely losing the students, too. Perhaps there’s a Golden Confusion Level Let them struggle and think and try, then give a little boost up.
- After each problem, ask for a little feedback. “What was the hardest wall to get over?”
- Then the partners switch roles for the next problem.

HW#6 and HW#6_answ_key

Here’s a Cuisenaire riddle/puzzle for CW, too, if there’s time: CW 10_2 Cuis_puz

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]]>The post Why Withhold the Fraction Algorithm? first appeared on The Pi Project.

]]>Here’s what we see:

When we require drawing, *every problem becomes a word* problem. In the problem below, all students recognized that 1/2 is 6 out of 12, visually. This is a “12-peak Toblerone”, so a total of 17 twelfths (by simply counting!) . Then this student imagined moving one 12th from the top row to make the 2nd row equal to one, leaving 5/12 on top. This shows *number sense! * Our students can do fraction addition and subtraction mentally. More importantly, visualization helps facilitate the transfer to word problems, as below.

Egyptian fractions: We spent a few days answering word problems by building fractions with Cuisenaire rods. Here, for example, is a TWELVE-WIDE wall:One fourth — the light green rod — is called one fourth because *four of them fit in a whole. *The purple rod is called one third because 3 of them fit, the red is 1/6, etc.

This student had no trouble finding a way to make 11/12 with Egyptian fractions:

After long exposure to physical representations, word problems become easier. This problem, for example, would be difficult to do with algorithms.

How about this problem: *Erin and Kana went shopping for groceries. Each of them had an equal amount of money at first. Then Erin spent $80 and Kana spent $128. After that Kana had 4/7 of what Erin had left. How much money did Erin have left after shopping? Solve by drawing a fraction model.*

This is very difficult to do without algebra. Try it yourself before looking at the answer here. Once you *see* the solution, it’ll make sense, and all of this will transfer to stronger algebra students in 3 years.

In the past, visual learners struggled with the algorithmic manner in which math was taught. (Challenge: randomly survey a couple dozen adults – we predict almost 1/3 of them will say they were ‘never very good at math’)

However, in the past, there were good middle class jobs available to high school graduates – jobs that are now disappearing. It is our duty to make math accessible to ALL students.

The good news is that requiring visualization of math also benefits the *innately abstract *math learners. Visualization skills helps students in Chemistry, Physics, Trigonometry, and other STEM subjects these students gravitate towards. Here’s an article about visualization in physics.

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