Activity #1: Moving digit cards
There was still confusion after the activity described in our last blog. Normal, but corrigible! We had students draw a Place Value Chart on an 11 x 17 piece of paper. Something like this:
We had digit cards already (from last year), but students could make those too. This student hs just laid out the number 41:
We wrote up one problem at a time, and students moved their digits right or left accordingly.
Here’s 21 x 10:
Here’s what’s hard: Don’t talk! Don’t tell them which way to move the cards, and how many spaces. So hard! Trust their thinking skills. They will eventually figure it out, and someone can come up and explain why they got what they did. It takes a long time, but they are connecting the dots, connecting neurons, and building confidence that math is “what they can SEE and talk through.”
You could make up as many or few problems as your class needs. For example,
21 x 10
21 ÷ 10
4.2 x 100
4.2 ÷ 100
Then move to 2-step problems. Here’s 41 ÷ 20.
Dividing by 20: First divide by 10, giving 4.1. Then divide by 2, but how to divide the 1 in half? WAIT. Let them talk it out. Think pair share… whatever works to get them thinking and talking.
Trust: someone will see this, and it will spread: One tenth is also 10 hundredths! So add a zero. Now you can divide 10 hundredths in half 🙂
2.4 ÷ 20
2.5 ÷ 20
2.4 x 20
2.4 x 300
Activity #2: Independent Practice
Here’s the worksheet we used next:
Answer Key: CW 1127 key
Here’s what most of us do WRONG 🙁 The concrete/pictorial basis of math is SO important, but we don’t spend enough time there. Children are not adults! They need a lot more time at that level than we think. At least a third of our students won’t learn math with any permanence if this stage is rushed. If we move on to the algorithm too quickly, they will give up trying to build meaning themselves at the concrete/pictorial level. They will switch to memorization of the algorithm, just to please the adults around them. But those algorithms are quickly forgotten, confused and to be frank, disliked.
So the first few problems on this Classwork (and on most HW) are review at the pictorial level. Fractions, patterns, and multiplication area models. One a day (our so-called daily vitamin) for weeks. We find that our fastest students don’t complain because these tasks are short; maybe one problem per concept. And the format is thoughtful; they do have to think for each problem.
Problem #5 finally goes on to today’s new practice, but it only has 4 parts. Again, we’ll practice this at the pictorial level for days before everyone has it. We can practice, not drill. We have time.
Page 4: A note about the challenges: Yes, they’re hard. It’s not fair to bore a handful of fast students just because everyone else is struggling. Your fastest students will sail through the first 5 problems, and then tackle the challenges with pride. Don’t tell them how to do the problems, either! If they get some, fine, if not, fine. They’re thinking. Neurons are connecting. They too should get to feel the pain and pleasure of solving a hard problem! It’s all about the Goldilocks spot.