Demonstrate: With 20 centicubes on the left side of your desk, count them as you move them to the right side. Demonstrate that a child‟s efficiency in this counting is enhanced when the counting uses multiples (skip counting), thereby building addition and multiplication facts. How’d They Do That? (Diagnostics) If a student generated the following, how would you sequence the thinking that must have taken place? Using The Calculator: What NSN concepts and skills does a child need to understand the entry of numbers into this device? Demonstrate: Regrouping so that the subtraction algorithm becomes manageable is a difficult concept for students to work through. The case for using concrete material becomes stronger. Using an Abacus, how could the disks be manipulated to demonstrate the following: 187 - 53 124 – 65 How’d They Do That? (Diagnostics): If a student generated the following, how would you sequence the thinking that must have taken place? Using The Calculator: Using the idea of decomposition of numbers, perform this subtraction on a calculator using at least three different subtraction steps. Demonstrate: Present the following sequence to a student and then have it demonstrated using manipulatives. What would you expect to see? 22 x4 88 20 + 2 x4 80 + 8 22 x4 88 How’d They Do That? (Diagnostics): If a student generated the following, how would you sequence the thinking that must have taken place? 14 x 12 8 20 40 100 168 7 x 14 = 7 x 10 = 70 = 7 x 4 = 28 98 Note: The idea is good but the formatting is improper. This is a very common error that needs to be corrected. 24 x 216 = 5184 20 x 200 = 4000 20 x 10 = 200 20 x 6 = 120 4 x 200 = 800 4 x 10 = 40 + 4 x 6 = 24 5184 Demonstrate: When a dividend contains a zero, students often falter with place value in the answer. In a question such as 607 5, how would you demonstrate the steps involved? How’d They Do That? (Diagnostics) If a student generated the following, how would you sequence the thinking that must have taken place? Does this teaching idea for division work for you? Suppose you want to share a large pile of candies with 26 people in your class. Instead of passing them out one at a time, you conservatively estimate that each person could get 6 pieces. So you hand out 6 to each. Now you find you have more than 26 left. Do you have everyone pass the 6 back and then give them 7 each? That would be silly. You‟d just pass out more. Ideas: 1. You intentionally underestimate how much can be shared. You can always pass out more. 2. If there‟s enough to share more…. Just do it. 3. How could you avoid overestimating? Using The Calculator: You want to „paper‟ three walls of your bedroom with shiny new Canadian „Loonies. How would you calculate how much this is going to cost? A True Story: A few months ago, I was waiting at the airport in Cincinnati, Ohio for a flight back home. I went to a Quizno‟s restaurant for a sandwich: the cost was $6.09 I gave the young cashier a $10.00 bill and he entered the value into the cash register. At the same time, I noticed that I had the 9¢ in change and placed this on the counter. The young man „froze‟ as if he were about to have a stoke; he had no idea what to do. What hadn‟t his educational system taught him? Introduction: 1. The concepts contained in the handling of money stem from an applications process. This makes it a HOTS activity. 2. Basic operations with money involve counting (skip counting forward and backward) using multiples such as 5¢ + 5¢ + 5¢ + 5¢ + 5¢ = 25¢ 3. Caution: Don‟t overuse money contexts. While many adults see this as very practical (and indeed it is), cultivate interests that are not always tied to our consumer society. A Money Model Key Issues: Hundreds 1. The money model is a value (symbolic) model, not an area model and this is a vital idea to build upon. The 100‟s, 10‟s and 1‟s value can be build on the place value mat as before; however, area isn‟t represented (it‟s non-proportional). The dime is 10 times the value of the penny; yet, is smaller in size. 2. Skill sequence: coin recognition, value of coins, using these values, counting sets of coins, making equivalent collections of coins such as showing 24¢ in 3 different ways, same and different amounts and making change (adding on the find a difference … the Quizno employee‟s problem) 3. If only we had the three coins … yet the nickel and quarter need to be included into the working system. Note how the place value system now has this extra „baggage‟. Tens Ones Fives TwentyFives Applying the New Ideas: What’s For Lunch? 2¢ 2¢ 5¢ Use the play money to show 3 ways to spend exactly 10¢. 1¢ 3¢ Introduction: 1. A mental computation is simply an invented strategy that is done mentally. The strategy for one student most often varies with another. 2. Initially, children should not be asked to do computations mentally as this puts a child at risk who is not at a direct modeling stage. 3. Solving questions „mentally‟ helps to force a student to focus on the relationships between numbers and the effect of number operations, as opposed to simply memorizing rules for performing computations. The Task: 1. Pair with another teacher. 2. One person role-plays the teacher; the other role-plays the student. 3. The teacher poses the first question: Calculate 15 groups of 27 (15 x 27). Do not write the question out so that no visual cues are provided. 4. The teacher writes the second question on paper: 17 x 21. The student solves using any method. 5. Discuss how the two computations were completed. How does the Front-end estimation strategy approach accurate mental computation? Isn’t All Mathematics Mental? Yes – it is; however, the set of strategies will often change when the „paper‟ becomes available to help us organize our thinking. Debate the worthiness of this statement using an example. Introduction: 1. Frequent opportunities to play with numerical values and the potential operations that can be included are superb HOTS directions. 2. These can be formally presented in large group, small group or activity centre designs. 3. Degree of difficulty can be simplistic (the way to begin) or quite complex as in the examples that we‟ll „play‟ with. 4. “Throw then a ball they can hit…..” so, for this exercise, pair up with another teacher to try to solve at least four of the eight number puzzles that you find on the accompanying page. Replace each „□’ with a whole number. Introduction: 1. For this activity, we‟ll use the SmartBoard to open a file that was uploaded by Ray Hoger, a Canadian teacher who is sharing this classroom idea. 2. The Notebook file was found on the Teacher Exchange which is part of the support section of www.smarttech.com . 3. Open the file by selecting: <File> <Open> 4. Choose the File <Division Fun>. 5. There are 8 pages to the exercise and these can be selected (in order if you wish) by touching the „thumbnail‟ of the page in the <Page Sorter>.