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Activated, But Not Resting, T Cells Can Be Recognized and Killed by Syngeneic NK Cells
Brian A. Rabinovich, Jennifer Li, John Shannon, Rose Hurren, Jan Chalupny, David Cosman and Richard G. Miller
J Immunol April 1, 2003, 170 (7) 3572-3576; DOI: https://doi.org/10.4049/jimmunol.170.7.3572
Brian A. Rabinovich
Department of Medical Biophysics, Ontario Cancer Institute, and Department of Immunology, Faculty of Medicine, University of Toronto, Toronto, Ontario, Canada; and
Jennifer Li
Department of Medical Biophysics, Ontario Cancer Institute, and
John Shannon
Rose Hurren
Jan Chalupny
Amgen, Seattle, WA 98101
David Cosman
Richard G. Miller
We demonstrate that IL-2-activated NK cells or lymphokine-activated killer cells recognize and kill syngeneic CD4+ and CD8+ T cells that have been activated by APCs. Induction with APC required TCR-specific Ag, and lysis was perforin mediated. Brefeldin A, which disrupts protein transport, inhibited the sensitivity induced by activation. In BALB/c, expression of NKG2D ligands correlated with lysis and could be inhibited by brefeldin A. As well, addition of anti-NKG2D mAb to a killing assay completely abrogated lysis. Transduction of mouse NKG2D into a human NK cell line, YTSeco, conferred upon it the ability to kill activated BALB/c T cells, indicating that NKG2D is necessary for recognition. Our data provide a basis for studying a role for NK cells in T cell regulation.
Natural killer cells are large granular lymphocytes that express multiple activation and inhibitory receptors. These receptors allow NK cells to rapidly survey their environment for danger. When an imbalance in signaling favors activation, secretion of cytokines and/or release of cytotoxic granules occur (1, 2). For example, the rapid secretion of IFN-γ by NK cells after viral infection (e.g., murine CMV (3), HSV (4, 5)) promotes a Th1 response necessary for the clearance of intracellular pathogens (3, 6). Activated NK cells also kill immature dendritic cells (DC) 3 (7), thus regulating T cell expansion and decreasing the risk of proinflammatory tissue damage. In this study, we demonstrate that NK cells can directly limit the survival of activated T cells.
NKG2D is a highly conserved (8, 9), homodimeric, lectin-like, type II transmembrane protein expressed on virtually all rodent and human NK cells (10, 11). It is highly promiscuous, and its ligands include a variety of distantly related MHC-I homologues that are often up-regulated on stressed and activated cells (12, 13). These include MICA and MICB (humans) (10), the ULBPs (humans) (14), Rae-1 (mice), and H-60 (BALB/c mice) (12, 13).
In this study, we demonstrate that activated T cells from BALB/c and B6 mice are killed by syngeneic activated NK cells or lymphokine-activated killer cells (LAK). This is mediated by NKG2D in BALB/c.
C57BL/6 (B6), BALB/c, BALB/c-dm2 (H-2Ld− mutant), and F1 (B6 × BALB/c) mice were purchased from The Jackson Laboratory (Bar Harbor, ME). B6 and BALB/c FcRIIIγ knockout mice were purchased from Taconic Labs (Germantown, NY). Perforin knockout mice (PKO), generously provided by H. Hentgartner (Zurich, Switzerland), were engineered on the B6 background (15) and bred at the Ontario Cancer Institute. D011.10 (D011) C.B-17-SCID TCR trangenic mice were engineered at the Ontario Cancer Institute by breeding D011 TCR transgenic mice (16) (The Jackson Laboratory) onto the C.B-17 SCID background. Most T cells (>99%) in these mice are CD4+ and specific for aa residues 323–339 of chicken OVA (in the context of I-Ad) (16). The 2C TCR transgenic mice were obtained from D. Y. Loh (Nippon Roche Research Centre, Kamakura, Japan) (17). The 2C T cell is CD8+ and specific for Ld (presenting the ubiquitous p2Ca mitochondrial peptide) (18, 19). Mice were kept in a specific pathogen-free animal colony in the Ontario Cancer Institute. Animal experimental protocols were monitored and approved by the Ontario Cancer Institute Animal Care Committee. In most experiments, 5- to 7-wk-old mice were used.
Hybridomas 2.4G2 (rat anti-mouse FcγRIII-α) and 1B2 (rat anti-mouse 2C TCR) were obtained from the American Type Culture Collection (ATCC, Manassas, VA) and mAbs purified from tissue culture supernatants, as previously described (20). Fluorescently labeled mAbs anti-NK1.1 (PK136), anti H2-Kb, anti-B220, anti-CD4, anti-CD8, and anti-TCRβ were purchased from BD PharMingen (San Diego, CA). An anti-mouse NKG2D mAb (M315) was made from splenocytes of Lewis rats immunized with mouse NKG2D-Fc fusion proteins (D. Cosman, unpublished observations). Cosman et al. (14) have previously reported that in a murine system, ULBP2-Fc binds only to NKG2D. We found that ULBP2-Fc completely blocked binding of the M315 mAb to murine NK cells and vice versa. The Fc fusion protein, ULBP2-Fc, and a leucine zipper ULBP2 fusion (ULBP2-LZ) were made, as described (14). Labeling of ULBP2-Fc was performed with EZ-link biotin from Pierce (Rockford, IL). Labeling of purified Abs with FITC was performed, as previously described (20). Brefeldin A (BFA), Con A, PMA, and ionomycin were purchased from Sigma-Aldrich (St. Louis, MO). Flow cytometric acquisition and analysis were performed using a BD Biosciences (San Jose, CA) FACSCalibur. FCS Express software (Novosoft; Toronto, Ontario, Canada) was used to analyze flow cytometry data.
Generation of mouse NKG2D-Fc fusion protein
Mouse NKG2D-Fc (mNKG2D-Fc) was made in the mammalian expression vector pDC409 (14) by cloning of mNKG2D cDNA in frame with the carboxyl terminus of a human IgG1 Fc region to generate a type II fusion protein. Plasmids encoding the Fc fusion proteins were transfected into CV-1/EBNA (ATCC CRL-10478) cells, and the fusion proteins were purified from culture supernatants by chromatography on a protein A-Poros column (Applied Biosystems, Foster City, CA), as described (14).
Preparation of LAK cells
NK cells were prepared from spleen cell suspensions from normal mice via the removal of B and T cells, as previously described (20). Purified NK cells (3 × 106) were suspended in 5 ml α-MEM/10% FCS/150 μM 2-ME containing human rIL-2 (Proleukin; Chiron, Emeryville, CA) at a concentration equivalent to 350 mouse IL-2 U/ml and plated in a single well of a six-well flat-bottom plate (Nunc, Scarborough, Ontario, Canada). A conversion factor of 1 mouse unit/21 human IU was obtained by generating a titration curve for CTLL-2 tritiated thymidine uptake over a range of Proleukin concentrations and calculating the concentration for which 50% maximum tritiated thymidine uptake occurred. NK cell-enriched splenocytes were expanded at 37°C/7% CO2 for 4–5 days before use in the cytotoxicity assay. Activated NK cell cultures were always greater than 90% NK1.1+ (B6) or NKG2D+ (BALB/c) and less than 5% TCR+.
T cell activation
T cells were isolated from splenocyte suspensions via the removal of B cells with anti-mouse B220 Dynabeads (Dynal, Oslo, Norway). For activation of 2C T cells, 2–3 × 106 T cells were suspended in 5 ml complete medium containing 5–15 mouse units of Proleukin/ml and plated onto BALB/c or F1 splenic adherent cells irradiated with 20 Gy γ-radiation in one well of a six-well plate. Plates were spun for 3 min at 700 rpm and incubated for various times at 37°C, 7% CO2. If APC were allogeneic to the NK used in the killing assay, they were removed following T cell activation by staining with mAbs anti-H2-Dd (HB87; ATCC) and anti-H2-Kd (K9.18.10s; kind gift of W. Jefferies, University of British Columbia, Vancouver, British Columbia, Canada) and negative sorting with anti-mouse IgG Dynabeads (Dynal). For activation of D011 T cells, BALB/c plastic adherent spleen cells were first pulsed overnight with 1 mg/ml chicken OVA before addition of T cells in the same manner as that described above for 2C. Activation of polyclonal T cells with Con A was performed, as previously described (20), and cells were washed with 200 mM α-methyl-mannoside/α-methyl-glucoside before the cytotoxicity assay. To inhibit protein transport in T cells during activation so that APC were not affected, we used the same method described below for addition of BFA to the cytotoxicity assay.
Cytotoxicity assay
Chromium release assays (CRA) were performed, as described previously (20). For assays in which Abs or Fc-fusion proteins were added that recognized ligands on the target cells, Ab-dependent cellular cytotoxicity was controlled by coating LAK with anti-FcRII/III Fab, or via the generation of LAK from FcRIIIγ−/− animals (21). The use of BFA to disrupt the Golgi apparatus in target cells without affecting LAK effector function was performed, as previously described (22). Briefly, targets were first incubated in a high concentration of BFA (5 μg/ml for 45 min), a dose sufficient to disrupt the Golgi. Subsequent manipulation of targets and the cytotoxicity assay was performed at 0.5 μg/ml BFA. This concentration prevents the Golgi from reforming in the target cells, but does not affect the ability of LAK to kill. Conversion to lytic units (LU) was performed according to the procedure described by Bryant et al. (23).
H-60 RT-PCR
Total RNA was prepared using RNeasy (Qiagen, Venlo, The Netherlands) and then reverse transcribed in a 20-μl reaction using an oligo(dT)12–18 primer and SuperScript II (Life Technologies, Carlsbad, CA). Ten percent of the cDNA reaction was PCR amplified using the Expand High Fidelity PCR System (Roche, Laval, QC). The following primers were used: H60-F (5′-GTGTGATGACGATTTGTTGAG-3′) and H60-R (5′-ATTGATGGATTCTGGGCCATA-3′) (24). The β-actin control primer pair was purchased from BioSource International (Camarillo, CA). Cycling was completed as follows: 94°C for 30 s, and then 30 cycles of 94°C for 30 s, 52°C for 30 s, and 72°C for 45 s.
Transduction of YTSeco with murine NKG2D
Murine NKG2D, corresponding to GenBank accession AF030313, was cloned into pMXs (generous gift of H. Arase and L. Lanier, University of California, San Francisco, CA) (25), a variant of pMX (26). This construct was transfected into the packaging cell line Plat-E (kind gift of T. Kitamura, University of Tokyo, Tokyo, Japan) (27) using Lipofectamine Plus (Life Technologies). Supernatants were harvested 48 h later and used to infect the human NK cell line YTSeco (generous gift of J. C. Zuniga-Pflucker, University of Toronto, Toronto, Ontario, Canada, and G. Cohen, MIT, Cambridge, MA) (28). This infection was conducted in the presence of polybrene at 10 μg/ml (Sigma-Aldrich). Infected cells were stained with PE-labeled αNKG2D mAb, and the positive fraction (YT-NKG2D) was sorted on a FACSVantage (BD Biosciences). The sequence of the integrated cDNA was verified using PCR primers specific for pMXs, as described (25).
All experiments performed in this study were reproduced at least twice.
T cell activation results in a transient sensitivity to syngeneic LAK
Both CD4+ and CD8+ T cells of known specificity were separately activated by incubation for varying times on splenic adherent APCs and examined for their susceptibility to lysis by syngeneic LAK in standard CRA. OVA-specific CD4+ transgenic T cells were isolated from D011.10 (D011) TCR transgenic mice on the C.B-17 SCID background (identical with BALB/c, except for IgH locus). Similarly, transgenic CD8+ T cells specific for H-2Ld were isolated from 2C trangenic mice on the B6 background. Resting T cells (both CD4+ and CD8+) were highly resistant to lysis by syngeneic LAK at all E:T tested. However, upon activation on APC for 15 h, both CD4+ and CD8+ T cells became highly sensitive to syngeneic LAK (Fig. 1⇓A). When 2C T cells were activated on BALB/c APC for varying numbers of days, sensitivity was high on day 1, declined by day 4, and was nearly absent by day 7 (Fig. 1⇓B). If these T cells were subsequently reactivated on fresh APC for 24 h, sensitivity re-emerged (Fig. 1⇓C) and was quantitatively similar to that observed on day 1 of primary activation (≥14 LU for both 1° and 2°, day 1).
Following activation, T cells are rendered sensitive to syngeneic LAK-mediated lysis. A, CD4+ D011 T cells (circles) or CD8+ 2C T cells (squares) were activated on BALB/c APC with or without OVA, respectively. At the indicated times, T cells were harvested and used as targets in a standard CRA for BALB/c and B6 LAK, respectively. The graph shows a representative experiment with LUs on the y-axis and time on the x-axis. B, 2C T cells were activated as in A for 0 (circles), 1 (squares), 4 (triangles), or 7 days (crossed squares). Cells were harvested at these time points and used as targets for B6 LAK in a CRA. E:T from 3:1 to 30:1 are shown. LUs are also shown beside each titration. C, 2C T cells were activated as in A for 0 (circles), 1 (squares), or 10 days (crossed squares) of primary culture or reactivated on day 10 for 1 day of secondary culture (□). Cells were harvested at these time points and used as targets for B6 LAK in a CRA, as in B. D, 2C T cells were activated, as in A, for 0 (•) or 6 h (squares) or for 6 h on DM2 (Ld−) APC (□). Cells were harvested at these time points and used as targets for B6 LAK in a CRA, as in B. E, 2C T cells were activated, as in A, for 0 (circles) or 20 h (squares). Cells were harvested at these time points and used as targets for B6 LAK (filled symbols) or B6 PKO LAK (open symbols) in a CRA, as in B. F, 2C T cells were activated, as in A, for 0 (circles) or 20 h (squares) in medium alone (filled symbols) or in the presence of BFA (open symbols). Cells were harvested at these time points and used as targets for B6 LAK in a CRA, as in B.
Acquisition of sensitivity to lysis appeared to require TCR engagement: when 2C T cells were activated on APC from DM2 mice that lack H-2Ld so that they could not be signaled through the 2C TCR, there was, at 6 h, no difference in sensitivity between these and resting T cells. In contrast, 2C T cells activated on wild-type H-2Ld+ APC (BALB/c) were 5-fold more sensitive to syngeneic LAK-mediated killing than resting controls (Fig. 1⇑D). The 2C T cells incubated on DM2 APC showed signs of viability loss after 6 h, so the experiment was not extended beyond this time point. D011.10 CD4+ T cells incubated on BALB/c APC that had not been pulsed with OVA (therefore no Ag) were also not lysed (unpublished observations).
MHC class I is well described as a pivotal NK inhibitory ligand (29). We therefore measured H2-Kb expression on resting and activated 2C T cells to test whether acquisition of sensitivity to lysis correlated with reduced expression of MHC-I. We found that MHC-I mean fluorescent intensity (MFI) increased from a relative value of 100 on day 0 to 257, 297, 248, and 214 on days 1, 2, 3, and 4 after activation, respectively.
LAK normally kill their targets by the perforin-dependent granule exocytosis pathway. To test whether killing of activated T cells was perforin dependent, we generated LAK from PKO mice (B6/H2b) and used them as effectors against activated 2C T cells. Although lacking perforin, PKO LAK activate normally and have the capacity to kill through Fas (30). As predicted from previous experiments, 2C T cells activated for 20 h were >22 times more sensitive to syngeneic B6 LAK than resting controls. In contrast, lysis of these activated T cells by PKO LAK was only ∼1.2 times greater than for resting controls (Fig. 1⇑E).
To this point, we have shown that activation of T cells of known specificity on APC carrying the appropriate Ag induces sensitivity to lysis by syngeneic LAK. To address whether wild-type T cells would behave similarly, we activated T cells from B6 mice with Con A, PMA/ionomycin, or immobilized anti-TCR mAb. Compared with resting T cells (LU ≤0.5), T cells activated for 24 h with Con A, PMA/ionomycin, or immobilized anti-TCR mAb were highly sensitive to lysis (LU ≥18, 20, and 32, respectively). We found that coating of T cells with Con A resulted in very minor lectin-mediated killing, and this was reversed by standard treatment with α-methyl-mannoside (unpublished observations).
We hypothesized that sensitization might be due to transport of an NK activation structure to the cell surface through the Golgi apparatus. To test this directly, we blocked protein trafficking to the plasma membrane during activation of 2C T cells with the Golgi disrupting reagent, BFA, using a method that affected T cells, but neither APC nor LAK (22) (see Materials and Methods). After 20 h of activation, T cells activated in the presence of BFA were at least 2-fold less sensitive to lysis by syngeneic LAK than activated T cell controls (Fig. 1⇑F).
Characterization of the receptor(s) and ligand(s) involved in recognition of activated T cells
The activation receptor NKG2D is known to recognize several different inducible nonclassical MHC-I molecules and has been implicated in the lysis of syngeneic cells bearing normal levels of classical MHC-I. We tested D011 (BALB/c) T cells for staining with a soluble mouse NKG2D/human IgG1-Fc fusion protein. As seen in Fig. 2⇓, A–C, they stained weakly when resting and intensely (>10-fold increase in MFI) after 24 h of activation, the latter largely inhibited when the cells were activated in the presence of BFA, consistent with Fig. 1⇑F. Staining intensity with NKG2D-Fc correlated with susceptibility to lysis by BALB/c LAK for D011 cells activated for 0 or 24 h in the absence or presence of BFA (Fig. 2⇓D). In particular, incubation with BFA reduced both sensitivity to lysis and binding of NKG2D-Fc. When D011 T cells were activated for up to 10 days, staining with NKG2D-Fc correlated with killing and predicted the transient nature of sensitivity (unpublished observations).
Involvement of NKG2D in recognition of activated T cells. A–C, D011 T cells were activated on BALB/c splenic adherent cells pulsed with OVA. After activation for 0 (A), 20 (B), or 20 h with BFA (C), as described in Materials and Methods, cells were labeled with mouse NKG2D-Fc (filled histograms) or, as control, ULBP3-Fc (open histograms) plus PE anti-human IgG-Fc. D, D011 T cells were activated, as in B, for 0 or 24 h either in medium alone (circles) or with BFA (squares), as described in Materials and Methods. T cells were stained with mouse NKG2D-Fc plus PE anti-human IgG-Fc (MFI displayed on left y-axis; filled symbols) and also tested for sensitivity to day 4 BALB/c LAK by CRA (LU are displayed on right y-axis; open symbols). E, Resting (filled symbols) or 24-h-activated, as in B (open symbols), D011 T cells were assessed for sensitivity to lysis by syngeneic BALB/c LAK in a standard 4-h CRA. ULBP2-LZ (circles), anti-NKG2D mAb (triangles), and ULBP3-Fc (squares) were titrated into the CRA, as indicated on the x-axis. Specific lysis values at an E:T of 10:1 are shown. F, D011 T cells were activated for 24 h, as in B, labeled with radioactive chromium, and tested for sensitivity to killing by YTSeco (circles) and YT-NKG2D (squares) with (open symbols) or without (filled symbols) the addition of anti-NKG2D mAb (0.5 μg/well). This is a representative CRA with E:T from 3:1 to 30:1. G, Total RNA was prepared from D011 T cells, either resting or after activation for 24 h, as in B. After reverse transcription to cDNA, PCR was performed with H-60 and β-actin-specific primers. PCR products are shown as indicated in sets of four lanes (2–5, 6–9, 10–13, 14–17), which represent 15, 20, 25, and 30 cycles of amplification.
To test directly whether NKG2D was responsible for recognition of activated D011 T cells, we added anti-NKG2D mAb or the soluble NKG2D ligand, ULBP2 (14), to the CRA to test whether they would block lysis. ULBP2 was added conjugated to a leucine zipper motif (ULBP2-LZ). ULBP3-Fc was used as a negative control protein because it does not bind to mouse NKG2D (data not shown). Fig. 2⇑E illustrates the results obtained using syngeneic BALB/c LAK as effectors and D011 T cells, activated on Ag-pulsed APC (24 h), as targets at an E:T = 10:1. Activated T cells were highly sensitive to lysis when nothing was added to the CRA and at all concentrations of ULBP3-Fc. When either anti-NKG2D mAb or ULBP2-LZ was added to the CRA, lysis of activated T cells was abrogated. Resting T cells were highly resistant to lysis under all conditions.
As an even more direct test of whether mouse NKG2D was responsible for recognition of activated D011 T cells, we expressed a mouse NKG2D gene (GenBank accession AF030313) in the human NK cell line, YTSeco, using a pMX-derived retroviral vector, pMXs (25, 26). Independent experiments determined that YTSeco transduced with mouse NKG2D (YT-NKG2D) stained intensely with anti-mouse NKG2D mAb (MFI = 411), whereas parental YTSeco did not (MFI = 5). Fig. 2⇑F shows that YT-NKG2D lysed activated D011 T cells >25 times more effectively than the parental cell line. Addition of anti-NKG2D mAb to the CRA abolished lysis.
A known NKG2D ligand, H-60, can be expressed in BALB/c (but not B6). To test for its expression, we performed semiquantitative RT-PCR on resting and activated D011 T cells. As seen in Fig. 2⇑G, activated D011 T cells expressed higher transcript levels of H-60 compared with resting controls. Densitometric analysis of this and a duplicate experiment indicated that the level of H-60 message was 2- to 3-fold higher in activated cells.
We conclude that NKG2D is the NK receptor that recognizes activated BALB/c D011 cells and that H-60 may be its ligand. However, the situation is not so simple for lysis of activated B6 2C cells. B6 LAK express NKG2D (31) and stain with our anti-NKG2D mAb similarly to BALB/c, but addition of the mAb into the lysis assay had little effect (unpublished observations). Furthermore, addition of mAb against NKR-P1C, a known B6 NK activation receptor, had no effect (unpublished observations). Similarly, YT-NKG2D could not lyse activated 2C cells (unpublished observations). See Discussion.
In this study, we report that IL-2-activated NK cells recognize and lyse syngeneic T cells in a perforin-dependent manner within 15 h of activation on APC in both primary and secondary cultures. Sensitivity persisted for up to 4 days, required T-specific Ag, and reverted to resistance by day 7. For studies with APC, we used T cells from TCR transgenic animals specific for OVA (CD4) or H-2Ld (CD8) on the C.B-17 (identical with BALB/c, except for IgH locus) and B6 backgrounds, respectively. T cells became maximally sensitive to lysis before signs of polarization to the Th1 or Th2 phenotypes were detectable, and we have no data suggesting that this polarization is involved in the phenomenon. When nontransgenic T cells from BALB/c and B6 mice were activated with Con A, anti-TCR mAb, or PMA/ionomycin, sensitivity to lysis was acquired with similar temporal kinetics.
Lysis of activated T cells could be due to either loss of an inhibitory ligand or gain of an activating ligand. We first checked whether expression of MHC-I, the major inhibitory ligand for NK, decreased on activated T cells, but instead found slightly elevated levels relative to resting controls. This suggested that up-regulation of an activation structure was more likely responsible for lysis than down-regulation of an unknown inhibitory ligand. Consistent with transport of an NK target structure to the cell surface in response to activation, we found that inhibition of protein transport with BFA during T cell activation, in a manner that did not affect APC or LAK (see Materials and Methods) (22), largely inhibited the acquisition of LAK sensitivity (12-fold lower than activated control).
In several cases, it has been found that NK cells can kill targets expressing normal levels of MHC-I. Under these circumstances, activation signals received through NKG2D have been reported as responsible for tipping the balance toward lysis (10, 13). We found that staining of T cells from BALB/c (D011 or wild type) with a soluble mouse NKG2D-Fc fusion protein increased markedly on activation, correlated with sensitivity to lysis, and was inhibited by BFA. When we added either anti-NKG2D mAb or the soluble NKG2D ligand, ULBP2-leucine zipper fusion protein, to the CRA, lysis of activated T cells was reduced to levels observed for the resting controls, suggesting strongly that NKG2D is in fact the sole receptor responsible for killing of activated T cells in BALB/c, although we cannot rule out that other receptors may still be involved in conjugation (e.g., adhesion molecules) and augmentation of the activation signal (e.g., CD28). As further confirmation for the role of NKG2D, we transduced a mouse NKG2D gene (32) into a human NK cell line (YTSeco) with a pMX-based retrovirus (25, 26). Only cells transduced with mouse NKG2D killed activated D011 T cells, confirming directly that NKG2D can recognize activated BALB/c T cells.
We attempted to identify the NKG2D ligand involved in this system. The MHC-I homologue, H-60, was a good candidate because: 1) it is the only NKG2D ligand reported to be expressed on adult tissues, and 2) its expression has only been found in BALB/c (13). We measured H-60 transcripts by RT-PCR in resting and activated D011 T cells (Fig. 2⇑G). We found transcripts in both populations with an approximate 2- to 3-fold increase after activation. This may not be enough to explain the observed sensitivity to lysis because a 10-fold increase in NKG2D-Fc staining was seen after activation. We cannot rule out that other NKG2D ligands are involved.
Activated B6 T cells, 2C or wild type, were also extremely sensitive to lysis by syngeneic LAK. We tested whether NKG2D plays a role in the B6 system. Although H-60 is not expressed in B6, Carayannopoulos et al. (33) have recently described a novel ligand for NKG2D in B6, MULT1, murine ULBP-like transcript-1. We found that addition of neither anti-NKG2D mAb nor ULBP2-Fc to the CRA had an appreciable effect. Moreover, NKG2D-Fc did not bind B6-activated T cells, indicating that MULT1 is not involved. Nevertheless, NKG2D is expressed by B6 LAK at levels comparable to BALB/c (unpublished observations) and mediates redirected lysis of the FcR+ target P815 (unpublished observations). This suggests that B6 and BALB/c LAK use different receptors to recognize activated T cells. This may not be as unusual as it seems. NK receptor use differs broadly among inbred mouse strains in a manner that does not obviously correlate with the distribution of their known ligands. The apparent paradox between B6 and BALB/c, found in this study, becomes just one more of a growing number.
We observed that neither resting NK cells nor NKG2D+CD8+ T cells from either B6 or BALB/c could lyse syngeneic activated T cells (unpublished observations). This may be explained by the recent findings of Diefenbach et al. (34), who found that resting splenic NK cells predominantly express a long splice variant of NKG2D that associates exclusively with DAP10 and is not capable of activating NK cells without coligation of a second NK activation receptor (e.g., NKRP-1C in B6). A second, short splice variant of NKG2D was found to be expressed by poly(I:C) and IL-2-activated NK cells. This short form of NKG2D can associate not only with DAP10, but also DAP12, and appears to act as a bona fide activation receptor. In NKG2D+CD8+ T cells, the absence of DAP12 may render NKG2D unable to activate these cells without coligation of the TCR (34).
If a T cell can become a LAK target following activation, how might this affect an adaptive immune response? The answer to this question may lie within the realm of tissue distribution. In a recent report by Dokun et al. (35), the location of NK cells was examined during an immune response to murine CMV. NK cells were found excluded from T cell zones in the spleen and lymph nodes. This should protect T cells during and immediately after activation. In the liver and thymus, NK and T cells can potentially interact. This is interesting because following infection, NK cell numbers decrease in the spleen and peak in the liver after day 5 (35). Furthermore, the liver has been reported to trap activated CD8+ (36, 37) and CD4+ T cells (38, 39), both of which then undergo apoptosis by a still undefined mechanism, which we would speculate is NK cell mediated.
Several reports have indicated that NK cells activated by DC go on to limit immune responses by killing immature DC (7, 40). We suggest that these same NK cells can also directly limit the T cell arm of the adaptive immune response. If this occurs in vivo, our work has important implications for autoimmune and lymphoproliferative diseases (e.g., graft-vs-host disease (41)). SCID or RAG-deficient animals, which receive syngeneic CD4+CD45RB+ T cells, develop a proinflammatory autoimmune-like colitis. As a strong in vivo correlate of our findings, when NK cells are depleted from these animals or when recipients are perforin deficient, the colitis score increases >5-fold (42).
In conclusion, we believe the data presented in this work provide a framework for the pursuit of further study in this new area of NK cell biology.
↵1 This work was supported by a research grant to R.G.M. from the National Cancer Institute of Canada and funds from the Ontario Graduate Student Program to B.A.R.
↵2 Address correspondence and reprint requests to Dr. Richard G. Miller, Department of Medical Biophysics, Ontario Cancer Institute, Room 9-305, 610 University Avenue, Toronto, Ontario M5G 2 M9, Canada. E-mail address: miller{at}oci.utoronto.ca
↵3 Abbreviations used in this paper: DC, dendritic cell; BFA, brefeldin A; CRA, chromium release assay; LAK, lymphokine-activated killer cell; LU, lytic unit; MFI, mean fluorescent intensity; PKO, perforin knockout mice.
Received October 28, 2002.
Accepted January 28, 2003.
Copyright © 2003 by The American Association of Immunologists
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The Journal of Immunology April 1, 2003, 170 (7) 3572-3576; DOI: 10.4049/jimmunol.170.7.3572
Innate Immunity Together with Duration of Antigen Persistence Regulate Effector T Cell Induction
Regulatory Roles of IL-2 and IL-4 in H4/Inducible Costimulator Expression on Activated CD4+ T Cells During Th Cell Development
Induction of CD4+ T Cell Apoptosis as a Consequence of Impaired Cytoskeletal Rearrangement in UVB-Irradiated Dendritic Cells
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WatsonMath.com | As a K-12 Math Consultant, this site is where I post ideas to support the teaching of mathematics.
As a K-12 Math Consultant, this site is where I post ideas to support the teaching of mathematics.
Here's the Answer. What's the Question?
Fawn Nguyen, in her blog Finding Ways, posted a problem that she found in Don Steward's blog. Both Fawn's and Don's blogs are way at the top of my go-to sites when I am looking for something creative to use in my work with math teachers. This problem is an example of why. It's a simple problem that is accessible to middle school students, but turned around in such a way that makes it ripe for creating great learning opportunities for students.
The problem makes clear the importance of using units as tools for understanding. Fawn's post discusses how she introduced the problem to her Grade 6 students and included their comments.
In order to make sense of the statements, student have to not only attach the labels, but also understand that labels are "acted upon" by operations. Another key understanding students need is that division translates in English to "per".
We're dividing a fraction by a fraction, so we invert the second fraction and multiply…even if the fractions are made up of words!
The answer of cakes per bag makes it clear what the question is.
Thanks, Fawn and Don, for providing us with this rich task!
Jo Boaler is one of my go to people for insight into best practices in math education. Educated in England, she is now a professor at Stanford University and supports a site called You Cubed that has high quality resources for K – 12 educators.� I participated in her online course on Fractions,� an area of mathematics that causes many students and teachers consternation.� What I like about Jo is her focus on conceptural understanding and number sense over memorization.
Here is a 15 minute video in which Jo Boaler discusses "Number Talks", which is a practice that helps students develop number sense.
they have a �closed beginning� meaning that they all start with the same initial problem.
they have a �closed end� meaning that they all end with the same answer.
they have an �open middle� meaning that there are multiple ways to approach and ultimately solve the problem.
Open middle problems require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. They support the Common Core State Standards and provide students with opportunities for discussing their thinking.
I like that they are all short problems, so they would be easy to do as a warm-up or assign as a homework assignment.� I also like that there is not always one answer.� This can lead to rich discussions.
There are problems for Kindergarten through High School, all aligned to the Common Core. Below are some examples from Grade 3, Grade 8, and High School.� When a student starts the problem, only the problem is shown.� The "Hint" and "Answer" are accessed by clicking.
Teachers can subscribe to the site to receive emails when new problems are added. Also, teachers are encouraged to submit their own problems.
Good stuff! Check it out!
I am a member of a team of math educators in Vermont that is working with the Vermont Agency of Education to help spread the word around the state about math education initiatives.� Vermont is a small, rural state and teachers are often isolated.� The Agency of Education employs 2 (actually 1.5) very dedicated educators to oversee mathematics for the whole state.� Tracy Watterson oversees K – 5 and Lara White oversees 6 – 12, but for only half of her time. Her other time is spent in another .5 Agency of Ed position.� This year, the 6 – 12 team is hosting bi-monthly meetings in 7 regions around the state to bring information directly to the teachers rather than the teachers having to come to a central location.� We're calling these gatherings Math Morsels … we bring food morsels as an enticement!� We meet from 4 to 6 PM, so teachers do not have to take time off during the school day.� We are in the midst of Round 2. The focus of this round has been to familiarize teachers with the Smarter Balanced (SBAC) assessment that will be administered for the first time this spring to Vermont students.
The SBAC is new to everyone, so this first year will most likely be full of much consternation and headaches for educators and students.� As in any new thing, my advice to teachers� is to go in as informed and prepared as you and your students can be, and accept the inevitable fact that issues will arise that you didn't even consider.� Don't beat up on yourself or your students. Learn from the experience and make changes to improve the process over time.� Also, There will be complaints about the sorry state of� education and how awful the Common Core is and there will be an appeal to go back to the basics, whatever they are/were.
# 1 Make sense of problems and persevere in solving them.
# 2 Reason abstractly and quantitatively.
# 3 Construct viable arguments and critique the reasoning of others.
# 4 Model with mathematics.
# 5 Use appropriate tools strategically.
# 6 Attend to precision.
# 7 Look for and make use of structure.
# 8 Look for and express regularity in repeated reasoning.
The most important change that we need to make as educators� in order to prepare our students for success on the SBAC is not the content that we teach. The most important instructional change we can make is to teach our students how to persevere in solving a problem.
As we looked through the test resources, it became clear that there are some problems that are asked in very different ways than we and the students are used to.� In order to correctly answer the questions, students need to be creative problem solvers.� Here's an example of� a 6th grade task. Students are shown a square and told that the area of the square is 324 square units.� The question asks for the length of a side.
This is a pretty straightforward question. The concept being assessed is that in a square the area is found by multiplying a side length by itself.� However, while a calculator is provided for many problems, there was not one available for this problem.� (Each problem is worked on a computer with one problem per page.) A teacher that I was sitting next to and I were surprised that there was no calculator, since we normally don't expect students to have the square root of 324 memorized. In fact, I myself didn't know it off the top of my head. However, I had some strategies that I have developed over the years that could lead me to the answer pretty quickly.
Students could of course guess and check 1 x 1, 2 x 2, until they found it,� This would be an example of # 1, making sense (knowing that a number times itself needs to have a product of 324).� It would also be an ad nauseum example of perseverance.
However, if the student added the abstract and quantitative reasoning expected by # 2, they might start with larger easy numbers, such as 10 x 10 and 20 x 20 to get a rough idea of the magnitude of the number they were looking for.� Once they looked at 100 and 400, they knew that they number they were looking for was closer to 400 and could guess and check 19 x 19, 18 x 18, and voila, they are there.
If they did the above, but instead of finding 19 x 19, and 18 x 18, they looked at the structure of 19 and realized that the ones place of 19 x 19 would be 1 (since 9 x 9 = 81), they wouldn't need to do that multiplication. They could move to 18 x 18, see that the ones place would be 4 (since 8 x 8 = 64), then test out 18 x 18, which would prove that a side length of 18 is the correct answer.
Another approach that was mentioned at the meeting that I had not thought of was to factor 324 and group the factors so that there were two sets of the same factors multiplied together.� 324 = 2 x 162, which is 2 x 2 x 81, which is 2 x 2 x 9 x 9, s0 324 could be expressed as (2 x 9) x (2 x 9), which is 18 x 18.� This is another example of using Practice Standards # 1, 2 and 7.
The problem was not an extremely difficult problem. It just had a number that a student might not have memorized as a perfect square. A student who had been given plenty of opportunities to make sense of problems and persevere,� who were comfortable with the structure of numbers and operations would have the tools to work through this problem fairly quickly. A student who expected a quick answer or a quick tool to figure it out would most likely skip it and move on.
The Practice Standards provide students with habits of mind that will serve them well as they work through non-routine problems.� Let's face it. All problems that our students will face in life are non-routine.� Let's prepare our students for a future in which they will persevere and expect that any problem worth solving may take some thought and can be solved using multiple strategies.� Doing this will mean changing the way we instruct students to approach problems.� It's not enough to have the 8 Practice Standards posted on the wall.� We must refer to them often, use them in our own practice,� and expect students to do the same.
Tracy Zager, in her blog Becoming the Math Teacher You'd Wish You Had shares a wonderful problem that looks simple.� However, scratching below the surface a bit, the problem opens up a lot of different approaches.� She aptly names the post A Problem with the Space Inside it to Learn.
Love the problem! As you illustrated, It is accessible to people with very different levels of mathematical expertise. The question "What do you think?" turns it into a sort of 3-Act problem. What I thought was "Are they parallel or not?" I simply counted the squares to notice that the top line had a slope of 1/5 and the bottom line had a slope of 1/4, so no, they weren't parallel.
�I was done…or so I thought until I started reading the comments. Someone else may care about where they will intersect. So I guess I should care, too. Julie went about it in a way that I never would have thought by generalizing the two equations. I would have not been as clever as Julie and just used the equations y = 1/5 x + 5 and y = 1/4 x – 4. Setting 1/5 x + 5 = 1/4 x – 4 and solving for x, and then y, I would get the point of intersection at (180,41), the same answer as Julie.
What if I don't know algebra yet and so can't find the point of intersection by setting the y=values of the two equations and solving for x? I graphed it and noticed that at x = 0, the lines were 9 vertical units apart. At x = 20, they were 8 vertical units apart. Hmmm….could the 20 come from 4 x 5? I continued the graph to x = 40 and they were 7 vertical units apart!
Practice Standard # 8 tells me to "look for and express regularity in repeated reasoning" and # 7 tells me to "look for and makes sense of structure." I'm pretty convinced that when x = 60 that the lines will be 6 vertical units apart. If I want to graph this, I can check out my hypotheses. At some point I will be convinced that for every increase of 20 units for x, the lines are 1 unit closer together. I fill out the following table. For the y-value of the upper line, I notice that for every increase of 20 units for x, there in an increase of 4. For the y-value of the lower line, I notice that for every increase of 20 units for x, there is an increase of 5! Wow! Another pattern! For every increase of 20 units, the y-value of the line with a slope of 1/4 increases by 5 and the y-value of the line with slope 1/5 increases by 4!
�So, the two lines intersect at (180, 41).
�I love problems that are accessible to a wide range of problem solvers. Thanks for sharing! I�m going to re-post this on my own blog www.watsonmath.com.
So, now I'm posting this on my own blog, which has lain fallow for a while.� I'm just starting to get my mojo back and have another post almost ready, which focuses on the Multiplication Machine at MoMath in NYC. Stay tuned!
Howard Phillips, on his blog� "Saving School Math" proposed a real life geometry problem that proved harder to solve than it first appeared. I love to banter about mathematics education with Howard.� We agree on some things about math education and disagree on other things, most likely because his background is in higher ed and mine is in K-12 ed. Nevertheless, the back and forth is always interesting.
Howard hails from from England and was a senior lecturer in mathematics and more at the University of Huddersfield.� He retired from teaching in 2004 and moved to Puerto Rico with his wife, who is the director of the Western Ballet Theater.
I suspect that this real life geometry problem was somehow related to his building a stage prop for the next ballet performance.
While designing a system for connecting �educational� cubes together I figured that the holes in the faces had to be positioned very carefully. To achieve what I wanted the holes had to be positioned with �length a equal to length b, and length c had to be twice the length a.
�SPOILER ALERT!� Stop here if you want to work this problem out by yourself without any hints!
My first solution was using trigonometry.
I took the challenge. After several messy hand-drawn diagrams, I created the shape on GeoGebra and messed around for a long time, getting nowhere.� I concluded that the length of the side of the square consisted of 6 copies of length "a" plus 2 copies of a length shorter than "a".� I couldn't figure out the value of the shorter length nor how it was related to "a" .� I was wondering if somehow phi was involved, but this was late at night, so visions of sugarplums and phi were dancing in my head. It would not make sense for phi to be involved with something so static and "non-growthy".� Phi is often involved in nature, but rarely in stage props.
To get an idea of how far flung my messing around took me, check out the image below of the GeoGebra mess. This is where I noticed the� 6 copies of length "a" plus 2 copies of a length shorter than "a".� If you look at the horizontal lines cutting the vertical sides of the square, you will see along the vertical sides the 6 equal lengths and the 2 smaller equal lengths. You will also notice the octagon that informed a later solution.
Howard wrote me back with a BIG HINT.� He used the fact that an angle bisector in a triangle divides the opposite side into lengths proportional to the two sides.� He wrote down the first few lines.� I would like to say that I immediately jumped on his idea and came up with my own solution. However, if I said that I would be lying!� I labored over hand-drawn diagrams.� Then I decided that I would again use GeoGebra to create a a simpler diagram than the one you see above, but one that I could print out and mess around with.
Creating the clean GeoGebra diagram was helpful in allowing me to really wrap myself around the problem.� It made it easier to follow Howard's approach and finally get the same answer that I got with trig.� Rather than let the side length of the square be x, I set the length of the side of the square to be 1 unit (whatever unit that may be).� This allowed me to not worry about yet another variable.
I had not forgotten about the octagon formed from the 8 holes, so decided to try out a solution using it. I got the same answer as I got using trig and using Howard's angle bisector: If the side of the square is 1 unit, then "a" = 0.14644 units.
While solving it this way, I was reminded of my discovery the night before.� I had noticed that the side of the square had a length that consisted of 6 copies of length "a" plus two shorter lengths, but I had no idea what the shorter lengths were or how they were related to a.
The relationship became clear when I broke up the top of the square into different lengths. It's easy to see on the top edge that there are 4 copies of length "a" plus 2 copies of length "(sq rt 2)a".� Square root 2 is approximately 1.414.� So the 2 copies could be expressed as having length "1.414 a" units, which could also be written as "(1 + .414) a", which is the same a "1a + .414a".� We have two of these lengths of "1a + .414a", so now we have the other 2 copies of "a" to make "6a" and we have the 2 copies of the shorter length, which we now know each have a length of .414 units.
This is a great example of showing how dead-ends like I thought I had during my late night playing around with GeoGebra actually informed my final geometric solution.� Not only that, but my finally geometric solution verified my observation that the side consisted of 6 copies of length "a" plus two shorter lengths.� I figured out the shorter lengths!
Now I am inspired to pull off the cobwebs of a post I started writing in 2013 about a "multiplication machine" at MoMath in NYC.� Thank you, Howard, for helping me get my blog post mojo back!
Geoff, in his emergent math blog, has posted a balanced critique of the Common Core Math Standards.
In my work as a mathematics consultant for the past six years, I have become convinced that a clear conceptual understanding of fractions is the biggest deficit that I run into with both teachers and students.� Fractions are the gateway to proportional reasoning and algebraic thinking. Any student who doesn't understand the structure of fractions and their operations will struggle in middle and high school mathematics.
The Progressions Documents for those of you that are not familiar with them are the original documents, each written by a group of talented educators, that were first step in developing the Common Core (CCSSM). The progressions documents were used to guide the three lead writers of the CCSSM in writing the individual standards.� The Common Core approach to fractions in Grades 3 – 5� focuses on conceptual understanding of fractions and their operations. There is a clear connection between fraction concepts and later algebraic concepts. The Progressions Document for Numbers and Operations: Fractions should be required reading for all math teachers grade 3 – 12.
�Recommendation 1 is to build on students� informal understanding of sharing and proportionality to develop initial fraction concepts. Learning is often most effective when it builds on existing knowledge, and fractions are no exception. By the time children begin school, most have developed a basic understanding of sharing that allows them to divide a region or set of objects equally among two or more people. These sharing activities can be used to illustrate concepts such as halves, thirds, and fourths, as well as more general concepts relevant to fractions, such as that increasing the number of people among whom an object is divided results in a smaller fraction of the object for each person. Similarly, early understanding of proportions can help kindergartners compare, for example, how one-third of the areas of a square, rectangle, and circle differ.
�Recommendation 2 is to ensure that students know that fractions are numbers that expand the number system beyond whole numbers, and to use number lines as a key representational tool to convey this and other fraction concepts from the early grades onward. Although it seems obvious to most adults that fractions are numbers, many students in middle school and beyond cannot identify which of two fractions is greater, indicating that they have cursory knowledge at best. Number lines are particularly advantageous for assessing knowledge of fractions and for teaching students about them. They provide a common tool for representing the sizes of common fractions, decimals, and percents; positive and negative fractions; fractions that are less than one and greater than one; and equivalent and nonequivalent fractions. Number lines also are a natural way of introducing students to the idea of fractions as measures of quantity, an important idea that needs to be given greater emphasis in many U.S. classrooms.
Recommendation 3 is to help students understand why procedures for computations with fractions make sense. Many U.S. students, and even teachers, cannot explain why common denominators are necessary to add and subtract fractions but not to multiply and divide them. Few can explain the �invert and multiply rule,� or why dividing by a fraction can result in a quotient larger than the number being divided. Students sometimes learn computational procedures by rote, but they also often quickly forget or become confused by these routines; this is what tends to happen with fractions algorithms. Forgetting and confusing algorithms occur less often when students understand how and why computational procedures yield correct answers.
Recommendation 4 involves focusing on problems involving ratios, rates, and proportions. These applications of fraction concepts often prove dif�ficult for students. Illustrating how diagrams and other visual representations can be used to solve ratio, rate, and proportion problems and teaching students to use them are important for learning algebra. Also useful is providing instruction on how to translate statements in word problems into mathematical expressions involving ratio, rate, and proportion. These topics include ways in which students are likely to use fractions throughout their lives; it is important for them to understand the connection between these applied uses of fractions and the concepts and procedures involving fractions that they learn in the classroom.
Recommendation 5 urges teacher education and professional development programs to emphasize how to improve students� understanding of fractions and to ensure that teachers have suf�ficient understanding of fractions to achieve this goal. Far too many teachers have diffi�culty explaining interpretations of fractions other than the part-whole interpretation, which is useful in some contexts but not others. Although many teachers can describe conventional algorithms for solving fractions problems, few can justify them, explain why they yield correct answers, or explain why some nonstandard procedures that students generate yield correct answers despite not looking like a conventional algorithm. Greater understanding of fractions, knowledge of students� conceptions and misconceptions about fractions, and effective practices for teaching fractions are critically important for improving classroom instruction.
The CCSSM writers, using the approach recommended by the research of�Case and Okamoto (1996),� introduces fractions as units on the number line starting in Grade 3. This is a new approach that is at present not understood and fully embraced by elementary teachers who don�t have a deep understanding of fractions on the number line, or of fractions as single numbers with a single value (magnitude). Fractions are often understood only as a comparison of part to whole.�While the part to whole interpretation of fractions is an important model to understand, it breaks downs when operations such as addition, subtraction, multiplication, and division are applied to two or more fractions.� One of the misconceptions that students have is that operations (addition, subtraction, multiplication, and division) on fractions have the same effect as operations on whole numbers. Note: In the following examples, we will only take into account positive rational numbers, since introducing negative values adds another layer of complexity to operations, which is not introduced until Grade 6 and also presents many misconceptions for students.
When two whole numbers are added, the resulting sum is larger than either addend. Adding two positive fractions will also result in a sum that is larger than either addend. Multiplying two whole numbers will result in a larger whole number.� However, multiplication does not have the same effect on fractions, which can be very confusing to students unless the misconception is recognized and explained with concrete examples.
When the two factors in a multiplication problem are fractions, each factor acts as a �scalar� on the other fraction.� In the case of 1/2 x 3/4, the product will be less than 1/2 (since we are finding 3/4 of 1/2) and the product will be less than 3/4 (since we are finding 1/2 of 3/4).
The operation of division also presents a different story than division with whole numbers.� When asked to divide a larger whole number by a smaller whole number, say 6 divided by 3, the answer is 2, which is less than 6.� When asked to divide 6 by 1/2, the answer is 12, which is more than 6.� This presents a lot of misunderstanding and rather than delve into the reasons that this happens (1/2 is smaller half as big as 1, so twice the number of 1/2s will fit into 6), most students simply memorize the �invert the second fraction and multiply� rule, but often forget whether that rule is for division or for multiplication, or which one they invert, or whether they have to get a common denominator when multiplying and dividing.
All of these misconceptions can be circumvented, but only if teachers anticipate the misunderstanding and provide guidance to students to understand how operations on numbers can affect the numbers differently, depending upon the value of the numbers being operated on.� Are both numbers less than one? Are both numbers more than one? Is one number more and one and another less than one?� And, of course, the introduction of integers in Grade 6 adds another layer of complexity.
Elementary students can grasp these ideas when they are introduced in a concrete way, using manipulatives, the number line and the area model. However, students are often given the rules and, since the rules are counter-intuitive to their understanding of how whole numbers work, they are often confused.
We count things using units. When using 1 as our unit, we count 1, 2, 3, 4, 5�By the same token, when we use 1/3 as our unit, we count 1/3, 2/3, 3/3, 4/3, 5/3�When we use apple as our unit, we count 1 apple, 2 apples, 3 apples, 4 apples, 5 apples. When we use x as our unit, we count 1x, 2x, 3x, 4x, 5x�The �introduction of fractions as units should help students understand why it is important to have a common denominator when adding and subtracting. The denominator expresses the size of the unit. The numerator tells how many units of the denominator exist.
Linking the understanding of units fractions to the number line, CCSSM first defines a fraction such as 3/5 as being the quantity formed by 3 parts, each with size 1/5. (3.NF.A.1). This idea is extended to the number line in 3.NF.A.2a.� The whole on the number line is first defined as the interval between 0 and 1. Then, if the interval is divided into, say, 4 equal parts, each part has size 1/4. The fraction 1/4 is then defined as the endpoint of the part based at 0. Finally, 3.NF.A.2b completes the number line story by explaining that the placement of 3/4 on the number line is achieved by starting at 0 and marking off 3 intervals each of length 1/4 to make an interval of size 3/4 and that the endpoint of that interval represents the number 3/4 on the number line.
These ideas are complex, not only for students, but for teachers as well.� The language in the standards (seen below) is not easy to comprehend. A significant amount of professional development for teachers needs to focus on how the concept of fraction unfolds between Grades 3 and 6.
CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
CCSS.Math.Content.3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
CCSS.Math.Content.3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
Students who intuitively know that you can�t add apples and oranges often add 2/3 + 4/5 and get 6/8.� If they understand 2/3 as 2 copies of 1/3 of a whole and 4/5 as 4 copies of 1/5 of a whole, it will be clear why they can�t be added in the form that they are presently in.
When adding fractions with unlike denominators in Grade 5, the CCSSM does not require that a lowest common denominator must be found, only that a common denominator must be found.
The most efficient common denominator is found by multiplying the denominators of the addends. This is how fractions are multiplied in algebraic expressions, when the actual numbers are not known.
The Grade 3 CCSSM standards and those in Grades 4 through 6 will hopefully result in students and elementary teachers having a conceptual understanding of fractions that will transfer to algebraic understanding in later years.� However, due to the pipeline effect, it will be several years before many students reach high school having learned about fractions using this approach. Even then, the understanding will most likely be fuzzy.� I highly recommend that middle and high school teachers reinforce the conceptual understanding of fractions outlined in the Progressions Document for Numbers and Operations: Fractions in their classes.� It will pay off.
Yesterday, in my post Fawn Nguyen's New "Math Talks" Site Enhances her "Visual Patterns" Site, I discussed how Fawn and I saw Pattern # 115 in her Visual Patterns site differently.� I also sent Fawn a pattern that fit my first erroneous way of interpreting Pattern # 115.� Shed added the pattern I sent to the site. It is Pattern # 116.� Under Pattern 116, were instructions to go to the link to get the equations.� The link, however, was to yesterday's Watsonmath post, which only included the equation to Pattern # 115, not # 116.� I'm creating this post so that if Fawn wants to change the link to #116, it will help students to see how two different people interpreted how the shape was constructed.
Below I will explain how I envision Pattern # 116 and how I guessed that Fawn might envision the same pattern, based on the different ways she and I interpreted Pattern # 115.
The fact that both formulas simplify to y = 8x – 1 does not mean that the simplified formula provides information about the shape.� The most efficient way to show the 10th pattern, which would have 8(10)-1 = 79 circles, would be to draw a large rectangular array of circles that has a vertical dimension of 8 and a horizontal dimension of 10, with one circle taken out somewhere.� The total number of circles is the same, but the pattern looks nothing like the Patterns 1, 2 and 3.
Using my formula y = 3(2x – 1) + 2 (x + 1), I can easily create the 10th pattern that has the same general "shape" as the original 3, but would have 79 circles.� I know that there would be a large rectangle that has a vertical dimension of 3 circles and a horizontal dimension of� 2(10) – 1 = 19 circles.� There would be 10 + 1 = 11 towers that are 2 circles tall that are evenly spaced along the top of the large rectangle. The outer 2 towers would be placed alongside the top row of the large rectangle.� The remaining 9 would be placed on top of the large rectangle and there will be a space that is one circle diameter wide between each tower.
1. Make sense of problem and persevere in solving them.
2. Reason abstractly and quantitatively.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
If you missed my previous post, make sure to check it out to see how Fawn uses Patterns in her classroom on a regular basis.
Finding Ways to Nguyen Students Over is Fawn's blog where you can get the gist of her educational outlook and where her unique personality reveals itself.� She manages to be both irreverent and passionate as she shares her day to day adventures as a middle school math teacher in California.� One doesn't have to teach middle school� to mine gems from her blog.� Her insights and issues resonate with upper elementary as well as high school math teachers.
Fawn must have been bored over Christmas/New Years vacation last year.� On December 27, she posted in her blog her idea for a new site Visual Patterns, created for the� "purpose of helping students develop algebraic thinking through visual patterns." The patterns on the site are created by Fawn, other math teachers, and students.�� The site is simple.� There are tabs on the top: 1 – 20, 21 – 40, …, 101 – 120. Each tab accesses 20 different patterns.� There are only 15 patterns on the 101 – 120 tab, with space for 5 more before she adds another tab.� There is a tab called Gallery, in which there are 11 patterns created by students.� Another tab, called Teachers, is one that I hadn't looked at yet, thinking that it was the patterns created by teachers. When I checked it out, I found a wonderful resource in which Fawn explains how she assigns patterns to students.
Fawn developed a system to randomly assign different patterns to students. For independent work that can be given for homework or classwork, click on the Teachers tab to see how Fawn assigns the patterns to her students.� She wanted students to be able to choose from 3 patterns and wanted each student to have their own 3 patterns to choose from.� To accomplish this, she used the free online Research Randomizer. When you go to this site, a blank form appears.� Fawn explained how she filled out the form for 39 students. At the time she posted this, there were 111 patterns.� Now there are 115.
Wanting to try out the Random Number generator and the form, I went to the site and followed Fawn's instructions.� Student #1 was assigned 64, 115, 32, Student # 2 was assigned 47,105, 53. I followed Student # 1 and worked through Patterns # 64 and #115 on the template.� There was plenty of room for my messy calculations that I did in the margin.
I emailed her and included my filled in template.� Within a couple of hours I received a reply from her and she very gently explained to me that I had counted the number of squares on my Step 2. I had visualized it as the large 3 x n rectangles on the bottom and the small 2 x 1 rectangles on the top.� My mistake was that after the first step, the 3 x n rectangles were overlapped by the 2 x 1 rectangles that were in the "middle" (not on the left end and right end). As a result, I said that there were 15 squares in # 2 when there were really only 14.� On Step 3, I had 2 too many squares and my error increased by 1 for each step.
She showed how she had visualized the pattern, which was totally different from how I had visualized it.� This is magic in these patterns.� For any given pattern, there are multiple ways that the equation can be created.� Her equation was Squares = 4(2n+1) – n – 4.
Here's how some teachers might want the equation to be simplified.
Once I see this simplified equation, I can go back to the figure and "see" it.� But I would never see y = 7x on my own.� In my opinion,the pattern just doesn't visually elicit that equation without some mental somersaults being turned.
I'm not sure how often Fawn gives students the 3 random patterns to choose from.� I can envision giving students 3 to choose from each week with the expectation that they hand in a sheet with 1 or 2 patterns filled in by the end of the week.
In her Visual Patterns Talks, she reinforces the idea that people see the patterns very differently. That is the beauty of working with patterns.� Students can be creative.
Yes, Fawn is my hero.� She not only is an excellent teacher, but takes the time to share her strategies with the world (or at least those of us interested in math education.) By the way, MTBoS stands for Math Twitter Blogosphere for those of you who haven't heard of it.� I attended the 2nd annual MTBoS Twitter Camp this summer in Philadelphia, where I met Fawn and many other talented, generous math teachers. It was a whirlwind of activity presented by and for math teachers from all over the country…and some from overseas.� I plan to attend the 3rd annual MTBoS Twitter Camp next summer, wherever it is held.
I highly recommend subscribing to Fawn's three sites…as well as the next site she is sure to dream up soon!